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--- author: - 'Robert J. Hardwick,' - 'Tommi Markkanen,' - Sami Nurmi bibliography: - 'RGstoch\_2.bib' title: | Renormalisation group improvement\ in the stochastic formalism --- IMPERIAL/TP/2019/TM/02 Introduction {#sec:intro} ============ A period of early cosmic inflation provides solutions to several cosmological conundrums [@Starobinsky:1980te; @Sato:1980yn; @Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi; @Linde:1983gd]. It is well known that perturbatively computed correlators of light scalar fields exhibit logarithmic infrared (IR) divergences during inflation. This is due to the secular growth of long-wavelength modes [@Weinberg:2005vy; @Seery:2010kh], sourced by quantum fluctuations continuously crossing the horizon. The infrared issues are closely connected to the non-existence of a physical propagator for a free, minimally coupled massless scalar in de Sitter space [@Allen:1985ux; @Allen:1987tz] and several non-perturbative resummation techniques for the interactive cases have been developed e.g., references [@Serreau:2011fu; @Prokopec:2011ms; @Arai:2012sh; @Garbrecht:2011gu; @Burgess:2009bs; @Herranen:2013raa]. The stochastic formalism [@Starobinsky:1982ee; @Starobinsky:1986fx; @Starobinsky:1994bd] is a widely used method for investigating the IR dynamics in the exponentially squeezed quantum state during inflation [@PhysRevLett.59.2555; @PhysRevD.46.1440; @PhysRevD.42.3413]. It is an approximative coarse-grained formalism where the impact of subhorizon fluctuations is modeled as stochastic noise. The dynamics of the long-wavelength field $\bar{\phi}$ is governed by the Langevin equation [@Starobinsky:1982ee; @Starobinsky:1986fx; @Starobinsky:1994bd] $$\label{eq:langevin} \frac{\partial }{\partial t}\bar{\phi} ({\bf x},t) = - \frac{1}{3H}\frac{\partial V{}}{\partial \bar{\phi}} + f ({\bf x},t) ~.$$ Here $f({\bf x},t)$ is the stochastic source term with correlation properties of white noise and the amplitude set by the solution of the linearised mode equation of the quantum field. The stochastic approach effectively performs a re-summation rendering the (quantum) correlators free of IR divergences. In specific examples it has been shown to correctly reproduce the leading-log order IR results derived by other means [@Tokuda:2017fdh; @Guilleux:2015pma; @Guilleux:2016oqv; @Arai:2011dd; @Moreau:2018ena; @Moreau:2018lmz; @Prokopec:2017vxx]. Recent works addressing foundations of the stochastic approach include references [@Moss:2016uix; @Rigopoulos:2016oko; @Tokuda:2018eqs; @Cruces:2018cvq; @Glavan:2017jye; @Hardwick:2017fjo; @Vennin:2015hra; @Grain:2017dqa; @Firouzjahi:2018vet]. During inflation light energetically subdominant scalars probe field values from the vacuum parametrically up to the Hubble scale $H$. Quantum corrections may induce significant running of couplings over this window and must be accounted in studying the dynamics. This feature in conjunction with the measured central values for couplings of the Standard Model (SM) of particle physics pointing to a metastable electroweak vacuum imply that Higgs fluctuations could easily have triggered a fatal transition to the true minimum in the early Universe [@Degrassi:2012ry], see reference [@Markkanen:2018pdo] for a recent review. The fact that this did not happen can be used to place a stringent constraint on the Higgs non-minimal coupling [@Herranen:2014cua; @Herranen:2015ima; @Markkanen:2018bfx], which is the last unknown parameter in the SM, and further can be used as a novel test to constrain SM extensions. In quantum field theory, the couplings of a Lagrangian depend on the renormalisation scale $\mu$. The choice of the scale is arbitrary and has no effect on physical quantities, which translates into a set of renormalisation group equations, which can be solved to obtain the renormalisation group improved theory. This renormalisation group improved theory corresponds to a resummation of infinite classes of perturbative corrections and therefore remains valid over a larger range of scales compared to the standard loop expansion at the same order in perturbation theory. It is not immediately obvious if the renormalisation group techniques can be applied as such in the approximative stochastic formalism, where the field is split by hand into a quantum UV part and a classical IR part, the UV part is substituted by the linearised solution of mode equations and the decaying quantum modes are dropped. Indeed, as we will discuss below, a simple replacement of the classical potential by the renormalisation group improved effective potential $V_{\rm eff}(\phi)$ in the stochastic Langevin equation  is not consistent in general. The field correlators computed in this way fail to obey the renormalisation group equations, i.e. the Callan-Symanzik (CS) equations [@Symanzik:1970rt; @Callan:1970yg] \[eq:CZforphin\] = - n|\^n , except in the limit of stationary solutions (here $\mu$ is the renormalisation scale and $\gamma$ is the field anomalous dimension). A violation of the CS equations is a sign of inconsistency, signalling a $\mu$ dependence of quantities which should be measurable. However, we will show that this problem is resolved when also the renormalisation of the kinetic term is correctly accounted for in the Langevin equation. Starting from the Wilsonian approach to the renormalisation group, we provide a reformulation of the stochastic formalism which is manifestly consistent with the renormalisation group improvement. We then move on to apply the formalism in the specific example of a Yukawa theory and discuss the qualitative effects of the renormalisation group corrections in this case. For earlier works addressing the Yukawa theory stochastically or renormalization group improvement during inflation see [@Miao:2006pn] and [@Bilandzic:2007nb], respectively. The paper is organised as follows. In section \[sec:Lange\] we briefly review the Wilsonian approach to renormalisation group and rewrite the stochastic Langevin equation into a form compatible with the renormalisation group improvement. In section \[sec:FP\] we investigate the corresponding Fokker-Planck equation and show that the correlation functions satisfy the correct CS equations. In section \[sec:examples\] we study the magnitude of RG effects in the stochastic formalism by considering the example of Yukawa theory on curved spacetime and, finally, in section \[sec:conclusions\] we conclude with a summary of our findings and prospects for future applications. Our sign conventions are $(-,-,-)$ according to reference [@Misner:1974qy]. The Renormalisation Group and the Langevin equation {#sec:Lange} =================================================== Wilsonian Renormalization ------------------------- The standard formulation of the stochastic approach see, e.g., references [@Garbrecht:2013coa; @Hardwick:2017fjo; @Starobinsky:1994bd], starts from the operator equation of motion of the form $(\Box+V')\hat{\phi} = 0$. The field is split into IR and UV parts and the UV part is substituted with the linearised solution. Dropping the decaying mode of the UV part, the approximative equation of motion for the IR part takes the form of a Langevin equation. Here we repeat the same procedure accounting for quantum corrections which in general affect both the kinetic term and the potential. To set up the framework, we follow the Wilsonian approach and introduce a hierarchy of scales \[murange\] H &lt; &lt;  , where $\Lambda$ is a UV cutoff and $H$ is the Hubble rate. The Wilsonian coarse-graining scale $\mu$ can be effectively thought as the renormalisation scale $\mu$ introduced when computing quantum corrections with an infinite cutoff $\Lambda\rightarrow \infty$. Integrating out modes in the momentum shell $\mu < k < \Lambda$ the generating functional takes the form Z\[J\] = e\^[-\^4 x\_[E]{} \[[L]{}\_[E]{}()+J\]]{} = e\^[i\^4 x \[[L]{}\_() +J\_\]]{}, where ${\cal L}_{\rm E}$ is the Euclidean action and ${\cal L}_{\mu}$ is the effective action of the coarse-grained theory describing modes with a cut-off in Euclidean momentum space $k_{\rm E} < \mu$. From now on we will drop the subscript “${\rm E}$” and it is to be understood that the cut-off is defined in Euclidean space. Neglecting terms with more than two derivatives, the effective action is given by the standard expression [@Peskin:1995ev] \[L(mu)\] [L]{}\_ = \^(Z\^[1/2]{}()())\_(Z\^[1/2]{}()()) - V\_[eff]{}(,(),{\_i()}) . The field operator $\phi(\mu)$ takes the form () = Z\^[-1/2]{}() () ,\[eq:fieldr\] where the factor $Z$ is due to the wave function renormalisation. Choosing $Z(\Lambda) = 1$, it is given by \[Z\] Z() = [exp]{} , where $\gamma$ is the anomalous field dimension defined by \[gamma\] = -() . Throughout this work, we will ignore all possible field dependent contributions in $Z(\mu)$, which is often called the local potential approximation [@Guilleux:2016oqv]. Since our approach only includes the very UV part in the effective action (in contrast to what is usually done) with the IR calculated stochastically this is expected to be a good approximation: the high UV physics does not suffer from secular IR effects, and can be addressed via standard perturbative quantum field in theory in curved space, where such terms have little impact, especially when close to de Sitter space where kinetic contributions are naturally suppressed. The effective potential $V_{\rm eff}$ in (\[L(mu)\]) contains in principle all quantum corrections, but as mentioned at the UV limit where only the modes $\mu < k <\Lambda $ are integrated over. Since $H < \mu$, the modes are at least marginally subhorizon. Any contribution from IR sensitive terms in $V_{\rm eff}$ should be small in this limit and is neglected here. We approximate the build-up of IR effects entirely using the stochastic approach. The UV effective potential $V_{\rm eff}$ obeys the Callan-Symanzik equation given by \[CZforV\] = (- + \_i\_i+ )V\_[eff]{}= 0 , where $\lambda_i$ denote couplings of the theory and the beta functions are defined as usual $\beta_i = {\rm d} \lambda_i/{\rm d}{\rm ln}\mu$. The scaling relations (\[Z\]), (\[gamma\]) and (\[CZforV\]) ensure that the effective action (\[L(mu)\]) does not depend on the choice of the renormalisation scale $\mu$. The quantum equation of motion for the field operator $\phi$ is obtained by varying the Lagrangian (\[L(mu)\]), which yields \[EOMchifull\] Z() [ ]{} () + V’\_[eff]{}(,(),\_i()) = 0 , where the prime denotes derivative with respect to the field $\phi(\mu)$. This quantum corrected equation is our starting point for setting up the stochastic formalism and studying its compatibility with the renormalisation group improvement. For reference, we list here the remaining Callan-Symanzik equations that will be needed in our analysis below. The derivatives of the effective potential scale as \[CZforVn\] = nV\^[(n)]{}\_[eff]{} . From the $\mu$ dependence of $\phi(\mu)$ given in (\[eq:fieldr\]) we can write the CS equation for an $n$-point function as $$\begin{aligned} \frac{\dd}{\dd \mu}\left[ \big( \sqrt{Z} \big)^{n}\langle\phi(x_1)\phi(x_2)\cdots\phi(x_n)\rangle\right]&=0\nonumber \\ \Leftrightarrow\quad\bigg( \mu \frac{\partial}{\partial \mu}+\beta_{\lambda_i} \frac{\partial}{\partial \lambda_i}+n\gamma\bigg) \langle\phi(x_1)\phi(x_2)\cdots\phi(x_n)\rangle&=0\,\label{eq:CZ0}.\end{aligned}$$ Here, and in what follows, we suspend explicit notation of the full set of arguments whenever this does not compromise definiteness. Running in the Langevin equation -------------------------------- Next, we follow precisely the standard steps in setting up the stochastic approach [@Garbrecht:2013coa; @Hardwick:2017fjo; @Starobinsky:1994bd] but use the full quantum corrected equation of motion (\[EOMchifull\]) with an arbitrary renormalisation scale $\mu$. We split the field operator into IR and UV parts with a sharp cutoff defined by a coarse graining parameter $\sigma\lesssim 1$ \[chisplit\] = (a H-)\_[**k**]{}e\^[i [**k**]{}]{}+(-a H) \_[**k**]{}e\^[i [**k**]{}]{} | +. It should be noted that the coarse graining scale $k/a = \sigma H$ is always below the renormalisation scale $\mu$ which throughout the work is chosen to lie in the UV window (\[murange\]). Substituting (\[chisplit\]) into the equation of motion (\[EOMchifull\]) and expanding in the UV field $\varphi$ yields \[0\] Z[ ]{} | + V’\_[eff]{}(|) + + [O]{}(\^2)= 0 . Following reference [@Starobinsky:1994bd], we substitute the UV part $\varphi$ with \_[**k**]{} = a\_[**k**]{} u\_[**k**]{} + a\^\_[-**k**]{} u\^[\*]{}\_[-**k**]{}, where the annihilation and creation operators satisfy the usual commutation relations and the mode functions $u_{\bf k}(t,\mu)$ are determined by the linearised mode equation \_[**k**]{}+ 3H \_[**k**]{}-u\_[**k**]{} + V”\_[eff]{} Z\^[-1]{}u\_[**k**]{} = 0 . Note that the effective mass term $V''_{\rm eff} (\mu,\bar{\phi}(\mu))/ Z(\mu)$ does not depend on the RG scale since the $\mu$ depencies of $V''_{\rm eff}$ and $Z(\mu)$ precisely cancel. Concentrating on the limit of light fields $V''_{\rm eff} \ll H^2$ and choosing the Bunch-Davies vacuum, the mode functions are just the usual Hankel functions[^1] \[chiuvHankel\] u\_[**k**]{}(,) = Z\^[-1/2]{}() (-)\^[3/2]{}H()(1-)H\_\^[(1)]{}(-) . Here ${\rm d}\tau = {\rm d} t/a$ defines the conformal time, $H_{\nu}^{(1)}$ is the Hankel functon of the first kind, and the index $\nu$ is determined by =  ,= - ,=  . Next one substitutes the linear solution (\[chiuvHankel\]) for $\varphi$ back to the full equation (\[EOMchifull\]) and drops terms $ {\cal O}(\varphi^2)$ [@Starobinsky:1994bd]. Dropping also the IR term $ \ddot{\bar{\phi}}$ which is subdominant on the slow roll attractor, one obtains \[QMlangevin\] + = aH\^2 (1-)(-a H)\_[**k**]{} e\^[i [**k**]{}]{} + [O]{} () . This is still an operator equation where $\bar{\phi}$ and $\varphi$ are quantum fields. However, the growing mode of $\dot{\varphi}$ commutes with the field $\varphi$ on superhorizon scales and the quantum state becomes exponentially squeezed [@Albrecht:1992kf; @Grishchuk:1990bj]. Therefore, choosing $\sigma \lesssim 1$ and neglecting the decaying mode, one can replace  with a classical, stochastic Langevin equation \[langevin\] + = G\^[1/2]{} . As usual, the IR field $\bar{\phi}$ is a classical stochastic variable, and the UV source behaves as white noise, hence $\langle\xi(t) \rangle = 0\,,$ $\langle\xi(t)\xi(t')\rangle = \delta(t-t')$ and the amplitude is \[noise\]  G(t, |(),) &=& H(1-) . ------------------------------------------------------------------------ |\_[= aH]{}\ && Z\^[-1]{}() . The equation (\[langevin\]) is the quantum corrected version of the standard Langevin equation in the stochastic formalism. It holds for any choice of the renormalisation scale $\mu$ which can be verified by a direct computation. Indeed, from  and the definition of the IR field in , it follows that \[dphidmu\] = -|() . Using this together with  and , and differentiating  with respect to $\mu$, we get \[dlangevindmu\] ( + - G\^[1/2]{}) = -G\^[1/2]{}- = 0 . In the last step we used the scaling relation \[dGdmu\] =-2 , which follows from . The RG scale independence (\[dlangevindmu\]) derives directly from the quantum equation of motion (\[EOMchifull\]) which is our starting point and by construction independent of the RG scale. It is now obvious that the renormalisation group improvement can be implemented in the Langevin equation (\[langevin\]) using the standard methods as we will show in section \[sec:examples\]. In the next section, we write down the corresponding Fokker-Planck equation for the probability distribution and discuss its properties. Renormalisation scale invariance of the Fokker-Planck equation {#sec:FP} ============================================================== The IR field $\bar{\phi}(\mu)$, obeying the Langevin equation (\[langevin\]), is a classical stochastic quantity whose correlation functions are determined by its probability distribution $P(\bar{\phi})$ through \[phimucorrelator\] |()\^n=  . From  and , changing $\mu \rightarrow \tilde{\mu}$ scales the IR field as $\bar{\phi}(\tilde{\mu}) = [Z(\tilde{\mu})/Z(\mu)]^{-1/2}\bar{\phi}(\mu)$ so that \[phimutilde\] |()\^n= \^[-n/2]{}|()\^n . Differentiating this with respect to the renormalisation scale $\tilde{\mu}$, one directly obtains the usual Callan-Symanzik equations for the field correlator (\[eq:CZ0\]). On the other hand, we can equally write \[phimutildecorrelator\] |()\^n&=&\ &=&\^[-n/2]{}  . Comparing this to  and  we find that they agree only if \[dPdmu\] P(|(),{\_i()},,t) = P(|(),{\_i()},,t) = 0 . This is similar to the RG scale invariance of the full QFT generating functional. It is merely a consequence of the requirement that we are free to renormalise the theory at any chosen scale and the choice does not affect the physical solutions. The probability distribution that corresponds to the Langevin process  obeys a Fokker-Planck equation which in the Itô interpretation[^2] reads \[FPIto\] = ( P)+ (GP) . In order to be consistent with the condition (\[dPdmu\]), the $\mu$ dependent entries in the Fokker-Planck equation must precisely cancel such that (\[FPIto\]) holds for any choice of the RG scale. This is similar to the RG scale independence of the Langevin equation (\[dlangevindmu\]) and again directly follows from our starting point (\[EOMchifull\]). However, to be fully explicit let us check this by direct computation. To this end, we differentiate separately each term of the Fokker-Planck equation (\[FPIto\]) with respect to $\mu$. Because $\phi$ is a test field, it does not affect the time evolution of the spacetime and consequently $t$ and $H$ have no dependence of the RG scale $\mu$. The time derivative therefore commutes with the $\mu$ derivative, and using  we get = = 0 . To compute the $\mu$ derivatives of the other two terms, we will repetitiously apply the relation \[chainrule\] = (-| + \_i\_i+ ) = + . Using ,  and , we then readily find ( P)&=& ( P)+( P)\ &=&( P- 2 P)+( P) = 0. In the same way, using also , we arrive at (GP)&=& (GP)+2(GP)\ &=&(-2GP )+2(GP) = 0. Combining the results, we find that = 0, verifying that the Fokker-Planck equation holds true for any choice of the RG scale and complies with the condition (\[dPdmu\]). This is the main result of our work. We reiterate that the renormalisation scale $\mu$ denotes a UV scale in the window (\[murange\]) which is always above the coarse graining scale of the stochastic approach. We have approximated all IR physics by the stochastic approach formulated with the UV limit of the effective action and the corresponding quantum equation of motion (\[EOMchifull\]) as the starting point. This should be distinguished from e.g. [@Guilleux:2015pma; @Guilleux:2016oqv; @Arai:2011dd; @Moreau:2018ena; @Moreau:2018lmz; @Prokopec:2017vxx] where functional renormalisation group techniques are used to gradually integrate over IR modes and scaling relations with respect to the IR cutoff are investigated. Our approach does not offer any information about this IR scaling but the RG scaling in this work refers to deep ultraviolet scaling only. Note that to maintain the correct RG scaling it was necessary to include the wave-function renormalisation $Z(\mu)$ (where the flow is normalized to start at $Z(\Lambda)=1$) in the drift terms of  and . Neglecting this and retaining only the running effective potential $V_{\rm eff}(\mu)$ would not be consistent with the RG scaling and correlators computed this way would fail to satisfy the Callan-Symanzik equations (\[eq:CZ0\]). However, in the special case of stationary limit in strict de Sitter space, $\partial P/\partial t = 0$ and $H=const.$, the solution of the Fokker-Planck equation (\[FPIto\]) takes the form \[stationarysol\] P\_[stat.]{}(|) = C [exp]{}(-|) , where the normalisation constant $C$ does not depend on the RG scale $\mu$. Since $G(\mu)\propto Z(\mu)^{-1}$ according to equation (\[noise\]), the wavefunction renormalisation $Z$ drops out from the equilibrium result. Hence, if one is interested only in the stationary limit, the correct RG scaling is obtained by dropping the explicit $Z$ dependence from the Fokker-Planck equation (\[FPIto\]) and choosing $G=H^3/(4\pi^2)$, as done for example in references [@Espinosa:2015qea; @Herranen:2014cua; @Espinosa:2007qp]. Away from the stationary limit however, the solution will depend on $Z$. This is especially important when not in strict de Sitter space, as then the stationary solution (\[stationarysol\]) does not necessarily coincide with the late time limit [@Prokopec:2015owa; @Cho:2015pwa]. In the next section we investigate Yukawa theory as a specific example and discuss quantitatively the effect of the $1/Z(\mu )$ for the two-point function. Example of Yukawa theory {#sec:examples} ======================== In this section we demonstrate the application of the renormalisation group improved stochastic formalism. As a specific example, we investigate a Yukawa theory with an energetically subdominant real scalar $\phi$ which can we treat as a test field in a classical background spacetime. We numerically solve for the two-point function of $\phi$ during slow roll inflation with a quadratic inflaton potential. Yukawa theory in curved spacetime --------------------------------- In section \[sec:Lange\] we defined the effective potential $V_{\rm eff}$ in the Langevin equation (\[langevin\]) as the Wilsonian integral over modes $H < k < \Lambda$, where $\Lambda$ is the UV cutoff. The included modes are at least marginally subhorizon and contributions from IR sensitive terms in $V_{\rm eff}$ should therefore be small. Here we neglect all the IR terms altogether and include only the deep UV part of $V_{\rm eff}$. Indeed, this is the very idea of the approximative stochastic approach. We follow reference [@Markkanen:2018pdo] and compute the UV part of the curved space effective potential using the resummed Heat Kernel approach [@Parker:1984dj; @Jack:1985mw] which is essentially an expansion around the local limit in configuration space. The UV expansion captures the curved space contributions in the renormalisation group running of the effective potential. [ This is not a Wilsonian approach but rather: in this expansion, no IR effects are included, allowing one to extend the momentum integration down to $k=0$ and make use of dimensional regularization when calculating $V_{\rm eff}$.]{} The method can be straightforwardly applied for the slow roll solution with a quadratic inflaton potential. However, the difference compared to de Sitter solution is quantitatively irrelevant within the precision of one-loop investigation and withing the limited range we will concentrate on. For simplicity, therefore, we compute the UV effective potential of the test field $\phi$ in a de Sitter space neglecting the time variation of the Hubble rate. The matter part of the action for our Yukawa example is given by [$$S =\int \dd^4x\sqrt{|g|}\,\bigg[{\frac{{1}}{{2}}}\nabla_\mu\phi\nabla^\mu\phi-{\frac{{1}}{{2}}}m^2\phi^2-{\frac{{\xi}}{{2}}}R\phi^2-{\frac{{\lambda}}{{4}}}\phi^4+i\bar \psi\nabla\!\!\!\!/\psi-g\phi\bar\psi\psi\bigg]\,.\label{eq:yu2}$$]{} Here $\phi$ is a real scalar field and the Dirac spinor $\psi$ contains $N_{\rm f}$ internal degrees of freedom — [ essentially leading to $N_{\rm f}$ copies of the same theory]{} — and $R =12H^2$ is the Ricci curvature scalar. The non-minimal curvature coupling $\xi \phi^2 R$ has a non-trivial renormalisation group running and therefore must be included in the action. Quantising the action (\[eq:yu2\]) in a classical de Sitter background unavoidably generates the gravitational operators [@ParkerToms; @Birrell:1982ix] [$$\begin{aligned} S_{g}&=-\int \dd^4x\,\sqrt{|g|}\bigg[V_{\Lambda}-\kappa R+\alpha_{1} R^2+\alpha_{2} R_{\mu\nu}R^{\mu\nu}+\alpha_{3} R_{\mu\nu\delta\eta}R^{\mu\nu\delta\eta}\bigg]\nonumber \\ &\overset{\rm dS}{\equiv}-\int \dd^4x\,\sqrt{|g|}\bigg[V_{\Lambda}-\kappa R+\alpha H^4\bigg]\label{eq:treecurve}\,,\end{aligned}$$]{} where $V_{\Lambda}$ is a constant and $\kappa$ and $\alpha_i$ are dimensionless couplings. In the second step we used that for the de Sitter solution the Ricci and Riemann tensors are given by [$$R^2=144H^4\,\qquad R_{\mu\nu}R_{\mu\nu}=36H^4\,,\qquad R_{\mu\nu\delta\eta}R^{\mu\nu\delta\eta}=24H^4\,,$$]{} and we have defined the common coupling $\alpha$ of the $\mathcal{O}(H^4)$ operators as [$$\alpha\equiv144\alpha_1+36\alpha_2+24\alpha_3\,.\label{eq:alp}$$]{} The gravitational backreaction of (\[eq:treecurve\]) is small and we will neglect it. However, this part will contribute to the effective potential through the $\phi$ dependent loop logarithms. The UV limit of the de Sitter effective potential for the theory (\[eq:yu2\]) at one loop level takes the form[^3] [@Markkanen:2018pdo] [$$\label{Yukawaeffpot} V_{\rm eff}(\phi)={\frac{{m^2}}{{2}}}\phi^2 +6{\xi}H^2\phi^2 + {\frac{{\lambda}}{{4}}}\phi^4+V_\Lambda-12\kappa H^2+\alpha H^4+V^{(1)}_\phi(\phi)+N_{\rm f}V^{(1)}_\psi(\phi)\,,$$]{} where [$$V^{(1)}_\phi(\phi)={\frac{{\mathcal{M}_\phi^4}}{{64\pi^2}}}\bigg[\ln \bigg({\frac{{|\mathcal{M}_\phi^2|}}{{\mu^2}}}\bigg)-{\frac{{3}}{{2}}}\bigg]-{\frac{{{\frac{{1}}{{15}}}H^4}}{{64\pi^2}}}\ln\bigg({\frac{{|\mathcal{M}_\phi^2|}}{{\mu^2}}}\bigg)\label{eq:curve3}\,,$$]{} and [$$\begin{aligned} V^{(1)}_\psi(\phi)=-{\frac{{4\mathcal{M}_\psi^4}}{{64\pi^2}}}\bigg[\ln\bigg({\frac{{|\mathcal{M}_\psi^2|}}{{\mu^2}}}\bigg)-{\frac{{3}}{{2}}}\bigg]+{\frac{{{\frac{{38}}{{15}}}H^4}}{{64\pi^2}}}\ln\bigg({\frac{{|\mathcal{M}_\psi^2|}}{{\mu^2}}}\bigg)\label{eq:curve4}\,.\end{aligned}$$]{} The effective masses, $\mathcal{M}^2_\phi$ and $\mathcal{M}^2_\psi$, are given by [$$\mathcal{M}^2_\phi\equiv m^2+3\lambda\phi^2+\left(\xi- {\frac{{1}}{{6}}}\right)12H^2\,,\qquad \mathcal{M}^2_\psi\equiv g^2\phi^2+H^2\,.\label{eq:effm2}$$]{} The couplings $m$, $\lambda$, $g$, $\xi$, $V_\Lambda$, $\kappa$ and $\alpha$ denote renormalised quantities with the renormalisation conditions set at the scale $\mu$. The effective potential does not depend on the renormalisation scale $\mu$ and obeys the Callan-Symanzik (\[CZforV\]) equation $$\begin{aligned} \frac{{\rm d} V_{\rm eff}}{{\rm d}{\rm ln}\mu}=\bigg(\mu \frac{\partial}{\partial \mu}+\beta_{m^2}\frac{\partial}{\partial m^2}+\beta_\lambda \frac{\partial}{\partial \lambda}+\beta_\xi\frac{\partial}{\partial \xi}+\beta_{V_{\Lambda}} \frac{\partial}{\partial {V_{\Lambda}}}+\beta_{\kappa}\frac{\partial}{\partial \kappa}+\beta_{\alpha}\frac{\partial}{\partial \alpha}-\gamma\phi \frac{\partial}{\partial\phi}\bigg)V_{\rm eff}=0\,.\label{eq:CZV2}\end{aligned}$$ The anomalous dimension $\gamma$ and the $\beta$ functions are given by $$\begin{aligned} 16\pi^2 \gamma &=2N_{\rm f}g^2 \label{Yukawagamma}\\ 16\pi^2 \beta_{m^2} &= m^2\left(6\lambda + 4N_{\rm f}g^2\right)\label{eq:la0} \\ 16\pi^2 \beta_{\lambda} &= 18 \lambda ^2+8N_{\rm f} g^2 \lambda-8N_{\rm f} g^4\label{eq:la} \\ \ 16\pi^2 \beta_{g} &=5g^3\,\label{eq:g}\\ 16\pi^2 \beta_{\xi} &= \left(\xi - {\frac{{1}}{{6}}} \right)\left(6\lambda + 4N_{\rm f}g^2\right)\,,\label{eq:2}\end{aligned}$$ and [$$\begin{aligned} {16\pi^2}\beta_{V_\Lambda}&={\frac{{m^4}}{{2}}}\,\label{eq:betaA}\\ {16\pi^2}\beta_\kappa&=-m^2\left( \xi-{\frac{{1}}{{6}}}\right)\, \label{eq:betaB}\\ {16\pi^2}\beta_{\alpha}&= 72\left( \xi-{\frac{{1}}{{6}}}\right)^2-{\frac{{11N_{\rm f} +1}}{{15}}}\,.\label{eq:betaC}\end{aligned}$$]{} For a different parametrization see, e.g., references [@Bando:1992wy; @Elizalde:1993qh; @Geyer:1996kg; @Geyer:1996wg]. From the above one may see that non-zero $\xi$ and $\lambda$ in  and  are unavoidable for a non-zero $g$: they are generated by running even if at some scale we set $\xi=0=\lambda$. In the following we will for simplicity set $m=0$. This allows us to neglect all dimensionful couplings of the theory since, as seen in equations (\[eq:la0\]), (\[eq:betaA\]) and (\[eq:betaB\]), $m=0, V_{\Lambda} =0, \kappa = 0$ is a fixed point of the renormalisation group flow. In equation (\[Yukawaeffpot\]) the couplings are evaluated at a fixed renormalisation scale $\mu$; their implicit $\mu$-dependence the $\mu$ dependence in the logarithms to satisfy (\[CZforV\]). The renormalisation group improved effective potential is obtained by solving for the running couplings and field $\phi(\mu)$ from equations (\[Yukawagamma\]) - (\[eq:betaC\]) and substituting them back into the effective potential  . This corresponds to resumming infinite sets of diagrams computed with a fixed $\mu$ and leads to improved convergence of the result [@Coleman:1973jx; @Kastening:1991gv]. We will make use of renormalisation group improvement in the sense of reference [@Ford:1992mv] by including the further step where the renormalisation scale $\mu$ is chosen to minimise the loop terms (\[eq:curve3\]) and (\[eq:curve4\]) over a range of values $\phi$ and $R$ so that the convergence of the one loop result is optimised, see reference [@Markkanen:2018bfx] for further discussion of the optimal scale choice in curved spacetime. Denoting the optimal choice by $\mu_*$, the renormalisation group improved potential (for $m=0,V_{\Lambda} =0, \kappa = 0$) is given by [$$V_{\rm RGI}(\phi)={\xi(\mu_*)}6H^2\phi(\mu_*)^2 + {\frac{{\lambda(\mu_*)}}{{4}}}\phi(\mu_*)^4+\alpha(\mu_*) H^4+ V^{(1)}_\phi(\phi(\mu_*),\mu_*)+ N_{\rm f}V^{(1)}_\psi(\phi(\mu_*),\mu_*) \,.\label{eq:effiSM}$$]{} The running field $\phi(\mu_*)$ can be written as (\_[\*]{}) = (\_0) = e\^[-\_[\_0]{}\^[\_\*]{} [d]{} ()/]{}(\_0) , where $\mu_0$ is a fixed reference scale at which we define the input values of the renormalised quantities. The wave function renormalisation factor $Z(\mu)$ is defined according to equation (\[Z\]) and $Z(\Lambda) = 1$. [In the following as the input for our analysis we will choose quantities renormalised at some reference scale $\mu_0$ in a frame with a canonical kinetic term. For this it is convenient to re-scale the field as $Z^{1/2}(\mu_0)\phi(\mu)=\tilde{\phi}(\mu)$. At the scale $\mu_0$ this will lead to a canonical kinetic term ${\frac{{1}}{{2}}}(\partial\tilde{\phi})^2$ and an effective potential that is a function of $\tilde{\phi}(\mu)$ and the scaled couplings \[inputvalues0\] () () ,()  ,() ,() , with the input values for the couplings set as \[inputvalues\] (\_0) \_0 , \_0  ,\_0 ,g\_0 . With the initial conditions as above the value for $Z(\mu_0)$ drops out, or rather, is absorbed in the initial condition in the frame with the re-scaled field $\tilde{\phi}$. Effectively, one may then perform the entire analysis with the unscaled field and simply set $Z(\mu_0)=1$ everywhere.]{} Explicit solutions for the running couplings are given in the appendix \[sec:analytic-runnings\]. We define the running scale $\mu_*(\phi, H)$ of the renormalisation group improved potential (\[eq:effiSM\]) by imposing the condition $$\begin{aligned} &\ln \mu_*=\nonumber \\ & \quad \frac{45\left(4N_{\rm f}{\cal M}_\psi^4-{\cal M}_\phi^4\right)+(30{\cal M}_\phi^4-4H^4)\ln {\cal M}_\phi^2+4N_{\rm f}(19H^4-30{\cal M}_\psi^4)\ln {\cal M}_\psi^2}{(19N_{\rm f}-1)8H^4+60({\cal M}_\phi^4-4N_{\rm f}{\cal M}_\psi^4)} \left. \rule{0pt}{5ex}\right |_{\mu_0} \,, \label{eq:opt-scale}\end{aligned}$$ where all the couplings on the right hand side are evaluated at the scale $\mu_0$. This serves to approximately minimise the one-loop logarithms in (\[eq:effiSM\]).[^4] If the right hand side is evaluated at $\mu_*$, equation (\[eq:opt-scale\]) is the condition for the loop logarithms in (\[eq:effiSM\]) to vanish exactly. This choice of the optimal scale was used in reference [@Markkanen:2018pdo]. Here we use equation (\[eq:opt-scale\]) instead because it is simpler to implement numerically. Numerical solution for the two-point function --------------------------------------------- We now proceed to numerically solving for the two-point function of the test field $\phi$ using the renormalisation group improved stochastic formalism. The UV effective potential $V_{\rm eff}$ and the wavefunction renormalisation $Z(\mu)$ are given by equations (\[eq:effiSM\]), (\[Z\]) and (\[Yukawagamma\]) respectively. As the background spacetime, we choose the slow roll solution corresponding to quadratic inflation, such that $$\frac{H^2}{H_{\rm end}^2} = 1 + 2(N_{\rm end}-N) \,.$$ where $H_{\rm end} = 10^{13}{\rm GeV}$. We start computation at $N = 0$ and set $N_{\rm end } = 500$. We choose to define the input parameter values at the scale $\mu_0 = 3.2\times 10^{14}{\rm GeV}$. We have explicitly checked that the one loop terms $V^{(1)}_\phi$ and $V^{(1)}_\psi$ in equation (\[eq:effiSM\]) remain small throughout the computation. We determine the two-point function from a large number of numerically generated realisations of the Langevin process (\[langevin\]). As the Langevin solver we use an adapted version of the [`nfield`](https://sites.google.com/view/nfield-py) python code, publicly available at: <https://github.com/umbralcalc/nfield>. The results are shown in figure \[fig:Yukawa-plot\] which depicts the time evolution of the two-point function for two parameter sets. ![\[fig:Yukawa-plot\] The amplitude of the two-point function in Yukawa theory in a quadratic inflation background, with the following parameters used in all panels: $\xi_0=10^{-2}$; $\lambda_0=10^{-2}$; $g_0=10^{-1}$ and $m_0=0$. Solid lines denote the full numerical solution and dashed lines represent the corresponding stationary process. Each process has been numerically generated by a Langevin solver with $10^4$ realisations. Colours of the lines correspond to: the full renormalised solution for the optimal scale $\langle \bar{\phi}^2(\mu_*)\rangle$ in red, and for a fixed scale $\langle \bar{\phi}^2(\mu_0)\rangle$ in green. The naive solution with $Z$ neglected throughout is shown in blue. The fractional difference between the full and naive solutions defined in equation (\[eq:delta-diff-2p\]) is plotted in black below each figure. The top row plots unzoomed (left) and zoomed (right) axes setting $N_{\rm f}=1$, while the bottom row plots the equivalent with $N_{\rm f}=10$. ](figs/Yukawa_theory_plot_Nf1.pdf "fig:"){width="48.00000%"} ![\[fig:Yukawa-plot\] The amplitude of the two-point function in Yukawa theory in a quadratic inflation background, with the following parameters used in all panels: $\xi_0=10^{-2}$; $\lambda_0=10^{-2}$; $g_0=10^{-1}$ and $m_0=0$. Solid lines denote the full numerical solution and dashed lines represent the corresponding stationary process. Each process has been numerically generated by a Langevin solver with $10^4$ realisations. Colours of the lines correspond to: the full renormalised solution for the optimal scale $\langle \bar{\phi}^2(\mu_*)\rangle$ in red, and for a fixed scale $\langle \bar{\phi}^2(\mu_0)\rangle$ in green. The naive solution with $Z$ neglected throughout is shown in blue. The fractional difference between the full and naive solutions defined in equation (\[eq:delta-diff-2p\]) is plotted in black below each figure. The top row plots unzoomed (left) and zoomed (right) axes setting $N_{\rm f}=1$, while the bottom row plots the equivalent with $N_{\rm f}=10$. ](figs/Yukawa_theory_plot_Nf1_zoom.pdf "fig:"){width="48.00000%"}\ ![\[fig:Yukawa-plot\] The amplitude of the two-point function in Yukawa theory in a quadratic inflation background, with the following parameters used in all panels: $\xi_0=10^{-2}$; $\lambda_0=10^{-2}$; $g_0=10^{-1}$ and $m_0=0$. Solid lines denote the full numerical solution and dashed lines represent the corresponding stationary process. Each process has been numerically generated by a Langevin solver with $10^4$ realisations. Colours of the lines correspond to: the full renormalised solution for the optimal scale $\langle \bar{\phi}^2(\mu_*)\rangle$ in red, and for a fixed scale $\langle \bar{\phi}^2(\mu_0)\rangle$ in green. The naive solution with $Z$ neglected throughout is shown in blue. The fractional difference between the full and naive solutions defined in equation (\[eq:delta-diff-2p\]) is plotted in black below each figure. The top row plots unzoomed (left) and zoomed (right) axes setting $N_{\rm f}=1$, while the bottom row plots the equivalent with $N_{\rm f}=10$. ](figs/Yukawa_theory_plot_Nf10.pdf "fig:"){width="48.00000%"} ![\[fig:Yukawa-plot\] The amplitude of the two-point function in Yukawa theory in a quadratic inflation background, with the following parameters used in all panels: $\xi_0=10^{-2}$; $\lambda_0=10^{-2}$; $g_0=10^{-1}$ and $m_0=0$. Solid lines denote the full numerical solution and dashed lines represent the corresponding stationary process. Each process has been numerically generated by a Langevin solver with $10^4$ realisations. Colours of the lines correspond to: the full renormalised solution for the optimal scale $\langle \bar{\phi}^2(\mu_*)\rangle$ in red, and for a fixed scale $\langle \bar{\phi}^2(\mu_0)\rangle$ in green. The naive solution with $Z$ neglected throughout is shown in blue. The fractional difference between the full and naive solutions defined in equation (\[eq:delta-diff-2p\]) is plotted in black below each figure. The top row plots unzoomed (left) and zoomed (right) axes setting $N_{\rm f}=1$, while the bottom row plots the equivalent with $N_{\rm f}=10$. ](figs/Yukawa_theory_plot_Nf10_zoom.pdf "fig:"){width="48.00000%"} For comparison we have also plotted the solution of the Langevin equation (\[langevin\]) with the wave function renormalisation factor $Z$ omitted. As discussed above, this leads to incorrect result for the two-point function and fails to satisfy the Callan-Symanzik equation (\[eq:CZ0\]) except in the case of stationary solutions for which the probability distribution is given by equation (\[stationarysol\]). The equilibrium solution for light spectator fields will differ from the stationary limit, depicted by the dashed line in figure \[fig:Yukawa-plot\], if its relaxation time is longer than the time scale associated to the evolution of the slow roll background [@Hardwick:2017fjo]. In the figure we have also shown the difference between the full solution and the solution without the $Z$ factor defined as $$\delta_{\langle \bar{\phi}^2\rangle}\equiv \frac{\langle \bar{\phi}^2\rangle - \langle \bar{\phi}^2\rangle_{Z=1}}{\langle \bar{\phi}^2 \rangle }\,. \label{eq:delta-diff-2p}$$ It can be seen that the two point function $\langle \bar{\phi}^2\rangle_{Z=1}$ computed without the $Z$ factor starts to differ from the full result $\langle \bar{\phi}^2\rangle$ as soon as the Langevin process departs from the stationary limit. The difference grows as number of fermionic fields $N_{\rm f}$ is increased, since the anomalous dimension scales proportional to $N_{\rm f}$ according to equation (\[Yukawagamma\]). The deviation from $\langle \bar{\phi}^2\rangle$ is a signal of $\langle \bar{\phi}^2\rangle_{Z=1}$ failing to obey the RG scaling relation (\[eq:CZ0\]). Conclusions {#sec:conclusions} =========== In this work we have investigated how the renormalisation group running can be incorporated in the stochastic approach to inflationary infrared dynamics of light test scalars [@Starobinsky:1994bd]. By making use of the Wilsonian picture of renormalisation and starting from the quantum action, we have reformulated the stochastic approach in terms renormalised ultraviolet quantities with a running renormalisation scale. Our main results are equations (\[langevin\]) and (\[FPIto\]) which define the renormalisation group improved versions of the Langevin and Fokker-Planck equations. They differ from the corresponding standard equations in two aspects: the classical potential is replaced with the renormalisation group improved effective potential $V_{\rm eff}$ and the field renormalisation $Z$ enters in the drift term and in the stochastic noise term. We have explicitly demonstrated that the renormalisation group improved stochastic approach results $n$-point functions which obey the correct Callan-Symanzik equations. The field renormalisation $Z$ plays a key role here; a simple replacement of the classical potential with $V_{\rm eff}$ in the standard stochastic equations, as was done for example in references [@Espinosa:2015qea; @Herranen:2014cua; @Espinosa:2007qp], is compatible with the renormalisation group scaling only in the limit of stationary solutions where $Z$ cancels out. Beyond this limit it is necessary to use (\[langevin\]) or (\[FPIto\]) to maintain the correct RG scaling. As shown in reference [@Hardwick:2017fjo], the true equilibrium may significantly deviate from the stationary solution whenever the spacetime is not exactly de Sitter. In situations where quantum corrections and running couplings are significant, it is therefore important to use the full equations (\[langevin\]) and (\[FPIto\]) with the correct behaviour under the renormalisation group instead of the naive replacement. As a specific example, we have investigated the Yukawa theory in an inflationary slow-roll background. We have numerically computed the two point function of the test scalar using the renormalisation group improved stochastic formalism and illustrated the difference compared to the naive approach with the field renormalisation term neglected. Two obvious situations come to mind where the renormalisation group compatible approach to the stochastic infrared dynamics is required for reliable results. The first is when investigating the Standard Model model vacuum metastability in the early universe which is fundamentally a quantum effect and sensitively depends on the RG running [@Markkanen:2018pdo]. The second arises when dealing with models with a large number of fields that couple to a stochastic spectator, which can exhibit significant running and/or quantum corrections. In general, any situation involving the stochastic approach to inflation but where results beyond the simple tree-level ones are needed requires one to incorporate renormalisation group effects into the problem which, as shown here, can be performed in a consistent manner. Our results apply to energetically subdominant test fields but it should be straightforward to generalise the formalism to the inflaton sector as well. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Vincent Vennin, Kimmo Kainulainen and Arttu Rajantie for useful discussions during the development of this work. RJH was supported by UK Science and Technology Facilities Council grant ST/N5044245 throughout the duration of this work. TM is supported by the U.K. Science and Technology Facilities Council grant ST/P000762/1 and by the Estonian Research Council via the Mobilitas Plus grant MOBJD323. For efficiency, some numerical computations were performed on the Sciama High Performance Compute (HPC) cluster which is supported by the ICG, SEPNet and the University of Portsmouth. Running for the Yukawa theory in curved spacetime {#sec:analytic-runnings} ================================================== It is possible to find analytic solutions for the running couplings for the theory described by (\[eq:yu2\]) and (\[eq:treecurve\]). For completeness we will give the results in a general background, but for the case $m=0$. The one loop quantum correction on a general background can be written as [@Markkanen:2018pdo] [$$V^{(1)}_\phi(\phi)={\frac{{\mathcal{M}_\phi^4}}{{64\pi^2}}}\bigg[\ln \bigg({\frac{{|\mathcal{M}_\phi^2|}}{{\mu^2}}}\bigg)-{\frac{{3}}{{2}}}\bigg]+{\frac{{{\frac{{1}}{{90}}}\left(R_{\mu\nu\delta\eta}R^{\mu\nu\delta\eta}-R_{\mu\nu}R^{\mu\nu}\right)}}{{64\pi^2}}}\ln\bigg({\frac{{|\mathcal{M}_\phi^2|}}{{\mu^2}}}\bigg)\label{eq:curve30}\,,$$]{} and [$$\begin{aligned} V^{(1)}_\psi(\phi)=-{\frac{{4\mathcal{M}_\psi^4}}{{64\pi^2}}}\bigg[\ln\bigg({\frac{{|\mathcal{M}_\psi^2|}}{{\mu^2}}}\bigg)-{\frac{{3}}{{2}}}\bigg]+{\frac{{{\frac{{1}}{{90}}}\left({\frac{{7}}{{2}}}R_{\mu\nu\delta\eta}R^{\mu\nu\delta\eta}+4R_{\mu\nu}R^{\mu\nu}\right)}}{{64\pi^2}}}\ln\bigg({\frac{{|\mathcal{M}_\psi^2|}}{{\mu^2}}}\bigg)\label{eq:curve40}\,,\end{aligned}$$]{} with effective masses [$$\mathcal{M}^2_\phi\equiv m^2+3\lambda\phi^2+\left(\xi- {\frac{{1}}{{6}}}\right)R\,,\qquad \mathcal{M}^2_\psi\equiv g^2\phi^2+R/12\,.\label{eq:effm20}$$]{} For this theory to one loop order the Callan-Symanzik equation (\[CZforV\]) gives [$$\begin{aligned} g^2(s)&={\frac{{g^2_0}}{{1-{\frac{{5g^2_0}}{{8\pi^2}}}s}}} \\{\frac{{\lambda(s)}}{{g^2(s)}}}&=\frac{-(10- 8N_{\rm f})\lambda_0-16N_{\rm f} g_0^2+ f(s)\left[8 N_{\rm f} g^2_0+\left(5-4N_{\rm f}+\sqrt{(1+4N_{\rm f})(25+4N_{\rm f})}\right)\lambda_0\right] }{(10- 8N_{\rm f})g_0^2-36\lambda_0+f(s)\left[18 \lambda_0+\left(-5+4N_{\rm f}+\sqrt{(1+4N_{\rm f})(25+4N_{\rm f})}\right)g^2_0\right]} \\ \xi(s)&=\left(\xi_0-{\frac{{1}}{{6}}}\right)\exp\bigg\{{\frac{{1}}{{16\pi^2}}}\int_{0}^s\left[ 6\lambda(s') + 4N_{\rm f}g^2(s')\right] \dd s'\bigg\}+{\frac{{1}}{{6}}} \,,\end{aligned}$$]{} where in the above we use the notation [$$s\equiv \ln \left( {\frac{{\mu}}{{\mu_0}}} \right)\,;\quad f(s)\equiv \bigg({\frac{{g_0}}{{g(s)}}}\bigg)^{{\frac{{2}}{{5}}}\sqrt{(1+4N_{\rm f})(25+4N_{\rm f})}}+1\,,$$]{} and where for clarity we have not written the analytic solution to the integral for $\xi$ as the result is very lengthy. For the purely gravitational operators one has [$$\begin{aligned} {V_\Lambda}&={\kappa}={\rm const.}\,\\ \alpha_1&={\frac{{1}}{{16\pi^2}}}\int_{0}^s\left\{ {\frac{{1}}{{2}}}{\left[ \xi (s')-{\frac{{1}}{{6}}}\right]^2-{\frac{{N_{\rm f}}}{{72}}}}\right\} \dd s'+\alpha_{1,0}\,\\ \alpha_2&={\frac{{4N_{\rm f}-1}}{{16\pi^2}}}{\frac{{s}}{{180}}}+\alpha_{2,0}\,\\ \alpha_3&={\frac{{7N_{\rm f}+2}}{{16\pi^2}}}{\frac{{s}}{{360}}}+\alpha_{3,0}\,,\end{aligned}$$]{} where the special case of de Sitter space comes via (\[eq:alp\]). [^1]: The origin of the $Z^{-1/2}$ factor here follows directly from the scaling of the 2-point function in . [^2]: One can confirm that the corresponding Stratonovich process will follow the same argumentation. [^3]: The leading quantum correction to the kinetic term is subdominant as one may see e.g. from Eq. (3.2) of [@Kirsten:1993jn], in accordance with our discussion in section \[sec:Lange\]. [^4]: Evaluating the RHS of (\[eq:opt-scale\]) at the scale $\mu_0$ amounts to neglecting the running from the loop correction, which is a next-to-leading effect in the loop expansion.

--- abstract: 'Consider the trilinear form for twisted convolution on $\R^{2d}$: $$\T_t(\bf):=\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(x,y)}dxdy,$$ where $\sigma$ is a symplectic form and $t$ is a real-valued parameter. It is known that in the case $t\neq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $\sigma$.' author: - 'Kevin O’Neill' title: A Sharpened Inequality for Twisted Convolution --- Introduction ============ Young’s convolution inequality states that for dimensions $d\geq1$ and functions $f\in L^p(\R^d), g\in L^q(\R^d)$, $$\label{eq:Young's} ||f*g||_{L^r}\leq \bA^d_{\bp}||f||_{L^p}||g||_{L^q},$$ where $p,q,r\in[1,\infty]$ with $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$. $\bA^d_{\bp}=\prod_{j=1}^3 C_{p_j}^d$ is the optimal constant, where $C_p=p^{1/p}/p'^{1/p'}$, and $p'$ is the conjugate exponent of $p$ [@MR0385456], [@MR0412366]. For the purpose of this paper, it is convenient to use the following, related trilinear form: $$\T(f_1,f_2,f_3)=\iint f_1(x)f_2(y)f_3(x+y)dxdy.$$ Through duality, one may rewrite as $$\label{eq:Young's trilinear} \left|\T(\bf)\right| \leq \bA^d_{\bp}\prod_{j=1}^3||f_j||_{L^{p_j}}$$ for all $\bf=(f_j\in L^{p_j}(\R^d):j=1,2,3)$, with $\sum p_j^{-1}=2$ and $\bp=(p_j:j=1,2,3)\in[1,\infty]^3$. From here on out, we take $p_j\in(1,\infty)$. In [@MR0412366], Brascamp and Lieb show that the maximizers of are precisely the triple of Gaussians $\bg =(e^{-\pi p_j'|x|^2}: j=1,2,3)$ and its orbit under the following symmetries. - $(f_1,f_2,f_3)\mapsto(af_1,bf_2,cf_3)$ for $a,b,c\neq0$. (Scaling) - $(f_1,f_2,f_3)\mapsto(M_{\xi}f_1,M_{\xi}f_2,M_{-\xi}f_3)$, where $M_{\xi}f(x)=e^{ix\cdot\xi}$ for $\xi\in\R^d$. (Modulation) - $(f_1,f_2,f_3)\mapsto(\tau_{v_1}f_1,\tau_{v_2}f_2,\tau_{v_1+v_2}f_3)$, where $\tau_vf(x)=f(x+v)$ for $v\in\R^d$. (Translation) - $(f_1,f_2,f_3)\mapsto(f_1\circ\psi,f_2\circ\psi,f_3\circ\psi)$, where $\psi$ is an invertible linear map on $\R^d$. (Diagonal Action of the General Linear Group) Note that these symmetries do not necessarily preserve $|\T(\bf)|$, but they do preserve $|\Phi(\bf)|$, where $\Phi(\bf):=\frac{\T(\bf)}{\prod_j||f_j||_{p_j}}$. Let $\scriptO_C(\bf)$ denote the orbit of the triple $\bf$ under the above symmetries. Define the distance from $\bg$ to $\scriptO_C(\bf)$ as $$\label{eq:define distance} \dist_\bp(\scriptO_C(\bf),\bg):=\inf_{\bh\in\scriptO_C(\bf)}\max_j||h_j-g_j||_{p_j}.$$ Note that the symmetries of an operator preserve the (normalized) distance of a triple from the manifold of maximizers. Christ [@ChristSY] proved the following quantitative stability theorem for Young’s convolution inequality. \[thm:ChristSY\] Let $K$ be a compact subset of $(1,2)^3$. Let $\bp$ satisfy $\sum_{j=1}^3p_j^{-1}=2$. For each $d\geq1$, there exists $c>0$ such that for all $\bp\in K$ and all $\bf\in L^\bp(\R^d)$, $$|\T(\bf)|\leq \left(\bA^d_\bp-c\dist_\bp(\scriptO_C(\bf),\bg)^2\right)\prod_j||f_j||_{p_j}.$$ One may instead state the above theorem in terms of the distance of a triple $\bf$ from the set of all triples of maximizers (that is, $\scriptO_C(\bg)$), as is done in [@ChristSY]. However, the distance defined in is more useful for analogy with our current analysis. It is also shown that the conclusion of Theorem \[thm:ChristSY\] is true for $\bp\in(1,2]^3$ provided one does not require the same $c$ for all $\bp$ in a region. (However it is not known if this uniformity fails.) Furthermore, the conclusion in this particular quantitative form is false whenever any $p_j=1$ or $p_j>2$. The purpose of this paper is to prove a similar quantitative stability result for twisted convolution. Let $t\geq0$ be a parameter and let $f_j\in L^{p_j}(\R^{2d})$, where $\R^{2d}$ is viewed as $\R^d\times\R^d=\{(x',x'''):x',x''\in\R^d\}$. Define the trilinear twisted convolution form with parameter $t$ as $$\label{eq:deftform} \T_t(\bf):=\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(x,y)}dxdy,$$ where $\sigma(x,y)=x'\cdot y''-x''\cdot y'$ is the symplectic form. It is often useful to write $\sigma(x,y)=x^tJy$, where $J$ is the matrix $$J=\begin{pmatrix} 0&I_d\\ -I_d&0 \end{pmatrix},$$ and $I_d$ is the $d\times d$ identity matrix. When $t=0$, becomes the trilinear form representing convolution. When $t\neq0$, it is obvious through the inequality $$\label{eq:abs values} |\T_t(\bf)|\leq\T(|\bf|)$$ that $\T_t$ is bounded for any triple $\bp$ of exponents for which $\T$ is bounded. It is also known that for $t\neq0$, $\T(\bf,t)$ is also bounded for $(p_1,p_2,p_3)=(2,2,2)$ and the full range of exponents implied by interpolation (see Chapter XII.4 of [@SteinHA], for instance). However, the particular conclusion we desire is false in the case $\sum_jp_j^{-1}\neq2$ since $\T_0=\T$ is unbounded. By , it is easy to see that $\T_t$ has norm at most $A_\bp^{2d}$, the optimal constant for Young’s convolution inequality. Furthermore, the optimal constant may be seen to equal $A_\bp^{2d}$ by taking a triple of Gaussians which optimize Young’s inequality and dilating them to concentrate at the origin so the oscillation of the twisting factor has negligible effect. However, no extremizers of $\T_t$ exist for fixed $t\neq0$. [@KleinRusso] One challenge to dealing with the above form directly arises because the symmetry group of $\T$ contains the general linear group $Gl(2d)$, while $\T_t$ does not; the only linear transformations which preserve $\sigma$ are the symplectomorphisms. To avoid this issue, it helps to introduce the following trilinear form: $$\label{eq:defLform} \T_A(\bf):=\iint f_1(x)f_2(y)f_3(x+y)e^{it\sigma(Ax,Ay)}dxdy,$$ where $A:\R^{2d}{\rightarrow}\R^{2d}$ is an arbitrary linear map. Replacing $x$ with $Lx$ and $y$ with $Ly$ for an invertible matrix $L$ sends $A$ to $A\circ L$, and the functional remains of the form . Boundedness properties of $\T_A$ follow directly from those of $\T_t$ and a change of coordinates. The symmetries of $\T_A$ are similar to the those of $\T$ with some slight modifications, though they reduce to the symmetries of $\T(\bf)$ when $A=0$. Here, the symmetries preserve $|\Phi(\bf,A)|$, where $\Phi(\bf,A)=\frac{\T(\bf,A)}{\prod_j||f_j||_{p_j}}$. - $(f_1,f_2,f_3,A)\mapsto(af_1,bf_2,cf_3,A)$, where $a,b,c\in\C$. (Scaling) - $(f_1,f_2,f_3,A)\mapsto(M_{\xi}f_1,M_{\xi}f_2,M_{-\xi}f_3,A)$. (Modulation) - $(f_1,f_2,f_3,A)\mapsto(M_{A^TJAv_2}\tau_{v_1}f_1,M_{-A^TJAv_1}\tau_{v_2}f_2,\tau_{v_1+v_2}f_3,A)$, where $A^T$ represents the transpose of the matrix $A$. (Translation/ Modulation Mix) - $(f_1,f_2,f_3,A)\mapsto\T(f_1\circ\psi,f_2\circ\psi,f_3\circ\psi,A\circ\psi)$, where $\psi\in Gl(d)$. (Diagonal Action of the General Linear Group) Note that only the last of these symmetries alters $A$. Let $\scriptO_{TC}(\bf,A)$ denote the orbit of $(\bf,A)$ under the above symmetries. Now, it is less obvious how to represent the distance of $A$ from the zero transformation than it was when our parameter was just a real number $t$. One may naively suggest that $||A||$ will play a role, but this approach ignores the role of the symplectic group. The real symplectic group $Sp(2d)$ is defined as the set of invertible $(2d)\times(2d)$ matrices $S$ such that $S^TJS=J$. Equivalently, $Sp(2d)$ may be viewed as the set of coordinate changes which preserve $\sigma$. Under this view, we see that $\sigma(Ax,Ay)=\sigma(SAx,SAy)$ for any $S\in Sp(2d)$. Thus, replacing $A$ with $S\circ A$ should not change our distance. With this in mind, define the distance from $\scriptO_{TC}(\bf,A)$ to $(\bg,0)$ by $$\label{eq:define TC distance} \dist_\bp(\scriptO_{TC}(\bf,A),(\bg,0))^2:=\inf_{(\bh,M)\in\scriptO_{TC}(\bf,A)}\left[\max_j||h_j-g_j||_{p_j}^2+||M^TJM||^2\right]$$ A useful fact in analyzing this distance is that $\inf_{S\in Sp(2d)}||S\circ A||^2=||A^TJA||$. (See Lemma 10.1 of [@ChristHeisenberg].) Define $||\bf||_\bp=\max_j||f_j||_{p_j}$. We now state our main theorem. \[theorem:main\] Let $K$ be a compact subset of $(1,2)^3$. For each $d\geq1$, there exists $c>0$ such that for all $\bp\in K$ with $\sum_{j=1}^3p_j^{-1}=2$, $\bf\in L^\bp(\R^{2d})$, and $(2d)\times(2d)$ matrices $A$, $$|\T_A(\bf)|\leq \left(\bA^{2d}_\bp-c\dist_\bp(\scriptO_{TC}(\bf,A),(\bg,0))^2\right)\prod_j||f_j||_{p_j}.$$ By setting $A=t^{1/2}I_{2d}$ in Theorem \[theorem:main\] (where $I_{2d}$ is the $(2d)\times(2d)$ identity matrix), one obtains the following corollary. However, one is cautioned that the orbit in this expression refers to the symmetries of $\T_A$, not those of $\T_t$. \[corollary:only\] Let $K$ be a compact subset of $(1,2)^3$. For each $d\geq1$, there exists $c>0$ such that for all $\bp\in K$ with $\sum_{j=1}^3p_j^{-1}=2$, $\bf\in L^\bp(\R^{2d})$, and $|t|\leq1$, $$|\T_t(\bf)|\leq \left(\bA^{2d}_\bp-c\dist_\bp(\scriptO_{TC}(\bf,t^{1/2}I_{2d}),(\bg,0))^2\right)\prod_j||f_j||_{p_j}.$$ The reason one uses $t^{1/2}I_{2d}$ rather than $tI_{2d}$ is so the $||M^TJM||^2$ term appearing in is proportional to $t^2$, rather than $t^4$. An alternative form of Corollary \[corollary:only\] states the function $\epsilon(\delta)$ in Theorem \[thm:perturbative1\] may be taken to be $C\sqrt{\delta}$ for some $C>0$. The methods in this paper follow the general approach found in [@ChristSY] and [@BE] in which one takes a Taylor-like expansion of the given operator and diagonalizes the resulting quadratic form. We will often use $C$ or $c$ to denote an arbitrary constant in $(0,\infty)$ which may change from line to line but always be independent of functions found in the equation. Reduction to Perturbative Case ============================== Our argument centers around an expansion of $\T(\bf,A)$ which requires a reduction to small perturbations. To this end, the following result from [@ChristHeisenberg] is essential. \[thm:perturbative1\] Let $d\geq 1$. Let $K$ be a compact subset of $(1,2)^3$ for which each $\bp\in K$ satisfies $\sum_{j=1}^3p_j^{-1}=2$. Then, there exists a function $\delta\mapsto\epsilon(\delta)$ (depending only on $K$ and $d$) satisfying $\lim_{\delta{\rightarrow}0}\epsilon(\delta)=0$ with the following property. Let $\bf\in L^\bp(\R^{2d})$ and suppose that $||f_j||_{p_j}\neq0$ for each $1\leq j\leq3$. Let $\delta\in(0,1)$ and suppose that $|\T(\bf,t)|\geq(1-\delta)\bA_\bp^{2d}\prod_j||f_j||_{p_j}$. Then there exist $S\in Sp(2d)$ and a triple of Gaussians $\mathbf{G}=(G_1,G_2,G_3)$ such that $G_j^\natural=G_j\circ S$ satisfy $$||f_j-G_j^\natural||_{p_j}<\epsilon(\delta)||f_j||_{p_j}$$ for $1\leq j\leq3$ and $$G_j(x)=c_je^{\pi p_j'|L(x-a_j)|^2}e^{ix\cdot v}e^{it\sigma(\tilde{a_j},x)}$$ where $v\in\R^{2d}, 0\neq c_j\in\C, a_1+a_2+a_3=0, \tilde{a_3}=0,\tilde{a_1}=a_2,\tilde{a_2}=a_1, L\in Gl(2d)$, and $$|t|\cdot||L^{-1}||^2\leq\epsilon(\delta).$$ Here is a rephrasing of Theorem \[thm:perturbative1\]. \[thm:perturbative2\] Let $d\geq 1$. Let $K$ be a compact subset of $(1,2)^3$ for which each $\bp\in K$ satisfies $\sum_{j=1}^3p_j^{-1}=2$. Then, there exists a function $\delta\mapsto\epsilon(\delta)$ (depending only on $K$ and $d$) satisfying $\lim_{\delta{\rightarrow}0}\epsilon(\delta)=0$ with the following property. Let $\bf\in L^\bp(\R^{2d})$ and suppose that $||f_j||_{p_j}\neq0$ for each $1\leq j\leq3$. Let $\delta\in(0,1)$ and suppose that $|\T_A(\bf)|\geq(1-\delta)\bA_\bp^{2d}\prod_j||f_j||_{p_j}$. Then, $$\label{eq:pert2conc} \dist_\bp(\scriptO_{TC}(\bf,A),(\bg,0))<\epsilon(\delta)$$ By a standard approximation argument, it suffices to prove Theorem \[thm:perturbative2\] for invertible maps $A$, as each noninvertible map is arbitrarily close to an invertible map. Suppose that $|\T(\bf,A)|\geq(1-\delta)\bA_\bp^{2d}\prod_j||f_j||_{p_j}$. Then invoking the symmetry of diagonal action of the general linear group, $$\label{eq:filler label} |\T(\bf\circ A^{-1},I_{2d})|\geq(1-\delta)\bA_\bp^{2d}\prod_j||f_j\circ A^{-1}||_{p_j},$$ where $\bf\circ A^{-1}=(f_j\circ A^{-1}:j=1,2,3)$. Applying Theorem \[thm:perturbative1\] under the case $t=1$, there exists $S_0\in Sp(2d)$ and a triple of Gaussians $\mathbf{G}=(G_1,G_2,G_3)$ such that $$\label{eq:norm control} ||f_j\circ A^{-1}-G_j\circ S_0||_{p_j}<\epsilon(\delta)||f_j\circ A^{-1}||_{p_j}$$ for $1\leq j\leq3$ and $$G_j(x)=c_je^{\pi p_j'|L(x-a_j)|^2}e^{ix\cdot v}e^{it\sigma(\tilde{a_j},x)}$$ where $v\in\R^{2d}, 0\neq c_j\in\C, a_1+a_2+a_3=0, \tilde{a_3}=0,\tilde{a_1}=a_2,\tilde{a_2}=a_1, L\in Gl(2d)$, and $$||L^{-1}||^2\leq\epsilon(\delta).$$ By a combination of translations, modulations, scalings, and compositions with invertible linear maps, becomes $$||h_j-g_j||_{p_j}<\epsilon(\delta)||h_j||_{p_j},$$ where $h_j$ is $f_j\circ A^{-1}$ composed with said operations. Since $\mathbf{G}$ was the composition of $\bg$ with the stated symmetries of $\T_A$, we see that $\bh$ is obtained by the composition of $\bf\circ A^{-1}$ with symmetries of $\T_A$ by the following reasoning. Three of these symmetries (scaling, modulation, and the diagonal action of the general linear group) may trivially be inverted by symmetries of the same form. To address the inversion of the translation/modulation mix, one observes that $\tau_{w_j}M_{B^TJB\tilde{w_j}}f=e^{iB^TJB\tilde{w_j}\cdot w_j}M_{B^TJB\tilde{w_j}}\tau_{w_j}f$ for matrices $B$ and vectors $w_j$. Hence, $\bh$ is obtained from $\bf\circ A^{-1}$ through the inverses of the symmetries applied initially to $\bg$ to obtain $\mathbf{G}$ but with an additional scaling symmetry. The only above symmetry which changes the matrix $B$ in $\T_B$ is the diagonal action of the general linear group. Following the use of this symmetry above, one obtains from that $(\bh,S_0^{-1}\circ L^{-1})\in\scriptO_{TC}(\bf,A)$. We now see that $$\begin{aligned} \dist_\bp(\scriptO_{TC}(\bf,A),(\bg,0))^2&\leq\max_j||h_j-g_j||_{p_j}^2+\inf_{S\in Sp(2d)}||S\circ S_0^{-1}\circ L^{-1}||^4\\ &\leq \epsilon(\delta)^2+||S_0S_0^{-1}\circ L^{-1}||^4\\ &\leq \epsilon(\delta)^2+||L^{-1}||^4\leq 2\epsilon(\delta)^2.\end{aligned}$$ As a corollary to Theorem \[thm:perturbative2\], it suffices to prove Theorem \[theorem:main\] in the case in which $\dist_\bp(\scriptO_{TC}(\bf,A),(\bg,0))<\delta_0$ for some $\delta_0>0$. Theorem \[thm:perturbative2\] guarantees that there are no sequences of $(\bf_n,A_n)$ at distance greater than $\delta_0$ such that $\T_{A_n}(\bf_n)/(\prod_j||f_{n,j}||_{p_j})$ converges to $A_\bp^{2d}$. Thus, for $(\bf,A)$ at distance at least $\delta_0$, $\T_A(\bf)$ must have a maximum strictly less than $A_\bp^{2d}$. While $||A^TJA||{\rightarrow}\infty$ for an appropriate sequence of matrices $A$, $\dist_\bp(\scriptO_{TC}(\bf,A),(\bg,0))$ remains bounded above as th symmetries of $\T_A$ ensure there exists $(\bh,M)\in\scriptO_{TC}(\bf,A)$ with $||M^TJM||\leq1$. Therefore, the conclusion of Theorem \[theorem:main\] holds for distances greater than $\delta_0$. Treating Some Terms of the Expansion ==================================== In this section, we consider $\T_A(\bg+\bf)$, where $A$ is a $(2d)\times(2d)$ matrix, $\bg =(g_j=e^{-\pi p_j'|x|^2}: j=1,2,3)$ and $\bf\in L^{\bp}(\R^{2d})$ are small perturbations. (This change in notation of $\bf$ from functions close to $\bg$ to the differences will continue for the remainder of the paper.) As in [@ChristSY], we may assume $\int g_j^{p_j-1}f_j=0$ via the scaling symmetry. In short, we will expand $\T(\bg+\bf,A)=\T_0(\bg+\bf)+(\T_A-\T_0)(\bg+\bf)$ and use the multilinearity of $\T_0$ and $\T_A$ to get sixteen terms of eight different types. Before writing out the expansion, we prove a few lemmas about its terms and describe a useful decomposition. Following [@ChristHY] and [@ChristSY], let $\eta>0$ be a small parameter to be chosen later (see Proposition \[prop:only\]). For each $1\leq j\leq3$, decompose $f_j=f_{j,\sharp}+f_{j,\flat}$, where $$\label{eq:def fsharp} f_{j,\sharp}=\left\{ \begin{array}{lr} f_j(x) \text{ if }|f_j(x)|\leq \eta g_j(x)\\ 0 \text{ otherwise, } \end{array} \right.$$ and $f_{j,\flat}=f_j-f_{j,\sharp}$. The purpose of this decomposition is twofold. First, it is used in the analysis of [@ChristSY] to analyze the quadratic form in the expansion with $L^2$ functions. Using the same decomposition allows us to borrow from that analysis in Proposition \[prop:only\], a version of Theorem \[thm:ChristSY\] with an additional favorable term. Second, the decomposition is used to reduce to the case of $f_j=f_{j,\sharp}$, which concentrates closer to the origin, allowing for control of the third order term in Lemma \[lemma:3rd order\]. \[lemma:trivial\] $(\T_A-\T_0)(\bf)=O(||\bf||_\bp^3).$ This claim follows trivially from the uniform boundedness of $\T_A$ and $\T_0$. The following lemma represents our main use of the $f_j=f_{j,\sharp}+f_{j,\flat}$ decomposition and the swapping of $f_j$ for $f_{j,\sharp}$ will be justified later. \[lemma:3rd order\] $(\T_A-\T_0)(f_{1,\sharp},f_{2,\sharp},g_3)=o(||\bf||_\bp^2+||A^TJA||^2)$ with decay rate depending only on $\eta$. Lemma \[lemma:3rd order\] also applies to the other two terms of this type. Note that the trivial bound $$\label{eq:trivial bound} |(\T_A-\T_0)(h_1,h_2,g_3)|=O\left(||h_1||_{p_1}||h_2||_{p_2}\right)$$ is insufficient to deal with the above term directly since it provides a second order control of a term which should heuristically be third order. However, still plays a useful role in the proof of Lemma \[lemma:3rd order\]. First, suppose that $||A^TJA||^3\geq||f_{1,\sharp}||_{p_1}||f_{2,\sharp}||_{p_2}$. Note that by our reduction to small perturbations in Theorem \[thm:perturbative2\], $||A^TJA||$ may be taken small enough that $||A^TJA||^3\leq||A^TJA||^2$. By , $$(\T_A-\T_0)(f_{1,\sharp},f_{2,\sharp},g_3)\leq C||f_{1,\sharp}||_{p_1}||f_{2,\sharp}||_{p_2}\leq||A^TJA||^3=o(||\bf||_\bp^2+||A^TJA||^2)$$ and we are done. So suppose that $||A^TJA||^3<||f_{1,\sharp}||_{p_1}||f_{2,\sharp}||_{p_2}$. Now, for $j=1,2$, write $f_{j,\sharp}=f_{j,\sharp,\leq M_j}+f_{j,\sharp,>M_j}$, where $f_{j,\sharp,\leq M_j}=f_{j,\sharp}\one_{B(0,M_j)}$ and $f_{j,\sharp,>M_j}=f_{j,\sharp}\one_{B(0,M_j)^c}$. In the above, $\one_E$ refers to the indicator function of the set $E$, $B(x_0,R)$ refers to the closed ball of radius $R$ centered at $x_0$, $E^c$ is the complement of the set $E$, and $M_j$ is chosen so that $$\label{eq:smallnorm} ||f_{j,\sharp,>M_j}||_{p_j}=||f_{j,\sharp}||_{p_j}^2.$$ Note that $M_j$ is dependent on $\eta$. We claim that $M_j\leq C\log(||f_{j,\sharp}||_{p_j}^{-1})$. To see this, observe that for given $\eta$ and $||f_{j,\sharp}||_{p_j}$ and varying $f_{j,\sharp}$, $M_j$ is maximized when $f_{j,\sharp}=\eta g_j$ on $B(0,M)^c$ and $f_{j,\sharp}=0$ on $B(0,M)$, where $M$ is the positive real number that leads to the appropriate value of $||f_{j,\sharp}||_{p_j}$. (Here, $M<M_j$ since $||f_{j,\sharp}||_{p_j}$ is small.) It suffices to find an upper bound for $M_j$ in this scenario. We integrate with respect to spherical coordinates to obtain $$\begin{aligned} ||f_{j,\sharp}||_{p_j}^2&=||f_{j,\sharp,>M_j}||_{p_j}\\ &=\int_{S^{d-1}}\left[\int_{M_j}^\infty \eta e^{-\pi p_j'r^2}r^{2d-1}dr\right]d\sigma(\theta)\\ &=C_d\eta\int_{M_j}^\infty \eta e^{-\pi p_j'r^2}r^{2d-1}dr\\ &=O(M_j^{2d-2}e^{-\pi p_j'M_j^2}).\end{aligned}$$ Thus, $||f_{j,\sharp}||_{p_j}\leq Ce^{-M_j}$, proving our claim. Expand $$\begin{gathered} (\T_A-\T_0)(f_{1,\sharp},f_{2,\sharp},g_3)=(\T_A-\T_0)(f_{1,\sharp,>M_1},f_{2,\sharp,>M_2},g_3)+(\T_A-\T_0)(f_{1,\sharp,>M_1},f_{2,\sharp,\leq M_2},g_3)\\ +(\T_A-\T_0)(f_{1,\sharp,\leq M_1},f_{2,\sharp,>M_2},g_3)+(\T_A-\T_0)(f_{1,\sharp,\leq M_1},f_{2,\sharp,\leq M_2},g_3)\end{gathered}$$ The first three of these terms may be treated by combining the trivial bound with . Let $R=B(0,M_1)\times B(0,M_2)\subset \R^{2d}\times\R^{2d}$. The absolute value of the remaining term is $$\begin{aligned} |(\T_A-\T_0)(f_{1,\sharp},f_{2,\sharp},g_3)|&\leq\iint_R|f_{1,\sharp}(x)|\cdot|f_{2,\sharp}(y)|\cdot g_3(x+y)\cdot|\sigma(Ax,Ay)|dxdy\\ &\leq C||f_{1,\sharp}||_{p_1}||f_{2,\sharp}||_{p_2}||g_3||_{p_3}||A^TJA||M_1M_2\\ &\leq C||f_{1,\sharp}||_{p_1}^{4/3}||f_{2,\sharp}||_{p_2}^{4/3}\log(||f_{1,\sharp}||_{p_1}^{-1})\log(||f_{2,\sharp}||_{p_2}^{-1})=o(||\bf||_\bp^2)\end{aligned}$$ \[lemma:onesigma\] For all $f\in L^{p_1}(\R^{2d})$ $$\iint f(x)g_2(y)g_3(x+y)\sigma(Ax,Ay)dxdy=0$$ The conclusion also applies to the same integral with $(g_1,f,g_3)$ or $(g_1,g_2,f)$ in place of $(f,g_2,g_3)$ (with $f\in L^{p_j}$ for the appropriate $j\in\{1,2,3\}$). Since $\sigma(Ax,Ay)=x^TA^TJAy$ is an antisymmetric bilinear form, we may diagonalize $A^TJA$ as $Q^T\Sigma Q$ for some orthogonal $Q$ and $$\Sigma=\begin{pmatrix} 0&a_1&...&0&0\\ -a_1&0&...&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&...&0&a_d\\ 0&0&...&-a_d&0 \end{pmatrix},$$ where $a_k\in\R$ and $\pm a_ki$ are the eigenvalues of $A^TJA$. Since $g_j(x)=e^{-\pi p_j'|x|^2}$, $g_2$ and $g_3$ remain unchanged under an orthogonal change of coordinates. Thus, the above is equal to and we may write the above as $$\iint f(Qx)g_2(y)g_3(x+y)\sum_{k=1}^d a_k(x_{2k-1}y_{2k}-x_{2k}y_{2k-1})dxdy.$$ Since $f(x)$ is an arbitrary function of $x$, $f(Qx)$ is also an arbitrary function of $x$, so it suffices to show that $$\int g_2(y)g_3(x+y)\sum_{k=1}^d a_k(x_{2k-1}y_{2k}-x_{2k}y_{2k-1})dy=0$$ for all $x\in\R^{2d}$. By linearity and permutation of coordinates, it suffices to show that $$\int g_2(y)g_3(x+y)(x_1y_2-x_2y_1)dy=0.$$ Writing $e^{-\pi p_j'|w|^2}=e^{-\pi p_j'(w_1^2+w_2^2)}e^{-\pi p_j'(w_3^2+...+w_{2d}^2)}$, the above integral factors into $$\int g_2(y_1,y_2)g_3(x_1+y_1,x_2+y_2)(x_1y_2-x_2y_1)dy_1dy_2\cdot\int g_2(\tilde{y})g_3(\tilde{x}+\tilde{y})d\tilde{y},$$ where $x=(x_1,x_2,\tilde{x}),y=(y_1,y_2,\tilde{y})$, and through abuse of notation, $g_j(w)=e^{-p_j'|w|^2}$ for $w$ in any dimension. It now suffices to show the first factor is zero. Expanding this factor gives $$\begin{gathered} x_1\int y_2g_2(y_2)g_3(x_2+y_2)dy_2\cdot\int g_2(y_1)g_3(x_1+y_1)dy_1\\-x_2\int y_1g_2(y_1)g_3(x_1+y_1)dy_1\cdot\int g_2(y_2)g_3(x_2+y_2)dy_2.\end{gathered}$$ An elementary computation shows that $g_2*g_3=Cg_1$ and $\int yg_2(y)g_3(x+y)dy=C'xg_1(x)$, hence the above becomes $$x_1\cdot C'x_2g_1(x_2)\cdot Cg_1(x_1)-x_2\cdot C'x_1g_1(x_1)\cdot Cg_1(x_2)=0.$$ If $S$ is a list of parameters, let $A\approx_S B$ mean there exists a $C>0$ depending only on elements of $S$ such that $A\leq CB$ and $B\leq CA$. \[lemma:twosigma\] For $\bg$ and $A$ as above, $$\iint g_1(x)g_2(y)g_3(x+y)\sigma^2(Ax,Ay)dxdy\approx_{d,\bp}||A^TJA||^2.$$ As in the proof of Lemma \[lemma:onesigma\], one may use an orthogonal change of coordinates to reduce to the computation of $$\iint g_1(x)g_2(y)g_3(x+y)\left[\sum_{k=1}^da_k(x_{2k-1}y_{2k}-x_{2k}y_{2k-1})\right]^2dxdy.$$ Expanding the square gives $$\sum_{j,k=1}^da_ja_k\iint g_1(x)g_2(y)g_3(x+y)(x_{2j-1}y_{2j}-x_{2j}y_{2j-1})(x_{2k-1}y_{2k}-x_{2k}y_{2k-1})dxdy.$$ By factoring the $g_j$ and computing the above integrals two coordinates at a time as in the proof of Lemma \[lemma:onesigma\], one finds that the cross terms are zero. Thus, the original integral is equal to a function to $d$ and $\bp$ alone times $\sum_{k=1}^da_k^2$. Recall that $\pm a_ki$ are the eigenvalues of $A^TJA$, so $||A^TJA||^2=\max_k|a_k|^2$ and the two expressions are equivalent. At this point, it is tempting to expand $\T_A(\bg+\bf)$, using the previous four lemmas to treat the $(\T_A-\T_0)$ terms (to get $-c||A^TJA||^2$) and Theorem \[thm:ChristSY\] to treat the $\T_0$ terms (and get $A_\bp^{2d}-c||\bf||_\bp^2$). However, Theorem \[thm:ChristSY\] may only be applied directly when the perturbative terms $f_j$ represent the projective distance from the orbit of the original functions to $\bg$. The subtle difference here is that the $f_j$ which represent the minimum value of $||\bf||_\bp^2$ may not be the same functions which represent the minimum value of $||\bf||^2+||A^TJA||^2$. For this reason, we will delve somewhat into the proof of Theorem \[thm:ChristSY\] and show that it is possible to obtain the same circumstances which lead to a $-c||\bf||^2$ decay. Balancing Lemma =============== For $t>0$ and $n=0,1,2,...,$ let $P_n^{(t)}$ denote the real-valued polynomial of degree $n$ with positive leading coefficient and $||P_n^{(t)}e^{-t\pi x^2}||_{L^2(\R)}=1$ which is orthogonal to $P_k^{(t)}e^{-t\pi x^2}$ for all $0\leq k<n$. For $d>1$, $\alpha=(\alpha_1,...,\alpha_d)\in\{0,1,2,...\}^d$, and $x=(x_1,...,x_d)\in\R^d$, define $$P_\alpha^{(t)}(x)=\prod_{k=1}^dP_{\alpha_k}^{(t)}(x_k).$$ Let $\tau_j=\frac{1}{2}p_jp_j'$ ($j=1,2,3$). In [@ChristSY], the following is proved en route to the main theorem. \[prop:only\] Let $\delta_0>0$ be sufficiently small. There exists $c,\tilde{c}>0$ and a choice of $\eta>0$ in the $f_j=f_{j,\sharp}+f_{j,\flat}$ decomposition such that the following holds. Suppose $||\bf||_\bp<\delta_0$ and $f_j$ satisfy the following orthogonality conditions: - $\langle Re(f_j),P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle=0$ whenever $\alpha=0$, $|\alpha|=1$ and $j\in\{1,2\}$, or $|\alpha|=2$ and $j=3$. - $\langle Im(f_j),P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle=0$ whenever $\alpha=0$ or $|\alpha|=1$ and $j=3$. Then, $$\label{eq:Christ conclusion} \frac{\T_0(\bg+\bf)}{\prod_j||g_j+f_j||_{p_j}}\leq A_\bp^{2d}-c||\bf||_\bp^2-\tilde{c}\sum_j||f_{j,\flat}||_{p_j}^{p_j}.$$ The above proposition is not stated as an explicit result of [@ChristSY]. However, is, in effect, the penultimate line of the proof of Theorem \[thm:ChristSY\] in Section 8 of [@ChristSY]. (The one difference is that $c||\bf||_\bp^2$ is replaced by $\sum_j||f_{j,\sharp}g_j^{(p_j-2)/2}||_2^2$ in the line in [@ChristSY], though it is shown the latter majorizes a constant multiple of the former.) We cite this particular intermediate result in order to take advantage of the $f_j=f_{j,\sharp}+f_{j,\flat}$ decomposition. The terms in Lemma \[lemma:3rd order\] involve $f_{j,\sharp}$ in place of $f_j$ so is used to deal with the case that $f_{j,\flat}$ makes up a significant portion of the $L^{p_j}$ norm of $f_j$. The goal of this section is to reduce to the situation in which the hypotheses of Proposition \[prop:only\] apply. This is done through the use of the following balancing lemma. Let $d\geq1$ and $\bp\in(1,2]^3$ with $\sum_jp_j^{-1}=2$. There exists $\delta_0>0$ such that if $||F_j-g_j||_{p_j}\leq\delta_0$, $||A^TJA||\leq\delta_0$, and $\langle F_j-g_j,g_j^{p_j-1}\rangle=0$, then there exist $v_j\in \R^{2d}$ satisfying $v_1+v_2+v_3=0$, $a_j\in\C$, $\xi\in\R^{2d}$, and a $(2d)\times(2d)$ matrix $\psi$ such that $$\sum_j(|v_j|+|a_j-1|)+||\psi-I_{2d}||+|\xi|\leq C\left[\left(\sum_j||f_j-g_j||_{p_j}\right)^2+||A^TJA||^2\right]$$ and the orthogonality conditions of Proposition \[prop:only\] hold for the functions $$\tilde{F_j}(x)=a_jF_j(\psi(x)+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x},$$ where $\tilde{v_1}=v_2, \tilde{v_2}=v_1$, and $\tilde{v_3}=0$. Begin by writing $h_j=g_j-F_j$ and $\tilde{h_j}=g_j-\tilde{F_j}$, where $\tilde{F_j}=a_jF_j(\psi(x)+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}$ and $\psi, v_j, a_j,\xi$ are to be determined. Letting $a_j=1+b_j$ and subbing in $f_j=g_j+h_j$, $$\begin{aligned} \tilde{h_j}(x)&=a_jF_j(\psi(x)+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}-g_j(x)\\ &=(1+b_j)(g_j(\psi(x)+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}+h_j(\psi(x)+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x})-g_j(x).\end{aligned}$$ Writing $\psi(x)=x+\phi(x)$ and taking the two terms involving $g_j$ from above, $$\begin{aligned} g_j(\psi(x)&+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}-g_j(x)\\ &=g_j(x)[g_j^{-1}(x)g_j(x+v_j+\phi(x))e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}-1]\\ &=g_j(x)(e^{-\pi p_j'[|x+v_j+\phi(x)|^2-|x|^2]}e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}-1)\\ &=g_j(x)x\cdot[-2p_j'(\phi(x)+v_j)+i\xi+iA^TJA\tilde{v_j}]+O((||\phi||+|\bv|+|\xi|)^2),\end{aligned}$$ where $O((|\phi||+|\bv|+|\xi|)^2)$ represents the $L^{p_j}$ norm of the remainder term. Substituting back into the initial expression for $\tilde{h_j}$, one finds $$\begin{gathered} \label{eq:hj expression} \tilde{h_j}(x)=a_jh_j(\psi(x)+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x}\\+g_j(x)x\cdot[-2p_j'(\phi(x)+v_j)+i\xi+iA^TJA\tilde{v_j}]+O((||\phi||+|\bv|+|\mathbf{b}|+|\xi|)^2).\end{gathered}$$ In computing $\langle \tilde{h_j},P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle$, we begin with the main term from . $$\begin{aligned} \langle a_jh_j(\psi(x)&+v_j)e^{ix\cdot\xi+iA^TJA\tilde{v_j}\cdot x},P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle\\ &=\langle h_j(\psi(x)+v_j),P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle +O((|\mathbf{b}|+|\xi|+|\bv|)||h_j||_{p_j})\\ &=|\det(\psi)|^{-1}\int h_j(y)P_\alpha^{(\tau_j)}(\psi^{-1}(y-v_j)g_j^{p_j-1}(\psi^{-1}(y-v_j)dy\\ &\hspace{.5in}+O((|\mathbf{b}|+|\xi|+|\bv|)||h_j||_{p_j})\\ &=\langle h_j,P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle +O((|\mathbf{b}|+|\xi|+||\phi||+|\bv|)||h_j||_{p_j}).\end{aligned}$$ Considering the full expression from , $$\begin{gathered} \langle \tilde{h_j},P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle=\langle h_j,P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle\\ +\langle g_j(x)x\cdot[b_j-2p_j'(\phi(x)+v_j)-i\xi-iA^TJA\tilde{v_j}],P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle\\ +O((||\phi||+|\bv|+|\mathbf{b}|+|\xi|)^2+(||\phi||+|\bv|+|\mathbf{b}|+|\xi|)||h_j||_{p_j}).\end{gathered}$$ In order to complete the proof via the Implicit Function Theorem, it suffices to show that the map $$\label{eq:invertible map} (\mathbf{b},\bv,\xi,\phi)\mapsto \langle g_j(x)[b_j-v_j\cdot x-i(\xi+A^TJA\tilde{v_j})\cdot x-2p_j'x\cdot\phi(x)],P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle$$ with $(j,\alpha)$ ranging over the indices specified in Proposition \[prop:only\] and taking the real or imaginary part as specified is invertible. Since $\{x\cdot\phi(x):\phi\text{ is a symmetric real }(2d)\times(2d)\text{ matrix}\}$ is precisely the set of symmetric, real, homogeneous, quadratic polynomials on $\R^{2d}$, the map $\phi\mapsto(\langle x\cdot\phi(x)g_3(x),P_\alpha^{(\tau_3)}g_3^{p_3-1}\rangle:|\alpha|=2)$ is invertible. These inner products vanish when $|\alpha|=0,1$. The contribution from the mapping $(\bv,\xi)$ with the constraint $v_1+v_2+v_3=0$ to $\langle g_j(x)[v_j\cdot x-i(\xi+A^TJA\tilde{v_j})\cdot x],P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle$ ranging over the indices of Proposition \[prop:only\] and taking the real and imaginary parts is also invertible. These products vanish when $|\alpha|=0,2$. Lastly, the contribution from $\langle g_j(x)b_j,P_\alpha^{(\tau_j)}g_j^{p_j-1}\rangle$ indexed over $j=1,2,3$ is in one-to-one correspondence with $\mathbf{b}$ and these inner products vanish when $|\alpha|=1,2$. Thus, the maps described in is invertible. Putting it All Together ======================= Let $(h_1,h_2,h_3,B)$ be a 4-tuple with $h_j\in L^{p_j}$ and $B$ an arbitrary $(2d)\times(2d)$ matrix such that $\dist_\bp(\scriptO_{TC}(\bh,B),(\bg,0))$ is sufficiently small. By the Balancing Lemma, there exists an element $(F_1,F_2,F_3,A)$ of the orbit of $(\bh,B)$ which satisfies the orthogonality conditions of Proposition \[prop:only\]. Let $f_j=F_j-g_j$. Since $$\dist_\bp(\scriptO_{TC}(\bh,B),(\bg,0))^2\leq \max_j||f_j||_{p_j}^2+||A^TJA||^2,$$ it suffices to prove that $$\frac{\T_A(\bg+\bf)}{\prod_j||g_j+f_j||_{p_j}}\leq A_\bp^{2d}-c\left[\max_j||f_j||_{p_j}^2+||A^TJA||^2\right].$$ By Proposition \[prop:only\], $$\frac{\T_0(\bg+\bf)}{\prod_j||g_j+f_j||_{p_j}}\leq A_\bp^{2d}-c||\bf||_\bp^2-\tilde{c}\sum_j||f_{j,\flat}||_{p_j}^{p_j}.$$ Thus, it suffices to show that $$\frac{(\T_A-\T_0)(\bg+\bf)}{\prod_j||g_j+f_j||_{p_j}}\leq -c||A^TJA||^2+O((||\bf||_\bp+||A^TJA||)^3).$$ We may ignore the product of norms in the denominator by appropriate modification of the constant $c$. Expanding $(\T_A-\T_0)(\bg+\bf)$ through the multilinearity of $\T_A-\T_0$, one obtains four types of terms. By Lemma \[lemma:onesigma\] and Lemma \[lemma:twosigma\], $$\begin{gathered} (\T_A-\T_0)(g_1,g_2,g_3)=\iint g_1(x)g_2(y)g_3(x+y)(e^{i\sigma(Ax,Ay)}-1)dxdy\hfill\\ \hfill=\iint g_1(x)g_2(y)g_3(x+y)(i\sigma(Ax,Ay)-\frac{1}{2}\sigma(Ax,Ay)^2+O(\sigma(Ax,Ay)^3))dxdy\\ \hfill=-c||A^TJA||^2+O(||A^TJA||^3).\end{gathered}$$ By similar application of Lemma \[lemma:onesigma\], $$\begin{aligned} (\T_A-\T_0)(f_1,g_2,g_3)&=\iint f_1(x)g_2(y)g_3(x+y)(i\sigma(Ax,Ay)+O(\sigma(Ax,Ay)^2))dxdy\\ &\leq 0+||A^TJA||^2\int f_1(x)x^2\left[\int y^2g_2(y)g_3(x+y)dy\right]dx\\ &=O(||f_1||_{p_1}||A^TJA||^2)\end{aligned}$$ and likewise for all other terms involving one $f_j$ and two $g_j$’s. The $(\T_A-\T_0)(f_1,f_2,f_3)$ term is negligible by Lemma \[lemma:trivial\], so only the terms with two $f_j$’s and one $g_j$ remain. Lemma \[lemma:3rd order\] only addresses the situation where the $f_j$ are replaced with $f_{j,\sharp}$. However, Proposition \[prop:only\] provides a $-\tilde{c}\sum_j||f_{j,\flat}||_{p_j}^{p_j}$ term which may be used here. Expanding further and applying Lemma \[lemma:3rd order\] and gives $$\begin{gathered} |(\T_A-\T_0)(f_1,f_2,g_3)|-\tilde{c}\sum_j||f_{j,\flat}||_{p_j}^{p_j}\\ \leq o(||\bf||_\bp^2+||A^TJA||^2)+ O(||f_{1,\sharp}||_{p_1}||f_{2,\flat}||_{p_2}+||f_{2,\sharp}||_{p_2}||f_{1,\flat}||_{p_1})-\tilde{c}\sum_j||f_{j,\flat}||_{p_j}^{p_j}.\end{gathered}$$ If $\sum_j||f_{j,\flat}||_{p_j}^{p_j}$ is small relative to $||\bf||_\bp^2$, then the above is negligible, as each $||f_{j,\flat}||_{p_j}$ is small. (Specifically, one may split into cases where $||f_{j,\flat}||_{p_j}\geq||f_j||_{p_j}^{(4-p_j)/2}$ for at least one $j$ or none of the $j$.) However, if $\sum_j||f_{j,\flat}||_{p_j}^{p_j}$ is large relative to $||\bf||_\bp^2$, then the last term dominates (as $p_j<2$), and the above is still negligible. This holds for the other terms involving two $f_j$’s and one $g_j$, thus completing the proof of the main theorem. [20]{} William Beckner. Inequalities in [F]{}ourier analysis. , 102(1):159–182, 1975. Gabriele Bianchi and Henrik Egnell A note on the Sobolev inequality. J. Funct. Anal. 100 (1991), no. 1, 18–24. Herm Jan Brascamp and Elliott H. Lieb. Best constants in [Y]{}oung’s inequality, its converse, and its generalization to more than three functions. , 20(2):151–173, 1976. Michael Christ. . preprint, math.CA arXiv:1406.1210 . Young’s inequality sharpened. . . . preprint, math.CA arXiv:1706.02005 Abel Klein and Bernard Russo. Sharp inequalities for [W]{}eyl operators and [H]{}eisenberg groups. , 235(2):175–194, 1978. Elias M. Stein. , volume 43 of [*Princeton Mathematical Series*]{}. Princeton University Press, Princeton, NJ, 1993.

--- abstract: | To each $\alpha\in(1/3,1/2)$ we associate the Cantor set $$\Gamma_{\alpha}:=\Big\{\sum_{i=1}^{\infty}\epsilon_{i}\alpha^i: \epsilon_i\in\{0,1\},\,i\geq 1\Big\}.$$ In this paper we consider the intersection $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ for any translation $t\in{\ensuremath{\mathbb{R}}}$. We pay special attention to those $t$ with a unique $\{-1,0,1\}$ $\alpha$-expansion, and study the set $$D_\alpha:=\{\dim_H(\Gamma_\alpha \cap (\Gamma_\alpha + t)):t \textrm{ has a unique }\{-1,0,1\}\,\alpha\textrm{-expansion}\}.$$We prove that there exists a transcendental number $\alpha_{KL}\approx 0.39433\ldots$ such that: $D_\alpha$ is finite for $\alpha\in(\alpha_{KL},1/2),$ $D_{\alpha_{KL}}$ is infinitely countable, and $D_{\alpha}$ contains an interval for $\alpha\in(1/3,\alpha_{KL}).$ We also prove that $D_\alpha$ equals $[0,\frac{\log 2}{-\log \alpha}]$ if and only if $\alpha\in (1/3,\frac {3-\sqrt{5}}{2}].$ As a consequence of our investigation we prove some results on the possible values of $\dim_{H}(\Gamma_\alpha \cap (\Gamma_\alpha + t))$ when $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ is a self-similar set. We also give examples of $t$ with a continuum of $\{-1,0,1\}$ $\alpha$-expansions for which we can explicitly calculate $\dim_{H}({\Gamma_\alpha}\cap({\Gamma_\alpha}+t)),$ and for which ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ is a self-similar set. We also construct $\alpha$ and $t$ for which $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ contains only transcendental numbers. Our approach makes use of digit frequency arguments and a lexicographic characterisation of those $t$ with a unique $\{-1,0,1\}$ $\alpha$-expansion. address: - 'Department of Mathematics and Statistics, Whiteknights, Reading, RG6 6AX, UK' - 'School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225002, People’s Republic of China' author: - Simon Baker - Derong Kong title: Unique expansions and intersections of Cantor sets --- introduction {#sec:1} ============ To each $\alpha\in(0,1/2)$ we associate the contracting similarities $f_0(x)=\alpha x$ and $f_{1}(x)=\alpha(x+1).$ The middle $(1-2\alpha)$ Cantor set $\Gamma_\alpha$ is defined to be the unique compact non-empty set satisfying the equation $$\Gamma_{\alpha} =f_0(\Gamma_\alpha)+f_1(\Gamma_\alpha).$$ It is easy to see that the maps $\{f_0,f_1\}$ satisfy the strong separation condition. Thus $\dim_{H}(\Gamma_{\alpha})=\dim_{B}(\Gamma_{\alpha})=\frac{\log 2}{-\log \alpha},$ where $\dim_{H}$ and $\dim_{B}$ denote the Hausdorff dimension and box dimension respectively. A natural and well studied question is “What are the properties of the intersection $\Gamma_\alpha\cap(\Gamma_\alpha+t)$?" This question has been studied by many authors. We refer the reader to [@Kenyon_Peres_1991; @Kraft_1992; @Li_Xiao_1998; @Kraft_2000; @Kong_Li_Dekking_2010] and the references therein for more information. As we now go on to explain, when $\alpha\in(0,1/3]$ the set $\Gamma_\alpha \cap (\Gamma_\alpha + t)$ is well understood, however when $\alpha\in(1/3,1/2)$ additional difficulties arise. Note that $\Gamma_\alpha\cap(\Gamma_\alpha+t)\ne\emptyset$ if and only if $t\in\Gamma_\alpha-\Gamma_\alpha$. Thus it is natural to investigate the difference set $\Gamma_{\alpha}-\Gamma_{\alpha},$ which is the self-similar set generated by the iterated function system $\{f_{-1},f_0,f_1\},$ where $f_{-1}(x)=\alpha(x-1)$. Alternatively, one can write $$\Gamma_{\alpha}-\Gamma_{\alpha}:=\Big\{\sum_{i=1}^{\infty}\epsilon_{i}\alpha^i: \epsilon_i\in\{-1,0,1\},\,i\geq 1\Big\}.$$ Importantly, for $\alpha\in(0,1/3)$ each $t\in \Gamma_{\alpha}-\Gamma_{\alpha}$ has a unique *$\alpha$-expansion* with *alphabet* $\{-1,0,1\}$, i.e., there exists a unique sequence $(t_i)\in\{-1,0,1\}^{\mathbb{N}}$ such that $t=\sum t_i \alpha^{i}.$ When $\alpha=1/3$ there is a countable set of $t$ with precisely two $\alpha$-expansions. These $t$ are well understood and do not pose any real difficulties, thus in what follows we suppress the case where t has two $\alpha$-expansions. For $\alpha\in(0,1/3]$ let $t\in \Gamma_\alpha - \Gamma_\alpha$ have a unique $\alpha$-expansion $(t_i)$. Then the sequence $(t_i)$ provides a useful description of the set $\Gamma_\alpha \cap (\Gamma_\alpha + t).$ Indeed, we can write (cf. [@Li_Xiao_1998]) $$\label{eq:11} {\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)=\Big\{\sum_{i=1}^{\infty}\epsilon_i\alpha^i:\epsilon_i\in \{0,1\}\cap (\{0,1\}+t_i)\Big\}.$$ With this new interpretation many questions regarding the set ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ can be reinterpreted and successfully answered using combinatorial properties of the $\alpha$-expansion $(t_i).$ The straightforward description of ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ provided by does not exist for $\alpha\in(1/3,1/2)$ and a generic $t\in {\Gamma_\alpha}-{\Gamma_\alpha}.$ The set ${\Gamma_\alpha}- {\Gamma_\alpha}$ is still a self-similar set generated by the transformations $\{f_{-1},f_0.f_1\},$ however this set is now equal to the interval $[\frac{-\alpha}{1-\alpha},\frac{\alpha}{1-\alpha}]$ and the good separation properties that were present in the case where $\alpha\in(0,1/3]$ no longer exist. It is possible that a $t\in {\Gamma_\alpha}- {\Gamma_\alpha}$ could have many $\alpha$-expansions. In fact it can be shown that Lebesgue almost every $t\in {\Gamma_\alpha}- {\Gamma_\alpha}$ has a continuum of $\alpha$-expansions (cf. [@Dajani_DeVries_2007; @Sidorov_2003; @Sidorov_2007]). Thus within the parameter space $(1/3,1/2)$ we are forced to have the following more complicated interpretation of ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ (cf. [@Li_Xiao_1998 Lemma 3.3]) $$\label{eq:12} {\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)=\bigcup_{\tilde{t}}\Big\{\sum_{i=1}^{\infty}\epsilon_i\alpha^i:\epsilon_i\in \{0,1\}\cap (\{0,1\}+\tilde{t}_i)\Big\},$$ where the union is over all $\alpha$-expansions $\tilde{t}=(\tilde{t}_i)$ of $t$. As stated above, for a generic $t$ this union is uncountable, this makes many questions regarding the set ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ intractable. In what follows we focus on the case where $t$ has a unique $\alpha$-expansion. For these $t$ the description of ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ given by simplifies to that given by . We now introduce some notation. For $\alpha\in(0,1/2)$ let $${\mathcal{U}}_\alpha:=\Big\{t\in {\Gamma_\alpha}-{\Gamma_\alpha}: t \textrm{ has a unique }\alpha\textrm{-expansion w.r.t. the aphabet}~{\left\{-1,0,1\right\}}\Big\}.$$ Within this paper one of our main objects of study is the following set $$D_{\alpha}:=\Big\{\dim_H({\Gamma_\alpha}\cap({\Gamma_\alpha}+t)): t\in {\mathcal{U}}_\alpha\Big\}.$$ In particular we will prove the following theorems. \[th:11\] There exists a transcendental number $\alpha_{KL}\approx 0.39433\ldots$ such that: 1. For $\alpha\in (\alpha_{KL},1/2)$ there exists $n^*\in\mathbb{N}$ such that $$D_{\alpha}=\Big\{0, \frac{\log2}{-\log \alpha}\Big\}\cup\Big\{\frac{\log 2}{ \log\alpha} \sum_{i=1}^{n}\Big(\frac{-1}{2}\Big)^{i}:1\leq n\leq n^*\Big\}.$$ 2. $$D_{\alpha_{KL}}=\Big\{0, \frac{\log2}{-\log \alpha_{KL}},\frac{\log 2}{-3\log \alpha_{KL}}\Big\}\cup\Big\{\frac{\log 2}{ \log\alpha_{KL}} \sum_{i=1}^{n}\Big(\frac{-1}{2}\Big)^{i}:1\leq n< \infty \Big\}.$$ 3. $D_{\alpha}$ contains an interval if $\alpha\in(1/3,\alpha_{KL}).$ In [@Li_Xiao_1998] it was asked “When $\alpha\in(1/3,1/2)$ what are the possible values of $\dim_H({\Gamma_\alpha}\cap({\Gamma_\alpha}+t))$ for $t\in {\Gamma_\alpha}- {\Gamma_\alpha}$?" The following theorem provides a partial solution to this problem. \[th:12\] 1. If $\alpha\in(1/3,\frac {3-\sqrt{5}}{2}]$ then $D_{\alpha}=[0,\frac{\log 2}{-\log \alpha}].$ 2. If $\alpha\in (\frac{3-\sqrt{5}}{2}, 1/2)$ then $D_{\alpha}$ is a proper subset of $[0,\frac{\log 2}{-\log \alpha}].$ Amongst ${\Gamma_\alpha}-{\Gamma_\alpha}$ a special class of $t$ are those for which ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ is a self-similar set. Determining whether ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ is a self-similar set is a difficult problem for a generic $t$ with many $\alpha$-expansions, thus we consider only those $t\in {\mathcal{U}}_{\alpha}$. Let $$S_{\alpha}:=\Big\{t\in {\mathcal{U}}_{\alpha}: {\Gamma_\alpha}\cap({\Gamma_\alpha}+t) \textrm{ is a self-similar set} \Big\}.$$ We prove the following result. \[th:13\] 1. If $\alpha\in(1/3,\frac{3-\sqrt{5}}{2}]$ then $\{\dim({\Gamma_\alpha}\cap({\Gamma_\alpha}+t) ): t\in S_\alpha\}$ is dense in $[0,\frac{\log 2}{-\log \alpha}].$ 2. If $\alpha\in(\frac{3-\sqrt{5}}{2},1/2)$ then $\{\dim({\Gamma_\alpha}\cap({\Gamma_\alpha}+t) ): t\in S_\alpha\}$ is not dense in $[0,\frac{\log 2}{-\log \alpha}].$ What remains of this paper is arranged as follows. In Section $2$ we recall the necessary preliminaries from expansions in non-integer bases, and recall an important result of [@Li_Xiao_1998] that connects the dimension of ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ with the frequency of $0$’s in the $\alpha$-expansion $(t_i).$ In Section $3$ we prove Theorem \[th:11\], and in Section $4$ we prove Theorem \[th:12\] and Theorem \[th:13\]. In Section $5$ we include some examples. We give two examples of an $\alpha\in(1/3,1/2),$ and $t\in {\Gamma_\alpha}-{\Gamma_\alpha}$ with a continuum of $\alpha$-expansions, for which we can explicitly calculate $\dim_{H}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))$. The techniques used in our first example can be applied to the more general case where $\alpha$ is the reciprocal of a Pisot number and $t\in \mathbb{Q}(\alpha)$. Our second example demonstrates that it is possible for $t$ to have a continuum of $\alpha$-expansions and for ${\Gamma_\alpha}\cap({\Gamma_\alpha}+t)$ to be a self-similar set. Moreover, both of these examples show that it is possible to have $$\dim_{H}({\Gamma_\alpha}+({\Gamma_\alpha}+t))>\sup_{\tilde{t}}\dim_{H}\Big(\Big\{\sum_{i=1}^{\infty}\epsilon_i\alpha^i:\epsilon_i\in \{0,1\}\cap (\{0,1\}+\tilde{t}_i)\Big\}\Big).$$ Our final example demonstrates the existence of $\alpha\in(1/3,1/2)$ and $t\in {\Gamma_\alpha}-{\Gamma_\alpha}$ for which ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ contains only transcendental numbers. Preliminaries ============= Let $M\in \mathbb{N}$ and $\alpha\in [\frac{1}{M+1},1).$ Given $x\in I_{\alpha,M}:=[0,\frac{M\alpha}{1-\alpha}]$ we call a sequence $(\epsilon_{i})\in \{0,\ldots, M\}^{\mathbb{N}}$ an *$\alpha$-expansion* for $x$ with alphabet ${\left\{0, \cdots,M\right\}}$ if $$x=\sum_{i=1}^{\infty}\epsilon_i\alpha^i.$$ This method of representing real numbers was pioneered in the early $1960$’s in the papers of Rényi [@Renyi_1957] and Parry [@Parry_1960]. One aspect of these representations that makes them interesting is that for $\alpha\in(\frac{1}{M+1},1)$ a generic $x\in I_{\alpha,M}$ has many $\alpha$-expansions (cf. [@Dajani_DeVries_2007; @Sidorov_2003; @Sidorov_2007]). This naturally leads researchers to study the set of $x\in I_{\alpha,M}$ with a unique $\alpha$-expansion, the so called *univoque set*. We define this set as follows $${\mathcal{U}}_{\alpha,M}:=\Big\{x\in I_{\alpha,M}: x \textrm{ has a unique }\alpha\textrm{-expansion w.r.t. the aphabet}~{\left\{0,1,\cdots,M\right\}}\Big\}.$$ Accordingly, let ${\widetilde{\mathcal{U}}}_\alpha$ denote the set of corresponding expansions, i.e., $${\widetilde{\mathcal{U}}}_{\alpha,M}:=\Big\{(\epsilon_i)\in\{0,\ldots,M\}^{\mathbb{N}}: \sum_{i=1}^{\infty}\epsilon_i\alpha^i\in {\mathcal{U}}_{\alpha,M}\Big\}.$$The sets ${\mathcal{U}}_{\alpha,M}$ and ${\widetilde{\mathcal{U}}}_{\alpha,M}$ have been studied by many authors. For more information on these sets we refer the reader to [@Erdos_Joo_Komornik_1990; @Darczy_Katai_1995; @Glendinning_Sidorov_2001; @DeVries_Komornik_2008; @Komornik_2011; @Komornik_Kong_Li_2015_1] and the references therein. Before continuing with our discussion of the sets ${\mathcal{U}}_{\alpha,M}$ and ${\widetilde{\mathcal{U}}}_{\alpha,M}$ we make a brief remark. In the introduction we were concerned with $\alpha$-expansions with digit set $\{-1,0,1\}$, not with a digit set $\{0,\ldots,M\}$. However, all of the result that are stated below for a digit set $\{0,\ldots,M\}$ also hold for any digit set of $M+1$ consecutive integers $\{s,\ldots,s+M\}.$ In particular, statements that are true for the digit set $\{0,1,2\}$ translate to results for the digit set $\{-1,0,1\}$ by performing the substitutions $0\to -1$, $1\to 0$, $2\to 1$. We now define the lexicographic order and introduce some notations. Given two finite sequences $\omega=(\omega_1,\ldots,\omega_n),\,\omega'=(\omega_1',\ldots,\omega_n')\in\{0,\ldots,M\}^n,$ we say that $\omega$ is less than $\omega'$ with respect to the lexicographic order, or simply write $\omega\prec \omega'$, if $\omega_1<\omega_1'$ or if there exists $1\leq j<n$ such that $\omega_i=\omega_i'$ for $1\leq i\leq j$ and $\omega_{j+1}<\omega_{j+1}'$. One can also define the relations $\preceq, \succ, \succeq$ in the natural way, and we can extend the lexicographic order to infinite sequences. We define the *reflection* of a finite/infinite sequence $(\epsilon_i)$ to be $(\overline{\epsilon_i})=(M-\epsilon_i),$ where the underlying $M$ should be obvious from our context. For a finite sequence $\omega=(\omega_1,\ldots,\omega_n)$ we define the finite sequence $\omega^{-}$ to be $(\omega_1,\ldots, \omega_n -1).$ Moreover, we denote the concatenation of $\omega$ with itself $n$ times by $\omega^n$, we also let $\omega^{\infty}$ denote the infinite sequence obtained by indefinitely concatentating $\omega$ with itself. Given $x\in I_{\alpha,M}$ we define the *greedy* $\alpha$-expansion of $x$ to be the lexicographically largest sequence amongst the $\alpha$-expansions of $x$. We define the *quasi-greedy* $\alpha$-expansion of $x$ to be the lexicographically largest infinite sequence amongst the $\alpha$-expansions of $x$. Here we call a sequence $(\epsilon_i)$ *infinite* if $\epsilon_i\neq 0$ for infinitely many $i$. When studying the sets ${\mathcal{U}}_{\alpha,M}$ and ${\widetilde{\mathcal{U}}}_{\alpha,M}$ a pivotal role is played by the quasi-greedy $\alpha$-expansion of $1$. In what follows we will denote the quasi-greedy $\alpha$-expansion of $1$ by $(\delta_i(\alpha))$. The importance of the sequence $(\delta_i(\alpha))$ is well demonstrated by the following technical lemma proved in [@Parry_1960] (see also, [@Erdos_Joo_Komornik_1990; @DeVries_Komornik_2008]). \[lem:21\] A sequence $(\epsilon_i)$ belongs to ${\widetilde{\mathcal{U}}}_{\alpha,M}$ if and only if the following two conditions are satisfied: $$\begin{aligned} &(\epsilon_{n+i}) \prec (\delta_i(\alpha)) \textrm{ whenever } \epsilon_1\ldots\epsilon_n \neq M^n\\ &(\overline{\epsilon_{n+i}}) \prec (\delta_i(\alpha)) \textrm{ whenever } \epsilon_1\ldots\epsilon_n \neq 0^n\end{aligned}$$ Lemma \[lem:21\] provides a useful characterisation of the set ${\widetilde{\mathcal{U}}}_{\alpha,M}$ in terms of the sequence $(\delta_i(\alpha))$. The following lemma describes the sequences $(\delta_i(\alpha))$. \[lem:22\] Let $M\in\mathbb{N},$ $\alpha\in[\frac{1}{M+1},1)$ and $(\delta_i(\alpha))$ be the quasi-greedy $\alpha$-expansion of $1$. The map $\alpha \to (\delta_i(\alpha))$ is a strictly decreasing bijection from the interval $[\frac{1}{M+1},1)$ onto the set of all infinite sequences $(\delta_i) \in\{0,\ldots, M\}^{\mathbb{N}}$ satisfying $$\delta_{k+1}\delta_{k+2} \cdots \preceq\delta_1\delta_2\cdots \textrm{ for all }k \geq 0.$$ The following technical result was proved in [@Li_Xiao_1998 Theorem 3.4] for $\alpha\in(0,1/3],$ where importantly every $t$ has a unique $\alpha$-expansion, except for $\alpha=1/3$ where countably many $t$ have two $\alpha$-expansions. The proof translates over to the more general case where $\alpha\in(1/3,1/2)$ and $t\in {\mathcal{U}}_{\alpha}.$ \[lem:23\] Let $\alpha\in(1/3,1/2)$ and $t\in {\mathcal{U}}_{\alpha},$ then $$\dim_{H}({\Gamma_\alpha}\cap({\Gamma_\alpha}+t))=\frac{\log2}{-\log \alpha}\underline{d}((t_i)),$$ where $$\underline{d}((t_i)):=\liminf_{n\to \infty} \frac{\#\{1\leq i \leq n: t_i=0\}}{n}.$$ Lemma \[lem:23\] will be a vital tool in proving Theorems \[th:11\] and \[th:12\]. This result allows us to reinterpret Theorems \[th:11\] and \[th:12\] in terms of statements regarding the frequency of $0$’s that can occur within an element of ${\widetilde{\mathcal{U}}}_\alpha.$ In what follows, for an infinite sequence $(t_i)\in\{-1,0,1\}^{\mathbb{N}}$ we will use the notation $$\overline{d}((t_i)):=\limsup_{n\to \infty} \frac{\#\{1\leq i \leq n: t_i=0\}}{n}.$$ When this limit exists, i.e., $\underline{d}((t_i))=\overline{d}((t_i))$, we simply use $d((t_i)).$ For a word $ t_1 \ldots t_n \in\{-1,0,1\}^n$ we will use the notation $$d(t_1\cdots t_n):=\frac{\#\{1\leq i \leq n: t_i=0\}}{n}.$$ Proof of Theorem \[th:11\] ========================== In this section we prove Theorem \[th:11\]. We start by defining the Thue-Morse sequence and its natural generalisation. Let $(\tau_i)_{i=0}^{\infty}\in\{0,1\}^{\mathbb{N}}$ denote the classical Thue-Morse sequence. This sequence is defined iteratively as follows. Let $\tau_0=0$ and if $\tau_i$ is defined for some $i\geq 0$, set $\tau_{2i}=\tau_i$ and $\tau_{2i+1}=1-\tau_i.$ Then the sequence $(\tau_i)_{i=0}^{\infty}$ begins with $$0110\, 1001\, 1001\, 0110\, 1001\, 0110 0110\ldots$$ For more on this sequence we refer the reader to [@Allouche_Shallit_1999]. Within expansions in non-integer bases the sequence $(\tau_i)_{i=0}^{\infty}$ is important for many reasons. In [@Komornik_Loreti_1998] Komornik and Loreti proved that the unique $\alpha$ for which $(\delta_{i}(\alpha))=(\tau_i)_{i=1}^{\infty}$ is the largest $\alpha\in(1/2,1)$ for which $1$ has a unique $\alpha$-expansion. This $\alpha$ has since become known as the *Komornik-Loreti constant*. Interesting connections between the size of ${\mathcal{U}}_{\alpha}$ and the Komornik Loreti constant were made in [@Glendinning_Sidorov_2001]. Using the Thue-Morse sequence we define a new sequence $(\lambda_i)\in \{-1,0,1\}^{\mathbb{N}}$ as follows $$(\lambda_i)_{i=1}^{\infty}=(\tau_i-\tau_{i-1})_{i=1}^{\infty}.$$ We denote the unique $\alpha\in(1/2, 1)$ for which $\sum_{i=1}^{\infty}(1+\lambda_i)\alpha^i=1$ by $\alpha_{KL}$. Our choice of subscript is because the constant $\alpha_{KL}$ is a type of generalised Komornik-Loreti constant. This number is transcendental (cf. [@Komornik_Loreti_2002]) and is approximately $0.39433.$ This sequence satisfies the property $$\label{eq:31} \begin{split} &\lambda_1=1,\quad\quad\lambda_{2^{n+1}}=1-\lambda_{2^n};\\ &\lambda_{2^n+i}=- \lambda_i \quad \textrm{ for any}~~ 1\leq i<2^n. \end{split}$$ This property can be deduced directly from [@Komornik_Loreti_2002 Lemma 5.2]. So, the sequence $(\lambda_i)_{i=1}^\infty$ starts at $$10\,(-1)1\,(-1)010\;(-1)01(-1)\,10(-1)1\cdots.$$ It will be useful when it comes to determining the frequency of zeros within certain sequences. To each $n\in\mathbb{N}$ we associate the finite sequence $w_n=\lambda_1\cdots \lambda_{2^{n}}.$ By (\[eq:31\]) the following property of $\omega_n$ can be verified. $$\label{eq:32} w_{n+1}^{-}=w_n\overline{w_n}.$$ Here the reflection of $w_n$ w.r.t. the digit set $\{-1,0,1\}$ is defined by $\overline{w_n}:=(-\lambda_1)(-\lambda_2)\cdots(-\lambda_{2^n}).$ We now prove two lemmas that allow us to prove statements $(1)$ and $(2)$ from Theorem \[th:11\]. \[lem:31\] For $n\geq 2$ the following inequalities hold: $$\label{eq:33} \#\{1\leq i\leq 2^n: \lambda_i=0\}=2\#\{1\leq i\leq 2^{n-1}: \lambda_i=0\} - 1 \textrm{ if }n\textrm{ is even;}$$ $$\label{eq:34} \#\{1\leq i\leq 2^n: \lambda_i=0\}=2\#\{1\leq i\leq 2^{n-1}: \lambda_i=0\} + 1 \textrm{ if }n\textrm{ is odd}.$$ Moreover $$\label{eq:35} d(w_n)=-\sum_{i=1}^{n}\Big(\frac{-1}{2}\Big)^{i}$$ for all $n\in \mathbb{N}.$ We begin by observing that $w_1=10,$ so $d(w_{1})=1/2$ and (\[eq:35\]) holds for $n=1$. We now show that and imply via an inductive argument. Let us assume is true for odd $N\in\mathbb{N}$. Then $$\begin{aligned} d(w_{N+1})&= \frac{\#\{1\leq i\leq 2^{N+1}: \lambda_{i}=0\}}{2^{N+1}}\\ &= \frac{2\#\{1\leq i\leq 2^{N}: \lambda_i=0\} - 1}{2^{N+1}}\\ &=d(w_N)-\frac{1}{2^{N+1}}\\ &=-\sum_{i=1}^{N+1}\Big(\frac{-1}{2}\Big)^i\end{aligned}$$In our second equality we used . The case where $N$ is even is done similarly. Proceeding inductively we may conclude that holds assuming and . It remains to show and hold. For $n=1$ we know that $w_1=10,$ therefore implies that the last digit of $w_2$ equals $1.$ What is more, repeatedly applying we see that the last digit of $w_n$ equals $0$ if $n$ is odd, and equals $1$ if $n$ is even. Property implies that $\lambda_{2^n+i}=0$ if $\lambda_{i}=0$ for any $1\le i<2^n$. Therefore, when $n$ is even we see that $$\begin{aligned} \#\{1\leq i\leq 2^n: \lambda_i=0\}&=\#\{1\leq i\leq 2^{n-1}: \lambda_i=0\}+\#\{2^{n-1}+1\leq i\leq 2^n: \lambda_i=0\}\\ & = \#\{1\leq i\leq 2^{n-1}: \lambda_i=0\} + \#\{1\leq i\leq 2^{n-1}: \lambda_i=0\} -1 \\ &= 2\#\{1\leq i\leq 2^{n-1}: \lambda_i=0\} -1.\end{aligned}$$ Thus is proved. Equation is proved similarly. Lemma \[lem:31\] determines the frequency of $0$’s within the finite sequences $w_n$. For our proof of Theorem \[th:11\] we also need to know the frequency of $0$’s within the sequence $(\lambda_i)_{i=1}^{\infty}.$ \[lem:32\] $$d((\lambda_i))=-\sum_{i=1}^{\infty}\Big(\frac{-1}{2}\Big)^{i}=\frac{1}{3}.$$ Let us begin by fixing $\varepsilon>0.$ Let $N\in\mathbb{N}$ be sufficiently large such that $$\label{eq:36} \Big|\frac{-\sum_{i=1}^{n}(-1/2)^i}{1/3}-1\Big|<\varepsilon$$ for all $n\geq N$. Now let us pick $N'\in\mathbb{N}$ large enough such that $$\label{eq:37} \frac{\sum_{j=0}^{N-1} 2^j}{N'}<\varepsilon$$ Let $n\geq N'$ be arbitrary and write $n=\sum_{j=0}^{k}\epsilon_{j}2^{j},$ where we assume $\epsilon_k=1$. By splitting $(\lambda_i)_{i=1}^n$ into its first $2^{k}$ digits, then the next $2^{k-1}$ digits, then the next $2^{k-2}$ digits, etc, we obtain: $$\begin{aligned} \label{eq:38} \frac{\#\{1\leq i\leq n:\lambda_i=0\}}{n}&=\frac{\#\{1\leq i\leq 2^{k}:\lambda_i=0\}}{n}\\ &+\sum_{l=0}^{k-1}\frac{\#\{\sum_{j=k-l}^{k}\epsilon_j 2^j+1 \leq i \leq \sum_{j=k-l-1}^{k}\epsilon_j 2^j:\lambda_i=0\}}{n}.\nonumber\end{aligned}$$ By repeatedly applying we see $$\label{eq:39} \begin{split} &\quad~\#\{1 \leq i \leq \epsilon_{k-l-1} 2^{k-l-1}:\lambda_i=0\}\\ &= \#\{\epsilon_{k-l}2^{k-l}+1 \leq i \leq \epsilon_{k-l}2^{k-l}+ \epsilon_{k-l-1} 2^{k-l-1}:\lambda_i=0\} \\ & =\cdots \\ & = \#\Big\{\sum_{j=k-l}^{k}\epsilon_j 2^j+1 \leq i \leq \sum_{j=k-l-1}^{k}\epsilon_j 2^j:\lambda_i=0\Big\} \end{split}$$ Substituing into we obtain $$\begin{aligned} \frac{\#\{1\leq i\leq n:\lambda_i=0\}}{n}&=\frac{\#\{1\leq i\leq 2^{k}:\lambda_i=0\}}{n}\\ &+\sum_{l=0}^{k-1}\frac{\#\{1 \leq i \leq \epsilon_{k-l-1} 2^{k-l-1}:\lambda_i=0\}}{n}.\nonumber\end{aligned}$$ By ignoring lower order terms and applying Lemma \[lem:31\], and we obtain the lower bound $$\begin{aligned} \label{lowerbound} \frac{\#\{1\leq i\leq n:\lambda_i=0\}}{n}&\geq \frac{\#\{1\leq i\leq 2^{k}:\lambda_i=0\}}{n}+\sum_{l=0}^{k-N-1}\frac{\#\{1 \leq i \leq \epsilon_{k-l-1} 2^{k-l-1}:\lambda_i=0\}}{n}\\ & \geq \frac{(1-\varepsilon)}{3}\Big(\frac{ 2^{k}}{n}+\sum_{l=0}^{k-N-1}\frac{\epsilon_{k-l-1} 2^{k-l-1}}{n}\Big)\\ &= \frac{(1-\varepsilon)}{3}\Big(\frac{\sum_{j=0}^{k}\epsilon_j2^j-\sum_{j=0}^{N-1}\epsilon_j2^j}{n}\Big)\\ &\ge \frac{(1-\varepsilon)}{3}\Big(1-\frac{\sum_{j=0}^{N-1}2^j}{n}\Big)\\ & \geq \frac{(1-\varepsilon)^2}{3}.\end{aligned}$$ As $\varepsilon>0$ was arbitrary this implies $\underline{d}((\lambda_i))\ge 1/3.$ By a similar argument it can be shown that $\overline{d}((\lambda_i))\le 1/3.$ Thus $d((\lambda_i))=1/3.$ Statements $(1)$ and $(2)$ from Theorem \[th:11\] follow from Lemma \[lem:23\], Lemma \[lem:31\], and Lemma \[lem:32\], when combined with the following results from [@Kong_Li_Dekking_2010 Lemma 4.12]. \[lem:33\] Let $\alpha\in(\alpha_{KL},1/2),$ then there exists $n^{*}\in\mathbb{N}$ such that every element of ${\widetilde{\mathcal{U}}}_{\alpha}\setminus{\left\{(-1)^{\infty}, 1^{\infty}\right\}}$ ends with one of $$(0)^{\infty}, (w_1\overline{w_1})^{\infty}, \ldots, (w_{n^{*}}\overline{w_{n^{*}}})^{\infty}.$$ \[lem:34\] Each element of ${\widetilde{\mathcal{U}}}_{\alpha_{KL}}\setminus{\left\{(-1)^\infty, 1^\infty\right\}}$ is either eventually periodic with period contained in $$(0)^{\infty}, (w_1\overline{w_1})^{\infty}, (w_2\overline{w_2})^\infty, \ldots,$$ or ends with a sequence of the form $$(w_0\overline{w_{0}})^{k_{0}}(w_0\overline{w_{i_1'}})^{k_{0}'}(w_{i_1}\overline{w_{i_1}})^{k_{1}}(w_{i_{1}}\overline{w_{i_{2}'}})^{k_{1}'}\cdots(w_{i_n}\overline{w_{i_{n}}})^{k_{n}}(w_{i_n}\overline{w_{i_{n+1}'}})^{k_{n}'}\cdots,$$ and its reflection, where $k_n\geq 0,$ $k_n'\in\{0,1\}$ and $$0<i_1'\leq i_1<i_2'\leq i_2<\cdots \leq i_n< i_{n+1}' \leq i_{n+1}<\cdots.$$ By Lemmas \[lem:23\], \[lem:31\] and \[lem:33\] we may conclude $$D(\alpha)=\Big\{0, \frac{\log2}{-\log \alpha}\Big\}\cup\Big\{\frac{\log 2}{ \log \alpha} \sum_{i=1}^{n}\Big(\frac{-1}{2}\Big)^i:1 \leq n \leq n^*\Big\}$$ for some $n^*\in\mathbb{N}$ for $\alpha\in(\alpha_{KL},1/2)$. Whilst at the constant $\alpha_{KL}$ by Lemmas \[lem:23\], \[lem:31\], \[lem:32\] and \[lem:34\] we have $$D(\alpha_{KL})=\Big\{0, \frac{\log2}{-\log \alpha_{KL}},\frac{\log 2}{-3\log\alpha_{KL}}\Big\}\cup\Big\{\frac{\log 2}{ \log \alpha_{KL}} \sum_{i=1}^{n}\Big(\frac{-1}{2}\Big)^i:1 \leq n < \infty\Big\}.$$ Thus statements $(1)$ and $(2)$ from Theorem \[th:11\] hold. It remains to prove statement $(3).$ We start by introducing the following finite sequences. Let $$\label{eq:310} \zeta_{n}=0\lambda_1\cdots \lambda_{2^{n}-1}\textrm{ and }\eta_{n}=(-1)\lambda_{1}\ldots\lambda_{2^{n}-1}.$$ The following result was proved in [@Kong_Li_Dekking_2010]. \[lem:35\] Let $\alpha\in(1/3,\alpha_{KL}),$ then there exists $n\in\mathbb{N}$ such that ${\widetilde{\mathcal{U}}}_{\alpha}$ contains the subshift of finite type over the alphabet $\mathcal{A}=\{\zeta_n,\eta_n,\overline{\zeta_n},\overline{\eta_n\}}$ with transition matrix $$A=\left( \begin{array}{cccc} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array} \right).$$ Let $\alpha\in (1/3,\alpha_{KL})$ and let $n$ be as in Lemma \[lem:35\]. So ${\widetilde{\mathcal{U}}}_{\alpha}$ contains the subshift of finite type determined by the alphabet $\mathcal{A}$ and the transition matrix $A$. On closer examination we see that this subshift of finite type allows the free concatentation of the words $\omega_1=\zeta_n \overline{\zeta_n}$ and $\omega_2=\zeta_n \eta_n\overline{\zeta_n}.$ Importantly $d(\omega_1)< d(\omega_2)$ by . For any $c\in[ d(\omega_1),d(\omega_2)]$ we can pick a sequence of integers $k_{1}, k_{2},\ldots$ such that the sequence $(\epsilon_i)=\omega_{1}^{k_1}\omega_{2}^{k_{2}}\omega_{1}^{k_{3}}\omega_2^{k_{4}}\ldots$ satisfies $d((\epsilon_i))=c$. Thus by Lemma \[lem:23\] the set $D(\alpha)$ contains the interval $[\frac{\log 2}{-\log \alpha}d(\omega_1),\frac{\log 2}{-\log \alpha}d(\omega_2)]$ and our proof is complete. Appealing to standard arguments from multifractal analysis we could in fact show that for any $c\in(\frac{\log 2}{-\log \alpha}d(\omega_1),\frac{\log 2}{-\log \alpha}d(\omega_2))$ there exists a set of positive Hausdorff dimension within ${\mathcal{U}}_{\alpha}$ with frequency $c$. Proof of Theorem \[th:12\] and Theorem \[th:13\] ================================================ We start this section by proving Theorem \[th:12\]. Theorem \[th:13\] will follow almost immediately as a consequence of the arguments used in the proof of Theorem \[th:12\]. To prove Theorem \[th:12\] we rely on the lexicographic description of ${\widetilde{\mathcal{U}}}_{\alpha}$ and $(\delta_i(\alpha))$ given in Section $2$. We take this opportunity to again emphasise that the preliminary results that hold in Section $2$ for the alphabet $\{0,1,2\}$ have an obvious analogue that holds for the digit set $\{-1,0,1\}$. It is instructive here to state our analogue of the quasi greedy $\alpha$-expansion of $1$ when $\alpha=\frac{3-\sqrt{5}}{2}$ and our digit set is $\{-1,0,1\}$. A straightforward calculation proves that this analogue satisfies $$\label{eq:41} \Big(\delta_i\Big(\frac{3-\sqrt{5}}{2}\Big)\Big)= 1(0)^{\infty}.$$ We split our proof of Theorem \[th:12\] into two lemmas. \[lem:41\] Let $\alpha\in(\frac{3-\sqrt{5}}{2},1/2),$ then there exists $n\in \mathbb{N}$ such that any element of ${\widetilde{\mathcal{U}}}_\alpha$ cannot contain the sequence $1(0)^{n}$ or $(-1)(0)^{n}$ infinitely often. Suppose $\alpha\in(\frac{3-\sqrt{5}}{2},1/2)$. Then by Lemma \[lem:22\] and (\[eq:41\]) we have $$\label{eq:42} (\delta_i(\alpha))\prec (1(0)^{\infty}).$$ For any $\alpha\in(1/3,1/2)$ we have $\delta_1(\alpha)=1$. Therefore by there exists $k\geq 0$ such that $(\delta_i(\alpha))$ begins with the word $1(0)^{k}(-1)$. If a sequence $(\epsilon_i)\in {\widetilde{\mathcal{U}}}_\alpha$ contained the sequence $1(0)^{k+1}$ infinitely often, then it is a consequence of Lemma \[lem:21\] for the digit set $\{-1,0,1\}$ that the following lexicographic inequalities would have to hold $$\label{eq:43} (-1)(0)^k1\preceq 1(0)^{k+1} \preceq 1(0)^{k}(-1).$$ Clearly the right hand side of does not hold, therefore $1(0)^{k+1}$ cannot occur infinitely often. Similarly, one can show that $(-1)(0)^{k+1}$ cannot occur infinitely often by considering the left hand side of . \[lem:42\] If $\alpha\in(1/3,\frac{3-\sqrt{5}}{2}]$ then for any sequence of natural numbers $(n_i)$ the sequence $$(1(-1))^{n_{1}} \,0 ^{n_{2}}\,(1(-1))^{n_{3}}\,0^{n_4}\cdots$$ is contained in ${\widetilde{\mathcal{U}}}_{\alpha}$. Fix a sequence of natural numbers $(n_i)$. It is a consequence of Lemma \[lem:21\] and Lemma \[lem:22\] that ${\widetilde{\mathcal{U}}}_{\frac{3-\sqrt{5}}{2}}\subset {\widetilde{\mathcal{U}}}_{\alpha}$ for all $\alpha\in(1/3,\frac{3-\sqrt{5}}{2}).$ Therefore it suffices to show that the sequence $$(\epsilon_i)_{i=1}^{\infty}:=(1(-1))^{n_{1}}\,0^{n_{2}}\,(1(-1))^{n_{3}}\,0^{n_{4}}\cdots$$ is contained in ${\widetilde{\mathcal{U}}}_{\frac{3-\sqrt{5}}{2}}.$ For all $n\geq 0$ the following lexicographic inequalities hold $$(-1)(0)^{\infty}\prec (\epsilon_i)_{i=n+1}^{\infty}\prec 1(0)^{\infty}.$$ Applying Lemma \[lem:21\] we see that $(\epsilon_i)\in {{\widetilde{\mathcal{U}}}}_{\frac{3-\sqrt{5}}{2}}$ and our proof is complete. Let $\alpha\in(\frac{3-\sqrt{5}}{2},1/2)$ and let $N\in \mathbb{N}$ be as in Lemma \[lem:41\]. Now let us pick $a\in (\frac{N}{N+1},1).$ Any $(t_i)\in {\widetilde{\mathcal{U}}}_\alpha$ with $\underline{d}((t_i))=a$ must contain either the sequence $1(0)^N$ infinitely often or $(-1)(0)^{N}$ infinitely often. By Lemma \[lem:41\] this is not possible. Thus by Lemma \[lem:23\] the set $D_\alpha$ is a proper subset of $[0,\frac{\log2}{-\log \alpha}]$ and statement $(2)$ of Theorem \[th:12\] holds. By Lemma \[lem:23\] it remains to show that for any $\alpha\in(1/3,\frac{3-\sqrt{5}}{2}]$ and $a\in [0,1]$ there exists $(t_i)\in {\widetilde{\mathcal{U}}}_{\alpha}$ such that $\underline{d}((t_i))=a.$ The existence of such a $(t_i)$ now follows from Lemma \[lem:42\] by making an appropriate choice of $(n_i).$ We now prove Theorem \[th:13\]. To prove this theorem we require the following technical characterisation of $S_{\alpha}$ from [@Kong_Li_Dekking_2010 Theorem 3.2]. We recall that an infinite sequence $(\omega_i)\in\{0,1\}^{\mathbb{N}}$ is called *strongly eventually periodic* if $(\omega_i)=IJ^\infty$, where $I,J$ are two finite words of the same length and $I\preceq J.$ Clearly, a periodic sequence is strongly eventually periodic. \[prop:43\] $t\in S_{\alpha}$ if and only if $(1-|t_i|)_{i=1}^{\infty}$ is strongly eventually periodic. Statement $(2)$ of Theorem \[th:13\] follows from the proof of Theorem \[th:12\]. It is a consequence of our proof that for $\alpha\in(\frac{3-\sqrt{5}}{2},1/2)$ there exists $\epsilon>0$ such that $\underline{d}((t_i))\notin(1-\epsilon,1)$ for all $(t_i)\in {\widetilde{\mathcal{U}}}_{\alpha}$. This statement when combined with Lemma \[lem:23\] implies statement $(2)$ of Theorem \[th:13\]. To prove statement $(1)$ we remark that for any $\alpha\in(1/3,\frac{3-\sqrt{5}}{2}]$ and $n_1,\ldots, n_j\in\mathbb{N},$ the sequence $$(t_i)=((1(-1))^{n_{1}}\,0^{n_{2}}\,(1(-1))^{n_3}\cdots (1(-1))^{n_{j-1}}\,0^{n_{j}})^{\infty}$$ is contained in ${\widetilde{\mathcal{U}}}_\alpha.$ The sequence $(1-|t_i|)$ is strongly eventually periodic, therefore by Proposition \[prop:43\] the corresponding $t$ is contained in $S_{\alpha}.$ For any $a\in [0,1]$ and $\epsilon >0,$ we can pick $n_{1},\ldots,n_j\in\mathbb{N}$ such that $|d((t_i))-a|<\epsilon.$ Applying Lemma \[lem:23\] we may conclude that statement (1) of Theorem \[th:13\] holds. Examples ======== We end our paper with some examples. We start with two examples of an $\alpha\in(1/3,1/2),$ and a $t\in {\Gamma_\alpha}-{\Gamma_\alpha}$ with a continuum of $\alpha$-expansions for which the Hausdorff dimension of ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ is explicitly calculable. The approach given in the first example applies more generally to $\alpha$ the reciprocal of a Pisot number and $t\in \mathbb{Q}(\alpha).$ Our second example demonstrates that it is possible for $t$ to have a continuuum of $\alpha$-expansions and for ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ to be a self-similar set. Let $\alpha=0.449\ldots$ be the unique real root of $2x^3+2x^2+x-1=0.$ Consider $t=\sum_{i=1}^{\infty}(-\alpha)^i.$ For this choice of $\alpha$ the set of $\alpha$-expansions of $t$ is equal to the allowable sequences of edges in Figure \[fig1\] that start at the point $((-1)1)^{\infty}$. (190,45)(0,-10) (20,2) (50,2) (80,2) (110,2) (140,2) (170,2) (102,-8)[$(1(-1))^{\infty}$]{} (72,-8)[$((-1)1)^{\infty}$]{} (135,-8)[$(110)^{\infty}$]{} (165,-8)[$(101)^{\infty}$]{} (5,-8)[$((-1)(-1)0)^{\infty}$]{} (37,-8)[$((-1)0(-1))^{\infty}$]{} (80,5)(95,45)(110,5) (94,23.5)[$>$]{} (110,5)(140,45)(170,5) (140,23.5)[$>$]{} (20,5)(50,45)(80,5) (47,23.5)[$<$]{} (23,2)[(1,0)[24]{}]{} (53,2)[(1,0)[24]{}]{} (107,2)[(-1,0)[24]{}]{} (137,2)[(-1,0)[24]{}]{} (167,2)[(-1,0)[24]{}]{} (31,6)[$-1$]{} (61,6)[$-1$]{} (93,6)[$1$]{} (124,6)[$1$]{} (155,6)[$1$]{} (91,30)[$-1$]{} (140,30)[$0$]{} (48,30)[$0$]{} Using we see that ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ coincides with those numbers $\sum_{i=1}^{\infty}\epsilon_i\alpha^i$ where $(\epsilon_i)$ is a sequence of allowable edges in Figure \[fig2\] that start at $((-1)1)^{\infty}$. (190,45)(0,-10) (20,2) (50,2) (80,2) (110,2) (140,2) (170,2) (102,-8)[$(1(-1))^{\infty}$]{} (72,-8)[$((-1)1)^{\infty}$]{} (134,-8)[$(110)^{\infty}$]{} (163,-8)[$(101)^{\infty}$]{} (7,-8)[$((-1)(-1)0)^{\infty}$]{} (37,-8)[$((-1)0(-1))^{\infty}$]{} (80,5)(95,45)(110,5) (94,23.5)[$>$]{} (110,5)(140,45)(170,5) (140,23.5)[$>$]{} (20,5)(50,45)(80,5) (47,23.5)[$<$]{} (23,2)[(1,0)[24]{}]{} (53,2)[(1,0)[24]{}]{} (107,2)[(-1,0)[24]{}]{} (137,2)[(-1,0)[24]{}]{} (167,2)[(-1,0)[24]{}]{} (33,6)[$0$]{} (63,6)[$0$]{} (93,6)[$1$]{} (125,6)[$1$]{} (155,6)[$1$]{} (93,30)[$0$]{} (138,30)[$0/1$]{} (46,30)[$0/1$]{} We let $$C_n:=\Big\{(\epsilon_i)_{i=1}^n\in\{0,1\}^n: \Big[\sum_{i=1}^{n}\epsilon_i\alpha^i,\sum_{i=1}^{n}\epsilon_i\alpha^i+ \frac{\alpha^{n+1}}{1-\alpha}\Big]\bigcap ({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))\neq\emptyset\Big\}.$$ Given $\delta_1\cdots\delta_m\in C_m$ we let $$C_{n}(\delta_1\cdots\delta_m):=\Big\{(\epsilon_i)_{i=1}^{m+n}\in C_{m+n}: (\epsilon_1,\ldots,\epsilon_m)=(\delta_1,\ldots, \delta_m)\Big\}.$$ Making use of standard arguments for transition matrices it can be shown that there exists $c>0$ such that $$\label{eq:51} \frac{\lambda^{n}}{c}\leq \#C_{n}\leq c \lambda^n \textrm{ and }\frac{\lambda^{n}}{c}\leq \#C_{n}(\delta_1\cdots\delta_m)\leq c \lambda^n,$$ for any $\delta_1\cdots\delta_m\in C_m.$ Here $\lambda\approx 1.69562\ldots$ is the unique maximal eigenvalue of the matrix $$A=\left( \begin{array}{cccccc} 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 2 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 2\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ \end{array} \right).$$ In the following we will show that $$\label{eq:52} \dim_H(\Gamma_\alpha\cap(\Gamma_\alpha+t))=\frac{\log\lambda}{-\log\alpha}\approx 0.644297.$$ In fact we show that $0<\mathcal{H}^{\frac{\log \lambda}{-\log \alpha}}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))<\infty.$ By (\[eq:51\]) the upper bound follows from the following straightforward argument: $$\begin{aligned} \mathcal{H}^{\frac{\log \lambda}{-\log \alpha}}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))&\leq \liminf_{n\to \infty}\sum_{(\epsilon_i)\in C_n} Diam\Big(\big[\sum_{i=1}^{n}\epsilon_i\alpha^i,\sum_{i=1}^{n}\epsilon_i\alpha^i+ \frac{\alpha^{n+1}}{1-\alpha}\big]\Big)^{\frac{\log \lambda}{-\log \alpha}}\\ &\leq c\lambda^n \Big(\frac{\alpha^{n+1}}{1-\alpha}\Big)^{\frac{\log \lambda}{-\log \alpha}}\\ & <\infty\end{aligned}$$In what follows we use the notation $\mathcal{I}_n$ to denote the basic intervals corresponding to the elements of $C_n,$ and $\mathcal{I}_{n}(\delta_1\cdots\delta_m)$ to denote the basic intervals corresponding to elements of $C_{n}(\delta_1\cdots\delta_m).$ The proof that $\mathcal{H}^{\frac{\log \lambda}{-\log \alpha}}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))>0$ is based upon arguments given in [@Baker_2014_1] and Example $2.7$ from [@Falconer_1990]. Let $\{U_j\}_{j=1}^{\infty}$ be an arbitrary cover of ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t).$ Since ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ is compact we can assume that $\{U_j\}_{j=1}^{p}$ is a finite cover. For each $U_j$ there exists $l(j)\in\mathbb{N}$ such that $\frac{\alpha^{l(j)+1}}{1-\alpha}<Diam(U_j)\leq \frac{\alpha^{l(j)}}{1-\alpha}.$ This implies that $U_j$ intersects at most two elements of $ \mathcal{I}_{l(j)} $. This means that for each $j$ there exists at most two codes $(\epsilon_{1},\ldots,\epsilon_{l(j)}),(\epsilon_{1}',\ldots,\epsilon_{l(j)}')\in C_{l(j)}$ such that $$U_j\cap \Big[\sum_{i=1}^{l(j)}\epsilon_i \alpha^i, \sum_{i=1}^{l(j)}\epsilon_i \alpha^i +\frac{\alpha^{l(j)}}{1-\alpha}\Big]\neq \emptyset\quad\textrm{ and}\quad U_j\cap \Big[\sum_{i=1}^{l(j)}\epsilon_i' \alpha^i, \sum_{i=1}^{l(j)}\epsilon_i' \alpha^i +\frac{\alpha^{l(j)}}{1-\alpha}\Big]\neq \emptyset.$$ Without loss of generality we may assume that $U_j$ always intersects at least one element of $\mathcal{I}_{l(j)}$. Since $\{U_i\}_{i=1}^p$ is a finite cover there exists $J\in\mathbb{N}$ such that $\alpha^{J}<Diam(U_i)$ for all $i$. By (\[eq:51\]) the following inequalities hold by counting arguments: $$\begin{aligned} \frac{\lambda^J}{c}\leq \#C_J & \leq \sum_{j=1}^p \#\Big\{(\epsilon_i)\in C_J: \Big[\sum_{i=1}^{J}\epsilon_i\alpha^i,\sum_{i=1}^{J}\epsilon_i\alpha^i+ \frac{\alpha^{J+1}}{1-\alpha}\Big]\cap U_j \neq \emptyset\Big\}\\ &\leq \sum_{j=1}^p \#C_{J-l(j)}(\epsilon_{1}\cdots\epsilon_{l(j)}) +\sum_{j=1}^m \#C_{J-l(j)}(\epsilon_{1}'\cdots\epsilon'_{l(j)}) \\ &\leq 2c\sum_{j=1}^{p}\lambda^{J-l(j)} \\ &\leq 2c\sum_{j=1}^{p}\lambda^J\cdot \alpha^{-l(j)\frac{\log \lambda}{-\log \alpha}}.\end{aligned}$$ Cancelling through by $\lambda^J$ we obtain $(2c^2)^{-1} \leq \sum_{j=1}^{p} \alpha^{-l(j)\frac{\log \lambda}{-\log \alpha}}.$ Since $Diam(U_j)$ is $\alpha^{l(j)}$ up to a constant term we may deduce that $\sum_{j=1}^{p}Diam(U_j)^{\frac{\log \lambda}{-\log \alpha}}$ can be bounded below by a strictly positive constant that does not depend on our choice of cover. This in turn implies $\mathcal{H}^{\frac{\log \lambda}{-\log \alpha}}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))>0.$ By we know that $$\label{eq:53} \dim_{H}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)) \geq \sup_{\tilde{t}} \dim_{H}\Big\{ \sum_{i=1}^{\infty}\epsilon_i\alpha^i:\epsilon_i\in \{0,1\}\cap (\{0,1\}+\tilde{t}_i)\Big\},$$ where the supremum is over all $\alpha$-expansions of $t$. If $t$ has countably many $\alpha$-expansions, then by the countable stability of the Hausdorff dimension we would have equality in . In the case where $t$ has a continuum of $\alpha$-expansions it is natural to ask whether equality persists. This example shows that this is not the case. Upon examination of Figure \[fig1\] we see that any $\alpha$-expansion of $((-1)1)^{\infty}$ satisfies $\underline{d}((t_i))\leq 1/3.$ In which case the right hand side of can be bounded above by $\frac{1}{3}\frac{\log 2}{-\log \alpha}\approx 0.281914.$ However by (\[eq:52\]) this quantity is strictly less than our calculated dimension $\frac{\log \lambda}{-\log \alpha}\approx 0.644297.$ Let $\alpha= \sqrt{2}-1 $ and $t=\frac{1}{\alpha(\alpha^3-1)}+\frac{1}{\alpha^2(1-\alpha^3)}.$ Then a simple calculation demonstrates that the set of $\alpha$-expansions of $t$ is precisely the set $\{ 0(-1)(-1),(-1)10\}^{\mathbb{N}}.$ Applying we see that $${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)=\Big\{\sum_{i=1}^{\infty}\epsilon_i\alpha^i:(\epsilon_i)\in \{100,000,010,011\}^{\mathbb{N}}\Big\}$$ This last set is clearly a self-similar set generated by four contracting similitudes of the order $\alpha^3$. This self-similar set satisfies the strong separation condition. So $$\dim_{H}({\Gamma_\alpha}\cap ({\Gamma_\alpha}+t))=\frac{\log 4}{-3\log \alpha}.$$ Each $\alpha$-expansion of $t$ satisfies $d((t_i))=1/3$. Thus the right hand side of can be bounded above by $\frac{\log 2}{-3\log \alpha}.$ Thus this choice of $\alpha$ and $t$ gives another example where we have strict inequality within . We now give an example of an $\alpha\in(1/3,1/2)$ and $t\in {\Gamma_\alpha}-{\Gamma_\alpha}$ for which ${\Gamma_\alpha}\cap ({\Gamma_\alpha}+t)$ contains only transcendental numbers. For $\alpha\in (0,1/3]$ examples are easier to construct, however, when $\alpha\in(1/3,1/2)$ the problem of multiple codings arises and a more delicate approach is required. Our examples arise from our proof of Theorem \[th:12\] and make use of ideas from the well known construction of Liouville. We call a number $x\in\mathbb{R}$ a Liouville number if for every $\delta>0$ the inequality $$|x-p/q|\leq q^{-(2+\delta)}$$ has infinitely many solutions. An important result states that every Liouville number is a transcendental number [@Bugeaud_2004]. This result will be critical in what follows. Let $p/q\in(1/3,\frac{3-\sqrt{5}}{2})$. Then there exists $t\in {\mathcal{U}}_{p/q}$ such that ${\Gamma_\alpha}\cap({\Gamma_\alpha}+t)$ only contains Liouville numbers. For any sequences of integers $(n_k)_{k=1}^{\infty}$ the sequence $$(t_i)=(1(-1))^{n_{1}}\,0\,(1(-1))^{n_{2}}\,0\cdots$$ is contained in ${\widetilde{\mathcal{U}}}_{p/q}.$ Now let $(n_k)$ be a rapidly increasing sequence of integers such that $$\label{eq:54} \Big(\frac{q}{p}\Big)^{2n_{1}+\cdots+2n_{k+1}+k+1}\geq q^{k(2n_1+\cdots 2n_k+k+3)}$$ Let $x\in \Gamma_{p/q}\cap (\Gamma_{p/q}+t),$ then $x=\sum_{i=1}^{\infty}\epsilon_i\Big(\frac{p}{q}\Big)^{i}$ where $\epsilon_i=1$ if $t_{i}=1,$ $\epsilon_i=0$ if $t_i=-1,$ and $\epsilon_i\in{\left\{0,1\right\}}$ if $t_i=0.$ It follows from our choice of $(t_i)$ that $$(\epsilon_i)=(10)^{n_{1}}\epsilon_{2n_{1}+1}(10)^{n_{2}}\epsilon_{2n_{1}+2n_{2}+2}\cdots.$$ For each $k\in\mathbb{N}$ we consider the rational $$\label{eq:55} \frac{p_k}{q_k}:=\sum_{i=1}^{2n_{1}+\cdots +2n_{k}+k}\epsilon_i\Big(\frac{p}{q}\Big)^{i}+\Big(\frac{p}{q}\Big)^{2n_{1}+\cdots +2n_{k}+k+1}\sum_{i=0}^{\infty}(p/q)^{2i},$$ where $p_k$ and $q_k$ are coprime. Either the block $00$ or $11$ occurs infinitely often within $(\epsilon_i)$. So $p_{k}/q_{k}\neq x$. Importantly, if we expand the right hand side of (\[eq:55\]) we can bound the denominator by $$\label{denominator bound} q_{k}\leq q^{2n_{1}+\cdots +2n_k+k+3}.$$The $p/q$-expansion on $p_{k}/q_{k}$ agrees with that of $x$ upto the first $(2n_{1}+\cdots +2n_{k+1}+k)$ position. Therefore $$\label{eq:56} |x-p_k/q_k|\leq c\cdot \Big(\frac{p}{q}\Big)^{2n_{1}+\cdots +2n_{k+1}+k+1}$$ for some constant $c$. Combining , , and we see that for each $k\in \mathbb{N}$ $$|x-p_k/q_k|\leq c q_{k}^{-k}.$$ Therefore $x$ is a Liouville number. Since $x$ was arbitrary, every $x\in \Gamma_{p/q}\cap(\Gamma_{p/q}+t)$ is Liouville. Acknowledgements {#acknowledgements .unnumbered} ================ The authors are grateful to Wenxia Li for being a good source of discussion and for his generous hospitality. [10]{} Jean-Paul Allouche and Jeffrey Shallit, *The ubiquitous [P]{}rouhet-[T]{}hue-[M]{}orse sequence*, Sequences and their applications ([S]{}ingapore, 1998), Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 1999, pp. 1–16. Simon Baker, *On universal and periodic [$\beta$]{}-expansions, and the [H]{}ausdorff dimension of the set of all expansions*, Acta Math. Hungar. **142** (2014), no. 1, 95–109. Yann Bugeaud, *Approximation by algebraic numbers*, Cambridge Tracts in Mathematics, vol. 160, Cambridge University Press, Cambridge, 2004. Karma Dajani and Martijn de Vries, *Invariant densities for random [$\beta$]{}-expansions*, J. Eur. Math. Soc. (JEMS) **9** (2007), no. 1, 157–176. Zolt[á]{}n Dar[ó]{}czy and Imre K[á]{}tai, *On the structure of univoque numbers*, Publ. Math. Debrecen **46** (1995), no. 3-4, 385–408. Martijn de Vries and Vilmos Komornik, *Unique expansions of real numbers*, Adv. Math. **221** (2009), no. 2, 390–427. Paul Erdős, István Joó, and Vilmos Komornik, *Characterization of the unique expansions $1=\sum_{i=1}^\infty q^{-n_i}$ and related problems*, Bull. Soc. Math. France **118** (1990), 377–390. Kenneth Falconer, *Fractal geometry*, John Wiley & Sons Ltd., Chichester, 1990, Mathematical foundations and applications. Paul Glendinning and Nikita Sidorov, *Unique representations of real numbers in non-integer bases*, Math. Res. Lett. **8** (2001), 535–543. Richard Kenyon and Yuval Peres, *Intersecting random translates of invariant [C]{}antor sets*, Invent. Math. **104** (1991), no. 3, 601–629. Vilmos Komornik, *Expansions in noninteger bases*, Integers **11B** (2011), Paper No. A9, 30. Vilmos Komornik, Derong Kong, and Wenxia Li, *Hausdorff dimension of univoque sets and devil’s staircase*, arXiv:1503.00475 (2015). Vilmos Komornik and Paola Loreti, *Unique developments in non-integer bases*, Amer. Math. Monthly **105** (1998), no. 7, 636–639. , *Subexpansions, superexpansions and uniqueness properties in non-integer bases*, Period. Math. Hungar. **44** (2002), no. 2, 197–218. Derong Kong, Wenxia Li, and F. Michel Dekking, *Intersections of homogeneous [C]{}antor sets and beta-expansions*, Nonlinearity **23** (2010), no. 11, 2815–2834. Roger Kraft, *Intersections of thick [C]{}antor sets*, Mem. Amer. Math. Soc. **97** (1992), no. 468, vi+119. Roger L. Kraft, *Random intersections of thick [C]{}antor sets*, Trans. Amer. Math. Soc. **352** (2000), no. 3, 1315–1328. Wenxia Li and Dongmei Xiao, *On the intersection of translation of middle-[$\alpha$]{} [C]{}antor sets*, Fractals and beyond ([V]{}alletta, 1998), World Sci. Publ., River Edge, NJ, 1998, pp. 137–148. William Parry, *On the $\beta$-expansions of real numbers*, Acta Math. Acad. Sci. Hungar. **11** (1960), 401–416. Alfred Rényi, *Representations for real numbers and their ergodic properties*, Acta Math. Acad. Sci. Hungar. **8** (1957), 477–493. Nikita Sidorov, *Almost every number has a continuum of [$\beta$]{}-expansions*, Amer. Math. Monthly **110** (2003), no. 9, 838–842. , *Combinatorics of linear iterated function systems with overlaps*, Nonlinearity **20** (2007), no. 5, 1299–1312.

--- abstract: 'A simple idea of relating the LQG and LQC degrees of freedom is discussed in context of toroidal Bianchi I model. The idea is an expansion of the construction originally introduced by Ashtekar and Wilson-Ewing and relies on explicit averaging of certain sub-class of spin-networks over the subgroup of the diffeomorphisms remaining after the gauge fixing used in homogeneous LQC. It is based on the set of clearly defined principles, thus is a convenient tool to control the emergence and behavior of the cosmological degrees of freedom in studies of dynamics in canonical LQG. Relating the proposed LQG-LQC interface with some results on black hole entropy suggests a modification to the *area gap* value currently used in LQC.' author: - Tomasz bibliography: - 'p-embedding.bib' title: Observations on interfacing loop quantum gravity with cosmology --- introduction ============ Loop Quantum Gravity [@t-lqg; @r-qg; @al-lqg-rev] (LQG) –one of the leading attempts to provide a solid framework unifying General Relativity with quantum aspects of reality– have matured over the past years to the level, in which extracting the concrete dynamical predictions out of it became a technically feasible task [@gt-aqg4; @*dgkl-gq; @hp-lqg-prl; @*s-comm; @*gt-matref]. Despite this success the concrete results regarding the evolution of quantum spacetime in LQG are yet to appear. On the other hand, in past ten years a series of dynamical predictions have been made (with various level of rigor) within the symmetry reduced framework originating from LQG known as Loop Quantum Cosmology [@b-livrev; @*bcmb-rev; @*as-rev] (LQC). There, the set of already known results ranges from establishing a singularity resolution [@b-sing] through qualitative changes of the standard early universe dynamics picture (found on the genuine quantum level) [@aps-prl; @*aps-det; @*apsv-spher; @*acs-aspects; @*bp-negL; @*pa-posL; @*v-open; @*hp-lqc-let; @*ppwe-radiation; @*mbmmp-b1-evo] to predictions of the behavior of cosmological perturbations [@bhks; @*aan-pert1; @*aan-pert2; @*aan-pert3; @*cmbg; @*lcb; @*we-pert1; @*we-pert2; @*fmmmo-pert1; @*fmmmo-pert2] and (in some cases) nonperturbative inhomogeneities [@bmp-geff-ftc; @*bmp-geff-det]. The results obtained within LQC framework cannot be treated as final, as LQC was never derived from LQG in any systematic way. Instead, it is a stand-alone theory constructed by applying the methods of LQG to cosmological models [@abl-lqc] further enhanced via parachuting some results and properties of LQG on the phenomenological level [@aps-imp]. Therefore, it is not a priori clear to what extent (if any) the predictions mentioned above reflect the true features of full LQG. Addressing this issues has brought a considerable interest within the loop community. Its pioneering studies [@bck-ref] were aimed towards controlling the so called *inverse volume corrections* in LQC and as tools to control the heuristic effective descriptions of inhomogeneous extensions of LQC [@bhkss-str]. Presently, the attempts to provide a precise connection between LQG and LQC are directed in three main areas: $(i)$ direct embedding of LQC framework within LQG one, $(ii)$ approximation of cosmological (symmetric) solutions in LQG, and $(iii)$ emergence of cosmological (LQC) degrees of freedom in appropriate scenarios within LQG. The approach $(i)$, realized on the level of mathematical formalism, focuses on embedding of the elements of the LQC formalism (for example straight holonomies) as a proper subclass of their analogs in LQG. So far however the most natural ways of constructing such embeddings have proved to lead to inconsistencies and resulted in several *no-go* statement [@bf-config; @*f-symm; @*bf-nonembed]. The main problem encountered in the attempts is the inherent diffeomorphism-invariance of LQG [@almmt-diffinv] and the fact that the symmetries characterizing the cosmological solutions are a subgroup of the diffeomorphism group. In consequence the formalism of the theory is (by construction) insensitive to the very components distinguishing the cosmological spacetimes. On the other hand, recently, extending the standard LQC holonomy-flux algebra by all holonomies along piecewise analytic curves and imposing the symmetries on the classical level led to a viable embedding of LQC in LQG [@e-embed]. To overcome the problems of $(i)$ another route \[listed as point $(ii)$\] was explored. There, instead of encoding the symmetries in the formalism one considered “the operational approach” – defining the symmetries through relations between spacetime quantities provided by the LQG observables. The idea has been realized by constructing the coherent states peaked about symmetric spacetimes (see for example [@e-symm]). Provided the evolution of such coherent states in LQG can be controlled on the level of full theory this description can provide a definition of a (n effective) reduced formalism on its own. Such formalism will have much stronger foundation in it’s LQG origin than LQC but applying it to probe for predictions is still a matter of the future. On the other hand, that formalism would loose its anchor in LQC, thus it would not be able to provide a solid connection with existing formulation of LQC or justification for the particular constructions implemented in it. As a consequence the utility of this approach as LQC test would be limited. To mend this gap one needs to construct a precise dictionary between objects of LQG and LQC formalism, while keeping both the theories autonomous (the approach $(iii)$). Such dictionary should associate selected (possibly emergent) degrees of freedom of LQG with cosmological ones – working as fundamental for LQC. In that aspect it is not necessary to restrict to symmetric or near symmetric states in LQG. Building this dictionary would be just a process of extracting some (global) degrees of freedom and as such should be well defined for *any* quantum spacetime (or at least for a family of such which is sufficiently large to cover physically interesting scenarios). A good example of such procedure is a simple toy model used in [@awe-b1] to fix certain ambiguities in quantizing the Bianchi I spacetimes in LQC. There, as an example of the spin network one considered a regular (cubic) lattice of which edges have been excited to the first state above the ground one. Much simpler models attempting to mimic cosmology, playing the role analogous to the one above, are also being constructed in Spin Foam formalism (often considered as the covariant formulation of LQG) [@rv-sfc1; @*rv-sfc2; @*hrv-sfc; @*bdpg-un; @*bfgl-un]. It is also worth remembering, that outside of the *top-down* approaches presented above there is a considerable literature on *bottom-up* approaches, where the components of LQC are cast onto LQG structures. An example of such is the so called *lattice LQC* [@we-lat], where the structure of degrees of freedom and elementary operators inherent to perturbative cosmology are defined on the regular lattice, again playing the role of an example of an LQG spin network. Another, bit more distant example is testing the BKL conjecture in context of LQC [@ahs-bkl]. The construction presented here is strongly motivated by the Bianchi I toy model mentioned two paragraphs above. it shares with this model the choice of spin network topology. Outside of this initial choice however we will keep the construction as general as possible (avoiding the dependence on particular prescription in either LQG or LQC) at the same time keeping full control over the assumptions entering the construction. In particular, as the unreduced side of the interface we will use *genuine* LQG without any simplifications. The dictionary will be provided by objects having a precise physical interpretation and well defined in both theories. To have such a simple tool available becomes recently more than just a convenience, as there is an increasing amount of effort towards making the preliminary dynamical predictions of either conservative LQG [@t-qsd3; @t-qsd5], its modifications [@gt-aqg1; @gt-aqg2; @*gt-aqg3] or simplifications (in particular the simplifications of $SU(2)$ gauge to $U(1)^3$ [@acr-red; @ac-red; @*ac-let; @b-b1]) via semiclassical approximations. One can thus confront the results of these projects with the predictions of LQC to either verify or falsify the latter. Before proceeding with the construction of the LQG-LQC interface let us briefly recall those element of both theories which will be relevant for its construction. Elements of LQC and LQC ======================= Since for all practical purposes both LQG and LQC are independent theories, just sharing common quantization methodology [@almmt-diffinv; @abl-lqc] we proceed with presenting them separately. Let us start with LQG. Loop quantum gravity -------------------- LQG is a quantization of canonical general relativity, owing a lot of its mathematical components to Young-Mills theories on the lattice. It’s starting point is a $3+1$ canonical splitting, with the phase space coordinatized by the Ashtekar-Barbero variables: $su(2)$ valued connection $A^i_a$ and the densitized triad $E^a_i$, where $A^i_a$ is a combination of the Levi-Civita connection and exterior curvature $A^i_a = \Gamma^i_a + \gamma K^i_a$ (with $\gamma$ being the Barbero-Immirzi parameter) [@b-var]. As any representation of GR it is a constrained theory with the algebra of constraints generated by: the Gauss constraint, the spatial diffeomorphisms and the Hamiltonian constraint. To deal with them the Dirac program is implemented: theory is quantized without constraints (so called kinematical level), which are next solved on the quantum level (with solutions forming the *physical* sector of the theory). The basic objects of the theory are the holonomies of $A$ along the piecewise analytic curves $U_\gamma(A) = {\mathcal{P}}\exp(\int_{\gamma}A_a^i\tau_i{{\rm d}}x^a)$ and the fluxes of $E^a_i$ across surfaces $K^i = \int_S E^{ai}{{\rm d}}\sigma_a$. Together they form form the holonomy-flux algebra, which is the fundamental object in constructing quantum theory. An application of the GNS (Gelfand-Naimark-Segal) construction to this algebra leads to the kinematical Hilbert space ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQG}}}$ spanned by the cylindrical functions supported on the graphs embedded in $3$-dimensional differential manifold. These functions are conveniently labeled by the $su(2)$ representations (on each edge of the graph + the internal edges within the graph vertices) – enumerated by half-integers. The quantum representation of holonomy-flux algebra provided by this construction is unique [@lost-uniq; @*f-uniq]. This space is non-separable, and defining a separable physical Hilbert space structure in the further steps of Dirac program requires a nontrivial effort (see for example the discussion in [@bpv-osc2] and references therein). On ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQG}}}$ the constraints are solved in hierarchy (in the order: Gauss, diffeomorphism, Hamiltonian). The Gauss constraints distinguishes (through the kernel of an operator corresponding to it) a subspace by selecting out the *gauge invariant* kinematical basis elements. These elements are characterized by the restrictions on the representation labels on the edges converging on a vertex (for each vertex of the graph), which restrictions can be thought of as analogs of the angular momentum addition rules in quantum mechanics. Next the diffeomorphism constraint is solved by procedure of averaging [@almmt-diffinv] over group of finite diffeomorphisms, which act on the basis elements by modifying the embedding of the graphs supporting them, but without modifying the topology of the graphs or their quantum labels. In particular, in case the framework used involves a single particular graph the group averaging would simply lift the graph from embedded to the abstract one (see for example [@gt-aqg1]). The diffeomorphism-invariant Hilbert space ${\mathcal{H}}_{{{\rm diff}}}^{{{\rm LQG}}}$ provided by this procedure serves next as a main “kinematical” space, on which the (diffeomorphism invariant) observables are defined and Hamiltonian constraint is solved. Action of many geometry observables is explicitly known. An interesting property of many of them, among others the area, volume, angle or length operators is that their spectra are purely discrete, composed of (generically) isolated points. In this meaning it is often stated, that in LQG the space(time) is discrete. These operators are however not physical observables since one more constraint remains – the Hamiltonian one. In the last step of Dirac program one identifies the physical Hilbert space as a kernel of the Hamiltonian constraint operator and builds the physical observables out of the diffeomorphism-invariant kinematical ones. The latter is achieved through the so called *partial observable* framework [@d-obs; @*r-obs]. In solving the Hamiltonian constraint several approaches are explored. Among them two approaches are considered as the most promising: *the Master program* [@dt-master1; @*dt-master2; @*dt-master3; @*dt-master4; @*dt-master5] and the matter deparametrization. In the first approach, to avoid mathematical complications related with the structure of the constraint algebra one constructs a single non-negative definite operator out of all the constraints – the so called master constraint. Then the physical Hilbert space is again given as a kernel of this operator. The last step was however not completed due to complicated mathematical structure of the constraint. In the second approach one uses the matter reference frames to provide (missing in the formalism) time variable [@gt-aqg4; @*dgkl-scalar; @hp-lqg-prl; @*s-comm; @*gt-matref]. This allows to reformulate the (originally constrained) theory as the free theory with a true Hamiltonian and where the original diffeomorphism-invariant Hilbert space itself or its proper subspace (depending on matter frame used) becomes the physical Hilbert space. The Hamiltonian is either former Hamiltonian constraint operator [@hp-lqg-prl; @*s-comm; @*gt-matref] or its square root [@gt-aqg4; @*dgkl-scalar]. In the former case its action on ${\mathcal{H}}_{{{\rm phy}}}^{{{\rm LQG}}}\equiv{\mathcal{H}}_{{{\rm diff}}}^{{{\rm LQG}}}$ is explicitly known and feasible to compute [@hp-mg13]. In this formulation, the diffeomorphism-invariant kinematical observables become the physical ones. Among the components of the theory two particular objects will play the crucial role in constructing our dictionary. These are, the diffeomorphism-invariant area operator and the Euclidean part of the Hamiltonian constraint. let us focus our attention on the area first. ### \[sec:lqg-ar\]The area operator The properties of this operator are known in detail (see for example [@t-lqg; @al-lqg-rev]). Its action on ${\mathcal{H}}_{{{\rm diff}}}^{{{\rm LQG}}}$ (i.e. on its spin network basis elements) is relatively simple [@flr-area]: the area of chosen (arbitrary) $2$-surface $S$ depends only on the $j$-labels of the edges of a spin-network reaching (or intersecting) this surface $$\label{eq:area-lqg} {\rm Ar}(S)\Psi[A] = 4\pi\gamma{\ell_{{{\rm Pl}}}}^2 \left[ \sum_{e^+} \sqrt{j_{e^+}(j_{e^+}+1)} + \sum_{e^-} \sqrt{j_{e^-}(j_{e^-}+1)} \right] ,$$ where $e^+$ are the incoming edges of the graph supporting $\Psi[A]$ which terminate on the surface, $e^-$ are the ones starting at the surface. $j_e^{\pm}$ are their respective $su(2)$ representation labels. The edges intersecting (piercing) the surface are counted as both incoming and outgoing (i.e. in that case a trivial $2$-valent node is temporarily introduced on the surface), thus their contribution is twice the terminating ones. The form of immediately implies, that the spectrum of ${\rm Ar}(S)$ is discrete. In particular the first non-zero value of the area is isolated from zero and equals $$\label{eq:lqg-area-min} A_1 = 2\sqrt{3}\pi \gamma {\ell_{{{\rm Pl}}}}^2.$$ ### \[sec:H-euclid\]The Euclidean part of Hamiltonian Classically the Hamiltonian constraint (or, more precisely, its part corresponding to gravity) is of the form $$C = \int {{\rm d}}^3 x \sqrt{-q}\, \mathcal{C} , \qquad \mathcal{C} = \frac{\gamma^2}{2\sqrt{\det E}} E^a_i E^b_j [ \epsilon^{ij}{}_k F^k_{ab} + 2(1-\gamma^2) K^i_{[a} K^j_{b]} ]$$ where the field strength $F_{ab}$ is the curvature of the connection $A$. The first term in $\mathcal{C}$ is the so called *Euclidean term* of the Hamiltonian density. The loop quantization procedure, requires to express $\mathcal{C}$ in terms of the holonomies and fluxes – the process known as Thiemann regularization [@t-qsd3]. In particular to express the curvature term $F$ one implements the known classical identity $$\label{eq:F-approx} F^i_{ab}X^a Y^b(x) = \lim_{{\rm Ar}(\triangle(x))\to 0} \frac{U_{\triangle(x)}-1}{{\rm Ar}(\triangle(x))}$$ where $\triangle(x)$ is the closed, piecewise analytic loop such that the vectors $X, Y$ are tangent to it at the point $x$, $U_{\triangle(x)}$ is the holonomy along this loop and ${\rm Ar}(\triangle(x))$ is the physical area of the loop. The right-hand side of this identity can be quantized directly in LQG, as the operators corresponding to the holonomy and the area are well defined. The particular implementation of this identity (and the action of the resulting regularized $\hat{F}$ operator) depends on the specific construction (or prescription) of the Hamiltonian constraint. In the original construction by Thiemann [@t-qsd3] the (Euclidean part of) Hamiltonian constraint added at the vertices the spin network the small triangular loops via introducing on the edges converging to it two $3$-valent nodes connected by an edge labeled by fundamental $su(2)$ representation ($j=1/2$). Then the limit of shrinking this triangular loop to a point (at the node) was taken in the sense of the embedding, which nonetheless led to a diffeomorphism invariant result due to the nature of operator components in the approximation of $F$ (for details, see [@t-qsd5]). In the alternative construction (see for example [@gt-aqg1]), where Hamiltonian constraint does not generate new edges, the loop has to be formed by existing edges of the spin network. It is defined by a requirement to form a plaquet – the minimal closed surface, of interior not intersected by any edges. In particular, when the spin networks are supported on the regular lattice, these loops are the minimal squares. The particular construction of the Hamiltonian constraint also affects critically the structure of the physical Hilbert space. Below we will briefly discuss the issues related to it. ### \[sec:Hphys\]Physical Hilbert space structure Since in the Master program ${\mathcal{H}}_{{{\rm phy}}}^{{{\rm LQG}}}$ is defined only abstractly (formally) to probe its properties we will focus on the deparametrization picture. To start with, we note that the kinematical Hilbert space which arises from the GNS construction is non-separable. The reason behind it is, that the induced inner product on ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQG}}}$ makes the states supported on disjoint graphs orthogonal. Each subspace of states supported on chosen graph is separable, with a discrete inner product, however the complete ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQG}}}$ contains a continuum of (disjoint) graphs. Unfortunately, the gauge invariant and diffeomorphism invariant Hilbert spaces retain this property: The former puts only the restrictions on the labels of the graph without significantly decreasing the possible graph structures. The latter still allows for the continuum of distinct (orthogonal) graphs with a discrete inner product between them. Since in the deparametrization picture the space ${\mathcal{H}}_{{{\rm diff}}}^{{{\rm LQG}}}$ becomes the physical one, this deficiency is transmitted directly to the physical sector. As the non-separability can affect significantly the construction and properties of the coherent states and the statistical ensembles (see [@bpv-osc1] for discussion of these issues on a simple quantum-mechanical example), this problem requires certain amount of care. The particular treatment depends on whether the action of the (true in the deparametrization picture) Hamiltonian changes the graph topology. If the graph is fixed (see for example [@gt-aqg1]) the Hamiltonian distinguishes subspaces invariant with respect to its action (supported on an unchanging graph). If the relevant observable operators are defined carefully and also preserve these subspaces, they become the superselection sectors, each of them being separable (as supported on one specific graph). The standard treatment calls then for a restriction of the studies to just one such sector. If the Hamiltonian is graph changing, this procedure becomes less straightforward, although often superselection sectors can be distinguished due to the fact, that (in specific prescriptions) the Hamiltonian changes the graph in a specific controlled way. This happens for example in case of the original construction of [@t-qsd3]. However, those superselection sectors can become already non-separable. On the other hand, our experience from LQC shows (see the discussion in [@bpv-osc2]), that for certain models such restriction might be insufficient to provide a sufficiently large semiclassical sector reproducing General Relativity dynamics in small gravitational field regimes. In that case an alternative construction may be needed. Such alternative is provided for example in [@bpv-osc2][^1]. There one makes use of available Lebesgue[^2] measure on the space of superselection sectors. Then the inner product is defined as the integral with respect to that measure of inner products $\langle\cdot|\cdot\rangle_{\epsilon}$ on the single superselection spaces ${\mathcal{H}}_{\epsilon}$ (with $\epsilon$ being an abstract superselection sector label) $$\label{eq:Hinteg} \forall \psi,\phi\in{\mathcal{H}}:\ \langle\psi|\phi\rangle = \int {{\rm d}}\mu(\epsilon) \langle\psi_{\epsilon}|\phi_{\epsilon}\rangle_{\epsilon} ,$$ where $\psi_{\epsilon}, \phi_{\epsilon} \in {\mathcal{H}}_{\epsilon}$ are the restrictions (projections) of the states to the single sector[^3]. Action of the operators preserving the sectors extends in a straightforward way. The integral Hilbert space ${\mathcal{H}}$ is again separable. \[sec:lqc\]Loop quantum cosmology --------------------------------- LQC, even when applied to the description of the inhomogeneous spacetimes, always relies on the reorganization of the geometry and matter degrees of freedom onto the quasi-global ones, for example the Fourier or spherical harmonic modes of the inhomogeneities/gravitational waves/matter (see [@mbgmm-G1; @fmmmo-pert1]). As a consequence in this description there are always distinguished degrees of freedom corresponding to the “background” homogeneous spacetime. This distinction is achieved by partial gauge fixing, which is naturally distinguished in the cases of homogeneous spacetimes and in perturbative approaches. The remaining (inhomogeneous) degrees of freedom are then treated as the objects “living” on that homogeneous background on the equal footing with the matter fields. Thus, in all of the models a proper handling of the homogeneous spacetimes is an essential first step. Here, we focus on the simplest model representing such spacetime – the model of Bianchi I universe. For the most of the paper we further fix the topology of its spatial slices to $3$-torus. The precise mathematical formulation of the LQC quantization of this model has been presented in [@abl-lqc] (isotropic spacetimes) and [@awe-b1] (actual quantization of the model following the procedures of [@abl-lqc]). It is performed via direct repetition of the procedure developed for LQG, although here the symmetries distinguish additional structure, which plays an essential role in the process. First, the homogeneity distinguishes the natural partial gauge, in which the spacetime metric takes the form $$\label{eq:g-b1} g = -N^2(t){{\rm d}}t^2 + a_1^2(t){{\rm d}}x + a_2^2(t) {{\rm d}}y + a_3^2(t) {{\rm d}}z$$ where $N(t)$ is the lapse function, $(a_1, a_2, a_3)$ are the scale factors in three orthogonal directions (in which the metric is diagonal) and ${{{}^o\! q}}= {{\rm d}}x^2 + {{\rm d}}y^2 + {{\rm d}}z^2$ is the isotropic fiducial metric constant in comoving coordinates $(x, y, z)$. This choice fixes all the spacetime diffeomorphisms up to: a global time reparametrization, and the (rigid in ${{{}^o\! q}}$ metric) global spatial translations. Similarly to the general GR case, we select the Ashtekar-Barbero variables, although here the fiducial metric distinguishes the orthonormal triad ${{{}^o\! e}}^a_i$ of vectors pointing in eigendirections of the physical metric and preserving the spatial symmetries of the system. That structure again allows to partially gauge-fix the variables through selecting $$\label{eq:AE-lqc} A_i^a = c^i (L_i)^{-1} {}^o\!\omega^i_a , \qquad E^i_a = p_i L_i V_o^{-1} \sqrt{{{{}^o\! q}}} {{{}^o\! e}}^a_i ,$$ where ${}^o\!\omega^i_a$ is a co-triad dual to ${{{}^o\! e}}^a_i$, $V_o$ is the fiducial (with respect to ${{{}^o\! q}}$) volume of the homogeneous spatial slices and $L_i$ are their (also fiducial) linear dimensions. The global coefficients $c^i$ and $p_i$ are the so called connection and triad coefficients. They form the canonical set with Poisson bracket $\{c^i,p_j\} = 8\pi G\gamma\delta^i_j$. In the next step one constructs the holonomy-flux algebra. Here however one notices, that upon the choice one can just select the subalgebra of holonomies $U_i^{(\lambda)}$ along the straight edges in direction ${{{}^o\! e}}_i$ and the fluxes $S_i$ along the unit squares orthogonal to ${{{}^o\! e}}_i$ as they suffice to separate the phase space points. Further, the fluxes $S_i$ can be associated with the triad coefficients themselves as $$\label{eq:S-p} S_i = p_i.$$ On such restricted (subalgebra of the) holonomy-flux algebra one implements the GNS construction, arriving to the unique quantum representation [@a-notes; @*ac-uniq]. The kinematical Hilbert space resulting from this construction is a product of the square summable functions on the Bohr compactification of the real line $${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQC}}} = \left[ \Sigma^2(\bar{{\mathbb{R}}}_{{{\rm Bohr}}}, {{\rm d}}\mu_{{{\rm Bohr}}}) \right]^3 ,$$ each copy of the space corresponding to one direction of ${{{}^o\! e}}_i^a$. The basic operators –quantum counterparts of the holonomy-flux algebra elements– are the holonomy operators $\hat{U}_i^{(\lambda)}$ and the unit flux operators $\hat{p}^i = \hat{S}^i$. the latter are known in the literature as the *LQC triad operators* due to the simple classical relation . We will implement the same naming policy here. One has to remember however, that these operators represent *the fluxes*, not the triads. As in full LQG, in LQC the operators corresponding to the holonomies and triads themselves *do not exist*. In fact, the relation between $p^i$ the scale factors $a_i$ in (see [@awe-b1]) shows, that the operators $\hat{p}^i$ measure *the area of the maximal surface orthogonal to ${{{}^o\! e}}_i^a$*. The kinematical states ${{| \psi \rangle}} \in {\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQC}}}$ automatically satisfy the Gauss and diffeomorphism constraints. The only gauge transformations left after the partial gauge fixing – the global spatial translations act on the elements of ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQG}}}$ as an identity. The only nontrivial constraint remaining is the Hamiltonian one. To construct the operator representing the (gravitational term of the) Hamiltonian constraint one repeats the Thiemann construction of LQG, partially discussed in sec. \[sec:H-euclid\]. For the Bianchi I model, the Lorentzian (exterior curvature) term is proportional to the Euclidean one, thus only the quantization of the latter is needed. For that one again has to deal with the field strength term, which is approximated by holonomies along a closed loop via . Since only the holonomies along the diagonal directions of $q$ are available the loop is a rectangle oriented in directions of ${{{}^o\! e}}^a_i$. Here however we see an important difference with respect to LQG. In the full theory either (depending on the formulation) one could move (shrink) the loop in the embedding manifold and the transformation has not modified the physical area of the loop. In LQC, the presence of the *background fiducial metric* ${{{}^o\! q}}$ –an object responsible for the rigid relation between $a_i$ and $p_i$– fixes a unique relation between the embedding (fiducial) area of the loop and its *physical area*. For a loop spanned by (holonomies along) vectors ${{{}^o\! e}}_j$, ${{{}^o\! e}}_k$ we have $${\rm Ar}(\square_{jk}) = \epsilon^{i}{}_{jk} p_i \lambda_j\lambda_k$$ where $\lambda_i$ are the fiducial lengths of the straight edges in direction ${{{}^o\! e}}^a_i$ forming the loop. This relation plays a crucial role in fixing the action of the Hamiltonian constraint in LQC [@aps-imp]. In principle in LQC one can set the fiducial edge lengths $\lambda_i$ freely which would allow to construct the loop or arbitrarily small areas. On the other hand in LQG the spectrum of the area operator is purely discrete and the first non-zero value of the area is determined by the theory. Since the goal of formulating LQC was the construction of the simplified settings approximating or mimicking the full theory as close as possible, this particular property (discreteness of the area) has been *parachuted from LQG*. The fiducial lengths $\lambda_i$ are fixed by the requirement that the physical area of the loop equals $$\label{eq:lqc-area} {\rm Ar}(\square) =: \Delta = 2 A_1 ,$$ where $A_1$ is provided by . Loop of these dimensions is then considered as the minimal loop realized in LQC. The reason, why one takes as the minimal area $2A_1$ instead of $A_1$ [@awe-cov] is that it should correspond to the area of the surface *pierced* by the edge, not just with the edge terminating on it. This requirement follows from the semi-heuristic lattice construction in [@awe-b1] and will became apparent in the process of constructing the dictionary in further sections. The particular way in which the requirement fixes the lengths $\lambda_i$ is construction dependent and in the past led to several distinct prescriptions in quantizing the Bianchi I model in LQC (see for example [@ch-b1]). Subsequently one choice has been distinguished by the construction in[@awe-b1] and by certain invariance requirements in application of the framework to the noncompact universes. At present the construction introduced in [@awe-b1] is considered to be the unique consistent prescription. With the lengths of holonomies fixed and the remaining components in the Hamiltonian constraint regularized via Thiemann construction, one arrives to the Hamiltonian constraint operator, which is a difference operator acting on the domain of elements of ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQC}}}$ supported on the finite number of points ${{| p_1,p_2,p_3 \rangle}}$ \[see eq. $(3.35)$-$(3.37)$ in [@awe-b1]\][^4]. Given that, after coupling with appropriate matter fields the dynamical sector of the theory is determined by either group averaging [@almmt-gave; @*m-gave1; @*m-gave2; @*m-gave3; @*m-gave4] (see also [@aps-det; @klp-gave] for applications in context of LQC) and partial observable formalism or via deparametrization. Here however we encounter the same problem as in full LQG: the kinematical Hilbert space ${\mathcal{H}}_{{{\rm kin}}}^{{{\rm LQC}}}$ is non-separable (due to the discrete inner product). Since in the deparametrization picture it becomes the physical space, the latter is also non separable. In isotropic LQC the procedure of dealing with this problem makes use of the fact, that the Hamiltonian (or Hamiltonian constraint) distinguishes certain subsets (“lattices”) invariant under its action. The subspaces of states supported on those sets form then superselection sectors, each being separable. Subsequently one choses just one sector to describe the dynamics. An extension of this approach to the anisotropic LQC is nontrivial, since for example in Bianchi I case the superselection sector “lattices” are formed out of families of sets dense on surfaces of codimension $1$ in the configuration space [@mbmmwe-b1]. Furthermore, it is not at all obvious, that a single sector would admit a proper semiclassical regime, where low energy dynamics conforms to GR. In order to deal with such difficulties one can again implement the integral Hilbert space construction [@bpv-osc2] following . In the case of known isotropic models [@aps-imp; @apsv-spher] the results following from implementing this construction are (up to minor corrections) equivalent to the ones provided by treatment involving just one superselection sector. \[sec:dict\]The dictionary ========================== The review in the previous section shows clearly, that both LQG and LQC describe the geometry via very distinct sets of degrees of freedom. The principal difference between these frameworks is the presence of background (fiducial) geometry in LQC and absence of such in LQG. This LQC background structure is distinguished by the symmetries of the theory on the classical level. By its very construction, LQG does not “detect” these symmetries as there the symmetry transformations are just a specific class of finite diffeomorphisms, to which the framework is insensitive. As a consequence, building an LQG state representing the cosmological spacetime is a nontrivial task [@e-symm]. Our goal here is constructing the relation between the frameworks on the kinematical level, that is we do not address the matter of agreement of the dynamics between these two frameworks. Answering the question whether the LQC dynamics of a state representing a universe is a good approximation to the dynamics of the state representing the same universe in LQG, is beyond the scope of this article. Instead, we explore the interplay between the structures in both theories and the consequence the consistency requirements of one framework impose on the other. In the process we try to keep the full control over the initial assumptions entering the construction and to determine the freedom left after making these assumptions. Our (principal) point of departure is the observation, that the basic quantities characterizing the state in LQC –the areas $p_i$ of maximal surfaces orthogonal to the basis triad vectors– correspond to observable quantities well defined for any physical LQG state. To see that, let us fix the type of spacetime represented and the approach to constructing the dynamical sector. - Our objects of studies will be physical states corresponding to the (homogeneous but not isotropic) Bianchi I universe, of which the spatial slices have a $3$-torus topology (although the results can be easily generalized to the noncompact flat case). Thus the embedding manifold for the kinematical spin-networks will be topologically $T^3$. - Since the dictionary we are constructing involves the correspondence between the geometry degrees of freedom only and does not employ the dynamics of the system, we can safely assume, that it will not depend on the type of matter content coupled to gravity. Thus we can work with chosen particular type of matter and the results will automatically generalize to other types of matter. For that purpose we select the construction of the dynamical sector through the deparametrization with respect to the irrotational pressureless dust serving as the time frame in both LQG [@hp-lqg-prl] and LQC [@hp-lqc-let]. While there is no consensus in the area as to whether the particular matter content selected above is well motivated physically, this choice provides simple and precise framework, circumventing the difficulties present in other approaches. In particular the diffeomorphism invariant Hilbert space in LQG and the kinematical Hilbert space of LQC automatically (and without any modifications or corrections) became the physical Hilbert spaces of their respective theories. Same applies to the area operators in LQG specified in sec. \[sec:lqg-ar\] and the “triad” operators $\hat{p}^i$ in LQC – they become physical observables. As a consequence they can be used directly, when comparing *physical* areas. Given that one can associate with each physical state ${{| \Psi \rangle}}\in{\mathcal{H}}_{{{\rm phy}}}^{{{\rm LQG}}}$ (where we do not require this state to be supported on just one spin network) a cosmological state ${{| \Phi \rangle}}\in{\mathcal{H}}_{{{\rm phy}}}^{{{\rm LQC}}}$ such that the expectation values of the area operators along the “flat and orthogonal” maximal surfaces agree with the expectation values ${{\langle \Phi |}}\hat{p}^i{{| \Phi \rangle}}$. Despite lack of background metric, the notion of flatness and orthogonality[^5] can be made precise in terms of the expectation values. Below we present one of possible constructions. it will not be used in construction of LQG-LQC interface and it is presented solely as an example that making the abovementioned association between LQG and LQC states is possible. One of many ways to define such construction is: 1. First one chooses on the embedding manifold a point $p$ and a triad of vectors anchored on it. 2. The angle operator in LQG is well defined [@t-lqg], thus one can distinguish (also in the embedding manifold) a triple of $2$-surfaces of $T^2$ topology –sections of the embedding manifold– such that the respective pairs of distinguished triad vectors are tangent to them (at their intersection). Their relative orientation is then fixed by the requirement that the expectation value of the angle operator corresponds to normal angles. 3. Finally, the “flatness” of the surfaces is enforced by requirement that the surfaces minimize their physical area (again in terms of the expectation values of the area operator). Then the area of the distinguished surfaces is associated with respective areas $\langle \hat{p}_i \rangle$ in LQC. This association between LQG and LQC states is far from unique: $(i)$ it is fixed just by relation between expectation values of three observables, which is obviously insufficient to determine the state, $(ii)$ the areas of the distinguished surfaces may depend on the point $p$ and the chosen vector triad, and $(iii)$ there may not be a global minimum of the areas in the point $3.$ of the above construction. At this moment however we do not look for the uniqueness of the association. We just want to show that it can be made. It may not be particularly useful for constructing the LQC limit of LQG, especially because so far we have not introduced any notion of symmetry. Literally, any (even very inhomogeneous) physical LQG state can be used in this construction. On top of that deficiency, at present we are lacking any relation with the auxiliary structures in LQC, which is necessary to really understand the relation between the frameworks. Therefore in further studies we are going to restrict the space of possible physical states, by selecting off a very specific (yet sufficiently large to accommodate the physically interesting spacetimes) set of spin networks supporting the states. \[sec:lat\]The lattice spin-network ----------------------------------- Our construction is heavily inspired by the semi-heuristic construction introduced in [@awe-b1]. There, one equipped the embedding ${\mathbb{R}}^3$ manifold with a fiducial metric ${{{}^o\! q}}$ and used it to define a regular lattice in it. This lattice has been next used to construct a specific spin-network by associating with each edge (link) of the lattice a $j$-label ($j=1/2)$ corresponding to the fundamental $su(2)$ representation – a minimal non-zero value allowed by the theory. Then, the gauge-invariant state has been distinguished as supported on this spin network only. Given that state, one could introduce a so called fiducial cell – a compact region of space acting as the infrared regulator of the theory. It was chosen such that its edges were parallel (in the sense of ${{{}^o\! q}}$) to the edges of the lattice. Due to fixing of the $j$-labels the area of each face of the cell was then proportional to the number of the lattice edges piercing it. With these areas one subsequently associated the values of $p_i$ in LQC. Given that association, the regularity of the lattice allowed in turn to associate with each plaquet (minimal square loop of the lattice) a “physical”[^6] area. Finally the requirement that all these areas equal $\Delta$ fixed the fiducial lengths of the edges of the plaquets, which in this construction are the curves along which the fundamental holonomies are taken. In this article we expand on this idea, dropping however most of the assumptions made in [@awe-b1]. To start with, we consider a single spin network, embedded in the $T^3$ manifold defined above. We further assume that our spin-network is topologically equivalent to the regular lattice[^7] or is a proper sub-graph of such. This graph is next equipped with the $su(2)$ $j$-labels on the graph edges (and internal edges at each node)*where in particular the value $j=0$ is allowed.* If the original graph is the proper subgraph of the (topologically) regular lattice, it is completed to the lattice by adding appropriate edges with $j=0$ and appropriate vertices. We further assume that the lattice is minimal: since two edges of $j=0$ entering $2$-valent vertex can be always replaced with one single edge, given a lattice spin-network we perform such reduction, whenever it does not destroy the regular lattice topology. Such construction of a spin-network, although abstract instead of embedded is used for example to formulate the *Algebraic Quantum Gravity* [@gt-aqg1] framework. To introduce the cosmological background structure we note, that a large class of spatial diffeomorphic gauge fixings can be implemented via equipping the embedding manifold with a metric tensor. For every spin network one can define a fiducial isotropic metric ${{{}^o\! q}}={{\rm d}}x^2 + {{\rm d}}y^2 + {{\rm d}}z^2$ such that ${\rm x}_i:=(x,y,z)$ are the functions defined along the edges of the graph, preserved by the discrete symmetries of the graph: for a cyclic permutation of the nodes along one lattice direction $i$[^8] the function ${{\rm x}}_i(\vec{x})$ changes as follows $${{\rm x}}_i \mapsto {{\rm x}}_i+\lambda_i , \qquad \lambda_i := 1/n_i$$ where $n_i$ is the number of graph edges forming a closed loop in direction $i$. The coordinates on the graph are next extended smoothly (non-uniquely) to the whole embedding manifold. It is worth reiterating, that neither the partial gauge fixing introduced here nor the auxiliary structure play any role in describing the physics. All the physical geometry observables are insensitive to this choice, as their action depends only on the topology of the graph and its quantum labels. Given the regular lattice defined above, one can precisely implement the construction of the LQG$\leftrightarrow$LQC dictionary specified at the beginning of sec. \[sec:dict\]. For that we choose the constancy surfaces $S_i$ of the coordinates ${{\rm x}}_i$ (thus orthogonal to ${{{}^o\! e}}_i^a$). The areas of these surfaces (expectation value of the operator ) are then associated with LQC “triad” or “flux” coeffcients $$\langle{\rm Ar}(S_i)\rangle = p_i := \langle\hat{p}_i\rangle .$$ The form of implies immediately, that these areas do not depend on the way the coordinates ${{\rm x}}_i$ have been completed between the elements of the graph. The values of $p_i$ depend however on the $j$-labels of the edges intersecting, terminating and contained within $S_i$ which may differ depending on the particular choice of the surface.[^9] Thus, this association is not unique. To emphasize this fact, we further denote these values as $p_{i,{{\rm x}}_i}$ and the surfaces themselves as $S_{i,{{\rm x}}_i}$. At this point we have to remember however, that there is some residual diffeomorphism freedom left in the system: the rigid (with respect to ${{{}^o\! q}}$) translations in ${{\rm x}}_i$. We will exploit this freedom in the next subsection to complete the dictionary construction. Before doing so however, we have to address one issue: since the dictionary will rely on the auxiliary structure, it is critical to check how it will be affected by the dynamics. In case, the LQG Hamiltonian is graph-preserving (like for example in [@gt-aqg1]) there is no problem: the only elements affected are the spin labels. The graph itself does not change, thus its embedding in the manifold can be assumed to be constant in time. This in turn allows to keep the auxiliary structure constant. The situation complicates a bit when the Hamiltonian is *graph changing*. There the preservation of the structure of the graph depends on particular form of that Hamiltonian. For example in original canonical LQG construction of [@t-qsd3] the Hamiltonian adds edges with $j=1/2$ labels forming a triangular loops with existing ones. This can be easily implemented in the construction considered here, if instead of triangle we add a square loop with two new $j=1/2$ edges. The new spin network can be then easily completed to a (topologically) regular lattice by adding $j=0$ edges. The auxiliary structure can then be easily rebuilt, essentially in tow ways: 1. The coordinates $x_i$ and fiducial metric can be redefined so the new lattice becomes regular in them. This corresponds to discontinuously shifting the vertices of the graph to new positions on the embedding manifold (passive diffeomorphism). The discontinuity is however not a problem, as the Hamiltonian flow is not used in the construction of the dictionary and the auxiliary structure can be defined at each time slice independently. 2. \[it:latex-dense\] The new nodes can be placed in the center (with respect to ${{{}^o\! q}}$ of existing plaquets. Then, the lattice would be a particular realization of the *dense spin-network* [@ag-dense]. It would however loose the regularity. The latter could in principle pose a problem as the fiducial length of the edges will be an essential component of the dictionary. In that case however one should average any quantity evaluated on the graph over the diffeomorphisms changing the fiducial lengths of the edges but preserving the directions (with respect to metric ${{{}^o\! q}}$) of the edges of the graph. For quantities which are averages weighted by the fiducial lengths this averaging procedure yields exactly the same results as the “uniformization” defined in the point above (see Appendix \[app:stat\]). It is important to note that the regularity assumption can be replaced with the average over (passive) diffeomorphisms preserving the directions of the edges (with respect to ${{{}^o\! q}}$). Indeed one can represent the (relevant for the graph) passive diffeomorphisms as random distributions of the vertices coordinates over the interval $[0,1]$, see Appendix \[app:stat\]. \[sec:avg\]The averaging procedure ---------------------------------- At present a relevant difference remains between the LQG state constructed previously and its LQC analog. The LQC state is constructed with implicit assumption of representing the homogeneous spacetime, whereas the LQG one can a priori be highly inhomogeneous. This implies that the association of the values of $p_i$ has to involve some kind of averaging (over the inhomogeneities) procedure. In LQG the lack of background structure makes the definition of such averaging difficult. Here however the choice of the spin network graph and partial gauge fixing allowed to construct the necessary background structure. To employ it, we now consider the remaining rigid translations as the *active transformations* shifting the surfaces $S_{i,{{\rm x}}_i}$ along the graph and define the values $p_i$ as the averages with respect to this translation group. In the mathematically precise sense the variables $p_i$ are chosen to equal the expectation values of the area operator (of each surface $S_i$) *averaged over the rigid spatial translation group*. A simple calculation using shows then, that $$\label{eq:p-dict} p_i = \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_i} \sum_{e\in \{e\}_i} \sqrt{j_e(j_e+1)} =: \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_i} \Sigma_i,$$ where $\{e\}_i$ is the set of all the edges of the graph which point in direction of ${{{}^o\! e}}_i$ and $n_i:=1/\lambda_i$ is the number of graph edges pointing in direction of ${{{}^o\! e}}_i$ and forming a close loop. Here, the edges terminating on the surface or contained within it do not contribute, as that would require tuning the translations and such translations are a zero measure set within the distinguished translation group. Next component needed in our dictionary are the (physical) areas of the minimal loops (plaquets) needed to approximate the field strength operator. To evaluate these areas we proceed exactly as in the case of the surfaces $S_i$: we average the relevant area operators over the rigid (active) translations group. Again, a simple calculation yields (here we denote these areas by $\sigma^i$) $$\label{eq:sigma1} \sigma_i = \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \sum_{e\in \{e\}_i} \sqrt{j_e(j_e+1)} = p_i \frac{\lambda_i}{\lambda_1\lambda_2\lambda_3} .$$ As a consequence, the ratio $\sigma_i/p_i$ (no summation over $i$) does not depend on the $j$-labels of the spin network. It depends *only on the number of edges of the graph* which is an expected result since (once the averaging is implemented) each surface $S_i$ can be simply composed out of $n_1n_2n_3/n_i$ plaquets. It is important to note, that, even though we are permitting the edges with $j=0$, the numbers $n_j$ are invariant due to gauge invariance and the requirement that the lattice is minimal. The standard procedure implemented in LQC would call in the next step for an association $\sigma_i=\Delta$, where $\Delta$ is defined via . This would fix $\lambda_i$ as the functions of the phase space, leading exactly to the dependence found in [@awe-b1]. Here however we do not implement this step, expecting in turn that the value of $\sigma_i$ should follow from the properties of the spin network. Thus, the only property at our disposal is the relation of $\sigma_i$ with the average (over the graph) of $j$-labels associated with edges in direction ${{{}^o\! e}}_i$ $$\label{eq:sigma-inv} \sigma_i = 8\pi\gamma{\ell_{{{\rm Pl}}}}^2 \overline{[ \sqrt{j(j+1)} ]_{e_i}} =: \Delta_i,$$ where the symbol $\overline{[ \cdot ]}$ denotes the average over the graph. For the models aimed to reproduce the cosmological spacetime via specific semiclassical states the value of such average depends on the details of the model and may in principle differ significantly from the value $\Delta_i = \sqrt{3/2}$ consistent with $\sigma_i=\Delta$ (see for example [@ac-red]). In such models, the values $\sigma_i$ do not need to be fixed by any fundamental constant and a priori may depend on the state. ### \[sec:trans-alt\]Alternative averaging procedure The results of the averaging procedure implemented above can be easily understood on the intuitive level if we introduce a convenient decomposition of the group of rigid translations. First, one can introduce a discrete group ${\mathbb{Z}}^3_{n_1,n_2,n_3}:={\mathbb{Z}}_{n_1}\times{\mathbb{Z}}_{n_2}\times{\mathbb{Z}}_{n_3}$ of cyclic permutations of the graph vertices. The quotient of the group of translations over ${\mathbb{Z}}^3_{n_1,n_2,n_3}$ is the group of translations over the distances $\delta_i\in [0,\lambda_i)$. The averaging procedure can now be split onto two steps: - Averaging over ${\mathbb{Z}}^3_{n_1,n_2,n_3}$, which simply replaces the $j$-label of a single edge with the average $\overline{[j]}_{e_i}$ of all the edges parallel to it. - Averaging over the quotient group. The result of this step follows directly from the observation that upon the action of the quotient group (with exception of the group neutral element which forms a zero measure set) each plaquet of the graph is intersected by exactly one edge (now carrying the averaged $j$) orthogonal to it. The above procedure leads immediately to and and further, after reassembling the surfaces $S_i$ out of the plaquets, to . The potential application of the studies performed here to the models, where the cosmological spacetime is defined by the semiclassical (often chosen to be coherent) state, lead to another complication. So far we have considered the single graph. Why such choice is perfectly fine to define a basis of a Hilbert (sub)space, it may be insufficient for such models. Therefore one needs to extend the dictionary to incorporate large number of such spin-network “superselection sectors”. We provide such extension below, using the integral Hilbert space construction presented in sec \[sec:Hphys\]. \[sec:int\]The integral extension --------------------------------- As in the case of a single lattice space, here we are going to define some subspace of ${\mathcal{H}}_{{{\rm phy}}}^{{{\rm LQG}}}$. 1. We start with a single lattice spin network defined in sec. \[sec:lat\] (without introducing the background metric ${{{}^o\! q}}$). 2. \[it:max\_s\] The plaquets of the spin network define three classes of surfaces of topology $T^2$ (maximal surfaces on the embedding manifold).[^10] such that within one class the surfaces do not intersect each other and all the intersections of the surfaces of distinct classes are of $S^1$ topology. 3. \[it:surf\] The discrete classes of surfaces are next completed to congruences of the embedding manifold (using the surfaces of the same topology) keeping the requirement, that intersections between representants of different classes are $S^1$. This (non-unique) extension always exists. 4. We extend the original lattice spin network to a class of disjoint spin networks of which edges are intervals of the intersections of surfaces defined above. This class is selected in such a way, that 1. each point of the embedding manifold is a node of exactly one spin network in the class, and 2. given two graphs of the set, the maximal $T^2$ surfaces of point \[it:max\_s\] are interlaced, that is within each class of surfaces, between two surfaces of one graph there is exactly one surface of the other graph. 5. Each spin network is completed to a (topologically) regular lattice by adding edges with $j=0$. In its essence this method produces a continuum of (topologically) regular lattices of which edges are “parallel”. They define a distinguished coordinate system ${{\rm x}}_i$ where the coordinates are functions constant on the $T^2$ surfaces from point \[it:surf\]. Given this coordinate system, one can equip the manifold with a background fiducial metric ${{{}^o\! q}}= \sum_{i=1}^3 ({{\rm d}}{{\rm x}}_i)^2$. This construction is of course quite restrictive, however it allows to preserve the well defined notion of principal (diagonal) directions of the single lattice. One way to produce the specific example of such set of spin-networks is to start with one regular lattice, equip the embedding manifold with the fiducial metric ${{{}^o\! q}}$ (as in sec. \[sec:lat\] and then act with the active rigid translations defined in sec. \[sec:avg\]. The family of possible continuous sets defined in points $1.-5.$ is however much bigger. In particular, the lattices do not need to be regular with respect to the metric ${{{}^o\! q}}$. An important property of the selected set of spin networks is that it admits a well defined Lebesgue measure ${{\rm d}}\sigma$ induced by the Lebesgue measure of a minimal cube of any (chosen arbitrarily) lattice within the set. This measure can now be used to construct the integral Hilbert space via via setting ${{\rm d}}\mu(\epsilon)={{\rm d}}\sigma$. Choosing different minimal cubes will lead to unitarily equivalent spaces. Given the new (integral) Hilbert (sub)space we proceed with defining averaged quantities $p^i$, $\sigma^i$ exactly as in sec. \[sec:avg\]. The only difference is an additional integration over the selected set of lattices. The calculations yield $$\begin{aligned} p_i &= \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_i} \int {{\rm d}}\sigma(\vec{\epsilon}) \sum_{e\in \{e\}^i_{\Gamma(\vec{\epsilon})}} \sqrt{j_e(j_e+1)} =: \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_i} \overline{\Sigma}_i, \\ \sigma_i &= \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \int {{\rm d}}\sigma(\vec{\epsilon}) \sum_{e\in \{e\}^i_{\Gamma(\vec{\epsilon})}} \sqrt{j_e(j_e+1)} = p_i \frac{\bar{\lambda}_i}{\bar{\lambda}_1\bar{\lambda}_2\bar{\lambda}_3} = 8\pi\gamma{\ell_{{{\rm Pl}}}}^2 \overline{[ \sqrt{j(j+1)} ]}_{e_i,\vec{\epsilon}} =: \bar{\Delta}_i, \label{eq:avg-sigma}\end{aligned}$$ where $\vec{\epsilon}$ labels the superselection sectors and $\Gamma(\vec{\epsilon})$ is the graph corresponding to superselection sector $\vec{\epsilon}$. We see, that the quantities $\lambda_i$, $\Delta_j$ are now simply replaced by their averages $\bar{\lambda}_i$, $\bar{\Delta_j}$ over the superselection sectors. The quantities $\bar{\lambda_i}$ are now *defined* by and do not necessary correspond to the average over fiducial lengths of the edges (which in turn may not be the constants of the graph). The average $j$-label, is however a proper average $$\bar{\Delta}_j = \int{{\rm d}}\sigma(\vec{\epsilon})\Delta_j(\vec{\epsilon}) .$$ The above result has been found under assumption of a specific compact topology of the embedding manifold. However in LQC a consistency restrictions that actually do fix the theory originate in models of the noncompact universes, some of them already in isotropic sector. Therefore it is prudent to extend our dictionary to such case and to incorporate in it the notion of isotropy. For simplicity we will consider single lattice states only, although the generalization to the integral states of sec. \[sec:int\] is not difficult. The noncompact extension and the isotropy ----------------------------------------- In order to keep the model as simple as possible we consider its extension to the flat space, assuming $R^3$ topology. ### The extension to ${\mathbb{R}}^3$ In LQC the standard method of dealing with infinities due to Cauchy slice noncompactness is selection of a compact region –the so called fiducial cell– which then becomes the infrared regulator of the model. The physical predictions are then extracted within the regulator removal limit. Well definiteness of that limit is the first consistency condition imposed on LQC and is precisely the origin of the so called *improved dynamics* prescription [@aps-imp]. Here we follow the same idea: We start with the construction of a single lattice spin network, as specified in sec. \[sec:lat\], although now the lattice is open and infinite. We then introduce the background structure exactly as in the compact case and distinguish the regulator – a rectangular cube of edges pointing in directions of ${{{}^o\! e}}_i$. The expansion of the regulator is well defined in terms of the number of edges forming the interval of “straight lines” which is contained within the cube (or ,equivalently, the number of elementary cells stacked along the edge of the cube). The remaining spatial gauge freedom of the model is the same as in the compact case: the rigid translations. Now however we encounter a technical difficulty: the translation group is noncompact and does not preserve the regulator structure. We sidestep this problem, introducing the periodic boundary conditions on the faces of the regulator, thus restricting to certain compact “cyclic translation” group. The new “translations” are obviously not elements of the original translation group, however the proposed “trick” is well motivated on the heuristic level: as $(i)$ their action will equal that of the original translations group elements on those spin networks which are composed of the set of identical “copies” of the portion contained within the regulator cell, and $(ii)$ in the regulator removal limit we recover the original translation group. On the technical level the above construction brings us exactly to a compact setting $T^3$ topology considered in previous subsections. Thus we proceed with construction of the dictionary exactly as before. The results - remain true here. In the regulator removal limit the values $p_i$ reach infinity, however the plaquets areas are well defined $$\label{eq:sigma-r3} \sigma_i = \lim_{n_1,n_2,n_3\to\infty} \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \sum_{e\in \{e\}^i} \sqrt{j_e(j_e+1)} = 8\pi\gamma{\ell_{{{\rm Pl}}}}^2 \overline{[ \sqrt{j(j+1)} ]}_{e_i} =: \Delta_i ,$$ provided that the limit on the right-hand side exists. it is however a reasonable expectation if we consider asymptotically homogeneous state. As a consequence the relation extends to the noncompact case. ### The isotropic sector In comparizon to the homogeneous non-isotropic spacetimes considered so far, the isotropic ones admit an additional symmetry class (subgroup) – the rotations. Implementing these symmetries in the compact $T^3$ case is not possible, thus our starting point is the noncompact setting of the previous sub-subsection. Here we make one additional initial assumption: we restrict the auxiliary metric by requiring that the fiducial lengths of the edges in all three directions are the same. the flat metric ${{{}^o\! q}}$ now defines the group of rigid rotations. As in the case of the translations, we consider them as active diffeomorphisms and average over them the observables used to define the dictionary. Due to noncompactness, the only meaningful element of the dictionary is the average area of plaquets $\sigma_i$. The rotational transformation of the spin-network can be easily parametrized by the Euler angles. As in the case of translations, here we can distinguish a discrete group if the rotations by proper angles, which due to insensitivity of the area operator to the orientation of the edges can be replaced by a group $\Sigma^3$ of permutations of ${{{}^o\! e}}_i$. We can then distinguish a quotient group $SO(3)/\Sigma^3$. Let us firsat consider the averageing over (the group of translations and) $\Sigma^3$. It follows immediately from that $$\label{eq:sigma-iso1} \sigma_{\Sigma} := \sigma_i = \lim_{n_1,n_2,n_3\to\infty} \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \sum_{e\in E(\Gamma)} \sqrt{j_e(j_e+1)} = 8\pi\gamma{\ell_{{{\rm Pl}}}}^2 \overline{[ \sqrt{j(j+1)} ]}_{E(\Gamma)} = \frac{1}{3} [ \Delta_1 + \Delta_2 + \Delta_3 ] =: \Delta_{\star} ,$$ where this time, we sum over *all* the edges of the spin network. This result is identical with the FRW limit of Bianchi I geometry (in LQC) studied in [@awe-b1]. The averaging over (translations and) full $SO(3)$ is slightly more involved. $$\label{eq:sigma-iso-gen} \sigma_R := \sigma_i = \lim_{n_1,n_2,n_3\to\infty} \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \int_{SO(3)} {{\rm d}}\sigma_{SO(3)} \sum_{e\in E(\Gamma)} |{{{}^o\! q}}_{ab}n^a {{{}^o\! e}}(e)_i^b| \sqrt{j_e(j_e+1)} ,$$ where $n^a$ is a unit (in ${{{}^o\! q}}$) vector orthogonal to the plaquet and ${{{}^o\! e}}_i^a$ is the fiducial triad element tangent to the graph edge $e$. The factor ${{{}^o\! q}}_{ab}n^a {{{}^o\! e}}(e)_i^b$ is a consequence of averaging over spatial tranlations as only the orthogonal (to ${{{}^o\! e}}_i$) “crossection” of the plaquet will contribute to the average over the translations along ${{{}^o\! e}}_i$. Using the known chart of $SO(3)$ defined by Euler angles (convention $Z(\alpha)X(\beta)Z(\gamma)$)[^11] we get $$\label{eq:sigma-iso-gen2}\begin{split} \sigma_R &= \lim_{n_1,n_2,n_3\to\infty} \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \frac{4}{\pi^2} \int_{\alpha\beta,\gamma\in [0,\ldots,\pi/2]} \hspace{-1cm}{{\rm d}}\alpha \sin(\beta){{\rm d}}\beta {{\rm d}}\gamma\, \frac{1}{3} [ \sin(\beta)(\cos(\gamma)+\sin(\gamma))+\cos(\beta) ] \sum_{e\in E(\Gamma)} \sqrt{j_e(j_e+1)} \\ &= 4\pi\gamma{\ell_{{{\rm Pl}}}}^2 \overline{[ \sqrt{j(j+1)} ]}_{E(\Gamma)} = \frac{3}{2}\Delta_{\star} , \end{split}$$ where to write the first equality we split the rotation group onto $\Sigma^3$ and the quotient $SO(3)/\Sigma^3$. As we can see, the contribution from the general angle rotations increases the physical plaquet area by a factor $3/2$. This discrepancy is unexpected, since the critical energy density (upper bound of the matter energy density operator spectrum) is a bijective function of the area gap and both Bianchi I and FRW spacetime models in LQC provide the same value of that quantity (see [@cs-b1] versus [@acs-aspects]). To address this discrepancy we will investigate a bit closer the rotation group used in the averaging process: the rotations in fiducial metric ${{{}^o\! q}}$. To do so, let us consider a large (classical size) cube of fiducial size $L$. Denote its surface area (averaged over the translation group and $\Sigma^3$ to mimic an isotropic spacetime as close as possible without rotating the graph) in the case when its edges are oriented along the triad vectors ${{{}^o\! e}}_i$, by $A_{\square}$. By repeating the same calculation as in one can show, that upon rotating this cube by Euler angles $(\alpha,\beta,\gamma)$ the surface area changes as follows: $$A_{\square}(\alpha,\beta,\gamma) = A_{\square} [ \sin(\beta)(\cos(\gamma)+\sin(\gamma))+\cos(\beta) ] .$$ This implies in particular, that even if the $j$-label distributions in all the directions are the same this surface area *is not invariant under the rotations*. The direct consequence of it is the no-go statement: one can not build the isotropic spacetime using lattice spin network with the edges oriented in directions of one particular vector triad. For that, the large set of spin networks oriented in random directions would be needed. Such construction is quite easy, if for example one defines the integral Hilbert space structure (discussed in sec. \[sec:Hphys\]) using the integral measure of $SO(3)$ group. In that case however, even the anisotropic universe model would attain the modification due to integrating over that structure. Indeed, in that case one can rotate the original triad ${{{}^o\! e}}_i$ and associate with each rotation a lattice state (orthogonal with respect to the original lattice state). The set of states supported on new (rotated) lattices forms now a set of superselection sectors (provided the inner product is introduced as in ). Then $$\sigma_i = \int_{SO(3)}{{\rm d}}\sigma_{SO(3)} {{{}^o\! q}}_{ab}{{{}^o\! e}}_i^a M(g)^b{}_c {{{}^o\! e}}_j^c \lim_{n_1,n_2,n_3\to\infty} \frac{8\pi\gamma{\ell_{{{\rm Pl}}}}^2}{n_1 n_2 n_3} \int {{\rm d}}\sigma(\vec{\epsilon}) \sum_{e\in \{e\}^j_{\Gamma(\vec{\epsilon},g)}} \sqrt{j_e(j_e+1)}$$ where $g$ is the finite rotation (parametrized by Euler angles) and $M(g)$ is the rotation matrix corresponding to it. The symbol $\Gamma(\vec{\epsilon},g)$ denotes here graphs oriented along the fiducial triad ${{{}^o\! e}}_i$ rotated by $g$ (that is a support of a superselection sector of lattices oriented in the rotated triad) and belonging to the superselection sector labeled by $\vec{\epsilon}$. Using again the $SO(3)$ chart defined by Euler angles gives then $$\sigma_i = \frac{\gamma{\ell_{{{\rm Pl}}}}^2}{\pi} \int_{0}^{2\pi} {{\rm d}}\alpha \int_{0}^{\pi} \sin(\beta){{\rm d}}\beta \int_{0}^{2\pi} {{\rm d}}\gamma\, \sum_{k=1}^{3} (e_i \cdot M(\alpha,\beta,\gamma) e_k) \overline{[ \sqrt{j(j+1)} ]}_{e_k,\vec{\epsilon}}(\alpha,\beta,\gamma) =: \tilde{\Delta}_i,$$ where the rotation matrix $M$ are now expressed in terms of Euler angles and the averages over the translation superselection sectors of $j$-labels (originally defined in sec. \[sec:int\]) are now defined separately for each rotation superselection sector labeled by $(\alpha,\beta,\gamma)$. In the case, the averages of $j$-labels of edges in the same direction over the graph (within a single superselection sector) are equal, a simple calculation shows that $$\sigma_i = 4\pi\gamma{\ell_{{{\rm Pl}}}}^2 \sum_{i=1}^3 \overline{[ \sqrt{j(j+1)} ]}_{e_i,\vec{\epsilon}} = \frac{1}{2} \sum_{i=1}^3 \bar{\Delta}_i ,$$ where $\bar{\Delta}_i$ is defined in . This leads exactly to the correction of the Bianchi I plaquet area by a multiplicative factor $3/2$ restoring the consistency with the isotropic limit.[^12] The physical consequences ------------------------- For both construction of separable physical Hilbert space (the single superselection sector and the integral one) we reached the same conclusion. Given a “lattice” LQG state specified in sections \[sec:lat\] or \[sec:avg\] (as well as its extension discussed in previous subsection) the Bianchi I LQC state mimicking it has to have the single plaquet area is proportional to the average of $j$-labels (in appropriate direction) of the original LQG state. The time dependence of the latter depends in turn on $(i)$ the choice of the (initial) LQG state and $(ii)$ on the statistics of the particular Hamiltonian used to generate the time evolution. In particular, any model following from strictly graph preserving Hamiltonian will have $$\sigma_i \propto p_i$$ which will lead to original Bojowald’s prescription in LQC [@b-homo1; @*b-homo2; @*bdv-homo]. On the other hand, the studies of the noncompact model, show that in low energy limit $\sigma_i$ should be constant. As a consequence, for the class of states considered in this article the average $j$-label, more precisely $\Delta_j$ should approach constant in low curvature limit, thus the expansion of the spacetime in the process of dynamical evolution should follow from increasing the number of spin-network nodes rather than the $j$-labels. This conclusion is supported by our intuitive understanding to the physical distance. Essentially, by examining the definitions of meter and second one sees, that the definition of a distance unit can be recast as a certain number of the spatial oscillations of the electromagnetic field corresponding to the photon of certain energy (defined in turn by particular spontaneous emission process). On the other hand, the coupling of matter to gravity in LQG leads to theory, where the matter degrees of freedom are represented by quantum labels living on the nodes (vertices) or the edges of the spin network (depending on the type of matter). This leads to an intuition, that the physical distance should be proportional to the “number of inhomogeneities” given interval is able to accommodate, thus should be proportional to the number $n_i$ of the spin-network edges. The observations from the noncompact extensions and the presented intuitions imply, that from physically viable models mimicking the cosmological spacetime by LQG semiclassical state one should expect the average $j$-label to remain constant at low curvatures at least in the leading order. This consistency however does not fix the asymptotic value of $\Delta_i$, which however is still subject to the constraints following from observations in high energy particle physics, although the upper bounds following from it are too huge to be useful. At present it seems that the more precise values can be provided only by statistical analysis of the dynamical evolution within specific frameworks of (or approximations to) LQG. One of the ways of determining that value from genuine LQG is provided by the interface of Chern-Simons theory with LQG used to evaluate black hole entropy [@abck-entr; @*abk-entr; @dl-entr; @cdpfb-entr; @*cdpfb-entr-det; @abfbdpv-prl; @bv-entr; @epn-entr; @bl-entr]. Indeed, the comparizon of two statistical calculations in [@abck-entr; @*abk-entr] and [@dl-entr] shows, that the edges with $j>1/2$ provide significant contribution to the surface areas, thus the average “area gap” $\Delta_i$ should be detectably larger than $\Delta$. On the other hand, more detailed combinatorial analysis of the relevant statistics [@abfbdpv-count] shows that for small areas the BH entropy features a “stair-like” structure which on heuristic level can be interpreted as the existence of “quantum of the area”. The numerical simulations [@cdpfb-entr; @*cdpfb-entr-det] determined it to approximately equal $\Delta' \approx 7.565{\ell_{{{\rm Pl}}}}^2$, which would give the average $\overline{\sqrt{j(j+1)}} \approx 1.267$ (roughly corresponding to average $\overline{j}\approx 0.86$). The stair-like entropy structure dissipates for larger area due to dispersion in $j$ and the complicated nature of the spectrum of area operator [@bv-degen], however these are the low areas (small number of graph edges intersecting the area) where any effect of $j$ distribution, especially any “peakness” of it, should be the most easy to observe. While heuristic, the above argument provides a strong indication that, while the LQC area gap will not correspond exactly to $\Delta$, it will remain of the same order. Given an interface constructed here it further suggests a specific correction to that area gap[^13]. Its value can be determined more precisely via use of the same statistical methods originally applied to find the distribution of the magnetic spins [@abck-entr; @*abk-entr; @dl-entr; @m-entr]. This is however beyond the scope of this article. Conclusions =========== We have considered the relation between the loop quantum cosmology physical states and the physical states of full loop quantum gravity possibly representing homogeneous universe on the kinematical level (that is without controlling the consistency of dynamical predictions of LQG and LQC frameworks). The LQG has been applied at its genuine level, without any simplifications. Using the observable quantities well defined both in loop quantum gravity and cosmology we constructed a precise interface between these two frameworks for the class of Bianchi I spacetimes of toroidal spatial topology. To define this interface we used the specific subspace of genuine LQG states, distinguished by minimal selection criteria allowing to define the necessary components of the LQC auxiliary structure for these states and motivated by construction originally proposed in [@awe-b1] In the case the physical state is supported on one spin network we assumed that the spin networks supporting the state are topologically equivalent to a sub-networks of the regular lattices. When the state is composed out of a continuum of states on distinct spin networks we further provided a notion of congruence of the embedding manifold by parallel lattices. This criterion, being the sole restrictive condition on the LQG physical Hilbert space allowed to provide a precise notion of global orthogonal[^14] directions on the spatial manifold. This structure has proven to be sufficient to define all the remaining auxiliary LQC structure necessary to construct a dictionary. The embedding manifold has been equipped with LQC fiducial background metric via partial gauge fixing. Upon this fixing the spin network supporting the state become regular lattice. The choice of (metric) regularity of the lattice was however not relevant in further construction of the dictionary, being instead a matter of convenience. Indeed, it was shown, that averaging over the diffeomorphisms preserving the global orthogonal directions leads to the same results. Given the selected class of spin network and the auxiliary structure provided by the partial gauge fixing the precise dictionary was constructed, where to identify the states in two frameworks we used two classes of observables: areas of global $T^2$ slices of the spatial (embedding) manifold – the maximal surfaces and the areas of the square plaquets defined by the minimal loops of the LQG spin network. This was achieved by averaging the LQG area operators corresponding to these surfaces over the diffeomorphism transformations remaining after the gauge fixing –the rigid translations– which were considered as active transformations. In the process no restrictions regarding the distribution of the quantum numbers ($j$-labels) on the spin network graph have been made. The result of this identification associated the LQC area of each plaquet with the average of $j$-labels (orthogonal to the plaquet) of the LQG spin network. No restrictions stronger that this relation , have been found. From there we concluded, that the particular association of the values of these areas as the LQC phase space functions (known as LQC prescription choice) depend solely on the statistical properties of the Hamiltonian (constraint) generating the evolution of the spin network and are not restricted by the principles imposed in construction the the LQG-LQC interface. In particular both the Bojowald’s $\mu_o$ prescriptions and the so called *improved dynamics* can be a priori realized. The results were further extended to the case of flat Bianchi I universe of topologically ${\mathbb{R}}^3$ spatial slices, where the relation found in $T^3$ case persisted unmodified . this relation and the consistency requirements on the LQC framework (existence of well defined infrared regulator removal limit) implies some restrictions on $j$-label statistics of LQG: for the class of states used to build the interface the averages of $j$-labels have to remain constant on low energy (gravitational field) limit. The studies were further extended to the case of isotropic flat FRW universe, where the plaquet area operators were further averaged over additional symmetries admitted by these classes of spacetimes – the rotations. This process related the (now unique) plaquet area with the average of $j$-labels over *all* the edges of the spin network graph . Two levels of implementation of the symmetries were considered: averaging over a discrete group of rotations by proper angles and the full $SO(3)$. Studies of the former case lead to the FRW limit of Bianchi I cosmology consistent with the analogous limit found in [@awe-b1]. In the latter it was found, that due to contributions of all the graph $j$-labels in the case the rotation angles differ from (multiples of) proper ones the area of the plaquet is larger by a factor $3/2$. Due to apparent discrepancy of the above result with the FRW limit of Bianchi I found in [@awe-b1] the changes of areas in considered model have been investigated up close. Consequently, it was found, that the set of lattices oriented in directions of one distinguished triad is insufficient to support the states accurately reproducing an isotropic spacetime. Consequently, an extension of the Hilbert space using the integral structure defined by the group of rotations was proposed. It was further shown that on the extended space the discrepancy is cured and the single plaquet area in the models of Bianchi I universes is also increased by a factor $3/2$. The found results have been finally confronted with the heuristic estimates of the $j$-statistics following from studies of black hole entropy in LQG. The dictionary constructed in this article indicates that associating the plaquet areas with the minimal nonzero LQG area is accurate only in cases when the spin network statistics makes $j>1/2$ non-generic (zero measure contribution). On the other hand, the (known in literature) heuristic results following from numerical analysis of black hole entropy provide a natural (from the point of view of the constructed dictionary) estimate on the $j$-label averages. This estimate leads again to the *constant area gap* principle of improved dynamics. it indicates however a slightly different value of this area gap, corresponding to the LQC critical energy density $\rho_c\approx 0.19\rho_{{{\rm Pl}}}$. This value, while lower than the original LQC critical energy density ($\approx 0.41\rho_{{{\rm Pl}}}$) remains at the same level of magnitude. The general purpose for constructing the above dictionary is to provide a viable tool of analyzing the cosmological limit of more advanced models aimed towards controlling or approximating the LQG dynamics (see of example [@we-lat; @ac-red; @*ac-let]. Since $(i)$ the (restrictive) selection criteria are precisely controlled here and the formalism remains relatively general, and $(ii)$ the formalism is adaptable to majority of prescriptions in defining the Hamiltonian (constraint) in LQG, it can be applied to a wide variety of models. It allows to extract the cosmological degrees of freedom out of such models in a precise way, further providing a tool for validating the initial assumptions selected in their construction (like for example the statistical averages of $j$ distributions). Through the consistency conditions on LQC in case of noncompact universes it also provides a tool for consistency control of the LQG models. At this point it is necessary to remember, that the interface relies on quite strong restriction of the spin-network graph topology. In principle, no such restrictions should be made in order for the results to be completely robust. Any generalization however, for example using the random graphs [@s-rand] is extremely difficult, as in such cases the LQC auxiliary structure (being a relevant part of the interface) has to emerge on the physical level (via observables) and may strongly depend on the $j$-label statistics of the physical states, which in turn is decided by the details in Hamiltonian (constraint) construction. Author thanks Edward Wilson-Ewing for discussions and helpful comments and especially to Emanuele Alesci for extensive discussions and the encouragement to write this article. This work has been supported in parts by the Chilean FONDECYT organization under regular project 1140335 and the National Center for Science (NCN) of Poland research grant 2012/05/E/ST2/03308 as well as by UNAB via internal project DI-562-14/R. \[app:stat\]Averaging over diffeomorphisms ========================================== Consider a $1$-dimensional lattice ($n$ edges) spanned across the interval $[0,1]$ with uniform random distribution of the vertices and each edge equipped with value $x_j$. Consider further the “average” of some function $f(x_i)$ weighted by the length of each edge $$\overline{f} = \sum_{i=1}^n f(x_i) l_i$$ The probabilistic space of the edge length distribution is the $n$ dimensional romboid $$\sum_{i=1}^n l_i = 1 ,\qquad \forall i\in\{1,\ldots,n\}:\ l_i>0 ,$$ with the measure ${{\rm d}}\sigma = {{\rm d}}l_1 \ldots {{\rm d}}l_n$. The volume of this romboid is $V_n = 1/n!$. The average value $\langle l_i\rangle$ of $l_i$ is the ratio of the volume of the romboid over the $n-1$ dimensional volume ot its base, which is $V_{n-1}$. As a consequence we have $$\langle\overline{f}\rangle = \sum_{i=1}^n f(x_i) \langle l_i\rangle = \frac{1}{n}\sum_{i=1}^n f(x_i) ,$$ which corresponds precisely to the case, where the vertices of the lattice are distributed uniformly. [^1]: The construction there is presented on the example of the simple quantum mechanical system – an harmonic oscillator, however the applications to LQG are also discussed there. [^2]: The construction can also be extended to many cases with a singular measure. [^3]: This projections are known in the literature as the so called *shadow states*. [^4]: For technical reasons (simplicity of the constraints) in [@awe-b1] different labeling of the kinematical basis states is used. [^5]: We use these terms since for homogeneous spacetimes (of diagonal spatial metric) they coincide with the standard meaning of flatness and orthogonality. This agreement may however not extend beyond that class of spacetimes. [^6]: On the formal level this value cannot be associated with the expectation value of the LQG area operator since no edge intersect such plaquet. It can be however made precise with use of the so called *dual graph* – a technique often applied in the spin-foam approaches. [^7]: To be mathematically precise, we define the graph which admits a set of discrete symmetries of the regular (closed) lattice on $T^3$. [^8]: The *direction* is defined here by topology of the graph: each direction is the set of the classes of equivalence of minimal closed loops not shrinkable to a point on the embedding manifold. [^9]: We remind that no restriction is made on the distribution of the values of $j$-labels on the graph. [^10]: One can introduce a notion of parallel edges terminating in a $6$-valent node as the pair not being the edge of a single plaquet and next build the surface by selecting a plaquet and extending the surface by including plaquets whose least two edges are parallel to edges contained already by the surface. [^11]: Instead of rotating the plaquet, we keep it fixed at $z=0$ and rotate the spin network. [^12]: In this case the averaging over rotations can be performed as averaging of $j$-labels between the rotation superselection sectors. [^13]: The idea that the value $\Delta'$ should replace the LQC area gap has been originally suggested [@dpfb-oral] by authors of [@cdpfb-entr]. It was however subsequently abandoned due to lack (at that time) of justification for such choice. [^14]: The meaning of orthogonality has been defined subsequently by no longer restrictive partial gauge fixing.

--- author: - The Planck team title: '*Planck* 2015 results. XIII. Cosmological parameters' ---

--- abstract: 'Unidirectional reflectionless propagation (or transmission) is an interesting wave phenomenon observed in many $\mathcal{PT}$-symmetric optical structures. Theoretical studies on unidirectional reflectionless transmission often use simple coupled-mode models. The coupled mode theory can reveal the most important physical mechanism for this wave phenomenon, but it is only an approximate theory, and it does not provide accurate quantitative predictions with respect to geometric and material parameters of the structure. In this paper, we rigorously study unidirectional reflectionless transmission for two-dimensional (2D) $\mathcal{PT}$-symmetric periodic structures sandwiched between two homogeneous media. Using a scattering matrix formalism and a perturbation method, we show that real zero-reflection frequencies are robust under $\mathcal{PT}$-symmetric perturbations, and unidirectional reflectionless transmission is guaranteed to occur if the perturbation (of the dielectric function) satisfies a simple condition. Numerical examples are presented to validate the analytical results, and to demonstrate unidirectional invisibility by tuning the amplitude of balanced gain and loss.' author: - Lijun Yuan - Ya Yan Lu title: 'Unidirectional Reflectionless Transmission for Two-Dimensional $\mathcal{PT}$-symmetric Periodic Structures' --- Introduction ============ In recent years, $\mathcal{PT}$-symmetry has attracted considerable attention in the optics and photonics community [@bender98; @feng17; @longhi17; @miri19]. A $\mathcal{PT}$-symmetric optical structure is usually realized by a complex dielectric function with a symmetric real part and an anti-symmetric imaginary part (i.e. a balanced gain and loss). The $\mathcal{PT}$-symmetry provides a fertile and feasible tool to manipulate lightwaves. Many interesting wave phenomena have been observed on $\mathcal{PT}$-symmetric optical structures. Noticeable examples include unidirectional reflectionless propagation [@lin11; @longhi11; @kalish12; @mostaf13; @sarisa17; @fu16; @rivolta16; @rege12; @feng13; @huang17; @horsley15; @yang16; @sarisa18], single-mode lasing [@miri12; @liu17], and simultaneous lasing and coherent perfect absorption [@longhi10; @chong11]. Unidirectional reflectionless propagation (or transmission) is the phenomenon wherein the reflection is zero for an incident wave coming from one side and nonzero for an incident wave coming from the other side. For lossless dielectric structures with certain symmetry, it is well known that zero reflection and zero transmission can really occur [@popov86; @shipman12; @chesnel18], but unidirectional reflectionless transmission is impossible, because the unitarity of the scattering matrix implies that zero reflections for left and right incident waves must occur at the same frequency. This is not the case for $\mathcal{PT}$-symmetric structures, since the scattering matrix is no longer unitary [@chong11; @ge12]. A particularly interesting case of unidirectional reflectionless transmission is unidirectional invisibility, for which the transmitted wave is identical to that without the local structure [@lin11]. Unidirectional reflectionless transmission has been studied on various $\mathcal{PT}$-symmetric optical structures, including one-dimensional (1D) structures [@lin11; @longhi11], planar layered structures [@kalish12; @mostaf13; @sarisa17; @yang16; @sarisa18], planar inhomogeneous structure [@horsley15], two-dimensional (2D) closed waveguides [@fu16], and 2D coupled waveguide resonator systems [@rivolta16]. Experimental demonstrations have been reported for $\mathcal{PT}$-symmetric photonic lattices [@rege12] and microscale SOI waveguides [@feng13]. It should be mentioned that unidirectional reflectionless transmission can also occur in non-$\mathcal{PT}$-symmetric optical structures [@feng14; @huang15; @shen14; @gu17; @zhao19]. Existing studies on unidirectional reflectionless transmission typically employ 1D Helmholtz equations or coupled-mode models. In this paper, we consider 2D $\mathcal{PT}$-symmetric periodic structures sandwiched between two homogeneous media, and find exact conditions under which unidirectional reflectionless transmission is guaranteed to occur. More specifically, assuming the $\mathcal{PT}$-symmetric structure is a small perturbation of a lossless dielectric structure and the dielectric structure has a simple (i.e. nondegenerate) real zero-reflection frequency, we show that real zero-reflection frequencies continue to exist, and they are different for left and right incident waves if the perturbation satisfies a simple condition. The continual existence (i.e. robustness) of real zero-reflection frequencies is proved using properties of the scattering matrix. A perturbation method is used to estimate the real zero-reflection frequencies and show that unidirectional reflectionless transmission occurs at arbitrarily small perturbations. Numerical examples are presented to validate our analytical results, and show that unidirectional invisibility can be obtained by tuning the amplitude of balanced gain and loss. The rest of this paper is organized as follows. In Sec. \[sec:Smatrix\], we recall some properties of the scattering matrix for general structures and discuss zero reflections for lossless dielectric structures with different symmetries. In Sec. [\[sec:PTsym\]]{}, we show that real zero-reflection frequencies are robust under $\mathcal{PT}$-symmetric perturbations. In Sec. [\[sec:perturbation\]]{}, we use a perturbation method to estimate the shifts of the zero-reflection frequencies for left and right incident waves and derive a condition to guarantee unidirectional reflectionless transmission. In Sec. [\[sec:Numerical\]]{}, numerical examples are presented to illustrate unidirectional reflectionless transmission and unidirectional invisibility. Scattering matrix {#sec:Smatrix} ================= We consider two-dimensional (2D) structures that are invariant in $z$, periodic in $y$ with period $L$, bounded in the $x$ direction, and surrounded by vacuum, where $\left\{ x, y, z \right\}$ is a Cartesian coordinate system. The dielectric function for such a structure and the surrounding media satisfies $$\label{eq:period_eps} \epsilon(x,y+L) = \epsilon(x,y)$$ for all $(x,y)$ and $\epsilon(x,y) = 1$ for $|x| > D$, where $D$ is a given constant. For the $E$-polarization, the $z$-component of the electric field, denoted by $u$, satisfies the following 2D Helmholtz equation: $$\label{eq:helm} \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + k_0^2 \epsilon\, u = 0,$$ where $k_0=\omega/c$ is the free space wavenumber, $\omega$ is the angular frequency, and $c$ is the speed of light in vacuum. In the left and right homogeneous media, we specify two incident plane waves $$\label{eq:incident_left} u^{(i)}_l(x,y) = a_l e^{i[ \alpha (x + D) + \beta y]} \quad \mbox{for} \quad x < -D$$ and $$\label{eq:incident_right} u^{(i)}_r(x,y) = a_r e^{ - i [ \alpha (x -D) - \beta y]} \quad \mbox{for} \quad x > D,$$ where $a_l$ and $a_r$ are the amplitudes of the incident waves, $(\pm \alpha, \beta)$ are the incident wave vectors, $\beta$ is real, and $\alpha^2 + \beta^2 = k_0^2$. Since the structure is periodic and the medium is homogeneous for $|x| > D$, the solution of Eq. (\[eq:helm\]) can be written as $$\label{eq:scat_solution_xlessD} u(x,y)= a_l e^{i[ \alpha (x + D) + \beta y]}+ \sum\limits_{j=-\infty}^{+\infty} b_{lj} e^{- i [ \alpha_j (x + D) - \beta_j y]},$$ for $x < - D$ and $$\label{eq:scat_solution_xlargeD} u(x,y) = a_r e^{ - i [ \alpha (x -D) - \beta y]} + \sum\limits_{j=-\infty}^{+\infty} b_{rj} e^{i [ \alpha_j (x - D) + \beta_j y]},$$ for $x > D$, where $\{b_{lj} \}$ and $\{ b_{rj} \}$ are the amplitudes of the out-going plane waves, and $$\label{eq:alpha_beta} \beta_j = \beta + 2 j \pi/ L, \quad \alpha_j = \sqrt{k_0^2 - \beta_j^2} ,$$ for $ j = 0, \pm 1, \pm 2, \ldots$. Notice that $\alpha_0 = \alpha$ and $\beta_0 = \beta$. If $\beta$ is real, $\beta \in \left[-\pi/L, \pi/L \right]$, $\omega$ is real, and $k_0 $ satisfies $$\label{eq:one_order} | \beta | < k_0 < 2 \pi / L - |\beta|,$$ then $\alpha_0$ is real and all $\alpha_j$ for $j \neq 0$ are pure imaginary with positive imaginary parts. In that case, all out-going plane waves for $j \neq 0$ decay to zero exponentially as $|x| \to \infty$, and the only out-going propagating plane waves are those for $j \neq 0$, i.e. the plane waves in Eqs. (\[eq:scat\_solution\_xlessD\]) and (\[eq:scat\_solution\_xlargeD\]) with coefficients $b_{l0}$ and $b_{r0}$. Let $S = S(\omega, \beta)$ be the $2 \times 2$ scattering matrix satisfying $$\label{eq:Smatrix} \left[ \begin{matrix} b_{l0} \\ b_{r0} \end{matrix} \right] = S(\omega, \beta) \left[ \begin{matrix} a_{l} \\ a_{r} \end{matrix} \right] = \left[ \begin{matrix} r_L(\omega, \beta) & t_R(\omega, \beta) \\ t_L(\omega, \beta) & r_R(\omega, \beta) \end{matrix} \right] \left[ \begin{matrix} a_{l} \\ a_{r} \end{matrix} \right],$$ where $r_L$ and $r_R$ ($t_L$ and $t_R$) are the reflection (transmission) coefficients for left and right incident waves, respectively, and $R_L = |r_L|^2$, $R_R = |r_R|^2$, $ T_L = |t_L|^2$ and $T_R = |t_R|^2$ are the corresponding reflectance and transmittance. Although $\omega$ is generally real, it is possible to study the diffraction problem for a complex frequency. If $\omega$ is allowed to have a small (positive or negative) imaginary part, and the real part of $k_0 = \omega / c$ satisfies Eq. ([\[eq:one\_order\]]{}), then $k_0^2 - \beta^2$ is close to the positive real axis, and $k_0^2 - \beta_j^2$ (for $j \neq 0$) are close to the negative real axis. In order to define $\alpha_j =\sqrt{k_0^2 - \beta_j^2}$ that depends continuously on the imaginary part of $\omega$, we can use a complex square root with a branch cut on the negative imaginary axis. That is, if $\eta = |\eta| e^{i \psi}$ for $-\pi/2 < \psi \leq 3 \pi /2 $, then $\sqrt{\eta} = \sqrt{|\eta| } e^{i \psi/2}$. Using this square root, Eqs. (\[eq:scat\_solution\_xlessD\]) and (\[eq:scat\_solution\_xlargeD\]) are still valid for $|x| > D$, the out-going waves are still dominated by plane waves with coefficients $b_{l0}$ and $b_{r0}$, and the scattering matrix can be defined as in Eq. (\[eq:Smatrix\]). If for a real frequency $\omega$ and a real wavenumber $\beta$, we have $r_L = 0$ and $r_R \neq 0$ (or $r_L \neq 0$ and $r_R = 0$), then we say unidirectional reflectionless transmission occurs at $(\omega, \beta)$ for a left (or right) incident wave. The scattering matrix has some important properties. The reciprocity gives rise to $$\label{eq:Smatrix_reciprocal} S(\omega, -\beta) = S^{\textsf{T}}(\omega, \beta),$$ where the superscript “" denotes matrix transpose, and $S(\omega, -\beta)$ is the scattering matrix for incident waves with $e^{- i \beta y} $ dependence. Equation (\[eq:Smatrix\_reciprocal\]) is a general result valid for complex dielectric function $\epsilon$ and complex $\omega$. For easy reference, we give a proof in Appendix A. Equation (\[eq:Smatrix\_reciprocal\]) gives rise to $r_L(\omega, \beta) = r_L(\omega, -\beta)$, $r_R(\omega, \beta) = r_R(\omega, -\beta)$ and $t_L(\omega,-\beta) = t_R(\omega,\beta)$. It is clear that if unidirectional reflectionless transmission occurs at a pair $(\omega, \beta)$, then it also occurs at $(\omega, -\beta)$. If $\beta = 0$, i.e. for normal incident waves, then $t_L = t_R$ for any $\omega$. For structures with some symmetries, the scattering matrix can be simplified. If the structure has an inversion symmetry, i.e. $$\label{eq:eps_rotationeven} \epsilon(x,y) = \epsilon(-x,-y) \quad \mbox{for all} \quad (x,y),$$ then the mapping $(x,y) \to (-x,-y)$ changes $\beta $ to $-\beta$, swaps $a_r$ with $a_l$ and $b_{l0}$ with $b_{r0}$. This leads to $$\left[ \begin{matrix} {b}_{r0} \\ {b}_{l0} \end{matrix} \right] = S(\omega, -\beta) \left[ \begin{matrix} {a}_{r} \\ {a}_{l} \end{matrix} \right].$$ Therefore, $$S(\omega, \beta) = P S(\omega, -\beta) P,$$ where $$P = \left[ \begin{matrix} 0 & 1 \\ 1 & 0\end{matrix} \right].$$ From Eq. (\[eq:Smatrix\_reciprocal\]), we obtain $$\label{eq:Smatrix_yeven} S(\omega, \beta) = P S^{\textsf{T}}(\omega, \beta) P.$$ This implies that $r_L= r_R$ for all $\omega$ and $\beta$. Therefore, unidirectional reflectionless transmission is impossible for structures with the inversion symmetry. If the structure has a reflection symmetry in the $y$ direction, i.e. $$\label{eq:eps_yeven} \epsilon(x,y) = \epsilon(x,-y) \quad \mbox{for all} \quad (x,y),$$ then the mapping $y \to -y$ changes $\beta$ to $-\beta$. Thus $$\label{eq:Smatrix_yeven} S(\omega, -\beta) = S(\omega, \beta).$$ From Eq. (\[eq:Smatrix\_reciprocal\]), we have $$\label{eq:S_12} S^{\small \textsf{T}}(\omega, \beta) = S(\omega, \beta).$$ This imples that $t_R = t_L$ for all $\omega$ and $\beta$. If the structure has a reflection symmetry in the $x$ direction, i.e. $$\label{eq:eps_xeven} \epsilon(x,y) = \epsilon(-x,y) \quad \mbox{for all} \quad (x,y),$$ then the mapping $x\to -x$ swaps $a_r$ with $a_l$ and $b_{l0}$ with $b_{r0}$. Thus $$\label{eq:Smatrix_xeven} P S(\omega, \beta) P^{\small \textsf{T}} = S(\omega, \beta).$$ It implies that $r_L = r_R$ and $t_L = t_R$ for all $\omega$ and $\beta$. Clearly, unidirectional reflectionless is again impossible in this case. Notice that the symmetry in the $x$ direction gives more constrains than each of the other two symmetries studied above. When $\epsilon$ and $\omega$ are real, the power (per period) carried by the incident wave must equal to the power radiated out by the out-going waves. This leads to the condition that $S$ must be unitary. A more general result for real $\epsilon$ and complex $\omega$ is $$\label{eq:Smatrix_Unitarity} S^*(\bar{\omega}, \beta) S(\omega, \beta) = I,$$ where the superscript “$^*$” denotes conjugate transpose, $\bar{\omega}$ is the complex conjugate of $\omega$, and $I$ is the identity matrix. A proof of Eq. (\[eq:Smatrix\_Unitarity\]) is given in Appendix B. For a real $\omega$, the unitarity of $S$ gives $$\begin{aligned} |r_L|^2 + |t_L|^2 = 1 \quad \mbox{and} \quad \bar{r}_{L} t_{R} + \bar{t}_{L} r_{R} = 0.\end{aligned}$$ Thus, if $r_{L} = 0$, then $|t_L| = 1$ and $r_R = 0$. Similarly, if $r_{R} = 0$, we must have $r_{L} = 0$. Therefore, unidirectional reflectionless transmission is impossible for lossless dielectric structures. If $\epsilon$ is real and symmetric in the $x$ direction, it is known that there could be real frequencies with zero reflections and zero transmissions. Popov [*et al.*]{} [@popov86] first studied a variant of this problem based on the scattering matrix. Shipman and Tu [@shipman12] analyzed this problem assuming the structure supports a bound state in the continuum for a nearby $\beta$ and a nearby $\omega$. If one assumes that $\omega^{(r)}$ is a simple (i.e. nondegenerate) zero of $r_L$ (as an analytic function of $\omega$), and it is the only zero in a domain $\mathcal{W}$ (of the complex $\omega$ plane) containing both $\omega^{(r)}$ and it complex conjugate, then Eq. (\[eq:Smatrix\_Unitarity\]) allows us to show that $\bar{\omega}^{(r)}$ is a zero of $r_R$. Since $r_L = r_R$ when $\epsilon$ is symmetric in $x$, and $\mathcal{W}$ contains only one zero of $r_L$, we conclude that $\omega^{(r)}$ must be real. $\mathcal{PT}$-symmetric structures {#sec:PTsym} =================================== We are interested in a class of $\mathcal{PT}$-symmetric structures for which the dielectric function $\epsilon$ is complex and satisfies $$\label{eq:PTsymm} \epsilon(x,y) = \bar{\epsilon}(-x,y).$$ In other words, the real part of $\epsilon$ is symmetric and the imaginary part is anti-symmetric in the $x$ direction. In Appendix C, we show that the scattering matrix satisfies $$\label{eq:Smatrix_PT} P S^*(\bar{\omega}, \beta) P S(\omega, \beta) = I,$$ where $P$ is the $2 \times 2$ matrix given in Sec. \[sec:Smatrix\] and $I$ is the identity matrix. A more general result (without the parameter $\beta$) on the scattering matrix of $\mathcal{PT}$-symmetric structures is given in [@ge12]. For a real $\omega$, Eq. (\[eq:Smatrix\_PT\]) gives $$\begin{aligned} \label{eq:generalized_unitarity1} && r_L \bar{r}_R + \bar{t}_R t_L = 1, \\ \label{eq:generalized_unitarity2} && t_R \bar{r}_R + \bar{t}_R r_R = 0, \\ \label{eq:generalized_unitarity3} && t_L \bar{r}_L + \bar{t}_L r_L = 0. \end{aligned}$$ Let $\phi_1 $ and $\phi_2$ be the phases of $t_R$ and $t_L$, respectively. Equation (\[eq:generalized\_unitarity2\]) implies that the phase of $r_R$ is either $\phi_1 + \pi/2$ or $\phi_1 - \pi/2$. Equation (\[eq:generalized\_unitarity3\]) implies that the phase of $r_L$ is either $\phi_2 + \pi/2$ or $\phi_2 - \pi/2$. Thus the phase of $r_R \bar{r}_L$ is $\phi_1 - \phi_2$, or $\phi_1 - \phi_2 \pm \pi$. For all cases, Eq. (\[eq:generalized\_unitarity1\]) leads to $\phi_1 = \phi_2$. Therefore $t_R \bar{t}_L$ and $r_L \bar{r}_R$ are real. It is can be verify that $\lambda_1 \lambda_2 = \mbox{det}(S) = - e^{2 i \phi_1}$, where $\lambda_1$ and $\lambda_2$ are the eigenvalues of $S$. If $t_R \bar{t}_L < 1$, $r_L \bar{r}_R$ is positive, thus the phases of $r_L$ and $r_R$ are identical, and Eq. (\[eq:generalized\_unitarity1\]) leads to $$\sqrt{R_L R_R} = 1 - \sqrt{T_L T_R}.$$ The above is the generalized energy conservation law [@ge12]. If $t_R \bar{t}_L > 1$, $r_L \bar{r}_R$ is negative, there is a $\pi$ difference between the phases of $r_L$ and $r_R$, and Eq. (\[eq:generalized\_unitarity1\]) leads to $$\sqrt{R_L R_R} = \sqrt{T_L T_R} - 1.$$ If $t_R \bar{t}_L = 1$, then at least one of $r_L$ and $r_R$ is zero. If only one of them is zero, we have unidirectional reflectionless transmission. For $\beta = 0$ or $\beta \neq 0$ but the structure has an additional reflection symmetry in the $y$ direction, then it is easy to show that $t_L = t_R$. This special case has been extensively studied before [@ge12]. We are interested in $\mathcal{PT}$-symmetric structures that are perturbations of a lossless dielectric structure (with a real dielectric function). If the unperturbed structure has a simple real zero-reflection frequency $\omega_L$ which is the only zero of $r_L$ contained in a domain $\mathcal{W}$ ($\mathcal{W}$ can be chosen as a small disk of the complex $\omega$ plane and centered at $\omega_L$), we expect $r_L$ of the perturbed $\mathcal{PT}$-symmetric structure still has only one simple zero $\tilde{\omega}_L$ in $\mathcal{W}$. We show that $\tilde{\omega}_L$ must still be real. Equation (\[eq:Smatrix\_PT\]) can be written down explicitly as $$\begin{aligned} t_R(\omega, \beta) \bar{t}_{L}(\bar{\omega}, \beta) + r_{R}(\omega, \beta) \bar{r}_{L}(\bar{\omega}, \beta) & =& 1, \\ r_{L}(\omega, \beta) \bar{r}_{L}(\bar{\omega}, \beta) + t_{L}(\omega, \beta) \bar{t}_{L}(\bar{\omega}, \beta) & =& 1, \\ t_{R}(\omega, \beta) \bar{r}_{R}(\bar{\omega}, \beta) + r_{R}(\omega, \beta) \bar{t}_{R}(\bar{\omega}, \beta) & =& 0, \\ r_{L}(\omega, \beta) \bar{t}_{L}(\bar{\omega}, \beta) + t_{L}(\omega, \beta) \bar{r}_{L}(\bar{\omega}, \beta) & =& 0. \end{aligned}$$ Since $r_L = 0$ at $\omega = \tilde{\omega}_L$, the last equation above gives either $r_{L} \left( \bar{\tilde{\omega}}_L, \beta \right) = 0$ or $t_{L} \left( \tilde{\omega}_L , \beta \right) = 0$. But the second equation above indicates that $t_{L} \left( \tilde{\omega}_L, \beta \right) $ can not be zero, thus $r_{L} \left( \bar{\tilde{\omega}}_L, \beta \right) = 0$. Since the domain $\mathcal{W}$ only contains only one zero of $r_L$, we must have $\tilde{\omega}_L = \bar{\tilde{\omega}}_{L}$, i.e. $\tilde{\omega}_{L}$ is real. Similarly, if we make proper assumptions about a right real zero-reflection frequency, $\omega_R$, of the unperturbed structure, we can show that there must be a real zero, $\tilde{\omega}_R$, of the right reflection coefficient $r_R$ for the perturbed $\mathcal{PT}$-symmetric structure. Perturbation analysis {#sec:perturbation} ===================== The previous section establishes the continual existence of real zero-reflection frequencies for both the left and right incident waves under $\mathcal{PT}$-symmetric perturbations. In this section, we use a perturbation method to show that these two frequencies are different in general, and thus unidirectional reflectionless transmission indeed occur. First, we consider a lossless dielectric structure with a real dielectric function $\epsilon$ that is also symmetric in the $x$ direction, and assume that there is a real wavenumber $\beta$ and real zero-reflection frequency $\omega_L = \omega_R$ for both left and right incident waves. Let $u_L$ and $u_R$ be the corresponding diffraction solutions for left and right incident plane waves with unit amplitude, respectively. The symmetry of the structure in $x$ implies that $u_R(x,y) = u_L(-x,y)$. At $x = \pm D$, these two solutions can be written down as $$\begin{aligned} \label{eq:uL_xpmD_fourier} && u_R (D,y) = u_L(- D,y) = e^{i \beta y} + \sum\limits_{j=-\infty}^{\infty} b^{-}_j e^{i \beta_j y}, \\ && u_R(-D,y) = u_L( D,y) = \sum\limits_{j=-\infty}^{\infty} b^{+}_j e^{i \beta_j y},\end{aligned}$$ where $\{ b^{+}_j \}$ are the Fourier coefficients of $u_L( D, y)$, and $\{ b^{-}_j \}$ are the Fourier coefficients of $u_L( - D, y) - e^{i \beta y}$. Since the reflection is zero and energy is conserved, we have $$b^-_0 = 0, \quad \left| b^+_0 \right| = 1.$$ Next, we consider a perturbed $\mathcal{PT}$-symmetric structure with a dielectric function $$\label{eq:eps_perturbation} \tilde{\epsilon} = \epsilon + \delta F(x,y),$$ where $\delta$ is a small real number, $F$ is a complex $O(1)$ function satisfying $$F(x,y) = \bar{F}(-x,y),$$ and $F(x,y) = 0$ for $|x| > D$. According to Sec. [\[sec:PTsym\]]{}, there must be real frequencies $\tilde{\omega}_L$ and $\tilde{\omega}_{R}$ such that $r_L = 0$ and $r_R = 0$, respectively, for the perturbed structure $\tilde{\epsilon}$ and the fixed $\beta$. Let $\tilde{k}_L = \tilde{\omega}_L/ c$ and $\tilde{k}_R = \tilde{\omega}_R / c$, where $c$ is the speed of light in vacuum. We expand $\tilde{k}_L$ and $\tilde{k}_R$ in power series of $\delta$: $$\begin{aligned} \label{eq:exp_freq1} \tilde{k}_L &= & k_0 + k^{(L)}_{1} \delta + k^{(L)}_{2} \delta^2 + \ldots, \\ \label{eq:exp_freq2} \tilde{k}_R & = & k_0 + k^{(R)}_{1} \delta + k^{(R)}_{2} \delta^2 + \ldots,\end{aligned}$$ where $k_0 = \omega_L / c = \omega_R/c$. In Appendix D, we show that the coefficients $k_1^{(L)}$ and $k_1^{(R)}$ are given by $$\label{eq:k1_left} k^{(L)}_1 = \dfrac{ - k_0 \int_{\Omega_D} F u_L \bar{u}_R dxdy}{ \dfrac{i L}{\alpha} (b^+_0 - \bar{b}^+_0) + i L \sum\limits_{\substack{j=-\infty \\ j \neq 0}}^{\infty} \dfrac{\bar{b}^-_j b^+_j + \bar{b}^-_j b^+_j}{\alpha_j} + 2 \int_{\Omega_D} \epsilon u_L \bar{u}_R dxdy },$$ and $$\label{eq:k1_right} k^{(R)}_1 = \dfrac{ - k_0 \int_{\Omega_D} F \bar{u}_L u_R dxdy}{ \dfrac{i L}{\alpha} (b^+_0 - \bar{b}^+_0) + i L \sum\limits_{\substack{j=-\infty \\ j \neq 0}}^{\infty} \dfrac{\bar{b}^-_j b^+_j + \bar{b}^-_j b^+_j}{\alpha_j} + 2 \int_{\Omega_D} \epsilon \bar{u}_L u_R dxdy },$$ where $\Omega_D $ is the rectangle given by $|x| < D$ and $|y| < L/2$. The coefficients $\{ \alpha_j \}$ are defined in Sec. \[sec:Smatrix\], $\omega_L$ and $\beta$ are assumed such that $\alpha = \alpha_0$ is real, and all other $\alpha_j$ for $j \neq 0$ are pure imaginary. It is easy to verify that $\bar{u}_R u_L$ is a $\mathcal{PT}$-symmetric function satisfying the same Eq. (\[eq:PTsymm\]). Using this result and the symmetry of $\epsilon$ and $F$, it is straightforward to show that $k^{(L)}_1$ and $k^{(R)}_1$ are real, and the denominators of Eqs. (\[eq:k1\_left\]) and (\[eq:k1\_right\]) are identical. Therefore, if $F(x,y)$ satisfies $$\label{eq:cond_F} \int_{\Omega_D} \left[ F(x,y) - F(-x,y) \right] u_L \bar{u}_R dxdy \neq 0,$$ then $k^{(L)}_1 \neq k^{(R)}_1$. This implies that as far as $F$ satisfies Eq. (\[eq:cond\_F\]), $\tilde{\omega}_L \neq \tilde{\omega}_R$ for arbitrarily small $\delta$, and unidirectional reflectionless transmission occurs at frequencies $\tilde{\omega}_L $ and $\tilde{\omega}_R$ for left and right incident waves, respectively. Numerical examples {#sec:Numerical} ================== In this section, we present numerical results to illustrate the unidirectional reflectionless transmission phenomenon, validate the perturbation results of Sec. \[sec:perturbation\], and show examples of unidirectional invisibility. We consider a periodic array of identical circular cylinders with period $L$ in the $y$ direction and surrounded by air as shown in Fig. [\[fig:fig2\]]{}(a). The coordinates are chosen so that the centers of the cylinders are on the $y$ axis and the center of one cylinder is at the origin. The first example is for cylinders with a $y$-independent dielectric function given by $$\label{eq:dielectric_cylinder} \epsilon(x,y) = \epsilon_1 + i \delta \sin\left(\dfrac{\pi x}{2 a} \right),$$ where $a = 0.3 L$ is the radius of the cylinders, $\epsilon_1 = 10$, and $ \delta$ is a real parameter. If $\delta \neq 0$, the structure is $\mathcal{PT}$-symmetric with respect to the reflection in the $x$ direction. The diffraction problem for a given incident wave can be solved by many different numerical methods. We use a mixed Fourier-Chebyshev pseudospectral method [@tref00] to discretize the Helmholtz equation inside the disk of a radius $a$ (corresponding to the cross section of the central cylinder), and use cylindrical and plane wave expansions outside the cylinders [@huang06]. In Fig. [\[fig:fig2\]]{}(b), we show the reflection and transmission spectra in logarithmic scale for left and right normal incident waves. For $\delta = 0$, $\epsilon$ is real and symmetric in both $x$ and $y$ directions, thus the reflection and transmission spectra are identical for left and right incident waves, i.e. $R_L = R_R$ and $T_L = T_R$. In Fig. [\[fig:fig2\]]{}(b), a dip can be observed in the reflection spectrum for $\delta =0$. Presumably, the reflection coefficient is exactly zero at normalized frequency $\omega L /(2\pi c) \approx 0.5882$. To use the perturbation results of Sec. \[sec:perturbation\], we assume $D = L/2$, then $\Omega_D$ is a square of side length $L$ centered at the origin. Using the numerical solutions to evaluate Eqs. (\[eq:k1\_left\]) and (\[eq:k1\_right\]), we obtain $k^{(L)}_1 \approx 0.018 L^{-1}$ and $k^{(R)}_1 \approx -0.018 L^{-1}$. This implies that when $\delta$ is increased from zero, the zero-reflection frequencies for left and right incident waves will increase and decrease, respectively. The numerical results for $\delta = 0.1$ are also shown in Fig. [\[fig:fig2\]]{}, and they indicate that the normalized zero-reflection frequency $\omega L/(2\pi c)$ is increased to $ 0.5885$ for left incident waves, and is decreased to $0.5879$ for right incident waves. According to our perturbation theory, the normalized zero-reflection frequency to the left incident waves is approximately $0.5882 + \delta k^{(L)}_1 L /(2\pi)$. Keeping four significant digits, this value is also $0.5885$. Therefore, the numerical and perturbation results agree very well with each other. At the zero-reflection frequency for left (or right) incident waves, the reflection coefficient for right (or left) incident waves is non-zero, therefore, we have unidirectional reflectionless transmissions. Since the structure is symmetric in $y$, the transmission coefficients for left and right incident waves are identical. From Eq. (\[eq:generalized\_unitarity1\]), it is clear that the transmittance is exactly $1$ at the zero-reflection frequencies. ![(a) A periodic array of identical circular cylinders with incident waves from left and right. (b) Example 1: reflection and transmission spectra for normal incident waves. []{data-label="fig:fig2"}](Fig1a.pdf "fig:") ![(a) A periodic array of identical circular cylinders with incident waves from left and right. (b) Example 1: reflection and transmission spectra for normal incident waves. []{data-label="fig:fig2"}](Fig2_2.pdf "fig:") In Fig. [\[fig:fig3\]]{}(a) we show zero-reflection frequencies for different $\delta$. The solid blue curve and the dashed red curve correspond to left and right incident waves, respectively. For $\delta=0$, the zero-reflection frequencies for left and right incident waves are identical. For $\delta \neq 0$, they are different in general, thus unidirectional reflectionless transmission can occur. Overall, the curves for left and right zero-reflection frequencies are mirror images of each other. At a zero-reflection frequency for a left incident wave, the transmission coefficient $r_L$ has unit magnitude, and we can define a phase $\theta$ relative to the incident wave (extended to the whole space) by $$e^{i \theta} = t_L e^{- 2 i \alpha D}.$$ In Fig. \[fig:fig3\](b), we show the relative phase $\theta$ for all zero-reflection frequencies on the blue solid curves in Fig. [\[fig:fig3\]]{}(a). Point $A$ in Figs. [\[fig:fig3\]]{}(a) and (b) is a point with a zero relative phase. It is obtained at $\omega L /(2\pi c) = 0.5959$ for $ \delta = 2.8$. In Figs. [\[fig:fig4\]]{}(a) and (c), we show the diffraction solutions for left and right normal incident waves for point $A$. For a left incident wave (i.e. $a_l=e^{- i \alpha D}, a_r=0$), the total wave is identical to the plane wave $e^{i k_0 x}$ (shown in Fig. [\[fig:fig4\]]{}(b)) away from the cylinders. This implies that the cylinders are invisible to left incident waves. For a right incident wave (i.e. $a_l=0, a_r= e^{i \alpha D}$), the transmitted wave also has unit amplitude and zero relative phase, the reflected wave has a magnitude about $6.93$. Although the $\mathcal{PT}$-symmetric structure has a balanced gain and loss profile, energy does not need to be conserved. For this case, a strong reflected wave is produced thanks to the gain medium. ![(a) Zero-reflection frequencies for different $\delta$ in the first example. Blue solid line: left incident wave, red dashed line: right incident waves. (b) Relative phases $\theta$ of the transmitted waves at zero-reflection frequencies for left incident waves. []{data-label="fig:fig3"}](Fig3a.pdf "fig:") ![(a) Zero-reflection frequencies for different $\delta$ in the first example. Blue solid line: left incident wave, red dashed line: right incident waves. (b) Relative phases $\theta$ of the transmitted waves at zero-reflection frequencies for left incident waves. []{data-label="fig:fig3"}](Fig3b2.pdf "fig:") ![Real part of the diffraction solutions, i.e. Re$(u)$, corresponding to point $A$ in Fig.[\[fig:fig3\]]{}. (a) Left incident wave with $a_l=e^{-i \alpha D }$ and $a_r=0$. (b) Plane wave $e^{i k_0 x}$ propagating in air. (c) Right incident wave with $a_l=0$ and $a_r= e^{i \alpha D}$. Red circles denote the cylinders. The wave fields are capped from $-1$ to $1$. []{data-label="fig:fig4"}](Fig4.eps) In order to study $\mathcal{PT}$-symmetric structures without the reflection symmetry in $y$, we consider another example. The structure is again a periodic array of identical circular cylinders with their centers on the $y$ axis, but the dielectric function of the cylinder centered at the origin is given by $$\epsilon(x,y) = \epsilon_1 + 2 \sin\left(\dfrac{\pi y}{2 a} \right) + 2 i \sin\left(\dfrac{\pi x}{2 a} \right),$$ where $\epsilon_1$ and $a$ are the same as the first example. The structure is $\mathcal{PT}$-symmetric with respect to a reflection in the $x$ direction, but it is not symmetric in $y$. For $\beta \neq 0$, the transmission coefficients for left and right incident waves are different in general. In Fig. [\[fig:fig\_example2\]]{}, we show the reflection and transmission spectra for left and right incident waves with $\beta L/(2 \pi)= 0.2$. At $\omega L/(2\pi c) = 0.3569$, the reflection is zero for left incident waves and nonzero for right incident waves, thus unidirectional reflectionless transmission occurs. The corresponding transmittance are $T_L = 0.9488$ and $T_R = 1.0539$. From Sec. \[sec:PTsym\], we know that $t_R \bar{t}_L$ is aways real. To verify this, we show the value of $t_R \bar{t}_L - 1$ as a function of the frequency in Fig. [\[fig:fig\_example2\_2\]]{}(a). Clearly, $t_R \bar{t}_L = 1 $ at zero-reflection frequency $\omega L/(2\pi c) = 0.3569$. In Fig. [\[fig:fig\_example2\_2\]]{}(b), we show the phases of $r_L$ and $r_R$. Notice that the phase of $r_L$ has a jump discontinuity of $\pi$ at the the zero-reflection frequency. ![Example 2: reflection and transmission spectra for $\beta L /(2\pi) = 0.2$. Insert is the logarithmic scale plot.[]{data-label="fig:fig_example2"}](Fig_Example2.pdf) ![ (a) Real (solid line) and imaginary (dashed line) parts of $t_R \bar{t_R} - 1$. (b) Normalized phases of $r_L$ (solid line), $r_R$ (dashed line) and $t_L$ (dashdot line).[]{data-label="fig:fig_example2_2"}](Fig_example2_3.pdf "fig:") ![ (a) Real (solid line) and imaginary (dashed line) parts of $t_R \bar{t_R} - 1$. (b) Normalized phases of $r_L$ (solid line), $r_R$ (dashed line) and $t_L$ (dashdot line).[]{data-label="fig:fig_example2_2"}](Fig_example2_4.pdf "fig:") Conclusions =========== In this paper, we studied 2D $\mathcal{PT}$-symmetric periodic structures sandwiched between two homogeneous media. Using a scattering matrix formalism we showed that the real zero-reflection frequencies are robust under $\mathcal{PT}$-symmetric perturbations. A simple condition on the perturbed dielectric function was derived by a perturbation method to guarantee that the real zero-reflection frequencies for left and right incident waves are different. Therefore, as far as the original unperturbed dielectric structure is symmetric and has a real non-degenerate zero-reflection frequency, unidirectional reflectionless transmission is certain to occur for almost any $\mathcal{PT}$-symmetric perturbations. Numerical examples are presented for periodic arrays of circular cylinders with $\mathcal{PT}$-symmetric dielectric functions. The numerical results confirmed the perturbation theory, and illustrated unidirectional invisibility and other interesting wave phenomena. Both the scattering matrix formalism and the perturbation analysis can be easily extended to unperturbed structures that are themselves $\mathcal{PT}$-symmetric. The scattering matrix formalism allows us to conclude that real non-degenerate zero-reflection frequencies are protected by the $\mathcal{PT}$-symmetry, in the sense that these frequencies remain real for any $\mathcal{PT}$-symmetric perturbations. The perturbation theory gives quantitative results on the changes of the real zero-reflection frequencies caused by perturbations. It is also straightforward to consider non-$\mathcal{PT}$-symmetric perturbation that could move the real zero-reflection frequencies to the complex plane. Our study enhances the theoretical understanding on the zero-reflection frequencies, unidirectional reflectionless transmission, and unidirectional invisibility for $\mathcal{PT}$-symmetric structures, and provides a solid foundation for further studies on these wave phenomena and for exploring their potential applications. Acknowledgments {#acknowledgments .unnumbered} =============== The authors acknowledge support from the Science and Technology Research Program of Chongqing Municipal Education Commission, China (Grant No. KJ1706155), and the Research Grants Council of Hong Kong Special Administrative Region, China (Grant No. CityU 11304117). Appendix {#appendix .unnumbered} ======== Appendix A: Reciprocity {#appendix-a-reciprocity .unnumbered} ----------------------- To derive Eq. (\[eq:Smatrix\_reciprocal\]), we consider the diffraction problem for incident waves $\tilde{a}_{l} e^{i [ \alpha (x +D) - \beta y]}$ and $\tilde{a}_r e^{-i[\alpha (x - D) + \beta y]}$ with frequency $\omega$ and wavenumber $-\beta$. The diffraction solution $\tilde{u}$ can be written as $$\tilde{u}(x,y) = \tilde{a}_l e^{i [\alpha (x + D) - \beta y]} + \sum\limits_{j=-\infty}^{+\infty} \tilde{b}_{lj} e^{- i [ \tilde{\alpha}_j (x + D) - \tilde{\beta}_j y]},$$ for $ x < - D$ and $$\tilde{u}(x,y) = \tilde{a}_r e^{- i [ \alpha (x - D) + \beta y]} + \sum\limits_{j=-\infty}^{+\infty} \tilde{b}_{rj} e^{i [ \tilde{\alpha}_j (x - D) + \tilde{ \beta}_j y]},$$ for $x > D$, where $$\tilde{\beta}_j = - \beta + 2 j \pi / L, \quad \tilde{\alpha}_j = \sqrt{k_0^2 - \tilde{\beta}_j^2},$$ for $ j=0, \pm1, \pm 2, \ldots$. Notice that $$\tilde{\beta}_j = - \beta_{-j}, \quad \tilde{\alpha}_j = \alpha_{-j}, \quad \mbox{for} \quad j=0, \pm1, \pm 2, \ldots.$$ The coefficients $\tilde{b}_{l0}$ and $\tilde{b}_{r0}$ are related to the incident coefficients $\tilde{a}_l$ and $\tilde{a}_r$ by scattering matrix $S(\omega, - \beta)$ as $$\left[ \begin{matrix} \tilde{b}_{l0} \\ \tilde{b}_{r0} \end{matrix} \right] = S(\omega, -\beta) \left[ \begin{matrix} \tilde{a}_{l} \\ \tilde{a}_{r} \end{matrix} \right].$$ From the governing equations of $u$ and $\tilde{u}$, we have $$0 = \tilde{u} (\Delta u + k^2_0 \epsilon u) - u (\Delta \tilde{u} + k_0^2 \epsilon \tilde{u}) = \nabla \cdot (\tilde{u} \nabla u) - \nabla \cdot ({u} \nabla \tilde{u}),$$ where $\Delta = \partial^2_x + \partial^2_y$ is the Laplace operator. Integrating the above equation on domain $\Omega_D$, we have $$\int_{\partial \Omega_D} \left( \tilde{u} \dfrac{\partial u}{\partial \nu} - {u} \dfrac{\partial \tilde{u}}{\partial \nu} \right) ds= 0.$$ The line integrals on the two edges of $\Omega_D$ at $y = \pm L/2$ cancel out. From the expressions of $u$ (i.e. Eqs. (\[eq:scat\_solution\_xlessD\]) and (\[eq:scat\_solution\_xlargeD\])) and $\tilde{u}$ at $x = \pm D$, and the relations of $\beta_j$ and $\tilde{\beta}_j$, we obtain $$\tilde{a}_l b_{l0} + \tilde{a}_r b_{r0} = a_l \tilde{b}_{l0} + a_r \tilde{b}_{r0}.$$ Therefore, $$\left[ a_l, a_r \right] \left[ S^{\small \textsf{T}}(\omega, \beta) - S(\omega, -\beta) \right] \left[ \begin{matrix} \tilde{a}_{l} \\ \tilde{a}_r \end{matrix} \right] = 0$$ for any complex $a_l, a_r, \tilde{a}_l$ and $\tilde{a}_r$. This leads to Eq. (\[eq:Smatrix\_reciprocal\]). Appendix B: Unitarity for lossless dielectric structures {#App:unitarity .unnumbered} -------------------------------------------------------- Assume $\epsilon$ is real and $\omega$ is complex with a small (positive and negative) imaginary part. We consider the diffraction problem for two incident waves $\hat{a}_l e^{i [\alpha(\bar{\omega}) (x + D)+ \beta y]}$ and $\hat{a}_r e^{-i [\alpha(\bar{\omega}) (x - D) - \beta y]}$ with frequency $\bar{\omega}$ and wavenumber $\beta$, where $\alpha(\bar{\omega}) = \sqrt{\bar{k}_0^2 - \beta^2}$ and $k_0 = \omega / c$. Here the square root is defined with a branch cut on the negative imaginary axis as shown in Sec. \[sec:Smatrix\]. The diffraction problem is governed by the Helmholtz equation $$\begin{aligned} \Delta w + \bar{k}_0^2 \epsilon w = 0.\end{aligned}$$ The solution $w$ can be written as $$w(x,y)= \hat{a}_l e^{i[ \alpha(\bar{\omega}) (x + D) + \beta y]} + \sum\limits_{j = -\infty}^{+\infty} \hat{b}_{lj} e^{-i [ \alpha_j(\bar{\omega}) (x + D) - {\beta}_j y ]},$$ for $x < - D$ and $$w(x,y)= \hat{a}_r e^{- i [ \alpha(\bar{\omega}) (x - D) - \beta y]} + \sum\limits_{j=-\infty}^{+\infty} \hat{b}_{rj} e^{i [ \alpha_j(\bar{\omega}) (x - D) + { \beta}_j y]},$$ for $x > D$, where $$\alpha_j(\bar{\omega}) = \sqrt{\bar{k}_0^2 - \beta_j^2}, \quad \mbox{for} \quad j = 0, \pm 1, \pm 2, \ldots.$$ The relations between $\alpha_j(\bar{\omega})$ and $\alpha_j(\omega)$ are $$\alpha_j(\bar{\omega}) = \left\{ \begin{array}{ll} \bar{\alpha}_j(\omega), & j = 0 \\ - \bar{\alpha}_j(\omega), & j = \pm 1, \pm 2, \ldots. \end{array} \right.$$ The coefficients $\hat{b}_{l0}$ and $\hat{b}_{r0}$ are related to the incident coefficients $\hat{a}_l$ and $\hat{a}_r$ by scattering matrix $S(\bar{\omega}, \beta)$ as $$\label{eq:Smatrix_complexFreq} \left[ \begin{matrix} \hat{b}_{l0} \\ \hat{b}_{r0} \end{matrix} \right] = S(\bar{\omega}, \beta) \left[ \begin{matrix} \hat{a}_{l} \\ \hat{a}_{r} \end{matrix} \right].$$ From the governing equations of $u$ and $w$, we have $$0 = \bar{w} (\Delta u + k^2_0 \epsilon u) - u (\Delta \bar{w} + k_0^2 \epsilon v) = \nabla \cdot (\bar{w} \nabla u) - \nabla \cdot ({u} \nabla \bar{w}).$$ Integrating the above equation on domain $\Omega_D$, we have $$\int_{\partial \Omega_D} \left( \bar{w} \dfrac{\partial u}{\partial \nu} - {u} \dfrac{\partial \bar{w}}{\partial \nu} \right) ds= 0.$$ The line integrals on the two edges of $\Omega_D$ at $y = \pm L/2$ cancel out. From the expressions of $u$ and $w$ at $x = \pm D$, and the relations of $\alpha_j$ and $\alpha_j(\bar{\omega})$, we obtain $$a_{l}\bar{\hat{a}}_l + a_{r} \bar{\hat{a}}_r = b_{l0}\bar{\hat{b}}_{l0} + b_{r0}\bar{\hat{b}}_{r0} .$$ Therefore, $$\left[ \bar{\hat{a}}_{l} , \bar{\hat{a}}_r \right] \left[ S^*(\bar{\omega}, \beta) S(\omega, \beta) - I \right] \left[ \begin{matrix} a_{l} \\ a_r \end{matrix} \right] = 0$$ for any complex $a_l, a_r, \hat{a}_l$ and $\hat{a}_r$. This leads to Eq. (\[eq:Smatrix\_Unitarity\]). Appendix C: Scattering matrix for $\mathcal{PT}$-symmetric structures {#appendix-c-scattering-matrix-for-mathcalpt-symmetric-structures .unnumbered} --------------------------------------------------------------------- Let $\epsilon$ satisfy the $\mathcal{PT}$-symmetric condition Eq. (\[eq:PTsymm\]), $\omega$ be complex with a small (positive or negative) imaginary part, $w$ be the diffraction solution defined in Appendix B and $v = \bar{w}(-x,y)$, then $$\Delta v + k_0^2 \epsilon v = 0.$$ From the governing equations of $u$ and $v$, we have $$0 = v (\Delta u + k^2_0 \epsilon u) - u (\Delta v + k_0^2 \epsilon v) = \nabla \cdot (v \nabla u) - \nabla \cdot ({u} \nabla v).$$ Integrating the above equation on domain $\Omega_D$, we have $$\int_{\partial \Omega_D} \left( v \dfrac{\partial u}{\partial \nu} - {u} \dfrac{\partial v}{\partial \nu} \right) ds= 0.$$ The line integrals on the two edges of $\Omega_D$ at $y = \pm L/2$ cancel out. Using the expressions of $u$ and $v$ (notice that $v = \bar{w}(-x,y)$), we have $$a_{l}\bar{\hat{a}}_r + a_{r} \bar{\hat{a}}_l = b_{l0}\bar{\hat{b}}_{r0} + b_{r0}\bar{\hat{b}}_{l0} .$$ From Eqs. (\[eq:Smatrix\]) and (\[eq:Smatrix\_complexFreq\]), we obtain $$\left[ \bar{\hat{a}}_{r} , \bar{\hat{a}}_l \right] \left[ P S^*(\bar{\omega}, \beta) P S(\omega, \beta) - I \right] \left[ \begin{matrix} a_l \\ a_r \end{matrix} \right] = 0$$ for any complex $a_l, a_r, \hat{a}_l$ and $\hat{a}_r$. This leads to Eq. (\[eq:Smatrix\_PT\]). Appendix D: Perturbation analysis {#appendix-d-perturbation-analysis .unnumbered} --------------------------------- To carry out the perturbation analysis, we first formulate a boundary value problem for the diffraction problem. Notice that a diffraction solution for incident waves given in Eqs. (\[eq:incident\_left\]) and (\[eq:incident\_right\]) has the general expansions given in Eqs. (\[eq:scat\_solution\_xlessD\]) and  (\[eq:scat\_solution\_xlargeD\]) for $|x| > D$. If we define a linear operator $\mathcal{T}$ such that $$\label{eq:T} \mathcal{T} e^{i \beta_j y} = i \alpha_j e^{i \beta_j y}, \quad j=0, \pm 1, \pm2, \ldots,$$ then the diffraction solution $u$ satisfies the following boundary conditions [@bao95] $$\label{eq:BCxmpD} \left\{ \begin{array}{ll} \dfrac{\partial u}{\partial x} = - \mathcal{T} u + 2 i \alpha a_l e^{i \beta y}, & x = -D, \\ \dfrac{\partial u}{\partial x} = \mathcal{T} u - 2 i \alpha a_r e^{i \beta y}, & x = D. \end{array} \right.$$ In the $y$-direction, $u$ satisfies the quasi-periodic conditions $$\begin{aligned} \label{eq:BCquasi} && u(x,L/2) = e^{i \beta L} u(x,-L/2), \\ && \dfrac{\partial u}{\partial y} (x, L/2) = e^{i \beta L} \dfrac{\partial u}{\partial y} (x, -L/2). \end{aligned}$$ Thus the diffraction problem is a boundary value problem of Helmholtz equation (\[eq:helm\]) with boundary conditions Eqs. (\[eq:BCxmpD\]) and (\[eq:BCquasi\]). Function $u_L$ ($u_R$) is a solution of the boundary value problem with $a_l =1 $ and $a_r = 0$ ($a_l=0$ and $a_r=1$). To derive Eq. (\[eq:k1\_left\]), we let $\tilde{u}_L$ be the diffraction solution of the perturbed structure $\tilde{\epsilon}$ at zero-reflection frequency $\tilde{\omega}_L$ (i.e. $\tilde{k}_L = \tilde{\omega}_L/c$) for a left incident wave with unit amplitude. Then $\tilde{u}_L$ is a solution of the boundary value problem with $k_0$ replaced by $\tilde{k}_L$, $a_l=1$ and $a_r = 0.$ We expand $\tilde{u}_L$, $\tilde{\mathcal{T}}$ and $\tilde{\alpha}_j$ for $\tilde{k}_L$ in power series of $\delta$: $$\begin{aligned} \label{eq:exp_uL} \tilde{u}_L & = & u_L + u_{1} \delta + u_{2} \delta^2 + \ldots, \\ \label{eq:exp_T} \tilde{\mathcal{T}} &= &\mathcal{T} + \mathcal{T}_1 \delta + \mathcal{T}_2 \delta^2 + \ldots,\\ \label{eq:exp_alpha} \tilde{\alpha}_j &=& \sqrt{\tilde{k}_L^2 - \beta_j^2} = \alpha_j + \gamma_{1j} \delta + \gamma_{2j} \delta^2 + \ldots,\end{aligned}$$ for $j = 0, \pm1, \pm2, \ldots.$ Similar to the definition of $\mathcal{T}$ in Eq. (\[eq:T\]), the actions of $\tilde{\mathcal{T}}$, $\mathcal{T}_1$ and $\mathcal{T}_2$ on $e^{i \beta_j y}$ are simply $e^{i \beta_j y}$ multiplied by $i \tilde{\alpha}_j$, $i \gamma_{1j}$ and $i \gamma_{2j}$, respectively. Using expansion Eq. (\[eq:exp\_freq1\]), we have $$\label{eq:gamma} \gamma_{1j} = \dfrac{ k_0 k_{1}}{\alpha_j}, \quad \mbox{for} \quad j = 0, \pm1, \pm2, \ldots.$$ Substituting expansions (\[eq:eps\_perturbation\]), (\[eq:exp\_freq1\]) and (\[eq:exp\_uL\])-(\[eq:exp\_alpha\]) into the boundary value problem for $\tilde{u}_L$, and comparing the coefficient of $\delta$, we have $$\label{eq:equ_uL1} \left\{ \begin{array}{ll} \Delta u_{1} + k_0^2 \epsilon u_{1} = - (2 k_0 k_1 \epsilon + k_0^2 F) u_L, & (x,y) \in \Omega_D \\ \dfrac{\partial u_{1}}{\partial x} = - \left( \mathcal{T} u_{1} + \mathcal{T}_1 u_L \right) + 2 i \gamma_{10} e^{i \beta y}, & x = -D, \\ \dfrac{\partial u_{1}}{\partial x} = \mathcal{T} u_{1} + \mathcal{T}_1 u_L , & x = D, \end{array} \right.$$ and $u_{1}$ satisfies the quasi-periodic condition Eq. (\[eq:BCquasi\]) in the $y$ direction. Let $$\begin{aligned} && \label{eq:tilde_u_xpmD_fourier} \tilde{u}_L(- D,y) = e^{i \beta y} + \sum\limits_{j=-\infty}^{\infty} \tilde{b}^{-}_j e^{i \beta_j y}, \\ && \tilde{u}_L( D,y) = \sum\limits_{j=-\infty}^{\infty} \tilde{b}^{+}_j e^{i \beta_j y},\end{aligned}$$ where $\{ \tilde{b}^{+}_j \} $ are the Fourier coefficients of $\tilde{u}( D, y)$ and $\{ \tilde{b}^{-}_j \}$ are the Fourier coefficients of $\tilde{u}( - D, y) - e^{i \beta y}$, then $$\tilde{b}^-_0 = 0, \quad \left| \tilde{b}^+_0 \right| = 1.$$ Let $$u_1(\pm D,y) = \sum\limits_{j=-\infty}^{\infty} c^{\pm}_j e^{i \beta_j y},$$ where $\{ c_j^{\pm} \}$ are the Fourier coefficients of $u_1(\pm D, y)$, then we must have $c^-_0 = 0. $ From the governing equations of $u_R$ and $u_1$, we have $$\begin{aligned} && - (2 k_0 k_1 \epsilon + k_0^2 F) u_L \bar{u}_R = \bar{u}_R (\Delta u_1 + k^2_0 \epsilon u_1) \\ && - u_1 (\Delta \bar{u}_R + k_0^2 \epsilon \bar{u}_R) = \nabla \cdot (\bar{u}_R \nabla u_1) - \nabla \cdot ({u_1} \nabla \bar{u}_R).\end{aligned}$$ Integrating the above equation on domain $\Omega_D$, we obtain $$\begin{aligned} \label{eq:k1} && \int_{\partial \Omega_D} \left( \bar{u}_R \dfrac{\partial u_1}{\partial \nu} - {u_1} \dfrac{\partial \bar{u}_R}{\partial \nu} \right) ds \nonumber \\ && = - \int_{\Omega_D} (2 k_0 k_1 \epsilon + k_0^2 F) u_L \bar{u}_R dxdy. \end{aligned}$$ Due to the quasi-periodic condition (\[eq:BCquasi\]), the line integrals on the two edges of $\Omega_D$ at $y = \pm L/2$ cancel out. Furthermore, by using the boundary conditions of $u_1$ at $x = \pm D$ and expansions of $u_L, u_R$ and $u_1$ at $x = \pm D$, the left-hand side of Eq. (\[eq:k1\]) can be reduced to $$i L \gamma_{10} (b^+_0 - \bar{b}^+_0) + i L \sum\limits_{\substack{j=-\infty \\ j \neq 0}}^{\infty} \gamma_{1j} ( \bar{b}^-_j b^+_j + \bar{b}^-_j b^+_j ).$$ In the above, the conditions $b_0^- = c_0^- = 0$ are used. Substituting the above into Eq. (\[eq:k1\]) and noticing the formula of $\gamma_{1j}$ (i.e. Eq. (\[eq:gamma\])), we obtain Eq. (\[eq:k1\_left\]). Equation (\[eq:k1\_right\]) can be similarly derived. [99]{} C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathcal{PT}$-symmetry,” Phys. Rev. Lett. **80**, 5243-5246 (1998). L. Feng, R. EI-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity-time symmetry,” Nature Photonics **11**, 752-762 (2017). S. Longhi, “Parity-time symmetry meets photonics: A new twist in non-Hermitian optics,” EPL **120**, 64001 (2017). M.-A. Miri and A. Alú, “Exceptional points in optics and photonics,” Science **363**, 7709 (2019). Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, And D. N. Christodoulides, “Unidirectional invisibility induced by $\mathcal{PT}$-symmetric periodic structures,” Phys. Rev. Lett. **106**, 213901 (2011). S. Longhi, “Invisibility in $\mathcal{PT}$-symmetric complex crystals,” J. Phys. A: Math. Theor. **44**, 485302 (2011). Y. Huang, Y. Shen, C. Min, S. Fan, and G. Veronis, “Unidirectional reflectionless light propagation at exceptional points,” Nanophotonics **548**, 192 (2017). S. Kalish, Z. Lin, and T. Kottos, “Light transport in random media with $\mathcal{PT}$ symmetry,” Phys. Rev. A **85**, 055802 (2012). A. Mostafazadeh, “Invisibility and $\mathcal{PT}$-symmetry,” Phys. Rev. A **87**, 012103 (2013) . E. Yang, T. Lu, Y. Wang, Y. Dai, and P. Wang, “Unidirectional reflectionless phenomenon in periodic ternary layered material,” Opt. Express **24**, 14311-14321 (2016). M. Sarisaman, “Unidirectional reflectionlessness and invisibility in the TE and TM modes of a $\mathcal{PT}$-symmetric slab system,” Phys. Rev. A **95**, 013806 (2017). M. Sarisaman and M. Tas, “Unidirectional invisibility and $\mathcal{PT}$ symmetry with graphene,” Phys. Rev. B **97**, 045409 (2018). S. A. R Horsley, M. Artoni, and G. C. La Rocca, “Spatial Kramers-Kronig relations and the reflections of waves,” Nat. Photon. **9**, 436-439 (2015). Y. Fu, Y. Xu, and H. Chen, “Zero index metamaterials with $\mathcal{PT}$ symmetry in a waveguide system,” Opt. Express **24**, 1648-1657 (2016). N. X. A. Rivolta, B. Maes, “Side-coupled resonators with parity-time symmetry for broadband unidirectional invisibility,” Phys. Rev. A **94**, 053854 (2016). A. Regensburger, C. Bersch, et al, “Parity-time synthetic photonic lattics,” Nature **488**, 167-171 (2012). L. Feng, Y. Xu, W. S. Fegadolli, et al, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. **12**, 108-113 (2013). M. -A. Miri, P. Likamwa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. **37**, 764-766 (2012). W. Liu [et al.]{}, “An integrated parity-time symmetric wavelength-tunable single-mode microring laser,” Nat. Commun. **8**, 15389 (2017). S. Longhi, “$\mathcal{PT}$-symmetric laser absorber,” Phys. Rev. A **82**, 031801(R) (2010). Y. D. Chong, L. Ge, and A. D. Stone, “$\mathcal{PT}$-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. **106**, 093902 (2011). E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Optica Acta **33**(5), 607-619 (1986). S. P. Shipman and H. Tu, “Total Resonant Transmission and Reflection by Periodic Structures,” SIAM J. Appl. Math. **72**, 216-239 (2012). L. Chesnel and S. A. Nazarov, “Non reflection and perfect reflection via Fano resonance in waveguides,” Commun. Math. Sci. **16**, 1779-1800 (2018). L. Ge, Y. D. Chong, and A. D. Stone, “Conservation relations and anisotropic transmission resonances in one-dimensional $\mathcal{PT}$-symmetric photonic heterostructures,” Phys. Rev. A **85**, 023802 (2012). Y. Shen, X. Deng, L. Chen, “Unidirectional invisibility in a two-layer non-$\mathcal{PT}$-symmetric slab,” Opt. Express **22**, 19440-19447 (2014). Y. Huang, G. Veronis, and C. Min, “Unidirectional reflectionless propagation in plasmonic waveguide-cavity systems at exceptional points,” Opt. Express **23**, 29882-29895 (2015). X. Gu, R. Bai, et al, “Unidirectional reflectionless propagation in a non-ideal parity-time metasurface based on far field coupling,” Opt. Express **25**, 11778-11787 (2017). F. Zhao, T. Dai, C. Zhang, R. Bai, Y. Q. Zhang, X. R. Jin, and Y. Lee, “Dual-band unidirectional reflectionlessness at exceptional points in a plasmonic waveguide system based on near-field coupling between two resonantors,” Nanotechnology **30**, 045205 (2019). L. Feng, X. Zhu, S. Yang, et al., “Demonstration of a large-scale optical exceptional point structure,” Opt. Express **22**, 1760-1767 (2014). G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A **12**, 1029 (1995). L. N. Trefethen, “Spectral methods in MATLAB,” Society for Industrial and Applied Mathematics (2000). Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. **24**, 3448-3453 (2006).

--- abstract: 'Proof theory provides a foundation for studying and reasoning about programming languages, most directly based on the well-known Curry-Howard isomorphism between intuitionistic logic and the typed lambda-calculus. More recently, a correspondence between intuitionistic linear logic and the session-typed pi-calculus has been discovered. In this paper, we establish an extension of the latter correspondence for a fragment of substructural logic with least and greatest fixed points. We describe the computational interpretation of the resulting infinitary proof system as session-typed processes, and provide an effectively decidable local criterion to recognize mutually recursive processes corresponding to valid circular proofs as introduced by Fortier and Santocanale. We show that our algorithm imposes a stricter requirement than Fortier and Santocanale’s guard condition, but is local and compositional and therefore more suitable as the basis for a programming language.' address: - 'Philosophy Department, Carnegie Mellon University, Pittsburgh, PA, 15213 USA' - 'Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, 15213 USA' author: - Farzaneh Derakhshan - Frank Pfenning title: | Circular Proofs as Session-Typed Processes:\ A Local Validity Condition --- Introduction {#intro .unnumbered} ============ Proof theory provides a solid ground for studying and reasoning about programming languages. This logical foundation is mostly based on the well-known Curry-Howard isomorphism [@Howard69] that establishes a correspondence between natural deduction and the typed $\lambda$-calculus by mapping propositions to types, proofs to well-typed programs, and proof reduction to computation. More recently, Caires et al. [@Caires10concur; @Caires16mscs] introduced a correspondence between intuitionistic linear logic [@Girard87tapsoft] and the session-typed $\pi$-calculus that relates linear propositions to session types, proofs in the sequent calculus to concurrent processes, and cut reduction to computation. In this paper, we expand the latter for a fragment of intuitionistic linear logic called *subsingleton logic* in which the antecedent of each sequent consists of at most one formula. We consider the sequent calculus of subsingleton logic with least and greatest fixed points added to the underlying type system and their corresponding rules added to the calculus [@DeYoung16aplas; @DeYoung19phd]. We closely follow Fortier and Santocanale’s [@Fortier13csl] development in singleton logic, where the antecedent consists of exactly one formula. Fortier and Santoconale [@Fortier13csl; @Santocanale02ita] extend the sequent calculus for singleton logic with rules for least and greatest fixed points. A naive extension, however, loses the cut elimination property (even when allowing infinite proofs) so they call derivations *pre-proofs*. *Circular pre-proofs* are distinguished as a subset of derivations which are [*regular*]{} in the sense that they can be represented as finite trees with loops. Fortier and Santocanale then impose a validity condition (which we call the *FS guard condition*) on pre-proofs to single out a class of pre-proofs that satisfy cut elimination. Moreover, they provide a cut elimination algorithm and show that it locally terminates on derivations that satisfy the guard condition. In addition, Santocanale and Fortier [@Fortier13csl; @Santocanale02ita; @Santocanale02fossacs] introduced a categorical semantics for interpreting cut elimination in singleton logic. In a related line of research, Baelde et al. [@Baelde16csl; @Baelde12tocl] add least and greatest fixed points to the sequent calculus for the multiplicative additive fragment of linear logic ($\textit{MALL}$) that results in losing the cut elimination property. They also introduced a validity condition to distinguish circular proofs from infinite pre-proofs in $\textit{MALL}$. Using Büchi automata, Doumane [@Doumane17phd] showed that the validity condition for identifying circular proofs in $\textit{MALL}$ with fixed points is <span style="font-variant:small-caps;">PSPACE</span> decidable. Nollet et al. [@Nollet18csl] introduced a polynomial time algorithm for locally identifying a stricter version of Baelde’s condition in $\textit{MALL}$ with fixed points. In this paper, we study (mutually) recursive session-typed processes and their correspondence with circular pre-proofs in subsingleton logic with fixed points. We introduce an algorithm to check a stricter version of the FS guard condition. Our algorithm is local in the sense that we check validity of each process separately, and it is stricter in the sense that it accepts a proper subset of the proofs recognized by the FS guard condition. We further introduce a synchronous computational semantics of cut reduction in subsingleton logic with fixed points in the context of session types, based on a key step in Fortier and Santocanale’s cut elimination algorithm which is compatible with prior operational interpretations of session-typed programming languages [@Toninho13esop]. We show preservation and a strong version of the progress property that ensures that each valid process communicates along its left or right interface in a finite number of steps. A key aspect of our type system is that validity is a compositional property (as we generally expect from type systems) so that the composition of valid programs defined over the same signature are also valid and therefore also satisfy strong progress. In the session type system, a singleton judgment $A \vdash B$ is annotated as $x:A \vdash \mathtt{P} :: (y:B)$ which is understood as: process $\mathtt{P}$ uses a service of type $A$ offered along channel $x$ by a process on its left and provides a service of type $B$ along channel $y$ to a process on its right [@DeYoung16aplas]. The left and right interfaces of a process in the session type system inherit the symmetry of the left and right rules in the sequent calculus. Each process interacts with other processes along its pair of left and right interfaces that are corresponding to the left and right sides of a sequent. For example, two processes $\mathtt{P}$ and $\mathtt{Q}$ with the typing $ x:A \vdash \mathtt{P} :: (y:B)$ and $y:B \vdash \mathtt{Q} :: (z:C)$ can be composed so they interact with each other using channel $y$. Process $\mathtt{P}$ provides a service of type $B$ and offers it along channel $y$ and process $\mathtt{Q}$ uses this service to provide its own service of type $C$. This interaction along channel $y$ can be of two forms: (i) process $\mathtt{P}$ sends a message to the right, and process $\mathtt{Q}$ receives it from the left, or (ii) process $\mathtt{Q}$ sends a message to the left and process $\mathtt{P}$ receives it from the right. In the first case, the session type $B$ is a *positive* type, and in the second case it is a *negative* type. Least fixed points have a positive polarity while greatest fixed points are negative [@Lindley16icfp]. As we will see in Sections \[alg1\] and \[priority\], due to the interactive nature of computation some types that would be considered “empty” (that is, have no closed values) may still be of interest here. DeYoung and Pfenning [@DeYoung16aplas; @Pfenning16lectures] provide a representation of Turing machines in the session-typed rule system of subsingleton logic with general equirecursive types. This shows that cut reduction on circular pre-proofs in subsingleton logic with equirecursive types has the computational power of Turing machines. Using this encoding on isorecursive types, we show that recognizing all programs that satisfy a strong progress property is undecidable, since this property can be encoded as termination of a Turing machine on a given input. However, with our algorithm, we can decide validity of a subset of Turing machines represented as session-typed processes in subsingleton logic with isorecursive fixed points. In summary, the principal contribution of our paper is to extend the Curry-Howard interpretation of proofs in subsingleton logic as communicating processes to include least and greatest fixed points. A circular proof is represented as a collection of mutually recursive process definitions. We develop a compositional criterion for validity of such programs, which is local in the sense that each process definition can be checked independently. Local validity in turn implies a strong progress property on programs and cut elimination on the circular proofs they correspond to. The structure of the remainder of the paper is as follows. In Section \[lfixed\] we introduce subsingleton logic with fixed points, and then examine it in the context of session-typed communication (Section \[pre\]). We provide a process notation with a synchronous operational semantics in Section \[operat\] and a range of examples in Section \[session\]. We then develop a local validity condition through a sequence of refinements in Sections \[alg1\]–\[modcut\]. We capture this condition on infinitary proofs in Section \[rules\] and reduce it to a finitary algorithm in Section \[alg\]. We prove that local validity implies Fortier and Santocanale’s guard condition (Section \[Guard\]) and therefore cut elimination. In Section \[semantics\] we explore the computational consequences of this, including the strong progress property which states that every valid configuration of processes will either be empty or attempt to communicate along external channels after a finite number of steps. We conclude by illustrating some limitations of our algorithm (Section \[negative\]) and pointing to some additional related and future work (Section \[conclusion\]). Subsingleton Logic with Fixed Points {#lfixed} ==================================== Subsingleton logic is a fragment of intuitionistic linear logic [@Girard87tapsoft; @Chang03tr] in which the antecedent and succedent of each judgment consist of at most one proposition. This reduces consideration to the additive connectives and multiplicative units, because the left or right rules of other connectives would violate this restriction. The expressive power of pure subsingleton logic is rather limited, among other things due to the absence of the exponential $!A$. However, we can recover significant expressive power by adding least and greatest fixed points, which can be done without violating the subsingleton restriction. We think of subsingleton logic as a laboratory in which to study the properties and behaviors of least and greatest fixed points in their simplest nontrivial form, following the seminal work of Fortier and Santocanale [@Fortier13csl]. The syntax of propositions then follows the grammar $$A,B ::= A \oplus B \mid 0 \mid A \& B \mid \top \mid 1 \mid \bot \mid t$$ where $t$ stands for propositions defined by least or greatest fixed points. Rather than including these directly as $\mu t.\, A$ and $\nu t.\, A$, we define them in a *signature* $\Sigma$ which records some important additional information, namely their relative *priority*. $$\Sigma ::= \cdot \mid \Sigma, t=^{i}_{\mu} A \mid \Sigma, t=^{i}_\nu A,$$ with the condition that if $ t=^{i}_{a} A \in \Sigma$ and $t=^{i}_b B \in \Sigma$, then $a=b$. For a fixed point $t$ defined as $ t=^{i}_{a} A$ in $\Sigma $, the subscript $a$ is the [*polarity*]{} of $t$: if $a=\mu$, then $t$ is a fixed point with [*positive*]{} polarity and if $a=\nu$, then it is of [*negative*]{} polarity. Finitely representable least fixed points (e.g., natural numbers and lists) can be represented in this system as defined type variables with positive polarity, while the potentially infinite greatest fixed points (e.g., streams and infinite depth trees) are represented as those with negative polarity. The superscript $i$ is the [*priority*]{} of $t$. Fortier and Santocanale interpreted the priority of fixed points in their system as the order in which the least and greatest fixed point equations are solved in the semantics [@Fortier13csl]. We use them syntactically as central information to determine local validity of circular proofs. We write $p(t)=i$ for the priority of $t$, and $\epsilon(i)= a$ for the polarity of type $t$ with priority $i$. The condition on $\Sigma$ ensures that $\epsilon$ is a well-defined function. The basic judgment of the subsingleton sequent calculus has the form $\omega \vdash_\Sigma \gamma$, where $\omega$ and $\gamma$ are either empty or a single proposition $A$ and $\Sigma$ is a signature. Since the signature never changes in the rules, we omit it from the turnstile symbol. The rules of subsingleton logic with fixed points are summarized in Figure \[fig:rule\], constituting a slight generalization of Fortier and Santocanale’s. When the fixed points in the last row are included, this set of rules must be interpreted as *infinitary*, meaning that a judgment may have an infinite derivation in this system. $$\begin{tabular}{c c c c} \multicolumn{2}{c}{\infer[{\mbox{\sc Id}}_A]{A \vdash A}{}} & \multicolumn{2}{c}{ \infer[{\mbox{\sc Cut}}_A] {\omega \vdash \gamma}{\omega \vdash A & A \vdash \gamma}} \\\\ \infer[\oplus R_1]{\omega \vdash A \oplus B}{\omega \vdash A} & \infer[\oplus R_2]{\omega \vdash A \oplus B}{\omega \vdash B} & \multicolumn{2}{c}{\infer[\oplus L]{A \oplus B \vdash \gamma}{ A \vdash \gamma & B \vdash \gamma}} \\\\ \multicolumn{2}{c}{\infer[\&R]{\omega \vdash A \& B}{\omega \vdash A & \omega \vdash B}} & \infer[\&L_1]{A \& B \vdash \gamma}{A \vdash \gamma} & \infer[\&L_2]{A \& B \vdash \gamma}{B \vdash \gamma} \\\\ \infer[1R]{\cdot \vdash 1}{} & \infer[1L]{1 \vdash \gamma}{\cdot \vdash \gamma} & \infer[{\bot}R]{\omega \vdash \bot}{\omega \vdash \cdot } & \infer[{\bot}L]{\bot \vdash \cdot}{} \\\\ \infer[\mu R]{\omega \vdash t}{\omega \vdash A & t=_{\mu} A \in \Sigma}& \infer[\mu L]{t \vdash \gamma}{A \vdash \gamma & t=_{\mu} A \in \Sigma}& \infer[\nu R]{\omega \vdash t}{\omega \vdash A & t=_{\nu} A \in \Sigma} & \infer[\nu L]{t \vdash \gamma}{A \vdash \gamma & t=_{\nu} A \in \Sigma}\\ \end{tabular}$$ Even a cut-free derivation may be of infinite length since each defined type variable may be unfolded infinitely many times. Also, cut elimination no longer holds for the derivations after adding fixed point rules. What the rules define then are the so-called *(infinite) pre-proofs* which include infinite derivations that do not necessarily enjoy the cut elimination property. In particular, we are interested in *circular pre-proofs*. Circular pre-proofs are those infinite pre-proofs that can be illustrated as finite trees with loops [@Doumane17phd]. As an example, the following circular pre-proof defined on the signature $\mathsf{nat}=^{1}_{\mu} \mathsf{1} \oplus \mathsf{nat}$ depicts an infinite pre-proof that consists of repetitive application of $\mu R$ followed by $\oplus R$: (2)[$\infer[\mu R]{\cdot \vdash \mathsf{nat}}{\infer[\oplus R]{\cdot \vdash \mathsf{1} \oplus \mathsf{nat}}{\cdot \vdash \mathsf{nat}}}$]{}; (-1,0.6).. controls (-3,.6) and (-3,-.7) .. (-1,-.6); It turns out not to be valid. On the other hand, on the signature $\mathsf{conat}=_{\nu}^{1} \mathsf{1}\ \& \ \mathsf{conat}$, we can define a circular pre-proof using greatest fixed points that is valid: (2)[$\infer[\nu R]{\cdot \vdash \mathsf{conat}}{\infer[\& R]{\cdot \vdash \mathsf{1} \&\ \mathsf{conat}}{\cdot \vdash \mathsf{1} & \cdot \vdash \mathsf{conat}}}$]{}; (1.1,0.6).. controls (3.1,0.6) and (3.2,-0.8) .. (1.1,-0.8); Fixed Points in the Context of Session Types {#pre} ============================================ Session types [@Honda93concur; @Honda98esop] describe the communication behavior of interacting processes. *Binary session types*, where each channel has two endpoints, have been recognized as arising from linear logic (either in its intuitionistic [@Caires10concur; @Caires16mscs] or classical [@Wadler12icfp] formulation) by a Curry-Howard interpretion of propositions as types, proofs as programs, and cut reduction as communication. In the context of programming, recursive session types have also been considered [@Toninho13esop; @Lindley16icfp], and they seem to fit smoothly, just as recursive types fit well into functional programming languages. However, they come at a price, since we abandon the Curry-Howard correspondence. In this paper we show that this is not necessary: we can remain on a sound logical footing as long as we (a) refine general recursive session types into least and greatest fixed points, (b) are prepared to accept circular proofs, and (c) impose conditions under which recursively defined processes are valid. General (nonlinear) type theory has followed a similar path, isolating inductive and coinductive types with a variety of conditions to ensure validity of proofs. In the setting of subsingleton logic, however, we find many more symmetries than typically present in traditional type theories that appear to be naturally biased towards least fixed points and inductive reasoning. Under the Curry-Howard interpretation a subsingleton judgment $A \vdash_\Sigma B$ is annotated as $$x:A \vdash_\Sigma \mathtt{P} :: (y:B)$$ where $x$ and $y$ are two different channels and $A$ and $B$ are their corresponding session types. One can understand this judgment as: process $\mathtt{P}$ provides a service of type $B$ along channel $y$ while using channel $x$ of type $A$, a service that is provided by another process along channel $x$ [@DeYoung16aplas]. We can form a chain of processes $\mathtt{P}_0, \mathtt{P}_1, \cdots, \mathtt{P}_{n}$ with the typing $$\cdot \vdash \mathtt{P}_0 :: (x_0:A_0),\quad x_0:A_0 \vdash \mathtt{P}_1 :: (x_1:A_1),\quad \cdots\quad x_{n-1}:A_{n-1} \vdash \mathtt{P}_{n} :: (x_{n}:A_{n})$$ which we write as $$\mathtt{P}_0 \mid_{x_0} \mathtt{P}_1 \mid_{x_1} \cdots\; \mid_{x_{n-1}} \mathtt{P}_n$$ in analogy with the notation for parallel composition for processes $P \mid Q$, although here it is not commutative. In such a chain, process $\mathtt{P}_{i+1}$ uses a service of type $A_i$ provided by the process $\mathtt{P}_{i}$ along the channel $x_i$, and provides its own service of type $A_{i+1}$ along the channel $x_{i+1}$. Process $\mathtt{P}_0$ provides a service of type $A_0$ along channel $x_0$ without using any services. So, a process in the session type system, instead of being reduced to a value as in the functional programming, interacts with both its left and right interfaces by sending and receiving messages. Processes $\mathtt{P}_i$ and $\mathtt{P}_{i+1}$, for example, communicate with each other along the channel $x_i$ of type $A_i$: If process $\mathtt{P}_{i}$ sends a message along channel $x_i$ to the right and process $\mathtt{P}_{i+1}$ receives it from the left (along the same channel), session type $A_i$ is called a *positive* type. And if process $\mathtt{P}_{i+1}$ sends a message along channel $x_i$ to the left and process $\mathtt{P}_{i}$ receives it from the right (along the same channel), session type $A_i$ is called a *negative* type. In Section \[session\] we show in detail that this symmetric behavior of left and right session types results in a symmetric behaviour of least and greatest fixed point types. In general, in a chain of processes, the leftmost type may not be empty. Also, strictly speaking, the names of the channels are redundant since every process has two distinguished ports: one to the left and one to the right, either one of which may be empty. Because of this, we may sometimes omit the channel name, but in the theory we present in this paper it is convenient to always refer to communication channels by unique names. For programming examples, it is helpful to allow any finite number of alternatives for internal ($\oplus$) and external ($\&$) choice. Finitary choices can equally well interpreted as propositions, so this is not a departure from the proofs as programs interpretation. \[signature\] We define [*session types*]{} with following grammar, where $L$ ranges over finite sets of labels denoted by $\ell$ and $k$. $$A::= {\oplus}\{\ell : A_\ell\}_{\ell \in L} \mid {\&}\{\ell : A_\ell\}_{\ell \in L} \mid 1 \mid \bot \mid t$$ where $t$ are type variables whose definition is given in a signature $\Sigma$ as before. The binary disjunction and conjunction are defined as $A \oplus B =\oplus\{\pi_1:A, \pi_2:B\}$ and $A \& B =\& \{\pi_1:A, \pi_2:B\}$. Similarly, we define $0=\oplus\{\}$ and $\top=\& \{\}$. As a first programming-related example, consider natural numbers in unary form ($\mathsf{nat}$) and infinite streams of natural numbers ($\mathsf{stream}$). \[natstream\] $$\begin{aligned} & \mathsf{nat}=^{1}_{\mu} \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}: \mathsf{nat}\}\\ & \mathsf{stream}=^{2}_{\nu} \& \{\mathit{head} : \mathsf{nat}, \ \ \mathit{tail}: \mathsf{stream}\} \end{aligned}$$ $\Sigma$, in this example, consists of two well-known, respectively, inductive and coinductive types: (i) the type of natural numbers ($\mathsf{nat}$) built using two constructors for ${\mathit{zero}}$ and $\mathit{successor}$, and (ii) the type of infinite streams ($\mathsf{stream}$) defined using two destructors for $\mathit{head}$ and $\mathit{tail}$ of a stream. Here, priorities of $\mathsf{nat}$ and $\mathsf{stream}$ are, respectively, $1$ and $2$, understood as “*$\mathsf{nat}$ has higher priority than $\mathsf{stream}$*”. As another example consider the signature with two types with positive polarity and the same priority: $\mathsf{std}$ and $\mathsf{pos}$. Here, $\mathsf{std}$ is the type of standard bit strings, i.e., bit strings terminated with $\$$ without any leading $0$ bits, and $\mathsf{pos}$ is the type of positive standard bit strings, i.e., all standard bit strings except $\$$. $$\begin{aligned} & \mathsf{std}=^{1}_{\mu} \oplus\{ \mathit{b0}:\mathsf{pos}, \mathit{b1}:\mathsf{std}, \$:\mathsf{1}\}\\ & \mathsf{pos}=^{1}_{\mu} \oplus \{\mathit{b0}:\mathsf{pos}, \mathit{b1}:\mathsf{std}\} \end{aligned}$$ \[bittype\] $$\begin{aligned} &\mathsf{bits}=^{1}_{\mu} \oplus\{ \mathit{b0}:\mathsf{bits},\ \mathit{b1}:\mathsf{bits}\} \\ &\mathsf{cobits}=^{2}_{\nu} \&\{ \mathit{b0}:\mathsf{cobits},\ \mathit{b1}:\mathsf{cobits}\}\end{aligned}$$ In the functional programming type system $\mathsf{cobits}$ is a greatest fixed point recognized as an infinite stream of bits, while $\mathsf{bits}$ is recognized as an empty type. However, in the session type system, we treat them in a symmetric way. $\mathsf{bits}$ is an infinite sequence of bits with positive polarity. And its dual type, $\mathsf{cobits}$, is an infinite stream of bits with negative polarity. In Examples \[bits\] and \[cobits\], in Section \[alg1\], we illustrate more the symmetry of these types by providing two recursive processes having them as their interfaces. Even though we can, for example, write transducers of type $\mathsf{bits} \vdash \mathsf{bits}$ inside the language, we cannot write a *valid* process of type $\cdot \vdash \mathsf{bits}$ that *produces* an infinite stream. A Synchronous Operational Semantics {#operat} =================================== The operational semantics for process expression under the proofs-as-programs interpretation of linear logic has been treated exhaustively elsewhere [@Caires10concur; @Caires16mscs; @Toninho13esop; @Griffith16phd]. We therefore only briefly sketch the operational semantics here. Communication is *synchronous*, which means both sender and receiver block until they synchronize. Asynchronous communication can be modeled using a process with just one output action followed by forwarding [@Gay10jfp; @DeYoung12csl]. However, a significant difference to much prior work is that we treat types in an isorecursive way, that is, a message is sent to unfold the definition of a type $t$. This message is written as $\mu_t$ for a least fixed point and $\nu_t$ for a greatest fixed point. The language of process expressions and their operational semantics presented in this section is suitable for general isorecursive types, if those are desired in an implementation. The resulting language then satisfies a weaker progress property sometimes called *local progress* (see, for example, [@Caires10concur]). \[process\] Processes are defined as follows over the signature $\Sigma$:\ \ $\begin{aligned} P,Q ::=\ \ & { y \leftarrow x} & identity\\& \mid ( x \leftarrow P_x ; Q_x) & cut \\ & \mid Lx.k; P \mid \mathbf{case}\, Rx\ (\ell {\Rightarrow}Q_{\ell})_{\ell \in L}& \& \{\ell:A_\ell\}_{\ell \in L}\\ & \mid Rx.k; P \mid \mathbf{case}\, Lx\ (\ell {\Rightarrow}Q_{\ell})_{\ell \in L} & \phantom{something} \oplus \{\ell:A_\ell\}_{\ell \in L} \\ & \mid \mathbf{wait}\, Lx\mid \mathbf{close}\, Rx ; Q & 1\\ & \mid \mathbf{close}\, Lx; Q \mid \mathbf{wait} \,Rx & \bot \\ & \mid Rx.\mu_{t}; P \mid \mathbf{case}\, Lx\ (\mu_{t} {\Rightarrow}P) & t=_{\mu}A\\ & \mid Lx. \nu_{t}; P \mid \mathbf{case}\, Rx\ (\nu_{t} {\Rightarrow}P) & t=_{\nu} A\\ & \mid { y \leftarrow X \leftarrow x} & \end{aligned}$\ \ where $X,Y, \ldots$ are [*process variables*]{}, and $x,y, \ldots$ are channel names. The left and right session types a process interacts with are uniquely labelled with channel names: $$x:A \vdash P :: (y:B).$$ We read this as [*Process $P$ uses channel $x$ of type $A$ and provides a service of type $B$ along channel $y$.*]{} However, since a process may not use any channel to provide a service along its right channel, e.g. $\cdot \vdash P :: (y: B),$ or it may not provide any service along its right channel , e.g.$ x:A \vdash Q:: (\cdot) $, the labelling of processes is generalized to be of the form: $$\bar{x}:\omega \vdash P::(\bar{y}:\gamma),$$ where $\bar{x}$ ($\bar{y}$) is either empty or $x$ ($y$), and $\omega$ ($\gamma$) is empty given that $\bar{x}$ ($\bar{y}$) is empty. Process definitions are of the form $\bar{x}:\omega \vdash X=P_{\bar{x},\bar{y}} ::(\bar{y}:B)$ representing that variable $X$ is defined as process $P$. A [*program*]{} $\mathcal{P}$ is defined as a pair $ \langle V,S \rangle$, where $V$ is a finite set of process definitions, and $S$ is the [*main*]{} process variable. Figure \[fig:annotrule\] shows the logical rules annotated with processes in the context of session types. This set of rules inherits the full symmetry of its underlying sequent calculus. These typing rules interpret *pre-proofs*: As can be seen in the rule ${\mbox{\sc Def}}$, the shown typing rules inherit the infinitary nature of deductions from the logical rules in Figure \[fig:rule\] and are therefore not directly useful for type checking. We obtain a finitary system to check *circular* pre-proofs by removing the first premise from the ${\mbox{\sc Def}}$ rule and checking each process definition in $V$ separately, under the hypothesis that all process definitions are well-typed. In order to also enforce validity, the rules need to track additional information (see rules in Figures \[fig:stp-order\] (infinitary) and \[fig:validity\] (finitary) in Sections \[rules\] and \[alg\]). All processes we consider in this paper provide a service along their right channel so in the remainder of the paper we restrict the sequents to be of the form $\bar{x}:\omega \vdash P::(y:A)$. We therefore do not need to consider the rules for type $\perp$ anymore, but the results of this paper can easily be generalized to the fully symmetric calculus. $$\begin{tabular}{c c} \infer[{\mbox{\sc Id}}]{x : A \vdash y \leftarrow x :: (y : A)}{} & \infer[{\mbox{\sc Cut}}^{w}]{ \bar{x} : \omega \vdash (w \leftarrow P_{w} ; Q_{w}) :: ( \bar{y} : \gamma)}{ \bar{x} : \omega \vdash P_{w} ::(w:A) & w: A \vdash Q_{w} :: (\bar{y}: \gamma)} \\\\ \infer[\oplus R]{\bar{x}:\omega \vdash Ry.k; P :: (y: \oplus\{\ell:A_\ell\}_{\ell \in L})}{\bar{x}: \omega \vdash P :: (y: A_{k}) \quad (k \in L)} & \infer[\oplus L]{x:\oplus\{ \ell:A_\ell \}_{ \ell \in L} \vdash \mathbf{case}\, Lx \ (\ell{\Rightarrow}P_{\ell}):: (\bar{y}: \gamma)}{\forall \ell\in L \quad x:A_{\ell} \vdash P_\ell :: (\bar{y}:\gamma)}\\\\ \infer[\& R]{\bar{x} : \omega \vdash \mathbf{case}\, Ry \ (\ell {\Rightarrow}P_\ell) :: (y : \& \{\ell:A_\ell\}_{\ell \in L})}{\bar{x} : \omega \vdash P_\ell :: (y :A_{\ell}) \quad \forall \ell\in L} & \infer[\& L]{x : \&\{ \ell:A_\ell \}_{ \ell \in L} \vdash Lx .k; P :: ( \bar{y} : \gamma)}{k\in L \quad x : A_{k} \vdash P :: ( \bar{y} : \gamma)}\\\\ \infer[1R]{\cdot \vdash \mathbf{close}\, Ry :: (y : 1)}{} & \infer[1L]{x : 1 \vdash \mathbf{wait}\, Lx;Q :: (\bar{y} : \gamma)}{ . \vdash Q :: (\bar{y} : \gamma) } \\\\ \infer[ {\bot} R]{\bar{x}:A \vdash \mathbf{wait} \, Ry; Q :: (y : \bot)}{\bar{x}:A \vdash Q:: \cdot} & \infer[{\perp}L]{x : \bot \vdash \mathbf{close} \, Lx :: \cdot}{} \\\\ \infer[\mu R]{ \bar{x} : \omega \vdash Ry .\mu_t; P_{y} :: (y:t)}{ \bar{x} : \omega \vdash P_{y} :: (y :A) & t=_{\mu}A } & \infer[\mu L]{x : t \vdash \mathbf{case}\, Lx\ (\mu_{t} {\Rightarrow}Q_{x }):: ( \bar{y} : \gamma)}{x : A \vdash Q_{x } :: ( \bar{y} : \gamma) & t=_{\mu} A }\\\\ \infer[\nu R]{\bar{x} : \omega \vdash \mathbf{case}\, Ry \ (\nu_t {\Rightarrow}P_{y }) :: (y : t)}{\bar{x} : \omega \vdash P_{y } :: (y : A) & t=_{\nu}A }& \infer[\nu L]{x : t \vdash Lx .\nu_{t}; Q_{x }:: ( \bar{y} : \gamma)}{ x : A \vdash Q_{x } :: ( \bar{y} : \gamma) & t=_{\nu} A }\\ \end{tabular}$$ $$\infer[{\mbox{\sc Def}}(X)]{\bar{x} : \omega \vdash \bar{y} \leftarrow X \leftarrow \bar{x} :: ( \bar{y} : \gamma)}{\bar{x} : \omega \vdash P_{\bar{x},\bar{y}} :: ( \bar{y} : \gamma) & \bar{u}:\omega \vdash X=P_{\bar{u},\bar{w}} :: (\bar{w}:\gamma) \in V }$$ The computational semantics is defined on configurations $$P_0 \mid_{\, x_1} \cdots \mid_{\, x_n} P_n$$ where $\mid$ is associative and has unit $(\cdot)$ but is not commutative. The following transitions can be applied anywhere in a configuration: $$\renewcommand{\arraystretch}{1} \begin{array}{lcll} P \mid_x (y \leftarrow x) \mid_y Q & \mapsto & ([z/x]P) \mid_z ([z/y]Q) & \mbox{($z$ fresh), forward} \\ (x \leftarrow P {\mathrel{;}}Q) & \mapsto & ([z/x]P) \mid_z ([z/x]Q) & \mbox{($z$ fresh), spawn} \\ (Rx.k {\mathrel{;}}P) \mid_x (\mathbf{case}\, Lx\, (\ell \Rightarrow Q_\ell)) & \mapsto & P \mid_x Q_k & \mbox{send label $k$ right} \\ \mathbf{case}\, Rx\, (\ell \Rightarrow P_\ell) \mid_x (Lx.k {\mathrel{;}}Q) & \mapsto & P_k \mid_x Q & \mbox{send label $k$ left} \\ \mathbf{close}\, Rx \mid_x (\mathbf{wait}\, Lx {\mathrel{;}}Q) & \mapsto & Q & \mbox{close channel right} \\ (\mathbf{wait}\, Rx {\mathrel{;}}P) \mid_x \mathbf{close}\, Lx & \mapsto & P & \mbox{close channel left} \\ (Rx.\mu_t {\mathrel{;}}P) \mid_x (\mathbf{case}\, Lx\, (\mu_t \Rightarrow Q)) & \mapsto & P \mid_x Q & \mbox{send $\mu_t$ unfolding message right} \\ \mathbf{case}\, Rx\, (\nu_t \Rightarrow P) \mid_x (Lx.\nu_t {\mathrel{;}}Q) & \mapsto & P \mid_x Q & \mbox{send $\nu_t$ unfolding message left} \\ \cdots \mid_{\bar x} \bar{y} \leftarrow X \leftarrow \bar{x} \mid_{\bar y} \cdots & \mapsto & \cdots \mid_{\bar x} [\bar{y}/\bar{w}, \bar{x}/\bar{u}] P \mid_{\bar y} \cdots & \mbox{where $\bar{u} : \omega \vdash X = P :: (\bar{w} : \gamma)$} \end{array}$$ The forward rule removes process $y \leftarrow x$ from the configuration and replaces both channels $x$ and $y$ in the rest of configuration with a fresh channel $z$. The rule for $x \leftarrow P\, ; Q$ spawns process $[z/x]P$ and continues as $[z/x]Q$. To ensure uniqueness of channels, we need $z$ to be a fresh channel. For internal choice, $Rx.k;P$ sends label $k$ along channel $x$ to the process on its right and continues as $P$. The process on the right, $\mathbf{case} \, Lx\, (\ell \Rightarrow Q_{\ell})$, receives the label $k$ sent from the left along channel $x$, and chooses the $k$th alternative $Q_k$ to continue with accordingly. The last transition rule unfolds the definition of a process variable $X$ while instantiating the left and right channels $\bar{u}$ and $\bar{w}$ in the process definition with proper channel names, $\bar{x}$ and $\bar{y}$ respectively. Examples of Session-Typed Processes {#session} =================================== In this section we motivate our local validity algorithm using a few examples. By defining type variables in the signature and process variables in the program, we can generate (mutually) recursive processes which correspond to circular pre-proofs in the sequent calculus. In Examples \[begen\]-\[block\], we provide such recursive processes along with explanation of their computational steps and their corresponding derivations. \[begen\] $$\Sigma_1:= \mathsf{nat}=^{1}_{\mu} \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\}.$$ We define a process $$\begin{aligned} \cdot \vdash \mathtt{Loop} :: (y:\mathsf{nat}) \\ \end{aligned},$$ where the variable $\mathtt{Loop}$ is defined as $$\begin{aligned} y \leftarrow \mathtt{Loop} \leftarrow \cdot = \ & Ry.\mu_{nat}; &&\phantom{lo space} \%\ \textit{send}\ \mu_{nat}\ \textit{to right} \tag{i}\\ & \phantom{low} Ry.s; && \phantom{lo space} \%\ \textit{send label}\ \mathit{s} \ \textit{to right} \tag{ii}\\ & \phantom{low s} y \leftarrow \mathtt{Loop} \leftarrow \cdot && \phantom{lo space} {\color{red} \%\ \textit{recursive call}}\ \tag{iii}\end{aligned}$$ $\mathcal{P}_1:=\langle \{\mathtt{Loop}\}, \mathtt{Loop} \rangle$ forms a program over the signature $\Sigma_1$. It (i) sends a [*positive*]{} fixed point unfolding message to the right, (ii) sends the label $\ \mathit{s}$, as another message corresponding to $\mathit{successor}$, to the right, (iii) calls itself and loops back to (i). The program runs forever, sending [*successor*]{} labels to the right, without receiving any fixed point unfolding messages from the left or the right. The process $\mathtt{Loop}$ corresponds to the following infinite derivation: (2)[$\infer[\mu R]{\cdot \vdash \mathsf{nat}}{\infer[\oplus R_s]{\cdot \vdash \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\}}{\cdot \vdash \mathsf{nat}}}$]{}; (-1,0.6).. controls (-3,.6) and (-3,-.7) .. (-1,-.6); \[block\] Define process $$\begin{aligned} x:\mathsf{nat} \vdash \mathtt{Block} :: (y:\mathsf{1}) \\ \end{aligned}$$ over the signature $\Sigma_1$ as $$\begin{aligned} & y \leftarrow \mathtt{Block} \leftarrow x= && \notag \\ & \phantom{small} \mathbf{case}\, Lx\ (\mu_{nat} {\Rightarrow}\tag{i} && \% \ \textit{receive}\ \mu_{nat}\ \textit{from left} \\ & \phantom{small spac}\mathbf{case}\,Lx && \%\ \textit{receive a label} \ \textit{from left} \tag{ii}\\ & \phantom{small space ti} ( \ z \Rightarrow \mathbf{wait}\, Lx; && \% \ \textit{wait and close} \ x \tag{ii-a}\\ & \phantom{small space times two} \mathbf{close}\, Ry && \%\ \textit{close}\ y \notag\\ & \phantom{small space ti} \mid s \Rightarrow y \leftarrow \mathtt{Block} \leftarrow x)) && {\color{red} \% \ \textit{recursive call \tag{ii-b}}} \end{aligned}$$ $\mathcal{P}_2:=\langle \{\mathtt{Block}\}, \mathtt{Block} \rangle$ forms a program over the signature $\Sigma_1$:\ (i) $\mathtt{Block} $ [***waits***]{}, until it receives a [*positive*]{} fixed point unfolding message from the left, (ii) waits for another message from the left to determine the path it will continue with:\ (a) If the message is a $\ \mathit{z}\ $ label, (ii-a) the program waits until a closing message is received from the left. Upon receiving that message, it closes the left and then the right channel.\ (b) If the message is an $\ \mathit{s} \ $ label, (ii-b) the program calls itself and loops back to (i).\ Process $\mathtt{Block}$ corresponds to the following infinite derivation: (2)[$\infer[\mu L]{\mathsf{nat} \vdash \mathsf{1}}{\infer[\oplus L]{ \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\} \vdash \mathsf{1}}{\infer[\mathsf{1} L]{\mathsf{1} \vdash \mathsf{1}}{\infer[\mathsf{1}R]{\cdot \vdash \mathsf{1}}{}}& \mathsf{nat} \vdash \mathsf{1} }}$]{}; (1.6,0).. controls (3,0) and (3,-1.2) .. (1,-1.2); Derivations corresponding to both of these programs are cut-free. Also no internal loop takes place during their computation, in the sense that they both communicate with their left or right channels after finite number of steps. For process $\mathtt{Loop}$ this communication is restricted to sending infinitely many unfolding and successor messages to the right. Process $\mathtt{Block}$, on the other hand, receives the same type of messages after finite number of steps as long as they are provided by a process on its left. Composing these two processes as in $x \leftarrow \mathtt{Loop} \leftarrow \cdot \mid y \leftarrow \mathtt{Block} \leftarrow x$ results in an internal loop: process $\mathtt{Loop}$ keeps providing unfolding and successor messages for process $\mathtt{Block}$ so that they both can continue the computation and call themselves recursively. Because of this internal loop, the composition is not acceptable: It never communicates with its left (empty channel) or right (channel $y$). The infinite derivation corresponding to the composition $x \leftarrow \mathtt{Loop} \leftarrow \cdot \mid y \leftarrow \mathtt{Block} \leftarrow x$ therefore should be rejected as invalid: (-3,0.5).. controls (-4.5,0.5) and (-4.5,-1) .. (-3,-1); (2)[$\infer[{\mbox{\sc Cut}}_{nat}]{\cdot \vdash \mathsf{1}}{\infer[\mu R]{\cdot \vdash \mathsf{nat}}{\infer[\oplus R_s]{\cdot \vdash \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\}}{\cdot \vdash \mathsf{nat}}} & \infer[\mu L]{\mathsf{nat} \vdash \mathsf{1}}{\infer[\oplus L]{ \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\} \vdash \mathsf{1}}{\infer[\mathsf{1} L]{\mathsf{1} \vdash \mathsf{1}}{\infer[\mathsf{1}R]{\cdot \vdash \mathsf{1}}{}}& \mathsf{nat} \vdash \mathsf{1} }}}$]{}; (3.2,0.5).. controls (4.5,0.5) and (4.5,-1) .. (3,-1); The cut elimination algorithm introduced by Fortier and Santocanale uses a reduction function $\textsc{Treat}$ that may never halt. They proved that for derivations satisfying the guard condition $\textsc{Treat}$ is locally terminating since it always halts on guarded proofs [@Fortier13csl]. The above derivation is an example of one that does not satisfy the FS guard condition and the cut elimination algorithm does not locally terminate on it. The [*progress property*]{} for a configuration of processes ensures that during its computation it either (i) takes a step, or (ii) is empty, or (iii) communicates along its left or right channel. Without (mutually) recursive processes, this property is enough to make sure that computation never gets stuck. Having (mutual) recursive processes and fixed points, however, this property is not strong enough to restrict internal loops. The composition $x \leftarrow \mathtt{Loop} \leftarrow \cdot \mid y \leftarrow \mathtt{Block} \leftarrow x$, for example, satisfies the progress property but it never interacts with any other external process. We introduce a stronger form of the progress property, in the sense that it requires any of conditions (ii) or (iii) to hold after a finite number of computation steps. Like cut elimination, strong progress is not compositional. Processes $\mathtt{Loop}$ and $\mathtt{Block}$ both satisfy the strong progress property but their composition $x \leftarrow \mathtt{Loop} \leftarrow \cdot \mid y \leftarrow \mathtt{Block} \leftarrow x$ does not. We will show in Section \[semantics\] that FS validity implies strong progress. But, in contrast to strong progress, this condition is compositional in the sense that composition of two disjoint valid programs is also valid. FS validity is not local. Our goal is to construct a locally checkable validity condition that accepts (a subset of) programs satisfying strong progress and is compositional. In functional programming languages, a program is called [*terminating*]{} if it reduces to a value in a finite number of steps, and is called [*productive*]{} if every piece of the output is generated in finite number of steps (even if the program potentially runs forever). The theoretical underpinnings for terminating and productive programs are also least and greatest fixed points, respectively, but due to the functional nature of computation they take a different and less symmetric form than here (see, for example, [@Birkedal13lics; @Grathwohl16phd]). In our system of session types, least and greatest fixed points correspond to defined type variables with positive and negative polarity, respectively, and their behaviors are quite symmetric: As in Definition \[process\], an unfolding message $\mu_t$ for a type variable $t$ with positive polarity is received from the left and sent to the right, while for a variable $t$ with negative polarity, the unfolding message $\nu_t$ is received from the right and sent to the left. Back to Examples \[begen\] and \[block\], process $\mathtt{Loop}$ seems less acceptable than process $\mathtt{Block}$: process $\mathtt{Loop}$ does not receive any least or greatest fixed point unfolding messages and its circularity cannot be justified with either induction or co-induction. We want our algorithm to accept process $\mathtt{Block}$ rather than $\mathtt{Loop}$, since it cannot accept both. This motivates a unified definition of termination and productivity in the session types system based on its symmetry. A program is [*(finitely) reactive to the left*]{} if it receives every fixed point unfolding message $\mu_t$ from the [*left*]{}, if any, in a finite number of steps, and it stops in finite steps if there are no more fixed point unfolding messages to be received from the [*left*]{}.\ A program is [*(finitely) reactive to the right*]{} if it receives every fixed point unfolding message $\nu_t$ from the [*right*]{}, if any, in a finite number of steps, and it stops in finite steps if there are no more unfolding messages to be received from the [*right*]{}.\ A program is called [*(finitely) reactive*]{} if it is either [*reactive to the right*]{} or [*to the left*]{}. By this definition, process $\mathtt{Block}$ is reactive while process $\mathtt{Loop}$ is not. Although reactivity is not compositional we use it to explain the intuition behind our condition and construct it one step at a time. Later, in Sections \[Guard\] and \[semantics\] we prove that our algorithm ensures FS validity and strong progress. We conclude this section with an example of a reactive process $\mathtt{Copy}$. This process, similar to $\mathtt{Block}$, receives a natural number from left but instead of consuming it, sends it over to the right along a channel of type $\mathsf{nat}$. \[prex\] With signature to be $$\Sigma_1:= \mathsf{nat}=^{1}_{\mu} \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\}$$ we defined process $\mathtt{Copy}$ $$\begin{aligned} x:\mathsf{nat} \vdash \mathtt{Copy} :: (y:\mathsf{nat}) \\ \end{aligned}$$ as $$\begin{aligned} & y \leftarrow \mathtt{Copy} \leftarrow x= && \notag \\ & \phantom{small} \mathbf{case}\, Lx\ (\mu_{nat} {\Rightarrow}\tag{i} && \% \ \textit{receive}\ \mu_{nat}\ \textit{from left} \\ & \phantom{small spac}\mathbf{case}\, Lx && \%\ \textit{receive a label} \ \textit{from left} \tag{ii}\\ & \phantom{small space tim} (\ z{\Rightarrow}Ry.\mu_{nat}; && \%\ \textit{send}\ \mu_{nat}\ \textit{ to right} \tag{ii-a}\\ & \phantom{small space times two plu} Ry.z; && \%\ \textit{send label}\ \mathit{z} \ \textit{ to right} \notag\\ & \phantom{small space times two plus} \mathbf{wait}\, Lx; && \% \ \textit{wait and close} \ x \notag \\ & \phantom{small space times two plus on} \mathbf{close}\, Ry && \%\ \textit{close}\ y \notag\\ & \phantom{small space tix} \mid s \Rightarrow Ry.\mu_{nat}; && \% \ \textit{send}\ \mu_{nat}\ to \ \textit{right} \tag{ii-b}\\ & \phantom{small space times two plu} Ry.s; && \% \ \textit{send label}\ \mathit{s}\ \textit{to right} \notag \\ & \phantom{small space times two plus} y \leftarrow \mathtt{Copy} \leftarrow x)) && {\color{red} \% \ \textit{recursive call}} \notag\end{aligned}$$ This is an example of a recursive process, and $\mathcal{P}_3:=\langle \{\mathtt{Copy}\}, \mathtt{Copy} \rangle$ forms a [*reactive to the left*]{} program over the signature $\Sigma_1$:\ (i) It waits until it receives a [*positive*]{} fixed point unfolding message from the left, (ii) waits for another message from the left to determine the path it will continue with:\ (a) If the message is a $\ \mathit{z}\ $ label, (ii-a) the program sends a [*positive*]{} fixed point unfolding message to the right, following by a message of label $\ \mathit{z}\ $ to the right, and then waits until a closing message is received from the left. Upon receiving that message, it closes the right channel.\ (b) If the message is a $\ \mathit{s} \ $ label, (ii-b) the program sends a [*positive*]{} fixed point unfolding message to the right, following by a message of label $\ \mathit{s}\ $ to the right, and then calls itself and loops back to (i).\ The computational content of $\mathtt{Copy}$ is to simply copy a natural number given from the left to the right. Process $\mathtt{Copy}$ is cut-free and satisfies the strong progress property. This property is preserved when composed with $\mathtt{Block}$ as $y\leftarrow \mathtt{Copy} \leftarrow x\mid z \leftarrow \mathtt{Block} \leftarrow y$. Local Validity Algorithm: Naive Version {#alg1} ======================================= In this section we develop a first naive version of our local validity algorithm using Examples \[bits\]-\[cobits\]. \[bits\] Let the signature be $$\Sigma_2:=\\ \mathsf{bits}=^{1}_{\mu} \oplus\{ \mathit{b0}:\mathsf{bits},\ \mathit{b1}:\mathsf{bits}\}$$ and define the process $\mathtt{BitNegate}$ $$\begin{aligned} x:\mathsf{bits} \vdash \mathtt{BitNegate} :: (y:\mathsf{bits}) \\ \end{aligned}$$ with $$\begin{aligned} & y \leftarrow \mathtt{BitNegate} \leftarrow x= && \notag \\ & \phantom{small} \mathbf{case}\, Lx\ (\mu_{bits} {\Rightarrow}\tag{i} && \% \ \textit{receive}\ \mu_{bits}\ \textit{from left} \\ & \phantom{small spac}\mathbf{case}\, Lx && \%\ \textit{receive a label} \ \textit{from left} \tag{ii}\\ & \phantom{small space ti}(\ \mathit{b0}{\Rightarrow}Ry.\mu_{bits}; && \%\ \textit{send}\ \mu_{bits}\ \textit{ to right} \tag{ii-a}\\ & \phantom{small space times two plu} Ry.b1; && \%\ \textit{send label}\ \mathit{b1} \ \textit{ to right} \notag\\ & \phantom{small space times two plus} y\leftarrow \mathtt{BitNegate} \leftarrow x && {\color{red}\% \ \textit{recursive call} } \notag \\ & \phantom{small space ti} \mid \mathit{b1} \Rightarrow Ry.\mu_{bits}; && \% \ \textit{send}\ \mu_{bits} to \ \textit{right} \tag{ii-b}\\ & \phantom{small space times two plu} Ry.b0; && \% \ \textit{send label}\ \mathit{b0} \textit{ to right} \notag \\ & \phantom{small space times two plus} y \leftarrow \mathtt{BitNegate} \leftarrow x)) && {\color{red} \% \ \textit{recursive call}} \notag\end{aligned}$$ $\mathcal{P}_4:=\langle \{\mathtt{BitNegate}\}, \mathtt{BitNegate} \rangle$ forms a [*reactive to the left*]{} program over the signature $\Sigma_2$:\ (i) It waits until it receives a [*positive*]{} fixed point unfolding message from the left, (ii) waits for another message from the left to determine the path it shall continue with:\ (a) If the message is a $\ \mathit{b0}\ $ label, (ii-a) the program sends a [*positive*]{} fixed point unfolding message to the right, following by a message of label $\ \mathit{b1}\ $ to the right, and then calls itself recursively and loops back to (i).\ (b) If the message is a $\ \mathit{b0} \ $ label, similarly, (ii-b) the program sends a [*positive*]{} fixed point unfolding message to the right, following by a message of label $\ \mathit{b1}\ $ to the right, and then calls itself and loops back to (i). Computationally, $\mathtt{BitNegate}$ is a buffer with one bit capacity that receives a bit from the left and stores it until a process on its right asks for it. After that, the bit is negated and sent to the right and the buffer becomes free to receive another bit. \[cobits\] Dual to Example \[bits\], we can define $\mathtt{coBitNegate}$:\ Let the signature be $$\Sigma_3:=\\ \mathsf{cobits}=^{1}_{\nu} \&\{ \mathit{b0}:\mathsf{cobits},\ \mathit{b1}:\mathsf{cobits}\}$$ with process $$\begin{aligned} x:\mathsf{cobits} \vdash \mathtt{coBitNegate} :: (y:\mathsf{cobits}) \\ \end{aligned}$$ where $\mathtt{coBitNegate}$ is defined as $$\begin{aligned} & y \leftarrow \mathtt{coBitNegate} \leftarrow x= && \notag \\ & \phantom{small} \mathbf{case}\, Ry\ (\nu_{cobits} {\Rightarrow}\tag{i} && \% \ \textit{receive}\ \nu_{cobits}\ \textit{from right} \\ & \phantom{small spac}\mathbf{case}\, Ry&& \%\ \textit{receive a label} \ \textit{from right} \tag{ii}\\ & \phantom{small space ti} (\ \mathit{b0}{\Rightarrow}Lx.\nu_{cobits}; && \%\ \textit{send}\ \nu_{cobits}\ \textit{ to left} \tag{ii-a}\\ & \phantom{small space times two plu} Lx.b1; && \%\ \textit{send label}\ \mathit{b1} \ \textit{ to left} \notag\\ & \phantom{small space times two plus} y\leftarrow \mathtt{coBitNegate} \leftarrow x && {\color{red}\% \ \textit{recursive call} } \notag \\ & \phantom{small space ti} \mid \mathit{b1} \Rightarrow Lx.\nu_{cobits}; && \% \ \textit{send}\ \nu_{cobits} \ \textit{to left} \tag{ii-b}\\ & \phantom{small space times two plu} Lx.b0; && \% \ \textit{send label}\ \mathit{b0}\ \textit{to left} \notag \\ & \phantom{small space times two plus} y \leftarrow \mathtt{coBitNegate} \leftarrow x)) && {\color{red} \% \ \textit{recursive call}} \notag\end{aligned}$$ $\mathcal{P}_5:=\langle \{\mathtt{coBitNegate}\}, \mathtt{coBitNegate} \rangle$ forms a [*reactive to the right*]{} program over the signature $\Sigma_3$:\ (i) It waits until it receives a [*negative*]{} fixed point unfolding message from the right, (ii) waits for another message from the right to determine the path it shall continue with:\ (a) If the message is a $\ \mathit{b0}\ $ label, (ii-a) the program sends a [*negative*]{} fixed point unfolding message to the left, following by a message of label $\ \mathit{b1}\ $ to the left, and then calls itself recursively and loops back to (i).\ (b) If the message is a $\ \mathit{b1} \ $ label, similarly, (ii-b) the program sends a [*negative*]{} fixed point unfolding message to the left, following by a message of label $\ \mathit{b0}\ $ to the left, and then calls itself and loops back to (i). Computationally, $\mathtt{coBitNegate}$ is a buffer with one bit capacity. In contrast to $\mathtt{BitNegate}$ in Example \[bits\], its types have negative polarity: it receives a bit from the [*right*]{}, and stores it until a process on its [*left*]{} asks for it. After that the bit is negated and sent to the [*left*]{} and the buffer becomes free to receive another bit. \[wait\] The property that assures the reactivity of the previous examples lies in their step (i) in which the program [*waits*]{} for an unfolding message from the left/right, i.e., the program can only continue the computation if it [*receives*]{} a message at step (i), and even after receiving the message it can only take finitely many steps further before the computation ends or another unfolding message is needed. We first develop a naive version of our algorithm which captures the property explained in Remark \[wait\]: Associate an initial integer value (say $0$) to each channel and define the basic step of our algorithm to be *decreasing* the value associated to a channel *by one* whenever it *receives* a fixed point unfolding message. Also, for a reason that is explained later in Remark \[\*\], whenever a channel *sends* a fixed point unfolding message its value is *increased by one*. Then at each recursive call, the value of the left and right channels are compared to their initial value. For instance, in Example \[prex\], in step (i) where the process receives a $\mu_{nat}$ message via the left channel ($x$), the value associated with $x$ is decreased by one, while in steps (ii-a) and (ii-b) in which the process sends a $\mu_{nat}$ message via the right channel ($y$) the value associated with $y$ is increased by one: $$\begin{aligned} \phantom{y \leftarrow \mathtt{Copy} \leftarrow x=} & \phantom{caseLx (\mu_{nat}} & x &\phantom{space} y \\ y \leftarrow \mathtt{Copy} \leftarrow x= & \phantom{caseLx (\mu_{nat}} & 0 &\phantom{space} 0 \\ \phantom{X=}& \mathbf{case}\, Lx\, (\mu_{nat} {\Rightarrow}& {\color{red}-1} &\phantom{space} 0 \\ \phantom{X=} & \phantom{caseLx (\mu_{nat}} \mathbf{case}\, Lx\, ( \mathit{z} \Rightarrow Ry.\mu_{nat}; & -1 &\phantom{space} {\color{red}1} \\ \phantom{X=}& \phantom{caseLx (\mu_{nat} {\Rightarrow}caseLx ( z \Rightarrow} R.z; \mathbf{wait}\, Lx; \mathbf{close}\, Ry & -1 &\phantom{space} 1\\ \phantom{X=}& \phantom{caseLx (\mu_{nat} {\Rightarrow}cas} \mathit{s} {\Rightarrow}Ry.\mu_{nat}; & -1 &\phantom{space} {\color{red}1}\\ \phantom{X=}& \phantom{caseLx (\mu_{nat} {\Rightarrow}caseLx \mathit{s} {\Rightarrow}} Ry.s;y\leftarrow \mathtt{Copy} \leftarrow x)) & {\color{blue}-1} &\phantom{space} {\color{blue}1} \end{aligned}$$ When the recursive call occurs, channel $x$ has the value ${\color{blue}-1} < 0$, meaning that at some point in the computation it received a positive fixed point unfolding message. We can simply compare the value of the list $[x,y]$ lexicographically at the beginning and just before the recursive call: ${\color{blue}[-1,1]}$ being less than $[0,0]$ exactly captures the property observed in Remark \[wait\] for the particular signature $\Sigma_1$. Note that by the definition of $\Sigma_1$, $y$ never receives a fixed point unfolding message, so its value never decreases, and $x$ never sends a fixed point unfolding message, and its value never increases.\ The same criteria works for the program $\mathcal{P}_3$ over the signature $\Sigma_2$ defined in Example \[bits\], since $\Sigma_2$ also contains only one positive fixed point: $$\begin{aligned} \phantom{y \leftarrow \mathtt{BitNegate} \leftarrow} & \phantom{caseLx (\mu_{nat}} & x &\phantom{space} y \\ y \leftarrow \mathtt{BitNegate} \leftarrow x= & \phantom{caseLx (\mu_{nat}} & 0 &\phantom{space} 0 \\ \phantom{X=} & \mathbf{case}\, Lx\ (\mu_{bits} {\Rightarrow}& {\color{red} -1} &\phantom{space} 0 \\ \phantom{X=}& \phantom{caseLx} \mathbf{case}\, Lx\ (b0 \Rightarrow Ry.\mu_{bits};\ & -1 & \phantom{space} {\color{red} 1} \\ \phantom{X=}& \phantom{caseLx (\mu_{nat} \Rightarrow Ry.} Ry.b1; y\leftarrow \mathtt{BitNegate} \leftarrow x & {\color{blue}-1} & \phantom{space} {\color{blue} 1}\\ \phantom{X=} & \phantom{caseLx (\mu_{nat} \Rightarrow (} \mathit{b1} \Rightarrow Ry.\mu_{bits}; & -1 & \phantom{space} {\color{red} 1} \\ \phantom{X=} & \phantom{caseLx (\mu_{nat} \Rightarrow Ry.} Ry.b0; y \leftarrow \mathtt{BitNegate} \leftarrow x)) & {\color{blue} -1} & \phantom{space}{\color{blue} 1} \end{aligned}$$ At both recursive calls the value of the list $[x,y]$ is less than $[0,0]$: ${\color{blue}[-1,1]} < [0,0]$.\ However, for a program defined on a signature with a negative polarity such as the one defined in Example \[cobits\], this condition does not work: & & x & y\ y x= & & 0 & 0\ & Ry (\_[cobits]{} & [0]{} &\ & Ry (b0 Lx.\_[cobits]{};  & [ 1]{} &\ & Lx.b1; y x & [ 1]{} &\ & Lx.\_[cobits]{}; & [1]{} &\ & Lx.b0; y x)) & [ 1]{} & By the definition of $\Sigma_3$, $y$ only receives unfolding fixed point messages, so its value only decreases. On the other hand, $x$ cannot receive an unfolding fixed point from the left and thus its value never decreases. In this case the property in Remark \[wait\] is captured by comparing the initial value of the list $[y,x]$, instead of $[x,y]$, with its value just before the recursive call: ${\color{blue}[-1,1]}<[0,0]$. For a signature with only a single recursive type we can form a list by looking at the polarity of its type such that the value of the channel that receives the unfolding message comes first, and the value of the other one comes second. Priorities in the Local Validity Algorithm {#priority} ========================================== The property given in Remark \[wait\] of previous section is not strict enough, particularly when the signature has more than one recursive type. In that case not all programs that are waiting for a fixed point unfolding message before a recursive call are reactive. \[begen1\] Consider the signature $$\begin{aligned} \Sigma_4:=\ & \mathsf{ack}=^{1}_{\mu} \oplus\{ \mathit{ack}:\mathsf{astream}\},\\ & \mathsf{astream}=^{2}_{\nu} \& \{\mathit {head}: \mathsf{ack}, \ \ \mathit{tail}: \mathsf{astream}\},\\ & \mathsf{nat}=^{3}_{\mu}\oplus \{\mathit{z}:1,\ \ \mathit{s}: nat\} \end{aligned}$$ $\mathsf{ack}$ is a type with [*positive*]{} polarity that, upon unfolding, sends an [*acknowledgment*]{} message to the right (or receives it from the left). $\mathsf{astream}$ is a type with [*negative*]{} polarity of a potentially infinite stream where its $\mathit{head}$ is always followed by an acknowledgement. $\mathcal{P}_6:=\langle \{ \mathtt{Ping}, \mathtt{Pong}, \mathtt{PingPong}\}, \mathtt{PingPong} \rangle$ forms a program over the signature $\Sigma_4$ with the typing of its processes $$\begin{array}{l} x:\mathsf{nat} \vdash \mathtt{Ping} :: (w:\mathsf{astream}) \\ w:\mathsf{astream} \vdash \mathtt{Pong}:: (y:\mathsf{nat}) \\ x:\mathsf{nat} \vdash \mathtt{PingPong}:: (y:\mathsf{nat}) \end{array}$$ We define processes $\mathtt{Ping}$, $\mathtt{Pong}$, and $\mathtt{PingPong}$ over $\Sigma_4$ as: $$\begin{aligned} & y \leftarrow \mathtt{PingPong} \leftarrow x=&& \notag\\& \phantom{smal} w \leftarrow \mathtt{Ping} \leftarrow x ; \tag{i} && \% \ \textit{spawn process}\ \mathtt{Ping} \\& \phantom{small space} y \leftarrow \mathtt{Pong} \leftarrow w && {\color{red}\% \ \textit{continue with a tail call}}\notag\\ \notag\\ & y\leftarrow \mathtt{Pong} \leftarrow w=&& \notag\\ & \phantom{smal} Lw.\nu_{astream};&& \% \ \textit{send}\ \mathit{\nu_{astream}} \ \textit{to left} \tag{ii-$\mathtt{Pong}$}\\& \phantom{small s} Lw.\mathit{head};&& \% \ \textit{send label}\ \mathit{head} \ \textit{to right} \tag{iii-$\mathtt{Pong}$}\\ & \phantom{small sp} \mathbf{case}\, Lw\ (\mu_{ack} {\Rightarrow}&& \% \ \textit{receive}\ \mathit{\mu_{ack}} \ \textit{from left} \tag{iv-$\mathtt{Pong}$}\\ &\phantom{small space } \mathbf{case}\, Lw\ ( && \% \ \textit{receive a label from left} \notag \\ & \phantom{small space times two}ack \Rightarrow Ry.\mu_{nat};&&\% \ \textit{send}\ \mathit{\mu_{nat}} \ \textit{to right} \notag\\ & \phantom{small space times two plu} Ry.s; && \% \ \textit{send label}\ \mathit{s} \ \textit{to right} \notag \\ & \phantom{small space times two plus o} y \leftarrow \mathtt{Pong} \leftarrow w))&& {\color{red} \% \ \textit{recursive call}} \notag \end{aligned}$$ $$\begin{aligned} & w \leftarrow \mathtt{Ping} \leftarrow x= && \notag \\ & \phantom{smal} \mathbf{case}\, Rw\ (\nu_{astream} {\Rightarrow}&& \% \ \textit{receive } \mathit{\nu_{astream}} \textit{from right} \tag{ii-$\mathtt{Ping}$}\\ & \phantom{small sp} \mathbf{case}\, Rw\ ( &&\notag \% \ \textit{receive a label from right} \\ &\phantom{small space times} \mathit{head} \Rightarrow Rw.\mu_{ack};&& \% \ \textit{send}\ \mu_{ack} \ \textit{to right} \tag{iii-$\mathtt{Ping}$}\\ & \phantom{small space times two plus on }Rw.\mathit{ack}; \notag && \% \ \textit{send label}\ \mathit{ack}\ \textit{to right}\\ & \phantom{small space times two plus one an } w\leftarrow \mathtt{Ping}\leftarrow x && {\color{red} \% \ \textit{recursive call}} \notag \\ & \phantom{small space times } \mid \mathit{tail} \Rightarrow w \leftarrow \mathtt{Ping}\leftarrow x)) \notag&& {\% \ \textit{recursive call}}\end{aligned}$$ \(i) Program $\mathcal{P}_6$ starting from $\mathtt{PingPong}$, spawns a new process $\mathtt{Ping}$ and continues as $\mathtt{Pong}$:\ (ii-$\mathtt{Pong}$) Process $\mathtt{Pong}$ sends an $\mathsf{astream}$ unfolding and then a $\mathit{head}$ message to the left, and then (iii-$\mathtt{Pong}$) [*waits*]{} for an acknowledgment, i.e., $\mathit{ack}$, from the left.\ (ii-$\mathtt{Ping}$) At the same time process $\mathtt{Ping}$ [*waits*]{} for an $\mathsf{astream}$ fixed point unfolding message from the right, which becomes available after step (ii-$\mathtt{Pong}$). Upon receiving the message, it waits for receiving either $\mathit{head}$ or $\mathit{tail}$ from the right, which is also available from (ii-$\mathtt{Pong}$) and is actually a $\mathit{head}$. So (iii-$\mathtt{Ping}$) it continues with the path corresponding to $\mathit{head}$, and acknowledges receipt of the previous messages by sending an unfolding messages and the label $\mathit{ack}$ to the right, and then it calls itself (ii-$\mathtt{Ping}$).\ (iv-$\mathtt{Pong}$) Process $\mathtt{Pong}$ now receives the two messages sent at (iii-$\mathtt{Ping}$) and thus can continue by sending a $\mathsf{nat}$ unfolding message and the label $\mathit{s}$ to the right, and finally calling itself (ii-$\mathtt{Pong}$).\ Although both recursive processes $\mathtt{Ping}$ and $\mathtt{Pong}$ at some point [*wait*]{} for a fixed point unfolding message, this program runs infinitely without receiving any messages from the outside, and thus is not reactive. The back-and-forth exchange of fixed point unfolding messages between two processes in the previous example can arise when at least two mutually recursive types with different polarities are in the signature. This is why we need to incorporate priorities of the type variables into the validity checking algorithm. \[\*\] In Example $\ref{begen1}$, for instance, we can add the condition that the [*wait*]{} in step (ii-$\mathtt{Ping}$) on [*receiving*]{} an unfolding message $\nu_{astream}$ for a type variable with priority $2$ is not valid anymore after step (iii-$\mathtt{Ping}$), since the process [*sends*]{} an unfolding message $\mu_{ack}$ for a type variable with a higher priority (priority $1$) at step (iii-$\mathtt{Ping}$). To include such a condition in our algorithm we form a list for each process. This list stores the information of the fixed point unfolding messages that the process received and sent before a recursive call for each type variable in their order of priority. \[c\] Consider the signature and program $\mathcal{P}_6$ as defined in Example \[begen1\]. For the process $x:\mathsf{nat} \vdash w \leftarrow \mathtt{Ping} \leftarrow x:: (w:\mathsf{astream})$ form the list $$[\mathsf{ack}-\mathit{received}, \mathsf{ack}-\mathit{sent}, \mathsf{astream}-\mathit{received}, \mathsf{astream}-\mathit{sent}, \mathsf{nat}-\mathit{received}, \mathsf{nat}-\mathit{sent} ].$$ Types with positive polarity, i.e., $\mathsf{ack}$ and $\mathsf{nat}$, receive messages from the left channel ($x$) and send messages to the right channel ($w$), while those with negative polarity, i.e., $\mathsf{astream}$, receive from the right channel ($w$) and send to the left one ($x$). Thus, the above list can be rewritten as $$[x_\mathsf{ack}, w_{\mathsf{ack}}, w_{\mathsf{astream}}, x_{\mathsf{astream}}, x_\mathsf{nat}, w_{\mathsf{nat}}].$$ To keep track of the sent/received messages, we start with $[0,0,0,0,0,0]$ as the value of the list, when the process $x:\mathsf{nat} \vdash \mathtt{Ping} :: (w:\mathsf{astream})$ is first spawned. Then, similar to what we had in the naive version of the algorithm, on the steps in which the process [*receives*]{} a fixed point unfolding message, the value of the corresponding element of the list is [*decreased by one*]{}. And on the steps it [*sends*]{} a fixed point unfolding message, the corresponding value is [*increased by one*]{}:\ $$\begin{aligned} & w \leftarrow \mathtt{Ping} \leftarrow x= & \phantom{caseRw} [0,0 \ ,0\ ,0, 0, 0]\\ & \phantom{\mathtt{Ping}=} \mathbf{case}\, Rw\ (\nu_{astream} {\Rightarrow}& [0,0,{\color{red}-1},0, 0 ,0] \\ & \phantom{caseRw (\nu_{astream})} \mathbf{case}\, Rw\ ( \mathit{head} \Rightarrow Rw.\mu_{ack}; & [0,{\color{red}1},-1,0, 0, 0]\\ & \phantom{caseRw (\nu_{astream} {\Rightarrow}caseRw ( head \ \ } Rw.\mathit{ack}; w\leftarrow \mathtt{Ping} \leftarrow x & {\color{blue} [0,1,-1,0, 0, 0]}\\ & \phantom{caseRw (\nu_{astream} {\Rightarrow}casex} \mid\mathit{tail} {\Rightarrow}w\leftarrow \mathtt{Ping} \leftarrow x))& [0,0,-1,0, 0, 0] \end{aligned}$$ The two last lines are the values of the list on which process $\mathtt{Ping}$ calls itself recursively. The validity condition as described in Remark \[\*\] holds iff the value of the list at the time of the recursive call is less than the value the process started with, in the lexicographical order. Here, for example, ${\color{blue} [0,1,-1,0]} \not < [0,0,0,0]$, and the validity condition does not hold for this recursive call. The following definition captures the idea of forming lists described above. \[list\] For a process $$\bar{x}:\omega \vdash P :: (y:B),$$ over the signature $\Sigma$, define $list(\bar{x},y)= [v_i]_{\ i \le n}$ such that 1. $v_i= ( \bar{x}_i, y_i )$ if $\epsilon(i)= \mu$, and 2. $v_i= (y_i, \bar{x}_i)$ if $\epsilon(i)=\nu,$ where $n$ is the maximum priority in $\Sigma.$ In the remainder of this section we use $n$ to denote the maximum priority in $\Sigma$. \[easy\] Consider the signature $\Sigma_1$ and program $\mathcal{P}_3:= \langle \{\mathtt{Copy}\}, \mathtt{Copy} \rangle$, in Example \[prex\]:\ $ \Sigma_1 := \mathsf{nat}=^{1}_{\mu} \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\},$ and $$\begin{aligned} y \leftarrow \mathtt{Copy} \leftarrow x = \mathbf{case}\,Lx\ (\mu_{nat} {\Rightarrow}\mathbf{case}\, Lx\ &(\ z{\Rightarrow}Ry.\mu_{nat}; Ry.z; \mathbf{wait}\, Lx; \mathbf{close}\, Ry \\ & \mid s \Rightarrow Ry.\mu_{nat}; Ry.s; y \leftarrow \mathtt{Copy} \leftarrow x)) \end{aligned}$$ By Definition \[list\], for process $ x:\mathsf{nat} \vdash \mathtt{Copy} :: (y:\mathsf{nat})$, we have $n=1$, and $list(x,y)= [(x_1,y_1)]$ since $\epsilon(1)= \mu$. Just as for the naive version of the algorithm, we can trace the value of $list(x,y)$: $$\begin{aligned} y \leftarrow \mathtt{Copy} \leftarrow x= & \phantom{caseLx (\mu_{nat}} & [0,0]\\ \phantom{X=}& \mathbf{case}\, Lx\ (\mu_{nat} {\Rightarrow}& [{\color{red}-1},0] \\ \phantom{X=} & \phantom{caseLx (\mu_{nat}} \mathbf{case}\, Lx\ ( \mathit{z} \Rightarrow Ry.\mu_{nat}; & [-1,{\color{red}1}]\\ \phantom{X=}& \phantom{caseLx (\mu_{nat} {\Rightarrow}caseLx ( z \Rightarrow} R.z; \mathbf{wait}\, Lx; \mathbf{close}\, Ry & { [-1,1]}\\ \phantom{X=}& \phantom{caseLx (\mu_{nat} {\Rightarrow}case} \mid \mathit{s} {\Rightarrow}Ry.\mu_{nat}; & [-1,{\color{red}1}]\\ \phantom{X=}& \phantom{caseLx (\mu_{nat} {\Rightarrow}caseLx \mathit{s} {\Rightarrow}} Ry.s;y\leftarrow \mathtt{Copy} \leftarrow x & {\color{blue}[-1,1]} \end{aligned}$$ Here, ${\color{blue}[-1,1]} <[0,0]$ and the recursive call is classified as valid. Sometimes we are interested in a prefix of the list from Definition \[list\]. We give the following definition of $list(x,y,j)$ to crop $list(x,y)$ exactly before the element corresponding to a [*sent*]{} fixed point unfolding message of types with priority $j$. These prefixes are used later in Definition \[defmain\]. For a process $$\bar{x}:\omega \vdash P :: (y:B)$$ over signature $\Sigma$, and $0 \le j\le n$, define $list(\bar{x},y,j)$, as a prefix of the list $list(\bar{x},y)=[v_i]_{i \le n}$ by 1. $[]$ if $i=0$, 2. $[ [v_i]_{ i < j},\ (\bar{x}_j)]$ if $\epsilon(j)=\mu$, 3. $[[v_i]_{ i < j},\ (y_j)]$ if $\epsilon(j)=\nu$. Consider the signature introduced in Example \[begen1\] $$\begin{aligned} \Sigma_4 :=\ & \mathsf{ack}=^{1}_{\mu} \oplus\{ \mathit{ack}:\mathsf{astream}\},\\ & \mathsf{astream}=^{2}_{\nu} \& \{\mathit {head}: \mathsf{ack}, \ \ \mathit{tail}: \mathsf{astream}\},\\ & \mathsf{nat}=^{3}_{\mu}\oplus \{\mathit{z}:1,\ \ \mathit{s}: nat\} \end{aligned}$$ and program $\mathcal{P}_6:=\langle \{ \mathtt{Ping}, \mathtt{Pong}, \mathtt{PingPong}\}, \mathtt{PingPong} \rangle$.\ For process $x:\mathsf{nat} \vdash \mathtt{Ping} :: (w:\mathsf{astream})$: $$\begin{array}{l} list(x,w)=[(x_1,w_1),(w_2,x_2), (x_3,w_3)],\\ list(x,w,3)=[(x_1,w_1),(w_2, x_2), (x_3)],\\ list(x,w,2)=[(x_1,w_1),(w_2)],\\ list(x,w,1)=[(x_1)],\ \mathit{and}\\ list(x,w,0)=[]. \end{array}$$ To capture the idea of [*decreasing*]{}/[*increasing*]{} the value of the elements on $list (\_,\_)$ by [*one*]{}, as depicted in Example \[c\] and Example \[easy\], we assume that a channel transforms to a new generation of itself after sending or receiving a fixed point unfolding message. \[su\] Process $ x:\mathsf{nat} \vdash y\leftarrow \mathtt{Copy} \leftarrow x :: (y:\mathsf{nat})$ in Example \[easy\] starts its computation with the initial generation of its left and right channels: $$x^0:\mathsf{nat} \vdash y^0 \leftarrow \mathtt{Copy} \leftarrow x^0 :: (y^0:\mathsf{nat}).$$ The channels evolve as the process sends or receives a fixed point unfolding message along them: $$\begin{aligned} y^0 \leftarrow \mathtt{Copy}\leftarrow x^0 & = \phantom{caseL{x^0} (\mu_{nat}} & \\ \phantom{X=}& \mathbf{case}\, L{x^0}\ (\mu_{nat} {\Rightarrow}& {\color{red} x^0 \rightsquigarrow x^1} \\ \phantom{X=} & \phantom{caseL {x^1} (\mu_{nat}} \mathbf{case}\, L{x^1}\ (\mathit{z} \Rightarrow R{y^0}.\mu_{nat};& {\color{red} y^0 \rightsquigarrow y^1} \\ \phantom{X=}& \phantom{caseL{x^1} (\mu_{nat} {\Rightarrow}caseL{x^1} ( z \Rightarrow} R{y^1}.z; \mathbf{wait}\, L{y^1}; \mathbf{close}\, R{x^1}& \\ \phantom{X=}& \phantom{caseL{x^0} (\mu_{nat} {\Rightarrow}case} \mid \mathit{s} {\Rightarrow}R{y^0}.\mu_{nat}; & {\color{red} y^0 \rightsquigarrow y^1}\\ \phantom{X=}& \phantom{caseL{x^0} (\mu_{nat} {\Rightarrow}caseL{x^1} \mathit{s} {\Rightarrow}} R{y^1}.s;y^1 \leftarrow \mathtt{Copy} \leftarrow x^1)) \end{aligned}$$ On the last line the process $$x^1:\mathsf{nat} \vdash y^1 \leftarrow \mathtt{Copy} \leftarrow x^1 :: (y^1:\mathsf{nat})$$ is called recursively with a new generation of variables. In the inference rules introduced in Section \[rules\], instead of recording the absolute value of each element of the $list(\ \_,\_)$ as we did in Example \[c\] and Example \[easy\], we introduce an extra set $\Omega$ that stores the relation between different generations of a channel indexed by their priority of types. Generally speaking, $x^{\alpha+1}_{i}<x^{\alpha}_{i}$ is added to $\Omega$, when $x^\alpha$ [*receives*]{} a fixed point unfolding message of a type with priority $i$ and transforms to $x^{\alpha+1}$. This corresponds to the [*decrease by one*]{} in the previous examples. If $x^\alpha$ [*sends*]{} a fixed point unfolding message of a type with priority $i$ and evolves to $x^{\alpha+1}$, $x^{\alpha}_i$ and $x^{\alpha+1}_i$ are considered to be incomparable in $\Omega$. This corresponds to [*increase by one*]{} in the previous examples, since for the sake of comparing $list(\ \_,\_)$ at the [*first call*]{} of a process and just before a [*recursive call*]{} in a lexicographic order, there is no difference whether $x^{\alpha+1}$ is greater than $x^\alpha$ or incomparable to it. When $x^\alpha$ receives/sends a fixed point unfolding message of a type with priority $i$ and transforms to $x^{\alpha+1}$, for any type with priority $j\neq i$, the value of $x^{\alpha}_j$ and $x^{\alpha+1}_j$ must remain equal. In these steps, we add $x^{\alpha}_j = x^{\alpha+1}_j$ for $j\neq i$ to $\Omega$. A process in the formalization of the intuition above is therefore typed as $$x^\alpha:A \vdash_{\Omega} P :: (y^{\beta}:B),$$ where $x^\alpha$ is the $\alpha$-th generation of channel $x$. The relation between the channels indexed by their priority of types is built step by step in $\Omega$ and represented by $\le$. The reflexive transitive closure of $\Omega$ forms a partial order $\le_{\Omega}$. We extend $\le_{\Omega}$ to the [*list*]{} of channels indexed by the priority of their types considered lexicographically. We may omit subscript $\Omega$ from $\le_{\Omega}$ whenever it is clear from the context. Mutual Recursion in the Local Validity Condition {#mutual} ================================================ In examples of previous sections, the recursive calls were not *mutual*. In the general case, a process may call any other process variable in the program, and this call can be mutually recursive. In this section, we incorporate mutual recursive calls into our algorithm. \[simplem\] Recall signature $\Sigma_4$ from Example \[begen1\] $$\begin{aligned} \Sigma_4:=\ & \mathsf{ack}=^{1}_{\mu} \oplus\{ \mathit{ack}:\mathsf{astream}\},\\ & \mathsf{astream}=^{2}_{\nu} \& \{\mathit {head}: \mathsf{ack}, \ \ \mathit{tail}: \mathsf{astream}\},\\ & \mathsf{nat}=^{3}_{\mu}\oplus \{\mathit{z}:1,\ \ \mathit{s}: nat\} \end{aligned}$$ Define program $\mathcal{P}_7= \langle \{\mathtt{Idle}, \mathtt{Producer},\}, \mathtt{Producer} \rangle$, where $$\begin{array}{l} z:\mathsf{ack} \vdash w \leftarrow \mathtt{Idle}\leftarrow z :: (w:\mathsf{nat})\\ x:\mathsf{astream} \vdash y \leftarrow \mathtt{Producer} \leftarrow x :: (y:\mathsf{nat}), \end{array}$$ and processes $\mathtt{Idle}$ ($\mathtt{I}$) and $\mathtt{Producer}$ (or simply $\mathtt{P}$ ) are defined as: $$\begin{array}{l} w \leftarrow \mathtt{I}\leftarrow z = \mathbf{case}\, Lz^{\alpha}\ (\mu_{ack} {\Rightarrow}\mathbf{case}\, L{z^{\alpha+1}}\ ( \mathit{ack}{\Rightarrow}Rw^{\beta}.\mu_{nat}; Rw^{\beta+1}.\mathit{s}; y^{\beta+1} \leftarrow \mathtt{P}\leftarrow x^{\alpha+1})) \\[1ex] y \leftarrow \mathtt{P} \leftarrow x = Lx^{\alpha}.\nu_{astream}; Lx^{\alpha+1}. \mathit{head}; y^{\beta} \leftarrow \mathtt{I}\leftarrow x^{\alpha+1}. \end{array}$$ We calculusate $list(x,y)= [(x_1,y_1), (y_2,x_2), (x_3,y_3)]$ and $list(z,w)= [(z_1,w_1), (w_2,z_2), (z_3,w_3)]$ since $\epsilon(1)=\epsilon(3)= \mu$ and $\epsilon(2)= \nu$. By analyzing the behavior of this program step by step, we see that it is a [*reactive*]{} program that counts the number of acknowledgements received from the left. The program starts with the process $ x^0:\mathsf{astream} \vdash_{\emptyset} y^0 \leftarrow \mathtt{Producer} \leftarrow x^0 :: (y^0:\mathsf{nat}) $. It first sends one message to left to unfold the [*negative*]{} fixed point type, and its left channel evolves to a next generation. Then another message is sent to the left to request the $\mathit{head}$ of the stream and after that it calls process $y^0 \leftarrow \mathtt{Idle} \leftarrow x^1$. $$\begin{aligned} & y^0 \leftarrow \mathtt{Producer}\leftarrow x^0= & \phantom{Lx^0.\nu_{astream}; } \phantom{L.\nu } [0,0,0,0,0,0]& \phantom{L.\nu }\\ &\phantom{sma} Lx^0.\nu_{astream}; & \phantom{L.\nu } [0,0,0,1,0,0] & \phantom{L.\nu } x^1_1=x^0_1, x^1_3=x^0_3 \\ & \phantom{small sp} Lx^1.\mathit{head};y^0 \leftarrow \mathtt{Idle} \leftarrow x^1 &\phantom{L.\nu } {\color{blue} [0,0,0,1,0,0]}& \end{aligned}$$ Process $x^1:\mathsf{ack} \vdash y^0 \leftarrow \mathtt{Idle} \leftarrow x^1 :: (y^0:\mathsf{nat}) $, then [*waits*]{} to receive an acknowledgment from the left via a [*positive*]{} fixed point unfolding message for $\mathsf{ack}$ and its left channel transforms to a new generation upon receiving it. Then it waits for the label $\mathit{ack}$, and upon receiving it, it sends one message to the right to unfold the [*positive*]{} fixed point $\mathsf{nat}$ (and this time the right channel evolves). Then it sends the label $\mathit{s}$ to the right and calls $y^1 \leftarrow \mathtt{Producer}\leftarrow x^2$ recursively: $$\begin{aligned} & y^0 \leftarrow \mathtt{Idle}\leftarrow x^1= & \phantom{caseLx^1} {\color{blue}[0, 0,\ 0,\ 1,\ 0, 0]} & \phantom{L}\\ &\phantom{sma} \mathbf{case}\, Lx^1\ (\mu_{ack} {\Rightarrow}& [-1,0,0,1,0,0] & \phantom{L} x^2_1 < x^1_1, x^2_2=x^1_2, x^2_3=x^1_3\\ & \phantom{small sp} \mathbf{case}\, Lx^2\ ( \mathit{ack} \Rightarrow Ry^0.\mu_{nat}; & [-1,0,0,1,0,1] & \phantom{Lx^0.\nu} y^1_1=y^0_1, y^1_2=y^0_2\\ & \phantom{small space } Ry^1.\mathit{s}; y^1 \leftarrow \mathtt{Producer} \leftarrow x^2)) & {\color{red} [-1,0,0,1,0,1]} &\phantom{L.}\\\end{aligned}$$ Observe that the actual recursive call for $\mathtt{Producer}$ occurs at the red line above, where $\mathtt{Producer}$ eventually calls itself. At that point the absolute value of $list(x^2,y^1)$ is recorded as ${\color{red} [-1,0,0,1,0,1]}$, which is less than the absolute value of $list(x^0,y^0)$ when $\mathtt{Producer}$ was called for the first time: $${\color{red} [-1,0,0,1,0,1]} < {[0,0,0,0,0,0]}.$$ The same observation can be made by considering the relations introduced in the last column $${\color{red}list(x^2,y^1)}={\color{red} [(x^2_1,y^1_1), (y^1_2,x^2_2), (x^2_3,y^1_3)]}< [(x^0_1,y^0_1), (y^0_2,x^0_2), (x^0_3,y^0_3)]= list(x^0,y^0)$$ since $x^2_1 < x^1_1= x^0_1$. This recursive call is valid regardless of the fact that ${\color{blue} [0,0,0,1,0,0]} \not < {[0,0,0,0,0,0]},$ i.e. $${\color{blue}list(x^1,y^0)}={\color{blue} [(x^1_1,y^0_1), (y^0_2,x^1_2), (x^1_3,y^0_3)]}\not < [(x^0_1,y^0_1), (y^0_2,x^0_2), (x^0_3,y^0_3)]= list(x^0,y^0)$$ since $x^1_1= x^0_1$ but $x^1_2$ is incomparable to $x^0_2$. Similarly, we can observe that the actual recursive call on $\mathtt{Idle}$, where $\mathtt{Idle}$ eventually calls itself, is valid. To account for this situation, we introduce an order on [*process variables*]{} and trace the last seen variable on the path leading to the recursive call. In this example we define $\mathtt{Idle}$ to be less than $\mathtt{Producer}$ at position $2$ ($\mathtt{I} \subset_{2} \mathtt{P}$), i.e.: > We incorporate process variables $\mathtt{Producer}$ and $\mathtt{Idle}$ to the lexicographical order of $list(\_,\_)$ such that their values are placed exactly before the element in the list corresponding to the [*sent*]{} unfolding messages of the type with priority $2$. We now trace the ordering as follows: $$\begin{aligned} & y^0 \leftarrow \mathtt{Producer}\leftarrow x^0= & \phantom{L } [0,0,0,\mathtt{P},0,0,0] & \phantom{L.\nu } \\ & \phantom{small } Lx^0.\nu_{astream}; & \phantom{L } [0,0,0,\mathtt{P},1,0,0] & \phantom{Lx^0.\nu } x^1_1=x^0_1, x^1_3=x^0_3 \\ & \phantom{Lx^0.\nu_{strea}} Lx^1.\mathit{head};y^0 \leftarrow \mathtt{Idle} \leftarrow x^1 & \phantom{L } {\color{blue} [0,0,0,\mathtt{I},1,0,0]} & \phantom{L.\nu }\\ \\ & y^0 \leftarrow \mathtt{Idle} \leftarrow x^1= & \phantom{L} {\color{blue}[0,0,0,\mathtt{I},1,0,0]} & \phantom{L.\nu} \\ & \phantom{small} \mathbf{case} \, Lx^1 (\mu_{ack} {\Rightarrow}& [-1,0,0,\mathtt{I},1,0,0] & \phantom{L.\nu} x^2_1<x^1_1, x^2_2=x^1_2, x^2_3=x^1_3\\ & \phantom{small spa } \mathbf{case} \,Lx^2 ( \mathit{ack} \Rightarrow Ry^0.\mu_{nat}; & [-1,0,0,\mathtt{I},1,0,1] & \phantom{L.\nu} y^1_1=y^0_1, y^1_2=y^0_2 \\ & \phantom{caseeLx^1 ( label } Ry^1.\mathit{s}; y^1 \leftarrow \mathtt{Producer} \leftarrow x^2 & {\color{red} [-1,0,0,\mathtt{P},1,0,1]} & \\\end{aligned}$$ ${\color{red} [-1,0,0,\mathtt{P},1,0,1]} < {[0,0,0,\mathtt{I},1,0,0]}$ and ${\color{blue} [0,0,0, \mathtt{I}, 1,0,0]} < {[0,0,0,\mathtt{P},0,0,0]}$ hold, and both mutually recursive calls are recognized to be valid, as they are, without a need to search for actual recursive calls, i.e., where a process calls itself. However, not every relation over the process variables forms a partial order. For instance, having both $\mathtt{P} \subset_{2} \mathtt{I}$ and $\mathtt{I} \subset_{2} \mathtt{P}$ violates the antisymmetry condition. Introducing the position of process variables into the $list(\_,\_)$ is also a delicate issue. For example, if we have both $\mathtt{I} \subset_{1} \mathtt{P}$ and $\mathtt{I} \subset_{2} \mathtt{P}$, it is not determined where to insert the value of $\mathtt{Producer}$ and $\mathtt{Idle}$ on the $list(\_,\_)$. Definition \[subs\] captures the idea of Example \[simplem\], while ensuring that the introduced relation is always a well-defined order and it is determined on which position of $list(\_,\_)$ the process variables shall be inserted. Definition \[defmain\] gives the lexicographic order on $list(\_,\_)$ augmented with the $\subseteq$ relation. \[subs\] For the signature $\Sigma$ and each $0 \le i \le n $, let $\subseteq_{\ i}$ be a relation on process variables in $V$ that satisfies the following conditions, where (a) $X =_{\ i}Y$ iff $X \subseteq_{\ i} Y$ and $Y \subseteq_{\ i} X$, and (b) $X \subset_{\ i}Y$ iff $X\subseteq_{i}Y$ but $X \neq_{\ i} Y$: 1. $(\forall i\le n)\ (\forall F\in V)$ $F=_{i}F$, \[or1\] 2. $(\forall i\le n)\ (\forall F,G\in V)$ if $F\subseteq_{i}G$ and $G \subseteq_{i}F$, then $F=_{i}G$,\[or2\] 3. $(\forall i\le n)\ (\forall F,G\in V)$ if $F\subseteq_{i}G$ and $G \subseteq_{i}H$, then $F\subseteq_{i}H$, \[or3\] 4. $(\forall i,j \le n)\ (\forall F,G,H \in V)$ if $F \subseteq_{i} G$ and $G\subseteq_{j}H$, then $i=j$,\[4\] 5. $(\forall i,j \le n)\ (\forall F,G \in V)$ if $F \subset_{i} G$ and $F \subset_{j} G$ then $j=i$. \[3\] 6. $(\forall i\le n)\ (\forall F,G \in V)$ if $F =_{i} G$, then $(\forall j\le n)$ $F=_{j}G$,\[2\] 7. $\forall F, G \in V$, either $\exists i\le n,F\subseteq_{i} G $ or $\exists i\le n,G\subseteq_{i}F $.\[1\] Conditions \[or1\]-\[or3\], guarantee that $\subseteq_{i}$ is a partial order, and thus $\subset_{i}$ is the strict partial order defined upon $\subseteq_{i}$. Define $\subseteq$ to be $\bigcup_{i \le n} \subseteq_i$, i.e. $F \subseteq G$ iff $F \subseteq_{i} G$ for some $i \le n$. Observe that $\subseteq$ is a total (condition \[1\]) order: it is reflexive (conditions \[4\] and \[or1\]), antisymmetric (conditions \[4\] and \[or2\]), and transitive (conditions \[4\] and \[or3\]). And $\subset$ is the strict portial order derived from it. \[defmain\] Using $\subset$ and $\le$, we define a new combined order $(\subset, <)$ (used in the local validity condition in Section \[alg\]). $$F,list(\bar{x},y)\ (\subset, <)\ G,list(\bar{z},w)$$ iff 1. If $F \subset G$, i.e., $F \subset_i G$ for a unique $i$ (by condition \[3\] in Definition \[subs\]), then $list(\bar{x},y,i) \le list(\bar{z},w, i)$, otherwise, 2. if $G \subset F$, i.e. $G \subset_i F$ for a unique $i$ (by condition \[3\] in Definition \[subs\]), then $list(\bar{x},y,i) < list(\bar{z},w,i)$, otherwise 3. $list(\bar{x},y) < list(\bar{z},w)$. By conditions \[3\] and \[2\] in Definition \[subs\], $(\subset, <)$ is an irreflexive and transitive relation and thus a strict partial order. \[algn\] A Modified Rule for Cut {#modcut} ======================= There is a subtle aspect of local validity that we have not discussed yet. We need to relate a fresh channel, created by spawning a new process, with the previously existing channels. Process $(x\leftarrow P_x; Q_x)$, for example, creates a fresh channel $w^0$, spawns process $[w^0/x]P_x$ providing along channel $w^0$, and then continues as $[w^0/x]Q_x$. For the sake of our algorithm, we need to identify the relation between $w^0$, $x$, and $y$. Since $w^0$ is a fresh channel, a naive idea is to make $w^0$ incomparable to any other channel for any type variable $t\in \Sigma$.[^1] While sound, we will see in Example \[cutchannel\] that we can improve on this naive approach to cover more valid processes. \[cutchannel\] Define the signature $$\begin{aligned} \Sigma_5:=\ & \mathsf{ctr}=^{1}_{\nu} \& \{\mathit {inc}: \mathsf{ctr}, \ \ \mathit{val}: \mathsf{bin}\},\\ & \mathsf{bin}=^{2}_{\mu} \oplus\{ \mathit{b0}:\mathsf{bin}, \mathit{b1}:\mathsf{bin}, \mathit{\$}:\mathsf{1}\}.\\ \end{aligned}$$ which provides numbers in binary representation as well as an interface to a counter. We explore the following program $\mathcal{P}_8= \langle \{\mathtt{BinSucc},\mathtt{Counter},\mathtt{NumBits},\mathtt{BitCount}\}, \mathtt{BitCount} \rangle,$ where $$\begin{array}{l} x:\mathsf{bin} \vdash y \leftarrow \mathtt{BinSucc} \leftarrow x :: (y:\mathsf{bin})\\ x:\mathsf{bin} \vdash y \leftarrow \mathtt{Counter}\leftarrow x :: (y:\mathsf{ctr})\\ x:\mathsf{bin} \vdash y \leftarrow \mathtt{NumBits} \leftarrow x :: (y:\mathsf{bin}) \\ x:\mathsf{bin} \vdash y \leftarrow \mathtt{BitCount}\leftarrow x :: (y:\mathsf{ctr})\\ \end{array}$$ We define the relation $\subset$ on process variables as $\mathtt{BinSucc} \subset_{0} \mathtt{Counter} \subset_{0} \mathtt{NumBits} \subset_{0} \mathtt{BitCount}$. The process definitions are as follows, shown here already with their termination analysis. $$\begin{aligned} & w^{\beta} \leftarrow \mathtt{BinSucc}\leftarrow z^{\alpha}= & \phantom{caseLz^\alpha} {\color{blue}[0, 0,\ 0,\ 0]} & \phantom{L}\\ &\phantom{sma} \mathbf{case}\, Lz^{\alpha}\ (\mu_{bin} {\Rightarrow}& [0,0,-1,0] & \phantom{L} z^{\alpha+1}_1=z^\alpha_1, z^{\alpha+1}_2 < z^{\alpha}_2\\ & \phantom{small sp} \mathbf{case}\, Lz^{\alpha+1}\ ( \mathit{b0} \Rightarrow Rw^{\beta}.\mu_{bin}; & {[0,0,-1,1]} & { \phantom{Lz^0.\nu} w^{\beta+1}_1=w^{\beta}_1}\\ & \phantom{small space more than that} Rw^{\beta+1}.\mathit{b1}; w^{\beta+1} \leftarrow z^{\alpha+1} & {[0,0,-1,1]} & \\ & \phantom{small space morexx} \mid \mathit{b1} \Rightarrow Rw^{\beta}.\mu_{bin}; & {[0,0,-1,1]} & { \phantom{Lz^0.\nu} w^{\beta+1}_1=w^{\beta}_1}\\ & \phantom{small space more than that} Rw^{\beta+1}.\mathit{b0}; w^{\beta+1} \leftarrow \mathtt{BinSucc} \leftarrow z^{\alpha+1} & { \color{red}[0,0,-1,1]} & \\ & \phantom{small space morexx} \mid \mathit{\$}\Rightarrow Rw^{\beta}.\mu_{\mathsf{bin}}; Rw^{\beta+1}.b1;\ & { [0,0,-1,1]} & { \phantom{Lz^0.\nu} w^{\beta+1}_1=w^{\beta}_1}\\ & \phantom{small space more than that}Rw^{\beta+1}.\mu_{\mathsf{bin}}; Rw^{\beta+2}.\$;\ w^{\beta+2} \leftarrow z^{\alpha+1})) & { [0,0,-2,2]} & { \phantom{Lz^0.\nu} w^{\beta+2}_1=w^{\beta+1}_1}\\\\ & y^{\beta} \leftarrow \mathtt{Counter}\leftarrow w^{\alpha}= & \phantom{caseRy^\beta} {\color{blue}[0, 0,\ 0,\ 0]} & \phantom{L}\\ &\phantom{sma} \mathbf{case}\, Ry^{\beta}\ (\nu_{ctr} {\Rightarrow}& [-1,0,0,0] & \phantom{L} y^{\beta+1}_1<y^\beta_1, y^{\beta+1}_2 = y^{\beta}_2\\ & \phantom{small sp} \mathbf{case}\, Ry^{\beta+1}\ ( \mathit{inc} \Rightarrow z^{0} \leftarrow \mathtt{BinSucc} \leftarrow w^{\alpha}; & &{\mathtt{BinSucc} \subset_{0} \mathtt{Counter}} \\ & \phantom{small space more than that} y^{\beta+1} \leftarrow \mathtt{Counter} \leftarrow z^{0} & {\color{red} [-1,\ \infty,\ \infty ,0]} &\phantom{L.}\\ & \phantom{small space morexx} \mid \mathit{val} \Rightarrow y^{\beta+1} \leftarrow w^\alpha)) & { [-1,0,0,0]} &\\\\ & w^{\beta} \leftarrow \mathtt{NumBits}\leftarrow x^{\alpha}= & \phantom{caseLx^\alpha} {\color{blue}[0, 0,\ 0,\ 0]} & \phantom{L}\\ &\phantom{sma} \mathbf{case}\, Lx^{\alpha}\ (\mu_{bin} {\Rightarrow}& [0,0,-1,0] & \phantom{L} x^{\alpha+1}_1=x^\alpha_1, x^{\alpha+1}_2 < x^{\alpha}_2\\ & \phantom{small sp} \mathbf{case}\, Lx^{\alpha+1}\ ( \mathit{b0} \Rightarrow z^{0} \leftarrow \mathtt{NumBits} \leftarrow x^{\alpha+1}; & { \color{red}[?,0,-1,?]} & {\color{red} \phantom{Lx^0.\nu} z^0_1\stackrel{?}{=}w^{\beta}_1, z_2^0\stackrel{?}{=}w^{\beta}_2}\\ & \phantom{small space more than that} w^{\beta} \leftarrow \mathtt{BinSucc} \leftarrow z^{0} & & \mathtt{BinSucc} \subset_{0} \mathtt{NumBits}\\ & \phantom{small space morexx} \mid \mathit{b1} \Rightarrow z^{0} \leftarrow \mathtt{NumBits} \leftarrow x^{\alpha+1}; & { \color{red}[?,0,-1,?]} & {\color{red} \phantom{Lx^0.\nu} z^0_1\stackrel{?}{=}w^{\beta}_1, z_2^0\stackrel{?}{=}w^{\beta}_2}\\ & \phantom{small space more than thatxx} w^{\beta} \leftarrow \mathtt{BinSucc} \leftarrow z^{0} & &\mathtt{BinSucc} \subset_{0} \mathtt{NumBits}\\ & \phantom{small space morexx} \mid \mathit{\$}\Rightarrow Rw^{\beta}.\mu_{\mathsf{bin}}; Rw^{\beta+1}.\$;\ w^{\beta+1} \leftarrow x^{\alpha+1})) & { [0,0,-1,1]} & \\\\ & y^{\beta} \leftarrow \mathtt{BitCount} \leftarrow x^{\alpha} = w^{0}\leftarrow \mathtt{NumBits}\leftarrow x^{\alpha}; y^{\beta}\leftarrow \mathtt{Counter}\leftarrow w^{0} &\end{aligned}$$ The program starts with process $\mathtt{BitCount}$ which creates a fresh channel $w^0$, spawns a new process $w^0 \leftarrow \mathtt{NumBits} \leftarrow x^\alpha$, and continues as $y^\beta \leftarrow \mathtt{Counter} \leftarrow w^0$. Process $y^\beta \leftarrow \mathtt{Counter} \leftarrow w^\alpha$ as its name suggests works as a counter. It waits to receive a fixed point unfolding message from the right. When it receives the unfolding message, if the next message from the right is $\mathit{val}$, the process forwards the binary number offered by $w^\alpha$ to $y^{\beta+1}$. If the next message from the right is $\mathit{inc}$, it spawns a new process $\mathtt{BinSucc}$ to compute the successor of the binary number offered by channel $w^{\alpha}$. Then the incremented binary number offered by $z^0$ is passed to its continuation $y^{\beta+1} \leftarrow \mathtt{Counter} \leftarrow z^0$. Note that in this process, both (mutually) recursive calls pass the algorithm even with the naive approach. The naive approach also accepts process $w^\beta \leftarrow \mathtt{BinSucc}\leftarrow z^\alpha$. This process first receives a fixed point unfolding message from the left. If the next message from the left is bit $\mathit{b0}$, it sends bit $\mathit{b1}$ to the right and then forwards the rest of bits offered by $z^{\alpha+1}$ to $w^{\beta+1}$. If the next message is $\mathit{\$}$, signaling the end of the binary number offered by $z^{\alpha+1}$, it sends bit $\mathit{b1}$ to the right and ends the binary message provided along $w^{\beta+1}$. The only recursive call in this process happens when the next message from the left is bit $\mathit{b1}$. In this case, the process sends bit $\mathit{b0}$ to the right and calls $w^{\beta+1} \leftarrow \mathtt{BinSucc}\leftarrow z^{\alpha+1}$ recursively to compute successor of the rest of the binary number offered along $z^{\alpha+1}$. The process $w^\beta \leftarrow \mathtt{NumBits} \leftarrow x^\alpha$ works on a binary number provided along channel $x^\alpha$. It offers the number of bits of the binary along channel $w^\beta$ as another binary number: $\mathtt{NumBits}$ first receives a $\mu$ unfolding message from the left, then if the process receives an “end of number” message ($\$$), it sends the same message to the right. If it receives a bit (either $\mathit{b0}$ or $\mathit{b1}$), then it spawns a new process $z^0 \leftarrow \mathtt{NumBits} \leftarrow x^{\alpha+1}$ to count the remaining number of bits in the binary number. This number is then offered along the fresh channel $z^0$ to the continuation of the original process as $ w^ \beta \leftarrow \mathtt{BinSucc} \leftarrow z^0$. Process $\mathtt{NumBits}$ is a reactive one: It will receive every unfolding message from the left in finite steps and if there is no remaining unfolding messages to receive, it stops. However with our naive approach toward spawning a new process, the recursive calls have the value $[\infty, 0, -1, \infty] \not < [0,0,0,0]$, meaning that they do not satisfy the validity condition. Note that we cannot just define $z_1^0= w_1^\beta$ and $z_2^0=w_2^\beta$, or $z_1^0=z_2^0=0$. The value of $z^0$ depends on how this channel evolves in the process $w^\beta \leftarrow \mathtt{BinSucc} \leftarrow z^0$. But by definition of type $\mathsf{bin}$, no matter how $z^0:\mathsf{bin}$ evolves to some $z^\eta$ in process $\mathtt{BinSucc}$, it won’t be the case that $z^\eta: \mathsf{ctr}$. In other words, $\mathsf{ctr}$ is not visible from $\mathsf{bin}$ and $z^0$ never evolves to $z^\eta$ such that $z^\eta_1$ has a different value than $z^0_1$. So in this recursive call, the value of $z_1^\eta$ is not important anymore and we safely put $z_1^0=w^{\beta}_1$. In this improved version we have: $$\begin{aligned} & w^{\beta} \leftarrow \mathtt{NumBits}\leftarrow x^{\alpha}= & \phantom{caseLx^\alpha} {\color{blue}[0, 0,\ 0,\ 0]} & \phantom{L}\\ &\phantom{sma} \mathbf{case}\, Lx^{\alpha}\ (\mu_{bin} {\Rightarrow}& [0,0,-1,0] & \phantom{L} x^{\alpha+1}_1=x^\alpha_1, x^{\alpha+1}_2 < x^{\alpha}_2\\ & \phantom{small sp} \mathbf{case}\, Lx^{\alpha+1}\ ( \mathit{b0} \Rightarrow z^{0} \leftarrow \mathtt{NumBits} \leftarrow x^{\alpha+1}; & { \color{red}[0,0,-1,\infty]} & {\color{red} \phantom{Lx^0.\nu} z^0_1=w^{\beta}_1 }\\ & \phantom{small space more than that} w^{\beta} \leftarrow \mathtt{BinSucc} \leftarrow z^{0} & & \mathtt{BinSucc} \subset_{0} \mathtt{NumBits}\\ & \phantom{small space morexx} \mid \mathit{b1} \Rightarrow z^{0} \leftarrow \mathtt{NumBits} \leftarrow x^{\alpha+1}; & { \color{red}[0,0,-1,\infty]} & {\color{red} \phantom{Lx^0.\nu} z^0_1=w^{\beta}_1}\\ & \phantom{small space more than that} w^{\beta} \leftarrow \mathtt{BinSucc} \leftarrow z^{0} & &\mathtt{BinSucc} \subset_{0} \mathtt{NumBits}\\ & \phantom{small space morexx} \mid \mathit{\$}\Rightarrow Rw^{\beta}.\mu_{\mathsf{bin}}; Rw^{\beta+1}.\$;\ w^{\beta+1} \leftarrow x^{\alpha+1})) & { [0,0,-1,1]} & \end{aligned}$$ This version of the algorithm recognizes both recursive calls as valid. In the following definition we capture the idea of visibility from a type more formally. For type $A$ and a set of type variables $\Delta$, we define $\mathtt{c}(A; \Delta)$ inductively as: $$\begin{array}{l} \mathtt{c}(1; \Delta)=\emptyset,\\ \mathtt{c}(\oplus\{\ell:A_{\ell}\}_{\ell \in L}; \Delta)= \mathtt{c}(\&\{\ell:A_{\ell}\}_{\ell \in L}; \Delta)= \bigcup_{\ell \in L}\mathtt{c}(A_{\ell}; \Delta),\\ \mathtt{c}(t; \Delta)= \{t\} \cup \mathtt{c}(A; \Delta \cup \{t\})\ \text{if}\ t=_{a}\ A\ \text{and}\ t \not \in \Delta,\\ \mathtt{c}(t; \Delta)= \{t\}\ \text{if}\ t=_{a} A\ \text{and} \ t \in \Delta. \end{array}$$ We put priority $i$ in the set $\mathtt{c}(A)$ iff for some type variable $t$ with $i=p(t)$, $t \in \mathtt{c}(A; \emptyset)$. Priority $i$ is visible from type $A$ if and only if $i \in \mathtt{c}(A)$. Typing Rules for Session-Typed Processes with Channel Ordering {#rules} ============================================================== In this section we introduce inference rules for session-typed processes corresponding to derivations in subsingleton logic with fixed points. This is a refinement of the inference rules in Figure \[fig:annotrule\] to account for channel generations and orderings introduced in previous sections. The judgments are of the form $$\bar{x}^{\alpha}:\omega \vdash_{\Omega} P :: (y^{\beta}: A),$$ where $P$ is a process, and $x^\alpha$ (the $\alpha$-th generation of channel $x$) and $y^ \beta$ (the $\beta$-th generation of channel $y$) are its left and right channels of types $\omega$ and $A$, respectively. The order relation between the generations of left and right channels indexed by their priority of types is built step by step in $\Omega$ when reading the rules from the conclusion to the premises. We only consider judgments in which all variables $x^{\alpha'}$ occurring in $\Omega$ are such that $\alpha' \leq \alpha$ and, similarly, for $y^{\beta'}$ in $\Omega$ we have $\beta' \leq \beta$. This presupposition guarantees that if we construct a derivation bottom-up, any future generations for $x$ and $y$ are fresh and not yet constrained by $\Omega$. All our rules, again read bottom-up, will preserve this property. We fix a signature $\Sigma$ as in Definition \[signature\], a finite set of process definitions $V$ over $\Sigma$ as in Definition \[process\], and define $\bar{x}^{\alpha}:\omega \vdash_{\Omega} P :: (y^{\beta}: A)$ with the rules in Figure \[fig:stp-order\]. To preserve freshness of channels and their future generations in $\Omega$, the channel introduced by ${\mbox{\sc Cut}}$ rule must be distinct from any variable mentioned in $\Omega$. Similar to its underlying sequent calculus in Section \[operat\], this system is infinitary, i.e., an infinite derivation may be produced for a given program. However, as long as the set of all process variables $V$ is finite, we can map the infinite derivation of a finite program back into a circular pre-proof in the underlying sequent calculus. Programs derived in this system are all [*type checked*]{}, but not necessarily valid. It is, however, the basis on which we build our finite system of [*(local) validity*]{} in Section \[alg\]. $$\infer[{\mbox{\sc Id}}]{x^{\alpha}: A \vdash_{\Omega} y^{\beta} \leftarrow x^{\alpha} :: (y^{\beta}: A)}{}$$ $$\infer[{\mbox{\sc Cut}}^{w}]{ \bar{x}^{\alpha}: \omega \vdash_{\Omega} (w \leftarrow P_{w} ; Q_{w}) :: (y^{\beta}: C)}{ \deduce {\bar{x}^{\alpha}: \omega \vdash_{\Omega \cup \mathtt{r}(y^\beta)} P_{w^0} ::(w^0:A)}{\mathtt{r}(v)= \{w^{0}_{p(s)}=v_{p(s)}\mid p(s) \not \in \mathtt{c}(A)\}} & w^0: A \vdash_{\Omega \cup \mathtt{r}(\bar{x}^\alpha)} Q_{w^0} :: (y^{\beta}: C)}$$ $$\begin{tabular}{c c} \infer[\oplus R]{\bar{x}^{\alpha}:\omega \vdash_{\Omega} Ry^{\beta}.k; P :: (y^{\beta}: \oplus\{\ell:A_{\ell}\}_{\ell \in L})}{\bar{x}^{\alpha}: \omega \vdash_{\Omega} P :: (y^{\beta}: A_{k}) \quad (k \in L)} & \infer[\oplus L]{x^{\alpha}:\oplus\{ \ell:A_\ell \}_{ \ell \in L} \vdash_{\Omega} \mathbf{case}\, Lx^{\alpha} \ (\ell{\Rightarrow}P_{\ell}):: (y^{\beta}: C)}{\forall \ell\in L \quad x^{\alpha}:A_{\ell} \vdash_{\Omega} P_\ell :: (y^{\beta}:C)}\\\\ \infer[\& R]{\bar{x}^{\alpha}: \omega \vdash_{\Omega} \mathbf{case}\, Ry^{\beta}\ (\ell {\Rightarrow}P_\ell) :: (y^{\beta}: \& \{\ell:A_\ell\}_{\ell \in L})}{\forall \ell\in L \quad \bar{x}^{\alpha}: \omega \vdash_{\Omega} P_\ell :: (y^{\beta}:A_{\ell})} & \infer[\& L]{x^{\alpha}: \&\{ \ell:A_l \}_{ \ell \in L} \vdash_{\Omega} Lx^{\alpha}.k; P :: (y^{\beta}:C)}{k\in L \quad x^{\alpha}: A_{k} \vdash_{\Omega} P :: (y^{\beta}:C)}\\\\ \infer[1R]{. \vdash_{\Omega} \mathbf{close}\, Ry^\beta :: (y^{\beta}: 1)}{} & \infer[1L]{x^{\alpha}: 1 \vdash_{\Omega} \mathbf{wait}\, Lx^\alpha;Q :: (y^{\beta}: A)}{ . \vdash_{\Omega} Q :: (y^{\beta}: A) }\\ \end{tabular}$$ $$\infer[\mu R]{ \bar{x}^{\alpha}: \omega \vdash_{\Omega} Ry^{\beta}.\mu_t; P_{y^{\beta}} :: (y^{\beta}:t)}{\deduce{ \bar{x}^{\alpha}: \omega \vdash_{\Omega'} P_{y^{\beta+1}} :: (y^{\beta+1}:A)}{\Omega'= \Omega \cup \{(y^{\beta})_{p(s)} =(y^{\beta+1})_{p(s)} \mid p(s)\neq p(t)\}} & t=_{\mu}A & }$$ $$\infer[\mu L]{x^{\alpha}: t \vdash_{\Omega} \mathbf{case}\, Lx^{\alpha}\ (\mu_{t} {\Rightarrow}Q_{x^{\alpha}}):: (y^{\beta}: C)}{\deduce{x^{\alpha+1}: A \vdash_{\Omega'} Q_{x^{\alpha+1}} :: (y^{\beta}:C)}{ \Omega'=\Omega \cup \{x^{\alpha+1}_{p(t)} < x^{\alpha}_{p(t)}\} \cup \{x^{\alpha+1}_{p(s)} =x^{\alpha}_{p(s)} \mid p(s)\neq p(t)\} } & t=_{\mu} A }$$ $$\infer[\nu R]{\bar{x}^{\alpha}: \omega \vdash_{\Omega} \mathbf{case}\, Ry^{\beta} \ (\nu_t {\Rightarrow}P_{y^{\beta}}) :: (y^{\beta}: t)}{\deduce{\bar{x}^{\alpha}: \omega \vdash_{\Omega'} P_{y^{\beta+1}} :: (y^{\beta+1}: A)}{\Omega'= \Omega \cup \{y^{\beta+1}_{p(t)} < y^{\beta}_{p(t)}\} \cup \{y^{\beta+1}_{p(s)} = y^{\beta}_{p(s)} \mid p(s)\neq p(t)\}} & t=_{\nu}A }$$ $$\infer[\nu L]{x^{\alpha}: t \vdash_{\Omega} Lx^{\alpha}.\nu_{t}; Q_{x^{\alpha}}:: (y^{\beta}: C)}{\deduce{x^{\alpha+1}: A \vdash_{\Omega'} Q_{x^{\alpha+1}} :: (y^{\beta}: C)} {\Omega'=\Omega \cup \{(x^{\alpha+1})_{p(s)} =(x^{\alpha})_{p(s)} \mid p(s)\neq p(t) \}} & t=_{\nu} A }$$ $$\infer[{\mbox{\sc Def}}(X)]{\bar{x}^{\alpha}: \omega \vdash_{\Omega} y^\beta \leftarrow X \leftarrow \bar{x}^\alpha:: (y^{\beta}: C)}{\bar{x}^{\alpha}: \omega \vdash_{\Omega} P_{\bar{x}^\alpha, y^\beta} :: (y^{\beta}: C) & \bar{x}:\omega \vdash X=P_{\bar{x},y} :: (y:C) \in V }$$ A Local Validity Condition {#alg} ========================== In Sections \[session\] to \[algn\], using several examples, we developed an algorithm for identifying [*valid*]{} programs. Illustrating the full algorithm based on the inference rules in Section \[rules\] was postponed to this section. We reserve it for the next section to prove our main result that the programs accepted by this algorithm satisfy the guard condition introduced by Fortier and Santocanale [@Fortier13csl]. The condition checked by our algorithm is a *local* one in the sense that we check validity of each process definition in a program separately. The algorithm works on the sequents of the form $$\langle\bar{u}^\gamma,X,v^{\delta}\rangle; \bar{z}^\alpha:\omega \vdash_{\Omega, \subset} P:: (w^{\beta}:C),$$ where $\bar{u}^\gamma$ is the left channel of the process the algorithm started with and can be either empty or $u^\gamma$; it is empty if the process has no left channel. Similarly, $v^{\delta}$ is the right channel of the process the algorithm started with (that cannot be empty). And $X$ is the last process variable a definition rule has applied to (reading the rules bottom-up). Again, in this judgment the (in)equalities in $\Omega$ can only relate variables $z$ and $w$ from earlier generations to guarantee freshness of later generations. Generally speaking, when analysis of the program starts with $\bar{u}^\gamma:\omega \vdash v^\delta \leftarrow X \leftarrow \bar{u}^\gamma :: (v^\delta: B)$, a snapshot of the channels $\bar{u}^\gamma$ and $v^{\delta}$ and the process variable $X$ are saved. Whenever the process reaches a call $\bar{z}^\alpha:\_\vdash w^\beta \leftarrow Y \leftarrow \bar{z}^\alpha:: (w^\beta: \_)$, the algorithm compares $X,list(\bar{u}^\gamma, v^\delta)$ and $Y,list(\bar{z}^\alpha, w^\beta)$ using the $(\subset, <)$ order to determine if the recursive call is (locally) valid. This comparison is made by the ${\mbox{\sc Call}}$ rule in the rules in Figure \[fig:validity\]. A program $\mathcal{P}= \langle V,S \rangle$ over signature $\Sigma$ and a fixed order $\subset$ satisfying properties in Definition \[subs\], is [*locally valid*]{} iff for every $\bar{z}:A\vdash X=P_{\bar{z},w}:: (w:C) \in V$, there is a proof for $$\langle\bar{z}^0,X,w^0\rangle; \bar{z}^0:\omega \vdash_{\emptyset, \subset} P_{\bar{z}^0, w^0}:: (w^0:C)$$ in the rule system in Figure \[fig:validity\]. This set of rules is *finitary* so it can be directly interpreted as an algorithm. This results from substituting the ${\mbox{\sc Def}}$ rule with the ${\mbox{\sc Call}}$ rule. Again, to guarantee freshness of future generations of channels, the channel introduced by ${\mbox{\sc Cut}}$ rule is distinct from other variables mentioned in $\Omega$. The starting point of the algorithm can be of an arbitrary form $$\langle\bar{z}^\alpha,X,w^{\beta}\rangle; \bar{z}^\alpha:\omega \vdash_{\Omega, \subset} P_{z^{\alpha},w^{\beta}}::(w^{\beta}: C),$$ as long as $\bar{z}^{\alpha+i}$ and $w^{\beta+i}$ do not occur in $\Omega$ for every $i>0$. In both the inference rules and the algorithm, it is implicitly assumed that the next generation of channels introduced in the $\mu/\nu-R/L$ rules do not occur in $\Omega$. Having this condition we can convert a proof for $$\langle\bar{z}^0,X,w^0\rangle; \bar{z}^0:\omega \vdash_{\emptyset, \subset} P_{z^{0},w^{0}}:: (w^0:C),$$ to a proof for $$\langle\bar{z}^\alpha,X,w^{\beta}\rangle; \bar{z}^\alpha:\omega \vdash_{\Omega, \subset} P_{z^{\alpha},w^{\beta}}::(w^{\beta}: C),$$ by rewriting each $\bar{z}^{\gamma}$ and $w^{\delta}$ in the proof as $\bar{z}^{\gamma+\alpha}$ and $w^{\delta+\beta}$, respectively. This simple proposition is used in the next section where we prove that every locally valid process accepted by our algorithm is a valid proof according to the FS guard condition. \[subuv\] If there is a deduction of $$\langle\bar{z}^0,X,w^0\rangle; \bar{z}^0:\omega \vdash_{\emptyset, \subset} P_{z^{0},w^{0}}:: (w^0:C),$$ then there is also a deduction of $$\langle\bar{z}^\alpha,X,w^{\beta}\rangle; \bar{z}^\alpha:\omega \vdash_{\Omega, \subset} P_{z^{\alpha},w^{\beta}}::(w^{\beta}: C),$$ if for all $0<i$, $\bar{z}^{\alpha+i}$ and $w^{\beta+i}$ do not occur in $\Omega$. By substitution, as explained above. Local Validity and Guard Conditions {#Guard} =================================== Fortier and Santocanale [@Fortier13csl] introduced a [*guard condition*]{} for identifying valid circular proofs among all infinite pre-proofs in the singleton logic with fixed points. They showed that the pre-proofs satisfying this condition, which is based on the definition of left $\mu-$ and right $\nu-$ traces, enjoy the cut elimination property. In this section, we translate their guard condition into the context of session-typed concurrency and generalize it for subsingleton logic. It is immediate that the cut elimination property holds for a proof in subsingleton logic if it satisfies the generalized version of the guard condition. The key idea is that cut reductions for individual rules stay untouched in subsingleton logic and rules for the new constant $1$ only provide more options for a proof in this system to terminate. We prove that all locally valid programs in the session typed system, determined by the algorithm in Section \[alg\], also satisfy the guard condition. We conclude that our algorithm imposes a stricter but local version of validity on the session-typed programs corresponding to the circular pre-proofs. Here we adapt definitions of the [*left*]{} and [*right traceable*]{} paths, [*left $\mu$-*]{} and [*right $\nu$-traces,*]{} and then [*validity*]{} to our session type system. \[rt\] Consider path $\mathbb{P}$ on a program $\mathcal{Q}=\langle V, S\rangle$ defined on a Signature $\Sigma$: $${\infer{\bar{z}^\alpha: \omega \vdash_{\Omega} Q :: (w^{\beta}:C)}{ \infer{\vdots}{{\bar{x}^\gamma: \omega' \vdash_{\Omega'} Q' :: (y^{\delta}:C')}}} }$$ $\mathbb{P}$ is called [*left traceable*]{} if $\bar{z}$ and $\bar{x}$ are non-empty and $\bar{z}=\bar{x}$. It is called [*right traceable*]{} if $w=y$. \[tr\] A path $\mathbb{P}$ on a program $\mathcal{Q}=\langle V, S\rangle$ defined over Signature $\Sigma$ is a left $\mu$-trace if (i) it is left-traceable, (ii) there is a left fixed point rule applied on it, and (iii) the highest priority of its left fixed point rule is $i\le n$ such that $\epsilon(i)=\mu$. Dually, $\mathbb{P}$ is a right $\nu$-trace if (i) it is right-traceable, (ii) there is a right fixed point rule applied on it, and (iii) the highest priority of its right fixed point is $i \le n$ such that $\epsilon(i)=\nu$. \[guard\] A program $\mathcal{Q}=\langle V, S\rangle$ defined on Signature $\Sigma$ satisfies the FS guard condition if every cycle $\mathbb{C}$ $${\infer{\bar{z}^\alpha: \omega \vdash_{\Omega} w^\beta \leftarrow X \leftarrow \bar{z}^\alpha :: (w^{\beta}:C)}{ \infer{\vdots}{{\bar{x}^\gamma: \omega' \vdash_{\Omega'} y^\delta \leftarrow X \leftarrow \bar{x}^\gamma :: (y^{\delta}:C')}}} }$$ over $\mathcal{Q}$ is either a left $\mu$-trace or a right $\nu$-trace. Similarly, we say a single cycle $\mathbb{C}$ satisfies the guard condition if it is either a left $\mu$-trace or a right $\nu$-trace. Definitions \[rt\]-\[guard\] are equivalent to the definitions of the same concepts by Fortier and Santocanale using our own notation. As an example, consider program $\mathcal{P}_3:= \langle \{\mathtt{Copy}\}, \mathtt{Copy} \rangle$ over signature $\Sigma_1$, defined in Example \[prex\], where $\mathtt{Copy}$ has types $x:\mathsf{nat} \vdash \mathtt{Copy} ::(y:\mathsf{nat})$. $$\begin{aligned} \Sigma_1:= \mathsf{nat}=^{1}_{\mu} \oplus\{ \mathit{z}:\mathsf{1}, \mathit{s}:\mathsf{nat}\}\end{aligned}$$ $$\begin{aligned} y \leftarrow \mathtt{Copy} \leftarrow x =\mathbf{case}\, Lx\ (\mu_{nat} {\Rightarrow}\mathbf{case}\, Lx\ & (\ z{\Rightarrow}Ry.\mu_{nat}; Ry.z; \mathbf{wait}\, Lx; \mathbf{close}\, Ry \\ & \mid s \Rightarrow Ry.\mu_{nat}; Ry.s; y \leftarrow \mathtt{Copy} \leftarrow x))\end{aligned}$$ Consider the first several steps of the derivation of the program starting with $x^0:\mathsf{nat} \vdash_{\emptyset} y^0 \leftarrow \mathtt{Copy} \leftarrow x^0 ::(y^0:\mathsf{nat})$: $$\infer[{\mbox{\sc Def}}(\mathtt{Copy})]{\color{red}x^0:\mathsf{nat}\vdash_{\emptyset} y^0 \leftarrow \mathtt{Copy} \leftarrow x^0 :: (y^0:\mathsf{nat})}{ \infer[\mu L]{ x^0:\mathsf{nat} \vdash_{\emptyset} \mathbf{case}\, Lx^0\ (\mu_{nat}\Rightarrow \cdots):: (y^0:\mathsf{nat})}{ \infer[\oplus L]{ x^1:1\oplus \mathsf{nat} \vdash_{\{x^1_1<x^0_1\}} \mathbf{case}\, Lx^1\ (\cdots):: (y^0: \mathsf{nat})}{x^1:\mathsf{1} \vdash_{\{x^1_1<x^0_1\}} Ry^0.\mu_{nat}; \cdots :: (y^0: \mathsf{nat}) & \infer[\mu R]{x^1:\mathsf{nat} \vdash_{\{x^1_1<x^0_1\}} Ry^0.\mu_{nat}; \cdots :: (y^0: \mathsf{nat})}{\infer[\oplus R]{x^1:\mathsf{nat} \vdash_{\{x^1_1<x^0_1\}} Ry^1.s; \cdots :: (y^1: 1 \oplus \mathsf{nat})}{\color{blue} x^1:\mathsf{nat} \vdash_{\{x^1_1<x^0_1\}} y^1 \leftarrow \mathtt{Copy} \leftarrow x^1 :: (y^1: \mathsf{nat})}}}}}$$ The path between $${\color{red} x^0:\mathsf{nat} \vdash_{\emptyset} y^0\leftarrow \mathtt{Copy} \leftarrow x^0 :: (y^0: \mathsf{nat})}$$ and $${\color{blue} x^1:\mathsf{nat} \vdash_{\{x^1_1<x^0_1\}} y^1 \leftarrow \mathtt{Copy} \leftarrow x^1 :: (y^1: \mathsf{nat})}$$ is by definition both left traceable and right traceable, but it is only a left $\mu$-trace and not a right $\nu$-trace: The highest priority of a fixed point applied on the left-hand side on this path belongs to a positive type; this application of the $\mu L$ rule added $x^1_1<x^0_1$ to the set defining the $<$ order. However, there is no negative fixed point rule applied on the right, and $y_1^1$ and $y_1^0$ are incomparable to each other. This cycle, thus, satisfies the [*guard condition*]{} by being a left $\mu$-trace, and it is also accepted by our algorithm since $list(x^1,y^1)=[(x^1_1,y^1_1)]<[(x^0_1,y^0_1)]=list(x^0,y^0)$, as being checked in the ${\mbox{\sc Call}}$ rule. Here, we can observe that being a left $\mu$-trace coincides with having the relation $x^1_1<x^0_1$ between the left channels, and not being a right $\nu$-trace coincides with not having the relation $y^1_1<y^0_1$ for the right channels. We can generalize this observation to every path and every signature with $n$ priorities. \[listor\] A cycle $\mathbb{C}$ $${\infer{\bar{z}^\alpha: \omega \vdash_{\Omega} w^\beta \leftarrow X \leftarrow \bar{z}^\alpha :: (w^{\beta}:C)}{ \infer{\vdots}{{\bar{x}^\gamma: \omega' \vdash_{\Omega'} y^\delta \leftarrow X \leftarrow \bar{x}^\gamma :: (y^{\delta}:C')}}} }$$ on a program $\mathcal{Q}=\langle V, S\rangle$ defined over Signature $\Sigma$ is a [*left $\mu$-trace*]{} if $\bar{x}$ and $\bar{z}$ are non-empty and the list $[x^\gamma]=[x_1^\gamma, \cdots, x_n^\gamma]$ is lexicographically less than the list $[z^\alpha]=[z_1^\alpha, \cdots, z_n^\alpha]$ by the order $<_{\Omega'}$ built in $\Omega'$. Dually, it is a [*right $\nu$-trace*]{}, if the list $[y^\delta]=[y_1^\delta, \cdots, y_n^\delta]$ is lexicographically less than the list $[w^\beta]=[w_1^\beta, \cdots, w_n^\beta]$ by the strict order $<_{\Omega'}$ built in $\Omega'$ This theorem is a corollary of Lemmas \[ordtrace\] and \[dordtrace\] proved in Appendix A. We provide a few additional examples to elaborate Theorem \[listor\] further. Define a new program $\mathcal{P}_9:=\langle \{\mathtt{Succ},\mathtt{Copy},\mathtt{SuccCopy}\}, \mathtt{SuccCopy} \rangle$, over the signature $\Sigma_1$, using the process $\ w:\mathsf{nat} \vdash \mathtt{Copy} :: (y:\mathsf{nat})$ and two other processs: $x:\mathsf{nat} \vdash \mathtt{Succ} :: (w:\mathsf{nat})$ and $\ x:\mathsf{nat} \vdash \mathtt{SuccCopy}:: (y:\mathsf{nat})$. The processes are defined as $$\begin{aligned} w \leftarrow \mathtt{Succ} \leftarrow x= Rw.\mu_{nat};Rw.s; w\leftarrow x \end{aligned}$$ $$\begin{aligned} y \leftarrow \mathtt{Copy} \leftarrow w= \mathbf{case}\, Lw\ (\mu_{nat} {\Rightarrow}\mathbf{case}\, Lw\ & (\ \mathit{s} {\Rightarrow}Ry.\mu_{nat}; Ry.\mathit{s}; y \leftarrow \mathtt{Copy} \leftarrow w\\ & \mid \mathit{z} \Rightarrow Ry.\mu_{nat}; Ry.\mathit{z}; \mathbf{wait}\, Lw;\mathbf{close}\, Ry)) \end{aligned}$$ $$\begin{aligned} y \leftarrow \mathtt{SuccCopy}\leftarrow x= w \leftarrow \mathtt{Succ} \leftarrow x ; y \leftarrow \mathtt{Copy} \leftarrow w,\end{aligned}$$ Process $\mathtt{SuccCopy}$ spawns a new process $\mathtt{Succ}$ and continues as $\mathtt{Copy}$. The $\mathtt{Succ}$ process prepends an $\mathit{s}$ label to the beginning of the finite string representing a natural number on its left hand side and then forwards the string as a whole to the right. $\mathtt{Copy}$ receives this finite string, representing a natural number, on its left hand side, one element at a time, and *distributes* it to the right element by element.\ The only recursive process in this program is $\mathtt{Copy}$ that is discussed earlier in this section. So program $\mathcal{P}_9$, itself, does not have a further interesting point to discuss. We consider a bogus version of this program in Example \[bogus\] that provides further intuition for Theorem \[listor\]. \[bogus\] Define program $\mathcal{P}_{10}:=\langle \{\mathtt{Succ},\mathtt{BogusCopy},\mathtt{SuccCopy}\}, \mathtt{SuccCopy} \rangle$ over the signature $$\begin{aligned} \Sigma_1 := \mathsf{nat} =^{1}_{\mu}\oplus \{\mathit{z}:1,\ \ \mathit{s}: \mathsf{nat}\},\end{aligned}$$ The processes $x:\mathsf{nat} \vdash \mathtt{Succ} :: (w:\mathsf{nat})$, $\ \ w:\mathsf{nat} \vdash \mathtt{BogusCopy} :: (y:\mathsf{nat})$, and $\ x:\mathsf{nat} \vdash \mathtt{SuccCopy}:: (y:\mathsf{nat})$, are defined as $$\begin{aligned} w \leftarrow \mathtt{Succ} \leftarrow x= Rw.\mu_{nat};Rw.s; w \leftarrow x \end{aligned}$$ $$\begin{aligned} y\leftarrow \mathtt{BogusCopy}\leftarrow w = \mathbf{case}\, Lw\ (\mu_{nat} {\Rightarrow}\mathbf{case}\, Lw\ & (\ \mathit{s}\Rightarrow Ry.\mu_{nat}; Ry.\mathit{s}; y \leftarrow \mathtt{SuccCopy} \leftarrow w\\ & \mid \mathit{z} \Rightarrow Ry.\mu_{nat}; Ry.\mathit{z}; \mathbf{wait}\, Lw;\mathbf{close}\, Ry)) \end{aligned}$$ $$\begin{aligned} y \leftarrow \mathtt{SuccCopy} \leftarrow x = w \leftarrow \mathtt{Succ} \leftarrow x ; y \leftarrow \mathtt{BogusCopy}\leftarrow w\end{aligned}$$ Program $\mathcal{P}_{10}$ is a non-reactive *bogus* program, since $\mathtt{BogusCopy}$ instead of calling itself recursively, calls $\mathtt{SuccCopy}$. At the very beginning $\mathtt{SuccCopy}$ spawns $\mathtt{Succ}$ and continues with $\mathtt{BogusCopy}$ for a fresh channel $w$. $\mathtt{Succ}$ then sends a fixed point unfolding message and a [*successor*]{} label via $w$ to the right, while $\mathtt{BogusCopy}$ receives the two messages just sent by $\mathtt{Succ}$ through $w$ and calls $\mathtt{SuccCopy}$ recursively again. This loop continues forever, without any messages being received from the outside. The first several steps of the derivation of $x^0:\mathsf{nat} \vdash_{\emptyset} \mathtt{SuccCopy}:: (y^0:\mathsf{nat})$ in our inference system (Section \[rules\]) are given below. $$\small \infer[{\mbox{\sc Def}}]{{\color{red}x^0:\mathsf{nat}\vdash_{\emptyset} y^0 \leftarrow \mathtt{SuccCopy} \leftarrow x^0 :: (y^0:\mathsf{nat})}}{\infer[{\mbox{\sc Cut}}^w]{x^0:\mathsf{nat} \vdash_{\emptyset} w \leftarrow \mathtt{Succ}; y^0 \leftarrow \mathtt{BogusCopy} \leftarrow w:: (y^0:\mathsf{nat})}{\infer[{\mbox{\sc Def}}]{x^0:\mathsf{nat} \vdash_{\emptyset} {w^0} \leftarrow \mathtt{Succ}\leftarrow x^0:: (w^0: \mathsf{nat})}{\infer[\mu R]{x^0:\mathsf{nat} \vdash_{\emptyset} Rw^{0}.\mu_{nat}; \cdots :: (w^0: \mathsf{nat})}{\infer[\oplus R]{x^0:\mathsf{nat} \vdash_{\emptyset} Rw^1.\mathit{z};\cdots :: (w^1:1 \oplus \mathsf{nat})}{{x^0:\mathsf{nat} \vdash_{\emptyset} w^1 \leftarrow \mathtt{Succ}\leftarrow x^0 :: (w^1: \mathsf{nat})}}}} \hspace{-10pt} & {\infer[{\mbox{\sc Def}}]{ w^0: \mathsf{nat}\vdash_{\emptyset} y^0 \leftarrow \mathtt{BogusCopy} \leftarrow w^0:: (y^0:\mathsf{nat})}{\infer[\mu L]{w^0: \mathsf{nat}\vdash_{\emptyset} \mathbf{case}\, Lw^0\ (\mu_{nat} {\Rightarrow}\cdots) ::(y^0:\mathsf{nat})}{\infer[\oplus L]{w^1: 1\oplus \mathsf{nat}\vdash_{\{w^1_1<w^0_1\}} \mathbf{case}\, Lw^1\ (\cdots) ::(y^0:\mathsf{nat})}{\boldsymbol{\cdots} & {\color{blue} w^1:\mathsf{nat} \vdash_{\{w^1_1<w^0_1\}} y^0 \leftarrow \mathtt{SuccCopy}\leftarrow w^1 ::(y^0:\mathsf{nat})}}}}}}}$$ Consider the cycle between $$\color{red}x^0:\mathsf{nat}\vdash_{\emptyset} y^0 \leftarrow \mathtt{SuccCopy} \leftarrow x^0 :: (y^0:\mathsf{nat})$$ and $$\color{blue}w^1:\mathsf{nat}\vdash_{\emptyset} y^0 \leftarrow \mathtt{SuccCopy} \leftarrow w^1 :: (y^0:\mathsf{nat}).$$ By Definition \[tr\], this path is right traceable, but not left traceable. And by Definition \[rt\], the path is neither a right $\nu$-trace nor a left $\mu$-trace: 1. No negative fixed point unfolding message is received from the right and $y^0$ does not evolve to a new generation that has a smaller value in its highest priority than $y^0_1$. In other words, $y^0_1 \not < y^0_1$ since no negative fixed point rule has been applied on the right channel. 2. The positive fixed point unfolding message that is received from the left is received through the channel $w^0$, which is a fresh channel created after $\mathtt{SuccCopy}$ spawns the process $\mathtt{Succ}$. Although $w^1_1<w^0_1$, since $x^0_1$ is incomparable to $w^0_1$, the relation $w^1_1<x^0_1$ does not hold. This path is not even a left-traceable path. Neither $[w^1]= [w^1_1] < [x^0_1]= [x^0]$, nor $[y^0]= [y^0_1] < [y^0_1] =[y^0]$ hold, and this cycle does not satisfy the guard condition. This program is not locally valid either since $[w^1_1,y^0_1]\not < [x^0_1, y^0_1]$. As another example consider the program $\mathcal{P}_6=\{\mathtt{Ping},\mathtt{Pong},\mathtt{PingPong}\}, \mathtt{PingPong} \rangle$ over the signature $\Sigma_4$ as defined in Example \[begen1\]. We discussed in Section \[priority\] that this program is not accepted by our algorithm as locally valid. $$\begin{aligned} \Sigma_4 :=\ & \mathsf{ack}=^{1}_{\mu} \oplus\{ \mathit{ack}:\mathsf{astream}\},\\ & \mathsf{astream}=^{2}_{\nu} \& \{\mathit {head}: \mathsf{ack}, \ \ \mathit{tail}: \mathsf{astream}\},\\ & \mathsf{nat}=^{3}_{\mu}\oplus \{\mathit{z}:1,\ \ \mathit{s}: \mathsf{nat}\} \end{aligned}$$ Processes $$\begin{array}{l} x:\mathsf{nat} \vdash \mathtt{Ping} :: (w:\mathsf{astream}),\\ w:\mathsf{astream} \vdash \mathtt{Pong} :: (y:\mathsf{nat}),\\ x:\mathsf{nat} \vdash \mathtt{PingPong}:: (y:\mathsf{nat}) \end{array}$$ are defined as $$\begin{aligned} w \leftarrow \mathtt{Ping} \leftarrow x= \mathbf{case}\, Rw\ (\nu_{astream} {\Rightarrow}\mathbf{case}\, Rw\ & (\ \mathit{head} \Rightarrow Rw.\mu_{ack}; Rw.\mathit{ack}; w\leftarrow \mathtt{Ping} \leftarrow x\\ & \mid \mathit{tail} \Rightarrow w \leftarrow \mathtt{Ping} \leftarrow x)) \end{aligned}$$ $$\begin{aligned} y \leftarrow \mathtt{Pong} \leftarrow w = & Lw.\nu_{astream};Lw.\mathit{head}; \\ &\quad \mathbf{case}\, Lw\ (\mu_{ack} {\Rightarrow}\mathbf{case}\, Lw\ (ack \Rightarrow Ry.\mu_{nat}; Ry.s; y \leftarrow \mathtt{Pong} \leftarrow w))\end{aligned}$$ $$\begin{aligned} y \leftarrow \mathtt{PingPong} \leftarrow x= w \leftarrow \mathtt{Ping}\leftarrow x ; y \leftarrow \mathtt{Pong} \leftarrow w\end{aligned}$$ The first several steps of the proof of $x^0:nat \vdash_{\emptyset} \mathtt{PingPong}:: (y^0:nat)$ in our inference system (Section \[rules\]) are given below (with some abbreviations). $$\small \infer[{\mbox{\sc Def}}]{{x^0:\mathsf{nat}\vdash_{\emptyset} y^0 \leftarrow \mathtt{PingPong} \leftarrow x^0 :: (y^0:\mathsf{nat})}}{ \infer[{\mbox{\sc Cut}}]{ x^0:\mathsf{nat} \vdash_{\emptyset} w \leftarrow \mathtt{Ping}\leftarrow x^0 ; y^0 \leftarrow \mathtt{Pong} \leftarrow w:: (y^0:\mathsf{nat})}{ \infer[{\mbox{\sc Def}}]{ {\color{red} x^0:\mathsf{nat} \vdash_{\emptyset} w^0 \leftarrow \mathtt{Ping}\leftarrow x^0 :: (w^0: \mathsf{astream})}}{\infer[\nu R]{x^0:\mathsf{nat} \vdash_{\emptyset} \mathbf{case}\, Rw^0\ (\nu_{astream} \Rightarrow \cdots) :: (w^0: \mathsf{astream})}{ \infer[\& R]{x^0:\mathsf{nat} \vdash_{A} \mathbf{case}\, Rw^1\ (\cdots) :: (w^1:\mathsf{ack}\ \&\ \mathsf{astream})}{ \infer[\mu R]{x^0:\mathsf{nat} \vdash_{A} Rw^1. \mu_{ack}; \cdots ::(w^1:\mathsf{ack})}{ \infer[\oplus R]{x^0:\mathsf{nat} \vdash_{B} Rw^2. \mathit{ack}; \cdots ::(w^2:\oplus\{\mathsf{astream}\})}{ \color{blue} x^0:\mathsf{nat} \vdash_{B} w^2 \leftarrow \mathtt{Ping} \leftarrow x^0::(w^2:\mathsf{astream})}}\hspace{-10pt} & { x^0:\mathsf{nat} \vdash_{A} \boldsymbol{\cdots} ::(w^1:\mathsf{astream})}}}} \hspace{-35pt} & { w^0: \mathsf{astream}\vdash_{\emptyset} \boldsymbol{\cdots}:: (y^0:\mathsf{nat})}}}$$ where $A= {\{w^1_1=w^0_1, w^1_2<w^0_2, w^1_3=w^0_3\}}$, and $B=\{w^2_1=w^1_1=w^0_1, w^1_2<w^0_2, w^2_3=w^1_3=w^0_3,\}$. The cycle between the processes $${\color{red}x^0:\mathsf{nat} \vdash_{\emptyset}w^0 \leftarrow \mathtt{Ping} \leftarrow {x^0} :: (w^0: \mathsf{astream})}$$ and $${\color{blue} x^0:\mathsf{nat} \vdash_{B} w^2 \leftarrow \mathtt{Ping} \leftarrow x^0 ::(w^2:\mathsf{astream})}$$ is neither a left $\mu$-trace, nor a right $\nu$-trace: 1. No fixed point unfolding message is received or sent through the left channels in this path and thus $[x^0]= [x_1^0,x_2^0,x_3^0]\not < [x_1^0,x_2^0,x_3^0]=[x^0]$. 2. On the right, fixed point unfolding messages are both sent and received: (i) $w^0$ receives an unfolding message for a negative fixed point with priority $2$ and evolves to $w^1$, and then later (ii) $w^1$ sends an unfolding message for a positive fixed point with priority $1$ and evolves to $w^2$. But the positive fixed point has a higher priority than the negative fixed point, and thus this path is not a right $\nu$-trace either. This reasoning can also be reflected in our observation about the list of channels in Theorem \[listor\]: When, first, $w^0$ evolves to $w^1$ by receiving a message in (i) the relations $w^0_1=w^1_1$, $w^0_2<w^1_2$, and $w^0_3=w^1_3$ are recorded. And, later, when $w^1$ evolves to $w^2$ by sending a message in (ii) the relations $w^1_2=w^2_2$, and $w^1_3=w^2_3$ are added to the set. This means that $w^2_1$ as the first element of the list $[w^2]$ remains incomparable to $w^0_1$ and thus $[w^2]= [w_1^2,w_2^2,w_3^2] \not < [w_1^0,w_2^0,w_3^0]=[w^0]$ We are now ready to state our main theorem that connects the local validity algorithm introduced in Section \[alg\] to the FS guard condition. A locally valid program satisfies the FS guard condition. We give a sketch of the proof here. See Appendix \[proofapp\] for the complete proof. By Theorem \[listor\], a cycle $\mathbb{C}$ $${\infer{\bar{z}^\alpha: \omega \vdash_{\Omega}w^\beta \leftarrow X \leftarrow \bar{z}^\alpha :: (w^{\beta}:C)}{ \infer{\vdots}{{\bar{x}^\gamma: \omega' \vdash_{\Omega'} y^\delta \leftarrow X \leftarrow \bar{x}^\gamma :: (y^{\delta}:C')}}} }$$ is either a [*left $\mu$-trace*]{} [or]{} a [*right $\nu$-trace*]{} if either $[x^\gamma]<_{\Omega'} [z^\alpha]$ [**or**]{} $[y^\delta]<_{\Omega'} [w^\beta]$ holds. The FS validity condition requires this disjunctive condition for all cycles in the program. In our algorithm, however, we merge the lists of left and right channels, e.g. $[x^\gamma]$ and $[y^\delta]$ respectively, into a single list $list(x^{\gamma},y^{\delta})$. The values in $list(x^{\gamma},y^{\delta})$ from Definition \[list\] are still recorded in their order of priorities, but for the same priority the value corresponding to [*receiving*]{} a message precedes the one corresponding to [*sending*]{} a message. Roughly, but not exactly, instead of checking $[x^\gamma]<_{\Omega'} [z^\alpha]$ or $[y^\delta]<_{\Omega'} [w^\beta]$, our algorithm checks $list(x^{\gamma},y^{\delta}) <_{\Omega'} list(z^{\alpha},w^{\beta})$. To make it exact we should also note that our algorithm checks all calls even those that do not form a cycle in the sense of the FS guard condition (that is, when process $X$ calls process $Y\neq X$). By adding process variables to our validity check condition, as described in Definition \[defmain\], there is no need to search for every possible cycle in the program. Instead, our algorithm only checks the condition for the immediate recursive calls that a process makes. As this condition enjoys transitivity, it also holds for all possible non-immediate recursive calls, including the cycles. Note that checking a disjunctive condition for each cycle implies that we cannot rely on transitivity. Therefore, to check the FS guard condition we have to examine every possible cycle separately, even if it is composed of two previously checked cycles. Our local validity condition, however, works on a single transitive condition based on a single list of channels and one process variable. By merging the lists and checking the condition only for the immediate recursive calls, our algorithm is [*local*]{} but [*stricter*]{} than the FS guard condition. Computational Meta-theory {#semantics} ========================= Fortier and Santocanale [@Fortier13csl] defined a function $\textsc{Treat}$ as a part of their cut elimination algorithm. They proved that this function terminates on a list of pre-proofs fused by consecutive cuts if all of them satisfy their guard condition. In our system, function $\textsc{Treat}$ corresponds to computation on a configuration of processes. In this section we first show that usual preservation and progress theorems hold even if a program does not satisfy the validity condition. Then we use Fortier and Santocanale’s result to prove a stronger compositional progress property for (locally) valid programs. In Section \[operat\], we introduced process configurations $\mathcal{C}$ as a list of processes connected by the associative, noncommutative parallel composition operator $\mid_x$. $$\mathcal{C} ::= \cdot \mid \mathtt{P} \mid (\mathcal{C}_1 \mid_{x}\ \mathcal{C}_2)$$ with unit $(\cdot)$. The type checking judgements for configurations $\bar{x}: \omega \Vdash \mathcal{C}:: (y:B)$ are:\ $$\begin{tabular}{c c c c c} \infer[]{x:A \Vdash \cdot :: (x:A) }{} & & \infer[]{\bar{x}:\omega \Vdash P :: (y:B)}{\bar{x}:\omega \vdash P :: (y:B)} & & \infer[]{\bar{x}:\omega \Vdash C_1 |_{\ z} \ C_2 :: (y:B)}{\bar{x}:\omega \Vdash C_1:: (z:A) & z:A \Vdash C_2 :: (y:B)}\\\\ \end{tabular}$$ A configuration can be read as a list of processes connected by consecutive cuts. Alternatively, considering $\mathcal{C}_1$ and $\mathcal{C}_2$ as two processes, configuration $\mathcal{C}_1 \mid_{z} \mathcal{C}_2$ can be read as their composition by a cut rule $(z \leftarrow \mathcal{C}_1; \mathcal{C}_2)$. In section \[operat\], we defined an operational semantics on configurations using transition rules. Similarly, these computational transitions can be interpreted as cut reductions called “internal operations” by Fortier and Santocanale . The usual preservation theorem ensures types of a configuration are preserved during computation [@DeYoung16aplas]. \[preservation\] (Preservation) For a configuration $\bar{x}:\omega \Vdash \mathcal{C} :: (y:A) $, if $\mathcal{C} \mapsto \mathcal{C}'$ by one step of computation, then $\bar{u}:\omega \Vdash \mathcal{C}' : (y:A) $. This property follows directly from the correctness of cut reduction steps. The usual progress property as defined below ensures that computation makes progress or it attempts to communicate with an external process. (Progress) Configuration $\bar{x}:\omega \Vdash \mathcal{C}:: (y:A)$ satisfies *progress* if 1. either $\mathcal{C}$ can make a transition, 2. or $\mathcal{C}= (\cdot)$ is empty, 3. or $\mathcal{C}$ attempts to communicate either to the left or to the right. The proof is by structural induction on the configuration typing. In the presence of (mutual) recursion, this progress property is not strong enough to ensure that a program does not get stuck in an infinite inner loop. Since our local validity condition implies the FS guard condition, we can use their results for a stronger version of the progress theorem on valid programs. \[progress\](Strong Progress) Configuration $\bar{x}:\omega \Vdash \mathit{C}:: (y:A)$ of (locally) valid processes satisfies the progress property. Furthermore, after a finite number of steps, either 1. $\mathcal{C}= (\cdot)$ is empty, 2. or $\mathcal{C}$ attempts to communicate to the left or right. There is a correspondence between $\textsc{Treat}$ function’s internal operations and computational transitions introduced in Section \[operat\]. The only point of difference is the extra computation rule we introduced for constant $1$. Fortier and Santocanale’s proof for termination of cut elimination remains intact after extending <span style="font-variant:small-caps;">Treat</span>’s primitive operation with a reduction rule for constant $1$, since this reduction step only introduces a new way of closing a process in the configuration. Under this correspondence, termination of the function $\textsc{Treat}$ on valid proofs implies the strong progress property for valid programs. As a corollary to Theorem \[progress\], computation of a closed valid program $\mathcal{P}= \langle V,S\rangle$ with $\cdot \vdash S=P :: (y:1)$ always terminates by closing the channel $y$ (which follows by inversion on the typing derivation). We conclude this section by briefly revisiting sources of invalidity in computation. In Example \[begen\] we saw that process $\mathtt{Loop}$ is not valid, even though its proof is cut-free. Its computation satisfies the strong progress property as it attempts to communicate with its right side in finite number of steps. However, its communication with left and right sides of the configuration is solely by sending messages. Composing $\mathtt{Loop}$ with any process $y:\mathsf{nat} \vdash \mathtt{P} :: (z:1)$ results in exchanging an infinite number of messages between them. For instance, for ${\mathtt{Block}}$, introduced in Example \[block\], the infinite computation of $\cdot \Vdash y \leftarrow \mathtt{Loop} \mid_{\, y} z \leftarrow \mathtt{Block} \leftarrow y :: (z:1)$ without communication along $z$ can be depicted as follows: -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------- $y \leftarrow \mathtt{Loop} \mid_{\, y} z \leftarrow \mathtt{Block} \leftarrow y$ $\mapsto$ $Ry.\mu_{\mathsf{nat}}; Ry.\mathit{s}; y \leftarrow \mathtt{Loop} \mid_{\, y} z \leftarrow \mathtt{Block} \leftarrow y$ $ \mapsto$ $Ry.\mu_{\mathsf{nat}}; Ry.\mathit{s}; y \leftarrow \mathtt{Loop} \mid_{\, y} \mathbf{case}\, Ly\ (\mu_{\mathsf{nat}}{\Rightarrow}\mathbf{case}\, Ly\ \cdots)$ $\mapsto$ $Ry.s; y \leftarrow \mathtt{Loop} \mid_{\, y} \mathbf{case}\, Ly\ (\mathit{s} \Rightarrow z \leftarrow \mathtt{Block} \leftarrow y \ \mid \mathit{z} \Rightarrow \mathbf{wait}\, Ly;\mathbf{close}\, Rz)$ $ \mapsto$ $y \leftarrow \mathtt{Loop} \mid_{\, y} z \leftarrow \mathtt{Block} \leftarrow y$ $\mapsto$ $\mathbf{\cdots}$ -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------- In this example, the strong progress property of computation is violated. The configuration does not communicate to the left or right and a never ending series of internal communications takes place. This internal loop is a result of the infinite number of unfolding messages sent by $\mathtt{Loop}$ without any unfolding message with higher priority being received by it. In other words, it is the result of $\mathtt{Loop}$ not being valid. In general, strong progress of $\mathtt{P_1} \mid_{\, x} \mathtt{P_2}$ can be violated if either $\mathtt{P_1}$ or $\mathtt{P_2}$ are not valid. In the next section we will see an example of a program with strong progress property that seems to preserve this property after being composed with any other valid program. We will show that neither our local validity algorithm nor the FS guard condition identifies it as valid. In fact, Theorem \[progress\] implies that no effective procedure, including our algorithm, can recognize a maximal set $\Xi$ of programs with the strong progress property that is closed under composition. DeYoung and Pfenning showed that any Turing machine can be represented as a process in subsingleton logic with equirecursive fixed points [@DeYoung16aplas; @Pfenning16lectures]. Using this result and Theorem \[progress\], we can easily reduce the halting problem to identifying closed programs $\mathcal{P}:=\langle V,S \rangle$ with $\cdot \vdash S :: x:1$ that satisfy strong progress. Note that a closed program satisfying strong progress is always in the maximal set $\Xi$. A Negative Result {#negative} ================= In this section we provide a straightforward example of a program with the strong progress property that our algorithm cannot identify as valid. Intuitively, this program seems to preserve the strong progress property after being composed by other valid programs. We show that this example does not satisfy the FS guard condition, either. Define the signature $$\begin{aligned} \Sigma_5 :=\ & \mathsf{ctr}=^{1}_{\nu} \& \{\mathit {inc}: \mathsf{ctr}, \ \ \mathit{val}: \mathsf{bin}\},\\ & \mathsf{bin}=^{2}_{\mu} \oplus\{ \mathit{b0}:\mathsf{bin}, \mathit{b1}:\mathsf{bin}, \mathit{\$}:\mathsf{1}\}\\ \end{aligned}$$ and program $\mathcal{P}_{11}= \langle \{\mathtt{Bit0Ctr},\mathtt{Bit1Ctr}, \mathtt{Empty}\}, \mathtt{Empty} \rangle,$ where $$\begin{array}{l} x:\mathsf{ctr} \vdash y \leftarrow \mathtt{Bit0Ctr} \leftarrow x :: (y:\mathsf{ctr})\\ x:\mathsf{ctr} \vdash y \leftarrow \mathtt{Bit1Ctr}\leftarrow x :: (y:\mathsf{ctr})\\ \cdot \vdash y \leftarrow \mathtt{Empty}:: (y:\mathsf{ctr}) \end{array}$$ with $$\begin{aligned} & y^{\beta} \leftarrow \mathtt{Bit0Ctr}\leftarrow x^{\alpha}= & \phantom{case} {\color{blue}[0, 0,\ 0,\ 0]} & \phantom{L}\\ &\phantom{sm} \mathbf{case}\, Ry^{\beta}\ (\nu_{ctr} {\Rightarrow}& [-1,0,0,0] & \phantom{L} y^{\beta+1}_1<y^\beta_1, y^{\beta+1}_2 = y^{\beta}_2\\ & \phantom{small } \mathbf{case}\, Ry^{\beta+1}\ ( \mathit{inc} \Rightarrow \ y^{\beta+1} \leftarrow \mathtt{Bit1Ctr} \leftarrow x^{\alpha} & {\color{red} [-1,\ 0,\ 0 ,0]} &\phantom{L.}\\ & \phantom{small space mor} \mid \mathit{val} \Rightarrow Ry^{\beta+1}.\mu_{bin};Ry^{\beta+2}.\mathit{b0};Lx^\alpha.\nu_{ctr}; Lx^{\alpha+1}.val; y^{\beta+2} \leftarrow x^{\alpha+1})) & { [-1,1,0,1]} & \\\end{aligned}$$ $$\begin{aligned} & y^{\beta} \leftarrow \mathtt{Bit1Ctr}\leftarrow x^{\alpha}= & \phantom{c} {\color{blue}[0, 0,\ 0,\ 0]} & \phantom{L}\\ &\phantom{sm} \mathbf{case}\, Ry^{\beta}\ (\nu_{ctr} {\Rightarrow}& [-1,0,0,0] & \phantom{L} y^{\beta+1}_1<y^\beta_1, y^{\beta+1}_2 = y^{\beta}_2\\ & \phantom{small } \mathbf{case}\, Ry^{\beta+1}\ ( \mathit{inc} \Rightarrow \ Lx^{\alpha}.\nu_{ctr}; Lx^{\alpha+1}.inc; y^{\beta+1} \leftarrow \mathtt{Bit0Ctr} \leftarrow x^{\alpha+1} & {\color{red} [-1,\ 1,\ 0 ,0]} &\phantom{L.} x^{\alpha+1}_2 = x^{\alpha}_2\\ & \phantom{small space mor} \mid \mathit{val} \Rightarrow Ry^{\beta+1}.\mu_{bin};Ry^{\beta+2}.\mathit{b1};Lx^\alpha.\nu_{ctr}; Lx^{\alpha+1}.val; y^{\beta+2} \leftarrow x^{\alpha+1})) & { [-1,1,0,1]} & \\\end{aligned}$$ $$\begin{aligned} & y^{\beta} \leftarrow \mathtt{Empty}\leftarrow \cdot = & \phantom{caseRy^\beta} {\color{blue}[0, \_,\ \_,\ 0]} & \phantom{L}\\ &\phantom{sma} \mathbf{case}\, Ry^{\beta}\ (\nu_{ctr} {\Rightarrow}& [-1,\_,\_,0] & \phantom{L} y^{\beta+1}_1<y^\beta_1, y^{\beta+1}_2 = y^{\beta}_2\\ & \phantom{small sp} \mathbf{case}\, Ry^{\beta+1}\ ( \mathit{inc} \Rightarrow \ w^{0} \leftarrow \mathtt{Empty} \leftarrow\cdot ; & {\color{red} [\infty,\ \_,\ \_ ,\infty]} &\phantom{L.} \mathsf{ctr}, \mathtt{bin} \in \mathtt{c}(\mathsf{ctr})\\ & \phantom{small spaceMORE THAN that and} y^{\beta+1} \leftarrow \mathtt{Bit1Ctr} \leftarrow w^{0} & {\color{red} [-1,\ \infty, \infty ,0]} &\phantom{L.} \mathsf{ctr}, \mathtt{bin} \in \mathtt{c}(ctr)\\ & \phantom{small space more T} \mid \mathit{val} \Rightarrow Ry^{\beta+1}.\mu_{bin};Ry^{\beta+2}.\$ ; \mathbf{close}\, Ry^{\beta+2})) & { [-1,\_,\_,1]} & \\\end{aligned}$$ In this example we implement a counter slightly differently from Example \[cutchannel\]. We have two processes $\mathtt{Bit0Ctr}$ and $\mathtt{Bit1Ctr}$ that are holding one bit ($b0$ and $b1$ respectively) and an empty counter $\mathtt{Empty}$ that signals the end of the chain of counting processes. This program begins with an empty counter, if a value is requested, then it sends $\$$ to the right and if an increment is requested it adds the counter $\mathtt{Bit1Ctr}$ with $b1$ value to the chain of counters. Then if another increment is asked, $\mathtt{Bit1Ctr}$ sends an increment ($\mathit{inc}$) message to its left counter (implementing the carry bit) and calls $\mathtt{Bit0Ctr}$. If $\mathtt{Bit0Ctr}$ receives an increment from the right, it calls $\mathtt{Bit1Ctr}$ recursively. All (mutually) recursive calls in this program are recognized as valid by our algorithm, except the one in which $\mathtt{Empty}$ calls itself. In this recursive call, $y^\beta\leftarrow \mathtt{Empty}\leftarrow \cdot$ calls $w^0\leftarrow \mathtt{Empty}\leftarrow \cdot$, where $w$ is the fresh channel it shares with $y^{\beta+1} \leftarrow \mathtt{Bit1Ctr} \leftarrow w^0$. The number of increment unfolding messages $\mathtt{Bit1Ctr}$ can send along channel $w^0$ are always less than or equal to the number of increment unfolding messages it receives along channel $y^{\beta+1}$. This implies that the number of messages $w^0\leftarrow \mathtt{Empty}\leftarrow \cdot$ may receive along channel $w^0$ is strictly less than the number of messages received by any process along channel $y^{\beta}$. There will be no infinite loop in the program without receiving an unfolding message from the right. Indeed Fortier and Santocanale’s cut elimination for the cut corresponding to the composition $\mathtt{Empty} \mid \mathtt{Bit1Ctr}$ locally terminates. Furthermore, since no valid program defined on the same signature can send infinitely many increment messages to the left, $\mathcal{P}_{11}$ composed with any other valid program satisfies strong progress. This result is also a negative example for the FS guard condition. The path between $y^{\beta} \leftarrow \mathtt{Empty}\leftarrow \cdot$ and $w^{0} \leftarrow \mathtt{Empty}\leftarrow \cdot$ in the $\mathtt{Empty}$ process is neither left traceable not right traceable since $w \neq y$. By Definition \[guard\] it is therefore not a valid cycle. Concluding Remarks {#conclusion} ================== [**Related work.**]{} The main inspiration for this paper is work by Fortier and Santocanale [@Fortier13csl], who provided a validity condition for pre-proofs in singleton logic with least and greatest fixed points. They showed that valid circular proofs in this system enjoy cut elimination. Also related is work by Baelde et al. [@Baelde16csl; @Baelde12tocl], in which they similarly introduced a validity condition on the pre-proofs in multiplicative-additive linear logic with fixed points and proved the cut-elimination property for valid derivations. Doumane [@Doumane17phd] proved that this condition can be decided in $\mathit{PSPACE}$ by reducing it to the emptiness problem of Büchi automata. Nollet et al. [@Nollet18csl] introduced a local polynomial time algorithm for identifying a stricter version of Baelde’s condition. At present, it is not clear to us how their algorithm would compare with ours on the subsingleton fragment due to the differences between classical and intuitionistic sequents [@Laurent18lics], different criteria on locality, and the prevailing computational interpretation of cut reduction as communicating processes in our work. Cyclic proofs have also been used for reasoning about imperative programs with recursive procedures [@Rowe17cpp]. While there are similarities (such as the use of cycles to capture recursion), their system extends separation logic is therefore not based on an interpretation of cut reduction as computation. Reasoning in their logic therefore has a very different character from ours. DeYoung and Pfenning [@DeYoung16aplas] provide a computational interpretation of subsingleton logic with equirecursive fixed points and showed that cut reduction on circular pre-proofs in this system has the computational power of Turing machines. Their result implies undecidability of determining all programs with a strong progress property. [**Our contribution.**]{} In this paper we have established an extension of the Curry-Howard interpretation for intuitionistic linear logic by Caires et al. [@Caires10concur; @Caires16mscs] to include least and greatest fixed points that can mutually depend on each other in arbitrary ways, although restricted to the subsingleton fragment. The key is to interpret circular pre-proofs in subsingleton logic as mutually recursive processes, and to develop a locally checkable, compositional validity condition on such processes. We proved that our local condition implies Fortier and Santocanale’s guard condition and therefore also implies cut elimination. Analyzing this result in more detail leads to a computational strong progress property which means that a valid program will always terminate either in an empty configuration or one attempting to communicate along external channels. [**Implementation.**]{} We have implemented the algorithm introduced in Section \[alg\] in SML, which is publicly available [@Das19bitbucket]. The implementation collects constraints and uses them to construct a suitable priority ordering over type variables and a $\subset$ ordering over process variables if they exist, and rejects the program otherwise. It also supports an implicit syntax where the fixed point unfolding messages are synthesized from the given communication patterns. Our experience with a range of programming examples shows that our local validity condition is surprisingly effective. Its main shortcoming arises when, intuitively, we need to know that a program’s output is “smaller” than its input. [**Future work.**]{} The main path for the future work is to extend our results to full ordered or linear logic with fixed points and address the known shortcomings we learned about through the implementation. We would also like to investigate the relationship to work by Nollet et al. [@Nollet18csl], carried out in a different context. Acknowledgments {#acknowledgments .unnumbered} =============== The authors would like to acknowledge helpful comments by Stephanie Balzer, Ankush Das, Jan Hoffmann, and Siva Somayyajula on an earlier draft, discussions with Jonas Frey and Henry DeYoung regarding the subject of this paper, and support from the National Science Foundation under grant CCF-1718267 *Enriching Session Types for Practical Concurrent Programming*. [DCPT12]{} David Baelde. Least and greatest fixed points in linear logic. , 13(1):2:1–2:44, 2012. David Baelde, Amina Doumane, and Alexis Saurin. Infinitary proof theory: the multiplicative additive case. In J.-M. Talbot and L. Regnier, editors, [*25th Annual Conference on Computer Science Logic (CSL 2016)*]{}, pages 42:1–42:17, Marseille, France, August 2016. LIPIcs 62. Lars Birkedal and Rasmus Ejlers M[ø]{}gelberg. Intensional type theory with guarded recursive types qua fixed points on universes. In [*28th Annual Symposium on Logic in Computer Science (LICS 2013)*]{}, pages 213–222, New Orleans, LA, USA, June 2013. IEEE Computer Society. Bor-Yuh Evan Chang, Kaustuv Chaudhuri, and Frank Pfenning. A judgmental analysis of linear logic. Technical Report CMU-CS-03-131R, Carnegie Mellon University, Department of Computer Science, December 2003. Lu[í]{}s Caires and Frank Pfenning. Session types as intuitionistic linear propositions. In [*Proceedings of the 21st International Conference on Concurrency Theory (CONCUR 2010)*]{}, pages 222–236, Paris, France, August 2010. Springer LNCS 6269. Lu[í]{}s Caires, Frank Pfenning, and Bernardo Toninho. Linear logic propositions as session types. , 26(3):367–423, 2016. Special Issue on Behavioural Types. Henry DeYoung, Lu[í]{}s Caires, Frank Pfenning, and Bernardo Toninho. Cut reduction in linear logic as asynchronous session-typed communication. In P. C[é]{}gielski and A. Durand, editors, [*Proceedings of the 21st Annual Conference on Computer Science Logic (CSL 2012)*]{}, pages 228–242, Fontainebleau, France, September 2012. LIPIcs 16. Ankush Das, Farzaneh Derakhshan, and Frank Pfenning. Subsingleton. <https://bitbucket.org/fpfenning/subsingleton>, June 2019. An implementation of subsingleton logic with ergometric and temporal types. Henry DeYoung. . PhD thesis, Computer Science Department, Carnegie Mellon University, 2019. Forthcoming. Amina Doumane. . PhD thesis, Paris Diderot University, France, June 2017. Henry DeYoung and Frank Pfenning. Substructural proofs as automata. In A. Igarashi, editor, [*14th Asian Symposium on Programming Languages and Systems*]{}, pages 3–22, Hanoi, Vietnam, November 2016. Springer LNCS 10017. Invited talk. J[é]{}r[ô]{}me Fortier and Luigi Santocanale. Cuts for circular proofs: Semantics and cut-elimination. In Simona Ronchi Della Rocca, editor, [*22nd Annual Conference on Computer Science Logic (CSL 2013)*]{}, pages 248–262, Torino, Italy, September 2013. LIPIcs 23. Jean-Yves Girard and Yves Lafont. Linear logic and lazy computation. In H. Ehrig, R. Kowalski, G. Levi, and U. Montanari, editors, [ *Proceedings of the International Joint Conference on Theory and Practice of Software Development*]{}, volume 2, pages 52–66, Pisa, Italy, March 1987. Springer-Verlag LNCS 250. Hans Brugge Grathwohl. . PhD thesis, Department of Computer Science, Aarhus University, Denmark, September 2016. Dennis Griffith. . PhD thesis, University of Illinois at Urbana-Champaign, April 2016. Simon J. Gay and Vasco T. Vasconcelos. Linear type theory for asynchronous session types. , 20(1):19–50, January 2010. Kohei Honda. Types for dyadic interaction. In [*4th International Conference on Concurrency Theory*]{}, CONCUR’93, pages 509–523. Springer LNCS 715, 1993. W. A. Howard. The formulae-as-types notion of construction. Unpublished note. An annotated version appeared in: *To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism*, 479–490, Academic Press (1980), 1969. Kohei Honda, Vasco T. Vasconcelos, and Makoto Kubo. Language primitives and type discipline for structured communication-based programming. In [*7th European Symposium on Programming Languages and Systems (ESOP 1998)*]{}, pages 122–138. Springer LNCS 1381, 1998. Olivier Laurent. Around classical and intuitionistic linear logic. In A. Dawar and E. Gr[ä]{}del, editors, [*Proceedings of the 33rd Annual Symposium on Logic in Computer Science (LICS 2018)*]{}, pages 629–638, Oxford, UK, July 2018. ACM. Sam Lindley and J. Garrett Morris. Talking bananas: Structural recursion for session types. In J. Garrigue, G. Keller, and E. Sumii, editors, [*Proceedings of the 21st International Conference on Functional Programming*]{}, pages 434–447, Nara, Japan, September 2016. ACM Press. R[é]{}mi Nollet, Alexis Saurin, and Christine Tasson. Local validity for circular proofs in linear logic with fixed points. In D. Ghica and A. Jung, editors, [*27th Annual Conference on Computer Science Logic (CSL 2018)*]{}, pages 35:1–35:23. LIPIcs 119, 2018. Frank Pfenning. Substructural logics. Lecture notes for course given at Carnegie Mellon University, Fall 2016, December 2016. Reuben N. S. Rowe and James Brotherston. Automatic cyclic termination proofs for recursive procedures in separation logic. In Y. Bertot and V. Vafeiadis, editors, [*Proceedings of the 6th Conference on Certified Programs and Proofs (CPP 2017)*]{}, pages 53–65, Paris, France, January 2017. ACM. Luigi Santocanale. A calculus of circular proofs and its categorical semantics. In M. Nielsen and U. Engberg, editors, [*5th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS 2002*]{}, pages 357–371, Grenoble, France, April 2002. Springer LNCS 2303. Luigi Santocanale. -bicomplete categories and parity games. , 36(2):195–227, 2002. Bernardo Toninho, Lu[í]{}s Caires, and Frank Pfenning. Higher-order processes, functions, and sessions: A monadic integration. In M.Felleisen and P.Gardner, editors, [*Proceedings of the European Symposium on Programming (ESOP’13)*]{}, pages 350–369, Rome, Italy, March 2013. Springer LNCS 7792. Philip Wadler. Propositions as sessions. In [*Proceedings of the 17th International Conference on Functional Programming*]{}, ICFP 2012, pages 273–286, Copenhagen, Denmark, September 2012. ACM Press. {#proofapp} Here we provide the proof for the observations we made in Section \[Guard\]. We prove that every program accepted by our algorithm in Section \[alg\] corresponds to a valid circular proof in the sense of the FS guard condition. As explained in Section \[priority\], the reflexive transitive closure of $\Omega$ in judgment ${x^\gamma: \omega \vdash_{\Omega} P :: (y^{\delta}:C) }$ forms a partial order $\le_{\Omega}$. To enhance readability of proofs, throughout this section we use entailment $\Omega \Vdash x \le y$ instead of $x\le_{\Omega}y$. We first prove Lemmas \[ordtrace\] and \[dordtrace\]. Theorem \[listor\] is a direct corollary of these two lemmas. \[ordtrace\] Consider a path $\mathbb{P}$ on a program $\mathcal{Q}=\langle V, S\rangle$ defined on a Signature $\Sigma$ $${\infer{z^\alpha: \omega \vdash_{\Omega} P :: (w^{\beta}:C)}{ \infer{\vdots}{{x^\gamma: \omega' \vdash_{\Omega'} P' :: (y^{\delta}:C') }}} }$$ with $n$ the maximum priority in $\Sigma$. 1. For every $i \in \mathtt{c}(\omega') $ with $\epsilon_i=\mu$, if $\Omega' \Vdash x^\gamma_i \le z^\alpha_i$ then $x=z$ and $i \in \mathtt{c}(\omega) $. 2. For every $i<n$, if $\Omega' \Vdash x^\gamma_i < z^\alpha_i$, then $i \in \mathtt{c}(\omega) $ and a $\mu L$ rule with priority $i$ is applied on $\mathbb{P}$. 3. For every $c \le n$ with $\epsilon_c=\nu$, if $\Omega' \Vdash x^\gamma_c \le z^\alpha_c$ , then no $\nu L$ rule with priority $c$ is applied on $\mathbb{P}$. Proof is by induction on the structure of $\mathbb{P}$. We consider each case for last (topmost) step in $\mathbb{P}$. Case : $$\infer[{\mbox{\sc Def}}(X)]{x^\gamma:\omega' \vdash_{\Omega'} y^\delta \leftarrow X \leftarrow x^\gamma ::(y^{\delta}:C')}{x^\gamma: \omega' \vdash_{\Omega'} P' :: (y^{\delta}:C') & x:\omega' \vdash X=P' :: (y:C') \in V }$$ None of the conditions in the conclusion are different from the premise. Therefore, by the induction hypothesis, statements (a)-(c) hold. Case : $$\infer[{\mbox{\sc Cut}}^{y}]{ x^\gamma: \omega' \vdash_{\Omega''} (y \leftarrow P'_y ; Q_y) :: (v^{\theta}:C'')}{ x^\gamma: \omega' \vdash_{\Omega''\cup \mathtt{r}(v^\theta)} P'_{y^0} :: (y^0: C') & y^0: C' \vdash_{\Omega''\cup \mathtt{r}(x^\gamma)} Q_{y^0} ::(v^{\theta}: C'')}$$ where $\mathtt{r}(u)= \{y^{0}_{p(s)}=u_{p(s)}\mid p(s) \not \in \mathtt{c}(C')\}$ and $\Omega'=\Omega'' \cup r(v^{\theta})$. All conditions in the conclusion are the same as the premise. Therefore, by the induction hypothesis, statements (a)-(c) hold. Case : $$\infer[{\mbox{\sc Cut}}^{x}]{ u^\eta: \omega'' \vdash_{\Omega''} (x \leftarrow Q_x ; P'_x) :: (y^{\delta}:C')}{ u^\eta: \omega'' \vdash_{\Omega''\cup \mathtt{r}(y^\delta)} Q_{a^0} :: (x^0: A) & x^0: A \vdash_{\Omega''\cup \mathtt{r}(u^\eta)} P'_{x^0} ::(y^{\delta}: C'')}$$ where $\mathtt{r}(v)= \{x^{0}_{p(s)}=v_{p(s)}\mid p(s) \not \in \mathtt{c}(A)\}$ and $\Omega'=\Omega'' \cup r(u^\eta)$. 1. $\Omega'' \cup r(u^\eta) \Vdash x^0_i \le z^\alpha_i$ does not hold for any $i\in c(A)$ since $x$ is a fresh channel. Therefore, this part is vacuously true. 2. By freshness of $x$, if $\Omega'' \cup r(u^\eta)\Vdash x_i^{0} < z^{\alpha}_{i}$, then $x_i^{0} ={u^{\eta}_{i}} \in r(u^\eta)$ and $\Omega'' \Vdash u^{\eta}_{i}< z^{\alpha}_{i}$. By the induction hypothesis, $i \in \mathtt{c}(\omega)$ and a $\mu L$ rule with priority $i$ is applied on $\mathbb{P}$. 3. By freshness of $x$, if $\Omega'' \cup r(u^\eta)\Vdash x_c^{0} \le z^{\alpha}_{c}$, then $x_c^{0} ={u^{\eta}_{c}} \in r(u^\eta)$ and $\Omega'' \Vdash u^{\eta}_{c}\le z^{\alpha}_{c}$. By the induction hypothesis, no $\nu L$ rule with priority $c$ is applied on $\mathbb{P}$. Case : $$\infer[1L]{u^\eta: 1 \vdash_{\Omega'} \mathbf{wait}\, Lu^\eta;P' :: (y^{\delta}: C')}{ . \vdash_{\Omega'} P' :: (y^{\delta}: C') }$$ This case is not applicable since by the typing rules $\Omega' \not \Vdash .\le z^{\alpha}_i$ for any $i\le n$. Case : $$\infer[\mu R]{ x^\alpha: \omega' \vdash_{\Omega''} Ry^{\delta'}.\mu_t; P :: (y^{\delta'}:t)}{x^\alpha: \omega' \vdash_{\Omega'} P :: (y^{\delta'+1}:C') & t=_{\mu}C' & \Omega'= \Omega'' \cup \{(y^{\delta'})_{p(s)} =(y^{\delta'+1})_{p(s)} \mid p(s)\neq p(t)\}}$$ For every $i\le n$, if $\Omega'' \cup \{(y^{\delta'})_{p(s)} =(y^{\delta'+1})_{p(s)} \mid p(s)\neq p(t)\} \Vdash x^\gamma_i \le z^\alpha_i$, then $\Omega'' \Vdash x^\gamma_i \le z^\alpha_i $. Therefore, by the induction hypothesis, statements (a)-(c) hold. Case : $$\infer[\mu L]{x^{\gamma'}: t \vdash_{\Omega''} \mathbf{case}\, Lx^{\gamma'}\ (\mu_{t} {\Rightarrow}P'):: (y^{\delta}:C')}{ x^{\gamma'+1}: \omega' \vdash_{\Omega'} P' :: (y^{\delta}:C') & t=_{\mu} \omega' & \Omega'=\Omega'' \cup \{x^{\gamma'+1}_{p(t)} < x^{\gamma'}_{p(t)}\} \cup \{x^{\gamma'+1}_{p(s)} =x^{\gamma'}_{p(s)} \mid p(s)\neq p(t)\}}$$ By definition of $\mathtt{c}(x)$, we have $\mathtt{c}(\omega') \subseteq \mathtt{c}(t)$. By $\mu L$ rule, for all $i\le n$, $x^{\gamma'+1}_{i} \le x^{\gamma'}_i \in \Omega'$. But by freshness of channels and their generations, $x^{\gamma'+1}$ is not involved in any relation in $\Omega''$. 1. For every $i \in \mathtt{c}(\omega')$ with $\epsilon_{i}=\mu$, if $\Omega' \Vdash x^{\gamma'+1}_{i} \le z^{\alpha}_{i}$ then $\Omega'' \Vdash x^{\gamma'}_{i} \le z^{\alpha}_{i}$. By the induction hypothesis, we have $x=z$ and $i \in \mathtt{c}(\omega)$. 2. We consider two subcases: (1) If $\Omega' \Vdash x^{\gamma'+1}_i< z_i^{\alpha}$ for $i \neq p(t)$, then $\Omega'\Vdash x^{\gamma'+1}_i= x^{\gamma'}_i$ and $\Omega'' \Vdash x^{\gamma'}_i< z_i^{\alpha}$. Now we can apply the induction hypothesis. (2) If $\Omega' \Vdash x^{\gamma'+1}_{p(t)}< z_{p(t)}^{\alpha}$, then $\Omega' \Vdash x^{\gamma'+1}_{p(t)} < x^{\gamma'}_{p(t)}$ and $\Omega'' \Vdash x^{\gamma'}_{p(t)} \le z^{\alpha}_{p(t)}$. Since a $\mu L$ rule is applied in this step on the priority $p(t)$, we only need to prove that $p(t) \in \mathtt{c}(\omega)$. By definition of $\mathtt{c}$, we have $p(t) \in \mathtt{c}(t) $ and we can use the induction hypothesis on part (a) to get $p(t) \in \mathtt{c}(\omega)$. 3. For every $c \le n$ with $\epsilon_{c}= \nu$, $\Omega' \Vdash x^{\gamma'+1}_c= x^{\gamma'}_c$ as $c \neq p(t)$. Therefore, if $\Omega' \Vdash x^{\gamma'+1}_c= z^{\alpha}_c$, then $\Omega'' \Vdash x^{\gamma'}_c= z^{\alpha}_c$. By the induction hypothesis no $\nu L$ rule with priority $c$ is applied on $\mathbb{P}$. Case : $$\infer[\nu R]{x^\gamma: \omega' \vdash_{\Omega''} \mathbf{case}\, Ry^{\delta'} \ (\nu_t {\Rightarrow}P') :: (y^{\delta'}:t)}{ x^\gamma: \omega' \vdash_{\Omega'} P' :: y^{\delta'+1}: C'& t=_{\nu}C'& \Omega'= \Omega'' \cup \{y^{\delta'+1}_{p(t)} < y^{\delta'}_{p(t)}\} \cup \{y^{\delta'+1}_{p(s)} =y^{\delta'}_{p(s)} \mid p(s)\neq p(t)\} }$$ For every $i\le n$, if $\Omega'' \cup \{y^{\delta'+1}_{p(t)} < y^{\delta'}_{p(t)}\} \cup \{y^{\delta'+1}_{p(s)} =y^{\delta'}_{p(s)} \mid p(s)\neq p(t)\} \Vdash x^\gamma_i \le z^\alpha_i$, then $\Omega'' \Vdash x^\gamma_i \le z^\alpha_i $. Therefore, by the induction hypothesis, statements (a)-(c) hold. Case : $$\infer[\nu L]{x^{\gamma'}: t \vdash_{\Omega''} Lx^{\gamma'}.\nu_{t}; P':: (y^{\delta}:C')}{ x^{\gamma'+1}: \omega' \vdash_{\Omega'} Q :: (y^{\delta}:C') & t=_{\nu} \omega' & \Omega'=\Omega'' \cup \{x^{\gamma'+1}_{p(s)} =x^{\gamma'}_{p(s)} \mid p(s)\neq p(t)\} }$$ By definition of $\mathtt{c}(x)$, we have $\mathtt{c}(\omega') \subseteq \mathtt{c}(t)$. By $\nu L$ rule, for all $i\neq p(t)\le n$, $x^{\gamma'+1}_{i} = x^{\gamma'}_i \in \Omega'$. In particular, for every $i \le n $ with $\epsilon_i=\mu$, $x^{\gamma'+1}_{i} = x^{\gamma'}_i \in \Omega'$. But by freshness of channels and their generations, $x^{\gamma'+1}$ is not involved in any relation in $\Omega''$. 1. For every $i \in \mathtt{c}(\omega') $ with $\epsilon_i=\mu$, if $\Omega' \Vdash x^{\gamma'+1}_i \le z^\alpha_i$, then $\Omega' \Vdash x^{\gamma'+1}_i=x^{\gamma'}_i$ and $\Omega'' \Vdash x^{\gamma'}_i \le z^\alpha_i$. By the induction hypothesis $x=z$ and $i \in \mathtt{c}(\omega)$. 2. If $\Omega' \Vdash x^{\gamma'+1}_i< z_i^{\alpha}$, then by freshness of channels and their generations we have $i \neq p(t)$, $\Omega' \Vdash x^{\gamma'+1}_i=x^{\gamma'}_i$ and $\Omega'' \Vdash x^{\gamma'}_i< z_i^{\alpha}$. By the induction hypothesis $i \in \mathtt{c}(\omega)$ and a $\mu L$ rule with priority $i$ is applied on the path. 3. For every $c \le n$ with $\epsilon_{c}= \nu$ and $c\neq p(t)$, if $\Omega' \Vdash x^{\gamma'+1}_c \le z^\alpha_c$, then $\Omega'\Vdash x^{\gamma'+1}_c = x^{\gamma'}_c$ and $\Omega'' \Vdash x^{\gamma'} \le z^\alpha_c$. Therefore, by induction hypothesis, no $\nu L$ rule with priority $c$ is applied on the path. Note that $\Omega' \not \Vdash x^{\gamma'+1}_p(t) \le z^{\alpha}_p(t)$. Case : $$\infer[\& R]{x^\gamma: \omega \vdash_{\Omega'} \mathbf{case}\, Ry^\delta \ (\ell {\Rightarrow}Q_{\ell}) :: (y^{\delta}:\&\{\ell:A_{\ell}\}_{\ell \in L})}{ x^\gamma: \omega \vdash_{\Omega'} Q_k :: y^{\delta}: A_k & \forall k \in L }$$ None of the conditions in the conclusion are different from the premise. Therefore, by the induction hypothesis, statements (a)-(c) hold. Case : $$\infer[\& L]{x^\gamma: \&\{\ell:A_{\ell}\}_{\ell \in L} \vdash_{\Omega'} Lx^\gamma.k; Q:: (y^{\delta}:C')}{ x^{\gamma}: A_k \vdash_{\Omega'} Q :: (y^{\delta}:C') }$$ By definition of $\mathtt{c}(x)$, we have $\mathtt{c}(A_k) \subseteq \mathtt{c}(\&\{\ell:A_{\ell}\}_{\ell \in L})$. Therefore, statements (a)-(c) follow from the induction hypothesis. Cases : The statements are trivially true if the last step of the proof is either $1R$ or ${\mbox{\sc Id}}$ rules. \[dordtrace\] Consider a path $\mathbb{P}$ on a program $\mathcal{Q}=\langle V, S\rangle$ defined on a Signature $\Sigma$, $${\infer{\bar{z}^\alpha: \omega \vdash_{\Omega} P ::(w^{\beta}: C)}{ \infer{\vdots}{{\bar{x}^\gamma: \omega' \vdash_{\Omega'} P' :: ( y^{\delta}:C') }}} }$$ with $n$ the maximum priority in $\Sigma$. 1. For every $i \in \mathtt{c}(\omega')$ with $\epsilon_i=\nu$, if $\Omega' \Vdash y^{\delta}_i \le w^{\beta}_i$, then $y=w$ and $i \in \mathtt{c}(\omega)$. 2. If $\Omega' \Vdash y^{\delta}_i< w_i^{\beta}$, then $i \in \mathtt{c}(\omega) $ and a $\nu L$ rule with priority $i$ is applied on $\mathbb{P}$ . 3. For every $c\le n$ with $\epsilon_c=\mu$, if $\Omega'\Vdash y^{\delta}_c \le w^{\beta}_c$, then no $\mu R$ rule with priority $c$ is applied on $\mathbb{P}$ . Dual to the proof of Lemma \[ordtrace\]. \[basics\] Consider a path $\mathbb{P}$ on a program $\mathcal{Q}=\langle V, S\rangle$ defined on a Signature $\Sigma$, with $n$ the maximum priority in $\Sigma$. $${\infer{\bar{z}^\alpha: \omega \vdash_{\Omega} P ::(w^{\beta}: C)}{ \infer{\vdots}{{\bar{x}^\gamma: \omega' \vdash_{\Omega'} P' :: ( y^{\delta}:C') }}} }$$ $\Omega'$ preserves the (in)equalities in $\Omega$. In other words, for channels $u,v$, generations $\eta, \eta' \in \mathbb{N}$ and type priorities $i, j\le n$, 1. If $\Omega \Vdash u_i^\eta < v_j^{\eta'}$, then $\Omega' \Vdash u_i^\eta <v_j^{\eta'}$. 2. If $\Omega \Vdash u_i^\eta \le v_j^{\eta'}$, then $\Omega' \Vdash u_i^\eta \le v_j^{\eta'}$. 3. If $\Omega \Vdash u_i^\eta = v_j^{\eta'}$, then $\Omega' \Vdash u_i^\eta = v_j^{\eta'}$. Proof is by induction on the structure of $\mathbb{P}$. We consider each case for topmost step in $\mathbb{P}$. Here, we only give one non-trivial case. The proof of other cases is similar. Case : $$\infer[\mu R]{ x^\alpha: \omega' \vdash_{\Omega''} Ry^{\delta'}.\mu_t; P :: (y^{\delta'}:t)}{x^\alpha: \omega' \vdash_{\Omega'} P :: (y^{\delta'+1}:C') & t=_{\mu}C' & \Omega'= \Omega'' \cup \{(y^{\delta'})_{p(s)} =(y^{\delta'+1})_{p(s)} \mid p(s)\neq p(t)\}}$$ - If $\Omega \Vdash u_i^\eta < v_j^{\eta'}$, then by the inductive hypothesis, $\Omega'' \Vdash u_i^\eta < v_j^{\eta'}$. By freshness of channels and their generations, we know that $y^{\delta'+1}$ does not occur in any (in)equalities in $\Omega''$ and thus $y^{\delta'+1} \neq u^\eta, v^{\eta'}$. Therefore $\Omega' \Vdash u_i^\eta < v_j^{\eta'}.$ Following the same reasoning, we can prove statements (b) and (c). \[algtoinf\] Consider a finitary derivation (Figure \[fig:validity\]) for $$\langle\bar{u}, X, v\rangle; \bar{x}^\alpha: \omega \vdash_{\Omega} P ::(y^\beta:C),$$ on a locally valid program $\mathcal{Q}= \langle V, S \rangle$ defined on signature $\Sigma$ and order $\subset$. There is a (potentially infinite) derivation $\mathbb{D}$ for $$\bar{x}^\alpha: \omega \vdash_{\Omega} P ::(y^\beta:C),$$ based in the infinitary rule system of Figure \[fig:stp-order\]. Moreover, for every $\bar{w}^{\gamma}:\omega' \vdash_{\Omega'} z^{\delta} \leftarrow Y \leftarrow \bar{w}^{\gamma} :: (z^{\delta}:C')$ on $\mathbb{D}$, we have $$Y,list(\bar{w}^{\gamma},z^{\delta}) \mathrel{(\subset, <_{\Omega'})} X,list(\bar{x}^{\alpha},y^{\beta}).$$ We prove this by coinduction, producing the derivation of $\bar{x}^\alpha: \omega \vdash_{\Omega} P ::(y^\beta:C).$ We proceed by case analysis of the first rule applied on $\langle \bar{u}, X, v\rangle; \bar{x}^\alpha: \omega \vdash_{\Omega} P ::(y^\beta:C),$ in its finite derivation. Case : $$\infer[{\mbox{\sc Call}}(Y)]{\langle \bar{u}^\gamma, X , v^\delta \rangle; \bar{x}^{\alpha}:\omega \vdash_{\Omega, \subset} y^{\beta} \leftarrow Y \leftarrow \bar{x}^{\alpha} ::(y^{\beta}:C)}{Y, list(\bar{x}^\alpha, y^\beta) \mathrel{(\subset, <_{\Omega})} X, list(\bar{u}^\gamma, v^\delta) & \bar{x}:\omega \vdash Y=P'_{\bar{x}, y} :: (y:C) \in V }$$ By validity of program there is a finitary derivation for $$\langle\bar{x}^0; Y; y^0\rangle;\bar{x}^0: \omega \vdash_{\emptyset, \subset} P'_{\bar{x}^0, y^0} ::(y^0):C.$$ Having Proposition \[subuv\] and freshness of future generations of channels in $\Omega$, there is also a finitary derivation for $$\langle\bar{x}^{\alpha}; Y; y^{\beta}\rangle;\bar{x}^{\alpha}: \omega \vdash_{\Omega, \subset} P'_{\bar{x}^{\alpha}, y^{\beta}} ::(y^{\beta}:C).$$ We apply the coinductive hypothesis to get an infinitary derivation $\mathbb{D'}$ for $$\bar{x}^{\alpha}: \omega \vdash_{\Omega, \subset} P'_{\bar{x}^{\alpha}, y^{\beta}} ::(y^{\beta}:C),$$ and then produce the last step of derivation $$\infer[{\mbox{\sc Def}}(Y)]{ \bar{x}^{\alpha}:\omega \vdash_{\Omega} y^{\beta} \leftarrow Y \leftarrow \bar{x}^{\alpha} ::(y^{\beta}:C)}{\deduce{\bar{x}^{\alpha}:\omega \vdash_{\Omega} P'_{\bar{x}^{\alpha}, y^{\beta}} ::(y^{\beta}:C)}{\mathbb{D'}} & \bar{x}:\omega \vdash Y=P'_{\bar{x}, y} :: (y:C) \in V }$$ in the infinitary rule system. Moreover, by the coinductive hypothesis, we know that for every $$\bar{w}^{\gamma'}:\omega' \vdash_{\Omega'} z^{\delta'} \leftarrow W \leftarrow w^{\gamma'} :: (z^{\delta'}:C')$$ on $\mathbb{D'}$, we have $$W,list(w^{\gamma'},z^{\delta'}) \mathrel{(\subset, <_{\Omega'})} Y,list(\bar{x}^{\alpha},y^{\beta}).$$ By Lemma \[basics\], we conclude from $Y, list(\bar{x}^\alpha, y^\beta) \mathrel{(\subset, <_{\Omega})} X, list(\bar{u}^\gamma, v^\delta)$ that $$Y, list(\bar{x}^\alpha, y^\beta) \mathrel{(\subset, <_{\Omega'})} X, list(\bar{u}^\gamma, v^\delta).$$ By transitivity of $(\subset, <_{\Omega'}),$ we get $$W,list(w^{\gamma'},z^{\delta'}) \mathrel{(\subset, <_{\Omega'})} X, list(\bar{u}^\gamma, v^\delta).$$ This completes the proof of this case as we already know $Y, list(\bar{x}^\alpha, y^\beta) \mathrel{(\subset, <_{\Omega})} X, list(\bar{u}^\gamma, v^\delta)$. Case : $$\infer[{\mbox{\sc Cut}}^{z}]{\langle \bar{u}^\gamma, X , v^\delta \rangle; \bar{x}^\alpha: \omega \vdash_{\Omega, \subset} (z \leftarrow Q_z ; Q'_z) :: (y^{\beta}:C)}{ \langle \bar{u}^\gamma, X , v^\delta \rangle; \bar{x}^\alpha: \omega \vdash_{\Omega \cup \mathtt{r}(y^\beta), \subset} Q_{z^0} :: (z^0: C') & \langle \bar{u}^\gamma, X , v^\delta \rangle;z^0: C' \vdash_{\Omega\cup \mathtt{r}(x^\alpha), \subset} Q'_{z^0} ::(y^{\beta}: C)},$$ where $\mathtt{r}(w)= \{z^{0}_{p(s)}=w_{p(s)}\mid p(s) \not \in \mathtt{c}(A)\}.$ By coinductive hypothesis, we have infinitary derivations $\mathbb{D'}$ ad $\mathbb{D''}$ for $\bar{x}^\alpha: \omega \vdash_{\Omega \cup \mathtt{r}(y^\beta)} Q_{z^0} :: (z^0: C')$ and $z^0: C' \vdash_{\Omega\cup \mathtt{r}(x^\alpha)} Q'_{z^0} ::(y^{\beta}: C),$ respectively. We can produce the last step of the derivation as $$\infer[{\mbox{\sc Cut}}^{z}]{ \bar{x}^\alpha: \omega \vdash_{\Omega} (z \leftarrow Q_z ; Q'_z) :: (y^{\beta}:C)}{ \deduce{ \bar{x}^\alpha: \omega \vdash_{\Omega \cup \mathtt{r}(y^\beta)} Q_{z^0} :: (z^0: C')}{\mathbb{D'}} & \deduce{z^0: C' \vdash_{\Omega\cup \mathtt{r}(x^\alpha)} Q'_{z^0} ::(y^{\beta}: C)}{\mathbb{D''}}}$$ Moreover, by the coinductive hypothesis, we know that for every $$\bar{w}^{\gamma}:\omega' \vdash_{\Omega'} z^{\delta} \leftarrow W \leftarrow \bar{w}^{\gamma} :: (z^{\delta}:C')$$ on $\mathbb{D'}$ and $\mathbb{D''}$, and thus $\mathbb{D}$, we have $$W,list(\bar{w}^{\gamma},z^{\delta}) \mathrel{(\subset, <_{\Omega'})} X,list(\bar{x}^{\alpha},y^{\beta}).$$ Cases : The proof of the other cases are similar by applying the coinductive hypothesis and the infinitary system rules. [6]{} A locally valid program satisfies the FS guard condition. Consider a cycle $\mathbb{C}$ on a (potentially infinite) derivation produced from $\langle \bar{u}, Y, v \rangle; \bar{z}^\alpha: \omega \vdash_{\Omega} w^\beta \leftarrow X \leftarrow \bar{z}^\alpha :: (w^{\beta}:C)$ as in Lemma \[algtoinf\], $$\infer[{\mbox{\sc Def}}(X)]{\bar{z}^\alpha: \omega \vdash_{\Omega} w^\beta \leftarrow X \leftarrow \bar{z}^\alpha :: (w^{\beta}:C)}{\infer{\bar{z}^\alpha: \omega \vdash_{\Omega} P_{z^\alpha, w^\beta} :: (w^{\beta}:C)}{ \infer{\vdots}{\infer[{\mbox{\sc Def}}(X)]{\bar{x}^\gamma: \omega \vdash_{\Omega'} y^\delta \leftarrow X \leftarrow \bar{x}^\gamma :: (y^{\delta}:C) }{\bar{x}^\gamma: \omega \vdash_{\Omega'} P_{x^\gamma, y^\delta} :: (y^{\delta}:C) & \bar{z}:\omega \vdash X=P_{\bar{z},w} :: (w:C) \in V }}} &\hspace{-90pt} \bar{z}:\omega \vdash_ X=P_{\bar{z},w} :: (w:C) \in V}$$ By Lemma \[algtoinf\] we get $$X, list(\bar{x}^\gamma, y^{\delta})\ (\subset, <_{\Omega'})\ X, list(\bar{z}^\alpha,w^{\beta}),$$ and thus by definition of $(\subset, <_{\Omega'})$, $$list (\bar{x}^\gamma, y^{\delta}) <_{\Omega'} list (\bar{z}^\alpha,w^{\beta}).$$ Therefore, there is an $i\le n$, such that either 1. $\epsilon_i=\mu,$ $x^\gamma_i < z^\alpha_i,$ and $x^\gamma_l = z^\alpha_l$ for every $l<i$, having that $\bar{x}=x$ and $\bar{z}=z$ are non-empty, or 2. $\epsilon_i=\nu,$ $y^{\delta}_i < w^{\beta}_i,$ and $y^{\delta}_l = w^{\beta}_l$ for every $l<i$. In the first case, by part (b) of Lemma \[ordtrace\], a $\mu L$ rule with priority $i \in \mathtt{c}(\omega)$ is applied on $\mathbb{C}$. By part (a) of the same Lemma $x=z$, and by its part (c), no $\nu L$ rule with priority $c<i\ $ is applied on $\mathbb{C}$. Therefore, $\mathbb{C}$ is a left $\mu$- trace.\ In the second case, by part (b) of Lemma \[dordtrace\], a $\nu R$ rule with priority $i \in \mathtt{c}(\omega)$ is applied on $\mathbb{C}$. By part (a) of the same Lemma $y=w$ and by its part (c), no $\mu R$ rule with priority $c<i\ $ is applied on $\mathbb{C}$. Thus, $\mathbb{C}$ is a right $\nu$- trace. [^1]: To represent this incomparability in our examples we write “$\infty$” for the value of the fresh channel.

--- author: - 'Yu-Cheng <span style="font-variant:small-caps;">Lin</span>$^{1}$, Heiko <span style="font-variant:small-caps;">Rieger</span>$^{2}$ and Ferenc <span style="font-variant:small-caps;">Iglói</span>$^{3,4}$' title: Antiferromagnetic Heisenberg chains with bond alternation and quenched disorder --- Introduction ============ Low-dimensional quantum systems (antiferromagnetic spin chains, ladders, two-dimensional systems, etc.) are fascinating objects which have been investigated intensively in experimental and theoretical works. From the theoretical point of view these systems exhibit several unusual properties, for example quasi-long-range order, topological string order, a spin liquid state, quantum phase transitions, etc.. Many of these features can be illustrated by the $S=1/2$ antiferromagnetic Heisenberg chain, which is defined by the Hamiltonian: $$H=\sum_i 2 J_i {\mib S}_i \cdot {\mib S}_{i+1}\;, \label{hamilton}$$ in terms of the spin-$1/2$ operators, ${\mib S}_i$, at site $i$. The homogeneous model with $J_i=J$ is gapless and there is quasi-long-range order in the ground state, i.e. spin-spin correlations decay algebraically,[@lutherpeschel] $\langle S^{\tau}_i S^{\tau}_{i+r}\rangle \sim r^{-1}$. Here $\tau=x,y,z$ and $\langle \dots \rangle$ stands for the ground-state expectation value. Introducing bond alternation such that $J_i=J$ at $i={\rm even}$ and $J_i=\alpha_d J$, ($\alpha_d >0$) at $i={\rm odd}$, the system with $\alpha_d \ne 1$ is in the dimerized phase,[@crossfisher] in which spin-spin correlations are short-ranged and the excitation spectrum has a finite gap. In the bond alternating model the quantity $\delta_d=\ln \alpha_d$ serves as a quantum control parameter and the quantum critical point is located at $\delta_d^c=0$. Quenched (i.e. time independent) disorder has a profound effect on the low-temperature/low-energy properties of quantum systems,[@sg-review] both at the quantum critical point and in an extended region of the off-critical regime–the so called quantum Griffiths phase[@griffiths; @mccoy]. In the disordered version of the uniform $S=1/2$ antiferromagnetic Heisenberg chain defined in eq.(\[hamilton\]), the couplings, $J_i$, are independent and identically distributed random variables. A detailed study by Fisher[@fisher] using an asymptotically exact strong disorder renormalization group (SDRG) method[@MDH] revealed that the ground state of the random model is the so called random singlet phase, in which singlets are formed between spin pairs which could be arbitrarily far apart. Average spin-spin correlations, which are dominated by rare regions, are quasi-long-ranged: $[\langle S_i S_{i+r}\rangle]_{\rm av} \sim r^{-2}$, where $[ \dots ]_{\rm av}$ stands for averaging over quenched disorder. On the other hand, [*typical*]{} correlations are much weaker and decay asymptotically as $\ln(\langle S_i S_{i+r}\rangle_{\rm typ}) \sim r^{1/2}$. In addition, the dynamical scaling in the random singlet phase is anomalous. The length scale $L$ and the energy scale $\Delta$, measured by the lowest gap, are related logarithmically: $$\ln \Delta \sim L^{\psi}, \quad \psi=1/2\;. \label{gap_rs}$$ In the random bond alternating Heisenberg model bonds at even ($J_{\rm e}$) and odd ($J_{\rm o}$) sites are taken from different distributions and the quantum control parameter is defined as $$\delta=[\ln J_{\rm e}]_{\rm av}-[\ln J_{\rm o}]_{\rm av}\;. \label{delta}$$ For $\delta \ne 0$ the model is in the random dimer phase, in which spatial spin-spin correlations are short ranged.[@HYBG96] Within a range of finite dimerization $|\delta| < \delta_G>0$, dynamical correlations are however still quasi-long-ranged, which is related to the fact that the system is gapless. This region is called Griffiths phase. In a finite chain of length $L$ the typical gap scales asymptotically as: $$\Delta \sim L^{-z}\;. \label{gap_rd}$$ Here the dynamical exponent, $z$, is a continuously varying function of the control parameter, $\delta$. On the border of the Griffiths phase, $\delta=\delta_G$, the dynamical exponent vanishes, whereas close to the random singlet phase it diverges with $\delta\to0$ as[@fisher]: $$z \sim 1/\delta\;. \label{z_delta}$$ In the random singlet phase as well as in the Griffiths phase, thermodynamic quantities such as the low-temperature susceptibility, $\chi(T)$, the low-temperature specific heat, $C(T)$, and the low-field magnetization, $M(h)$, at zero temperature are singular. For example the susceptibility in the random singlet phase behaves as: $$\chi(T) \sim \frac{1}{T (\ln T)^2}\;, \label{chi_rs}$$ whereas in the Griffiths phase it is given by: $$\chi(T) \sim T^{-1+\beta},\quad \beta=1/z\;. \label{chi_rd}$$ Recent experimental studies on the compound ${\rm Cu Cl}_{2x}{\rm Br}_{2(1-x)}(\gamma-{\rm pic})_2$ show another type of competition between bond alternation and randomness.[@ajiro; @wakisaka] Here the $x=0$ compound, ${\rm Cu Br}_{2}(\gamma-{\rm pic})_2$, is a homogeneous $S=1/2$ antiferromagnetic Heisenberg chain in which the ${\rm Cu-Cu}$ bond is bi-bridged by two ${\rm Br}$ atoms: ${\rm Cu} < { {\rm Br} \atop {\rm Br}} > {\rm Cu}$ and the coupling constant is given by $J''=20.3{\rm K}$.[@ajiro] The $x=1$ compound, ${\rm Cu Cl}_{2}(\gamma-{\rm pic})_2$, is a bond alternating $S=1/2$ antiferromagnetic Heisenberg chain in which two kinds of ${\rm Cu} < { {\rm Cl} \atop {\rm Cl}} > {\rm Cu}$ bonds alternate along the chain.[@GJW82] The bond alternation is induced by a freezing transition of the rotational motion of the methyl-group at $50K$.[@GJW82] The experimentally measured coupling strengths are $J=13.2{\rm K}$ and $J \alpha$, with $\alpha=0.6$. In the mixed compound with a small finite concentration of Br ($x \ne 1$), atoms connected with alternating ${\rm Cu} < { {\rm Cl} \atop {\rm Cl}} > {\rm Cu}$ bonds form a cluster and different clusters are separated by bonds with ${\rm Br-Br}$ and/or ${\rm Br-Cl}$ bridges. The strength of the ${\rm Cu} < { {\rm Br} \atop {\rm Cl}} > {\rm Cu}$ bond has been estimated from the theoretically calculated magnetization curve as $J'=1.3J$.[@hida03] In ref.  the data for the low-temperature susceptibility of the diluted system show an algebraic temperature dependence, $\chi \sim T^{\beta-1}$, which is compatible with the expected behavior in the Griffiths phase as given by eq.(\[chi\_rd\]). For a wide range of the concentration, $x$, the measured effective exponent is about $\beta_{\rm exp} \approx 0.5 - 0.67$, and shows only a weak concentration dependence. Theoretical Model ================= A theoretical model for ${\rm Cu Cl}_{2x}{\rm Br}_{2(1-x)}(\gamma-{\rm pic})_2$ was presented and numerically studied in ref. . It is not obvious from the information one can extract from the experiments whether the rotational order of the methyl-group remains long-ranged in the presence of the dilution by ${\rm Cu}< { {\rm Br} \atop {\rm Cl}} >{\rm Cu}$ or ${\rm Cu} < { {\rm Br} \atop {\rm Br}} > {\rm Cu}$. Therefore two models were introduced in [@hida03]: i) The [*fixed parity*]{} model in which the rotational order is assumed to be perfectly long ranged, i.e. the $J$ ($\alpha J$) bonds stay in the same parity position, say at $i={\rm even}$ ($i={\rm odd}$), in any bond-alternating cluster. Introducing a parity parameter, $p_i$, given by $p_i=1(-1)$ if the ${\rm Cu} < { {\rm Cl} \atop {\rm Cl}} > {\rm Cu}$ bond has a value of $J$ ($\alpha J$), one can describe the parity correlations in the fixed parity model as $p_i p_{i+2l}=1$ and $p_i p_{i+2l-1}=-1$; ii) The [*random parity*]{} model in which the rotational long-range order between two clusters of alternating bonds is assumed to be completely destroyed by the dilution. In this case, if $i$ and $i+2l$ ($i+2l-1$) refer to different clusters, the average parity correlations vanish: $[p_i p_{i+2l}]_{\rm av}=[p_i p_{i+2l-1}]_{\rm av}=0$. In this paper we consider a more general model in which rotational long-range order is partially destroyed by dilution, so that correlations between parities in two different clusters are given asymptotically as $[p_i p_{i+2l}]_{\rm av} = - [p_i p_{i+2l-1}]_{\rm av}\sim l^{-\rho}$. Here we recover the fixed parity and the random parity models in the limits $\rho=0$ and $\rho \to \infty$, respectively. We note that some aspects of the effect of correlated disorder on quantum systems is studied in ref. . Here we summarize the values of the coupling constant in eq.(\[hamilton\]) in the following way: $$J_i=\left\{ \begin{array}{lll} J \alpha^{(1-p_i)/2} & \mbox{with prob. $x^2$}\\ J' & \mbox{with prob. $2x(x-1)$}\\ J'' & \mbox{with prob. $(x-1)^2$} \end{array} \right.\;. \label{couplings}$$ The parity correlations within a cluster of alternating bonds are given by $p_i p_{i+j}=(-1)^j$ and the parity correlations for bonds in different clusters are defined above for the different models. The theoretical results for the low-energy behavior of the fixed parity and the random parity models obtained from the density matrix renormalization group (DMRG) method[@hida03] are not fully consistent with the measured low-temperature properties of the ${\rm Cu Cl}_{2x}{\rm Br}_{2(1-x)}(\gamma-{\rm pic})_2$.[@ajiro] In ref.  the random parity model is found to be in the random singlet phase, in which the susceptibility exponent is $\beta=0$ and is much lower than the experimental value $\beta_{\rm exp} \approx 0.5 - 0.67$. For the fixed parity model the susceptibility exponent is found to be too large: $\beta \ge 1$. Another problem with the latter model is that the DMRG analysis could not be performed for $x \le 0.4$ due to strong finite size effects. In the present paper we revisit the models of the compound ${\rm Cu Cl}_{2x}{\rm Br}_{2(1-x)}(\gamma-{\rm pic})_2$. Our study is different from ref.  in two respects. Firstly, we consider a more general model in which the effect of disorder correlations are taken into account. Secondly, we use a numerical implementation of the SDRG method. This method usually gives very accurate results for the form of singularities, in particular for the dynamical exponent[@lkir]. This method has been successfully applied to clarify the low-energy singularities of more complicated systems, such as random spin ladders[@ladder] and two- and three-dimensional random Heisenberg antiferromagnets[@2d]. The major advantage of this method is that one can consider large systems $L \sim 1000-2000$ with good statistics compared with the DMRG method. Numerical results ================= Numerical method ---------------- The SDRG method for random AF spin chains proceeds as follows: During the renormalization process, one first identify the strongest bond, say $J_{23}=\Omega$, which connects sites $2$ and $3$. If the neighboring bonds are much weaker, $J_{23} \gg J_{12}, J_{34}$, then the spins $2$ and $3$ form an effective singlet and can be decimated out. The new coupling between the remaining sites, $1$ and $4$, is obtained within a second-order perturbation calculation as: $$\tilde{J}=\frac{1}{2} \frac{J_{12} J_{34}}{J_{23}}\;. \label{Jtilde}$$ By repeating this decimation process, we gradually reduce the energy scale, $\Omega$. At the fixed point, the energy scale is give by $\Omega \sim L^{-z}$ if only a small fraction $1/L$ of sites is active (i.e. not yet decimated). In our computations we consider finite periodic chains with $L={\rm even}$ sites and decimate until the last pair of sites having a gap $\Delta \sim \Omega$. From the distribution of the logarithms of the gap: $$P(\ln \Delta) \sim \Delta^{1/z}, \quad \Delta \to 0\;, \label{z}$$ we obtain the the dynamical exponent, $z$. In the random singlet phase, the dynamical exponent is formally infinite, as described in eq.(\[gap\_rs\]), and the appropriate scaling combination in a finite system is given by: $$\ln\left[ L^{\psi} P(\ln \Delta)\right] \simeq f(L^{-\psi}\ln \Delta)\;. \label{psi}$$ The SDRG method as outlined above is expected to provide asymptotically exact results both in the RS phase[@fisher] and in the Griffiths region[@ijl01]. In the present model the randomness is discrete and at the starting point of the renormalization there are several couplings with the same value of $\Omega$. In this degenerate situation we randomly choose the actual coupling to be decimated. After a sufficiently large number of renormalization steps we will have a (quasi)continuous distribution. To illustrate the correctness of the SDRG procedure for discrete randomness, we consider the random dimerized $XX$-chain[@XX] in which the couplings take the value $J$ or $\alpha J$ and a fraction, $x$, of the odd (even) couplings are $J$ ($\alpha J$). To be close to our model in eq.(\[couplings\]) we took $\alpha=0.6$ and $x=0.6$ for the $XX$-chain. In Fig. \[fig:xx\] the distribution of the gaps for different finite systems is shown in a log-log scale, the asymptotic slope of which agrees very well with the known [*exact*]{} result[@ijr00] for $1/z$ which is given by the formula $[(J_{\rm e}/J_{\rm o})^{(1/z)}]=1$, i.e. by the solution of the equation $(xJ^{(1/z)}+(1-x)(\alpha J)^{(1/z)}) \cdot ((1-x)J^{(-1/z)}+x(\alpha J)^{(-1/z)})=1$. Fixed parity model ------------------ Next, we turn to our model defined in eq. (\[couplings\]) and start with the fixed parity case. The distribution of the gaps for two different values of the concentration, $x=0.6$ and $0.8$, are shown in Fig. \[fig:fixed\]. Evidently, the exponent $\beta$, given by the asymptotic slope of the distributions, is finite and concentration dependent. It can also be extracted from the scaling behavior of the average gap via $[\ln \Delta]_{\rm av} \simeq {\rm const} + \beta^{-1} \ln L$. As seen in Fig. \[fig:fix\_lnE\] one can obtain a reliable estimate of $\beta$ for $x \ge 0.4$. For smaller concentrations, even the largest system size $L=2048$ is not yet in the asymptotic regime. The estimates for $\beta$ are depicted in Fig. \[fig:z\_x\]. They are systematically lower than the DMRG results in ref. ´, which is probably due to finite-size effects. Using the DMRG method, ref.  reports results up to $\ln L\le 5$, which is, even for comparatively large concentrations, not in the asymptotic regime according to Fig. \[fig:fix\_lnE\]. As can be seen in Fig. \[fig:z\_x\], for small concentrations there is a quadratic dependence: $\beta \sim x^2$, which can be understood as follows: In the fixed parity model the average value of the log-couplings in the odd and even positions are different, so that the dimer control parameter in eq. (\[delta\]) is $\delta \sim x^2 \ln \alpha \ne 0$. For small $x$, thus for small $\delta$ we can use eq. (\[z\_delta\]), which is compatible with the observed quadratic $x$-dependence of $\beta$. Random parity model ------------------- For the random parity model the gap distribution for an intermediate concentration, $x=0.8$, is shown in Fig. \[fig:random\](a). One observes that the distribution gets systematically broader with increasing system size, which is a characteristic of the RS phase. Indeed, the logarithmic scaling in eq. (\[psi\]) is well satisfied with $\psi=1/2$, as illustrated in Fig. \[fig:random\](b). The same logarithmic scaling with $\psi=1/2$ holds for other values of the concentration, too. These results can be understood by noting that odd and even couplings have on average of the same strength in the random parity model, consequently the (dimer) control parameter is $\delta=0$. The short-range correlations in the dimerized sequences, especially for large concentration, does not modify this large-scale asymptotic behavior. Our results for the random parity model are in agreement with the DMRG calculations in ref. . Correlated random parity model ------------------------------ Now we discuss the results for the correlated random parity model. We recall that according to scaling arguments[@WH; @ri99] the disorder correlations modify the critical behavior in the RS phase, provided the decay exponent is sufficiently small: $\rho < 2/\nu$, where $\nu=2$ is the correlation length exponent for uncorrelated disorder. To illustrate this relation we estimate the control parameter, $\delta$, in a finite chain of length, $L$. Introducing the notation $\epsilon_i=\ln(J_{2i}/J_{2i-1})$, we have $\delta^2=1/4 L^2\left[(\sum_{i=1}^{L-1} \epsilon_i)^2\right] \sim 1/L \int_1^L G(r) {\rm d} r \sim L^{-\rho}$, where $G(r)$ is the disorder correlator. Thus $\delta \sim L^{-\rho/2}$, for $\rho < 1$, whereas for $\rho \ge 1$ we have the non-correlated disorder result, $\delta \sim L^{-1/2}$. We can conclude from this consideration that for $\rho <1$ the correlated random parity model has a vanishing gap and is in the RS phase, however with $\rho$-dependent properties. The scaled gap distribution for four different correlation parameters, $\rho < 1$, is shown in Fig. \[fig:correlated\], in which the logarithmic scaling collapse in eq. (\[psi\]) can be obtained with different exponents, $\psi(\rho)$. A plot of the $\psi(\rho)$ data is given in Fig. \[fig:psi\_rho\] in which the $\rho$ dependence can be well fitted with the approximate formula: $$\psi(\rho)=\frac{3 - \rho}{4}, \quad \rho < 1\;. \label{psi_rho}$$ This result is exact at $\rho=1$, in which case the standard, non-correlated RS phase result, $\psi=1/2$ is recovered. For a small $\rho$ the exponent approaches $\psi \approx 3/4$, but at $\rho=0$ we jump to the fixed parity model. The linear dependence of $\psi(\rho)$ on $\rho$ in eq.(\[psi\_rho\]) is in the same form as in the exact expression for the $XX$-model[@ri99] with correlated randomness. The low-temperature singularity of the susceptibility of the correlated random parity model is given by the formula: $$\chi(T) \sim \frac{1}{T (\ln T)^{1/\psi}}\;, \label{chi_corr}$$ in which $\psi$ is $\rho$-dependent, as given in Eq.(\[psi\_rho\]). Discussion ========== We close our paper by comparing the experimentally measured low-temperature susceptibility of the compound ${\rm Cu Cl}_{2x}{\rm Br}_{2(1-x)}(\gamma-{\rm pic})_2$ with the results of our theoretical calculations. We recall that the measured susceptibility exponent is finite, $\beta_{\rm exp} \approx 0.5-0.67$, and has only a weak concentration dependence. These properties are partially compatible with the results for the fixed parity model, which is found to be in the Griffiths phase, where $\beta$ is finite for all values of the concentration. According to Fig. \[fig:z\_x\] there is a range of concentration around $x \approx 0.8$ in which our results for $\beta$ agree with the experimental values. However, this range is rather narrow and our values for $\beta$, which have a substantial concentration dependence, are generally significantly smaller than $\beta_{\rm exp}$. For models with non-strictly correlated parities, in which case $\beta$ is found formally zero, the agreement with the experiment is even less satisfactory. One possible explanation of this discrepancy between experiment and theory is that the experimentally measured $\beta_{\rm exp}$ are effective, temperature dependent values, which should approach the true behavior as $T \to 0$. Indeed, as seen in Fig. \[fig:fix\_lnE\] the local slopes of the curves that show $\ln \Delta$ vs. $\ln L$, which define the effective exponent $z(L)=1/\beta(L)$, have a strong size dependence. This can be converted into a temperature dependence through $\Omega \sim T \sim L^{-z}$. The corrections to $\beta(L)$ are particularly strong for small values of $x$. For moderately large sizes, $\ln L \approx 5$, the effective exponents have only a weak concentration dependence. The leading finite size corrections are of the form, $\beta(L) \simeq \beta + a/L$, which are compatible with a temperature correction as: $$\beta(T) \simeq \beta + c T^{\beta} + \dots.\quad \beta>0. \;, \label{beta_fixed}$$ In the random singlet phase with $\beta=0$, both for correlated and non-correlated parity, the effective exponents have a logarithmic temperature dependence: $$\beta(T) \simeq \frac{c_1}{|\ln T|} + \dots.\quad \beta=0. \;, \label{beta_rs}$$ which can be obtained by analyzing eqs.(\[chi\_rs\]) and (\[chi\_corr\]). Indeed, these finite temperature corrections are strong for the fixed parity model, particularly for small $\beta$, i.e. for a small concentration, $x$. We tried to estimate the effective exponent, $\beta(T)$, at the temperature of the experimental measurement $T=2 {\rm K}$. For this we performed the renormalization transformation down to an energy-scale, $\Omega=T$, and measured the fraction of non-decimated sites: $n(\Omega)=1/l(\Omega)$. Here the length-scale is given by $l(\Omega) \sim \Omega^{-\beta}$. Our results for $l$ as a function of the temperature for the fixed parity and the random parity model are presented in Fig. \[fig:beta\_T\] for different values of the concentration. In the log-log plot the local slope of the curves is just the effective exponent, $\beta(T)$, at $T=\Omega$. As can be seen in this figure the effective exponent, $\beta(T)$, approaches its limiting value only if the temperature is sufficiently low. At the temperature of the measurement, $T=2{\rm K}$, the asymptotic region seems to be still quite far. For the fixed parity model the effective exponent is about $\beta=0.4$, which is practically independent of the concentration. This result is consistent with the experimental results of ref. . For the random parity model, as shown in the inset, the effective exponent continuously vary with the temperature. Its value at $T=2 {\rm K}$ is about $\beta=0.36$, which is also consistent with the experiments. Therefore, using the available experimental data it is not possible to distinguish the type of parity correlations present in the compound, ${\rm Cu Cl}_{2x}{\rm Br}_{2(1-x)}(\gamma-{\rm pic})_2$ and the question, which type of model should be used for its theoretical description remains still open. Experimental measurements at lower temperatures are needed to clarify this point. Acknowledgment {#acknowledgment .unnumbered} ============== This work has been supported by a German-Hungarian exchange program (DAAD-MÖB), by the Hungarian National Research Fund under grant No OTKA TO34138, TO37323, MO45596 and M36803, by the Ministry of Education under grant No. FKFP 87/2001 and by the Centre of Excellence ICA1-CT-2000-70029. [99]{} A. Luther and I. Peschel, Phys. Rev. B **12** (1975) 3908. M.C. Cross and D.S. Fisher, Phys. Rev. B **19** (1979) 402. For a review see e.g.: H. Rieger and A.P. Young, *Quantum Spin Glasses*, Lecture Notes in Physics **492**, in “Complex Behaviour of Glassy Systems”, p. 254, ed. J.M. Rubi and C. Perez-Vicente (Springer Verlag, Berlin-Heidelberg-New$\,$York, 1997). R.B. Griffiths, Phys. Rev. Lett. **23** (1969) 17. B. McCoy, Phys. Rev. Lett. **23** (1969) 383. D.S. Fisher, Phys. Rev. B **50** (1994) 3799. S.K. Ma, C. Dasgupta, and C.-K. Hu, Phys. Rev. Lett. **43** (1979) 1434; C. Dasgupta and S.K. Ma, Phys. Rev. B **22** (1980) 1305. R.A. Hyman, K. Yang, R.N. Bhatt, and S.N. Girvin, Phys. Rev. Lett. **76** (1996) 839. Y. Ajiro, T. Wakisaka, H. Itoh, K. Watanabe, H. Satoh, Y. Inagaki, T. Asano, M. Mito, K. Takeda, H. Mitamura, and T. Goto, Physica B **329-333** (2003) 1004. T. Wakisaka, Master Thesis (Kyushu University), (2003). H.J.M. de Groot, L.J. de Jongh, and R.D. Willett, J. Appl. Phys. **53** (1982) 8038. K. Hida, J. Phys. Soc. Jpn. **72** (2003) 2627. H. Rieger and F. Iglói, Phys. Rev. Lett. **83** (1999) 3741. Y.-C. Lin, N. Kawashima, F. Iglói and H. Rieger, Prog. Theor. Phys. (Suppl.) **138** (2000) 470; D. Karevski, Y.-C. Lin, H. Rieger, N. Kawashima and F. Iglói, Eur. Phys. J. B **20** (2001) 267. R. Mélin, Y-C. Lin, P. Lajkó, H. Rieger, and F. Iglói:, Phys. Rev. B **65** (2002) 104415. Y.-C. Lin, R. Mélin, H. Rieger and F. Iglói, Phys. Rev. B **68** (2003) 024424. F. Iglói, R. Juhász, and P. Lajkó:, Phys. Rev. Lett. **86** (2001) 1343. F. Iglói, R. Juhász, and H. Rieger, Phys. Rev. B **61** (2000) 11552. For the random $XX$ model the decimation rule in Eq.(\[Jtilde\]) is modified by having a prefactor $1$ instead of $1/2$. A. Weinrib, and B.I. Halperin, Phys. Rev. B**27** (1983) 413.

--- abstract: 'We solve the twisted conjugacy problem on Thompson’s group $F$. We also exhibit orbit undecidable subgroups of ${\mathrm{Aut}}(F)$, and give a proof that ${\mathrm{Aut}}(F)$ and ${\mathrm{Aut}}_+(F)$ are orbit decidable provided a certain conjecture on Thompson’s group $T$ is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [@bomave2], we construct a family of free extensions of $F$ where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak-Fel’shtyn-Gonçalves in [@bleakfelshgonc1] showing that $F$ has property $R_\infty$, and which can be extended to show that Thompson’s group $T$ also has property $R_\infty$.' address: - 'Departament de Matemàtica Aplicada IV, Escola Politècnica Superior de Castelldefels, Universitat Politècnica de Catalunya, C/Esteve Torrades 5, 08860 Castelldefels, Barcelona, Spain' - 'Département de Mathématiques, Faculté des Sciences d’Orsay, Université Paris-Sud 11, Bâtiment 425, Orsay, France' - 'Departament Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Manresa, Catalunya' author: - José Burillo - Francesco Matucci - Enric Ventura bibliography: - 'go4.bib' title: '<span style="font-variant:small-caps;">The conjugacy problem in extensions of Thompson’s group $F$</span>' --- Introduction \[sec:intro\] ========================== Since Max Dehn formulated the three main problems in group theory in 1911, they have been a central subject of study in the theory of infinite groups. There now exists a large body of works devoted to the study of these problems. In this paper we focus on the conjugacy problem and a variant known as *the twisted conjugacy problem*. The conjugacy problem is known to be solvable for Thompson’s groups $F,T$ and $V$ by works of Guba and Sapir [@gusa1], Belk and the second author [@matucci9] and Higman [@hig]. Our interest arose in the study of extensions of the group $F$ where we find an unsolvability result. Even though Thompson himself used the groups $F,T,V$ in the construction of finitely presented groups with unsolvable word problem, to the best of our knowledge, the result that we obtain is a first in a direct generalization of the original Thompson groups. Moreover, we also look at property $R_\infty$ which has been under study recently and which is known to true for the group $F$ and one of its extensions. We now give a more detailed description of the results. Let $F$ be a group. We say that a subgroup $A\leqslant {\mathrm{Aut}}(F)$ has solvable *orbit decidability problem* (ODP) if it is decidable to determine, given $y,z \in F$, whether or not there is $\varphi \in A$ and $g \in F$ such that $$\varphi(z)= g^{-1} y g.$$ On the other hand, if $\varphi \in {\mathrm{Aut}}(F)$, we say that $F$ has solvable *$\varphi$-twisted conjugacy problem* (TCP$_\varphi$) if it is decidable to determine, given $y,z \in F$, whether or not they are $\varphi$-*twisted conjugated* to each other, i.e. whether there exists $g\in F$ such that $$\label{eq:TCP-equation} \numberwithin{equation}{section} z = g^{-1} y \varphi(g).$$ More generally, we say that the group $F$ has solvable *twisted conjugacy problem* (TCP) if (TCP$_\varphi$) is solvable for any given $\varphi \in {\mathrm{Aut}}(F)$. In their recent paper [@bomave2], Bogopolski, Martino and Ventura developed a criterion to study the conjugacy problem for some extensions of groups, and found a connection of this problem with the two problems mentioned above. Let $F,G,H$ be finitely presented groups and consider a short exact sequence $$\label{eq:usual-exact-sequence} \numberwithin{equation}{section} 1 \longrightarrow F \overset{\alpha}{\longrightarrow} G \overset{\beta}{\longrightarrow} H \longrightarrow 1.$$ In this situation, $\alpha(F) \unlhd G$ and so the conjugation map $\varphi_g$, for $g \in G$, restricts to an automorphism of $F$, $\varphi_g \colon F\to F$, $x\mapsto g^{-1}xg$, (which does not necessarily belong to ${\mathrm{Inn}}(F)$). We define the *action subgroup* of the sequence (\[eq:usual-exact-sequence\]) to be the group of automorphisms $$A_G =\{\varphi_g \mid g \in G\} \leqslant {\mathrm{Aut}}(F).$$ \[thm:bomave-extensions\] Let $$1 \longrightarrow F \overset{\alpha}{\longrightarrow} G \overset{\beta}{\longrightarrow} H \longrightarrow 1.$$ be an algorithmic short exact sequence of groups such that 1. $F$ has solvable twisted conjugacy problem, 2. $H$ has solvable conjugacy problem, and 3. for every $1 \ne h \in H$, the subgroup $\langle h \rangle$ has finite index in its centralizer $C_H(h)$, and we can compute a set of coset representatives of $\langle h\rangle$ in $C_H(h)$. Then, the conjugacy problem for $G$ is solvable if and only if the action subgroup $A_G =\{\varphi_g \mid g \in G\}\leqslant {\mathrm{Aut}}(F)$ is orbit decidable. Here, a short exact sequence is *algorithmic* if all the involved groups are finitely presented and given to us with an explicit finite presentation, and all the morphisms are given by the explicit images of the generators. Condition (3) is of more technical nature. It is clearly satisfied in free groups (where the centralizer of a non-trivial element $h$ is just the cyclic subgroup generated by its maximal root $\hat{h}$), and it is also true in torsion-free hyperbolic groups, see [@bomave2]. The goal of the present paper is to study the conjugacy problem in some extensions of Thompson’s group $F$ via Theorem \[thm:bomave-extensions\] (see [@bomave2; @ventura1] for references to similar applications of this same theorem into other families of groups). We will assume the reader is familiar with Thompson’s groups $F$ (also denoted by ${\mathrm{PL}}_2(I)$, where $I=[0,1]$ is the unit interval) and $T$ (also denoted by ${\mathrm{PL}}_2(S^1)$, where $S^1$ is the unit circle) and in any case, the comprehensive survey by Cannon, Floyd and Parry [@cfp] is an excellent source of information for Thompson’s groups. We will employ techniques on conjugacy in the Bieri-Thompson-Stein-Strebel groups used by Kassabov and the second author in [@matucci5] and a rephrasing by Belk and the second author in [@matucci9; @matucci3] of a conjugacy invariant of Brin and Squier [@brin2]. The idea is to assume that the twisted conjugacy equation has a solution and use this to determine necessary conditions that a twisted conjugator should satisfy. This allows one to build some candidate conjugators which must then be tested. With these techniques, we obtain the first result in the paper: \[thm:TCP-solvable\] Thompson’s group $F$ has solvable twisted conjugacy problem. Putting together Theorems \[thm:bomave-extensions\] and \[thm:TCP-solvable\], this opens us to the possibility of finding extensions of $F$ with solvable/unsolvable conjugacy problem, by detecting subgroups of ${\mathrm{Aut}}(F)$ which are orbit decidable/orbit undecidable: \[coro\] Consider Thompson’s group $F={\mathrm{PL}}_2(I)$, a torsion-free hyperbolic group $H$, and let $$1 \longrightarrow F \overset{\alpha}{\longrightarrow} G \overset{\beta}{\longrightarrow} H \longrightarrow 1.$$ be an algorithmic short exact sequence. The group $G$ has solvable conjugacy problem if and only if the action subgroup $A_G \leqslant {\mathrm{Aut}}(F)$ is orbit decidable. Using the previous result one can create extensions of $F$ with unsolvable conjugacy problem. \[thm:CP-extension-unsolvable\] There are extensions of Thompson’s group $F$ by finitely generated free groups, with unsolvable conjugacy problem. It is also possible to build some interesting extensions of $F$ with solvable conjugacy problem, provided that an open conjecture about $F$ is true. We study this in Section \[sec:ODP-solvable\]. A group $G$ has the *property $R_\infty$* if it has infinitely many distinct $\varphi$-twisted conjugacy classes, for any $\varphi \in {\mathrm{Aut}}(G)$. Thompson’s group $F$ was shown to have property $R_\infty$ by Bleak, Fel’shtyn and Gonçalves in [@bleakfelshgonc1]. We give an alternative proof, which can be extended to Thompson’s group $T$. \[thm:R-infty-T\] Thompson’s group $T$ has property $R_\infty$. The paper is organized as follows. In Section \[sec:twisted-problem\] we introduce the groups we will be working with, we restate the twisted conjugacy problem for $F$ and prove Theorems \[thm:TCP-solvable\] and \[coro\]. In Section \[sec:CP-extensions\] we construct orbit undecidable subgroups of ${\mathrm{Aut}}(F)$ and exhibit free extensions of $F$ with unsolvable conjugacy problem. In Section \[sec:ODP-solvable\] we consider orbit decidability and construct some interesting extensions of $F$, which happen to have solvable conjugacy problem assuming an open conjecture on $F$ is true. In Section \[ssec:R-infty\] we show that the groups $F$ and $T$ have property $R_{\infty}$ using ideas from Section \[sec:twisted-problem\]. Finally, in Section \[sec:generaltions-of-results\] we analyze the extent to which the techniques of this paper generalize to other families of Thompson-like groups. Acknowledgments {#acknowledgments .unnumbered} --------------- The authors would like to thank Matt Brin, Collin Bleak, Martin Kassabov, Jennifer Taback and Nathan Barker for helpful conversations about this work. \[sec:twisted-problem\] The twisted conjugacy problem for $F$ ============================================================= In this section we prove Theorem \[thm:TCP-solvable\]. The techniques developed for this purpose will be later used in Section \[ssec:R-infty\] to obtain a couple of byproducts. Thompson’s group and its automorphisms\[sec:thompson and autos\] ---------------------------------------------------------------- We will look at Thompson’s group $F$ from different perspectives. The standard one is to look at $F$ as the group ${\mathrm{PL}}_2(I)$ of orientation preserving piecewise-linear homeomorphisms of the unit interval $I=[0,1]$ with a discrete (and hence finite) set of breakpoints at dyadic rational points, and such that all slopes are powers of $2$ (the interval $I$ can be replaced to an arbitrary $[p,q]$ with $p,q$ being dyadic rationals and the resulting group is clearly isomorphic). We will also need to regard $F$ as a subgroup of a bigger group: consider the group ${\mathrm{PL}}_2(\mathbb{R})$ of all piecewise-linear homeomorphisms of $\mathbb{R}$ with a discrete set of breakpoints at dyadic rational points and such that all slopes are powers of $2$; and consider the subgroup of those elements $f$ which are eventually integral translations, i.e. for which there exist $m_-, m_+\in \mathbb{Z}$ and $L,R\in \mathbb{R}$ such that $f(x)=x+m_-$ for all $x\leqslant L$, and $f(x)=x+m_+$ for all $x\geqslant R$. It is straightforward to see that this subgroup of ${\mathrm{PL}}_2(\mathbb{R})$ is isomorphic to ${\mathrm{PL}}_2(I)$; see Proposition 3.1.1 in Belk and Brown [@bebr] for an explicit isomorphism (it is interesting to note that, through this isomorphism, $2^{m_-}$ is the slope at the right of 0, and $2^{m_+}$ the slope at the left of 1). Both copies of Thompson’s group will be denoted $F$, and it will be clear from the context which one are we talking about at any moment. Thompson’s group admits a finite presentation. The two generators are usually written $x_0$ and $x_1$, which represent the following maps on the real line: $$x_0(t)=t+1\qquad x_1(t)=\left\{\begin{array}{ll} t&\text{if }t<0\\ 2t&\text{if }0\leq t\leq 1\\ t+1&\text{if }t>1. \end{array}\right.$$ With these generators, $F$ admits a finite presentation with just two relations, which have lengths 10 and 14. See [@cfp] for details. Moreover, as we will need this later, we observe that when we regard $F$ as the group ${\mathrm{PL}}_2([0,1])$, the generator $x_0$ has this form: $$\theta(t):= \begin{cases} 2t & t \in \left[0,\frac{1}{4} \right] \\ t + \frac{1}{4}& t \in \left[\frac{1}{4},\frac{1}{2} \right] \\ \frac{t}{2}+\frac{1}{2} & t \in \left[\frac{1}{2},1 \right]. \end{cases}$$ We distinguish $x_0$ and $\theta$ to make it clear that the first one is seen as an element of ${\mathrm{PL}}_2(\mathbb{R})$ while the second is regarded as a map in ${\mathrm{PL}}_2([0,1])$. The *support* of an element $f\in {\mathrm{PL}}_2(\mathbb{R})$ is the collection of points where it is different from the identity, ${\operatorname{supp}}(f)=\{ t\in \mathbb{R}\mid f(t)\neq t\}$. {height="5.5cm"}\ {height="5.5cm"}\ *When talking about elements $f\in {\mathrm{PL}}_2 ({\mathbb{R}})$, we say that a property $\mathcal{P}$ holds *for $t$ positive sufficiently large* (respectively, *for $t$ negative sufficiently large*) to mean that there exists a number $R>0$ such that $\mathcal{P}$ holds for every $t\geqslant R$ (respectively, there exists a number $L<0$ such that $\mathcal{P}$ holds for every $t\leqslant L$). For example, $f\in F$ if and only if it is an integral translation for $t$ positive sufficiently large, and for $t$ negative sufficiently large.* *\[thm:remark-initial-slope\] Observe that, for $g\in F\leqslant {\mathrm{PL}}_2(\mathbb{R})$, the integer $m_-$ above (satisfying that $g(t)=t+m_-$ for $t$ negative sufficiently large) can also be obtained as the limit $m_- =\lim_{t\to -\infty} g(t)-t$. Similarly, $g(t)=t+m_+$ for $t$ positive sufficiently large, where $m_+ =\lim_{t\to +\infty} g(t)-t$. These two real numbers are called, respectively, the *initial slope* and the *final slope* of $g$ because, when regarded as an element of ${\mathrm{PL}}_2(I)$, the slopes of $g$ on the right of the point 0 and on the left of the point 1 are, precisely, $2^{m_-}$ and $2^{m_+}$, respectively.* Automorphisms and transitivity on dyadics ----------------------------------------- To deal with the $\varphi$-twisted conjugacy problem for $F$, we first need to understand what the automorphisms of Thompson’s group $F$ look like. They have all been classified by Brin in his Theorem 1 in [@brin5] (see also Theorem 1.2 in [@burcleary2] for a more explicit version). The key idea to understand ${\mathrm{Aut}}(F)$ is the fact that conjugation by elements from ${\widetilde{\mathrm{EP}}}_2$ preserves $F$, and these conjugations give precisely all automorphisms of $F$: \[thm:brin-thm\] For Thompson’s group $F$, the map $$\begin{array}{ccc} {\widetilde{\mathrm{EP}}}_2 & \longrightarrow & {\mathrm{Aut}}(F) \\ \tau & \mapsto & \begin{array}[t]{rcl} \gamma_{\tau}\colon F & \rightarrow & F \\ g & \mapsto & \tau^{-1}g\tau, \end{array} \end{array}$$ is well defined and it is a group isomorphism, so ${\mathrm{Aut}}(F) \simeq {\widetilde{\mathrm{EP}}}_2$. Furthermore, given $\varphi \in {\mathrm{Aut}}(F)$ by the images of the standard generators, one can algorithmically compute the (unique) $\tau\in {\widetilde{\mathrm{EP}}}_2$ such that $\varphi (g)=\tau^{-1}g\tau$ for all $g\in F$. *\[aut+\] We denote by ${\mathrm{Aut}}_+ (F)$ the group of automorphisms of $F$ given by conjugation by orientation preserving $\tau$’s (see Theorem \[thm:brin-thm\]); it is an index two subgroup ${\mathrm{EP}}_2 \simeq {\mathrm{Aut}}_+(F) <_2 {\mathrm{Aut}}(F)\simeq {\widetilde{\mathrm{EP}}}_2$.* *Theorem \[thm:brin-thm\], including its algorithmic contents, is crucial for the arguments of the present paper. Brin’s original theorem establishes the isomorphism and we can do the algorithmic determination of $\tau$ in the following form. Burillo and Cleary [@burcleary2] obtained a finite presentation for ${\mathrm{Aut}}(F)$ with nine generators $\varphi_1\,\ldots , \varphi_9$ all expressed in terms of the standard presentation of $F$, and as conjugations by suitable $\tau_1,\ldots ,\tau_9\in {\widetilde{\mathrm{EP}}}_2$, i.e. $\varphi_i=\gamma_{\tau_i}$ for $i=1,\ldots ,9$. Suppose $\varphi \in {\mathrm{Aut}}(F)$ is given by the images of $x_0$ and $x_1$. We can enumerate all formal words $w$ on letters $\varphi_1,\ldots ,\varphi_9$ and for each one compute the images of $x_0$ and $x_1$ by $w(\varphi_1,\ldots ,\varphi_9)$ until they match with $\varphi(x_0)$ and $\varphi(x_1)$ (here we need to use the word problem for $F$); this match will happen sooner or later because $\varphi_1,\ldots ,\varphi_9$ do generate ${\mathrm{Aut}}(F)$. Once we have this word, it is clear that $\tau=w(\tau_1,\ldots ,\tau_9)\in {\widetilde{\mathrm{EP}}}_2$ satisfies $\gamma_{\tau}=\gamma_{w(\tau_1,\ldots ,\tau_9)}=w(\gamma_{\tau_1},\ldots ,\gamma_{\tau_9})=w(\varphi_1,\ldots ,\varphi_9)=\varphi$.* The following is a result explaining how to build ${\mathrm{PL}}_2$-maps acting in a prescribed way on some given rational numbers. The first part gives an arithmetic condition for the existence of such a map. The second part expresses the flexibility of these groups: one can always “cut" the graphical representation of an element at a given dyadic rational, and freely “glue" the pieces to obtain new elements. This result will often be needed along the present paper. \[thm:rationals-coincide\] Let $\eta,\zeta$ be dyadic rationals, let $\alpha, \beta\in \mathbb{Q}\cap(\eta,\zeta)$ written in the form $\alpha=\frac{2^t m}{n}$ and $\beta=\frac{2^k p}{q}$ with $t,k\in \mathbb{Z}$ and $m,n,p,q$ odd integers such that $(m,n)=(p,q)=1$, and let $\eta <\alpha_1 <\cdots <\alpha_r <\zeta$ and $\eta <\beta_1 <\cdots <\beta_r <\zeta$ be two finite sequences of rational numbers. 1. The following are equivalent: - there exists $g\in {\mathrm{PL}}_2([\eta,\zeta ])$ such that $g(\alpha)=\beta$, - there exists $g\in {\mathrm{PL}}_2 (\mathbb{R})$ such that $g(\alpha)=\beta$, - there exists $g\in {\mathrm{EP}}_2$ such that $g(\alpha)=\beta$, - there exists $g\in F$ such that $g(\alpha)=\beta$, - $n=q$ and $p\equiv 2^Rm \pmod{n}$ for some $R\in \mathbb{Z}$. Moreover, there is an algorithm which constructs such elements $g$ if condition (e) is satisfied. 2. There exists $g\in F$ with $g(\alpha_i )=\beta_i$ if and only if for every $i=1,\ldots, r$ there exists $g_i \in F$ such that $g_i(\alpha_i) = \beta_i$. Moreover, if such a $g$ exists it can be constructed from the $g_i$’s. The following is a well known standard result (see for example [@matucci5] for a proof). \[thm:standard-folklore\] Let $p \in \mathbb{Q}$ and $g \in {\mathrm{PL}}_2([p,p+1])$. Let $u,v \in (p,p+1)$ be such that $u \not \in {\mathrm{Fix}}(g)$. Then there exists at most a unique integer $m$ such that $g^m(u)=v$, and one can algorithmically decide it (and compute such an $m$ if it exists). Restatement of the TCP \[sec:restatement-TCP\] ---------------------------------------------- Our goal in this section is to solve the twisted conjugacy problem in $F$: given $\varphi \in {\mathrm{Aut}}(F)$ and $y,z\in F$ (all in terms of the standard presentation of $F$, i.e. $\varphi (x_0),\, \varphi (x_1),\, y,\, z$ are given to us as words on $x_0,\, x_1$), we have to decide whether there exists $g\in F$ such that $$z=g^{-1} y \varphi(g).$$ Applying Theorem \[thm:brin-thm\], we can compute $\tau \in {\widetilde{\mathrm{EP}}}_2$ such that $\varphi (g)=\tau^{-1}g\tau$ for all $g\in F$, and the previous equation becomes $z=g^{-1} y (\tau^{-1}g\tau)$, that is $$z \tau^{-1} = g^{-1} (y \tau^{-1})g.$$ Relabeling ${\overline{y}}:=y\tau^{-1}\in {\widetilde{\mathrm{EP}}}_2$ and ${\overline{z}}:= z\tau^{-1}\in {\widetilde{\mathrm{EP}}}_2$ to get $$\label{eq:conjugacy}\numberwithin{equation}{section} {\overline{z}} = g^{-1} {\overline{y}} g,$$ the problem reduces to the standard conjugacy problem in ${\widetilde{\mathrm{EP}}}_2$, but with the conjugator $g$ forced to be chosen from $F\leqslant {\widetilde{\mathrm{EP}}}_2$. *Given two elements ${\overline{y}}, {\overline{z}}\in {\widetilde{\mathrm{EP}}}_2$, we write ${\overline{y}}\sim_F {\overline{z}}$ if they are conjugated by a conjugator in $F$, i.e. if there exists $g\in F$ such that ${\overline{z}} = g^{-1} {\overline{y}} g$.* Notice that if one of ${\overline{y}}$ and ${\overline{z}}$ is in ${\mathrm{EP}}_2$ and the other is not, then equation (\[eq:conjugacy\]) has no solution. Thus, we can split its study into two cases: the orientation preserving case, i.e. when ${\overline{y}},{\overline{z}} \in {\mathrm{EP}}_2$ (studied in Sections \[ssec:periodicity-boxes\], \[ssec:fixed-points\], \[ssec:reducing-to-squares\] and \[sec:rescaling-the-circle\]) and then the orientation reversing one, i.e. when ${\overline{y}},{\overline{z}}\in {\mathcal{R}\cdot {\mathrm{EP}}_2}$ (considered in Section \[sec:special-case\]). Finally, in Section \[ssec:solution-TCP\] we put all pieces together. Orientation preserving case of the TCP: periodicity boxes and building conjugators\[ssec:periodicity-boxes\] ------------------------------------------------------------------------------------------------------------ We now deal with the equation $z=g^{-1}yg$ for $y,z\in {\mathrm{EP}}_2$ and $g\in F$. The argument will make use of techniques and statements in [@matucci5] and refer often to that paper. Subsection 4.1 in [@matucci5] shows that, if $z=g^{-1}yg$ with $y,z,g \in {\mathrm{PL}}_2(I)$, then there exists $\varepsilon >0$ depending only on $y$ and $z$ such that $g$ is linear inside $[0,\varepsilon]^2$; the box $[0,\varepsilon]^2$ is called an *initial linearity box*. The goal of this section is to show an analog of this result inside suitable boxes $(-\infty,L]^2$ and $[R,\infty)^2$ where $y,z \in {\mathrm{EP}}_2$ are periodic. The following is a first necessary condition for two maps to be conjugate to each other. \[thm:identical-at-infinity\] Let $y,z \in {\mathrm{EP}}_2$ be such that $y\sim_F z$. Then there exist two numbers $L,R \in \mathbb{R}$ such that $y(t)=z(t)$ for all $t \in (-\infty,L] \cup [R,\infty)$. Let $g\in F$ be such that $g^{-1}y g=z$. For $t$ negative sufficiently large, we have $g(t)=t+m_-$, and so $$z(t)=g^{-1} y g(t)= g^{-1}y(t+m_-)=g^{-1}(y(t)+m_-)=y(t)+m_- -m_-=y(t).$$ Similarly for $t$ positive sufficiently large. We move on to prove the existence of periodicity boxes. \[ifpb\] For every pair of elements $y,z \in {\mathrm{EP}}_2^>(-\infty, p)$ (with $-\infty <p\leqslant +\infty$), there exists a computable constant $L\in \mathbb{R}$ (depending only on $y$ and $z$) such that every conjugator $g\in F$ between $y$ and $z$ must act as a translation inside the *initial periodicity box* $(-\infty,L]^2$. Similarly, for every pair of elements $y,z \in {\mathrm{EP}}_2^>(p, +\infty)$ (with $-\infty \leqslant p<+\infty$) and a *final periodicity box* $[R, +\infty)^2$. The exact same statement is true replacing ${\mathrm{EP}}_2^>$ to ${\mathrm{EP}}_2^<$. If $y$ and $z$ are not equal for $t$ positive and negative sufficiently large then, by Lemma \[thm:identical-at-infinity\], there is no possible conjugator $g\in F$ and there is nothing to prove. So assume they are and consider a negative sufficiently large $L\in \mathbb{R}$ such that $y(t)=z(t)$ and $y(t-1)=y(t)-1$ (and so, $z(t-1)=z(t)-1$), for every $t\leqslant L$ (clearly, such an $L$ is computable). We claim that every possible $g\in F$ satisfying $g^{-1}yg=z$ must be a translation for $t\leqslant L$. By the symmetry of $y$ and $z$ in the definition of $L$ and up to writing the conjugacy relation as $(g^{-1})^{-1}zg^{-1}=y$ (which changes the conjugator from $g$ to $g^{-1}$), we can assume that $g$ has non-positive translation at $-\infty$ (i.e. $g(t)=t+m_-$ for negative sufficiently large $t$, and with $m_-\leqslant 0$). Assume, by contradiction, that $g$ is not a translation map in $(-\infty,L]$. Then, there is $\lambda<L$ such that $$g(t)=\begin{cases} t+m_- & t \leqslant \lambda \\ \alpha(t-\lambda) + \lambda +m_- & \lambda \leqslant t<\mu \end{cases}$$ for some suitable real numbers $\alpha \neq 1$, $\lambda <\mu <L$. Since $z$ is increasing and strictly above the diagonal ${\mathrm{id}}(t)=t$, we can choose $r<\lambda <L$ such that $\lambda <z(r)<\mu <L$. By our choice of $r$, we have $y(r)=z(r)$, $y(t-1)=y(t)-1$ and $z(t-1)=z(t)-1$ for all $t\leqslant r$. Moreover, since $gz(t)=yg(t)$ for all $t \in \mathbb{R}$, we have $$\alpha(z(r)-\lambda) + \lambda +m_- =gz(r) =yg(r) =y(r+m_-) =y(r)+m_- =z(r)+m_-.$$ Rearranging the terms, we have $$\alpha(z(r)-\lambda)=z(r)-\lambda$$ and, since $z(r)-\lambda >0$, we get $\alpha=1$, a contradiction. Hence, $g(t)=t+m_-$ for every $t\leqslant L$ as claimed. The symmetric argument gives a constant $R$ establishing the final periodicity box $[R,+\infty)^2$. If $y,z\in {\mathrm{EP}}_2^<$, then we apply the previous argument to $y^{-1},z^{-1}$ and derive the same conclusion. *Note that, in the previous lemma, the constants $L$ and $R$ depend on $y$ and $z$ but not on the conjugator $g$. This will be crucial later.* We observe that the results of Subsection 4.2 in [@matucci5] and their proofs follow word-by-word in our generalized setting, and hence we do not reprove them. We restate Lemma 4.6 in [@matucci5] to give an example of how results appear in this context. Let $z\in {\mathrm{EP}}_2^<$. Let $C_{F}(z)=C_{{\mathrm{PL}}_2(\mathbb{R})}(z)\cap F$ be the set of elements in $F$ commuting with $z$. Then the map $\varphi_z:C_{F}(z) \to \mathbb{Z}$ defined by $$\varphi_z(g) = \lim_{t \to -\infty}g(t)-t$$ is an injective group homomorphism. A similar statement is true for ${\mathrm{EP}}_2^>$. Subsection 4.2 in [@matucci5] shows how to build a candidate conjugator $g$ between any two elements of $F$ after we have chosen the initial slope of $g$. A *unique candidate conjugator* $g$ between $y$ and $z$ with a given initial slope $q$, if it exists, is the unique function that one needs to test as a conjugator of $y$ and $z$ with initial slope $q$: if $g$ fails to satisfy $g^{-1}yg=z$, then there is no conjugator of $y$ and $z$ with initial slope $q$. The proof of Corollary 4.12 in [@matucci5] can be lifted verbatim and so we only restate it in our new case. \[thm:explicit-conjugator\] Let $y,z \in {\mathrm{EP}}_2^<$. Suppose there exist $L<R$ such that $y$ and $z$ coincide and are periodic on $(-\infty,L] \cup [R,+\infty)$, so that $(-\infty,L]^2$ is the initial periodicity box. Let $\ell \in \mathbb{Z}_{<0}$. 1. Let $g_0 \in F$ be a map which is affine inside $(-\infty,L)^2$ and such that $\lim_{t \to -\infty}g_0(t)-t=q$. Then the unique conjugator ${\widehat{g}} \in {\mathrm{PL}}_2({\mathbb{R}})$ between $y$ and $z$, which is affine inside $(-\infty,L)^2$ and such that $\lim_{t \to -\infty}{\widehat{g}}(t)-t=\ell$ is defined pointwise by $${\widehat{g}}(t)=\lim_{r\to +\infty} y^{-r}g_0z^r(t).$$ Moreover, the map ${\widehat{g}}$ is recursively constructible and $y$ and $z$ are always conjugate in ${\mathrm{PL}}_2(\mathbb{R})$ via ${\widehat{g}}$. 2. There exists an algorithm to decide whether or not there is $g \in F$ such that $\lim_{t \to -\infty}g(t)-t=\ell$ and $g^{-1}yg=z$. The above result has been stated, for simplicity, for two functions $y,z \in {\mathrm{EP}}_2^<$. However, the same result can be stated for $y,z \in {\mathrm{PL}}_2^<([p_1,p_2])$ for any $p_1, p_2 \in \mathbb{Q}$, or for $y,z \in {\mathrm{EP}}_2^<(p,+\infty)$. *The results of this subsection do not involve dyadic rationals and slopes that are powers of 2 and are, in fact, true for other classes of groups without restrictions on the breakpoints and the slopes (for example ${\mathrm{PL}}_+(\mathbb{R})$, the Bieri-Thompson-Stein-Strebel groups in $\mathbb{R}$ and the corresponding subgroups with eventually periodic tails). See [@matucci5] for more details.* Orientation preserving case of the TCP: fixed points\[ssec:fixed-points\] ------------------------------------------------------------------------- The goal of this Subsection is to reduce to the case where the sets $\partial {\mathrm{Fix}}(y)$ and $\partial {\mathrm{Fix}}(z)$ do coincide. Up to suitable special cases, this will allow us to reduce to looking for potential conjugators $g\in F$ such that $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z)\subseteq {\mathrm{Fix}}(g)$, thus restricting ourselves to studying conjugacy among the corresponding intervals of $y$ and $z$ between any two consecutive points $p$ and $q$ of $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z)$. On each such interval $y$ (and $z$) is either the identity, or has no fixed points apart from $p$ and $q$ and so they belong to either ${\mathrm{EP}}_2^<(p,q)$ or ${\mathrm{EP}}_2^>(p,q)$. Note that the sets $\partial {\mathrm{Fix}}(y)$ and $\partial {\mathrm{Fix}}(z)$ are discrete subsets of $\mathbb{Q}$, and their intersections with any finite interval $[L, R]$ are easily computable by just solving finitely many systems of linear equations. An apparent technical difficulty is that, since $y,z \in {\mathrm{EP}}_2$, the full sets $\partial {\mathrm{Fix}}(y)$ and $\partial {\mathrm{Fix}}(z)$ may be infinite; however, due to the periodicity, they are controlled by finite sets. \[thm:identify-fixed-points\] There is an algorithm which, given $y,z\in {\mathrm{EP}}_2$ being equal for $t$ negative sufficiently large and for $t$ positive sufficiently large, decides whether or not there exists some $g\in F$ such that $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$ and, in the affirmative case, it constructs such a $g$. For the given $y,z$ we can easily compute constants $L<R$ such that, for all $t\in (-\infty, L]$, $y(t)=z(t)$ and $y(t-1)=y(t)-1$, and such that, for all $t\in [R, +\infty)$, $y(t)=z(t)$ and $y(t+1)=y(t)+1$. Moving $L$ down and/or $R$ up if necessary, we can also assume that if $\partial {\mathrm{Fix}}(y) \neq \emptyset$ then it has at least one point in $[L, R)$ (and similarly for $z$). Now compute the finite sets of rational numbers $\partial {\mathrm{Fix}}(z)\cap [L, R)$, $\partial {\mathrm{Fix}}(y)\cap [L, R)$, $\partial {\mathrm{Fix}}(z)\cap [L-1, L)=\partial {\mathrm{Fix}}(y)\cap [L-1, L)$, and $\partial {\mathrm{Fix}}(z)\cap [R, R+1)=\partial {\mathrm{Fix}}(y)\cap [R, R+1)$; let $p,q,m,n\geqslant 0$ be their cardinals, respectively. By the periodicity of $y$ and $z$ outside $[L, R]$, these constitute full information about $\partial {\mathrm{Fix}}(y)$ and $\partial {\mathrm{Fix}}(z)$. Up to switching $y$ with $z$, we may assume that $p\leqslant q$. Clearly, $m=0$ if and only if $\partial {\mathrm{Fix}}(y)$ and $\partial {\mathrm{Fix}}(z)$ have a minimum element (as opposed to having infinitely many points approaching $-\infty$). Similarly, $n=0$ if and only if $\partial {\mathrm{Fix}}(y)$ and $\partial {\mathrm{Fix}}(z)$ have a maximum element. If either $\partial {\mathrm{Fix}}(y)$ or $\partial {\mathrm{Fix}}(z)$ is empty then there is nothing to prove. Assume $\partial {\mathrm{Fix}}(y)\neq \emptyset \neq \partial {\mathrm{Fix}}(z)$, i.e. $1\leqslant p\leqslant q$. We denote by $a_0$ (respectively, $b_0$) the smallest element in $\partial {\mathrm{Fix}}(z)\cap [L,R)$ (respectively $\partial {\mathrm{Fix}}(y)\cap [L,R)$) and we use it to enumerate in an order preserving way all the elements of the discrete set $\partial {\mathrm{Fix}}(z)$ (respectively, $\partial {\mathrm{Fix}}(y)$) as $a_i$ (respectively, $b_i$); the index $i$ will run over a finite, infinite or bi-infinite subset of $\mathbb{Z}$ depending on whether or not $m$ (and/or $n$) is zero. With this definition, $\partial {\mathrm{Fix}}(z)\cap [L, R)=\{ a_0<a_1<\cdots <a_{p-1} \}$ and $\partial {\mathrm{Fix}}(y)\cap [L, R)=\{ b_0<b_1<\cdots <b_{q-1} \}$. Note that any $g\in F$ satisfying $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$ must map all the $a_i$’s bijectively to all the $b_i$’s. In particular, if $m=0$ then $a_0$ must be mapped to $b_0$, and if $n=0$ then $a_{p-1}$ must be mapped to $b_{q-1}$ (and so $a_0$ to $b_{q-p}$). Hence, in the special case that either $m=0$ or $n=0$, the following claim completes the proof. *Claim 1:* For every $b_i\in \partial {\mathrm{Fix}}(y)$, we can algorithmically decide whether or not there exists some $g\in F$ such that $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$ and $g(a_0)=b_i$ and, in the affirmative case, the algorithm constructs one explicitly. The remaining case to study is when $m\neq 0\neq n$, so that $a_0$ potentially could be sent to any of the $b_i$’s by the map $g$. Let $\ell=\mathrm{lcm}(m,n)$ and let $[L-\ell/m, L)$ be the smallest interval to the left of $L$ to contain $\ell$ points of $\partial {\mathrm{Fix}}(z)$. Similarly, let $[R, R+\ell/n)$ be the corresponding interval to the right of $R$. Consider the following two finite sets: $$\begin{array}{c} A:=\partial {\mathrm{Fix}}(z) \cap \left[L-\frac{2\ell}{m}, R+\frac{2\ell}{n} \right), \\ \\ B:=\partial {\mathrm{Fix}}(y) \cap \left[L-\frac{2\ell}{m}, R+\frac{2\ell}{n} \right), \end{array}$$ and let $s_0$ be the rightmost point of $\partial {\mathrm{Fix}}(z) \cap \left[L-\frac{2\ell}{m}, L-\frac{\ell}{m} \right)$, and let $t_0$ be the leftmost point of $\partial {\mathrm{Fix}}(z) \cap \left[R+\frac{\ell}{n}, R+\frac{2\ell}{n} \right)$. We compute $A$, $B$, $s_0$ and $t_0$ explicitly. *Claim 2:* Suppose there exists a map $g\in F$ such that $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$ and $g(s_0)\in \left[R+\frac{k\ell}{n}, R+\frac{(k+1)\ell}{n}\right)$ for $k\geqslant 2$; then, there exists a $g'\in F$ such that $\partial {\mathrm{Fix}}(y)=g'(\partial {\mathrm{Fix}}(z))$ and $g'(s_0)\in \left[R+\frac{(k-1)\ell}{n}, R+\frac{k\ell}{n}\right)$. Similarly, if there exists $g\in F$ such that $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$ and $g(t_0)\in \left[L-\frac{(k+1)\ell}{n}, L-\frac{k\ell}{n}\right)$ for some $k\geqslant 2$, then there exists a $g'\in F$ such that $\partial {\mathrm{Fix}}(y)=g'(\partial {\mathrm{Fix}}(z))$ and $g'(t_0)\in \left[L-\frac{k\ell}{n}, L-\frac{(k-1)\ell}{n}\right)$. With the help of Claim 2 we can complete the proof in the following way. Suppose there exists $g\in F$ such that $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$. Since $p\leqslant q$ it cannot simultaneously happen that $g(s_0)<s_0$ and $t_0<g(t_0)$. Hence either $s_0\leqslant g(s_0)$ or $g(t_0)\leqslant t_0$ and, in either case, a repeated application of Claim 2 implies the existence of $g'\in F$ such that $\partial {\mathrm{Fix}}(y)=g'(\partial {\mathrm{Fix}}(z))$ and $g'(A)\cap B\neq \emptyset$. This gives finitely many possibilities for $g'(a_0)$ and so, applying Claim 1 finitely many times we can decide whether or not there exists a $g\in F$ satisfying $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$. Hence, it only remains to prove the above two claims. We will distinguish four cases. **Case 1: $m=0$ and $n=0$.** In this case, $\partial {\mathrm{Fix}}(z)=\{ a_0<a_1<\cdots <a_{p-1} \}$ and $\partial {\mathrm{Fix}}(y)=\{ b_0<b_1<\cdots <b_{q-1} \}$ and, clearly, $p=q$ and $g(a_0)=b_0$ are necessary conditions for such a $g$ to exist. If both conditions hold, then Proposition \[thm:rationals-coincide\] makes the decision for us. **Case 2: $m\geqslant 1$ and $n=0$.** This case is entirely symmetric to the next one. **Case 3: $m=0$ and $n\geqslant 1$.** In this case, $\partial {\mathrm{Fix}}(z)$ and $\partial {\mathrm{Fix}}(y)$ both have first elements $a_0$ and $b_0$ and infinitely many points approaching $+\infty$. As in case 1, $g(a_0)=b_0$ is a necessary condition for such a $g$ to exist. We have $\partial {\mathrm{Fix}}(z)\cap [R, R+1)=\{ a_{p}<a_{p+1}<\cdots <a_{p+(n-1)} \}$ and that the elements in $\partial {\mathrm{Fix}}(z) \cap [R+1,+\infty)$ are integral translations of these: for every $j\geqslant 0$, write $j=\lambda n+\mu$ with $\lambda,\mu\geqslant 0$ integers and $\mu =0,\ldots, n-1$, and we have $a_{p+j}=\lambda+a_{p+\mu}$. A similar argument for $y$ yields that $\partial {\mathrm{Fix}}(y)\cap [R, R+1)=\{ b_{q}<b_{q+1}<\cdots <b_{q+(n-1)} \}$ and that, for every $j\geqslant q$, we have $b_{q+j}=\lambda+b_{q+\mu}$. Moreover, from $a_p=b_q$ on, the two sequences coincide, i.e., for every $j\geqslant 0$, $$\lambda+a_{p+\mu}=a_{p+j}=b_{q+j}=\lambda+b_{q+\mu}.$$ Now if some $g\in F$ satisfies $g(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$, it must apply the points in an order preserving way, starting from the smallest ones, that is, $g(a_k)=b_k$ for any integer $k$. In particular, for $k\geqslant q\geqslant p$, we have $$g(\lambda_1+a_{p+\mu_1})=g(a_{p+(k-p)})=g(a_k)=b_k=b_{q+(k-q)}=\lambda_2+b_{q+\mu_2},$$ where $k-p=\lambda_1 n+\mu_1$ and $k-q=\lambda_2 n+\mu_2$. Since $g$ is of the form $g(t)=t+m_+$ with $m_+\in \mathbb{Z}$ for $t$ positive sufficiently large then, for large enough $k$, the above equation tells us that $$\lambda_1+a_{p+\mu_1}+m_+=g(\lambda_1+a_{p+\mu_1})=\lambda_2+b_{q+\mu_2}.$$ Therefore, $a_{p+\mu_1}-b_{q+\mu_2}=b_{q+\mu_1}-b_{q+\mu_2}$ must be an integer and so, $\mu_1=\mu_2$, which means that $k-p$ and $k-q$ are congruent modulo $n$, i.e. $q-p$ is multiple of $n$. Assume then this necessary condition, $q-p=\lambda n$ with $\lambda \in \mathbb{Z}$, and apply Proposition \[thm:rationals-coincide\] (2) to the sequences $a_0<\cdots <a_{p+\lambda n-1}$ and $b_0<\cdots <b_{q-1}$ (both with $q$ points). If there is no $g\in F$ sending the first list to the second then there is no $g$ such that $\partial {\mathrm{Fix}}(y) =g(\partial {\mathrm{Fix}}(z))$ and we are done. Otherwise, we get a $g$ matching these first $q$ points, $g(a_0)=b_0, \ldots ,g(a_{p+\lambda n-1})=b_{q-1}$, and, after a final small modification, we will see that it automatically matches the rest. Choose two dyadic numbers $a_{p+\lambda n-1}<\alpha <\beta <a_{p+\lambda n}$, choose $h\in F$ such that $h(\alpha )=g(\alpha)$ and $h(\beta )=\beta -\lambda$ (such an $h$ exists and is effectively computable by Proposition \[thm:rationals-coincide\] (2)), and let us consider the following map: $$\widetilde{g}(t)=\begin{cases} g(t) & t \leqslant \alpha \\ h(t) & \alpha \leqslant t\leqslant \beta \\ t-\lambda & \beta \leqslant t.\end{cases}$$ By construction, $\widetilde{g}$ is continuous, piecewise linear with dyadic breakpoints, and all slopes are powers of 2; furthermore $g\in F$ and $\widetilde{g}$ is an integral translation for $t\geqslant \beta$ so, $\widetilde{g}\in F$. On the other hand, $$\partial {\mathrm{Fix}}(y)\cap [L, b_{q-1}]=g(\partial {\mathrm{Fix}}(z)\cap [L,a_{p+\lambda n-1}])=\widetilde{g}(\partial {\mathrm{Fix}}(z)\cap [L,a_{p+\lambda n-1}]),$$ and $$\partial {\mathrm{Fix}}(y)\cap [b_{q}, +\infty)=\{ b_q, b_{q+1},\ldots \}=\{ a_{p+\lambda n}-\lambda, a_{p+\lambda n+1}-\lambda, \ldots \} =$$ $$=\widetilde{g}(\{ a_{p+\lambda n}, a_{p+\lambda n+1}, \ldots \})=\widetilde{g}(\partial {\mathrm{Fix}}(z)\cap [a_{p+\lambda n}, +\infty )).$$ Hence, $\partial {\mathrm{Fix}}(y)=\widetilde{g}(\partial {\mathrm{Fix}}(z))$ and we are done. **Case 4: $m\geqslant 1$ and $n\geqslant 1$.** The argument in this case is similar to that of case 3 but repeated twice, up and down (and with no restriction for $b_i$ because we have both infinitely many fixed points bigger and smaller than $b_i$). Following the notation above, the $m$ fixed points from $\partial {\mathrm{Fix}}(z)\cap [L-1, L)=\partial {\mathrm{Fix}}(y)\cap [L-1, L)$ are labeled and ordered as $a_{-m}<\cdots <a_{-1}$ and $b_{-m}<\cdots <b_{-1}$ (hence, $a_{-j}=b_{-j}$ for $j=1,\ldots ,m$). The elements from $\partial {\mathrm{Fix}}(z) \cap (-\infty,L-1)$ and $\partial {\mathrm{Fix}}(y) \cap (-\infty,L-1)$ are their integral translations to the left. Now if some $g\in F$ satisfies $g(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$ and $g(a_0)=b_i$, it must send the points $a_j$ to the $b_j$ in an order preserving way starting from $g(a_0)=b_i$, both up and down. Hence, two arguments exactly like in the previous case give us two necessary congruences among $p,q$ and $i$, modulo $n$ (close to $+\infty$) and modulo $m$ (close to $-\infty$). If one of them fails, then there is no such $g$ and we are done. If both are satisfied, then apply Proposition \[thm:rationals-coincide\] (2) to a long enough tuple of $a_j$’s and $b_j$’s: a negative answer tells us there is no such $g\in F$, and a positive answer provides a $g\in F$ which, after two local modifications like in the previous case (one close to $+\infty$ and the other close to $-\infty$), will finally give us a $g'\in F$ such that $g'(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$, and $g'(a_0)=b_i$. This completes the proof of Claim 1. We will prove the first part of the claim; the symmetric argument for the second part is left to the reader. Assume the existence of $g\in F$ such that $g(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$ and $g(s_0 )\in \left[R+\frac{k\ell}{n}, R+\frac{(k+1)\ell}{n}\right)$ for $k\geqslant 2$. To push $g(s_0)$ down, let us define the *reduction* map $g_-$ by $$g_-(t)= \begin{cases} g(t-\frac{\ell}{m}) & t<s_0 \\ g(t)-\frac{\ell}{n} & t \ge s_0. \end{cases}$$ To understand the map $g_-$, note that its graphical representation can be obtained from that of $g$ by performing the following operation: remove the graph within $[s_0-\ell/m,s_0]$, translate the graph of $g$ defined on $[s_0,+\infty)$ by the vector $(0,-\ell/m)$ and translate the graph of $g$ defined on $(-\infty,s_0-\ell/m]$ by the vector $(\ell/m,0)$. Hence, $g_-$ is the same as $g$ avoiding the piece over the interval $[s_0-\ell/m, s_0]$. It is obvious that the two parts of $g_-$ to the left and to the right of $s_0$ are both continuous, increasing, piecewise linear, with dyadic breakpoints, with slopes being powers of two, and being eventually translations (near $-\infty$ and $+\infty$, respectively). To check whether $g_-$ is in $F$ it only remains to analyze what happens around the point $s_0$. First of all, $g_-$ is continuous at $s_0$: observe that $s_0-\frac{\ell}{m}\in \partial {\mathrm{Fix}}(z)$ is exactly $\ell$ points to the left of $s_0$ in the discrete set $\partial {\mathrm{Fix}}(z)$; since $g(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$ and $g$ is an increasing function, $g(s_0-\frac{\ell}{m})$ must be exactly $\ell$ points to the left of $g(s_0)$ in the discrete set $\partial {\mathrm{Fix}}(y)$ that is, $g(s_0-\frac{\ell}{m})=g(s_0)-\frac{\ell}{n}$. Unfortunately, the slopes of $g_-$ to the left and to the right of $s_0$ (i.e. the slopes of $g$ to the left of $s_0-\ell/m$ and to the right of $s_0$) may be different; and $s_0$ may not be a dyadic rational number. If these two facts happen simultaneously then $g_-$ will not an element of $F$ because of having a breakpoint at a non-dyadic point, namely $s_0$. This technical difficulty will be fixed later by modifying the map $g_-$ in a suitably small neighborhood of $s_0$. Before doing this, let us check that $g_-$ fulfils our requirement. Since $g(s_0-\frac{\ell}{m})=g(s_0)-\frac{\ell}{n}$, the hypothesis $g(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$ implies that $$g_-(\partial {\mathrm{Fix}}(z)\cap (-\infty, s_0])=g\left(\partial {\mathrm{Fix}}(z)\cap \left(-\infty, s_0-\frac{\ell}{m}\right]\right)=$$ $$=\partial {\mathrm{Fix}}(y)\cap \left(-\infty, g(s_0)-\frac{\ell}{n}\right],$$ and $g_-(s_0)=g(s_0-\frac{\ell}{m})=g(s_0)- \frac{\ell}{n}$, and $$g_-(\partial {\mathrm{Fix}}(z)\cap [s_0, +\infty))=g(\partial {\mathrm{Fix}}(z)\cap [s_0, +\infty ))- \frac{\ell}{n} =\partial {\mathrm{Fix}}(y)\cap \left[g(s_0)-\frac{\ell}{n}, +\infty \right).$$ Hence, $g_-(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$ and $g_-(s_0)=g(s_0)-\frac{\ell}{n}\in \left[R+\frac{(k-1)\ell}{n}, R+\frac{k\ell}{n}\right)$, as we wanted. To complete the proof of Claim 1 we must be able to fix the above technical problem, by modifying $g_-$ in such a way that the resulting map belongs to $F$, but not changing the image of any point in $\partial {\mathrm{Fix}}(z)$; this will be achieved by changing $g_-$ only in a small enough neighborhood of $s_0$ not containing any other point of $\partial {\mathrm{Fix}}(z)$ (and, of course, not changing the image of $s_0$ itself). Let $\alpha_1$ be a dyadic point found strictly between $\alpha_2:=s_0$ and the point of $A$ immediately to the left of $s_0$; and let $\alpha_3$ be a dyadic point found strictly between $\alpha_2:=s_0$ and the point of $A$ immediately to the right of $s_0$. Now consider the points $$\beta_1:=g_-(\alpha_1)=g\left(\alpha_1-\frac{\ell}{m} \right),$$ $$\beta_2:=g_-(\alpha_2)=g\left(\alpha_2-\frac{\ell}{m} \right)=g(\alpha_2)-\frac{\ell}{n},$$ $$\beta_3:=g_-(\alpha_3)=g(\alpha_3)-\frac{\ell}{n}.$$ Since $\alpha_1<\alpha_2<\alpha_3$ and $\beta_1<\beta_2<\beta_3$ are rational points such that, for every $i=1,2,3$, $\beta_i$ is the image of $\alpha_i$ by some element in $F$, then we can apply Proposition \[thm:rationals-coincide\] (2) and construct a function $h\in F$ such that $\beta_i=h(\alpha_i)$. Finally, define $$g'(t)=\begin{cases} h(t) & t\in [\alpha_1,\alpha_3] \\ g_-(t) & t\not \in [\alpha_1,\alpha_3]. \end{cases}$$ Clearly, $g'\in F$, $g'(s_0)=g_-(s_0)$ and $g'(\partial {\mathrm{Fix}}(z))=g_-(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$. This completes the proof of Claim 2. This finishes the proof of Proposition \[thm:identify-fixed-points\]. \[thm:RTCP\] The decidability of the following two problems is equivalent: 1. For any two $y,z\in {\mathrm{EP}}_2$ we can determine whether or not there is $g\in F$ such that $g^{-1}yg=z$. 2. For any two $y,z\in {\mathrm{EP}}_2$ such that $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z)$ we can determine, whether or not there is $g\in F$ such that $g^{-1}yg=z$. Obviously, if (TCP) is decidable, then (RTCP) is decidable. Assume now that (RTCP) is decidable. By the discussion at the beginning of this subsection, if $y$ and $z$ are conjugate via $g \in F$, then $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$. By Theorem \[thm:identify-fixed-points\] we can decide whether or not there is a map $g \in F$ such that $\partial {\mathrm{Fix}}(y)=g(\partial {\mathrm{Fix}}(z))$. If there is no such map, then $y$ and $z$ are not conjugate. If there is such a $g\in F$ (and in this case Theorem \[thm:identify-fixed-points\] constructs it) then $\partial ({\mathrm{Fix}}(gzg^{-1}))=g(\partial {\mathrm{Fix}}(z))=\partial {\mathrm{Fix}}(y)$ and we can apply (RTCP) to the two maps $y$ and $gzg^{-1}$ to detect whether or not they are conjugate. Obviously, this is the same decision as the one we are interested in. By Lemma \[thm:RTCP\] we can restrict our focus to studying (RTCP). Orientation preserving case of the TCP: Reducing the problem to squares. \[ssec:reducing-to-squares\] ----------------------------------------------------------------------------------------------------- We can make $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z)$ as done in Proposition \[thm:identify-fixed-points\]. If $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z) = \emptyset$ we defer the discussion to Subsection \[sec:rescaling-the-circle\]. On the other hand, if $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z) \neq \emptyset$ and $g\in F$ is a conjugator between $y$ and $z$, the only thing we can say is that $g$ acts on $\partial {\mathrm{Fix}}(y)$ in an order preserving way. There are two possibilities: 1. ${\mathrm{Fix}}(g) \neq \emptyset$. 2. ${\mathrm{Fix}}(g) = \emptyset$. We can assume that $g \in F^>$. Case (2) can indeed happen as is shown by the following example: take $y=z$ to be a non-trivial periodic function of period $1$ with fixed points. Then the map $g(t)=t+1 \in F^>$ is a conjugator for $y$ and $z$ having no fixed points. We need to find if there is a conjugator $g$ between $y$ and $z$ such that $g \in F^>$. We can assume $y \ne {\mathrm{id}}\ne z$, otherwise our analysis becomes trivial. We can write the supports ${\operatorname{supp}}(y)={\operatorname{supp}}(z)$ as the union of the family $\{I _j\}$ of (possibly unbounded) intervals on which $y$ and $z$ have no fixed points ordered so that $I_j$ is to the left of $I_{j+1}$, for every $j$. If this family were finite, since we are assuming $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z) \neq \emptyset$, then it means that $\partial {\mathrm{Fix}}(y)$ is finite and so $g$ must fix the smallest element in $\partial {\mathrm{Fix}}(y)$ since $g$ is order preserving, hence ${\mathrm{Fix}}(g) \ne \emptyset$ and this would not be the case that we are studying now. Thus we must study the case of the following proposition. Let $y,z \in {\mathrm{EP}}_2$ be such that ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z)$ and that $\partial {\mathrm{Fix}}(y)$ has infinitely many points. Then there are only finitely many candidate conjugators $g \in F^>$. We give out only some relevant details of how to prove this proposition. This entails generalizations of many results of this paper and of Kassabov-Matucci  [@matucci5] and so we only explain how to achieve them. The main point here is noticing that we can develop a Stair Algorithm and bounding initial slopes of $g \in F$, even if at $-\infty$ the functions $y,z$ have no initial slope. By hypothesis, $\{I_j\}$ has infinitely many intervals and so $g$ “shifts” them, that is $g(I_j)=I_{j+k}$, for some fixed $k$. Let $t_j$ be the left endpoint of $I_j$. We make a series of observations: 1. We can build candidate conjugators (Theorem \[thm:explicit-conjugator\]) on each $I_j$, given a fixed initial slope at $t_j$, 2. The initial slope of $z$ on $I_j$ coincides with the initial slope of $y$ in the image interval $g(I_j)$, 3. There is an “initial” box for $g$ in $I_j$, 4. We can bound the “initial” slopes of $g$ on $I_j$, 5. We can bound the initial slope of $g$ at $-\infty$. \(1) and (2) are a straightforward calculation. (3) is a verbatim rewriting of the proof of Lemma 4.2 in  [@matucci5]. \(4) A standard trick from  [@matucci5] is observing that $$z=g^{-1}yg= g^{-1}y^{-r} y y^rg$$ and so the slope of $y^r g$ at $t_j$ is $(y'(t_{i+k}))^r g'(t_i)$ and $y^r g$ is a conjugator for $y$ and $z$ on $I_j$. On each $I_j$ there are only finitely many slopes for $g'(t_i^+)$ to be tested and on each one, we apply Theorem \[thm:explicit-conjugator\] to build candidate conjugators that we can test. \(5) Recall that a candidate conjugator $g$ pushes all the intervals in ${\operatorname{supp}}(y)$ in the same direction by the “same amount of intervals in ${\operatorname{supp}}(y)$”. In particular, the initial slope of $g$ determines the number $k$ such that $g(I_j)=I_{j+k}$ for every $j$. We use ideas similar to Claim 2 in Proposition \[thm:identify-fixed-points\]. Let us call $J_L$ the left open interval on which $y=z$ and they are periodic. A similar definition can be made for $J_R$. Let $J_C=\mathbb{R} \setminus (J_L \cup J_R)$ the remaining central piece. Assume that there is a conjugator $g$ between $y$ and $z$ which sends and interval $I_j$ inside $J_L$ to an interval $I_{j+k+1}$ with the requirement that $I_{j+k}$ is entirely contained into $J_R$. Using ideas similar to Claim 2 in Proposition \[thm:identify-fixed-points\] one can create a new conjugator $\overline{g}$ such that $\overline{g}(I_j)=I_{j+k}$. Therefore, similarly to Claim 2 in Proposition \[thm:identify-fixed-points\], this allows us to reduce the study to only finitely many candidate conjugators where $g(J_C) \cap J_C \ne \emptyset$ or where the rightmost interval $I_j$ inside $J_L$ goes to the leftmost interval $I_s$ of $J_R$ (or viceversa). This argument reduces the number of initial slopes of $g$ to be tested. To conclude we observe that there are only finitely many slopes for $g$ at $-\infty$ and finitely many “initial” slopes for $g$ on the finitely many intervals $I_j$ contained in $J_C$ and then we can apply Theorem \[thm:explicit-conjugator\] on each of these intervals building finitely many candidate conjugators $g \in F^>$ which we can then test one by one. The previous result allows one to restrict to the case of looking for conjugators $g$ with fixed points. \[tim:fix-of-z-is-inside-fix-g\] Let $y,z \in {\mathrm{EP}}_2$ be such that ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z) \ne \emptyset$ and let $g$ be a conjugator between $y$ and $z$ such that ${\mathrm{Fix}}(g) \ne \emptyset$. Then ${\mathrm{Fix}}(z) \subseteq {\mathrm{Fix}}(g)$. Let $a \in {\mathrm{Fix}}(g)$ and let $b$ be the the smallest point of $\partial {\mathrm{Fix}}(z)$ such that $a<b$. Since $g$ fixes ${\mathrm{Fix}}(z)$ set wise and is order-preserving, then $g(b)$ must also be the smallest point of $\partial {\mathrm{Fix}}(z)$ such $g(b)>a$, therefore $g(b)=b$ and so $g$ must fix all of ${\mathrm{Fix}}(z)$ pointwise. We need to show that (RTCP) of Lemma \[thm:RTCP\] is decidable. Lemma \[tim:fix-of-z-is-inside-fix-g\] tells us that we can restrict ourselves to solve the problem inside the closed intervals of ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z)$. As was done in [@matucci5] we observe that if $p \in \partial {\mathrm{Fix}}(y)$ is a non-dyadic rational point and $g$ is a conjugator between $y$ and $z$, then $g'(p^-)=g'(p^+)$ or, in other words, the slope of $g$ at one side of $p$ is completely determined by the slope on its other side. This implies that the important points of $\partial {\mathrm{Fix}}(y)$ are the dyadic rational ones (if they exist) as they are the ones where $g$ has freedom to have different slopes on the two sides and therefore the conjugator that we are attempting to build can be constructed by by gluing two conjugators on the two sides of a dyadic rational point of $\partial {\mathrm{Fix}}(y)$. In the case that $\partial {\mathrm{Fix}}(y)$ had no dyadic rational points, then we can compute a conjugator at a point $p \in \partial {\mathrm{Fix}}(y)$ and this uniquely determines the conjugator on the entire real line. Otherwise, there are dyadic rational points in $\partial {\mathrm{Fix}}(y)$ and we argue as following. Let $L<R$ are two integers chosen so that $y$ and $z$ coincide and are periodic inside $(-\infty,L] \cup [R,+\infty)$. The case when $\partial {\mathrm{Fix}}(y) \cap [L,R]$ contains no dyadic rational point is dealt with as above. Similarly, if there is only one dyadic point inside $\partial {\mathrm{Fix}}(y) \cap [L,R]$, then we have two instances of the previous case on the two sides of the dyadic point. Otherwise, we choose $p_1,p_2$ with the property of being dyadic and consecutive inside $\partial {\mathrm{Fix}}(y)$ and such that $[p_1,p_2] \subseteq [L,R]$. With these provisions, we can use the solution of the standard conjugacy problem inside ${\mathrm{PL}}_2([p_1,p_2])$ using the techniques in [@matucci5]. If there is no conjugator on any of those intervals, then $y$ and $z$ cannot be conjugate. Otherwise, we can glue the conjugators that we find on each such interval. We then need to understand what happens outside $[L,R]$. Let $p$ be the rightmost dyadic point of $\partial {\mathrm{Fix}}(y)\cap [L,R]$. If $y,z \in {\mathrm{EP}}_2^>(p,+\infty)$ (or $y,z \in {\mathrm{EP}}_2^<(p,+\infty)$), then we deal with this case in Subsection \[sec:rescaling-the-circle\]. Otherwise, let $q$ be the leftmost point of $\partial {\mathrm{Fix}}(y) \cap (R,+\infty)$. If $y(t)=z(t)=t$ on $[p,q]$, then we define $g(t)=t$ on $[p,+\infty)$ and this defines a conjugator for $y$ and $z$ on $[p,+\infty)$ which we can glue to the previous intervals. Otherwise, we apply the standard conjugacy problem on the interval $[p,q]$ with final slope $1$ at $q^-$ since the conjugator $g$ has to be the identity translation on $[R,+\infty)$. If the standard conjugacy problem on $[p,q]$ has no solution, then $y$ and $z$ cannot be conjugate. Otherwise, if $h$ is the conjugator on $[p,q]$ we define $$g(t):= \begin{cases} h(t) & t \in [p,q] \\ t & [q,+\infty) \end{cases}$$ which is a well-defined map of $F$, since $g'(q^-)=g'(q^+)=1$, regardless of whether or not $q$ is dyadic. The map $g$ defines a conjugator for $y$ and $z$ on $[p,+\infty)$ which we can glue to the previous intervals. A similar argument can be applied to the left of $L$. Orientation preserving case of the TCP: Mather invariants\[sec:rescaling-the-circle\] ------------------------------------------------------------------------------------- The procedure outlined in [@matucci5] to solve the conjugacy problem in Bieri-Thompson-Stein-Strebel groups requires various steps which we have studied already: (i) making ${\mathrm{Fix}}(y)$ and ${\mathrm{Fix}}(z)$ coincide (seen in Subsection \[ssec:fixed-points\]) and (ii) showing that, for a possible initial slope of a conjugator in $F$ (see Remark \[thm:remark-initial-slope\]), there exists at most one candidate and we can compute it through an algorithm (seen in Subsection \[ssec:periodicity-boxes\]). The next natural step is to bound the number of integers $\lim_{t \to -\infty}g(t)-t$ representing possible initial slopes for which we need to build a candidate conjugator. In order to do this, we will employ ideas to characterize conjugacy seen in [@matucci3], by taking very large powers of $y$ and $z$ and building a conjugacy invariant. In [@matucci3] a conjugacy class in $F$ has been described by a double coset $Ay^\infty B$ where $y^\infty$ is an element of Thompson’s group $T$ obtained by taking suitable high powers of $y$ and $A$ and $B$ are two finite cyclic groups (of rotations of the circle). In the case of the twisted conjugacy problem that we are studying, the Mather invariant will be essentially defined by a product $A y^\infty B$ where $A \cong B \cong \mathbb{Z}$. *Mather invariant construction.* In what follows, we will assume that $y,z \in {\mathrm{EP}}_2^>$, to simplify the notation. We can define Mather invariants in the two neighborhoods of infinity (that is on ${\mathrm{EP}}_2(-\infty,p)$ and ${\mathrm{EP}}_2(q,+\infty)$ for some suitable numbers $p,q$), while solving the conjugacy problem between any two consecutive dyadic points of $\partial {\mathrm{Fix}}(y)=\partial {\mathrm{Fix}}(z)$. Let $y,z \in {\mathrm{EP}}_2^>$ and assume that, on the intervals $(-\infty,L]\cup [R,+\infty)$, the maps $y$ and $z$ coincide and are periodic, for some integers $L \leqslant R$. Let $N \in \mathbb{N}$ be large enough so that $y^N((y^{-1}(L),L)) \subseteq (R,+\infty)$ and $z^N((y^{-1}(L),L)) \subseteq (R,+\infty)$. We look for an orientation preserving homeomorphism $H \in {\mathrm{PL}}_2(\mathbb{R})$ such that 1. $H(y^k(L))=k$, for any integer $k$, and 2. $H(y(t))=\lambda(H(t))=H(t)+1$, where $\lambda(t)=t+1$. To construct $H$, choose any ${\mathrm{PL}}_2$-homeomorphism $H_0:[y^{-1}(L),L] \to [-1,0]$ with finitely many breakpoints. Then we extend it to a map $H \in {\mathrm{PL}}_2(\mathbb{R})$ by defining $$H(t):=H_0(y^{-k}(t))+k \qquad \mbox{if} \; t \in [y^{k-1}(L),y^k(L)] \; \mbox{for some integer $k$}.$$ We make a series of remarks. - By construction, it is easy to see that $H(y(t))=\lambda(H(t))$ for any real number $t$. - If we define ${\overline{y}}:=Hy H^{-1},{\overline{z}}:=Hz H^{-1}$, we observe that, by construction, they both coincide with $\lambda(t)=t+1$ on the intervals $(-\infty,1]\cup [N,+\infty)$. It is also clear that ${\overline{y}}=\lambda$. - We notice that ${\overline{\lambda}}=H\lambda H^{-1} \in {\mathrm{EP}}_2$. To show this, let $t$ be positive sufficiently large so that $y$ is periodic of period $1$ and that all the calculations below make sense and define $\widetilde{t}=H_0^{-1}(t- k-1)$: $${\overline{\lambda}}(t+1)=H \lambda y^{k+2}H_0^{-1}\lambda^{-k-2}(t+1)= H\lambda y^{k+2}(\widetilde{t})= H(y^{k+2}(\widetilde{t})+1) =$$ $$\lambda^{k'}H_0 y^{-k'}(y^{k+2}(\widetilde{t}+1)) = \lambda^{k'-1}H_0 y^{-k'+1}(y^{k+1}(\widetilde{t}+1))+1 ={\overline{\lambda}}(t)+1,$$ where $k'$ are the jumps that $y$ must make to bring $y^{k+2}(\widetilde{t}+1)$ back to the domain of $H_0$. A similar argument can be shown for $t$ negative sufficiently large. We define $$C_0:= (-\infty,0)/\mathbb{Z} \qquad C_1:= (N,\infty)/\mathbb{Z}$$ and let $p_0:(-\infty,0) \to C_0$ and $p_1:(N,\infty) \to C_1$ be the natural projections. Then we define the map ${\overline{y}}^\infty:C_0 \to C_1$ by $${\overline{y}}^{\infty}([t]):=[{\overline{y}}^{N}(t)].$$ Similarly we can define the map ${\overline{z}}^{\infty}$. The maps ${\overline{y}}^{\infty}$ and ${\overline{z}}^{\infty}$ are well-defined and they do not depend on the specific $N$ chosen (the proof is analogous to the one in Section 3 in [@matucci3]). They are called the *Mather invariants* of ${\overline{y}}$ and ${\overline{z}}$ (compare this with the definitions in Section 3 in [@matucci3]). This induces the equation ${\overline{g}}{\overline{z}}^N={\overline{y}}^N{\overline{g}}$ which, following [@matucci3], passes to quotients and becomes $$\label{eq:mather-equivalence} v_1^k {\overline{z}}^{\infty} = {\overline{y}}^{\infty} v_0^{\ell}$$ since all the maps ${\overline{y}},{\overline{z}},{\overline{g}}$ are in ${\mathrm{EP}}_2$ and where $v_1 := p_1 {\overline{\lambda}} p_1^{-1}$ is an element of Thompson’s group $T_{C_1}$ defined on the circle $C_1$ and induced by ${\overline{\lambda}}$ on $C_1$ by passing to quotients via the map $p_1$, $v_0:=p_0 {\overline{\lambda}} p_0^{-1}$ and where $\ell,k$ are the initial and final slopes of $g$. The following result shows that the integer solutions of equation (\[eq:mather-equivalence\]) correspond to conjugators between $y$ and $z$. The proof is an extension of the proof of Theorem 4.1 in [@matucci3]. \[thm:mather-iff\] Let $y,z \in {\mathrm{EP}}_2^>$. Then $y$ and $z$ are conjugate through an element $g \in F$ if and only if there is a pair of integers $k,\ell$ that satisfy equation (\[eq:mather-equivalence\]). Clearly, if $g \in F$ conjugates $y$ to $z$, then equation (\[eq:mather-equivalence\]) is satisfied by the calculations above. Conversely, assume that we have a pair $(k,\ell)$ such that (\[eq:mather-equivalence\]) is satisfied. We use Theorem \[thm:explicit-conjugator\] to find a map $g \in {\mathrm{PL}}_2({\mathbb{R}})$ which is affine around $-\infty$, such that $\lim_{x\to -\infty}g(x)-x=\ell$ and that $yg=gz$. By conjugating via $H$ we see that ${\overline{y}}{\overline{g}}={\overline{g}}{\overline{z}}$. If $x$ is positive sufficiently large then ${\overline{y}}(x)={\overline{z}}(x)=x+1$ so $${\overline{g}}(x)+1={\overline{y}}{\overline{g}}(x)={\overline{g}}{\overline{z}}(x)={\overline{g}}(x+1).$$ Arguing similarly at $\infty$ we deduce that ${\overline{g}} \in {\mathrm{EP}}_2$ and so the equation ${\overline{y}}^N{\overline{g}}={\overline{g}}{\overline{z}}^N$ passes to quotients and becoming ${\overline{g}}_{\mathrm{ind}} {\overline{z}}^{\infty} = {\overline{y}}^{\infty} v_0^{\ell}$. By using our assumption we see that ${\overline{g}}_{\mathrm{ind}} {\overline{z}}^{\infty} = {\overline{y}}^{\infty} v_0^{\ell} = v_1^k {\overline{z}}^{\infty}$ and by cancellation we obtain ${\overline{g}}_{\mathrm{ind}} = v_1^k$. By taking the unique lift of ${\overline{g}}_{\mathrm{ind}}$ and $v_1^k$ defined on $[N,N+1)$ and passing through the point $(N,g(N))$, we see that ${\overline{g}}$ and ${\overline{\lambda}}^k$ coincide on $[N,N+1]$ and therefore they coincide on $[N,+\infty)$ since they are both in ${\mathrm{EP}}_2$. Thus, $g \in F$ since ${\overline{g}}(x)={\overline{\lambda}}(x)$ around $+\infty$. We relabel $t_0:={\overline{z}}^{\infty} v_0^{-1} ({\overline{z}}^{\infty})^{-1}$, $t_1:=v_1$ and and $t:={\overline{y}}^{\infty}({\overline{z}}^{\infty})^{-1}$ and we rewrite equation (\[eq:mather-equivalence\]) as $$\label{eq:thesis-case} \numberwithin{equation}{section} t_1^k t_0^\ell=t$$ where $t_0,t_1,t \in T_{C_1}$. To solve equation (\[eq:thesis-case\]) we will need Lemma 8.4 from [@matucci5] which we restate for the reader’s convenience. \[thm:matucci5-special-case\] Let $p \in \mathbb{Q}$ and let ${\mathrm{PL}}_2([p,p+1])$ be the group of piecewise-linear homeomorphisms of the interval $[p,p+1]$ with finitely many breakpoints which occur at dyadic rational points and such that all their slopes are powers of $2$. If $t_0,t_1,t \in {\mathrm{PL}}_2([p,p+1])$, there is an algorithm which outputs one of the following two mutually exclusive cases in finite time: 1. Equation (\[eq:thesis-case\]) has at most one solution and we compute a pair $(k,\ell)$ such that, if equation (\[eq:thesis-case\]) is solvable, then $(k,\ell)$ must be its unique solution. 2. Equation (\[eq:thesis-case\]) has infinitely many solutions which are given by the sequence of pairs $(k_j,\ell_j)$ where $k_j = a_1 j +b_1$ and $\ell_j=a_2 j + b_2$ for any $j \in \mathbb{Z}$ and for some integers $a_1,a_2,b_1,b_2$ which we can compute. Lemma \[thm:matucci5-special-case\] gives a solution for equation (\[eq:thesis-case\]) in the case that $t_0,t_1,t$ live in a copy of Thompson’s group ${\mathrm{PL}}_2([p,p+1])$ of functions over an interval. However, equation (\[eq:thesis-case\]) needs to be solved in a copy of Thompson’s group $T$ of functions over a circle, so we will need to adapt Lemma \[thm:matucci5-special-case\] to our needs. \[thm:thesis-case\] Let $T$ be Thompson’s group ${\mathrm{PL}}_2(S^1)$ and let $t_0,t_1,t \in T$. Then there is an algorithm which outputs one of the following two mutually exclusive cases in finite time: 1. Equation (\[eq:thesis-case\]) has at most finitely many solutions and we compute a finite set $S$ such that, if $(k,\ell)$ is a solution of equation (\[eq:thesis-case\]), then $\ell \in S$. 2. Equation (\[eq:thesis-case\]) has infinitely infinitely many solutions and we compute a sequence of solutions $(k_j,\ell_j)$ where $k_j = a_1 j +b_1$ and $\ell_j=a_2 j + b_2$ for any $j \in \mathbb{Z}$ and for some integers $a_1,a_2,b_1,b_2$. For a map $h \in T$, we denote by $\mathrm{Per}(h)$ the set of all periodic points of $h$. Obviously, ${\mathrm{Fix}}(h) \subseteq \mathrm{Per}(h)$. By a result of Ghys and Sergiescu [@GhysSergiescu] every element of $T$ has at least one periodic point. For $i=0,1$, we find a $q_i \in \mathrm{Per}(t_i)$ be a point of period $d_i$. If $d=\mathrm{lcm}(d_0,d_1)$, then both $t_0^d,t_1^d$ have fixed points and therefore $\mathrm{Per}(t_i^d)={\mathrm{Fix}}(t_i^d)$. Using the division algorithm we write $k=k'd+r$ and $\ell=\ell'd+s$ with $0 \leqslant r,s <d$ so that equation (\[eq:thesis-case\]) becomes $$\label{eq:family-of-equations} \numberwithin{equation}{section} (t_1^d)^{k'}(t_0^d)^{\ell' }=t_1^{-r}t t_0^{-s}.$$ By considering all possibilities for $0 \leqslant r,s <d$, equation (\[eq:family-of-equations\]) can be regarded as a family of $d^2$ equations in $T$. Equation (\[eq:thesis-case\]) is solvable if and only if at least one of the $d^2$ equations (\[eq:family-of-equations\]) is solvable. Up to renaming $t_0^d$ with $t_0$, $t_1^d$ with $t_1$ and $t_1^{-r}t t_0^{-s}$ with $t$, we observe that each of the equations (\[eq:family-of-equations\]) has the same form of equation (\[eq:thesis-case\]), therefore we have reduced ourselves to study equation (\[eq:thesis-case\]) with the extra assumption that both $t_0$ and $t_1$ have fixed points. We compute the full fixed point sets of $t_0$ and $t_1$. We now break the proof into two cases. *Case 1: There is a point $p \in \partial {\mathrm{Fix}}(t_1)$ such that $p \not \in {\mathrm{Fix}}(t_0)$.* Rewriting equation (\[eq:thesis-case\]) and applying it to $p$, we get $$\label{eq:thesis-cases-rewrite} \numberwithin{equation}{section} t_0^{-\ell}(p)= t^{-1}(p).$$ Since $p \not \in {\mathrm{Fix}}(t_0)$ and $t_0$ is orientation preserving, then there exists at most one number $\ell$ satisfying equation (\[eq:thesis-cases-rewrite\]) by Lemma \[thm:standard-folklore\]. *Case 2: There is a point $p \in \partial {\mathrm{Fix}}(t_1) \cap {\mathrm{Fix}}(t_0)$.* If $p \not \in {\mathrm{Fix}}(t)$, by particularizing at $p$ we see that equation (\[eq:thesis-case\]) is not solvable for any pair $(k,\ell)$. Otherwise, $p \in {\mathrm{Fix}}(t)$ and we can cut the unit circle open at $p \in \mathbb{Q}/\mathbb{Z}$ and regard $t_0,t_1,t$ as elements of ${\mathrm{PL}}_2([p,p+1])$. We can now finish the proof by applying Lemma \[thm:matucci5-special-case\]. *The proof of Lemma \[thm:thesis-case\] shows how to locate the pairs $(k,\ell)$. We need to find all periodic orbits and their periods and this can be effectively achieved by computing the Brin-Salazar revealing pairs of the tree pair diagrams of $T$, using the Brin-Salazar technology to compute neutral leaves and thus deducing the size of periodic orbits (see, for example, Section 4 in [@matucci8]).* *We observe that the construction of the Mather invariant can be carried out even when $y$ and $z$ are elements of ${\mathrm{EP}}_2^>(p,+\infty)$ or of ${\mathrm{EP}}_2^>(-\infty,p)$ for any rational number $p$. All the results of the current subsection can still be recovered. For this reason, in the following we will refer to the Mather invariant regardless of the ambient set where it will be built.* Orientation reversing case of the TCP\[sec:special-case\] --------------------------------------------------------- We now study the orientation reversing case of TCP, that is, we want to solve the equation $$\label{eq:conj}\numberwithin{equation}{section} z=g^{-1}yg,$$ where $y,z \in {\mathcal{R}\cdot {\mathrm{EP}}_2}\setminus \{{\mathrm{id}}\}$ and $g \in F$. The general idea that we will follow is to square the equation and attempt to solve $$z^2 = g^{-1} y^2 g$$ so that $y^2,z^2 \in {\mathrm{EP}}_2$ and we can appeal to the results of the previous subsections. Since $y,z$ are strictly decreasing and approach $\mp \infty$ when $t \to \pm \infty$ then both $y$ and $z$ have exactly one fixed point each. Moreover, all possible $g$’s fulfilling equation (\[eq:conj\]) must also satisfy $g({\mathrm{Fix}}(z))= {\mathrm{Fix}}(gzg^{-1})={\mathrm{Fix}}(y)$. By Proposition \[thm:rationals-coincide\](ii), one can algorithmically decide whether or not there is $g \in F$ mapping the point ${\mathrm{Fix}}(z)$ to the point ${\mathrm{Fix}}(y)$. If there is no such $g$, then equation (\[eq:conj\]) has no solution and we are done. Otherwise, compute such a $g\in F$ and, after replacing $z$ by $gzg^{-1}$, we can assume that ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z)=\{ p\}$, for some $p\in \mathbb{Q}$. We start with a special case and then move on to consider all orientation reversing maps. \[thm:orientation-reversing-special\] Let $y,z \in \mathcal{R}\cdot {\mathrm{EP}}_2$ be such that $y^2=z^2={\mathrm{id}}$ and $y(p)=z(p)=p$, for some $p\in \mathbb{Q}$. Then $y$ and $z$ are conjugate by an element of $F$ if and only if there exists $u\in \mathbb{Z}$ such that $y^{-1}z(t)=t+u$ for $t$ positive sufficiently large. The forward direction follows from a straightforward check of the behavior of $y$ and $z$ at neighborhoods of $\pm \infty$. For the converse, define the following map $$g(t):= \begin{cases} t & \; \mbox{if} \; \; \; \; \; t \in (-\infty,p] \\ y^{-1}z(t) & \; \mbox{if} \; \; \; \; \; t \in [p,+\infty). \end{cases}$$ If $t\leqslant p$, then $$g(t)=t=y^{-2}z^2(t)=y^{-1}(y^{-1}z)z(t)=y^{-1}gz(t)$$ since $y^2=z^2={\mathrm{id}}$ and $z(t)\geqslant p$. On the other hand, if $t\geqslant p$, then $$g(t)=y^{-1}z(t)= y^{-1}gz(t)$$ since $z(t) \leqslant p$. So $y$ and $z$ are conjugate to each other by the element $g\in {\mathrm{EP}}_2$. The final step is to observe that $g$ is, in fact, in $F$ because $g(t)=t$, for $t$ negative sufficiently large, and $g(t)=t+u$ by construction, for $t$ positive sufficiently large. We quickly recall and extend an argument from [@matucci5] to reduce the number of candidate conjugators to test. The trick is to reduce the number of initial slopes that we need to test. \[thm:solving-reverse-squared\] Let ${\overline{y}}, {\overline{z}} \in {\mathcal{R}\cdot {\mathrm{EP}}_2}^<(p,+\infty)$ and $g \in F(p,+\infty)$ and consider the equation $$\label{eq:standard-conjugacy-equation} \numberwithin{equation}{section} {\overline{z}} = x^{-1} {\overline{y}} x.$$ Then $x=g$ is a solution of (\[eq:standard-conjugacy-equation\]) if and only if there exists an integer $n$ such that $x={\overline{y}}^{2n}g \in {\mathrm{EP}}_2(p,+\infty)$ is the unique solution of equation (\[eq:standard-conjugacy-equation\]) such that $({\overline{y}}^{2n}g)'(p^+) \in [(y^2)'(p^+),1]$. This follows immediately by noticing that equation (\[eq:standard-conjugacy-equation\]) is equivalent to $${\overline{z}} = ({\overline{y}}^{2n}x)^{-1}{\overline{y}} ({\overline{y}}^{2n} x).$$ To show uniqueness, we observe that in Subsection \[sec:restatement-TCP\] we noticed that a solution of equation (\[eq:standard-conjugacy-equation\]) is also a solution of the squared equation $$\label{eq:squared-conjugacy-equation-first} \numberwithin{equation}{section} {\overline{z}}^2 = g^{-1} {\overline{y}}^2 g.$$ Uniqueness follows from Theorem \[thm:explicit-conjugator\] applied to the squared equation (\[eq:squared-conjugacy-equation-first\]). \[thm:orientation-reversing-general\] Let $y,z \in \mathcal{R}\cdot {\mathrm{EP}}_2$ be such that $y(p)=z(p)=p$, for some $p\in \mathbb{Q}$. We can decide whether or not $y$ and $z$ are conjugate by an element of $F$. If there exists a conjugator, we can construct one. If $y^2=z^2={\mathrm{id}}$, then we are done by Proposition \[thm:orientation-reversing-special\]. Moreover, if $y$ and $z$ are conjugate via an element of $F$, it is immediate that $y^{-1}z(t)=t+u$, for some integer $u$ and for any $t$ positive sufficiently large (as observed in the proof of Proposition \[thm:orientation-reversing-special\]). Thus we can assume that $y^{-1}z$ is a translation, for $t$ positive sufficiently large. Assume now $y^2\ne {\mathrm{id}}\ne z^2$. We can appeal to Proposition \[thm:identify-fixed-points\] and assume that ${\mathrm{Fix}}(y^2)={\mathrm{Fix}}(z^2)$, up to suitable conjugation. Moreover, if there exists a conjugator between $y$ and $z$, then it must fix ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z)=\{p\}$ and so $\{p\} \subseteq {\mathrm{Fix}}(y^2)={\mathrm{Fix}}(z^2) \subseteq {\mathrm{Fix}}(g)$. Let $L<R$ be two suitable integers so that $y^2$ and $z^2$ coincide and are periodic on the set $(-\infty,L] \cup [R,+\infty)$. If either $L$ or $R$ does not exist, then $y$ and $z$ cannot be conjugate. We can apply the techniques from [@matucci5] on any two consecutive dyadic rational points $p_1,p_2$ of $\partial {\mathrm{Fix}}(y^2) \cap [L,R]$ where $y^2|_{[p_1,p_2]} \ne {\mathrm{id}}|_{[p_1,p_2]}$ and $z^2|_{[p_1,p_2]} \ne {\mathrm{id}}|_{[p_1,p_2]}$ and find (if they exist) all the finitely conjugators between $y^2|_{[p_1,p_2]}$ and $z^2|_{[p_1,p_2]}$ with initial slopes within $(y^2)'(p^+)$ and $(y^{-2})'(p^+)$. Similarly we can do on $[a,+\infty)$ where $a$ is the rightmost dyadic rational point of $\partial {\mathrm{Fix}}(y^2) \cap [L,R]$ by applying Lemma \[thm:solving-reverse-squared\] in the case that $y^2$ and $z^2$ have no fixed points on $[R,+\infty)$ (to reduce the number of initial slopes on which we can apply Theorem \[thm:explicit-conjugator\]) or using the argument at the end of Subsection \[ssec:reducing-to-squares\] in case $y^2$ and $z^2$ have fixed points on $[R,+\infty)$. Thus in all cases, up to using the same trick of Lemma \[thm:solving-reverse-squared\] to reduce the slopes to test, we apply Theorem \[thm:explicit-conjugator\] (or its bounded version from [@matucci5]) to build finitely many functions between any two consecutive dyadic rational points $p_1,p_2$ of $\partial {\mathrm{Fix}}(y^2)$ (respectively, on an interval of the type $[p_1,+\infty)$) and such that $y^2 \ne {\mathrm{id}}$ on $[p_1,p_2]$ (respectively, on an interval of the type $[p_1,+\infty)$). We now test all these functions as conjugators between $y$ and $z$ in the respective intervals. If there is an interval such that none of these functions conjugates $y$ and $z$, then $y$ and $z$ cannot be conjugate via an element of $F$. Otherwise, on each such interval $U_s$ we fix a conjugator $g_s$ between $y$ and $z$. Now we will carefully glue all these conjugators with the function that we have built in Proposition \[thm:orientation-reversing-special\]. Assume that $(p,+\infty) \setminus {\mathrm{Fix}}(y^2)$ is a disjoint union of ordered intervals $I_i=(a_i,b_i)$ so that $a_i < a_j$, if $i<j$. Similarly, assume that $(-\infty,p) \setminus {\mathrm{Fix}}(y^2)$ is a disjoint union of ordered intervals $J_i=(d_i,c_i)$ such that $c_i>c_j$, if $i<j$. $$g(t):= \begin{cases} t & \; \mbox{if} \; \; \; \; \; t=p \text{ or } t \in {\mathrm{Fix}}(y^2) \cap (-\infty,p) \\ y^{-1}z(t) & \; \mbox{if} \; \; \; \; \; t \in {\mathrm{Fix}}(y^2) \cap (p,+\infty) \\ g_s(t) & \; \mbox{if} \; \; \; \; \; t \in U_s \end{cases}$$ Since $y$ acts on ${\mathbb{R}}$ in an order reversing way, it is immediate to verify that $y(a_i)=c_i=z(a_i)$, $y(c_i)=a_i=y(c_i)$, $y(b_i)=d_i=z(b_i)$ and $y(d_i)=b_i=z(d_i)$ and therefore the map $g$ is in $F$. It is straightforward to observe that this map is continuous and in $F$ and that it is a conjugator, by construction. For example, since $z([c_{i+1},d_i])=[b_i,a_{i+1}]$ and $y^2=z^2={\mathrm{id}}$ on $[c_{i+1},d_i]$ then it is clear that $$g(t)=t=y^{-2}z^2(t)=y^{-1}(y^{-1}z)z(t)=y^{-1}gz(t)$$ for any $t \in [c_{i+1},d_i]$. Solution of the TCP\[ssec:solution-TCP\] ---------------------------------------- We are now ready to prove Theorem \[thm:TCP-solvable\]. **Theorem \[thm:TCP-solvable\].** *Thompson’s group $F$ has solvable twisted conjugacy problem.* Given $y,z \in F$ and $\varphi \in {\mathrm{Aut}}(F)$, we need to establish whether or not there is a $g \in F$ such that $$\label{eq:recall-TCP} \numberwithin{equation}{section} z = g^{-1} y \varphi(g).$$ In Subsection \[sec:restatement-TCP\] we have shown that equation (\[eq:recall-TCP\]) is equivalent to the equation $$\label{eq:original-conjugacy-equation} \numberwithin{equation}{section} {\overline{z}} = g^{-1} {\overline{y}} g,$$ for ${\overline{y}},{\overline{z}} \in {\widetilde{\mathrm{EP}}}_2$ and $g \in F$. We describe a procedure to wrap up all work of the previous subsections: 1. If one of ${\overline{y}}$ and ${\overline{z}}$ belongs to ${\mathrm{EP}}_2$ and the other in ${\mathcal{R}\cdot {\mathrm{EP}}_2}$, then equation (\[eq:original-conjugacy-equation\]) has no solution for $g\in F$, since conjugation does not change the orientation of a function. 2. If both ${\overline{y}},{\overline{z}} \in {\mathrm{EP}}_2$, then we apply the results of Subsections \[ssec:periodicity-boxes\] through \[sec:rescaling-the-circle\] to solve equation (\[eq:original-conjugacy-equation\]). 3. If ${\overline{y}},{\overline{z}} \in {\mathcal{R}\cdot {\mathrm{EP}}_2}$, then we apply Theorem \[thm:orientation-reversing-general\] to solve equation (\[eq:original-conjugacy-equation\]). This ends the proof of Theorem \[thm:TCP-solvable\]. Extensions of $F$ with unsolvable conjugacy problem \[sec:CP-extensions\] ========================================================================= In this section we recall the necessary tools from [@bomave2] in order to construct extensions of Thompson’s group $F$ with unsolvable conjugacy problem (proving Theorem \[thm:CP-extension-unsolvable\]). As explained in the introduction, Bogopolski, Martino and Ventura give a criterion to study the conjugacy problem in extensions of groups (see Theorem \[thm:bomave-extensions\]). Applying it to the case we are interested in, let $F$ be Thompson’s group, let $H$ be any torsion-free hyperbolic group (for example, a finitely generated free group), and consider an algorithmic short exact sequence $$\label{eq:exact-sequence} \numberwithin{equation}{section} 1 \longrightarrow F \overset{\alpha}{\longrightarrow} G \overset{\beta}{\longrightarrow} H \longrightarrow 1.$$ We can then consider the *action subgroup* of the sequence, $A_G =\{\varphi_g \mid g \in G\} \leqslant {\mathrm{Aut}}(F)$, and Theorem \[coro\] tells us that $G$ has solvable conjugacy problem if and only if $A_G\leqslant {\mathrm{Aut}}(F)$ is orbit decidable. In the present section we will find orbit undecidable subgroups of ${\mathrm{Aut}}(F)$ and so, extensions of Thompson’s group $F$ with unsolvable conjugacy problem. A good source of orbit undecidable subgroups in ${\mathrm{Aut}}(F)$ comes from the presence of $F_2\times F_2$ via Theorem 7.4 from [@bomave2]: \[thm:bomave-unsolvable\] Let $F$ be a finitely generated group such that $F_2 \times F_2$ embeds in ${\mathrm{Aut}}(F)$ in such a way that the image $B$ intersects trivially with $\mathrm{Stab}^*(v)$ for some $v \in F$, where $$\mathrm{Stab}^*(v)=\{ \theta \in {\mathrm{Aut}}(F) \mid \theta(v) \text{ is conjugate to } v \text{ in } F\}.$$ Then ${\mathrm{Aut}}(F)$ contains an orbit undecidable subgroup. Let us first find a copy of $F_2\times F_2$ inside ${\mathrm{Aut}}(F)$ and then deal with the technical condition about avoiding the stabilizer. We can define two maps $\varphi_{-\infty},\, \varphi_{\infty} \colon {\mathrm{EP}}_2 \to T={\mathrm{PL}}_2(S^1)$ in the following way: given $f\in {\mathrm{EP}}_2$ we find a negative sufficiently large integer $L$ so that $f$ is periodic in $(-\infty,\, L]$; then we pass $f|_{(L-1,\, L]}$ to the quotient modulo $\mathbb{Z}$ to obtain an element from $T$ defined to be the image of $f$ by $\varphi_{-\infty}$. The map $\varphi_{+\infty}$ is defined similarly but looking at a neighborhood of $+\infty$. The maps $\varphi_{-\infty}$ and $\varphi_{+\infty}$ are clearly well-defined homomorphisms from ${\mathrm{EP}}_2$ to $T$. Note also that, for $f_1, f_2\in {\mathrm{EP}}_2$ and $k\in \mathbb{Z}$, if $f_1$ and $f_2+k$ agree for $t$ negative (resp. positive) sufficiently large, then $\varphi_{-\infty}(f_1)=\varphi_{-\infty}(f_2)$ (resp. $\varphi_{+\infty}(f_1)=\varphi_{+\infty}(f_2)$). We begin by showing that both $\varphi_{-\infty}$ and $\varphi_{+\infty}$ are surjective. \[thm:embed-F\_2\] For every $a\in T$ and every dyadic rational $p$, there exist preimages of $a$ by $\varphi_{-\infty}$ and $\varphi_{+\infty}$, respectively inside ${\mathrm{EP}}_2(-\infty,p) \leqslant {\mathrm{EP}}_2$ and ${\mathrm{EP}}_2(p,+\infty) \leqslant {\mathrm{EP}}_2$. We show the result for the case ${\mathrm{EP}}_2(p,+\infty)$ (the other case is completely analogous). Let $a\in T$ and choose ${\widetilde{a}}\in {\mathrm{EP}}_2$ to be any standard periodic lift of $a$ conveniently translated up so that $p<{\widetilde{a}}(p+1)$. By Proposition \[thm:rationals-coincide\], we can construct $g\in F$ such that $g(p)=p$ and $g(p+1)={\widetilde{a}}(p+1)$. Finally, consider $${\widehat{a}}(t)= \begin{cases} t & t\leqslant p \\ g(t) & p\leqslant t\leqslant p+1 \\ {\widetilde{a}}(t) & p+1\leqslant t, \end{cases}$$ which is clearly an element of ${\mathrm{EP}}_2(p,+\infty)$ such that $\varphi_{+\infty}({\widehat{a}}) =\varphi_{+\infty}({\widetilde{a}})=a$. The following Corollary is the key observation of the current subsection. \[thm:F\_2 x F\_2 in EP\_2\] The automorphism group of Thompson’s group $F={\mathrm{PL}}_2(I)$ contains a copy of the direct product of two free groups, $F_2 \times F_2 \leqslant {\mathrm{EP}}_2 \leqslant {\mathrm{Aut}}^+(F)$. It is well known that Thompson’s group $T={\mathrm{PL}}_2(S^1)$ contains a copy of $F_2$, the free group on two generators, say generated by $a,b\in T$. Apply Lemma \[thm:embed-F\_2\] to obtain preimages of $a$ and $b$ by $\varphi_{-\infty}$, say ${\widehat{a}}_-, {\widehat{b}}_- \in {\mathrm{EP}}_2(-\infty ,0)$, and preimages of $a$ and $b$ by $\varphi_{+\infty}$, say ${\widehat{a}}_+, {\widehat{b}}_+ \in {\mathrm{EP}}_2(0, +\infty)$. Since $\varphi_{-\infty}$ and $\varphi_{+\infty}$ are homomorphisms, we have again $\langle {\widehat{a}}_-, {\widehat{b}}_-\rangle \simeq F_2 \simeq \langle {\widehat{a}}_+, {\widehat{b}}_+\rangle$. And, on the other hand, by disjointness of supports, they commute to each other and so $F_2 \times F_2 \simeq \langle {\widehat{a}}_-, {\widehat{b}}_-, {\widehat{a}}_+, {\widehat{b}}_+\rangle \leqslant {\mathrm{EP}}_2 \simeq {\mathrm{Aut}}^+(F)$. We are finally ready to prove Theorem \[thm:CP-extension-unsolvable\]. **Theorem \[thm:CP-extension-unsolvable\].** *There are extensions of Thompson’s group $F$ by finitely generated free groups, with unsolvable conjugacy problem.* We need to redo the proof of Corollary \[thm:F\_2 x F\_2 in EP\_2\] in an algorithmic fashion and choosing our copy of $F_2\times F_2$ inside ${\mathrm{Aut}}^+(F)$ carefully enough so that it satisfies the technical condition in Theorem \[thm:bomave-unsolvable\]. Let $\Theta$ be the map obtained by repeating periodically the map $\theta$ defined in Subsection \[sec:thompson and autos\] inside each square $[k,k+1]^2$, for any integer $k$. Let $\alpha(t):= \Theta^2(t) \pmod{1} \in T$ and $\beta(t):=\Theta^2(t)+\frac{1}{2} \pmod{1} \in T$. By using the ping-pong lemma it is straightforward to verify that $\alpha$ and $\beta$ generate a copy of $F_2$ inside $T$. Now take $a=\alpha^2$, $b=\beta^2$, $c=\alpha \beta \alpha^{-1}$ and $d=\beta \alpha \beta^{-1}$, which generate a copy of the free group of rank four, $F_4 \simeq \langle a,b,c,d\rangle \leqslant T$. Using Lemma \[thm:embed-F\_2\], we can find preimages of $a,b\in T$ by $\varphi_{-\infty}$, denoted by ${\widehat{a}},\, {\widehat{b}}\in {\mathrm{EP}}_2(-\infty,\, 0) \leqslant {\mathrm{EP}}_2$, and preimages of $c,d\in T$ by $\varphi_{+\infty}$, denoted by ${\widehat{c}},\, {\widehat{d}} \in {\mathrm{EP}}_2(0,\, +\infty) \leqslant {\mathrm{EP}}_2$. Since $\langle a,b\rangle \cong F_2 \cong \langle c,d\rangle$ and $\varphi_{-\infty}$ and $\varphi_{+\infty}$ are both group homomorphisms, we get $\langle {\widehat{a}},{\widehat{b}}\rangle \cong F_2 \cong \langle {\widehat{c}},{\widehat{d}}\rangle$. Moreover, the disjointness of supports gives us that $F_2\times F_2 \cong \langle {\widehat{a}},{\widehat{b}},{\widehat{c}}, {\widehat{d}}\rangle \leqslant {\mathrm{EP}}_2$; this is the copy $B$ of $F_2\times F_2$ inside ${\mathrm{EP}}_2$ (though as positive automorphisms of $F$ via Brin’s Theorem) ready to apply Theorem \[thm:bomave-unsolvable\]. Additionally, note that, by construction, $\varphi_{-\infty}({\widehat{a}})=a$, $\varphi_{-\infty}({\widehat{b}})=b$, $\varphi_{+\infty}({\widehat{c}})=c$ and $\varphi_{+\infty}({\widehat{d}})=d$ but, at the same time, $\varphi_{+\infty}({\widehat{a}})=\varphi_{+\infty}({\widehat{b}})= \varphi_{-\infty}({\widehat{c}})=\varphi_{-\infty}({\widehat{d}})=1_T$. Let now $v\in F$ be the map $v(t)=t+1$, for all $t \in \mathbb{R}$. We will show that $B\cap \mathrm{Stab}^*(v)=\{ {\mathrm{id}}\}$. Let $\tau \in B \cap \mathrm{Stab}^*(v)$. On one hand, $\tau \in B$ and so $\tau(0)=0$ and $\tau = w_1({\widehat{a}},{\widehat{b}})w_2({\widehat{c}},{\widehat{d}})$ for some unique reduced words $w_1({\widehat{a}},{\widehat{b}}) \in \langle {\widehat{a}},{\widehat{b}} \rangle$ and $w_2({\widehat{c}},{\widehat{d}}) \in \langle {\widehat{c}},{\widehat{d}} \rangle$. On the other hand, $\tau \in \mathrm{Stab}^*(v)$ and so $\tau^{-1} v\tau = g^{-1}v g$ for some $g\in F$, which implies that $\tau g^{-1}$ commutes with $v$ in ${\mathrm{EP}}_2$. By definition of $v$, the map $\tau g^{-1}$ is periodic of period 1 on the entire real line, thus $\varphi_{-\infty}(\tau g^{-1})=\varphi_{+\infty}(\tau g^{-1})$ in $T$. On the other hand, since $g\in F$, there exist integers $m_-$ and $m_+$ such that, for negative sufficiently large $t$, $\tau g^{-1}(t)=\tau(t-m_-)=\tau(t)-m_-$, and for positive sufficiently large $t$, $\tau g^{-1}(t)=\tau(t-m_+)=\tau(t)-m_+$. Modding out these two equations by $\mathbb{Z}$ around $\pm \infty$, we get $$\varphi_{-\infty}(\tau g^{-1})=\varphi_{-\infty}(\tau )=\varphi_{-\infty} (w_1({\widehat{a}},{\widehat{b}}) w_2({\widehat{c}},{\widehat{d}}))=$$ $$=\varphi_{-\infty} (w_1({\widehat{a}},{\widehat{b}})) \varphi_{-\infty}(w_2({\widehat{c}},{\widehat{d}}))=w_1(a,b);$$ similarly, $\varphi_{+\infty}(\tau g^{-1})=w_2(c,d)$. Hence, $$w_1(a,b)=\varphi_{-\infty}(\tau g^{-1})=\varphi_{+\infty}(\tau g^{-1})=w_2(c,d),$$ an equation holding in $\langle a,b,c,d\rangle \leqslant T$. Since this is a free group on $\{ a,b,c,d\}$, we deduce that $w_1(a,b)$ and $w_2(c,d)$ are the trivial words and therefore $\tau ={\mathrm{id}}$. Having shown that $B\cap \mathrm{Stab}^*(v)=\{{\mathrm{id}}\}$, an application of Theorem \[thm:bomave-unsolvable\] gives us orbit undecidable subgroups of ${\mathrm{Aut}}^+(F)$, and Theorem \[coro\] concludes the proof. *The element $v$ chosen in the previous proof is actually $x_0$, the first generator of the standard finite presentation defined in Subsection \[sec:thompson and autos\].* The orbit decidability problem for $F$ \[sec:ODP-solvable\] =========================================================== In this section we study the orbit decidability problem for ${\mathrm{Aut}}(F)$ and ${\mathrm{Aut}}_+(F)$. We study two different cases and use techniques which are “dual” to those of Section \[sec:twisted-problem\]. As a consequence, provided that one knows the solvability of a certain decision problem, we can build nontrivial extensions of $F$ with solvable conjugacy problem. By using Theorem \[thm:brin-thm\] and following computations similar to those in Subsection \[sec:restatement-TCP\], the orbit decidability problem for ${\mathrm{Aut}}(F)$ can be restated as the following one: given $y,z \in F$ decide whether or not there exists a $g \in {\mathrm{EP}}_2$ such that either 1. $g^{-1}yg=z$, or 2. $g^{-1}(\mathcal{R}y\mathcal{R})g=z$. Notice that the first equation corresponds to orbit decidability for ${\mathrm{Aut}}_+(F)$. Up to renaming $\mathcal{R}y\mathcal{R}$ by $y$, both (i) and (ii) can be regarded as an instance of (i). Orbit decidability problem: fixed points\[ssec:ODP-fixed-points\] ----------------------------------------------------------------- It is immediate to adapt Lemma \[thm:identical-at-infinity\] to this setting, noticing that if $y \sim_{{\mathrm{EP}}_2} z$ then $y$ and $z$ coincide around $\pm \infty$. *\[thm:remark-fixed-points-coincide\] Since $y,z \in F$ have only finitely many intervals of fixed points, we can use the results of Subsection \[ssec:fixed-points\] and assume that ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z)$, up to conjugating by a $g \in F$. It can be shown that if there is no $g \in F$ such that ${\mathrm{Fix}}(y)=g({\mathrm{Fix}}(z))$, then there is no $h \in {\mathrm{EP}}_2$ such that ${\mathrm{Fix}}(y)=h({\mathrm{Fix}}(z))$.* \[thm:ODP-with-fixed-points\] Let $y,z \in F$ such that ${\mathrm{Fix}}(y)={\mathrm{Fix}}(z) \ne \emptyset$. It is decidable to determine whether or not there is a $g \in {\mathrm{EP}}_2$ such that $g^{-1}yg=z$. If $g \in {\mathrm{EP}}_2$ conjugates $y$ to $z$, then it must fix ${\mathrm{Fix}}(z)$ point wise. For any two consecutive points $p_1,p_2$ of $\partial {\mathrm{Fix}}(z)$ we can use the techniques in [@matucci5] to decide whether or not there is a $h_{p_1,p_2} \in {\mathrm{PL}}_2([p_1,p_2])$ conjugating $y|_{[p_1,p_2]}$ to $z|_{[p_1,p_2]}$. Let $R=\max {\mathrm{Fix}}(z)$. If $R=+\infty$, then there exists a rational number $p$ such that $y=z={\mathrm{id}}$ on $[p,+\infty)$ and so we can choose $g \in {\mathrm{EP}}_2(R,+\infty)$ to be $g={\mathrm{id}}$ to conjugate $y$ to $z$. Assume now that $R<+\infty$. By using the same idea seen in Subsection \[ssec:solution-TCP\] and rewriting the equation $z=g^{-1}yg=(y^n g)^{-1} y (y^n g)$ we restrict to looking for candidate conjugators with slopes at $R^+$ inside $[y'(R^+),1]$. For any power $2^\alpha$ within $[y'(R^+),1]$, we apply Theorem \[thm:explicit-conjugator\](ii) to build the unique conjugator $g \in {\mathrm{PL}}_2(R,+\infty)$ such that $g'(R^+) = 2^\alpha$. We find a finite number of conjugators $g_1,\ldots,g_s \in {\mathrm{PL}}_2(R,+\infty)$. Notice: by Theorem \[thm:explicit-conjugator\](ii) every $g_i$ conjugates $y$ to $z$, but it may not be true that $g_i \in {\mathrm{EP}}_2(R,+\infty)$. There exists a positive sufficiently large number $M$ such that, for any $t \geqslant M$, we have $y(t)=t+k=z(t)$ and that for any $i=1,\ldots,s$ and any $t \geqslant M$, we have: $$g_i(t)+k=yg_i(t)=g_iz(t)=g_i(t+k),$$ so that every $g_i$ is periodic of period $k$ on $[M,+\infty)$. To finish the proof, we only need to check if any of the $g_i$’s is in ${\mathrm{EP}}_2(R,+\infty)$. To do so, we check if $g_i(t+1)=g_i(t)+1$ on the interval $[M,M+k]$. If any of them is indeed periodic of period $1$, then we have found a valid conjugator, otherwise $y$ and $z$ are not conjugate. \[ssec:ODP-Mather\]Orbit decidability problem: Mather invariants ------------------------------------------------------------------ We assume that $y,z \in F^>$ and that there exist two integers $L < R$ such that $y(t)=z(t)=t+a$ for $t \leqslant L$ and $y(t)=z(t)=t+b$ for $t \geqslant R$, for suitable integers $a,b \geqslant 1$. Up to conjugation by a suitable $g \in F$, we can assume that $L=0$ and $R=1$. Define the two circles $$C_0:= (-\infty,0)/a\mathbb{Z} \qquad C_1:= (1,\infty)/b\mathbb{Z}$$ and let $p_0:(-\infty,0) \to C_0$ and $p_1:(1,\infty) \to C_1$ be the natural projections. As was done before, let $N$ be a positive integer large enough so that $y^N(-a,0) \subseteq (1,+\infty)$ and define the map $y^\infty:C_0 \to C_1$ by $$y^{\infty}([t]):=[{\overline{y}}^{N}(t)].$$ Similarly we define $z^{\infty}$ and call them the *Mather invariants* for $y$ and $z$. Arguing as in Subsection \[sec:rescaling-the-circle\] we see that, if $g^{-1}yg = z$ for $g \in {\mathrm{EP}}_2$, then $$\label{eq:ODP-mather} \numberwithin{equation}{section} v_1 z^{\infty}=y^{\infty} v_0$$ where $v_i$ is an element of Thompson’s group $T_{C_i}$ induced by $g$ on $C_i$, for $i=0,1$, and such that $v_i(t+1)=v_i(t)+1$. Recall that a group $G$ has solvable *$k$-simultaneous conjugacy problem* ($k$-CP) if, for any two $k$-tuples $(y_1, \ldots, y_k)$, $(z_1, \ldots, z_k)$ of elements of $G$, it is decidable to say whether or not there is a $g\in G$ so that $g^{-1} y_i g=z_i$, for all $i=1,\ldots, k$. Kassabov and the second author  [@matucci5] show that Thompson’s group $F$ has solvable $k$-CP. \[conj-T\] Thompson’s group $T$ has solvable $k$-CP. This conjecture is believed to be true, and partial results have been obtained by Bleak, Kassabov and the second author in Chapter 7 of the second author’s thesis [@matuccithesis]; it is work in progress to complete this investigation. \[thm:ODP-without-fixed\] Let $y,z \in F^>$. If the $2$-simultaneous conjugacy problem is solvable in Thompson’s group $T$, then it is decidable to determine whether or not there is a $g \in {\mathrm{EP}}_2$ such that $g^{-1}yg=z$. A straightforward extension of Theorem 4.1 in [@matucci3] yields that $y \sim_{{\mathrm{EP}}_2} z$ if and only if there exists $v_i \in T_{C_i}$ such that $v_i(t+1)=v_i(t)+1$, for $i=0,1$ and they satisfy equation (\[eq:ODP-mather\]). Since $v_0$ needs to be equal to $y^{-\infty}v_1 z^{\infty}$, our problem is reduced to deciding whether or not there is $v_1 \in T_{C_1}$ solving these equations: $$\label{eq:ODP-equations-1} \numberwithin{equation}{section} \begin{array}{cc} v_1(t+1)=v_1(t)+1, & \forall t \in C_1 \\ y^{-\infty}v_1 z^{\infty}(t+1)=y^{-\infty}v_1 z^{\infty}(t)+1, & \forall t \in C_0. \end{array}$$ Recalling that $C_0$ is a circle of length $a$ and $C_1$ is a circle of length $b$, we define $s_i:C_i \to C_i$ to be the rotation by $1$ in $C_i$, for $i=0,1$. The problem now becomes this: we need to decide whether or not there exists a map $v_1 \in T_{C_1}$ such that $$\label{eq:ODP-equations-2} \numberwithin{equation}{section} \begin{array}{c} v_1s_1=s_1v_1 \\ y^{-\infty} v_1 z^{\infty}s_0= s_0y^{-\infty}v_1 z^\infty. \end{array}$$ If we relabel $y^\infty s_0y^{-\infty}:=y^\ast$ and $z^{\infty}s_0 z^{-\infty}:=z^\ast$, equations (\[eq:ODP-equations-2\]) become $$\label{eq:simultaenous-equation} \begin{array}{c} v_1^{-1}s_1v_1 = s_1 \\ v_1^{-1}y^\ast v_1 =z^\ast. \end{array}$$ Equations (\[eq:simultaenous-equation\]) are an instance of $2$-CP which is solvable by assumption. Non-trivial extensions of $F$ with solvable conjugacy problem ------------------------------------------------------------- \[thm:ODP-solvable\] If Conjecture \[conj-T\] is true for $k=2$, then ${\mathrm{Aut}}(F)$ and ${\mathrm{Aut}}_+(F)$ are orbit decidable (as subgroups of ${\mathrm{Aut}}(F)$). In particular, assuming that such conjecture is true, every group $G$ in an algorithmic short exact sequence $$1 \longrightarrow F \overset{\alpha}{\longrightarrow} G \overset{\beta}{\longrightarrow} H \longrightarrow 1,$$ where $F={\mathrm{PL}}_2(I)$, $H$ is a torsion-free hyperbolic group, and the action subgroup $A_G$ is either ${\mathrm{Aut}}(F)$ or ${\mathrm{Aut}}_+(F)$, has solvable conjugacy problem. An application of Remark \[thm:remark-fixed-points-coincide\] and Lemmas \[thm:ODP-with-fixed-points\] and \[thm:ODP-without-fixed\] implies the solvability of orbit decidability for the groups ${\mathrm{Aut}}(F)$ and ${\mathrm{Aut}}_+(F)$. We verify the requirements of Theorem \[thm:bomave-extensions\]. By Theorem \[thm:TCP-solvable\], condition (1) is satisfied. It is well known (see, for example, Proposition 4.11(b) [@bomave2]) that if $H$ is a free group or a torsion-free hyperbolic group, conditions (2) and (3) from Theorem \[thm:bomave-extensions\] are satisfied. By Theorem \[thm:ODP-solvable\] we know that the action subgroup is orbit decidable, then Theorem \[thm:bomave-extensions\] implies that $G$ has solvable conjugacy problem. \[ssec:R-infty\] Property $R_\infty$ in Thompson groups $F$ and $T$ ===================================================================== In this section we show that Thompson groups $F$ and $T$ both have property $R_\infty$. We recall the definition of property $R_\infty$, for the reader’s convenience. *A group $G$ has property $R_\infty$ if for any $\varphi \in {\mathrm{Aut}}(G)$, there exists a sequence $\{z_i\}_{i \in \mathbb{N}}$ of pairwise distinct elements which are pairwise not $\varphi$-twisted conjugate. See also Section \[sec:intro\].* We know that an automorphism $\varphi$ of $F$ is obtained by conjugation in $F$ by an element $\tau\in{\widetilde{\mathrm{EP}}}_2$. Moreover, we have seen in Subsection \[sec:restatement-TCP\] that two elements $y,z\in F$ are $\varphi$-twisted conjugate if and only if the two elements $y\tau$ and $z\tau$ (now elements of ${\widetilde{\mathrm{EP}}}_2$) are conjugate by an element of $F$. Therefore, to prove that $F$ has property $R_\infty$ it is enough to show that, given $\tau\in{\widetilde{\mathrm{EP}}}_2$, there exists a family of elements $z_i\in F$, for all $i=1,2,\ldots,n,\ldots$ such that they are pairwise not $\varphi$-twisted conjugate, i.e., $z_i\tau$ and $z_j\tau$ are not conjugate by an element of $F$. Assume first that $\tau\in {\mathrm{EP}}_2$. If two elements are conjugate by an element of $F$ then their fixed point sets match each other. So to prove that $z_i\tau$ and $z_j\tau$ are not conjugate, it would be enough to construct the $z_i \in F$ in such a way that $z_i\tau$ has, say, a fixed point set with $i$ connected components so that the fixed point sets for all the $z_i\tau$ would be different and the elements cannot be conjugate. We observe that the fixed point set of $z_i\tau$ contains exactly the points $t\in\mathbb{R}$ such that $z_i(t)=\tau^{-1}(t)$. Thus, it is enough to construct a map $z_i\in F$ such that it has exactly $i$ disjoint intervals where $z_i(t)=\tau^{-1}(t)$, thus producing $i$ connected components for ${\mathrm{Fix}}(z_i\tau)$. A reader familiar with $F$ should be able to construct easily such family $z_i$. The proof above does not work if $\tau$ is orientation reversing. But it can be modified to solve this case too. Assume now that $\tau=\sigma\mathcal{R}$ with $\sigma\in {\mathrm{EP}}_2$. Construct the elements $z_i\in F$ similarly to the orientation preserving case using $\sigma$, but in such a way that the fixed point set for $z_i\sigma$ is symmetric with respect to the origin. More precisely, we can ensure that ${\mathrm{Fix}}(z_i\sigma)$ has $2i+1$ connected components given by $\{0\}$, $i$ connected components inside ${\mathbb{R}}_+$ and the opposite of these components in ${\mathbb{R}}_-$. Moreover, we can ensure that $z_i \sigma > 0$ if and only if $t>0$. Observe that by this symmetry, the map $\mathcal{R}z_i\sigma\mathcal{R}$ has the exact same fixed points as $z_i\sigma$ and so ${\mathrm{Fix}}((z_i\sigma\mathcal{R})^2)={\mathrm{Fix}}((z_i\sigma)^2)$. Using this family $z_i$, we see that if $z_i\tau$ and $z_j\tau$ were conjugate via an element of $F$, then $(z_i\sigma\mathcal{R})^2$ and $(z_j\sigma\mathcal{R})^2$ would also be, and these have a different number of connected components in their fixed-point sets, by construction, yielding a contradiction. The argument above shows that we can recover property $R_\infty$ for $F$, giving a new proof of the following result. \[thm:R-infty-F\] Thompson’s group $F$ has property $R_\infty$. *We notice that very recently Koban and Wong [@kowon] have shown that the group $F \rtimes \mathbb{Z}_2$ has property $R_\infty$.* Since we have a characterization for ${\mathrm{Aut}}(T)$ also in terms of conjugation by piecewise-linear maps, the method described above to prove property $R_\infty$ for $F$ can be used for $T$ as well. **Theorem \[thm:R-infty-T\].** *Thompson’s group $T$ has property $R_\infty$.* By Theorem 1 in [@brin5], the group ${\mathrm{Aut}}(T)$ can be realized by inner automorphisms and by conjugations by $\mathcal{R}$, the map which reverses the orientation. The process will consist on constructing maps with different fixed-point sets. Consider a piecewise-linear map on $[0,1]$ whose only fixed points are 0, $\frac{1}{2}$ and 1, and also such that the graph is symmetric respect to the point $[\frac{1}{2},\frac{1}{2}]$. Identify the endpoints to obtain a map on $S^1$ and hence an element of $T$. Call this map $h_1$ and consider its lift $\widetilde{h}_1 \in {\mathrm{PL}}_2({\mathbb{R}})$. From the way we have constructed $h_1$, we see that $\widetilde{h}_1$ is symmetric respect $[\frac{1}{2},\frac{1}{2}]$ inside the square $[0,1]^2$, and so $\widetilde{h}_1$ is invariant under $\mathcal{R}$, i.e., $\mathcal{R}\widetilde{h}_1\mathcal{R}=\widetilde{h}_1$ inside ${\mathrm{PL}}_2({\mathbb{R}})$. Therefore $\mathcal{R}h_1 \mathcal{R} = h_1$ in $T$. Now define inductively the map $h_i$ by subdividing the interval $[0,1]$ in its two halves and in each half define a scaled-down version of $\widetilde{h}_{i-1}$, by a factor of 2. Observe that if $i\neq j$, then $h_i$ and $h_j$ have different number of fixed points. For a fixed $\varepsilon \in \{0,1\}$, if $h_i\mathcal{R}^{\varepsilon}$ and $h_j\mathcal{R}^{\varepsilon}$ were conjugate in $T$, then $(h_i\mathcal{R}^{\varepsilon})^2$ and $(h_j\mathcal{R}^{\varepsilon})^2$ are also conjugate in $T$. We notice that $(h_i\mathcal{R})^2=h_i^2$ and that $h_i^2$ and $h_j^2$ have different number of fixed points, so they cannot be conjugate. Generalizations and some questions \[sec:generaltions-of-results\] ================================================================== In this section we make a series of observations about the extent to which the material of this paper generalizes and describe some natural related questions. Extensions of the Bieri-Thompson-Stein-Strebel groups ${\mathrm{PL}}_{S,G}(I)$ ------------------------------------------------------------------------------ It seems likely that the theory developed in this paper can be generalized to a certain extent to the Bieri-Thompson-Stein-Strebel groups ${\mathrm{PL}}_{S,G}(I)$, with the computational requirements described in [@matucci5]. We recall that ${\mathrm{PL}}_{S,G}(I)$ is the group of piecewise-linear homeomorphisms of the unit interval $I$ with finitely many breakpoints occurring inside $S \leqslant \mathbb{R}$, an additive subgroup of $\mathbb{R}$ containing $1$, and such that the breakpoints lie in $G \leqslant U(S)$, where $U(S)=\{g \in \mathbb{R}^* \mid gS=S \text{ and } g>0 \}$. Since our results rely on straightforward generalizations of those in [@matucci5] and [@matucci3], to generalize our algorithms to the groups ${\mathrm{PL}}_{S,G}(I)$ we need to observe a number of things: 1. We define the analogues ${\mathrm{PL}}_{S,G}(\mathbb{R}),{\widetilde{\mathrm{EP}}}_{S,G}, {\mathrm{EP}}_{S,G}$ and observe that the existence of periodicity boxes, the construction of conjugators and moving fixed points (Subsections \[ssec:periodicity-boxes\], \[ssec:periodicity-boxes\] and \[ssec:fixed-points\]) generalize immediately via the results in [@matucci5] (which are proved in ${\mathrm{PL}}_{S,G}(I)$). 2. To reduce the number of possible “initial slopes” we need to generalize Subsection \[sec:rescaling-the-circle\]. We can do this since the material in [@matucci3] can be generalized to ${\mathrm{PL}}_{S,G}(I)$. The second observation that is needed to reduce slopes is the one used in the proof of Theorem \[thm:TCP-solvable\], where we multiply a candidate conjugator $g$ by a power of $y^2$. This shows that we need to build candidate conjugators only for slopes in $[(y^2)'(p^+),1]$ and, by Lemma 5.4 in [@matucci5], we can show that the sets of slopes is discrete in $\mathbb{R}_+$, thereby giving us only finitely many slopes inside $[(y^2)'(p^+),1]$. Hence, this part generalizes too. 3. Brin’s Theorem \[thm:brin-thm\] has a non-trivial generalization in a result of Brin and Guzman [@bringuzman] which describes certain classes of automorphisms of the groups ${\mathrm{PL}}_{\mathbb{Z}[\frac{1}{n}],\langle n \rangle}(I)$. There exist elements in the automorphism group ${\mathrm{Aut}}({\mathrm{PL}}_{\mathbb{Z}[\frac{1}{n}],\langle n \rangle}(I))$ which are represented by conjugation via elements that are not in ${\widetilde{\mathrm{EP}}}_n$ (and that are called “exotic”). Therefore, we can only generalize results of the current paper by restricting the action subgroup being used. Instead of studying the full automorphism group ${\mathrm{Aut}}({\mathrm{PL}}_{S,G}(I))$, we can restrict to study conjugations by element of ${\widetilde{\mathrm{EP}}}_{S,G}$ so that we can adapt our results in a straightforward manner. *It should be noted that the tools of this paper are not generally sufficient to solve either the twisted conjugacy problem or the orbit decidability problem in any group ${\mathrm{PL}}_{S,G}(I)$ generalizing Thompson’s group $F$ (for example, in generalized Thompson’s groups $F(n)$). This is because the full automorphism group may contain conjugations via not piecewise-linear maps.* It is however possible to give suitable reformulations of Theorems  \[thm:TCP-solvable\],  \[thm:ODP-solvable\] and  \[thm:CP-extension-unsolvable\] in the setting of actions whose acting group is realized by conjugations by an element of ${\widetilde{\mathrm{EP}}}_{S,G}$. The restatement of Theorem  \[thm:ODP-solvable\] will need to assume that the $2$-simultaneous conjugacy problem is solvable for the groups $T_{S,G}$ and this is also work-in-progress as mentioned in Section  \[sec:ODP-solvable\]. Since the techniques used to study the twisted conjugacy problem for $F$ arise from those used in [@matucci5] to study the simultaneous conjugacy problem for $F$, it is natural to ask the following question: *Is the $k$-simultaneous twisted conjugacy problem solvable for $F$? More precisely, is it decidable to determine whether or not, given $\varphi \in {\mathrm{Aut}}(F)$ and $y_1,\ldots,y_k,z_1,\ldots,z_k \in F$, there exists a $g \in F$ such that $z_i=g^{-1}y_i \varphi(g)$?* Extensions of Thompson’s group $T$ ---------------------------------- As observed at the beginning of the proof of Theorem \[thm:R-infty-T\], if $\varphi \in {\mathrm{Aut}}(T)$, then there exists an $\varepsilon \in \{0,1\}$ such that $\varphi(\lambda)=\mathcal{R}^{\varepsilon} \tau^{-1} \alpha \tau \mathcal{R}^{\varepsilon}$, for all $\alpha \in T$. Arguing as in Subsection \[sec:restatement-TCP\], equation (\[eq:TCP-equation\]) can be rewritten as $$\label{eq:TCP-in-T-again} \numberwithin{equation}{section} g^{-1}(y\mathcal{R}^\varepsilon) g = z\mathcal{R}^\varepsilon$$ for $y,z,g \in T$ and $\varepsilon \in \{0,1\}$. To attack equation (\[eq:TCP-in-T-again\]), we can start by squaring it and initially reduce ourselves to solve the equation $$\label{eq:squared-TCP-in-T-again} \numberwithin{equation}{section} g^{-1}(y\mathcal{R}^\varepsilon)^2 g = (z\mathcal{R}^\varepsilon)^2.$$ The advantage of working with equation (\[eq:squared-TCP-in-T-again\]) is that $(y\mathcal{R}^\varepsilon)^2, (z\mathcal{R}^\varepsilon)^2 \in T$. The conjugacy problem in $T$ is solvable by the work of Belk and the second author in [@matucci9] and thus we can list all the conjugators in $T$ between $(y\mathcal{R}^\varepsilon)^2$ and $(z\mathcal{R}^\varepsilon)^2 $. However, there might be infinitely many of them and there is no obvious way to detect which of them will also be conjugators between $y\mathcal{R}^\varepsilon$ and $z\mathcal{R}^\varepsilon$. We cannot use the techniques of the current paper, since there is no uniqueness given by an the “initial slope” of elements of $T$ (although something similar may be feasible, as it was done in Chapter 7 in [@matuccithesis] to study centralizers in $T$). We are thus led to ask: *Is the twisted conjugacy problem solvable in Thompson’s group $T$?* To conclude, we mention that the orbit decidability problem for $T$ is solvable for ${\mathrm{Aut}}(T)$ and ${\mathrm{Aut}}_+(T)$. Let $T$ be Thompson’s group ${\mathrm{PL}}_2(S^1)$. Then ${\mathrm{Aut}}(T)$ and ${\mathrm{Aut}}_+(T)$ are orbit decidable. We need to decide whether or not, given $y,z \in T$, there exists an element $g \in T$ such that at least one of the two equalities $$\label{eq:ODP-in-T} \numberwithin{equation}{section} z=g^{-1}yg \qquad \text{or} \qquad z=g^{-1}(\mathcal{R}y\mathcal{R})g$$ holds. This amounts to study two distinct conjugacy problems for elements of $T$, each of which is solvable by the work [@matucci9].

--- abstract: 'The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find a closed formula for *all* the entangled states for which this state is a CSS. Our formula holds for multipartite states in all dimensions. For the bipartite case of two qubits our formula reduce to the one given in Phys. Rev. A **78**, 032310 (2008).' author: - Shmuel Friedland - Gilad Gour date: '1 October, 2010' title: Closed formula for the relative entropy of entanglement in all dimensions --- \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Proposition]{} Introduction ============ Immediately with the emergence of quantum information science (QIS), entanglement was recognized as the key resource for many tasks such as teleportation, super dense coding and more recently measurement based quantum computation (for review, see e.g. [@Hor09; @Ple07]). This recognition sparked an enormous stream of work in an effort to quantify entanglement in both bipartite and multipartite settings. Despite the huge effort, except the negativity [@Vid02] (and the logarithmic negativity [@Ple05]) closed formulas for the calculation of different measures of entanglement exist only in two qubits systems and, to our knowledge, only for the entanglement of formation [@Woo98]. Moreover, the discovery that several measures of entanglement and quantum channel capacities are not additive [@Has09; @Yard08], made it clear that formulas in lower dimensional systems, in general, can not be used to determine the asymptotic rates of different quantum information tasks. Hence, formulas in higher dimensional systems are quite essential for the development of QIS. Among the different measures of entanglement, the relative entropy of entanglement (REE) is of a particular importance. The REE is defined by [@Ved98]: $$\label{def} E_{R}(\rho)=\min_{\sigma'\in\mathcal{D}}S(\rho\|\sigma')=S(\rho\|\sigma)\;,$$ where $\mathcal{D}$ is the set of separable states or positive partial transpose (PPT) states, and $S(\rho\|\sigma)\equiv\text{Tr}\left(\rho\log\rho-\rho\log\sigma\right)$. It quantifies to what extent a given state can be operationally distinguished from the closest state which is either separable or has a positive partial transpose (PPT). Besides of being an entanglement monotone it also has nice properties such as being asymptotically continuous. The importance of the REE comes from the fact that its asymptotic version provides the unique rate for reversible transformations [@Hor02]. This property was demonstrated recently with the discovery that the regularized REE is the unique function that quantify the rate of interconversion between states in a reversible theory of entanglement, where all types of non-entangling operations are allowed [@Bra08]. The state $\sigma=\sigma(\rho)$ in Eq. (\[def\]) is called the closest separable state (CSS) or the closest PPT state. Recently, the inverse problem to the long standing problem [@Eis05] of finding the formula for the CSS $\sigma(\rho)$ was solved in [@MI] for the case of two qubits. In [@MI] the authors found a closed formula for the inverse problem. That is, for a given state $0<\sigma$ on the boundary of 2-qubits separable states, $\partial\mathcal{D}$, the authors found an explicit formula describing all entangled states for which $\sigma$ is the CSS. Quite astonishingly, we show here that this inverse problem can be solved analytically not only for the case of two qubits, but in fact in all dimensions and for any number of parties. We now describe briefly this formula. Denote by ${\mathrm{H}}_n$ the Hilbert space of $n\times n$ hermitian matrices, where the inner product of $X,Y\in{\mathrm{H}}_n$ is given by ${\mathop{\mathrm{Tr}}\nolimits}XY$. Denote by ${\mathrm{H}}_{n,+,1}\subset{\mathrm{H}}_{n,+}\subset {\mathrm{H}}_n$ the convex set of positive hermitian matrices of trace one, and the cone of positive hermitian matrices, respectively. Here $n=n_1n_2\cdots n_s$ so that the multi-partite density matrix $\rho\in{\mathrm{H}}_{n,+,1}$ can be viewed as acting on the $s$-parties Hilbert space $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}\otimes\cdots\otimes\mathbb{C}^{n_s}$. Let $0<\sigma\in{\mathrm{H}}_{n,+}$ (i.e. $\sigma$ is full rank). Then for any $\sigma'\in{\mathrm{H}}_n$ and a small real $\varepsilon$ we have the Taylor expansion of $$\log(\sigma+\varepsilon \sigma')=\log\sigma+\varepsilon L_{\sigma}(\sigma')+O(\varepsilon^2).$$ Here $L_{\sigma}:{\mathrm{H}}_n\to{\mathrm{H}}_n$ is a self-adjoint operator (defined in the next section), which is invertible, and satisfies $L_{\sigma}(\sigma)=I$. Assume now that $0<\sigma\in\partial\mathcal{D}$ (later we will extend the results for all $\sigma\in\partial\mathcal{D}$; i.e. not necessarily full rank). Then, from the supporting hyperplane theorem, $\sigma$ has at least one supporting hyperplane, $\phi\in{\mathrm{H}}_n$, of the following form: $$\label{phisuphy} {\mathop{\mathrm{Tr}}\nolimits}(\phi\sigma')\ge {\mathop{\mathrm{Tr}}\nolimits}(\phi\sigma)=0 \;\;\;\forall\;\;\; \sigma'\in \mathcal{D}\;,$$ where $\phi$ is normalized; i.e. ${\mathop{\mathrm{Tr}}\nolimits}\phi^2=1$. For each such $\phi$, we define the family of *all* entangled states, $\rho(x,\sigma)$, for which $\sigma$ is the CSS: $$\label{rhoformul} \rho(x,\sigma)=\sigma - x L_{\sigma}^{-1}(\phi),\quad 0<x\le x_{\max}.$$ Here, $x_{\max}$ is defined such that $\rho(x_{\max},\sigma)\in{\mathrm{H}}_{n,+,1}$ and $\rho(x_{\max},\sigma)$ has at least one zero eigenvalue. We also note that ${\mathop{\mathrm{Tr}}\nolimits}L_{\sigma}^{-1}(\phi)=0$. Moreover, for the case of two qubits, $\phi$ is unique and is given by $\phi=\left(|\varphi\rangle\langle\varphi|\right)^{\Gamma}$, where $\Gamma$ is the partial transpose, and $|\varphi\rangle$ is the unique normalized state that satisfies $\sigma^{\Gamma}|\varphi\rangle=0$. Hence, for the case of two qubits our formula is reduced to the one given in [@MI], by recognizing that for this case, the self-adjoint operator $L_{\sigma}^{-1}$ is given by the function $G(\sigma)$ of Ref. [@MI]. This paper is organized as follows. In the next section we discuss the definition of $L_{\sigma}$. In section III we find necessary and sufficient conditions for the CSS and in section IV we prove the main result for the case were the CSS is full rank. To illustrate how the formula can be applied, in section V we discuss the qubit-qudit $2\times m$ case. In section VI we discuss how to apply the formula for tensor products. In section VII we discuss the singular case, and show that the CSS state can be described in a similar way to the non-singular case. We end in section VIII with conclusions. Definition of $L_{\sigma}$ ========================== Let $0<\alpha\in{\mathrm{H}}_{n,+}$. Fix $\beta\in {\mathrm{H}}_n$. Let $t\in (-\varepsilon,\varepsilon)$ for some small $\varepsilon=\varepsilon(\alpha)>0$. Rellich’s theorem, , e.g. [@Kat80], yields that $\log(\alpha+t\beta)$ is analytic for $t\in (-\varepsilon,\varepsilon)$. So $$\label{analytic} \log(\alpha+t\beta)=\log \alpha+t{\mathrm{L}}_{\alpha}(\beta)+O(t^2).$$ Here ${\mathrm{L}}_{\alpha}:{\mathrm{H}}_n\to {\mathrm{H}}_n$ is the following linear operator. In the eigenbasis of $\alpha$, $\alpha={\rm diag}(a_1,\ldots,a_n)$ is a diagonal matrix, where $a_1,\ldots,a_n>0$. Then for $\beta=[b_{ij}]_{i,j=1}^n$ we have that $$\label{frstvarlog} [{\mathrm{L}}_{\alpha}(\beta)]_{kl} = b_{kl}\frac{\log a_k-\log a_{l}}{a_k-a_l}, \quad k,l=1,\ldots,n.$$ Here we assume that for a positive $a$, $\frac{\log a-\log a}{a-a}=\frac{1}{a}, \frac{a-a}{\log a -\log a}=a$. Equivalently, for a real diagonal matrix $\alpha=(a_1,\ldots,a_n)>0$ define the real symmetric matrix $$\begin{aligned} &\left[T(\alpha)\right]_{k,l=1}^n=\frac{\log a_k-\log a_{l}}{a_k-a_l}\nonumber\\ &\left[S(\alpha)\right]_{k,l=1}^n= \frac{a_k-a_l}{\log a_k-\log a_{l}}.\label{defTalpha} \end{aligned}$$ Then, $L_{\alpha}(\beta)=\beta\circ T(\alpha)$, where $\beta\circ \eta$ is the entrywise product of two matrices, sometimes called the Hadamard product of matrices. Note that $L_{\alpha}$ is an invertible operator, where $L_{\alpha}^{-1}(\beta)=\beta\circ S(\alpha)$. A necessary and sufficient condition for $\sigma(\rho)$ =======================================================  We start with a necessary and sufficient condition the CSS, $\sigma(\rho)$, must satisfy. \[ness\] Let $0<\rho\in{\mathrm{H}}_{n,+,1}\backslash\mathcal{D}$. The state $0<\sigma(\rho)\in\mathcal{D}$ is a solution to Eq. (\[def\]), if and only if $\sigma\equiv\sigma(\rho)$ satisfies $$\label{necmaxcon} \max_{\sigma'\in \mathcal{D}} {\mathop{\mathrm{Tr}}\nolimits}\sigma'L_{\sigma}(\rho)={\mathop{\mathrm{Tr}}\nolimits}\sigma L_{\sigma}(\rho)=1.$$ We will see later that the assumptions that $0<\rho$ and $0<\sigma$ are not necessary. First, note that $L_{\sigma}(\sigma)=I_n$, and $L_\sigma$ is a self-adjoint operator. Hence, $ {\mathop{\mathrm{Tr}}\nolimits}\sigma L_{\sigma}(\rho)={\mathop{\mathrm{Tr}}\nolimits}L_{\sigma}(\sigma)\rho={\mathop{\mathrm{Tr}}\nolimits}(\rho)=1$. Now, let $\sigma'\in \mathcal{D}$. Since $\mathcal{D}$ is a convex set, it follows that for every $t\in [0,1]$, $(1-t)\sigma+t\sigma'=\sigma+ t(\sigma'-\sigma)\in\mathcal{D}$. Thus, applying Rellich’s theorem for a small $t>0$ gives $$\label{nes1} \log(\sigma+t(\sigma'-\sigma))=\log\sigma+tL_{\sigma} (\sigma'-\sigma)+O(t^2).$$ If $\sigma$ is a solution to Eq.(\[def\]), we must have ${\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma\ge {\mathop{\mathrm{Tr}}\nolimits}\rho\log\left[\sigma+t(\sigma'-\sigma)\right]$ which together with Eq.(\[nes1\]) implies that for a small positive $t$, $t{\mathop{\mathrm{Tr}}\nolimits}\rho L_{\sigma}(\sigma'-\sigma)\le 0$. Dividing by $t$ gives ${\mathop{\mathrm{Tr}}\nolimits}\rho L_{\sigma}(\sigma')\le {\mathop{\mathrm{Tr}}\nolimits}\rho L_{\sigma}(\sigma)=1$. This completes the necessary part of the proof since $L_\sigma$ is self-adjoint. The sufficient part of the proof follows directly from the construction of $\rho(x,\sigma)$ in Theorem \[main\]. The proposition above leads to the following intuitive corollary: Let $0<\sigma$ be a CSS of an entangled state $\rho$. Then, $\sigma\in\partial\mathcal{D}$. Assume that the CSS $\sigma$ is an interior point of $\mathcal{D}$. Thus, for each $\sigma'$ separable $(1-t)\sigma+t\sigma'$ is separable for all small $|t|$, where $t$ is either positive or negative. Hence, instead of Eq.(\[necmaxcon\]) we get the identity ${\mathop{\mathrm{Tr}}\nolimits}L_{\sigma}(\rho)\sigma'=1$ for *all* separable states $\sigma'\in\mathcal{D}$. This yields that $L_{\sigma}(\rho)=I_n$. Hence $\rho=\sigma$ which is impossible since $\rho$ was assumed not to be separable. Main Theorem ============ In the following we prove the main theorem for the case where the CSS, $\sigma$, is full rank. Note that if $\rho$ is full rank then $\sigma$ also must be full rank. \[main\] **(a)** Let $0<\sigma\in\partial\mathcal{D}$, and let $\rho\in{\mathrm{H}}_{n,+,1}$. Then, $$E_{R}(\rho)=S(\rho\|\sigma)\;\;\;\left(\text{i.e.}\;\sigma\;\text{is the CSS of}\;\rho\right)$$ if and only if $\rho=\rho(x,\sigma)$, where $\rho(x,\sigma)$ is defined in Eq.(\[rhoformul\]).\ **(b)** If $\rho>0$ (i.e. full rank) than the CSS is unique. **(a)** We first assume that $\sigma$ is a CSS of $\rho$. Recall that any linear functional $\Phi$ on ${\mathrm{H}}_n$ is of the form $\Phi(X)={\mathop{\mathrm{Tr}}\nolimits}(\phi X)$ for some $\phi\in{\mathrm{H}}_n$. Now, since $\mathcal{D}$ is a closed convex subset of ${\mathrm{H}}_{n,+,1}$ it follows (from the supporting hyperplane theorem) that for each boundary point $\sigma\in\partial\mathcal{D}$ there exists a nonzero linear functional on $\Phi:{\mathrm{H}}_n\to {\mathbb{R}}$, represented by $\phi\in{\mathrm{H}}_n$, satisfying the following condition: $$\Phi(\sigma)\le \Phi(\sigma') \textrm{ for all } \sigma'\in\mathcal{D}.$$ Note that the equation above holds true if $\phi$ is replaced by $\phi-aI$ (this is because ${\mathop{\mathrm{Tr}}\nolimits}\sigma={\mathop{\mathrm{Tr}}\nolimits}\sigma'=1$). Moreover, since $\phi\neq 0$ we can normalize it. Therefore, there exists $\phi\in{\mathrm{H}}_n$ satisfying (\[phisuphy\]) and the normalization $$\label{norphi} {\mathop{\mathrm{Tr}}\nolimits}\phi^2=1.$$ For most $\sigma$ on the boundary $\partial\mathcal{D}$, the supporting hyperplane of $\mathcal{D}$ at $\sigma$ is unique (see Fig. 1). This is equivalent to say that $\phi\in{\mathrm{H}}_n$ satisfying the conditions in Eq. (\[phisuphy\]) and (\[norphi\]) is unique. However, for some special boundary points $\sigma\in\partial\mathcal{D}$, there is a cone of such $\phi$ of dimension greater than one satisfying (\[phisuphy\]) (see Fig. 1). Now, denote $\phi':=-(L_{\sigma}(\rho)-I)$. Since we assume that $\sigma$ is a CSS of $\rho$ we get from Eq. (\[necmaxcon\]) the condition ${\mathop{\mathrm{Tr}}\nolimits}(\phi'\sigma')\ge {\mathop{\mathrm{Tr}}\nolimits}(\phi'\sigma)=0$. Recall that $L_{\sigma}(\sigma)=I$. Hence $\phi'=-L_{\sigma}(\rho-\sigma)$. Since $\rho\ne \sigma$ and $L_{\sigma}$ is invertible, it follows that $\phi'\ne 0$. Hence, $\phi'$ can be normalized such that $\phi'=x\phi$, where $\phi$ satisfies Eq. (\[norphi\]) and $x>0$. Apply $L_{\sigma}^{-1}$ to $\phi$ to deduce (\[rhoformul\]). We remark that ${\mathop{\mathrm{Tr}}\nolimits}L_{\sigma}^{-1}(\phi)=0$. Indeed $$0={\mathop{\mathrm{Tr}}\nolimits}\phi \sigma={\mathop{\mathrm{Tr}}\nolimits}\phi L_{\sigma}^{-1}(I)={\mathop{\mathrm{Tr}}\nolimits}L_{\sigma}^{-1}(\phi) I= {\mathop{\mathrm{Tr}}\nolimits}L_{\sigma}^{-1}(\phi).$$ Assume now that $0<\sigma\in\partial\mathcal{D}$, and let $\phi$ be a supporting hyperplane at $\sigma$, satisfying Eq. (\[phisuphy\]) and (\[norphi\]). Set $\rho\equiv\rho(x,\sigma)$ as in Eq.(\[rhoformul\]). We want to show that for this $\rho$, $E_R(\rho)=S(\rho\|\sigma)$. Recall first that the relative entropy $S(\rho\|\sigma'):={\mathop{\mathrm{Tr}}\nolimits}(\rho\log\rho)-{\mathop{\mathrm{Tr}}\nolimits}(\rho\log\sigma')$ is jointly convex in its arguments [@NC Thm 11.12]. By fixing the first variable $\rho$ we deduce that ${\mathop{\mathrm{Tr}}\nolimits}(\rho\log\sigma')$ is concave on $\mathcal{D}$. Consider the function $$f(t):={\mathop{\mathrm{Tr}}\nolimits}(\rho\log((1-t)\sigma+t\sigma')), \quad t\in[0,1],$$ where $\rho\ge 0$ is given by Eq. (\[rhoformul\]) and $x>0$. The joint convexity of the relative entropy implies that $f(t)$ is concave. Now, to show that the minimum of $S(\rho\|\sigma')$ is obtained at $\sigma'=\sigma$, it is enough to show that $f(0)\ge f(1)$ for each $\sigma'\in \mathcal{D}$. To see that, we first show show that $f'(0)\le 0$, which then, combined with concavity of $f$, implies that $f(0)\ge f(1)$. For small $t$ we have $$\log(\sigma+t(\sigma-\sigma'))=\log\sigma+tL_{\sigma}(\sigma'-\sigma)+O(t^2).$$ Hence, $$\begin{aligned} \label{compf'0} f'(0) & ={\mathop{\mathrm{Tr}}\nolimits}(\rho L_{\sigma}(\sigma'-\sigma))=\nonumber\\ & ={\mathop{\mathrm{Tr}}\nolimits}\left[L_{\sigma}(\rho)(\sigma'-\sigma)\right]={\mathop{\mathrm{Tr}}\nolimits}\left[(I-x\phi)(\sigma'-\sigma)\right]\nonumber\\ &=x{\mathop{\mathrm{Tr}}\nolimits}(\phi\sigma-\phi\sigma')=-x{\mathop{\mathrm{Tr}}\nolimits}(\phi\sigma')\le 0. \end{aligned}$$ This completes the proof of part **(a)**. Moreover, $E_{R}(\rho)={\mathop{\mathrm{Tr}}\nolimits}(\rho\log\rho)-{\mathop{\mathrm{Tr}}\nolimits}(\rho\log\sigma)>0$, since $\rho\ne \sigma$. Hence $\rho$ is entangled. **(b)** This part follows from the strong concavity of $\log\sigma$ (see appendix \[strong\]). Therefore, from Corollary \[conclog\] of appendix A, it follows that for a fixed entangled state $\rho>0$, the function ${\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma$ is a strict concave function on the open set of all strictly positive Hermitian matrices in ${\mathrm{H}}_n$. Hence, if both $\sigma$ and $\sigma'$ are CSS of $\rho$, then both are full rank and we have ${\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma={\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma'$. Hence, for $t\in(0,1)$ we set $\sigma''\equiv t\sigma+(1-t)\sigma'$ and from the strong concavity $${\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma''>t{\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma+(1-t){\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma'={\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma\;,$$ in contradiction with the assumption that $\sigma$ is a CSS. Let $0<\rho\in{\mathrm{H}}_{n,+,1}$ be entangled state and let $\sigma$ be a CSS of $\rho$. Then, $\sigma$ is also the CSS of $\rho(t)\equiv t\rho+(1-t)\sigma$ for all $t\in [0,t_{\max}]$, where $t_{\max}>1$ is the maximum $t$ such that $\rho(t)\geq 0$. Since $\sigma$ is the CSS of $\rho$, from theorem \[main\] we have $\rho=\rho(x,\sigma)$ for some $x$. Hence $\rho(t)=\rho(tx,\sigma)$ is of the same form. From theorem \[main\] $\sigma$ is a CSS of $\rho(t)$. A weaker version of the corollary above was proved in [@Ved98]; note that here $t$ can be greater than one. Bipartite partial transpose analysis ==================================== In the following we show how to apply Theorem \[main\] to specific examples. In particular, we focus on the bipartite case (i.e. $n=n_1n_2$) and we will assume that $\mathcal{D}$ in Eq.(\[def\]) is the set of PPT states. In the $2\times2$ and $2\times3$ case, $\mathcal{D}$ is also the set of separable states [@Hor95]. The boundary of the PPT states is simple to characterize. If $\sigma\in\mathcal{D}$ satisfies $\sigma>0$ and also $\sigma^{\Gamma}>0$, where $\Gamma$ is the partial transpose, then $\sigma$ must be an interior point of $\mathcal{D}$. If on the other hand $\sigma$ or $\sigma^{\Gamma}$ are singular, then $\sigma$ must be on the boundary of $\mathcal{D}$. We therefore have: $$\partial\mathcal{D}=\left\{\sigma\in\mathcal{D} \;\Big|\; det(\sigma^{\Gamma}\sigma)=0\right\}\;.$$ Suppose now that $0<\sigma\in\partial\mathcal{D}$. Hence, $\sigma^{\Gamma}$ has at least one zero eigenvalue. Let $|\varphi\rangle$ be a normalized eigenstate corresponding to an eigenvalue zero and define an Hermitian matrix $\phi=(|\varphi\rangle\langle\varphi|)^{\Gamma}$. Since the partial transpose is self-adjoint with respect to the inner product $\langle\rho,\rho'\rangle={\mathop{\mathrm{Tr}}\nolimits}(\rho\rho')$, it follows that $\phi$ satisfies Eq.(\[phisuphy\]) and is normalized (i.e. ${\mathop{\mathrm{Tr}}\nolimits}\phi^2=1$). That is, $\phi$ represents the supporting hyperplane at $\sigma$. Note that if $\sigma^{\Gamma}$ has more than one zero eigenvalue than clearly $\phi$ is not unique and in fact there is a cone of supporting hyperplanes of $\mathcal{D}$ at $\sigma$ (see points C and E in Fig.1). To illustrate this point in more details, we discuss now the case where $n_1=2$ (i.e. the first system is a qubit) and $n_2\equiv m$. In the $2\times m$ case, we can write any state $\sigma\in{\mathrm{H}}_{2m,+,1}$ using the block representation of $$\label{sigform} \sigma=\left[\begin{array}{cc}A&B\\\ B^\dag&C\end{array}\right]\in {\mathbb{C}}^{(2m)\times (2m)}, \quad A,B,C\in{\mathbb{C}}^{m\times m},$$ and $A^\dag=A$, $C^\dag=C$. The partial transpose of $\sigma$ is given by (here the partial transpose corresponds to the transpose on the first qubit system; i.e. it is the left partial transpose): $\sigma^{\Gamma}:=\left[\begin{array}{cc}A&B^\dag\\B&C\end{array}\right]$. The following theorem shows that $\sigma^{\Gamma}$ can have more then one zero eigenvalue. \[pt2theo\] Let $m\ge 2$. If $\sigma>0$ and $\sigma^{\Gamma}\ge 0$ then ${\mathrm{rank\;}}\sigma^{\Gamma}\ge m+1$. Furthermore, for each $k=0,\ldots,m-1$ there exist strictly positive hermitian matrices $\sigma\in{\mathrm{H}}_{2m,+,1}$ such that $\sigma^{\Gamma}\ge 0,{\mathrm{rank\;}}\sigma^{\Gamma}=2m-k$. Recall that since $\sigma\in {\mathrm{H}}_{2m,+,1}$ is strictly positive definite we have $A>0$. Hence, $\sigma$ and $\sigma^{\Gamma}$ are equivalent to the following block diagonal hermitian matrices $$\begin{aligned} \hat \sigma & =\left[\begin{array}{cc}A&0\\0&C-B^\dag A^{-1}B\end{array}\right]\\ & = \left[\begin{array}{cc}I&0\\-B^\dag A^{-1}&I\end{array}\right] \left[\begin{array}{cc}A&B\\B^\dag&C\end{array}\right] \left[\begin{array}{cc}I&0\\-B^\dag A^{-1}&I\end{array}\right]^\dag\\ \tilde \sigma & =\left[\begin{array}{cc}A&0\\0&C-BA^{-1}B^\dag\end{array}\right]\\ & = \left[\begin{array}{cc}I&0\\-BA^{-1}&I\end{array}\right] \left[\begin{array}{cc}A&B^\dag\\B&C\end{array}\right] \left[\begin{array}{cc}I&0\\-BA^{-1}&I\end{array}\right]^\dag\;, \end{aligned}$$ respectively. Hence $$\begin{aligned} & \sigma>0\iff C-B^\dag A^{-1}B>0\\ & \sigma^{\Gamma}\ge 0\iff C-BA^{-1}B^\dag\ge 0 \end{aligned}$$ Note first that $C\neq BA^{-1}B^\dag$. Otherwise, we get that $BA^{-1}B^\dag>B^\dag A^{-1}B$, and since $B^\dag A^{-1}B\ge 0$ it follows that each eigenvalue of $BA^{-1}B^\dag$ must be positive and the $i-th$ eigenvalue of $BA^{-1}B^\dag$ must be strictly greater then the $i-th$ eigenvalue of $B^\dag A^{-1}B$. This can not be true since $\det BA^{-1}B^\dag=\det B^\dag A^{-1}B$. Hence ${\mathrm{rank\;}}\sigma^{\Gamma}\geq m+1$. This complete the first part of the theorem. Next, let $E\ge 0$. Then to satisfy the condition $\sigma^{\Gamma}\geq 0$ of the above inequality we define $C$ by $$\label{defC22} C= BA^{-1}B^\dag+E,\;\Rightarrow\; {\mathrm{rank\;}}\sigma^{\Gamma}= m+{\mathrm{rank\;}}E.$$ With the above identity the condition that $\sigma>0$ is equivalent to $$\label{Econd} BA^{-1}B^\dag-B^\dag A^{-1}B+E>0.$$ We first show that one can choose $A>0$ and $B$ and $E\ge 0$ such that ${\mathrm{rank\;}}E=1$ and Eq. (\[Econd\]) hold. For this purpose, we will see that it is enough to find $A>0$ and $B$ such that $ G:=BA^{-1}B^\dag- B^\dag A^{-1}B$ has exactly $m-1$ strictly positive eigenvalues and one negative eigenvalue. Let $F\ge 0$ given. Then $F$ has the spectral decomposition $F=U\Lambda U^\dag$, where $U$ is unitary and $\Lambda\ge 0$ is a diagonal matrix with the diagonal entries equal to the nonnegative eigenvalues of $F$. Choose $B=U\Lambda^{\frac{1}{2}}$ and $A=I$. Then $G=F-\Lambda$. We claim that we can choose $F$ such that $G$ has $m-1$ positive eigenvalues and one negative eigenvalue. Fix $H=[h_{ij}]\in {\mathrm{H}}_{2m}$ with zero diagonal, i.e $h_{ii}=0$ for $i=1,\ldots,n$, and a diagonal $D=\diag(d_1,\ldots,d_m)$. Assume that $d_1>\ldots>d_m>0$. Choose $t\gg 1$ and consider $H(t)=tD+H=t(D+\frac{1}{t}H)$. Set $z=\frac{1}{t}$ and recall that $D(z)=D+zH$ has analytic eigenvalues for small $z$. Since the eigenvalues of $D$ are simple, and $D{\mathbf{e}}_i=d_i{\mathbf{e}}_i$, where ${\mathbf{e}}_i=(\delta_{1i},\ldots,\delta_{ni}){^\top}$ it follows that the eigenvalues $\lambda_1(z),\ldots,\lambda_m(z)$ of $D(z)$ have the Taylor expansion $$\lambda_i(z)=d_i+O(z^2), \quad i=1,\ldots,m$$ since $H$ has zero diagonal, (see e.g. [@Kat80]). By choosing $F=H(t), t\gg 1$ we deduce that $G(t):=H(t)-\Lambda(t)=H+O(\frac{1}{t})$. It is left to show that there exist hermitian $H$ with zero diagonal entries and $m-1$ positive eigenvalues. Let $\lambda_1\ge \ldots \ge \lambda_m$. It is known (Schur’s theorem, e.g. [@HJ99 (5.5.8)]) that the sequence $(\lambda_1,\ldots,\lambda_m)$ must majorize the sequence of the diagonal entries $(0,\ldots,0)$ of $H$. $$\label{majcon} \sum_{i=1}^r \lambda_i\ge \sum_{i=1}^r 0=0, r=1,\ldots,m-1, \;\; \sum_{i=1}^m \lambda_i=\sum_{i=1}^m 0 =0.$$ Furthermore, if $\lambda_1\ge\ldots\ge \lambda_n$ satisfies the above conditions, then there exists a real symmetric matrix $H$ with zero diagonal and the eigenvalues $\lambda_1,\ldots,\lambda_m$ (see Theorem 4.3.32 in [@HJ88]). Choose $\lambda_1\ge\ldots\ge\lambda_{m-1}>0$ and $\lambda_m=-\sum_{i=1}^{m-1} \lambda_i$. Then (\[majcon\]) holds. Thus there exists $H$ with zero diagonal and $m-1$ strictly positive eigenvalues. Hence for $t\gg 1$ $G(t)$ has $m-1$ strictly positive eigenvalues. Choose $t_0\gg 1$ and set $G=G(t_0)$. Let $G|u\rangle=\lambda_m|u\rangle$, where $\lambda_m<0$. Let $E_0=-2\lambda_m |u\rangle\langle u|$. So $G+E_0>0$ and ${\mathrm{rank\;}}E_0=1$. For $k>1$ let $E_1\ge 0$ such that ${\mathrm{rank\;}}(E_0+E_1)=k$. Then $E=E_0+E_1$. Note that from Theorem \[main\] it follows that we can rewrite the expression of the relative entropy of $\rho$ similar to the formula (7) of Ref. [@MI]. That is, $$\label{relentrfor} E_R(\rho)={\mathop{\mathrm{Tr}}\nolimits}(\rho\log\rho)-{\mathop{\mathrm{Tr}}\nolimits}(\sigma\log\sigma)+x{\mathop{\mathrm{Tr}}\nolimits}(L_{\sigma}^{-1}(\phi)\log\sigma).$$ From the theorem above it follows that for the case $m=2$, if $\sigma>0$ then $\sigma^{\Gamma}$ can have at most one zero eigenvalue. Hence, for this case $\phi$ is unique as pointed out in [@MI]. For $m=3$ it follows from the theorem that there exists $\sigma>0$ such that $\sigma^{\Gamma}$ has two independent eigenstates corresponding to zero eigenvalue. Here is an example of such a state $\sigma$ of the form (\[sigform\]) $$\sigma=\frac{1}{229}\left[\begin{array}{cccccc}1&0&0&0&6&8\\0&1&0&1&0&0\\0&0&1&0&0&0\\0&1&0&100&0&0\\ 6&0&0&0&46&60\\8&0&0&0&60&80 \end{array}\right].$$ Tensor Products =============== We now show briefly how to extend the results presented in this paper to tensor product of separable states. For this purpose we denote by $\mathcal{D}^{A}$ and $\mathcal{D}^{B}$ the set of separable states in Alice’s lab and Bob’s lab, respectively. We also denote by $\mathcal{D}^{AB}$ the set of separable states of the composite system. First observe that if $\sigma_a\in \partial\mathcal{D}^A$ then for any separable state $\sigma_{b}'\in\mathcal{D}^{B}$ the state $\sigma_a\otimes\sigma_{b}'\in \partial\mathcal{D}^{AB}$. Furthermore, let $\phi_a\in{\mathrm{H}}_{n}^{A}$ be a supporting hyperplane of $\mathcal{D}^{A}$ at $\sigma_a$ of the form given in Eq. (\[phisuphy\]) and (\[norphi\]). Let $\phi_{b}'\in{\mathrm{H}}_{n'}^{B}$, which is nonnegative on $\mathcal{D}^{B}$, i.e. ${\mathop{\mathrm{Tr}}\nolimits}(\phi_{b}'\sigma_{b}')\ge 0$ for all $\sigma_{b}'\in\mathcal{D}^{B}$ (i.e. $\phi_{b}'$ is an entanglement witness in Bob’s lab). Assume the normalization ${\mathop{\mathrm{Tr}}\nolimits}((\phi_{b}')^2)=1$. Then it is straightforward to show that $\phi:=\phi_a\otimes\phi_{b}'$ satisfies Eq. (\[phisuphy\]) and (\[norphi\]) for any $\sigma'\in\mathcal{D}^{AB}$ and $\sigma=\sigma_a\otimes\sigma_{b}'$. Assume first that $\sigma_a>0,\sigma_{b}'>0$. Then we can use $\phi=\phi_a\otimes \phi_{b}'$ in the formula of Eq. (\[rhoformul\]) to find the corresponding entangled state $\rho\in {\mathrm{H}}_{nn',+,1}$. If $\sigma_a>0$ and $\sigma_{b}'$ is singular, we can still use the formula in Eq. (\[rhoformul\]), where $\phi_{b}'\ge 0$ on $\mathcal{D}^B$ and $\phi_b'{\mathbf{x}}={\mathbf{0}}$ if $\sigma_{b}'{\mathbf{x}}={\mathbf{0}}$. If $\sigma_a$ is singular then we can use the formula given in the next section. The case of singular CSS {#singular} ======================== If the entangled state $\rho$ is not full rank then the CSS $\sigma$ can be singular (i.e. not full rank). More precisely, if ${\mathbf{x}}$ is an eigenvector of $\sigma$ corresponding to zero eigenvalue then ${\mathbf{x}}$ must also be an eigenvector of $\rho$ corresponding to zero eigenvector. For the singular $\sigma$ we work below with the basis where $\sigma$ is diagonal $$\label{singdiag} \sigma=\diag(s_1,\ldots,s_n)\;,$$ where $s_1\ge\ldots \ge s_r> 0=s_{r+1}=\ldots=s_n$ and $1\le r<n$. Here $r={\mathrm{rank\;}}\sigma<n$ since $\sigma$ is singular. Note that in this basis $\rho$ has the following block diagonal form $$\label{bldiagf} \rho=\left[\begin{array}{cc} \rho_{11}&0\\0&0\end{array}\right], \textrm{where } \rho_{11}\in {\mathrm{H}}_{r,+,1}.$$ With this eigen-basis of $\sigma$, we define the matrices $T(\sigma),\;S(\sigma)$ on the support of $\sigma$ just as in Eq. (\[defTalpha\]), and zero outside the support (i.e. the last $n-r$ rows and columns of $T(\alpha),S(\alpha)$ are set to zero). Note that with this definition $$T(\sigma)\circ S(\sigma)=S(\sigma)\circ T(\sigma)= P_{\sigma}=\diag(\underbrace{1,\ldots,1}_r,0,\ldots,0),$$ where $P_{\sigma}$ the projection to the support of $\sigma$. Define now the linear operators $L_{\sigma}, L_{\sigma}^{\ddag}:{\mathrm{H}}_n\to{\mathrm{H}}_n$ $$\label{defopMN} L_{\sigma}(\xi):=T(\sigma)\circ \xi, \quad L_{\sigma}^{\ddag}(\xi):=S(\sigma)\circ\xi.$$ Then $L_{\sigma}$ and $L_{\sigma}^{\ddag}$ are selfadjoint and $$\label{lsigrel} L_{\sigma}L_{\sigma}^{\ddag}=L_{\sigma}^{\ddag}L_{\sigma}=P_{\sigma}.$$ Note that $L_{\sigma}^{\ddag}$ is the *Moore-Penrose* inverse of $L_{\sigma}$, and that if $\sigma>0$ then $L_{\sigma}^{\ddag}=L_{\sigma}^{-1}$. Note also that $L_{\sigma}(\sigma)=P_{\sigma}$. With the above definition for $L_\sigma$, Eq. (\[analytic\]) can be generalized to the singular case: \[sinexp\] Let $\sigma\in{\mathrm{H}}_{n,+}$ be a nonzero singular matrix. Let $\xi\in {\mathrm{H}}_n$ be positive on the eigenvector subspace of $\sigma$ corresponding to the zero eigenvalue. (${\mathbf{x}}^\dagger \xi {\mathbf{x}}> 0$ if $\sigma {\mathbf{x}}={\mathbf{0}}$ and ${\mathbf{x}}\ne {\mathbf{0}}$.) Assume that $\rho\in {\mathrm{H}}_n$ is nonzero and $\rho{\mathbf{x}}=0$ if $\sigma{\mathbf{x}}=0$. Then there exists $\varepsilon>0$ such that for any $t\in (0,\varepsilon)$ the following hold. $$\label{varfor} {\mathop{\mathrm{Tr}}\nolimits}(\rho\log(\sigma+t\xi))={\mathop{\mathrm{Tr}}\nolimits}(\rho\log\sigma)+t{\mathop{\mathrm{Tr}}\nolimits}(\rho L_{\sigma}(\xi))+O(t^2|\log t|).$$ Without a loss of generality we may assume that $\sigma$ and $\rho$ of the form (\[singdiag\]) and (\[bldiagf\]). (However we do not need the assumption that $\rho_{11}\ge 0$.) Then there exists $\varepsilon>0$ such that for $\sigma(t)>0$ for $t\in(0,\varepsilon)$. Rellich’s theorem yields that the eigenvalues and the eigenvectors $\sigma(t):=\sigma+t\xi$ are analytic in $t$ for $|t|<\varepsilon$. Let $s_1(t),\ldots,s_n(t)$ be the analytic eigenvalues of $\sigma(t)$ such that $$s_i(t)=s_i+b_it +\sum_{j=2}b_{ij}t^j,\quad i=1,\ldots,n.$$ The positivity assumption on $\xi$ implies that $b_i>0$ for $i=r+1,\ldots,n$. (Note that $s_i=0$ for $i>r$.) Rellich’s theorem also claims that the eigenvectors of $\sigma(t)$ can parameterized analytically. So there exists a unitary $U(t), t\in [0,\varepsilon)$, depending analytically on $t$, for $|t|<\varepsilon$, such that the following conditions hold. $$\begin{aligned} & \sigma(t)=U(t)\diag(s_1(t),\ldots,s_n(t))U(t)^\dagger\\ & U(t)U^\dagger (t)=I_n,\; U(t)=\sum_{j=0}^{\infty} t^j U_j,\; U_0=I_n. \end{aligned}$$ Hence $$\begin{aligned} &\log(\sigma+t\xi) =U(t)\diag(\log s_1(t),\ldots,\log s_n(t))U^\dagger(t)\\ & =U(t)\diag(0,\ldots,0,\log s_{r+1}(t),\ldots \log (s_n(t)))U^\dagger (t)\\ & + U(t)\diag(\log s_1(t),\ldots, \log s_r(t),0,\ldots,0)U^\dagger (t). \end{aligned}$$ Note that the last term in this expression in analytic it $t$ for $|t|<\varepsilon$. Clearly, $\log s_i(t)= \log (b_it) $ + analytic term. Observe next that $$\begin{aligned} &U(t)\diag(0,\ldots,0,\log s_{r+1}(t),\ldots \log (s_n(t)))U^\dagger (t)\\ &=\diag(0,\ldots,0,\log s_{r+1}(t),\ldots \log (s_n(t))) \\ &+ t U_1\diag(0,\ldots,0,\log s_{r+1}(t),\ldots \log (s_n(t))) \\ &+t\diag(0,\ldots,0,\log s_{r+1}(t),\ldots \log (s_n(t)))U_1^\dagger +O(t^2|\log t|). \end{aligned}$$ Using the standard fact that ${\mathop{\mathrm{Tr}}\nolimits}XY={\mathop{\mathrm{Tr}}\nolimits}YX$ and the form of $\rho$ given by (\[bldiagf\]) we deduce that $$\begin{aligned} &{\mathop{\mathrm{Tr}}\nolimits}(\rho U(t)\diag(0,\ldots,0,\log s_{r+1}(t),\ldots \log (s_n(t)))U^\dagger (t))\\ & = O(t^2|\log t|). \end{aligned}$$ Hence $$\begin{aligned} &{\mathop{\mathrm{Tr}}\nolimits}(\rho\log(\sigma+t\xi))\\ &={\mathop{\mathrm{Tr}}\nolimits}(\rho U(t)\diag(\log s_1(t),\ldots, \log s_r(t),0,\ldots,0)U^\dagger (t))\\ & +O(t^2|\log t|). \end{aligned}$$ Similar expansion result hold when we replace $\sigma$ by a a diagonal $\alpha>0$ as in the beginning of this section. Combine these results to deduce the validity of (\[varfor\]). From the lemma above it follows that Proposition \[ness\] holds true also for singular $\rho$ and singular CSS $\sigma$. To see that, let $\sigma$ be singular CSS of an entangled state $\rho$, and suppose first that $\sigma'>0$. Define also $\sigma(t)\equiv (1-t)\sigma+t\sigma'=\sigma+t(\sigma'-\sigma)$. Note that $\xi:=\sigma'-\sigma$ satisfies the assumptions of Lemma \[sinexp\]. Thus, the arguments in Proposition \[ness\] yield that ${\mathop{\mathrm{Tr}}\nolimits}\rho L_{\sigma}(\sigma')\le {\mathop{\mathrm{Tr}}\nolimits}\rho L_{\sigma}(\sigma)=1$. Using the continuity argument, we deduce that this inequality hold for any $\sigma'\in\mathcal{D}$. In Eq. (\[phisuphy\]) we defined the supporting hyperplane in terms of Hermitian matrix $\phi$ satisfying that ${\mathop{\mathrm{Tr}}\nolimits}\sigma\phi=0$. As we will see below, it will be more convenient to represent the supporting hyperplane of $\mathcal{D}$ at $\sigma$ in terms of $\psi\equiv I-\phi$. That is, the supporting hyperplane will be described by the linear functional $\Psi:{\mathrm{H}}_n\to {\mathbb{R}}$, defined by $\Psi(\xi)={\mathop{\mathrm{Tr}}\nolimits}(\psi\xi)$, where $\psi$ satisfies: $$\label{propsi} {\mathop{\mathrm{Tr}}\nolimits}(\psi\sigma')\leq{\mathop{\mathrm{Tr}}\nolimits}(\psi\sigma)=1\;,\;\text{for all}\;\sigma'\in\mathcal{D}.$$ Note also that ${\mathop{\mathrm{Tr}}\nolimits}P_{\sigma}\sigma'\leq 1$ and therefore for any $x\in[0,1]$, $\psi(x)\equiv x\psi+(1-x)P_{\sigma}$ also satisfies the same condition ${\mathop{\mathrm{Tr}}\nolimits}(\psi(x)\sigma')\leq{\mathop{\mathrm{Tr}}\nolimits}(\psi(x)\sigma)=1$. We now ready to prove the main theorem for the case of singular CSS. \[mainsing\] Let $\sigma\in\partial\mathcal{D}$ be a singular matrix in the boundary of $\mathcal{D}$, and let $\rho\in{\mathrm{H}}_{n,+,1}$ be an entangled state. Then, $$E_{R}(\rho)=S(\rho\|\sigma)\;\;\;\left(\text{i.e.}\;\sigma\;\text{is the CSS of}\;\rho\right)$$ if and only if $\rho$ is of the form $$\label{rhosingxfrm} \rho(x,\sigma)=(1-x)\sigma+xL_{\sigma}^{\ddag}\left(\psi\right)\;,\;\;0<x_{\max}\leq 1.$$ Here $x_{\max}\le 1$ is the maximum value of $x$ not greater than 1 such that $\rho(x,\sigma)\in{\mathrm{H}}_{n,+,1}$. The supporting hyperplane of $\mathcal{D}$ at $\sigma$ is represented by $\psi\ne P_{\sigma}$, that satisfies the conditions in Eq.(\[propsi\]) and is zero outside the support of $\sigma$ (i.e. if $\sigma{\mathbf{x}}={\mathbf{0}}$ then $\psi{\mathbf{x}}={\mathbf{0}}$). Suppose first that $\rho=\rho(x,\sigma)$. We want to prove that $\sigma$ is the CSS of $\rho$. First, observe that $${\mathop{\mathrm{Tr}}\nolimits}(L_{\sigma}^{\ddag}(\psi))={\mathop{\mathrm{Tr}}\nolimits}(L_{\sigma}^{\ddag}(\psi)P_{\sigma})={\mathop{\mathrm{Tr}}\nolimits}(\psi L_{\sigma}^{\ddag}(P_{\sigma}))={\mathop{\mathrm{Tr}}\nolimits}(\psi\sigma)=1.$$ Hence $\rho(x,\sigma)\in{\mathrm{H}}_{n,+,1}$ for $x\in[0,x_{\max}]$, where we assume that $x_{\max}\le 1$. It is left to show that ${\mathop{\mathrm{Tr}}\nolimits}(\rho(x,\sigma)\log\sigma')\le {\mathop{\mathrm{Tr}}\nolimits}(\rho(x,\sigma)\log\sigma)$ for any $\sigma'\in\mathcal{D}$. From the continuity argument, it is enough to show this inequality for all $\sigma'>0$. Let $\sigma(t)=(1-t)\sigma+\sigma't$. Let $f(t)={\mathop{\mathrm{Tr}}\nolimits}(\rho(x,\sigma)\log\sigma(t))$. Using the equality (\[varfor\]), similar to Eq. (\[compf’0\]), we get for $\rho=\rho(x,\sigma)$: $$\begin{aligned} f'(0) & ={\mathop{\mathrm{Tr}}\nolimits}(\rho L_{\sigma}(\sigma'-\sigma))={\mathop{\mathrm{Tr}}\nolimits}\left[(\sigma'-\sigma)L_{\sigma}(\rho)\right]\nonumber\\ &={\mathop{\mathrm{Tr}}\nolimits}\left[\big((1-x)P_{\sigma}+x\psi\big)(\sigma'-\sigma)\right]=\nonumber\\ &=(1-x){\mathop{\mathrm{Tr}}\nolimits}(\sigma' P_{\sigma})+x{\mathop{\mathrm{Tr}}\nolimits}(\sigma'\psi)-1\leq 0\;,\end{aligned}$$ where we have used that $L_{\sigma}(\rho)=(1-x)P_{\sigma}+x\psi$ and $x\in(0,1]$. (Note that ${\mathop{\mathrm{Tr}}\nolimits}\sigma'P_{\sigma}\le {\mathop{\mathrm{Tr}}\nolimits}\sigma'=1$.) Hence $f'(0)\le 0$, and since $f(t)$ is concave we have $f(0)\geq f(1)$, which implies that ${\mathop{\mathrm{Tr}}\nolimits}\left[\rho(x,\sigma)\log\sigma\right]\ge {\mathop{\mathrm{Tr}}\nolimits}\left[\rho(x,\sigma)\log\sigma'\right]$. This completes the second direction of the theorem. Moreover, note that $E_{R}(\rho)={\mathop{\mathrm{Tr}}\nolimits}(\rho\log\rho)-{\mathop{\mathrm{Tr}}\nolimits}(\rho\log\sigma)>0$, since $\rho\ne \sigma$. Hence $\rho$ is entangled. Assume now that $\sigma$ is a singular CSS of an entangled state $\rho'$. Without a loss of generality we may assume that $\sigma$ and $\rho'$ of the form (\[singdiag\]) and (\[bldiagf\]). Hence, from Proposition \[ness\], when applied to the singular case (see the discussion above), we get $${\mathop{\mathrm{Tr}}\nolimits}\sigma' L_{\sigma}(\rho')\le {\mathop{\mathrm{Tr}}\nolimits}\sigma L_{\sigma}(\rho')=1$$ for all $\sigma'\in\mathcal{D}$. Denote $\psi'\equiv L_{\sigma}(\rho')$. Note that $\psi'\neq P_{\sigma}$ since $\rho'\ne\sigma$. Hence, with this notations the equation above reads ${\mathop{\mathrm{Tr}}\nolimits}\psi'\sigma'\leq{\mathop{\mathrm{Tr}}\nolimits}\psi'\sigma=1$. Moreover, by definition $\psi'$ is zero outside the support of $\sigma$. Then $\rho'=L_{\sigma}^{\ddagger}(L_{\sigma}(\rho'))=L_{\sigma}^{\ddagger}(\psi)$. Hence Eq. (\[rhosingxfrm\]) holds for $\rho=\rho', \psi=\psi'$ and $x=1$. Note that if we define $\psi$ by $$\psi=t\psi'+(1-t)P_{\sigma}$$ for some $t\in(0,1]$, then $\psi$ also satisfies the requirements of the theorem. By taking $L_{\sigma}^{\ddag}$ on both sides of the equation above, we get $\rho=t\rho'+(1-t)\sigma$. This implies Eq. (\[rhosingxfrm\]) $x=t$. Note that from the first part of the proof we indeed conclude that $\sigma$ is the CSS to $\rho$ Hence Eq. (\[rhosingxfrm\]) holds for any $x\in (0,1]$. This completes the proof of the theorem. Recall that Theorem \[main\] claimed that for $\rho>0$ the corresponding CSS is unique. This is no longer true if $\rho$ semi-positive definite [@MI]. The reason for that is quite simple. \[finitcond\] Let $\rho\in {\mathrm{H}}_{n,+,1}$ and assume that $\rho$ is singular. Denote $f_{\rho}(\xi)={\mathop{\mathrm{Tr}}\nolimits}\rho \log\xi$ for $\xi\in{\mathrm{H}}_{n,+}$. Then $f_{\rho}(\xi)>-\infty$ for $\xi\in{\mathrm{H}}_{n,+}$ if and only if one of the following condition holds. 1. \[finitcond1\] $\xi>0$. 2. \[finitcond2\] $\rho{\mathbf{x}}={\mathbf{0}}$ if $\xi{\mathbf{x}}={\mathbf{0}}$. Equivalently, assume that $\rho\in{\mathrm{H}}_{n,+}$ is in the block diagonal form (\[bldiagf\]) where $\rho_{11}>0$. There exists a unitary matrix $U$ of order $n-r$ such that $\xi=\diag(I_r,U)^\dagger \diag(\xi_2,0)\diag(I_r,U)$, where $0<\xi_2\in{\mathrm{H}}_{p,+}$ and $p\in[r,n-1]$. Denote by ${\mathrm{H}}_{n,+}(\rho)$ the set of all $\xi\in{\mathrm{H}}_{n,+}$ such that $f_{\rho}(\xi)>-\infty$. Then ${\mathrm{H}}_{n,+}(\rho)$ is a convex set. The function $f_{\rho}:{\mathrm{H}}_{k,+}(\rho)\to {\mathbb{R}}$ is concave but not strictly concave. Clearly, if $\xi>0$ then $f_{\rho}(\xi)>-\infty$. More precisely, Assume that ${\mathbf{x}}_1,\ldots,{\mathbf{x}}_n$ is an orthonormal system of eigenvectors of $\xi$ with the corresponding eigenvalues $x_1\ge\ldots \ge x_n>0$. Then $f_{\rho}(\xi)=\sum_{i=1}^n (\log x_i) {\mathbf{x}}_i^{\dagger} \rho {\mathbf{x}}_i$. Using the continuity argument we deduce that this formula remains valid for $\xi$ semipositive definite. Hence $f_{\rho}(\xi)>-\infty$ if and only if ${\mathbf{x}}^{\dagger} \rho{\mathbf{x}}=0$ for each eigenvector ${\mathbf{x}}$ in the null space of $\xi$. Since $\rho\in{\mathrm{H}}_{n,+}$ it follows that ${\mathbf{x}}^{\dagger}\rho{\mathbf{x}}=0\iff \rho{\mathbf{x}}={\mathbf{0}}$. This proves the first part of \[finitcond2\]. The second part of \[finitcond2\] follows straightforward from this condition. We now show that ${\mathrm{H}}_{n.+}(\rho)$ is a convex set. Let $\zeta\in {\mathrm{H}}_{n,+}$ and $s>0$. Then $g(s)=f_{\rho}(\zeta+sI_k)$ is a strictly increasing function on $(0,\infty)$. (Choose an eigenbase of $\zeta$.) Assume that $\xi,\eta \in{\mathrm{H}}_{n,+}(\rho)$. Then for any $s>0,t\in (0,1)$, the concavity of $\log \zeta$ on $\zeta>0$ yields that $\log (t(\xi+sI_k)+(1-t)(\eta+sI_k))\ge t\log (\xi+sI_k)+(1-t)\log (\eta+sI_k),$ which implies $f_{\rho} (t(\xi+sI_k)+(1-t)(\eta+sI_k))\ge tf_{\rho} (\xi+sI_k)+(1-t)f_{\rho} (\eta+sI_k)$. Letting $s\searrow 0$ and using the assumption that $f_{\rho} (\xi), f_{\rho}(\eta)>-\infty$ we deduce that $f_{\rho} (t\xi+(1-t)\eta)>-\infty$. Hence ${\mathrm{H}}_{n,+}(\rho)$ is convex. The above arguments show also that $f_{\rho}$ is a concave function on ${\mathrm{H}}_{n.+}(\rho)$. It is left to show that $f_{\rho}$ is not strictly concave on ${\mathrm{H}}_{n.+}(\rho)$. Let $\xi=\diag(\beta,\xi_2), \eta=\diag(\beta,\eta_2)$, where $0<\beta\in {\mathrm{H}}_{r,+}, \xi_2,\eta_2\in{\mathrm{H}}_{n-r,+}$ and $\xi_2\ne \eta_2$. Clearly, $f_{\rho}(t\xi+(1-t)\eta)={\mathop{\mathrm{Tr}}\nolimits}\rho_{11} \log \beta$ for all $t\in [0,1]$. Let $\rho\in {\mathrm{H}}_{n,+,1}$ and assume that $\rho$ is singular. Then the set of CSS to $\rho$ is a compact convex on the boundary of ${\mathcal{D}}$, which may contain more then one point. $\;$ conclusions =========== To conclude, given a state $\sigma$ on the boundary of separable or PPT states, we have found a closed formula for *all* entangled states for which $\sigma$ is a CSS. We have also shown that if $\sigma$ is full rank, than it is unique. Quite remarkably, our formula holds in all dimensions and for any number of parties. As an illustrating example, we have analyzed the case of qubit-qudit systems and described how to apply the formula for this case. *Acknowledgments:—* GG research is supported by NSERC. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. **81**, 865 (2009). M. B. Plenio and S. Virmani, Quant. Inf. Comp. **7**, 1 (2007). G. Vidal, and R. F. Werner, Phys. Rev. A **65**, 032314 (2002). M. B. Plenio, Phys. Rev. Lett. 95, 090503 (2005). W. K. Wootters, Phys. Rev. Lett. **80**, 2245 (1998). M. B. Hastings, Nature Physics **5**, 255 (2009). Graeme Smith and Jon Yard, Science **321**, 1812 (2008). V. Vedral and M. B. Plenio, Phys. Rev. A **57**, 1619 (1998). M. Horodecki, J. Oppenheim, and R. Horodecki, Phys. Rev. Lett. **89**, 240403 (2002). F. G. S. L. Brandao and M. B. Plenio, Nature Physics 4, 873 (2008). J. Eisert, e-print arXiv:quant-ph/0504166v1. A. Miranowicz and S. Ishizaka, Phys. Rev. A **78**, 032310 (2008). R.A. Horn and C.R. Johnson, *Topics in Matrix Analysis*, Cambridge University Press 1999. R.A. Horn and C.R. Johnson, *Matrix Analysis*, Cambridge University Press 1988. T. Kato, [*Perturbation Theory for Linear Operators*]{}, Springer-Verlag, 2nd ed., New York 1980. A. Nielsen and I.L. Chuang, *Quantum Computation and Quantum Information*, Cambridge University Press 2000. R. Horodecki, P. Horodecki, and M. Horodecki, Phys. Lett. A **200**, 340 (1995). The strong concavity of $\log A$ {#strong} ================================ \[concaop\] For an interval ${\mathrm{int}}\subset {\mathbb{R}}$ let ${\mathrm{H}}_n({\mathrm{int}})$ be the set of all $n\times n$ hermitian matrices whose eigenvalues are in ${\mathrm{int}}$. (Here ${\mathrm{int}}$ can be open, closed, half open, half infinite or infinite.) Let $f:{\mathrm{H}}_n({\mathrm{int}})\to {\mathrm{H}}_{n}$ be a continuous function. $f$ is called *monotone*, *strict monotone* and *strong monotone* if for any $C,A\in {\mathrm{H}}_n({\mathrm{int}})$ the corresponding conditions hold respectively: $C\ge A \Rightarrow f(C)\ge f(A)$, $C> A\Rightarrow f(C)> f(A)$, $C\gneq A\Rightarrow f(C)\gneq f(A)$. $f$ is called *concave*, *strict concave* and *strong concave* if for any $C,A\in {\mathrm{H}}_n({\mathrm{int}})$ the corresponding conditions hold respectively: $f((1-s)A+sB)\ge (1-s)f(A)+sf(B)$, $f((1-s)A+sB)> (1-s)f(A)+sf(B)$ if ${\mathrm{rank\;}}(A-B)=n$ and $s\in (0,1)$, $f((1-s)A+sB)\gneq (1-s)f(A)+sf(B)$ if $A\ne B$ and $s\in (0,1)$. A well known result is that the functions $f_t(A):=A^t, t\in (0,1)$ and $\log A$ are strictly concave and strictly monotone on $H_{n}((0,\infty))$. See [@HJ99 §6.6]. In this section we show that $\log A$ is strongly concave on $H_{n}((0,\infty))$. This implies that ${\mathop{\mathrm{Tr}}\nolimits}\rho\log\sigma$ is strictly concave for a fixed $\rho>0$ and all $\sigma>0$. Hence the CSS $\sigma$ to an entangled $\rho>0$ is strictly positive and unique. We need also to consider ${\mathbf{x}}^{\dagger}(\log A){\mathbf{x}}$, where ${\mathbf{x}}$ is a nonzero column vector in ${\mathbb{C}}^n$, and $A$ is singular and positive. Then it makes sense only to consider only those ${\mathbf{x}}\in{\mathbf{U}}_+(A)\subset{\mathbb{C}}^n$, where ${\mathbf{U}}_+(A)$ is the subspace spanned by eigenvectors of $A$ corresponding to positive eigenvalues. For ${\mathbf{x}}\in{\mathbf{U}}_+(A)$ we have that ${\mathbf{x}}^\dagger(\log A) {\mathbf{x}}>-\infty$. We also agree that for each ${\mathbf{x}}\in{\mathbb{C}}^n\backslash {\mathbf{U}}_+(A)$ ${\mathbf{x}}^\dagger(\log A) {\mathbf{x}}=-\infty$. Then for each ${\mathbf{x}}\in{\mathbb{C}}^n$, the function ${\mathbf{x}}^\dagger(\log A){\mathbf{x}}$ is concave on ${\mathrm{H}}_{n,+}$. we agree here that $$t(-\infty)=-\infty=-\infty-\infty=-\infty+{\mathbb{R}}={\mathbb{R}}-\infty \textrm{ for any } t>0.$$ \[strongconcav\] Let $A,B\in {\mathrm{H}}_{n,+}$. Then $$\label{RAB} R(A,B):=A+B-(A^{\frac{1}{2}}B^{\frac{1}{2}}+B^{\frac{1}{2}}A^{\frac{1}{2}})\ge 0.$$ Furthermore one has the identity $$\label{basid} ((1-s)A^{\frac{1}{2}}+sB^{\frac{1}{2}})^2=(1-s)A+sB-(1-s)sR(A,B)).$$ Hence for $t\in (0,\frac{1}{2}]$ $$\begin{aligned} & (1-s)A^{t}+sB^{t} \le ((1-s)A+sB\nonumber\\ & -(1-s)s R(A,B))^{t} \le((1-s)A+sB)^{t},\label{strongtconc}\\ & (1-s)\log A+s\log B \le \log\Big[(1-s)A+sB\nonumber\\ &-(1-s)s R(A,B)\Big]\le \log ((1-s)A+sB). \label{stronglogconcav1} \end{aligned}$$ Let $A,B\in{\mathrm{H}}_{n,+}$ and assume that $R(A,B)$ is defined by (\[RAB\]). We claim that $R(A,B)\ge 0$. This is a straightforward consequence of the Cauchy-Schwarz and the arithmetic-geometric inequalities $$\begin{aligned} |{\mathbf{x}}^\dagger A^{\frac{1}{2}}B^{\frac{1}{2}}{\mathbf{x}}|\le (({\mathbf{x}}^\dagger A{\mathbf{x}})({\mathbf{x}}^\dagger B{\mathbf{x}}))^{\frac{1}{2}}\le \frac{1}{2}({\mathbf{x}}^\dagger A{\mathbf{x}}+{\mathbf{x}}^\dagger B{\mathbf{x}}),\\ |{\mathbf{x}}^\dagger B^{\frac{1}{2}}A^{\frac{1}{2}}{\mathbf{x}}|\le (({\mathbf{x}}^\dagger B{\mathbf{x}})({\mathbf{x}}^\dagger A{\mathbf{x}}))^{\frac{1}{2}}\le \frac{1}{2}({\mathbf{x}}^\dagger B{\mathbf{x}}+{\mathbf{x}}^\dagger A{\mathbf{x}}). \end{aligned}$$ Furthermore, $R(A,B)=0$ if and only if $A=B$. Clearly $R(A,A)=0$. Suppose that $R(A,B)=0$. Then the above arguments yield that we must have the equalities in the Cauchy-Schwarz inequalities, and equalities in the arithmetic-geometric mean for each ${\mathbf{x}}$. So $A^{\frac{1}{2}}{\mathbf{x}}=B^{\frac{1}{2}}{\mathbf{x}}$ for each ${\mathbf{x}}$. Hence $A^{\frac{1}{2}}=B^{\frac{1}{2}}\Rightarrow A=B$. A straightforward calculation shows the validity of (\[basid\]). Hence $$\begin{aligned} \left((1-s)A^{\frac{1}{2}}+sB^{\frac{1}{2}}\right)^2 & =(1-s)A+sB-(1-s)sR(A,B))\\ & \le (1-s)A+sB \textrm{ for } s\in (0,1). \end{aligned}$$ Since $A^{\frac{1}{2}}$ is monotone, we deduce from the above inequality the inequality (\[strongtconc\]) for $t=\frac{1}{2}$. The inequality (\[strongtconc\]) for $t=\frac{1}{2^m}$ follows by induction. Hence (\[stronglogconcav1\]) follows from (\[strongtconc\]) for $t=\frac{1}{2^m}$. Assume that $t\in (0,\frac{1}{2})$. By assuming that $A^{2t}$ is concave and order preserving we deduce from the above inequality (\[strongtconc\]) for $t\in(0,\frac{1}{2}]$. \[conclog\] For each $\rho\in{\mathrm{H}}_{n}((0,\infty))$ the function ${\mathop{\mathrm{Tr}}\nolimits}(\rho\log\sigma)$ is a strict concave function on ${\mathrm{H}}_{n}((0,\infty))$. Let $\sigma,\eta\in {\mathrm{H}}_{n}((0,\infty))$ and assume that $\sigma\ne\eta$. Then $R(\eta,\sigma)\gneq 0$. Hence for $s\in (0,1)$ $$\begin{aligned} & \log((1-s)\sigma +s\eta-(1-s)s R(\sigma,\eta))\lneq \log((1-s)\sigma +s\eta)\\ & \Rightarrow\;\; (1-s)\log\sigma+s\log\eta\lneq \log((1-s)\sigma +s\eta)\\ & \Rightarrow {\mathop{\mathrm{Tr}}\nolimits}(\rho((1-s)\log\sigma+s\log\eta))< {\mathop{\mathrm{Tr}}\nolimits}(\rho\log((1-s)\sigma +s\eta)). \end{aligned}$$ Obviously the above corollary does not hold if $\rho\ge 0$ has at least one zero eigenvalue. If $\sigma,\eta>0$ has same eigenvalues and eigenvectors which span the range of $\rho$, i.e. $\sigma {\mathbf{U}}_+(\rho)=\eta{\mathbf{U}}_+(\rho)={\mathbf{U}}_+(\rho), \sigma=\eta|{\mathbf{U}}_+(\rho)$, then ${\mathop{\mathrm{Tr}}\nolimits}(\rho((1-s)\log\sigma+s\log\eta))$ is constant for $s\in [0,1]$.

--- abstract: 'In this paper we introduce the concept of metasurfaces which are fully transparent when looking from one of the two sides of the sheet and have controllable functionalities for waves hitting the opposite side (one-way transparent sheets). We address the question on what functionalities are allowed, considering limitations due to reciprocity and passivity. In particular, we have found that it is possible to realize one-way transparent sheets which have the properties of a twist-polarizer in reflection or transmission when illuminated from the other side. Also one-way transparent sheets with controllable co-polarized reflection and transmission from the opposite side are feasible. We show that particular non-reciprocal magneto-electric coupling inside the sheet is necessary to realize lossless non-active transparent sheets. Furthermore, we derive the required polarizabilities of constituent dipole particles such that the layers composed of them form one-way transparent sheets. We conclude with design and simulations of an example of a nonreciprocal one-way transparent sheet functioning as an isolating twist-polarizer.' author: - 'Y. Ra’di, V. S. Asadchy, and S. A. Tretyakov' title: 'One-way transparent sheets' --- Introduction ============ Many novel and elegant device designs in antenna engineering and optics will become possible if we will be able to realize electrically (optically) thin layers which have the application-required reflection and transmission coefficients. This need is addressed by inventing various metasurfaces (see reviews in [@Holloway; @metasurfaces]). Metasurfaces, usually realized as electrically and/or magnetically polarizable composite layers, can shape reflected and transmitted wavefront for required functionalities. Sheets with angle-stable reflection and transmission [@Dienstfrey], absorbing sheets [@Shvets; @Watts; @absorption], high-impedance surfaces, including artificial magnetic conductors [@Sievenpiper], and various polarization-transforming devices [@semchenko; @euler10; @alu; @Teemu] are examples of metasurfaces. Recently, thin functional sheets have attracted considerable attention also in optics, since the possibilities to control optical transmission, reflection, and refraction using nonuniform sheets have been understood [@Gaburro; @metasurfaces; @Huygens1; @Huygens2]. In majority of studies, the main focus has been on tailoring the transmitted wave while the reflection is kept as low as possible. Here, we study sheets that are totally transparent from one of the two sides (*one-way transparent sheets*) for normally incident plane waves. The main goal of this study is to find out what functionalities are possible if one-way transparency is required. Can we make the layer fully reflecting or act as a twist-polarizer or as a phase shifter for plane waves coming from the opposite, non-transparent side? The second related question is what kind of physical properties the constituents of these sheets must have in order to ensure the desired functionalities. Finally, we present a practical design example of a one-way transparent sheet. Clearly, one-way transparent sheets can have multiple applications due to their “invisibility” for excitations from one side. Metasurfaces are microscopically structured layers (usually periodic), where the (average) distance between inclusions is smaller than the wavelength in the surrounding media, ensuring that the surfaces do not generate diffraction lobes. For an observer in the far zone the response is that of effectively homogeneous current sheets. Thus, for layers of electrically negligible thickness (metasurfaces) illuminated by normally incident plane waves, the reflected and transmitted waves are plane waves created by the surface-averaged electric and magnetic current sheets with the surface current densities $\_J_{\rm e}$ and $\_J_{\rm m}$. In composite sheets, the layer has a complicated microstructure, usually containing some electrically small but resonant inclusions (such as complex-shape patches or split rings or small helices). The surface-averaged current densities can be related to the electric and magnetic dipole moments $\_p$, $\_m$ induced in each unit cell as $\_J_{\rm e}={j\omega \_p\over S}$, $\_J_{\rm m}={j\omega \_m\over S}$. Here $S$ is the unit-cell area, and we use the time-harmonic convention $\exp(j\omega t)$. The higher-order multipoles induced in the inclusions do not contribute to the radiated plane-wave fields of the infinite array and we do not need to consider them explicitly. For realizing a one-way transparent sheet we will need to find such structures, where the induced surface-averaged current densities equal zero for illumination from one of the sheet side but have non-trivial and controllable values if the incidence direction is reversed. In the next section we will introduce the model for polarizations induced in the unit cells of metasurfaces which we will study here. Transparent arrays of bi-anisotropic unit cells =============================================== Effective polarizability dyadics of particles in periodic arrays ---------------------------------------------------------------- In order to reveal the most general possible functionalities of one-way transparent sheets, we assume the most general linear relations between the induced polarizations and the fields, the bi-anisotropic relations. It is convenient to write these relations as the linear relations between the dipole moments of the unit cells and the incident electromagnetic fields, which is equivalent to relating the induced surface current densities to the incident fields: $$\left[ \begin{array}{c} \mathbf{p} \\ \mathbf{m}\end{array} \right] =\left[ \begin{array}{cc} \overline{\overline{\widehat{\alpha}}}_{\rm ee}& \overline{\overline{\widehat{\alpha}}}_{\rm em}\\\overline{\overline{\widehat{\alpha}}}_{\rm me}& \overline{\overline{\widehat{\alpha}}}_{\rm mm} \end{array} \right]\cdot \left[ \begin{array}{c} \mathbf{E}_{\rm inc} \\ \mathbf{H}_{\rm inc}\end{array} \right] . \label{eq:h}$$ The effective polarizabilities include the effects of lattice interactions and higher-order multipole excitations in unit cells. In this paper, we consider isotropic (in the plane) thin sheets. The uniaxial symmetry ensures the isotropic response of the metasurfaces for normally-incident plane waves of arbitrary polarizations. The orientation of the layer in space is defined by the unit vector $\_z_0$, orthogonal to its plane. The uniaxial symmetry allows only isotropic response and rotation around the axis $\_z_0$. Thus, we can write all the polarizabilities in (\[eq:h\]) in the forms: $$\begin{array}{c}\overline{\overline{\widehat{\alpha}}}_{\rm ee}=\widehat{\alpha}_{\rm ee}^{\rm co}\overline{\overline{I}}_{\rm t}+\widehat{\alpha}_{\rm ee}^{\rm cr}\overline{\overline{J}}_{\rm t},\qquad \displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm mm}=\widehat{\alpha}_{\rm mm}^{\rm co}\overline{\overline{I}}_{\rm t}+\widehat{\alpha}_{\rm mm}^{\rm cr}\overline{\overline{J}}_{\rm t},\vspace{.1cm}\\\displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm em}=\widehat{\alpha}_{\rm em}^{\rm co}\overline{\overline{I}}_{\rm t}+\widehat{\alpha}_{\rm em}^{\rm cr}\overline{\overline{J}}_{\rm t},\qquad\displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm me}=\widehat{\alpha}_{\rm me}^{\rm co}\overline{\overline{I}}_{\rm t}+\widehat{\alpha}_{\rm me}^{\rm cr}\overline{\overline{J}}_{\rm t}, \end{array}\label{eq:j}$$ where indices ${\rm co}$ and ${\rm cr}$ refer to the symmetric and antisymmetric parts of the corresponding dyadics, respectively. Here, $\overline{\overline{I}}_{\rm t}=\=I-\_z_0\_z_0$ is the two-dimensional unit dyadic, and $\overline{\overline{J}}_{\rm t}=\mathbf{z}_0\times\overline{\overline{I}}_{\rm t}$ is the vector-product operator. In the last set of relations it is convenient to separate the coupling coefficients responsible for reciprocal and non-reciprocal coupling processes [@basic]: $$\begin{array}{c} \overline{\overline{\widehat{\alpha}}}_{\rm em}=(\widehat\chi-j\widehat\kappa)\overline{\overline{I}}_{\rm t}+(\widehat V+j\widehat \Omega)\overline{\overline{J}}_{\rm t},\vspace{.1cm}\\\displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm me}=(\widehat\chi+j\widehat \kappa)\overline{\overline{I}}_{\rm t}+(-\widehat V+j\widehat \Omega)\overline{\overline{J}}_{\rm t}. \end{array}\label{eq:hof}$$ There are two reciprocal classes (chiral $\widehat\kappa$ and omega $\widehat \Omega$) and two non-reciprocal classes (“moving” $\widehat V$ and Tellegen $\widehat\chi$). Note also that for reciprocal particles the electric and magnetic polarizabilities are always symmetric dyadics. The four main types of magneto-electric coupling are summarized in Table \[ta:main-classes\]. [|p[0.22]{}|p[0.22]{}|]{}\ Omega & Chiral\ $\begin{array}{c} \displaystyle \overline{\overline{\alpha}}_{\rm em}=\overline{\overline{\alpha}}_{\rm me}=j\Omega \overline{\overline{J}}_{\rm t}\\ \end{array}$ & $\begin{array}{c} \displaystyle \overline{\overline{\alpha}}_{\rm em}=-\overline{\overline{\alpha}}_{\rm me}=j\kappa\overline{\overline{I}}_{\rm t}\\ \end{array}$\ Moving & Tellegen\ $\begin{array}{c}\displaystyle \overline{\overline{\alpha}}_{\rm em}=-\overline{\overline{\alpha}}_{\rm me}=V\overline{\overline{J}}_{\rm t}\\ \end{array}$ & $\begin{array}{c}\displaystyle \overline{\overline{\alpha}}_{\rm em}=\overline{\overline{\alpha}}_{\rm me}=\chi\overline{\overline{I}}_{\rm t}\\ \end{array}$\ \[ta:main-classes\] The imaginary units in these notations are introduced in order to ensure that all the polarizability components are purely real for lossless particles. Reflection and transmission of plane waves from uniaxial bi-anisotropic arrays ------------------------------------------------------------------------------ We consider array properties for normally incident plane waves. In the following theory of one-way transparent layers, we need to distinguish between illuminations of the sheet from two opposite sides. In the rest of the paper, we will use double signs for these two cases, where the top and bottom signs correspond to the incident plane wave propagating in $-\mathbf{z}_0$ and $\mathbf{z}_0$ directions, respectively. In the incident plane wave, the electric and magnetic fields satisfy $$\mathbf{H}_{\rm inc}=\mp\frac{1}{\eta_0}\_z_0\times \mathbf{E}_{\rm inc}=\mp\frac{1}{\eta_0}\overline{\overline{J}}_{\rm t}\.\mathbf{E}_{\rm inc}, \label{eq:k}$$ where $\eta_0=\sqrt{\mu_0/\epsilon_0}$ is the wave impedance in the isotropic background medium (possibly free space). In terms of the effective polarizabilities, the dipole moments in (\[eq:h\]) can be written as $$\displaystyle \left[ \displaystyle\begin{array}{c} \mathbf{p} \\ \mathbf{m}\end{array} \right] =\left[\displaystyle \begin{array}{c}\displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm ee}\mp\frac{1}{\eta_0}\overline{\overline{\widehat{\alpha}}}_{\rm em}\cdot\overline{\overline{J}}_{\rm t}\vspace{.1cm}\vspace*{.2cm}\\\displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm me}\mp\frac{1}{\eta_0}\overline{\overline{\widehat{\alpha}}}_{\rm mm}\cdot\overline{\overline{J}}_{\rm t} \end{array}\right]\.\begin{array}{c}\mathbf{E}_{\rm inc} \end{array}. \label{eq:l}$$ Knowing the dipole moments induced in each unit cell and substituting the polarizabilities from (\[eq:j\]), we can now write the amplitudes of the reflected and transmitted plane waves as [@Teemu] $$\begin{array}{l} \displaystyle \mathbf{E}_{\rm r}=-\frac{j\omega}{2S}[\eta_0\mathbf{p}\mp \mathbf{z}_0\times\mathbf{m}] \vspace*{.2cm}\\\displaystyle \hspace*{.5cm}=-\frac{j\omega}{2S}\left\{\left[\eta_0\widehat{\alpha}_{\rm ee}^{\rm co}\pm 2j\widehat\Omega -\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co}\right]\overline{\overline{I}}_{\rm t}\right.\vspace*{.2cm}\\\displaystyle \hspace*{2cm}\left.+\left[\eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}\mp 2\widehat\chi -\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}\right]\overline{\overline{J}}_{\rm t}\right\}\cdot\mathbf{E}_{\rm inc}, \end{array} \label{eq:p}$$ $$\begin{array}{l} \displaystyle \mathbf{E}_{\rm t}=\mathbf{E}_{\rm inc}-\frac{j\omega}{2S}[\eta_0\mathbf{p}\pm\mathbf{z}_0\times\mathbf{m}]\vspace*{.2cm}\\\displaystyle \hspace*{.5cm} =\left\{\left[1-\frac{j\omega}{2S}\left(\eta_0\widehat{\alpha}_{\rm ee}^{\rm co}\pm 2\widehat V +\frac{1}{\eta_0}\widehat{\alpha}_{\rm mm}^{\rm co}\right)\right]\overline{\overline{I}}_{\rm t}\right.\vspace*{.2cm}\\\displaystyle \hspace*{2cm}\left. -\frac{j\omega}{2S} \left[\eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}\mp 2j\widehat \kappa+\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}\right] \overline{\overline{J}}_{\rm t}\right\}\cdot\mathbf{E}_{\rm inc}, \end{array}\label{eq:q}$$ in which, $S$ is the unit-cell area. Using these relations, we will next study transparent layers. General conditions for totally transparent layers ------------------------------------------------- By definition, a transparent layer must not change the amplitude and phase of incident waves, that is, $$\begin{array}{c} \mathbf{E}_{\rm r}=0,\qquad\mathbf{E}_{\rm t}=\mathbf{E}_{\rm inc}. \end{array}\label{eq:r}$$ From (\[eq:p\]) and (\[eq:q\]) we find the necessary conditions for a transparent array of particles in the form $$\begin{array}{c} \displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}\pm 2j\widehat\Omega -\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co}=0, \vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}\mp 2\widehat\chi -\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=0, \vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}\pm 2\widehat V +\frac{1}{\eta_0}\widehat{\alpha}_{\rm mm}^{\rm co}=0,\vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}\mp 2j\widehat \kappa+\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=0. \end{array}\label{eq:s}$$ As above, the double signs correspond to the two opposite incidence directions. Clearly, these conditions ensure that the surface-averaged induced electric and magnetic current densities equal zero (see (\[eq:p\]) and (\[eq:q\])). First, it is obvious that the conditions for two-way transparency allow only trivial solution: in this case all the polarizability components must equal zero. Indeed, if we demand that conditions (\[eq:s\]) are satisfied for both choices of the $\pm $ sign, so that the sheet looks transparent from both sides, then all the magnetoelectric coefficients must be zero. Next, we see immediately that in that case all the other polarizabilities must also vanish. We stress that this does not imply that the sheet is simply absent: zero dipole moments in each unit cell mean only that the *surface averaged* electric and magnetic current densities are zero. For example, a low-loss frequency-selective surface is transparent from both sides at the parallel resonance of the unit cell, although strong currents are induced in the structure. It is quite simple but interesting result. In particular, it implies that sheets which are fully transparent only from one side must exhibit electromagnetic coupling inside inclusions, or the sheet will be transparent from both sides. Note that this general conclusion holds also for non-reciprocal sheets. What will be the properties of an array which is transparent from one side for waves coming from the other side? We expect that using different particles (reciprocal and non-reciprocal), we should be able to control the response seen from the other side of the sheet. Suppose, that a grid of particles is set to be transparent for $+\mathbf{z}_0$-directed incident waves. Then, using (\[eq:s\]), we can express the electric and magnetic polarizabilities in terms of the magneto-electric parameters: $$\begin{array}{c} \displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co} -\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co}= 2j\widehat\Omega, \vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr} -\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=-2\widehat\chi, \vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co} +\frac{1}{\eta_0}\widehat{\alpha}_{\rm mm}^{\rm co}= 2\widehat V,\vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}+\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=- 2j\widehat \kappa. \end{array}\label{plus}$$ From (\[eq:p\]) and (\[eq:q\]), the reflected and transmitted fields for the wave coming from the non-transparent side ($-\mathbf{z}_0$-directed wave) can be expressed in terms of the magneto-electric parameters only: $$\begin{array}{l} \displaystyle \mathbf{E}_{\rm r}=\frac{j2\omega}{S}\left(-j\widehat\Omega\overline{\overline{I}}_{\rm t} +\widehat \chi \overline{\overline{J}}_{\rm t}\right)\cdot\mathbf{E}_{\rm inc}, \end{array} \label{eq:u}$$ $$\begin{array}{l} \displaystyle \mathbf{E}_{\rm t}=\left[ \left(1 - \frac{j2\omega}{S}\widehat V\right)\overline{\overline{I}}_{\rm t}-\frac{2\omega}{S} \widehat \kappa\overline{\overline{J}}_{\rm t}\right]\cdot\mathbf{E}_{\rm inc}. \end{array}\label{eq:v}$$ Now, we can study possible responses of reciprocal and non-reciprocal one-way transparent sheets. Reciprocal one-way transparent sheets ------------------------------------- Let us first consider arrays of reciprocal unit cells (omega and chiral bi-anisotropic coupling). For reciprocal particles, the electric and magnetic polarizabilities are symmetric dyadics ($\widehat \alpha_{\rm ee}^{\rm cr}=0$, $\widehat \alpha_{\rm mm}^{\rm cr}=0$), and the parameters of the non-reciprocal magneto-electric coupling vanish ($\widehat \chi=\widehat V=0$). The last relation in (\[plus\]) tells that the chirality parameter $\widehat\kappa$ is also zero. This ensures that the transmission coefficient from the other side (\[eq:v\]) equals unity, as it should be due to reciprocity. From the other relations (\[plus\]) we find that $$\begin{array}{c} \displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}=j\widehat \Omega=-\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co} \end{array}\label{eq:w}$$ and, using (\[eq:u\]), the reflected field for the waves coming from the other side can be written as $$\begin{array}{l} \displaystyle \mathbf{E}_{\rm r}=\frac{2\omega}{S}\widehat \Omega \, \mathbf{E}_{\rm inc}. \end{array} \label{eq:x}$$ If the array is passive, then the absolute value of this reflection coefficient must equal zero, because the transmission coefficient is unity from both sides (due to reciprocity). Thus, the omega coupling coefficient in passive reciprocal one-way transparent sheets must be zero for all non-zero frequencies, and we end up with the trivial solution when all the polarizabilities are zero. However, if the inclusions can be active, we see that reciprocal layers can be transparent from one side, while the co-polarized reflection from the other side can be controlled by the value of the omega coupling. Note that this is the only possible functionality even for active inclusions: The requirement of reciprocity is very limiting, because it sets the transmission coefficient to be unity from both sides. In particular, this does not allow chirality in the particles, and, thus, no polarization transformation is possible in isotropic reciprocal one-way transparent sheets. Non-reciprocal one-way transparent sheets ----------------------------------------- As is evident from equations (\[eq:u\]) and (\[eq:v\]), the use of non-reciprocal particles in principle allows full control over co- and cross-polarized reflection and transmission coefficients of one-way transparent sheets (passivity limitations are discussed below). To find out what polarizabilities are required for any desired functionality one can start from the required reflection and transmission coefficients and find the corresponding magneto-electric parameters. For example, if we would like to realize a one-way transparent twist-polarizer in transmission ($\_E_{\rm t}=\_z_0\times \_E_{\rm inc}=\=J_{\rm t}\cdot \_E_{\rm inc}$), the required values of the coupling parameters read $$\widehat\kappa=-{S\over 2\omega},\qquad \widehat V=-j{S\over 2\omega},\qquad \widehat \Omega=\widehat\chi=0.$$ The corresponding electric and magnetic polarizabilities follow from (\[plus\]): $$\begin{array}{c} \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}=\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co}=\widehat V=-j{S\over 2\omega},\vspace{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}=\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=-j \widehat\kappa =j{S\over 2\omega}. \end{array}\label{eq:one1}$$ Note that the magnitudes of all the required normalized polarizabilities are equal, and this provides one of the examples of extreme response of balanced bi-anisotropic particles [@balanced]. It is easy to check from (\[eq:l\]) that in this case of zero reflection the induced electric and magnetic dipoles of each unit cell form Huygens pairs, radiating only in the forward direction, as it should be for any non-reflecting sheet (see examples in [@Teemu; @Huygens1; @Huygens2]). As a second example, let us consider the one-way transparent twist-polarizer in reflection. Now we set $\widehat \Omega=0$, so that the reflected field is twist-polarized. Choosing the value of the Tellegen parameter to be $\widehat \chi=-j{S\over 2\omega}$, the amplitude and phase of the reflected cross-polarized field are equal to those of the incident field. The first two equations in (\[plus\]) tell that the normalized co-polarized electric and magnetic polarizabilities are equal and that we should have at least one of the antisymmetric components in electric and magnetic polarizabilities non-zero. If the device is passive, the amplitude of the transmitted field must be zero (because the reflected field has the same amplitude as the incident field). This determines the chirality parameter $\widehat \kappa=0$ and the velocity parameter $\widehat V=-j{S\over 2\omega}$. This gives a unique solution for the electric and magnetic polarizabilities: $$\begin{array}{c} \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}= \frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co}=\widehat V=-j{S\over 2 \omega}, \vspace{.5cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}=-\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=-\widehat \chi=j{S\over 2 \omega}. \end{array}$$ It is interesting to notice that also in this case the values of all normalized parameters are equal (another example of balanced bi-anisotropic particles). As is already clear from the above, the requirement of passivity of the particles imposes certain limitations on achievable response. For example, if we assume that only the Tellegen parameter $\widehat \chi$ is non-zero while all the other coupling coefficients are zero, we see that the transmission coefficient from both sides equals unity, and conclude that in this case it is possible to control the cross-polarized reflection only if the particles are active. An interesting case is the case of a “moving” grid ($\widehat{V}\neq 0$). We can set $\widehat \Omega=\widehat \chi=0$, so that the reflection coefficient is zero and the induced current sheets form a Huygens’ pair. Upon substitution of $\widehat \Omega=\widehat \chi=0$ in (\[plus\]), we see that the electric and magnetic polarizabilities are balanced: $$\begin{array}{c} \displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}= \frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co},\vspace{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}= \frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}. \end{array}\l{movco}$$ We conclude that we can fully control the transmission coefficient choosing the values of $\widehat V$ and $\widehat \kappa$ (each of these two parameters will uniquely define the values in ), maintaining the property of zero reflection (Huygens’ layer). The only limitation on the transmission coefficient values comes from passivity: The total amplitude of the transmitted field should be smaller than the amplitude of the incident field. One of the interesting limiting cases is the case of non-chiral “moving” arrays. Setting $\widehat \kappa=0$, we find the required effective polarizabilities as $$\begin{array}{c} \displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm co}= \widehat V=\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co},\vspace*{.2cm}\\\displaystyle \eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}=\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=0 \end{array} \label{eq:aa}$$ and, using (\[eq:u\]) and (\[eq:v\]), the reflection and transmission coefficients for the wave coming from the other side can be written as $$\begin{array}{l} \displaystyle \mathbf{E}_{\rm r}=0,\qquad\displaystyle \mathbf{E}_{\rm t}=\left(1 -\frac{j2\omega}{S}\widehat V\right){\_E}_{\rm inc}. \end{array} \label{eq:bb}$$ One can see that the sheet of “moving” particles can be designed to work as a completely transparent layer from one side and a partially transparent layer from the other side (with controllable amplitude and phase of the transmitted field). In the special case of a balanced and lossy layer the sheet is transparent from one side and acts as a perfect absorber from the other side [@absorption]. Requirements for individual polarizabilities of unit cells ========================================================== The above theory gives the required conditions for effective (collective) polarizabilities of unit cells forming one-way transparent sheets. These parameters connect the induced electric and magnetic dipole moments to the incident electric and magnetic fields, see (\[eq:h\]). Thus, the effective polarizabilities depend not only on the individual particles but also on electromagnetic coupling between particles in the infinite array. Here, we use the known theory of reflection and transmission in infinite dipole arrays (e.g., [@analytical]) to find the corresponding requirements on the polarizabilities of individual particles in free space. This is necessary to approach the problem of the particle design (finding the inclusion shape and sizes which provide the desired response of the whole array). To characterize individual particles, we consider their response to the local electromagnetic fields, which exist at the position of one reference particle: $$\left[ \begin{array}{c} \mathbf{p} \\ \mathbf{m}\end{array} \right] =\left[ \begin{array}{cc} \overline{\overline{\alpha}}_{\rm ee}& \overline{\overline{\alpha}}_{\rm em}\\\overline{\overline{\alpha}}_{\rm me}& \overline{\overline{\alpha}}_{\rm mm} \end{array} \right]\.\left[ \begin{array}{c} \mathbf{E}_{\rm loc} \\ \mathbf{H}_{\rm loc}\end{array} \right]. \label{eq:e}$$ Since the grid is excited by plane-wave fields which are uniform in the array plane (normal incidence), the induced dipole moments are the same for all particles. The local fields exciting the particles are the sums of the external incident fields and the interaction fields caused by the induced dipole moments in all other particles: $$\begin{array}{c} \mathbf{E}_{\rm loc}=\mathbf{E}_{\rm inc}+\beta_{\rm e}\, \mathbf{p}, \vspace*{.2cm}\\\displaystyle \mathbf{H}_{\rm loc}=\mathbf{H}_{\rm inc}+\beta_{\rm m}\,\mathbf{m}, \end{array}\label{eq:f}$$ where $\beta_{\rm e}$ and $\beta_{\rm m}$ are the interaction constants that describe the effect of the entire array on a single inclusion. These dyadic coefficients are proportional to the two-dimensional unit dyadic $\={I}_{\rm t}$. Explicit analytical expression for the interaction constants can be found in [@basic]. Because all of the dipoles are in the same plane, the induced magnetic dipoles do not produce any electric interaction field in the tangential plane, and vice versa (see [@yatsenko03]). Expressing the incident fields in (\[eq:h\]) in terms of the local fields and the interaction constants (\[eq:f\]) we can find find the polarizabilities of the individual unit cells in terms of the required collective polarizabilities. In the general case of uniaxial polarizabilities these expressions can be found in [@Teemu]. As an example, let us study the case of one-way transparent non-chiral “moving” sheet (the required effective polarizabilities are given by (\[eq:aa\])). In this case the relations between the individual and collective polarizabilities can be written as $$\begin{array}{c} \displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm ee}=\frac{\aeeo-\beta_{\rm m}(\aeeo\ammo-V^2)}{1-(\aeeo\beta_{\rm e}+\ammo\beta_{\rm m})+\beta_{\rm e}\beta_{\rm m}(\aeeo\ammo-V^2)}\=I_{\rm t}, \vspace*{.2cm}\\\displaystyle \overline{\overline{\widehat{\alpha}}}_{\rm mm}=\frac{\ammo-\beta_{\rm e}(\aeeo\ammo-V^2)}{1-(\aeeo\beta_{\rm e}+\ammo\beta_{\rm m})+\beta_{\rm e}\beta_{\rm m}(\aeeo\ammo-V^2)}\=I_{\rm t}, \vspace*{.2cm}\\\displaystyle \widehat{V} \hspace{1mm} \overline{\overline{J}}_{\rm t}=\frac{V} {1-(\aeeo\beta_{\rm e}+\ammo\beta_{\rm m})+\beta_{\rm e}\beta_{\rm m}(\aeeo\ammo-V^2)}\overline{\overline{J}}_{\rm t}. \end{array}\label{eq:ff}$$ The necessary conditions for the polarizabilities of the single moving particle can be found substituting (\[eq:ff\]) in conditions (\[eq:aa\]) as $$\begin{array}{c} \displaystyle \eta_0\aeeo=V=\frac{1}{\eta_0} \ammo, \vspace*{.2cm}\\\displaystyle \aeer= \ammr=0. \end{array} \label{eq:gg}$$ These conditions are the balance conditions for individual particles. Interestingly, conditions (\[eq:gg\]) were obtained in paper [@Joni] as the conditions for zero total scattering from small single uniaxial nonreciprocal particles. Finally, we note that the required reciprocal magneto-electric coupling in particles can be realized using proper shaping of metal or dielectric inclusions (helical shapes for chiral particles and, for example, the shape of the letter $\Omega$ for omega coupling). Non-reciprocal coupling requires the use of non-reciprocal components, such as magnetized ferrite or plasma, or active components, e.g., amplifiers. For details we refer to [@basic; @lindell94] and references therein. Example of a one-way transparent sheet composed of non-reciprocal bi-anisotropic particles ========================================================================================== In this section, we present a realizable design of a one-way transparent sheet acting as a twist-polarizer from the non-transparent side. We verify the operation of the sheet by full-wave simulations using the Ansoft High Frequency Structure Simulator (HFSS). As a non-reciprocal particle, we use the particle possessing “chiral-moving" coupling (bi-anisotropy parameters $\widehat \Omega=\widehat \chi=0$) presented in [@particle]. As it is shown in the Fig. \[ris:particle\], the particle includes a ferrite sphere magnetized by external bias field and coupled to metal elements. Recently, polarizabilities of this particle were extracted analytically and numerically [@Sajad]. As it was noted above, for the one-way transparency operation, the sheet must have balanced effective polarizabilities. We use the numerical method introduced in [@Fanyaev] which allows us to extract polarizabilities of an arbitrary polarizable particle. Using this method, parameters of the particle are optimized. The optimized dimensions of the particle (the target frequency is about $2$ GHz) read: , , , and the radius of the wire is . Material of the metal elements is copper and the ferrite material is yttrium iron garnet. The properties of the ferrite material are: the relative permittivity $\epsilon_r=\, 15$, the dielectric loss tangent $\tan\delta= 10^{-4}$, saturation magnetization $M_S=1780$ G and the full resonance linewidth $\Delta H=0.2$ Oe (measured at $9.4$ GHz). The internal bias field is $H_b=9626$ A/m, corresponding to the desired resonance frequency. Simulated individual polarizabilities of the optimized particle and effective polarizabilities of the grid (area cell $S=1482.25$ mm$^2$) are shown in Figs. \[fig:individual\] and \[fig:effective\], respectively. ![Simulated polarizabilities of individual particle.[]{data-label="fig:individual"}](fig2.eps){width="\columnwidth"} ![Simulated effective polarizabilities of the grid.[]{data-label="fig:effective"}](fig3.eps){width="\columnwidth"} These figures exhibit fairly balanced electric and magnetic response in terms of individual and effective polarizabilities. Also it can be seen that at the resonance frequency the effective polarizabilities do not satisfy the conditions (\[eq:one1\]) but they satisfy the following similar conditions: $$\eta_0\widehat{\alpha}_{\rm ee}^{\rm co}=\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm co}=\widehat V=-j{S\over 2\omega}, \label{eq:one11}$$ $$\eta_0\widehat{\alpha}_{\rm ee}^{\rm cr}=\frac{1}{\eta_0} \widehat{\alpha}_{\rm mm}^{\rm cr}=-j \widehat\kappa =-{S\over 2\omega}. \label{eq:two22}$$ Conditions (\[eq:one11\]) and (\[eq:two22\]) are related to the case of a one-way transparent sheet which acts as twist-polarizer with additional $90^\circ$ phase shift for the non-transparent side ($\_E_{\rm t}=j \=J_{\rm t}\cdot \_E_{\rm inc}$). To calculate reflection and transmission of the sheet, periodic boundary conditions. Simulated co- and cross-polarized reflection and transmission for the wave incident from the transparent and non-transparent sides are shown in Fig. \[fig:RT\]. ![Simulated reflection and transmission (in terms of intensity) for the sheet when the incident wave propagates along (a) $+\mathbf{z}_0$-axis and (b) $-\mathbf{z}_0$-axis.[]{data-label="fig:RT"}](fig4.eps){width="0.95\columnwidth"} As is seen from Fig. \[fig:RT\], the designed composite sheet transmits 87% of the incident wave power propagating along the transparent direction. At the same time, it transmits 75% of the wave power (in cross polarization) when the layer is illuminated from the non-transparent side. Thus, the sheet acts as a one-way transparent layer and as twist-polarizer from the non-transparent side, as it was predicted theoretically. Non-ideal magnitude of the cross-polarized transmitted wave can be explained by inevitable absorption loss inside the ferrite sphere and copper wires (about 13%). Also, some small reflection exists due to parasitic weak omega and Tellegen coupling effects in the particle (about 12%). The designed sheet is a non-reciprocal analogy of the device proposed in [@Teemu] which consists of reciprocal chiral particles and acts as a twist-polarizer for both directions of incidence. Non-reciprocal electromagnetic coupling allows us to obtain dramatically different response for the opposite incident directions. The proposed composite sheet based on “chiral-moving" particles exhibits the target electromagnetic properties of a one-way transparent sheet and has realistic parameters allowing practical realizations. Conclusion ========== Although it is not possible to realize a fully transparent sheet except the trivial case of zero averaged induced surface currents, we have shown that it is possible to realize one-way transparent sheets. In these structures, the polarizabilities of unit cells are different from zero, but they are balanced in such a way, that the averaged induced currents are zero for illumination from one of the two sides of the sheet. However, the response to plane waves illuminating the opposite side of the sheet is non-trivial and can be controlled by design of the metasurface microstructure. Electromagnetic coupling (bi-anisotropy) inside unit cells of the metasurface is a necessary condition for one-way transparent layers. If we are limited to the case of lossless sheets with passive particles, then one-way transparency necessarily requires non-reciprocal coupling and is impossible with chiral and omega particles. It was shown that presence of “moving” coupling is necessary to make a lossless non-active one-way transparent sheet. In particular, it has been shown that non-reciprocal coupling effects allow to realize a one-way transparent sheet which acts as a twist-polarizer in reflection or transmission when illuminated from non-transparent side. Another possible device is a one-way transparent phase-shifting sheet. If active particles are allowed, our possibilities to control electromagnetic response from the opposite side of the sheet are extended: There is no restriction on the amplitude of reflection and transmission for the wave coming from the non-transparent side. Also omega coupling becomes allowed and makes it possible to realize a one-way transparent sheet with controllable co-polarized reflection from the opposite side. Required effective and individual polarizabilities of bi-anisotropic particles as components of a one-way transparent layer have been derived. Finally, we have shown a realistic design of a non-reciprocal one-way transparent sheet and simulated its performance parameters. [00]{} C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials," [*IEEE Antennas Propag. Mag.,*]{} vol. 54, no.2, pp. 10-35, 2012. A. V. Kildishev, A. Boltasseva, V. M. Shalaev, “Planar photonics with metasurfaces," [*Science,*]{} vol. 339, p. 1232009, 2013. J. A. Gordon, C. L. Holloway, and A. Dienstfrey, “A physical explanation of angle-independent reflection and transmission properties of metafilms/metasurfaces," [*IEEE Antennas Wireless Propag. Lett.,*]{} vol. 8, pp. 1127-1130, 2009. Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial," [*Phys. Rev. B,*]{} vol. 79, p. 045131, 2009. C. M. Watts, X. Liu , and W. J. Padilla, “Metamaterial electromagnetic wave absorbers," [*Adv. Mater.,*]{} vol. 24, pp. OP98-OP120, 2012. Y. Ra’di, V. S. Asadchy, and S. A. Tretyakov, “Total absorption of electromagnetic waves in ultimately thin layers," [*IEEE Trans. Antennas Propag.,*]{} vol. 61, no. 9, pp. 4606-4614, 2013. D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovich, “High-impedance electromagnetic surfaces with a forbidden frequency band," [*IEEE Trans. Microw. Theory Tech.,*]{} vol. 47, no. 11, pp. 2059-2074, 1999. I. V. Semchenko, S. A. Khakhomov, and A. L. Samofalov, “Transformation of the polarization of electromagnetic waves by helical radiators," [*Journal of Communications Technology and Electronics,*]{} vol. 52, pp. 850-855, 2007. M. Euler, V. Fusco, R. Cahill, and R. Dickie, “Comparison of frequency-selective screen-based linear to circular split-ring polarisation convertors," [*IET Microwaves, Antennas & Propag.,*]{} vol. 4, pp. 1764-1772, 2010. Y. Zhao and A. Alù, “Manipulating light polarization with ultrathin plasmonic metasurfaces," [*Physical Review B,*]{} vol. 84, p. 205428, 2011. T. Niemi, A. Karilainen, and S. Tretyakov, “Synthesis of polarization transformers," [*IEEE Trans. Antennas Propag.,*]{} vol. 61, no. 6, pp. 3102-3111, 2013. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. -P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction," [*Science,*]{} vol. 334, no. 6054, pp. 333-337, 2011. C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets," [*Phys. Rev. Lett.,*]{} vol. 110, p. 197401, 2013. F. Monticone, N. M. Estakhri, and A. Alù, “Full control of nanoscale optical transmission with a composite metascreen," [*Phys. Rev. Lett.,*]{} vol. 110, p. 203903, 2013. A. N. Serdyukov, I. V. Semchenko, S. A. Tretyakov, and A. Sihvola, [*Electromagnetics of bi-anisotropic materials: Theory and applications,*]{} Amsterdam: Gordon and Breach Science Publishers, 2001. Y. Ra’di and S. A. Tretyakov, “Balanced and optimal bi-anisotropic particles: Maximizing power extracted from electromagnetic fields," [*New J. Phys.,*]{} vol. 15, p. 053008, 2013. S. A. Tretyakov, [*Analytical modeling in applied electromagnetics,*]{} Norwood, MA: Artech House Publishers, 2003. V. V. Yatsenko, S. I. Maslovski, S. A. Tretyakov, S. L. Prosvirnin, and S. Zouhdi, “Plane-wave reflection from double arrays of small magnetoelectric scatterers," [*IEEE Trans. Antennas Propag.,*]{} vol. 51, pp. 2-11, 2003. J. Vehmas, Y. Ra’di, A. O. Karilainen, and S. A. Tretyakov, “Eliminating electromagnetic scattering from small particles," [*IEEE Trans. Antennas Propag.,*]{} vol. 61, no. 7, pp. 3747-3756, 2013. I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, [*Electromagnetic waves in chiral and Bi-isotropic media,*]{} Norwood, MA: Artech House, 1994. S. A. Tretyakov, “Nonreciprocal composite with the material relation of the transparent absorbing boundary," [*Microwave Opt. Technol. Lett.,*]{} vol. 19, no. 5, pp. 365-368, 1998. M. S. Mirmoosa, “Polarizabilities of nonreciprocal bi-anisotropic particles," [*Master’s thesis,*]{} Department of Radio Science and Engineering, Aalto University, 2013. I. A. Faniayeu, V. S. Asadchy, I. V. Semchenko, and S. A. Khakhomov, “Calculation and analysis of the tensors of electric, magnetic and chiral polarizabilities of the helices with optimal shape," [*The Sixth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics,*]{} St. Petersburg, Russia, pp. 324-326, 2012.

--- abstract: 'A “sandpile” cellular automaton achieves complex temporal correlations, like a $1/f$ spectrum, if the position where it is perturbed diffuses slowly rather than changing completely at random, showing that the spatial correlations of the driving are deeply related to the intermittent activity. Hence, recent arguments excluding the relevance of self-organized criticality in seismicity and in other contexts are inaccurate. As a toy model of single fault evolution, and despite of its simplicity, our automaton uniquely reproduces the scaling form of the broad distributions of waiting times between earthquakes.' author: - Marco Baiesi - Christian Maes title: Realistic time correlations in sandpiles --- Many complex natural phenomena exhibit quiet periods with a slow loading of the system separated by events in which a rapid evolution takes place [@bak_book; @sornette_book]. Some examples are earthquakes [@bak02:_unified_old; @corral03:_unified], solar flares [@BPS_05], creep experiments with cellular glasses [@maes98:_glass] and even with piles of rice [@aegerter03:_rice-exper], and rain falls [@peters_rain]. In these and in other natural and social systems one finds that the intensity of the events and the spatial and temporal scales separating them may vary over broad ranges and often are scale-invariant. The similar phenomenologies and the nonequilibrium nature suggest that a common mechanism is emerging in different systems. An interesting candidate is self-organized criticality (SOC) [@bak_book; @sornette_book; @btw], a theory essentially focussing on the occurrence of fast avalanches of nonlinearities, which have been well visualized in simple cellular automata (CA) called “sandpiles”. CA and their probabilistic versions are model-systems that have been used quite extensively in the field of complex systems [@gutowitz91; @frisch86]. Good reasons include the possibility of reliable simulations and the conceptual simplicity of the dynamical rules. Furthermore, one has strong indications that the microscopic details of nature are not all-important in deciding certain macroscopic features. Symmetry properties, conservation laws and considerations on the right scale of description matter much more. CA incorporate these essential ingredients and express in the most strongest form we know today how emergent behavior can be very rich, varied and complex even if based on few simple dynamical rules on a discrete architecture. A feature distinguishing several sandpile models from other CA is the scale-free distribution of their avalanche sizes. This resembles the distribution of energy released by earthquakes, for example. On the other hand, temporal correlations between avalanches have been far less investigated, until Boffetta [*et al.*]{} noticed that realistic correlations are not found in time series of some sandpiles [@boffetta99]. Indeed, their avalanches have interoccurrence times distributed exponentially, and the time series appears as a Poisson process of uncorrelated events. This fact was used to argue that SOC cannot be an interpretation of solar flares activity [@boffetta99] and, more recently, that SOC is not related to earthquakes [@yang2004]. After the critique to SOC by Boffetta [*et al.*]{} [@boffetta99] there has been a wave of investigations on the temporal properties of sandpiles and other SOC models. Nowadays we know that there are models with interesting temporal correlations, like the $1/f$ decay in power spectra [@zhang00:_1f; @davidsen-paczuski02:_1f; @woodard05:_building], or like power-law tails in distributions of waiting times between events [@norman01:_model_waiting; @sanchez02:_wait-SOC; @fragos04:_model_Ellerman; @hedges05:_OFC_waiting; @lippiello05:_SOC_memory; @maya05], foreshocks and aftershocks like for earthquakes [@hergarten02:_OFC_aftershock] or even features typical of turbulence [@btw_turb; @fabio]. Correlated drivings can give rise to non-exponential waiting time distributions between events [@sanchez02:_wait-SOC], which e.g. appeared in models of solar activity. In one case the strength of the driving perturbation in the Lu-Hamilton model was slowly varied at random [@norman01:_model_waiting]. In another model of “Ellerman bombs” [@fragos04:_model_Ellerman], directed percolation was one mechanism increasing the number of driving points. These examples further confirm and illustrate the coexistence of criticality and complex time correlations but it is likely that the correlations intrinsic in these driving mechanisms are simply inherited and show up directly in the output of the automaton. In this Letter we stress that a completely random driving of sandpiles is not expected to be a realistic feature but rather an artificial feature put by hand. In other words, random drivings can be an arbitrary, [*a priori*]{} limitation of the automata. Usually sandpiles are driven either at completely random positions or from a fixed single site, but none of these cases reflects the forms of slow loading in real systems, like the crust of the Earth or the solar corona. In seismicity, for example, one observes a rich scaling picture for the waiting time distributions [@bak02:_unified_old; @corral03:_unified], involving location, temporal occurrence, and magnitude of events. In order to have satisfactory CA models of earthquakes, it is therefore desirable to remove excessive randomness from the properties of their driving. Below, we show how extremely simple spatial correlations in the input drive can generate a complex time series as an output. Thus, the focus is on the [*spatial*]{} properties of the nucleating point of the avalanches. As far as we know, we provide the first example of $1/f$ noise in the avalanche activity of a non-running, classical sandpile. Furthermore, we observe broad distributions of waiting times between events also of small sizes. Most importantly, these distributions display a scaling behavior with intensity thresholds, analogous to those recently found for earthquakes [@corral03:_unified] and for solar flares [@BPS_05]. This feature makes at present this new model unique, and corroborates its interpretation in geophysical terms. The dynamics of the model strongly resembles the one of the sandpile originally discussed by Kadanoff [*et al.*]{} [@kadanoff89]: each site $i$ of a linear lattice of side $L$ holds an integer number $h_i$. In the sandpile jargon, $h_i$ is normally called “number of grains”, but this distribution as well may be thought as the potential energy profile in an object mainly extended in one dimension, like a fault. Stable configurations fulfill a local stability condition on the discrete gradient between every pair $(i, j)$ of nearest neighbors, $$h_i - h_j < H \label{eq:inst}$$ where constant $H$ is a threshold. A local instability not fulfilling (\[eq:inst\]) is resolved by a toppling, which consists in moving $\alpha$ grains from $i$ to $j$, i.e., $h_i\to h_i - \alpha$ and $h_j \to h_j + \alpha$. In the geophysical interpretation, the toppling thus models a redistribution of energy that takes place after some local elastic threshold within a fault is overtaken. Stress increase appears as a local increase of potential energy during time step $t$ at position $i'$, causing $h_{i'}(t) = h_{i'}(t-1)+1$. This eventually leads to a violation of (\[eq:inst\]) between site $i'$ and one or more of its nearest neighbors $j$. Thus, a toppling takes place between each unstable pair $(i',j)$, and $j$ in turn may also become unstable. One then iterates this procedure by making all topplings in parallel as long as some pairs of sites violating (\[eq:inst\]) are present [@note0]. The whole set of updates giving rise to a new stable configuration represents an avalanche (earthquake) at time $t$, with an area $a$ equal to the number of sites that toppled at least once and a size $s$ (energy released) equal to the total number of toppling. As in the standard sandpile scenario, the long time scales of the driving are thus completely separated from the ones of the avalanches. This is the appropriate limit for models of earthquakes, while mixing of time scales is present in solar activity [@BPS_05], and other models of SOC are possible [@maya05]. ![(a) Snapshot of a sandpile profile ($L=256$) around its average height. The arrow points to the position $i'$ where grain addition was taking place. (b) Example of time series of the avalanche sizes for a sandpile with $L=2048$. \[fig:1\]](fig_EPL_1a.eps "fig:"){width="5.5cm"} 0.3truecm ![(a) Snapshot of a sandpile profile ($L=256$) around its average height. The arrow points to the position $i'$ where grain addition was taking place. (b) Example of time series of the avalanche sizes for a sandpile with $L=2048$. \[fig:1\]](fig_EPL_1b.eps "fig:"){width="5.5cm"} A novel feature distinguishing the model proposed here from previous ones is that the site where a grain is added coincides with the position of a random walk: $i'(t)$ with equal probability is drawn from one of the nearest neighbors of $i'(t-1)$. This feature again has a counterpart in seismicity, where it is known that epicenters are spatially clustered and that aftershocks slowly diffuse after large events [@scholz_book]. The particular driving introduced here makes the model critical also with periodic boundary conditions (BC) [@note_BC]. We adopted periodic BC because there cannot be effects related to a fluctuating distance between a predefined boundary and the point where grains are added. For our purposes it is enough to show results on the sandpile in one dimension, with $H=4$ and $\alpha=2$. According to our data, the versions with different $H$ or $\alpha$ give similar results. The same remark applies to choices of different mobility; as long as the walker diffuses slowly, no change occurs in the basic aspects of what follows. A typical sandpile profile is plotted in Fig. \[fig:1\](a). Configurations like that are reached after an appropriate period of fueling of an empty lattice. That period was then not considered in the following statistical analysis. The sandpile profiles observed with periodic BC usually are an alternated sequence of uphills and downhills, thus alternating local minima and maxima. Each patch with the same trend is like a profile of a classical sandpile with open BC, which oscillates around a typical slope. However, grain addition and avalanches make patches to dynamically evolve, merging them or splitting them into shorter pieces. A time series of the avalanche sizes in the stationary regime is shown in Fig. \[fig:1\](b). It has an intermittent behavior with bursts of activity and temporal clustering of large avalanches, features due to time correlations, as shown below. The probibility density of sizes for several $L$’s, $P_L(s)$, are shown in Inset (a) of Fig. \[fig:Pa\] and display multiscaling [@tebaldi99:_multisc_BTW] while the area probability density $P_L(a)$ (Fig. \[fig:Pa\]) obey simple finite size scaling $$P_L(a) \simeq a^{-\tau} F\left( a / L^D \right) \label{eq:Pa}$$ as confirmed by a collapse onto a single scaling function $F(x)$ in the plot of $a P_L(a)$ vs $a/L$ (see the Inset (b) of Fig. \[fig:Pa\]). The model thus has a critical behavior, with $\tau \simeq D \simeq 1$ [@note1]. ![Probibility density of the area of avalanches, for lattices with $L=256$, $512$, $\ldots$, $4096$, from left to right. The small secondary tail of $P(a)$ is coming from rare avalanches with $a>L/2$ involving grains falling onto two sides of a mountain. Insets: (a) Size distributions for the same set of $L$ (curves are smoothed). (b) Collapse of $P_L(a)$, according to Eq. (\[eq:Pa\]) with $\tau=D=1$, onto a scaling function $F$. \[fig:Pa\]](fig_EPL_2.eps){width="7.7cm"} Complex temporal correlations and long memory are revealed by the nontrivial form of the power spectrum $S(f)$ of the time series of avalanche sizes $s(t)$ (with $s(t)=0$ if at time $t$ there were no topplings). Fig. \[fig:Power-s\](a) shows that at quite low frequencies $S(f) \sim 1/f^{\gamma}$ with $\gamma = 1.00(1)$. In Fig. \[fig:Power-s\](b) we plot $S(f)$ vs $f L$ to show that the $1/f$ region becomes broader when $L$ in increased: it is bounded on the right by a bump at $f \sim 1/L$, while the crossover frequency to a flat spectrum at very low $f$ scales approximately as $1/L^2$. Almost equal spectra are observed for the time series of avalanche areas, confirming that the basic origin of that $1/f$ spectrum is in the temporal correlations of events rather than in their actual intensities. We remind that a $1/f$ spectrum is observed in many natural phenomena [@montroll84:_1f], and it is regarded as one of the most complex forms of time correlation. It is remarkable that the long-range memory in the model arises by the non-trivial, self-organized structure encoded in the sandpile profile, in agreement with a basic idea of SOC theory, namely that nonequilibrium extended systems can self-organize to complex regimes even if their dynamical rules are simple. ![Log-log plot of the power spectrum of the time series of avalanche sizes, for $L=64$, $128$, $256$, $512$, and $1024$, (a) vs the frequency and (b) vs $f L$. In (a) curves have been offset half decade from each other, and a straight line indicates a $1/f$ scaling. For the estimates of $S(f)$, we used time windows with $2^{21}$ data. \[fig:Power-s\]](fig_EPL_3.eps){width="7.8cm"} The other form of temporal correlation concerns the waiting time distribution between events which has a power-law tail, rather than an exponential decay as for uncorrelated events. The waiting time distributions $P_s(t_w)$ between avalanches of size $\ge s$ in a sandpile with $L=4096$ are plotted in Fig. \[fig:Ptw\](a) for some thresholds $s$ ranging from $2^8=64$ to $2^{17}=131072$. Already for a small fixed $s$, the distributions have a power-law tail, which does not disappear when larger $L$ are considered. In the Inset of Fig. \[fig:Ptw\](a), for example, one can see that $P_{s=64}(t_w)$ has the same power-law tail for $L=256$ and $L=4096$. Interestingly, in fact when $s\gtrsim L/2$ (steepest part of $P_L(s)$, see Inset (a) of Fig. \[fig:Pa\]) in $P_s(t_w)$ there are two regimes with two different power law decays, as one observes in the analysis of waiting times of earthquakes [@corral03:_unified] and for solar flares [@BPS_05]. As in these cases, we find it appropriate to rescale times by multiplying them by rates of events $\ge s$, denoted as $R(s)$[@note_Rs]. ![(a) Log-log plot of distributions of waiting times between events with size $\ge s$, for $L=4096$ and various $s$. Inset: the cases ($L=256$, $s=64$; $\bullet$) and ($L=4096$, $s=64$; $\circ$) are compared. (b) Log-log plot of the same distributions, but rescaled with the rate of events $\ge s$. \[fig:Ptw\]](fig_EPL_4a.eps "fig:"){width="5.7cm"} ![(a) Log-log plot of distributions of waiting times between events with size $\ge s$, for $L=4096$ and various $s$. Inset: the cases ($L=256$, $s=64$; $\bullet$) and ($L=4096$, $s=64$; $\circ$) are compared. (b) Log-log plot of the same distributions, but rescaled with the rate of events $\ge s$. \[fig:Ptw\]](fig_EPL_4b.eps "fig:"){width="5.5cm"} The rescaled times $T = t_w R(s)$ appear to be the natural ones, because $P_s(t_w) / R(s)$ vs $t_w R(s)$ collapse quite well onto a single scaling curve $G(T)$ \[see Fig. \[fig:Ptw\](b)\]. Since $G(T) dT = P_s(t_w) d t_w$, we see that $G(T)$ is the probability density of $T$, not depending anymore on thresholds. Hence, a unifying scheme can be applied to all thresholds also in our sandpile model, showing that there is a basic self-similar mechanism taking place. For short rescaled times $G(T)\sim T^{-\beta_1}$ with $\beta_1 = 0.8(1)$, while $G(T)\sim T^{-\beta_2}$ with $\beta_2 = 2.0(2)$ for $T\gg 1$. (for earthquakes Corral found $\beta_1 = 0.9(1)$ and $\beta_2 = 2.2(1)$ [@corral03:_unified]). The crossover between the two power-law regimes is around $T^*\approx 0.7$. Thus, for $t_w$ much smaller than the average occurrence time $\overline t(s) = R(s)^{-1}$ of events with size $\ge s$ (i.e. $T \ll 1$), the correlations between events are qualitatively different from the ones at $t_w \gg \overline t(s)$. A mechanism giving rise to strong temporal correlations between avalanches should be their spatial overlap, since the memory of past activity is stored in the height profile of the sandpile. By dropping grains at a point slowly diffusing, as we do, a fresh part of the profile is almost always found, probably enhancing the correlation with past activity. In some classical models driven at random points, scale-free waiting time distributions are observed between sufficiently large avalanches [@sanchez02:_wait-SOC], which indeed have a consistent probability to overlap with previous ones. On the other hand, that argument does not really explain why the $1/f$ part of the spectrum of our sandpile extends to very low frequencies. We have mainly discussed SOC in CA, but a different class of SOC models also supports the view that space and time correlations are related to each other. We are referring to models with “extremal dynamics”. In these models each unit carries a continuum variable and the unit closer to instability is always the one relaxing (toppling). The Olami-Feder-Christensen (OFC) model of earthquakes [@olami92:_OFC] is representative of extremal dynamics. Whether criticality in the OFC model is only apparent or persists in the limit $L\to \infty$ is an open issue. Nevertheless, the OFC model has complex patterns in time [@hergarten02:_OFC_aftershock; @hedges05:_OFC_waiting] and space: indeed, by its nature, the OFC map self-organizes also the position of the epicenters (driving points), which form sequences deeply linked to past activity [@peixoto04:_epic_OFC]. Similarly, in the Bak-Sneppen model of evolution, distances between extremal sites follow a Levy flight, and $1/f^{\gamma}$ spectra are found with $0<\gamma<1$ [@paczuski96:_bigPRE]. Recently, Lippiello [*et al.*]{} introduced a hybrid sandpile/extremal dynamics model of seismicity that also display waiting time distributions with power-law tails [@lippiello05:_SOC_memory]. In summary, we have studied a sandpile cellular automaton exhibiting self-organized criticality (even without open boundaries). A critical stationary state with correlated avalanches and intermittent transport takes place because the model is perturbed at a position slowly diffusing in space, a process leading to clustered nucleation points and motivated by the observation of natural phenomena like earthquakes, where epicenters are correlated. Indeed, the distributions of waiting times between avalanches larger than given thresholds have striking and unique similarities with the ones found in real seismicity. We can thus reproduce some of the complexity of natural critical phenomena, involving jointly energetic, spatial and temporal aspects. Our results suggest that the logic that led to the present model is right, namely that spatial correlations should not be arbitrarily removed by imposing a random drive in models of critical phenomena like sandpiles, which are by themselves already very simplified objects. The positive comparison of our results with earthquake activity leads to propose that a rugged landscape as depicted in Fig. \[fig:1\](a) may be related to the stress distribution along a fault. The absence of a single slope profile (as observed in sandpiles with open boundaries) implies that system-wide events are not always possible, with potential interesting implications about predictability of large events. We thank S. Boettcher, M. De Menech, M. Paczuski, A. L. Stella and C. Vanderzande for useful discussions. M. B. acknowledges support from a FWO position (Flanders). [10]{} (Copernicus, New York) 1996. (Springer, Heidelberg) 2000. and [Scanlon T.,]{} , [**88**]{} (2002) 178501. , , [**68**]{} (2003) 035102(R). and [Stella A. L.]{}, , [**96**]{} (2006) 051103. and [Strauven H.]{}, , [**57**]{} (1998) 4987–4990. and [Wijngaarden R. J.]{}, , [**67**]{} (2003) 051306. and [Christensen K.]{}, , [**88**]{} (2002) 018701. and [Wiesenfeld K.]{}, , [**59**]{} (1987) 381–384. (Ed.), [*Cellular Automata: Theory and Experiment*]{} (MIT Press, Cambridge) 1991. and [Pomeau Y.,]{} [*Phys. Rev. Lett.*]{}, [**56**]{} (1986) 1505–1508. and [ A.]{}, , [**83**]{} (1999) 4662–4665. and [Ma J.]{}, , [**92**]{} (2004) 228501. , , [**61**]{} (2000) 5983–5986. and [Paczuski M.]{}, , [**66**]{} (2002) 050101(R). and [Carreras B. A.]{}, cond-mat/0503159 preprint, 2005. and [Liu H.-L.]{}, , [**557**]{} (2001) 891–896. and [Carreras B. A.]{}, , [**88**]{} (2002) 068302. and [Vlahos L.]{}, , [**420**]{} (2004) 719–728. and [Takacs G.]{}, physics/0505015 preprint, 2005. and [Godano C.]{}, , [**72**]{} (2005) 678–684. and [Baiesi M.]{}, , [**95**]{} (2005) 181102. and [Neugebauer H. J.]{}, , [**88**]{} (2002) 238501. [Helmstetter A., Hergarten S.]{} and [Sornette D.]{}, [*Phys. Rev. E*]{}, [**70**]{} (2004) 046120. and [Stella A. L.]{}, , [**309**]{} (2002) 289. and [Baiesi M.]{}, , [**96**]{} (2006) 105005. , *et al.*, [*Phys. Rev. A*]{}, [**39**]{} (1989) 6524. We chose not to use sequential updates because the model is not Abelian. , *The Mechanics of Earthquakes and Faulting*, Cambridge University Press (Cambridge, 2002), [2nd]{} ed. With periodic BC, no grains escape from the sandpile, and its average height increases as $t/L$. and [Stella A. L.]{}, , [**83**]{} (1999) 3952–3955. The etsimate of $D$ is easily obtained by matching all the cutoffs of the distributions on a single value, while the value $\tau=1$ is deduced by the good data collapse and by the fact that a direct fit of the slopes of the distribution on Fig. \[fig:Pa\] would give $\tau\approx 0.9$ while an analysis of the momenta $\langle a^q \rangle$ yields $\tau \gtrsim 1$. Hence, a consistent conclusion is that $\tau\simeq1 $ and that its direct estimates are slightly contrasting due to finite size corrections to scaling. and [Shlesinger M. F.]{}, In [*Studies in Statistical Mechanics*]{}, J. Lebowitz and E. Montroll, Eds., vol. 11. (North-Holland, Amsterdam) 1984, p. 1. $R(s)$ is the total number of events with size $\ge s$ divided by the total number of added grains. and [Christensen K.]{}, , [**68**]{} (1992) 1244–1247. and [Prado C. P. C.]{}, , [**69**]{} (2004) 025101(R). and [Bak P.]{}, , [**53**]{} (1996) 414–443.

--- abstract: 'The Deep Graph Library ([$\mathsf{DGL}$]{}) was designed as a tool to enable structure learning from graphs, by supporting a core abstraction for graphs, including the popular Graph Neural Networks ([$\mathsf{GNN}$]{}). [$\mathsf{DGL}$]{} contains implementations of all core graph operations for both the [$\mathsf{CPU}$]{} and [$\mathsf{GPU}$]{}. In this paper, we focus specifically on [$\mathsf{CPU}$]{} implementations and present performance analysis, optimizations and results across a set of [$\mathsf{GNN}$]{} applications using the latest version of [$\mathsf{DGL}$]{} (0.4.3). Across 7 applications, we achieve speed-ups ranging from $1.5\times$-$13\times$ over the baseline [$\mathsf{CPU}$]{} implementations.' author: - | Sasikanth Avancha\ Parallel Computing Lab, Intel Labs\ Intel Corporation\ Bangalore, India\ `[email protected]`\ Vasimuddin Md.\ Parallel Computing Lab, Intel Labs\ Intel Corporation\ Bangalore, India\ `[email protected]`\ Sanchit Misra\ Parallel Computing Lab, Intel Labs\ Intel Corporation\ Bangalore, India\ `[email protected]`\ Ramanarayan Mohanty\ Parallel Computing Lab, Intel Labs\ Intel Corporation\ Bangalore, India\ `[email protected]` bibliography: - 'references.bib' title: Deep Graph Library Optimizations for Intel x86 Architecture --- Introduction {#sec:intro} ============ Graph Neural Networks ([$\mathsf{GNN}$]{}) [@hamilton17nips; @Kipf:2016tc; @velickovic2018graph; @xu2018how; @kipf18rgcn] are a very important class of Neural Network algorithms for learning the structure of large, population-scale graphs. Often, [$\mathsf{GNN}$]{}s are combined with traditional graph structure discovery algorithms via traversal (e.g., Breadth-First Search ($\mathsf{BFS}$), Depth-First Search ($\mathsf{DFS}$), RandomWalk) to achieve higher accuracy in learning their structure. Given a graph $G = (\mathcal{V}, \mathcal{E})$, the neural network formulation in [$\mathsf{GNN}$]{}s implies that, unlike graph traversal algorithms, they attempt to learn structure of $G$ via low-dimensional representations associated with $\mathcal{V}$ or $\mathcal{E}$ or both. [$\mathsf{GNN}$]{} algorithms, broadly, learn these representations in two parts: in feature vectors $F_{v}$ and/or $F_{e}$ associated with $\mathcal{V}$ and $\mathcal{E}$, respectively and a set of graph-wide, shared parameters $W$. Via a recursive process called Aggregation, [$\mathsf{GNN}$]{}s encode multi-hop neighborhood representations in $F_{v}$ and/or $F_{e}$. Depending upon the specific algorithm and the task (e.g., node classification, link prediction etc.), feature vectors aggregation precedes or succeeds a shallow neural network, typically consisting one or more linear transforms followed by a classification or regression model etc; some algorithms additionally employ a self-attention mechanism. Given that aggregation is the core operation in all [$\mathsf{GNN}$]{} training and inference algorithms, our focus in this paper is on accelerating aggregation performance on Intel  Xeon  high-performance [$\mathsf{CPU}$]{}s. Let $t = (u,v,e)$ be a tuple, where $e$ is the edge, and $u \in U$ and $v \in V$ are the source and destination vertices, respectively. Inherently, the aggregation operation involves [*message passing*]{} between any two entities in $t$. [$\mathsf{DGL}$]{} implements the aggregation operation via two basic primitives: ${\tt send}(x)$ and ${\tt recv}(y, \oplus)$, where $x, y \in t$ and $\oplus$ is a reduction operation. We observed that [$\mathsf{DGL}$]{} fuses send and recv into a single primitive ${\tt fused\_sr}(x,y,\oplus)$ when aggregation consists of a simple arithmetic operation (as described in [@wang2019dgl]); [$\mathsf{DGL}$]{} implements built-in fused aggregation primitive in such cases. [$\mathsf{DGL}$]{} implements unary aggregation primitives (e.g. $\mathsf{copy\_u}$ and $\mathsf{copy\_e}$) which reduce a set of source features into the destination feature. In [$\mathsf{DGL}$]{} parlance, the unary aggregation primitive is called [*Copy-Reduce*]{}. Similarly, [$\mathsf{DGL}$]{} implements binary aggregation primitives (e.g. $\mathsf{u\_mul\_e\_add\_v}$), which first perform an element-wise binary operation on two input feature vectors, and reducing the result into the destination feature via message passing. In [$\mathsf{DGL}$]{} parlance, the binary aggregation primitive is called [*Binary-Reduce*]{}. We discuss these primitives in greater detail in the next section. Given that Aggregation is a core operation in all [$\mathsf{GNN}$]{} training and inference algorithms, our focus in this paper is on Aggregation performance on Intel  Xeon  high-performance [$\mathsf{CPU}$]{}s, along with other primitives. Inherently, aggregating $F_{v}$ ($F_{e}$[[[)]{}]{}]{} in a graph involve the process of [*message passing*]{}, i.e., propagating them between source and destination nodes in $V$. [$\mathsf{DGL}$]{} [@wang2019dgl] implements Aggregation via two primitives: $send(E, \otimes^{e})$ and $recv(V, \oplus, \otimes^{v})$, which trigger message passing on edges and nodes, respectively. Here, $\otimes^{v}$ and $\otimes^{e}$ refer to the neural network functions applied to nodes and edges, respectively, and $\oplus$ is the Aggregation operator. As Wang et al. described in [@wang2019dgl], and from our own experience, [$\mathsf{DGL}$]{} fuses $send$ and $recv$ into a $send\_and\_recv(E, \otimes^{e}, V, \oplus, \otimes^{v})$ operation when it detects the use of simple operations and employs built-in functions for them. For example, [$\mathsf{DGL}$]{} provides unary functions such as $copy\_u$ and $copy\_e$ to copy $F_{v}$ and $F_{e}$ to the output message field, respectively. Similarly, it provides binary functions such as $u\_add\_v$ which has the following definition at <https://docs.dgl.ai/generated/dgl.function.u_add_v.html>: “Builtin message function that computes a message on an edge by performing element-wise add between features of src and dst if the features have the same shape; otherwise, it first broadcasts the features to a new shape and performs the element-wise operation”. As Wang et al. describe in [@wang2019dgl], the [DGLGraph]{} interface hides the details of the graph data structure (e.g., $\mathsf{CSR}$ or $\mathsf{COO}$) from the programmer to enable better productivity. However, this implies that the application performance using [$\mathsf{DGL}$]{} for [$\mathsf{GNN}$]{} training and inference depends on how well the graph data structure and its associated operations have been optimized for the underlying architecture. Our analysis of various applications using [$\mathsf{DGL}$]{} revealed that the aggregation operation is implemented using sub-optimal primitives (such as serialization, explicit buffer copies prior to reduction), resulting in low performance on the Intel  Xeon  processor family. In this paper, we optimized the aggregate primitives in [$\mathsf{DGL}$]{} on [$\mathsf{CPU}$]{} and demonstrated the per epoch time speedups as high as $13\times$ on [$\mathsf{DGL}$]{} [$\mathsf{GNN}$]{} applications. The rest of the paper is organized as follows. Section \[sec:dglprimitives\] describes the aggregate primitives used by [$\mathsf{DGL}$]{}. Section \[sec:dglopt\] describes the optimizations applied to binary-reduce and copy-reduce primitives. Section \[sec:other\_primitives\] discusses primitives implemented in the PyTorch framework that impact GNN performance, and their optimizations. In Section \[sec:results\], we discuss the results of our optimizations and show performance improvements across various [$\mathsf{GNN}$]{} applications on Intel  Xeon  processors. Section \[sec:conclusion\] concludes the paper. Aggregation Primitives {#sec:dglprimitives} ====================== Our understanding and analysis of the aggregation primitives in [$\mathsf{DGL}$]{} lead to the conclusion that these operations can be represented as a sequence of linear algebraic expressions involving node and edge features, along with the appropriate operators. The expression is sufficient to describe the aggregation over the complete graph or sub-graph upon which the operation executes. Binary-Reduce ([$\mathsf{BR}$]{}) --------------------------------- As discussed in section \[sec:intro\], [$\mathsf{BR}$]{} consists of a sequence of two operations – an element-wise binary operation between a pair of feature vectors, and an element-wise operation that [*reduces*]{} the intermediate feature vector into the output feature vector. When applied over the whole graph, the operands may be considered as multi-dimensional tensors representing nodes and edges. Equation \[eq:br\] shows a mathematical representation of [$\mathsf{BR}$]{}, with operators $\otimes$ (element-wise binary operator) and $\oplus$ (element-wise reduction operator), and feature vector operands $x$, $y$ and $z$. $x$ and $y$ are inputs to $\otimes$, $\otimes(x,y)$ and $z$ are inputs to $\oplus$; the final result is in $z$. $$\begin{gathered} BR(x, y, \otimes, \oplus, z): \oplus (\otimes(x, y), z), \label{eq:br} \\ \forall x, y, z \in G(\mathcal{V}, \mathcal{E}) \nonumber\end{gathered}$$ ![Example directed subgraph showing aggregation direction. For node $v_{00}$, directed edges from all $u_{2x}$ to $u_{1x}$ to $v_{00}$ will be part of [$\mathsf{BR}$]{} operation.[]{data-label="fig:brcr"}](figures/subgraph.jpg) Figure \[fig:brcr\] shows an example subgraph induced on a (possibly) larger directed graph, and rooted at node $v_{00}$. In the example, all nodes labeled $u_{1x}$ are 1-hop neighbors of $v_{00}$; similarly nodes $u_{2x}$ are its 2-hop neighbors. Now, for example, $u\_mul\_e\_add\_v$ [$\mathsf{BR}$]{} between nodes $u_{20}$ and $u_{12}$, with feature vectors F$u_{20}$ and F$u_{12}$, respectively and edge-feature vector F$e_{2012}$ would have the following expression: $$\begin{gathered} BR_{21}(Fu_{20}, Fe_{2012}, \times, +, u_{12}): \nonumber \\ t \leftarrow Fu_{20} \times Fe_{2012} \nonumber \\ u_{12} \leftarrow u_{12} + t \label{eq:bre}\end{gathered}$$ Further, we observe that nodes $u_{11}$ and $u_{13}$ will [*not*]{} be part of [$\mathsf{BR}$]{} on the subgraph rooted at $v_{00}$ because there are no edges [*from*]{} them [*to*]{} $v_{00}$. If the size and shape of the input feature vectors are not equal, and if one of them has size $1$, then [$\mathsf{BR}$]{} [*broadcasts*]{} the smaller feature vector to the dimension of the larger one; in all other cases, [$\mathsf{BR}$]{} will fail to execute. Copy-Reduce ([$\mathsf{CR}$]{}) ------------------------------- [$\mathsf{DGL}$]{} implements the [$\mathsf{CR}$]{} operation separately from [$\mathsf{BR}$]{} as it is widely used in [$\mathsf{GNN}$]{} applications without any binary operation. Therefore, we view it as a special class of [$\mathsf{BR}$]{}. [$\mathsf{CR}$]{} takes only one input operand associated with the source (node or edge) and passes the feature vector as a message (i.e., [*copies*]{} it to the destination (node or edge), where it is reduced onto the latter. As shown in Equation \[eq:cr\], [$\mathsf{CR}$]{} can be mathematically represented using [$\mathsf{BR}$]{} syntax with y replaced with $\mathsf{NULL}$ or $\phi$, resulting in $\otimes(x,y)$ becoming a [*unary*]{} operation [copy(x)]{}: $$\begin{gathered} BR(x, \phi, \otimes, \oplus, z): \oplus (\otimes(x, \phi), z) \nonumber \\ \otimes(x, \phi) = {\tt copy}(x) \nonumber \\ BR(x, \phi, \otimes, \oplus, z) \Rightarrow CR(x, {\tt copy}, \oplus, z) \nonumber \\ CR(x, {\tt copy}, \oplus, z): \oplus ({\tt copy}(x), z), \forall x, z \in G(\mathcal{V}, \mathcal{E}) \label{eq:cr}\end{gathered}$$ For example, in Figure \[fig:brcr\], with sum (+) as reduction operation, [$\mathsf{CR}$]{} between $u_{10}$ and $v_{00}$ can be expressed as: $$\begin{gathered} CR_{21}(u_{10}, {\tt copy}, +, v_{00}): \nonumber \\ t \leftarrow {\tt copy}(u_{10}) \nonumber \\ v_{00} = v_{00} + {\tt copy}(u_{10}) \label{eq:cre}\end{gathered}$$ Configurations of Binary-Reduce and Copy-Reduce ----------------------------------------------- Various configurations of [$\mathsf{BR}$]{} arise as a result of multiple candidates for each input operand and reduction destination. Here, we present the comprehensive list of configurations of [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} primitives implemented in [$\mathsf{DGL}$]{}. (Table \[tab:configs\]). **[$\mathsf{BR}$]{}** **[$\mathsf{CR}$]{}** --------------------------------------------------------------------------- -------------------------------------------------------------- $\mathsf{u\_\otimes\_v\_\oplus\_v}$, $\mathsf{v\_\otimes\_u\_\oplus\_v}$, $\mathsf{u\_copy\_\oplus\_v}$, $\mathsf{e\_copy\_\oplus\_v}$ $\mathsf{u\_\otimes\_v\_\oplus\_e}$, $\mathsf{v\_\otimes\_u\_\oplus\_e}$, $\mathsf{u\_\otimes\_e\_\oplus\_v}$, $\mathsf{e\_\otimes\_u\_\oplus\_v}$, $\mathsf{u\_\otimes\_e\_\oplus\_e}$, $\mathsf{e\_\otimes\_u\_\oplus\_e}$, $\mathsf{v\_\otimes\_e\_\oplus\_v}$, $\mathsf{e\_\otimes\_v\_\oplus\_v}$, $\mathsf{v\_\otimes\_e\_\oplus\_e}$, $\mathsf{e\_\otimes\_v\_\oplus\_e}$ : Various configurations of [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} primitives inbuilt in [$\mathsf{DGL}$]{}. []{data-label="tab:configs"} [$\mathsf{DGL}$]{} has built-in support for a set of configurations in which $\otimes \in \{\mathsf{add}, \mathsf{sub}, \mathsf{mul}, \mathsf{div}, \mathsf{dot}\}$ and $ \oplus \in \{\mathsf{add}, \mathsf{max}, \mathsf{min}, \mathsf{mul},\mathsf{div},\mathsf{copy} \}$. In practice, [$\mathsf{DGL}$]{} showed that these configuration are enough to support a large majority of applications. Our evaluation showed that even with these simple operations, the [$\mathsf{BR}$]{} primitive executes for a majority of the run-time across various applications (described in the Section \[sec:results\]). We profiled $7$ [$\mathsf{GNN}$]{} applications (total $8$ instances) and the [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} primitives used by them (Table \[tab:cr\_apps\]). **Application** **[$\mathsf{BR}$]{} Configurations** ---- ------------------- ------------------------------------------------------------------- -- 1. GCN ($\mathsf{u\_copy\_add\_v}$) 2. GCN-Sampled ($\mathsf{u\_copy\_add\_v}$) 3. GraphSAGE ($\mathsf{u\_copy\_add\_v}$) 4. GraphSAGE-Sampled ($\mathsf{u\_copy\_add\_v}$) 5. GCMC ($\mathsf{u\_copy\_add\_v}$), ($\mathsf{u\_dot\_v\_add\_e}$) 6. Line Graph ($\mathsf{u\_copy\_add\_v}$) 7. Monet ($\mathsf{u\_mul\_e\_add\_v}$) ($\mathsf{e\_copy\_add\_v}$), ($\mathsf{e\_copy\_max\_v})$, ($\mathsf{u\_add\_v\_copy\_e}$), ($\mathsf{e\_sub\_v\_copy\_e}$), ($\mathsf{e\_div\_v\_copy\_e}$), ($\mathsf{u\_mul\_e\_add\_v})$, ($\mathsf{v\_mul\_e\_copy\_e}$) 9. RGCN-Hetero ($\mathsf{u\_copy\_add\_v}$) : [$\mathsf{GNN}$]{} applications and [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} configurations used by them.[]{data-label="tab:cr_apps"} Baseline Implementations of [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} in [$\mathsf{DGL}$]{} {#sec:basecr} ----------------------------------------------------------------------------------------- The graph adjacency matrix in [$\mathsf{DGL}$]{} is in Compressed Sparse Row (CSR) format. The [$\mathsf{CPU}$]{} implementation first loads the features of $u$, $v$ and/or $e$, as required, for each row offset (representing the source node) and corresponding column indices (representing destination nodes). Using these feature vectors, it performs [$\mathsf{BR}$]{} or [$\mathsf{CR}$]{} for the tuple $(u, v, e)$. Specifically, to execute [$\mathsf{CR}$]{}, [$\mathsf{DGL}$]{} implements a [*push*]{} model. By [*push*]{}, we mean that in $\oplus({\tt copy}(x_k), z_{k-1})$ executes [*from*]{} hop $k$ [to]{} $k-1$, so $x \in$ hop $k$ and $z \in$ hop $k-1$. To achieve good performance for [$\mathsf{CR}$]{} on the [$\mathsf{CPU}$]{}, the implementation parallelizes the loop over rows of the $\mathsf{CSR}$ matrix (i.e., the source nodes, $x_k$). Since [$\mathsf{CR}$]{} is an integral part of [$\mathsf{BR}$]{}, we first focus on the problems associated with parallel execution in [$\mathsf{CR}$]{}. As shown in Equation \[eq:br\], the binary operation $\otimes$ is straightforward, executes before $\oplus$ and can be parallelized easily, with the result stored in some temporary feature $t$. Algorithm \[algo:push\] describes the baseline push model in [$\mathsf{DGL}$]{}’s [$\mathsf{CR}$]{} implementation. copy\_u(u, out) $F_{v} \leftarrow F_{v} \oplus out$ When different nodes $u$ share neighbor $v$ and if the [$\mathsf{CR}$]{} destination is nodes $v$, then the [$\mathsf{CR}$]{} operation results in race condition among the threads. [$\mathsf{DGL}$]{} employs serialization using critical sections to resolve the race condition. The serialization significantly impacts the performance, leading to slower application run times. Also, the [push]{} approach to [$\mathsf{CR}$]{} is [*scatter-heavy*]{} given that the graph adjacency matrix is more than 99.9% sparse; this bounds [$\mathsf{CR}$]{} performance by memory access latency to randomly scattered destination node addresses in memory. We also observed via profiling and analysis that there is a potential reuse proportional to the average node degree. However, the [push]{} model fails to make use of this reuse because it simply scatters the feature vector to different addresses. This results in a significant amount of wasted memory bandwidth. Optimizing Aggregation Primitives {#sec:dglopt} ================================= [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} account for a majority of the run-time in [$\mathsf{GNN}$]{} applications. In this section, we describe techniques we have created to optimize their implementations within [$\mathsf{DGL}$]{}. As discussed in Section \[sec:basecr\], achieving high performance for [$\mathsf{CR}$]{} is critical to [$\mathsf{BR}$]{} performance as well; it is also potentially harder to achieve. Therefore, we first focus on [$\mathsf{CR}$]{} optimizations. Copy-Reduce ----------- To avoid the problems associated with the default [push]{} model, [$\mathsf{DGL}$]{} provides a way to [pull]{} messages from nodes $u$ and reduce them into nodes $v$. Now, parallelizing the [$\mathsf{CR}$]{} operation by distributing $v$ across OpenMP threads will not result in collisions because only one thread *owns* all feature vector vectors $F_{v}$ at each destination node $v$ and reduces each *pulled* source feature vectors into $F_{v}$  (Algorithm \[algo:pull\]). copy\_u(u, out) $F_{v} \leftarrow F_{v} \oplus out$ While Algorithm \[algo:pull\] solves the collision problem of Algorithm \[algo:push\], it still does not solve either the feature vector reuse problem (to reduce wasted memory bandwidth) or the memory latency problem due to random access pattern of source addresses. It turns from a [*scatter-heavy*]{} algorithm to a [*gather-heavy*]{} one. To solve the problems associated with both [push]{} and [pull]{}, we further optimize this algorithm and implement a variant of the Sparse-Dense Matrix Multiply operation that Wang et al. [@wang2019dgl] allude to. The neighborhood graph is represented as a sparse matrix, the adjacency matrix in CSR format. The dense matrix consists of the feature vectors $F_{u}$ or $F_e$ associated with source nodes $u$ or edges $e$, respectively. Algorithm \[algo:cropt\] shows the details of our optimized [$\mathsf{CPU}$]{} implementation of [$\mathsf{CR}$]{}, for $\mathsf{u\_copy\_add\_v}$ configuration. $A$ - Matrix of size $M \times K$ in CSR format $B$ - Dense matrix of size $K \times N$ $C$ - Dense matrix of size $M \times N$ Reduction-operator: $\oplus$ $N$ = length($F_{u}$), $kb$ = block-size on $K$ dimension, $nb$ = block-size on $N$ dimension $C \leftarrow 0$ $B[c,\dots,c+kb] \leftarrow {\tt RadixSort}(B[c,\dots,c+kb])$ $C[r][n] \leftarrow += B[c][n]$ The critical part of this formulation is that the rows and columns of the sparse matrix A represent the destination (M) and source (K) nodes, respectively and the dense matrix B consists of the source node feature vectors $F_{u}$. Thus, the output matrix C consists of feature vectors of destination nodes $F_{v}, v \in {\tt Neighborhood}(u)$, reduced from multiple source nodes $F_{u}$. Given that A is an adjacency matrix in $\mathsf{CSR}$ format, each row (i.e., $v$) only consists of column indices of connected source nodes $u$. So, in effect, the [*matrix multiply*]{} operation is to select those rows (i.e., source nodes $u$) of feature vectors $F_{u}$ from B that reduce into rows (i.e., destination node $v$) feature vectors $F_{v}$ in C. To achieve high performance, Algorithm \[algo:cropt\] contains two primary optimizations: 1. Parallelizes over rows of A (and C). This optimization is similar to that in Algorithm \[algo:pull\], where threads own destination nodes, and thus, there is no collision problem that we observe in Algorithm \[algo:push\]. 2. Takes advantage of the [*reuse*]{} present in the graph, and avoids random gathers by: 1. Blocking the K dimension of A and B, ensuring that all threads work on one block of ${\tt kb}$ source nodes at a time, 2. Sort the block of rows in B according to row-id using Radix Sort, and 3. Block the N dimension of B and C to process ${\tt nb}$ feature vector elements at a time Due to 2(a), any feature vector in B read by some thread $t$ could be in the L2 cache of the [$\mathsf{CPU}$]{} if/when some other thread $t'$ reads the same feature vector. Due to 2(b), accesses of source node feature vectors from DRAM are not completely random, but in ascending order of addresses - which should help reduce DRAM access latency. Due to 2(c), all threads work only on a block of C of size ${\tt M} \times {\tt nb}$ at a time, where ${\tt nb}$ is the block size. We use a value of ${\tt nb}$ such that the block of C stays in the Last Level Cache ($\mathsf{LLC}$) of the [$\mathsf{CPU}$]{} until it is completely processed. Binary-Reduce ------------- We focus now on optimizing the binary operation within [$\mathsf{BR}$]{}, applying the optimized Algorithm \[algo:cropt\] to handle the [$\mathsf{CR}$]{} part. Algorithms \[algo:bropt1\], \[algo:bropt2\], \[algo:bropt3\] describe the optimized BR for different configurations of input and output operations. Our optimizations consist of three major steps. 1. Of the two input operands, gather the features of the second operand corresponding to each instance of the first operand, as required by the binary operation. 2. Perform the element-wise binary operation ($\otimes$) on the two operands. \[step2\] 3. Reduce the dense matrix generated using $\oplus$. If the reduction destination is a node, then apply [$\mathsf{CR}$]{} on the node feature matrix. If the reduction destination is an edge, copy the result of Step \[step2\] to the dense edge feature matrix. To clarify the usage of various [$\mathsf{BR}$]{} configurations, we have shown three algorithms: (Node, Node, Any), (Node, Edge, Any) and (Edge, Node, Any) in Algorithms \[algo:bropt1\], \[algo:bropt2\] and \[algo:bropt3\], respectively. In Algorithm \[algo:bropt1\], for each source node $u$, we load feature $F_u$ and gather connected destination node features $F_v$ (line 4). Depending on whether the final destination of reduction or copy is $u$, $v$ or the edge between them $e$, lines 6, 8 and 11, scatter the result $F_u \otimes F_v$ to the node-feature matrix $V_f$ or $E_f$, respectively. In Algorithm \[algo:bropt2\], the second operand is the edge incident on $u$; therefore, we must first obtain the edge index $e$ from the incidence matrix $E^T$, gather its feature $F_e$ from edge-feature matrix $E_f$ and then perform $\otimes$ followed by reduction or copy on lines 7, 9, or 11, respectively, corresponding to the final destination. In Algorithm \[algo:bropt3\], the first operand is the set of all edges $E$; in line 2, we load each edge-feature $F_e$; the second operand is the set of nodes $V$ upon which $e$ is incident; therefore, in line 4, we gather node-features $F_u \forall u \in V$; again, depending on the final destination, we reduce or copy $F_e \otimes F_u$ to $V_f$ or $E_f$ in lines 6, 8 and 10, respectively. Matrix $A$ of size $M \times K$ in CSR format Matrix $E$ of size $M^2 \times K$ in CSR format (Incidence) Matrix $E^T$ is $K \times M^2$ in CSR format (Incidence) Feature matrix $V_f$ of size $M \times d$ Feature matrix $E_f$ of size $M^2 \times d$ Input operands: X (Nodes), Y (Nodes) Output operand: Z (Edges) Binary-operator: $\otimes$, Reduction-operator: $\oplus$ $F_u \leftarrow V_f[u]$ $F_v \leftarrow V_f[v]$ $V_f[u] \leftarrow F_u \oplus (F_u \otimes F_v)$ $V_f[v] \leftarrow F_v \oplus (F_u \otimes F_v)$ $e \leftarrow E^T[v]$ $E_f[e] \leftarrow F_u \otimes F_v$ Matrix $A$ of size $M \times K$ in CSR format Matrix $E$ of size $M^2 \times K$ in CSR format (Incidence) Matrix $E^T$ is $K \times M^2$ in CSR format (Incidence) Feature matrix $V_f$ of size $M \times d$ Feature matrix $E_f$ of size $M^2 \times d$ Input operands: X (Nodes), Y (Edges) Output operand: Z (Any) Binary-operator: $\otimes$, Reduction-operator: $\oplus$ $F_u \leftarrow V_f[u]$ $e \leftarrow E^T[v] $ $F_e \leftarrow E_f[e]$ $V_f[u] \leftarrow F_u \oplus (F_{u} \otimes F_e)$ $V_f[v] \leftarrow F_v \oplus (F_{u} \otimes F_e)$ $E_f[e] \leftarrow F_{u} \oplus F_e$ Matrix $A$ of size $M \times K$ in CSR format Matrix $E$ of size $M^2 \times K$ in CSR format (Incidence) Matrix $E^T$ is $K \times M^2$ in CSR format (Incidence) Feature matrix $V_f$ of size $M \times d$ Feature matrix $E_f$ of size $M^2 \times d$ Input operands: X (Edges), Y (Nodes) Output operand: Z (Any) Binary-operator: $\otimes$, Reduction-operator: $\oplus$ $F_e \leftarrow E_f[e]$ $F_u \leftarrow V_f[u]$ $V_f[u] \leftarrow F_u \oplus (F_e \otimes F_{u})$ $V_f[v] \leftarrow F_u \oplus (F_e \otimes F_{u})$ $E_f[e] \leftarrow F_e \oplus F_{u}$ Matrix A of size $M \times K$ in CSR format (Adjacency Matrix) Matrix E of size $M^2 \times V$ in CSR format (Incidence Matrix) Node feature matrix $V_f \times d$ Edge Feature matrix $E_f \times d$ Input operands: X, Y Output operand: Z Binary-operator: $\otimes$, Reduction-operator: $\oplus$ $T_1 \leftarrow$ {gather $F_e \in E_f$, $\forall e \in$ $E$} $T_2 \leftarrow \{F_e \in X, \forall e \in $ $A^T$\[$l$\].edges} $T_1 \leftarrow$ {scatter $F_r \in Y$ to $F_e$, $\forall e \in$ A\[$r$\].edges} $T_2 \leftarrow \{F_e \in Y, \forall e \in$ $A^T$\[$l$\].edges} $T \leftarrow T_1 \otimes T_2$ Copy-Reduce(A, T, Z, $\oplus$) $Z \leftarrow \mathsf{copy}(T)$ As can be seen, Algorithm \[algo:cropt\] is critical for the performance of both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{} operations. The algorithm is designed and optimized for small input matrices, usually occurring in applications that sample and batch the input graph for processing. However, the algorithm, right now, is not fully optimized for large input matrices, usually occurring in applications processing full graph in non-batched mode. Thus, for applications with full graph processing we make use of [mkl\_sparse\_?\_mm()]{} MKL matrix multiplication kernel. PyTorch Primitives {#sec:other_primitives} ================== We used PyTorch as the backend to execute [$\mathsf{DGL}$]{} and the neural network functions, e.g., Linear layer. Our application profiles indicated that a number of PyTorch primitives execute sub-optimally on the [$\mathsf{CPU}$]{}. Of these, [*BatchNorm1d*]{} and [*Embedding*]{} accounted for a significant amount of run-time in the Line Graph Neural Network ($\mathsf{LGNN}$) application. BatchNorm1d did not have an implementation within $\mathsf{MKLDNN}$ for PyTorch; therefore, we created an optimized version in a PyTorch extension by parallelizing across the samples and vectorizing across features per sample. The Embedding primitive in PyTorch is similar to Copy-Reduce in terms of operations: gather a set of feature vectors using index vectors and copy them into destination vectors in the Forward pass; scatter-reduce the gradients of Embedding weights in the Backward pass. Results {#sec:results} ======= In this section, we demonstrate the performance benefits of optimized aggregation and other primitives in various [$\mathsf{GNN}$]{} applications implemented in [$\mathsf{DGL}$]{}. Applications {#sec:applications} ------------ We analyzed and optimized seven applications that are implemented using [$\mathsf{DGL}$]{} and available within the [$\mathsf{DGL}$]{} Github repository <https://github.com/dmlc/dgl/>. We briefly discuss these applications. - GCN [@Kipf:2016tc] is a semi-supervised learning approach on graph-structured data that applies the notion of convolutions on graphs. In each layer, it applies linear transforms to regularized node features and normalizes them before aggregation. - GraphSAGE [@hamilton17nips] is a general inductive framework that uses node features to generate node embeddings for data unseen by the network. For each node $u$, it aggregates neighbor $v$ features $F_{v}$ and concatenates the aggregated $F_{v}$ to $F_{u}$ before applying a linear transform. - Relational GCN (R-GCN) [@kipf18rgcn] is a [$\mathsf{GNN}$]{} that applies the GCN framework to relational graphs. For each node $u$, it first aggregates linearly transformed neighbor feature $F_{v}$ [*under relation*]{} $r$ with $F_{u}$ and then aggregates them across all relations $r \in R$. - Line Graph Neural Network ($\mathsf{LGNN}$) [@chen2018supervised] is an instance of a [$\mathsf{GNN}$]{} that employs both node feature aggregation as well as edge-feature aggregation. Thus, there are two sequential aggregation steps that make this application particularly suitable for our optimization. - MoNet [@Monti_2017_CVPR] is a general framework for applying GCN to replace previous methods of learning on non-Euclidean spaces, such as Geodesic CNN and Anisotropic CNN. In the [$\mathsf{DGL}$]{} implementation, the core aggregation step is $\mathsf{u\_mul\_e\_add\_v}$ (transform node features multiplied by Gaussian weights on the edges) followed by a sum, mean of max operation on the resulting feature vectors. - Graph Convolutional Matrix Completion (GC-MC) [@berg2017graph] is a graph-based auto-encoder framework for matrix completion that uses GCN for recommender systems. In the [$\mathsf{DGL}$]{} implementation, the aggregation operation is $\mathsf{copy\_u(u, out)}$ followed by sum reduction. - Graph Attention Networks (GAT) leverage masked self-attentional layers by stacking layers in which nodes attend over their neighborhoods’ features. In this paper, we analyze GAT performance as applied to life-sciences applications such as molecules property prediction. Experimental Evaluation {#sec:exp} ----------------------- [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/graphsage_full_reddit.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/gcn_full.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/gat_full.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/gcmc_full.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/monet_full.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/lgnn_full.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/rgcn_hetero.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline on [$\mathsf{DGL}$]{} over seven different [$\mathsf{GNN}$]{} applications processing full graph in non-batched mode. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the run-time of all the remaining components. The performance numbers are averaged over 10 epochs, except for $\mathsf{LGNN}$, where we used 3 epochs. The datasets used for the experiments are mentioned in the charts. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}.[]{data-label="fig:dgl-0.4.3-cpu-performance"}](figures/graphsage_full_amzn.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline of GraphSAGE application with sampled graph processing on Amazon OGB-product and Reddit datasets. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the runtime of all the remaining components. The performance numbers are averaged over 10 epochs. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}. []{data-label="fig:dgl-0.4.3-cpu-performance-sampling"}](figures/graphsage_sampled_reddit.png "fig:"){width=".98\linewidth"} [.24]{} ![Training performance comparison of [$\mathsf{CPU}$]{} optimizations against [$\mathsf{CPU}$]{} baseline of GraphSAGE application with sampled graph processing on Amazon OGB-product and Reddit datasets. The speedup by optimized code is mentioned on the top of the optimized bar. The Misc. is the runtime of all the remaining components. The performance numbers are averaged over 10 epochs. Here, [$\mathsf{BR}$]{} primitive represents the time for both [$\mathsf{BR}$]{} and [$\mathsf{CR}$]{}. []{data-label="fig:dgl-0.4.3-cpu-performance-sampling"}](figures/graphsage_sampled_amzn.png "fig:"){width=".98\linewidth"} ### Experimental Setup {#sec:exp-setup} We performed all the experiments on Intel  Xeon  8280 [$\mathsf{CPU}$]{} @2.70GHz with 28 cores (single socket), equipped with 98 [GB]{} of memory per socket. The peak bandwidth to DRAM on this machine is 128 GB/s. We used gcc v7.1.0 compiler for compiling [$\mathsf{DGL}$]{} and the backend PyTorch neural network framework from source code. We used the latest release of [$\mathsf{DGL}$]{}v0.4.3 to demonstrate the performance enhancements due to our optimizations. We used Pytorch v1.6.0-rc1 as the backend for all our experiments. All the applications execute with default parameter settings. We used Pytorch autograd profiler to profile the applications. Datasets \#nodes \#edges \#features \#classes --------------------- ---------------------- ------------------------ ------------------------------ ----------------- Pubmed $\numprint{19717}$ $\numprint{44338}$ $\numprint{500}$ $\numprint{3}$ Reddit $\numprint{232965}$ $\numprint{11606919}$ $\numprint{602}$ $\numprint{41}$ Amazon OGB-Products $\numprint{2449029}$ $\numprint{123718280}$ $\numprint{100}$ $\numprint{47}$ BGS $\numprint{44333}$ $\numprint{227916}$ $\numprint{103}$ (Relations) $\numprint{2}$ : Benchmark graph dataset[]{data-label="tab:dataset"} Table \[tab:dataset\] shows the details of the datasets used in our experiments. Additionally, we used MovieLens-1M (ML-1M) dataset for GC-MC application and a synthetic dataset built using stochastic block model (SBM) for LGNN application. ML-1M is a benchmark dataset based on the user ratings for the movies; it consists of $ \numprint{6040}$ users, $\numprint{3706}$ movies, $ \numprint{1000209}$ ratings with rating levels $1,2,\ldots,5$. And, SBM is a synthetic dataset consists of random graph model with planted clusters. We used the default input parameters to generate the dataset. ### Performance Evaluation of [$\mathsf{DGL}$]{} We compared the performance of optimized [$\mathsf{DGL}$]{} against the baseline (i.e non-optimized) [$\mathsf{DGL}$]{}. We ran all the seven applications with non-batched (full graph) processing; moreover, we also experimented with GraphSAGE with batched graph processing (sampled) (Figure \[fig:dgl-0.4.3-cpu-performance\] and Figure \[fig:dgl-0.4.3-cpu-performance-sampling\]). We used the biggest of the benchmark datasets provided in the [$\mathsf{DGL}$]{} for these applications. For GraphSAGE, we also show performance results for a bigger dataset – the Amazon ogb-products dataset from <https://ogb.stanford.edu/docs/nodeprop/>. Overall, for applications with non-batched processing, we observed a speedup of $1.6\times - 12.8\times$ on per epoch time over the baseline [$\mathsf{DGL}$]{} on the CPU; specifically, we observe [$\mathsf{BR}$]{} speedup between $1.72\times - 34\times$ per epoch time compared to the [$\mathsf{DGL}$]{} baseline across the seven application (Figure \[fig:dgl-0.4.3-cpu-performance\]). Similarly, for GraphSAGE with batched processing, we see overall speedup of $1.5\times$-$1.7\times$ per epoch over [$\mathsf{DGL}$]{} baseline; specifically, we observe [$\mathsf{BR}$]{} speedup between $7.2\times - 10.6\times$ per epoch over [$\mathsf{DGL}$]{} baseline (Figure \[fig:dgl-0.4.3-cpu-performance-sampling\]). All our optimizations ensure the same accuracy as the baseline [$\mathsf{DGL}$]{}. Our optimizations of BatchNorm1d and Embedding PyTorch primitives (in LGNN application) resulted in $13\times$ and $76\times$ respectively. Together with these three optimized primitives optimized LGNN achieves $2\times$ speedup over baseline. The Misc. portion of the runtimes in Figure \[fig:dgl-0.4.3-cpu-performance\] is majorly contributed by other primitives – due to Pytorch framework – plus some [$\mathsf{DGL}$]{} framework overheads. These PyTorch primitives can be optimized on similar lines as 1D Native Batch Norm and Embedding primitives. Conclusions {#sec:conclusion} =========== Aggregation operations are critical to Graph Neural Network applications functionality. Via extensive application profiling and analysis of their implementations in the popular [$\mathsf{DGL}$]{}, we observed that aggregation primitives account for a majority of the run-time across applications. The Binary-Reduce abstraction in [$\mathsf{DGL}$]{} is the main aggregate operation. It is a memory-intensive operation with element-wise operations being the only compute; therefore, on CPU, the performance of this primitive is bound by the available memory-bandwidth. We optimized the sparse-dense matrix multiplication formulation of binary-reduce (and its special case, copy-reduce). We have demonstrated the benefits of the optimizations across a range of [$\mathsf{GNN}$]{} applications in [$\mathsf{DGL}$]{}.

--- abstract: 'This is a writeup of lectures on “statistics” that have evolved from the 2009 Hadron Collider Physics Summer School at CERN to the forthcoming 2018 school at Fermilab. The emphasis is on foundations, using simple examples to illustrate the points that are still debated in the professional statistics literature. The three main approaches to interval estimation (Neyman confidence, Bayesian, likelihood ratio) are discussed and compared in detail, with and without nuisance parameters. Hypothesis testing is discussed mainly from the frequentist point of view, with pointers to the Bayesian literature. Various foundational issues are emphasized, including the conditionality principle and the likelihood principle.' author: - | Robert D. Cousins[^1]\ Dept. of Physics and Astronomy\ University of California, Los Angeles\ Los Angeles, California 90095 bibliography: - 'statstheory.bib' date: 'August 4, 2018' title: | **Lectures on Statistics in Theory:\ Prelude to Statistics in Practice** --- Introduction ============ This is a writeup of slides that I first prepared for two hours of lectures at the Hadron Collider Physics Summer School (HCPSS) at CERN in 2009, and which eventually grew to about four hours of lectures within the CMS collaboration in summer of 2017. I will prepare a shortened version of the slides for lectures at the 2018 HCPSS at Fermilab. Unlike commonly available lectures by many of my colleagues on practical statistics for data analysis, my lectures focus on discussion of the foundational aspects, for which there is much less secondary literature written by physicists. I first got interested in the foundations of statistics in the late 1980’s, when I learned that deep issues of great importance to science, such as “Is there a statistically significant departure from expectations in my data?” were not at all settled. The issue is not merely, “How many sigma is a discovery?”, but rather, “Is the (equivalent) number of sigma even the right figure of merit for inferring the presence of a discovery?” The more I read about the Bayesian–frequentist debates in the primary statistics literature (which has the wonderful practice of including commentary and a rejoinder in many major papers), the more it became a “hobby” to browse this literature and (thanks to the PhyStat series organized by Louis Lyons and colleagues) to discuss these issues with preeminent statisticians and other interested physicists. The style here is rather terse, reflecting the origin in the slides [@cousins-slides], with each subsection drawing attention to a point of interest or controversy, including pointers to more literature. I do not assume much advanced statistics knowledge, but the reader will may well find the subject to be surprisingly [ *difficult*]{}. That is correct (!), and in fact is one of the main points of my lectures. Of course, a familiarity with examples of plots from various HEP analyses will be helpful. By concentrating on the “theoretical” underpinnings, I hope to provide the reader with [*what one must know in order to choose appropriate methods*]{} from the many possibilities. This includes the hope that by studying these topics, one will learn to avoid common pitfalls (and even silly statements) that can trip up professionals in the field. This is a dense writeup, and I do not expect one to pick it all up in a quick read-through. It should however be extremely useful to study the topics, referring to the references. I have also tried to put in enough sub-headings so that one can use it as a reference on specific topics. After some preliminaries, I begin with definitions and the Bayesian approach. That should help to understand what the frequentist approach (described next) is [*not*]{}! The frequentist discussion includes interval estimation, hypothesis testing, conditional frequentist estimation, and the much-debated issue of downward fluctuations in a search for an excess. After discussing likelihood-ratio intervals, I compare the three approaches, including major foundational issues such as the likelihood principle (Sections \[likelihoodprin\], \[intervalsummary\]). Finally I add nuisance parameters in the context of each of the approaches. I conclude with a word about current practice at the LHC. The appendices have a detailed worked example of a hypothesis test for two simple hypotheses; a brief discussion of the look-elsewhere effect; some further notes on Bayesian model selection; and some remarks on point estimation. All of these are are important topics that did not make it into my lectures. Preliminaries ============= Why foundations matter ---------------------- In the “final analysis”, we often make approximations, take a pragmatic approach, or follow a convention. To inform such actions, it is important to understand some foundational aspects of statistical inference. In Quantum Mechanics, we are used to the fact that for all of our practical work, one’s philosophical interpretation (e.g., of collapse of the wave function) does not matter. In statistical inference, however, [*foundational differences result in different answers*]{}: one cannot ignore them! The professional statistics community went through the topics of many of our discussions starting in the 1920’s, and revisited them in the resurgence of Bayesian methods in recent decades. I attempt to summarize some of the things that we should understand from that debate. [*Most importantly*]{}: One needs to understand both frequentist and Bayesian methods! Definitions are important ------------------------- As in physics, much confusion can be avoided by being precise about definitions, and much confusion can be generated by being imprecise, or (especially) by assuming every-day definitions in a technical context. By the end of these notes, you should see just as much confusion in these statements: 1. “The confidence level tells you how much confidence one has that the true value is in the confidence interval,” 2. “A noninformative prior probability density contains no information.” …As you have learned to see in the statement, “I did a lot or work today by carrying this big stone around the building and then putting it back in its original place.” Confusion is also possible because the statistics literature uses some words differently than does the HEP literature. A few examples are in Table \[jargon\], adapted from James [@james2006]. Here I tend to use words from both columns, with nearly exclusive use of the statisticians’ definition of “estimation”, as discussed below. Physicists say… Statisticians say… -------------------------- ---- -------------------- Determine, Measure Estimate Estimate (Informed) Guess Gaussian Normal Breit-Wigner, Lorentzian    Cauchy : Potential for confusion \[jargon\] Key tasks: Important to distinguish ----------------------------------- The most common tasks to be performed in statistical inference are typically classified as follows. [*Point estimation*]{}: What single “measured” value of a parameter do you report? While much is written about point estimation, in the end it is not clear what the criteria are for a “best” estimator. Decision Theory can be used to specify criteria and choose among point estimators. However, in HEP this is only implicitly done, and point estimation is usually not a contentious issue: typically the maximum-likelihood (point) estimator (MLE) serves our needs rather well, sometimes with a small correction for bias if desired. (See Appendix \[pointest\].) We generally put a “hat” (accent circumflex) over a variable to denote a point estimate, e.g., $\hat\mu$. [*Interval estimation*]{} : What interval (giving a measure of uncertainty of the parameter inference) do you report? This is crucial in HEP (and in introductory physics laboratory courses), and as discussed below, is deeply connected to frequentist hypothesis testing. In HEP it is fairly mandatory that there is a [ *confidence level*]{} that gives the [*frequentist coverage*]{} probability (Section \[sec-coverage\]) of a method, even if it is a Bayesian-inspired recipe. Point estimation and interval estimation can be approached consistently by insisting that the interval estimate contain the point estimate; in that case one can construct the point estimate by taking the limit of interval estimates as intervals get smaller (limit of confidence level going to zero). For many problems in HEP, there is reasonable hope of approximate reconciliation between Bayesian and frequentist methods for point and interval estimation, especially with large sample sizes. [*Hypothesis testing*]{}: There are many special cases, including a test of: [(a)]{} A given functional form (“model”) vs another functional form. Also known as “model selection”; [(b)]{} A single value of a parameter (say 0 or 1) vs all other values; [(c)]{} [*Goodness of Fit*]{}: A given functional form against all other (unspecified) functional forms (also known as “model checking”) Bayesian methods for hypothesis testing generally attempt to calculate the probability that a hypothesis is true. Frequentist methods cannot do this, and often lead to results expressed as $p$-values (Section \[pvalues\]). There is a large literature bashing $p$-values, but they are still deemed essential in HEP. [*Decision making*]{}: What action should I take (tell no one, issue press release, propose new experiment, …) based on the observed data? Decision making is rarely performed formally in HEP, but it is important to understand the outline of the formal theory, in order to avoid confusion with statistical inference that stops short of a decision, and to inform informal application. In frequentist statistics, the above hypothesis testing case (b) maps identically onto interval estimation. This is called the duality of “inversion of a hypothesis test to get confidence intervals”, and vice versa. I discuss this in more detail in Section \[duality\]. In contrast, in Bayesian statistics, hypothesis testing case (b) is an especially controversial form of case (a), model selection. The model with fixed value of the parameter is considered to be a lower-dimensional model in parameter space (one fewer parameter) than the model with parameter not fixed. I just mention this here to foreshadow a very deep issue. Because of the completely different structure of the approaches to testing, there be dramatic differences between frequentist and Bayesian hypothesis testing methods, with conclusions that apparently disagree, even in the limit of large data sets. Beware! See Appendix \[modelselection\] and my paper on the Jeffreys-Lindley paradox [@cousinsJL]. Probability =========== Definitions, Bayes’s theorem ---------------------------- Abstract mathematical probability $P$ can be defined in terms of sets and axioms that $P$ obeys, as outlined in Ref. [@cowan] (Chapter 1) and discussed in much more detail in Ref. [@persi] (Chapter 5). [*Conditional probabilities*]{} $P(B|A)$ (read “$P$ of $B$ given $A$”) and $P(A|B)$ are related by Bayes’s Theorem (or “Bayes’s Rule”): $$\label{eqn-bayes} P(B|A) = P(A|B) P(B) / P(A).$$ A cartoon illustration of conditional probabilities and a “derivation” of Bayes’s theorem is in Fig. \[bayes\_in\_pix\]. ![A cartoon explanation of probability, conditional probability, and Bayes’s Theorem, using picture arithmetic.[]{data-label="bayes_in_pix"}](figures/bayes_in_pix.png){width="95.00000%"} Two established incarnations of $P$ (still argued about, but to less practical effect) are: - [*Frequentist*]{} $P$: the limiting frequency in an ensemble of imagined repeated samples (as usually taught in Q.M.). $P(\textnormal{constant of nature})$ and $P(\textnormal{SUSY is true})$ do not exist (in a useful way) for this definition of $P$ (at least in one universe). - [*(Subjective) Bayesian $P$*]{}: subjective (personalistic) [ *degree of belief*]{} (as advocated by de Finetti [@definetti], Savage [@savagefoundations], and others). $P(\textnormal{constant of nature})$ and $P(\textnormal{SUSY is true})$ exist for You. (In the literature, “You” is often capitalized to emphasize the personalistic aspect.) This has been argued to be a basis for coherent personal decision-making (where “coherent” has a technical meaning). [*It is important to be able to work with either definition of $P$, and to know which one you are using!*]{} The descriptive word “Bayesian” applies to use of the above definition of probability as degree of belief! The adjective “Bayesian” thus also normally applies to the use of a probability density for any parameter (such as a constant of nature) whose true value is [*fixed but unknown*]{}, even in a context where the practitioner is not really aware of the degree-of-belief definition. In contrast, Bayes’s theorem applies to any definition of probability that obeys the axioms and for which the probabilities are defined in the relevant context! The distinction was noted by statistician Bradley Efron in his keynote talk at the 2003 PhyStat meeting at SLAC [@efronslac]: “Bayes’ rule is satisfying, convincing, and fun to use. But using Bayes’ rule does not make one a Bayesian; [*always*]{} using it does, and that’s where difficulties begin.” (Emphasis in original.) Clearly Bayes’s theorem has more applications for Bayesian $P$ than for frequentist $P$, since Bayesian $P$ can be used in more contexts. But one of the sillier things one sometimes sees in HEP is the use of a frequentist example of Bayes’s Theorem as a foundational argument for “Bayesian” statistics! Below I give a simple example for each definition of $P$. Example of Bayes’s theorem using frequentist $P$ {#btagsec} ------------------------------------------------ A b-tagging method is developed, and one measures: $P({\textnormal{b-tag}}| {\textnormal{b-jet}})$, i.e., efficiency for tagging b-jets $P({\textnormal{b-tag}}| {\textnormal{not~a}}~{\textnormal{b-jet}})$, i.e., efficiency for background $P({\textnormal{no~b-tag}}| {\textnormal{b-jet}}) = 1 - P({\textnormal{b-tag}}| {\textnormal{b-jet}})$ $P({\textnormal{no~b-tag}}| {\textnormal{not~a}}~{\textnormal{b-jet}}) = 1 - P({\textnormal{b-tag}}| {\textnormal{not~a}}~{\textnormal{b-jet}})$. [**Question:**]{} Given a selection of jets tagged as b-jets, what fraction of them is b-jets? I.e., what is $P({\textnormal{b-jet}}| {\textnormal{b-tag}})$ ? [**Answer:**]{} [*Cannot be determined from the given information!*]{} One needs in addition: $P({\textnormal{b-jet}})$, the true fraction of [*all*]{} jets that are b-jets. Then Bayes’s Theorem inverts the conditionality: $$P({\textnormal{b-jet}}| {\textnormal{b-tag}}) \propto P({\textnormal{b-tag}}|{\textnormal{b-jet}}) P({\textnormal{b-jet}}),$$ (where I suppress the normalization denominator). As noted, in HEP $P({\textnormal{b-tag}}| {\textnormal{b-jet}})$ is called the [*efficiency*]{} for tagging b-jets. Meanwhile $P({\textnormal{b-jet}}| {\textnormal{b-tag}})$ is often called the [ *purity*]{} of a sample of b-tagged jets. As this should be a conceptually easy distinction for experienced data analysts in HEP, it is helpful to keep it in mind when one encounters cases where it is perhaps tempting to make the logical error of equating $P(A|B)$ and $P(B|A)$. Example of Bayes’s theorem using Bayesian $P$ {#bayesbayes} --------------------------------------------- In a [*background-free*]{} experiment, a theorist uses a “model” to predict a signal with Poisson mean of 3.0 events. From the Poisson formula (Eqn. \[eqn-poisson\]) we know: $P(0~{\textnormal{events}}| {\textnormal{model}}~{\textnormal{true}}) = 3.0^0e^{-3.0}/0! = 0.05$ $P(0~{\textnormal{events}}| {\textnormal{model}}~{\textnormal{false}}) = 1.0$ $P(>0~{\textnormal{events}}| {\textnormal{model}}~{\textnormal{true}}) = 0.95$ $P(>0~{\textnormal{events}}| {\textnormal{model}}~{\textnormal{false}}) = 0.0$. The experiment is performed, and [*zero events are observed*]{}. [**Question:**]{} Given the result of the experiment, what is the probability that the model is true? I.e., What is $P({\textnormal{model}}~{\textnormal{true}}| 0~{\textnormal{events}})$? [**Answer:**]{} [*Cannot be determined from the given information!*]{} One needs in addition: $P({\textnormal{model}}~{\textnormal{true}})$, the degree of belief in the model prior to the experiment. Then Bayes’s Theorem inverts the conditionality: $$P({\textnormal{model}}~{\textnormal{true}}| 0~{\textnormal{events}}) \propto P(0~{\textnormal{events}}| {\textnormal{model}}~{\textnormal{true}}) P({\textnormal{model}}~{\textnormal{true}})$$ (again suppressing the normalization). It is instructive to apply Bayes’s Theorem in a little more detail, with the normalization. In Eqn. \[eqn-bayes\], let “$A$” correspond to “0 [[events]{.nodecor}]{}” and “$B$” correspond to “[[model]{.nodecor}]{} [[true]{.nodecor}]{}”. Similarly, with $P({\textnormal{not~}}B) = 1 - P(B)$, we can write a version of Bayes’s Theorem (replacing $B$ with “${\textnormal{not~}}B$”) as $$\label{bayesnotb} P({\textnormal{not~}}B|A) = P(A| {\textnormal{not~}}B) \times P({\textnormal{not~}}B) / P(A).$$ (As a check, we can add Eqns. \[eqn-bayes\] and \[bayesnotb\] and get unity, confirming that $P(A)$ is the correct normalization.) Solving Eqn. \[bayesnotb\] for $P(A)$ and substituting into Eqn. \[eqn-bayes\], and inserting numerical values from above, yields $P(B|A) = 0.05 P(B) / (1- 0.95 P(B))$, i.e., $$P({\textnormal{model}}~{\textnormal{true}}| 0~{\textnormal{events}}) = \frac{0.05 \times P({\textnormal{model}}~{\textnormal{true}})}{(1- 0.95 P({\textnormal{model}}~{\textnormal{true}}))}.$$ We can examine the limiting cases of strong prior belief in the model and very low prior belief. If we let the “model” be the Standard Model, then we could express our high prior belief as $P({\textnormal{model}}~{\textnormal{true}}) = 1 - \epsilon$, where $\epsilon \ll 1$. Plugging in gives, to lowest order, $$P({\textnormal{model}}~{\textnormal{true}}| 0~{\textnormal{events}}) \approx 1 - 20\epsilon.$$ This is still very high degree of belief in the SM. Unfortunately, one still finds (in the press and even among scientists) the fallacy that is analogous to people saying, “$P(0~{\textnormal{events}}| {\textnormal{model}}~{\textnormal{true}}) = 5$%, with 0 events observed, means there is 5% chance the SM is true.” (UGH!) In contrast, let the “model” be large extra dimensions, so that for a skeptic, the prior belief can be expressed as $P({\textnormal{model}}~{\textnormal{true}}) = \epsilon$, for some other small $\epsilon$. Then to lowest order we have, $$P({\textnormal{model}}~{\textnormal{true}}| 0~{\textnormal{events}}) \approx 0.05 \epsilon.$$ Low prior belief becomes even lower. More realistic examples are of course more complex. But this example is good preparation for avoiding misinterpretation of $p$-values in Section \[pvalues\]. A note re [*Decisions*]{} ------------------------- Suppose that as a result of the previous experiment, your degree of belief in the model is $P({\textnormal{model}}~{\textnormal{true}}| 0~{\textnormal{events}}) = 99$%, and you need to [*decide*]{} on an action (making a press release, or planning next experiment), based on the model being true. [**Question:**]{} What should you [*decide*]{}? [**Answer:**]{} [*Cannot be determined from the given information*]{}! One needs in addition: The [*utility*]{} function (or its negative, the [*loss*]{} function), which quantifies the relative costs (to You) of [*Type I error:*]{} declaring model false when it is true; and of [*Type II error:*]{} not declaring model false when it is false. Thus, Your [*decision*]{} requires two subjective inputs: Your prior probabilities, and the relative costs (or benefits) to You of outcomes. Statisticians often focus on decision-making. In HEP, the tradition thus far is to communicate experimental results (well) short of formal decision calculations. It is important to realize that frequentist (classical) “hypothesis testing” as discussed in Section \[hypotest\] below (especially with conventions like 95% [[C.L.]{.nodecor}]{} or 5$\sigma$) is [*not*]{} a complete theory of decision-making! One must always keep this in mind, since the traditional “accept/reject” language of frequentist hypothesis testing is too simplistic for “deciding”. Aside: What is the “whole space”? {#aside} --------------------------------- For probabilities to be well-defined, the “whole space” needs to be defined. This can be difficult or impossible for both frequentists and Bayesians. For frequentists, specification of the whole space may require listing the experimental protocol in detail, including the experimenters’ reaction to potentially unexpected results that did not occur! For Bayesians, normalization of probabilities of hypotheses requires enumerating all possible hypotheses and assigning degree of belief to them, including hypotheses not yet formulated! Thus, the “whole space” itself is more properly thought of as a conditional space, conditional on the assumptions going into the model (Poisson process, whether or not total number of events was fixed, etc.), and simplifying assumptions or approximations. Furthermore, it is widely accepted that restricting the “whole space” to a relevant (“conditional”) subspace can sometimes improve the quality of statistical inference. The important topic of such “conditioning” in frequentist inference is discussed in detail in Section \[conditioning\]. In general I do not clutter the notation with explicit mention of the assumptions defining the “whole space”, but some prefer to do so. In any case, it is important to keep them in mind, and to be aware of their effect on the results obtained. Probability, probability density, likelihood {#probpdf} ============================================ These are key building blocks in both frequentist and Bayesian statistics, and it is crucial distinguish among them. In the following we let $x$ be an observed quantity; sometimes we use $n$ if the observation is integer-valued and we want to emphasize that (to aging Fortran programmers). A [*“(statistical) model”*]{} is an equation specifying probabilities or probability densities for observing $x$. We use $\mu$ for parameters (sometimes vector-valued) in the model. In Bayesian statistics, the parameters themselves are considered to be “random variables”. The notation for such a general model is $p(x|\mu)$, where the vertical line (read “given”) means conditional probability, conditional on a particular value of $\mu$. (In the statistical literature, $\theta$ is more common.) In frequentist statistics, typically the dependence on $\mu$ is not a proper conditional probability, and thus many experts advocate using notation with a semi-colon: $p(x\/;\mu)$. The modern text by George Casella and Roger Berger  [@casellabergerbook] (p. 86) however uses the vertical line for “given” in the context where the parameter is not a random variable being conditioned on. I do not know of any examples where this causes trouble, so at the risk of offending some, I use a vertical line throughout for both Bayesian and frequentist models. Then the most common examples in HEP are: - Binomial probability for ${n_\textnormal{\scriptsize on}}$ successes out of ${n_\textnormal{\scriptsize tot}}$ trials (Section \[sec-binomial\]): $$\label{eqn-binomial} {\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}},{\rho}) = \frac{{n_\textnormal{\scriptsize tot}}!}{{n_\textnormal{\scriptsize on}}!({n_\textnormal{\scriptsize tot}}-{n_\textnormal{\scriptsize on}})!}\, {\rho}^{{n_\textnormal{\scriptsize on}}}\,(1-{\rho})^{({n_\textnormal{\scriptsize tot}}-{n_\textnormal{\scriptsize on}})}$$ - Poisson probability for $n$ events to be observed: $$\label{eqn-poisson} P(n|\mu) = \frac{\mu^n \textnormal{e}^{-\mu}}{n!}$$ - Gaussian probability [*density*]{} function (pdf): $$\label{eqn-gaussian} p(x| \mu,\sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \textnormal{e}^{-(x-\mu)^2/2\sigma^2},$$ so that $p(x|\mu,\sigma)dx$ is a differential of probability $dP$. (A typical course in statistical physics shows how the latter two can be viewed as limiting cases of the first.) The binomial and Poisson formulas are sometimes called probability [*mass*]{} functions in the statistics literature. In the Poisson case, suppose that $n=3$ is observed. Substituting this [*observed value*]{} $n=3$ into $P(n|\mu)$ yields the [*likelihood function*]{}, ${{\cal L}}(\mu) = \mu^3 \exp(-\mu)/3!$, plotted in Fig. \[pois\_likli\_3obs\]. ![The likelihood function ${{\cal L}}(\mu)$ after observing $n=3$ in the Poisson model of Eqn. \[eqn-poisson\]. The likelihood obtains its maximum (ML) at $\mu=3$.[]{data-label="pois_likli_3obs"}](figures/pois_likli_3obs.png){width="49.00000%"} It is tempting to consider the area under ${{\cal L}}$ as meaningful, but ${{\cal L}}(\mu)$ is [*not*]{} a probability density in $\mu$. The area under ${{\cal L}}$ (or parts of it) is meaningless! The Poisson example makes this particularly clear, since the definition of ${{\cal L}}(\mu)$ starts with a probability (not a probability density); it makes no sense to multiply ${{\cal L}}(\mu)$ by $d\mu$ and integrate. As we shall see, likelihood [*ratios*]{} ${{\cal L}}(\mu_1)/{{\cal L}}(\mu_2)$ are useful and frequently used. Change of observable variable (“metric”) $x$ in pdf $p(x|\mu)$ {#metricchange} -------------------------------------------------------------- For pdf $p(x|\mu)$ and a 1-to-1 change of variable (metric) from (vector) $x$ to (vector) $y(x)$, the volume element is modified by the Jacobian. For a 1D function $y(x)$, we have $p(y) |dy| = p(x) |dx|$, so that $$\label{jacobian} p(y(x)|\mu) = p(x|\mu)\ /\ |dy/dx|.$$ The Jacobian modifies the probability density in such a way to guarantee that $$P( y(x_1)< y < y(x_2) ) = P(x_1 < x < x_2 ),$$ (or equivalent with decreasing $y(x)$). That is, the Jacobian guarantees that [*probabilities are invariant under change of variable*]{} $x$. Because of the Jacobian in Eqn. \[jacobian\], the [*mode*]{} of probability [*density*]{} is in general [*not*]{} invariant under change of metric. That is, the value of $y$ for which $p(y(x))$ is a maximum is not trivially related to the value of $x$ for which $p(x)$ is a maximum. Another consequence of the Jacobian in Eqn. \[jacobian\] is that the likelihood function ${{\cal L}}(\mu)$ differs for different choices of the data variable. However, likelihood [*ratios*]{} such as ${{\cal L}}(\mu_1) /{{\cal L}}(\mu_2)$ [*are*]{} invariant under change of variable $x$ to $y(x)$. The Jacobian in the denominator cancels that in the numerator. Thus, for example, the value of $\mu$ that maximizes ${{\cal L}}(\mu)$ will be independent of the choice of data variable, but the value of ${{\cal L}}(\mu)$ at that maximum is different. The latter point also explains why the value of ${{\cal L}}(\mu)$ at its maximum is not an appropriate test statistic for goodness of it; useful tests based on the likelihood function use likelihood [ *ratios*]{} [@cousinsgoodness]. Change of parameter $\mu$ in pdf $p(x|\mu)$ {#invariantl} ------------------------------------------- The pdf for $x$ given parameter $\mu=3$ is the [*same*]{} as the pdf for $x$ given $1/\mu=1/3$, or given $\mu^2=9$, or given any specified function of $\mu$. They all imply the same $\mu$, and hence the same pdf for $x$. In slightly confusing notation, that is what we mean by changing the parameter from $\mu$ to $f(\mu)$, and saying that $$p(x|f(\mu)) = p(x|\mu).$$ Inserting an observed value of $x$, we see the important result that the likelihood ${{\cal L}}(\mu)$ is [*invariant*]{} (!) under reparameterization from parameter $\mu$ to $f(\mu)$: ${{\cal L}}( f(\mu) ) = {{\cal L}}(\mu)$. The absence of a Jacobian reinforces the fact that ${{\cal L}}(\mu)$ is [*not*]{} a pdf in $\mu$. Furthermore, the mode of ${{\cal L}}(\mu)$ is thus also invariant: if we let $\hat\mu$ be the value of $\mu$ for which ${{\cal L}}(\mu)$ is a maximum, and if we let $\hat f$ be the value of $f(\mu)$ for which ${{\cal L}}(f)$ is a maximum, then $$\hat f = f(\hat\mu).$$ This is an important property of the popular ML technique of point estimation. (See discussion in Ref. [@barlow], Section 5.3.1.) Probability integral transform ------------------------------ In 1938 Egon Pearson commented on a paper of his father Karl (of $\chi^2$ fame), noting that the [*probability integral transform*]{} “…seems likely to be one of the most fruitful conceptions introduced into statistical theory in the last few years” [@pearson1938]. Indeed this simple transform makes many issues more clear (or trivial). Given continuous $x \in (a,b)$, and its pdf $p(x)$, one transforms to $y$ given by $$y(x) = \int_a^x p(x^\prime) dx^\prime.$$ Then trivially $y \in (0,1)$, and with the chain rule and the fundamental theorem of calculus, it follows that the pdf for $y$ is $p(y) = 1$ (uniform) for all $y$ ! (If $x$ is discrete, there are complications.) [*So for continuous variables, there always exists a metric y in which the pdf is uniform*]{}. (As an aside, the inverse transformation can provide for efficient Monte Carlo (MC) generation of $p(x)$ starting from a pseudo-random number generator uniform on (0,1). See Section 40.2 in Ref. [@pdg2018].) Looking ahead to Section \[bayesintro\], I mention here that the specification of a Bayesian prior pdf $p(\mu)$ for parameter $\mu$ is thus equivalent to the choice of the metric $g(\mu)$ in which the pdf is uniform. This is a deep issue, not always recognized by users of uniform prior pdf’s in HEP! Bayes’s theorem generalized to probability [*densities*]{} ---------------------------------------------------------- Recall (Eqn. \[eqn-bayes\]) that $P(B|A) \propto P(A|B) P(B)$. For Bayesian $P$, continuous parameters such as $\mu$ are random variables with pdf’s. Let pdf $p(\mu|x)$ be the conditional pdf for parameter $\mu$, given data $x$. As usual $p(x|\mu)$ is the conditional pdf for data $x$, given parameter $\mu$. Then Bayes’s Theorem becomes $p(\mu|x) \propto p(x|\mu) p(\mu)$. Substituting in a particular set of observed data $x_0$, we have $p(\mu|x_0) \propto p(x_0|\mu) p(\mu)$. Recognizing the likelihood (variously written as ${{\cal L}}(x_0|\mu)$, ${{\cal L}}(\mu)$, or unfortunately even ${{\cal L}}(\mu|x_0)$), then $$\boxed{ p(\mu|x_0) \propto {{\cal L}}(x_0|\mu) p(\mu),}$$ where: $p(\mu|x_0) =$ the [*posterior pdf*]{} for $\mu$, given the results of this experiment, ${{\cal L}}(x_0|\mu) =$ the likelihood function of $\mu$ from this experiment, $p(\mu) =$ the [*prior pdf*]{} for $\mu$, before applying the results of this experiment. Note that there is one (and only one) probability [*density in*]{} $\mu$ on each side of the equation, consistent with ${{\cal L}}(x_0|\mu)$ [*not*]{} being a density in $\mu$! (Aside: occasionally someone in HEP refers to the prior pdf as “[*a priori*]{}”. This is incorrect, as is obvious when one considers that the posterior pdf from one experiment can serve as the prior pdf for the next experiment.) Bayesian analysis {#bayesintro} ================= All equations up until now are true for [*any*]{} definition of probability $P$ that obeys the axioms, including frequentist $P$, as long as the probabilities exist. For example, if $\mu$ is sampled from an ensemble with known “prior” pdf, then it has a frequentist interpretation. (This is however unusual.) The word “Bayesian” refers [*not*]{} to these equations, but to the choice of definition of $P$ as [*personal subjective degree of belief*]{}. For example, if $\mu$ is a constant of nature, its Bayesian pdf expresses one’s relative belief in different values. Bayesian $P$ applies to hypotheses and constants of nature (while frequentist $P$ does not), so there are many Bayesian-only applications of Bayes’s Theorem. Bayesian analysis is based on the Bayesian posterior pdf $p(\mu|x_0)$, as sketched here. - [*Point estimation*]{}: Some Bayesians use the posterior mode (maximum posterior density) as the point estimate of $\mu$. This has the problem that it is metric dependent: the point estimate of the mean lifetime $\tau$ will not be the inverse of the point estimate of the decay rate $\Gamma$. This is because the Jacobian moves the mode around under change of parameter from lifetime $\tau$ to decay rate $\Gamma=1/\tau$. (Recall Section \[metricchange\].) The posterior median can be used in 1D, and is metric-independent. There are also Bayesians (as well as frequentists) who think that emphasis on point estimation is misguided. - [*Interval estimation*]{}: The [*credibility*]{} of $\mu$ being in any interval $[\mu_1,\mu_2]$ can be calculated by integrating $p(\mu|x_0)$ over the interval. For reporting a default interval with, say, 68% credibility, one needs in addition a convention for which 68% to use (lower, upper, or central quantiles are common choices). It is preferable to refer to such intervals as “[ *credible intervals*]{}”, as opposed to “confidence intervals”, unless the Bayesian machinery is used just as a technical device to obtain valid (at least approximate) frequentist confidence intervals (as is often the case in HEP). - [*Hypothesis testing*]{}: Unlike frequentist statistics, testing credibility of whether or not $\mu$ equals a particular value $\mu_0$ is [*not*]{} performed by examining interval estimates (at least assuming a regular posterior pdf). One starts over with Bayesian model selection, as discussed in Appendix \[modelselection\]. (Dirac $\delta$-functions in the prior and posterior pdfs can however connect interval estimation to model selection, with its issues.) - [*Decision making*]{}: All [*decisions*]{} about $\mu$ require not only $p(\mu|x_0)$ but also further input: the utility function (or it negative, the loss function). See, e.g., Ref. [@james2006] (Chapter 6) and Ref. [@bergerdecision1985] (Chapter 2). Since Bayesian analysis [*requires*]{} a prior pdf, big issues in Bayesian estimation include: - What prior pdf to use, and how sensitive is the result to the prior? - How to interpret posterior probability if the prior pdf is not Your personal subjective belief? [*Frequentist tools can be highly relevant to both questions!*]{} Can “subjective” be taken out of “degree of belief”? ---------------------------------------------------- There are compelling arguments (Savage [@savagefoundations], De Finetti [@definetti] and others) that Bayesian reasoning with [ *personal subjective*]{} $P$ is the uniquely “coherent” way (with technical definition of coherent) of updating [*personal*]{} beliefs upon obtaining new data and making decisions based on them. (These foundational works are very heavy going. For a more accessible review by an outspoken subjectivist, with more complete references, see Lindley’s 2000 review [@lindley2000]. A 2018 book by Diaconis and Skyrms [@persi] is also very detailed and at a deep level that seems mostly comprehensible to physicists.) A huge question is: [*Can the Bayesian formalism be used by scientists to report the results of their experiments in an “objective” way (however one defines “objective”), and does any of the glow of coherence remain when subjective $P$ is replaced by something else?*]{} An idea vigorously pursued by physicist Harold Jeffreys [@jeffreys1961] in the mid-20th century is: [*Can one define a prior $p(\mu)$ that contains as little information as possible?*]{} The really [*really*]{} thoughtless idea (despite having a fancy name, “Laplace’s Principle of Insufficient Reason”), recognized by Jeffreys as such, but dismayingly common in HEP is: just choose prior $p(\mu)$ uniform in whatever metric you happen to be using! “Uniform prior” requires a choice of metric ------------------------------------------- Recall that the probability integral transform [*always*]{} allows one to find a metric in which $p$ is uniform (for continuous $\mu$). Thus the question, “What is the prior pdf $p(\mu)$?” is equivalent to the question, “For what function $g(\mu)$ is $p(g)$ uniform?” There is usually [*no reason*]{} to choose $g$ arbitrarily as $g(\mu) = \mu$ (!). Jeffreys’s choice of metric in which prior is uniform {#jeffprior} ----------------------------------------------------- The modern foundation of the vast literature on prior pdfs that one may hope (in vain) to be uninformative is the monograph by Harold Jeffreys [@jeffreys1961]. He proposes more than one approach, but the one that is commonly referred to as “Jeffreys Prior” (and considered the default “noninformative” prior by statisticians for estimation in 1-parameter problems) is [*derived from the statistical model*]{} $p(x|\mu)$. [*This means that the prior pdf depends on the measurement apparatus!*]{} For example, if the measurement apparatus has a resolution function that is Gaussian for mass $m$ (with $\sigma$ independent of $m$), then the Jeffreys prior pdf $p(m)$ for the mass is uniform in $m$. If a different measurement apparatus has a resolution function that is Gaussian for $m^2$, then the Jeffreys prior pdf $p(m^2)$ is uniform in $m^2$. In the latter case, by the rules of probability (Eqn. \[jacobian\]), the prior pdf $p(m)$ is not uniform, but rather proportional to $m$ (!). Jeffreys’s derivation of his eponymous prior is based on the idea that the prior should be uniform in a metric related to the Fisher information, calculated from curvature of the log-likelihood function averaged over sample space. Some examples are: Poisson signal mean $\mu$, no background: $p(\mu) = 1/\sqrt{\mu}$ Poisson signal mean $\mu$, mean background $b$: $p(\mu) = 1/\sqrt{\mu+b}$ Mean $\mu$ of Gaussian with fixed $\sigma$ (unbounded or bounded $\mu$): $p(\mu) = 1$ rms deviation $\sigma$ of a Gaussian when mean fixed: $p(\sigma) = 1/\sigma$ Binomial parameter $\rho$, $0 \le \rho \le 1$ : $p(\rho) = \rho^{-1/2}(1 - \rho)^{-1/2} = \textnormal{Beta}(1/2,1/2)$ If parameter $\mu$ is changed to $f(\mu)$, the recipe for obtaining the Jeffreys prior for $f(\mu)$ yields a different-looking prior that corresponds to the [*same choice of uniform metric*]{}. So if you use Jeffreys’s recipe to obtain a prior pdf for $\mu$, and your friend uses Jeffreys’s recipe to obtain a prior pdf for $f(\mu)$, then those pdfs will be correctly related by the appropriate Jacobian. (This is not true for some other rules, in particular if each of you takes a uniform prior in the metric you are using.) Thus [*probabilities*]{} (integrals of pdfs over equivalent endpoints) using Jeffreys prior are invariant under choices of different parameterizations. For a detailed modern review of Jeffreys’s entire book, including his prior, with discussion by six prominent statisticians (including outspoken subjectivist Lindley), and rejoinder, see Ref. [@robert2009]. ### Reference priors of Bernardo and Berger As Jeffreys noted, his recipe encounters difficulties with models having more than one parameter. José Bernardo [@bernardoRSS1979] and J.O. Berger [@bergerbern89; @bergerbern92] advocate an approach that they argue works well in higher dimensions, with the crucial observation one must choose an ordering of the parameters in order to well-define the multi-dimensional prior pdf. Their so-called “Reference prior” reduces to the Jeffreys prior in 1D, with a different rationale, namely the prior that leads to a posterior pdf that is most dominated by the likelihood. There are many subtleties. Beware! See also Bernardo’s talk at PhyStat-2011, which includes hypothesis testing, and discussion following [@bernardo2011phystat]. Demortier, Jain, and Prosper pioneered the use of Bernardo/Berger reference priors in a case of interest in HEP [@demortier2010], with about 30 citations thus far at inspirehep.net. What to call such non-subjective priors? ---------------------------------------- - “[*Non*]{}informative priors”? Traditional among statisticians, even though [*they know it is a misnomer*]{}. (You should too!) - “Vague priors”? - “Ignorance priors”? - “Default priors”? - “Reference priors”? (Unfortunately also refers to the specific recipe of Bernardo and Berger) - “Objective priors”? Despite the highly questionable use of the word, Jeffreys prior and its generalization by Bernardo and Berger are now widely referred to as “objective priors”. Kass and Wasserman [@kasswasserman] give the best (neutral) name in my opinion: Priors selected by “formal rules”. Their article is required reading for anyone using Bayesian methods! Whatever the name, the prior pdf in one metric determines it in all other metrics: be careful in the choice of metric in which it is uniform! For one-parameter models, the “Jeffreys Prior” is the most common choice among statisticians for a non-subjective “default” prior—so common that statisticians can be referring to the Jeffreys prior when they say “flat prior” (e.g., D.R. Cox in discussion at PhyStat 2005 [@coxdiscussion], p. 297). A key point: priors such as the Jeffreys Prior are based on the likelihood function and thus inherently derived from the [ *measurement apparatus and procedure*]{}, not from thinking about the parameter! This may seem strange, but does give advantages, particularly for frequentist (!) coverage (Section \[sec-coverage\]), as mentioned in Section \[pragmatism\] and emphasized to us by Jim Berger at the Confidence Limits Workshop at Fermilab in 2000 [@bergerclk]. Whatever you call them, non-subjective priors [*cannot*]{} represent ignorance! ------------------------------------------------------------------------------- Although some authors have claimed that various invariance principles can be invoked to yield priors that represent complete “ignorance”, I do not know of any modern statistician who thinks that this is possible. On the contrary, (subjectivist) Dennis Lindley wrote, [@lindley1990], “the mistake is to think of them \[Jeffreys priors or Bernardo/Berger’s reference priors\] as representing ignorance.” This Lindley quote is emphasized in the monograph by prominent Bayesian Christian Robert [@robert2007] (p. 29). Objectivist Jose Bernardo says, regarding his reference priors, “\[With non-subjective priors,\] The contribution of the data in constructing the posterior of interest should be ‘dominant’. Note that this does not mean that a non-subjective prior is a mathematical description of ‘ignorance’. Any prior reflects some form of knowledge.” Nonetheless, Berger [@bergerdecision1985] (p. 90) argues that Bayesian analysis with noninformative priors (older name for objective priors), such as those of Jeffreys and Bernardo/Berger, “[*is the single most powerful method of statistical analysis*]{}, in the sense of being the [*ad hoc*]{} method most likely to yield a sensible answer for a given investment of effort.” \[emphasis in original\]. Priors in high dimensions ------------------------- Is there a sort of informational “phase space” that can lead us to a sort of probability Dalitz plot? I.e., the desire is that structure in the posterior pdf represents information in the data, not the effect of Jacobians in the priors. This is a [*notoriously hard problem!*]{} Be careful: Uniform priors push the probability away from the origin to the boundary! (The volume element goes as $r^2dr$.) The state of the art for “objective” priors may be the “reference priors” of Bernardo and Berger, but multi-D tools have been lacking. Subjective priors are also very difficult to construct in high dimensions: human intuition is poor. Subjective Bayesian Michael Goldstein [@goldsteinphystat] told us in 2002 at Durham, “…meaningful prior specification of beliefs in probabilistic form over very large possibility spaces is very difficult and may lead to a lot of arbitrariness in the specification …”. Bradley Efron at PhyStat-2003 [@efronslac] concluded: “Perhaps the most important general lesson is that the facile use of what appear to be uninformative priors is a dangerous practice in high dimensions.” Sir David Cox [@cox2006], p. 46: “With multi-dimensional parameters…naive use of flat priors can lead to procedures that are very poor from all perspectives…”. Also p. 83: “…the notion of a flat or indifferent prior in a many-dimensional problem is untenable.” Types of Bayesians {#fivefaces} ------------------ The broad distinction between the subjective and objective Bayesians is far from the complete story. At PhyStat-LHC in 2007, Sir David Cox described “Five faces of Bayesian statistics” [@coxphystat]: - empirical Bayes: number of similar parameters with a frequency distribution - neutral (reference) priors: Laplace, Jeffreys, Jaynes, Berger and Bernardo - information-inserting priors (evidence-based) - personalistic priors - technical device for generating frequentist inference Currently in HEP, the main application is the last “face” on his list: we typically desire good frequentist properties for point or interval estimation when using a nominally “Bayesian” recipe. In particular, for [*upper limits*]{} on a Poisson mean, we use a uniform prior (i.e., not what objective statisticians recommend), for [ *frequentist*]{} reasons. (See Section \[intervalsummary\] and Ref. [@cousinsajp1995].) Cox’s third “face” also arises in HEP when likelihoods are incorporated from subsidiary measurements, and are used to provide evidence-based priors. Unfortunately, some people in HEP have in effect also added to Cox’s list a 6th “face”: - Priors uniform in arbitrary variables, or in “the parameter of interest”. I know of no justification for this in modern subjective or objective Bayesian theory. It is an “ignorance” prior only in the sense that it betrays ignorance of the modern Bayesian literature! Do “Bayesians” care about frequentist properties of their results? ------------------------------------------------------------------ Another claim that is dismaying to see among some physicists is the blanket statement that “Bayesians do not care about frequentist properties”. While that may be true for pure subjective Bayesians, most of the pragmatic Bayesians that we have met at PhyStat meetings do use frequentist methods to help calibrate Bayesians statements. That seems to be essential when using “objective” priors to obtain results that are used to communicate inferences from experiments. A variety of opinions on this topic are in the published comments in the inaugural issue of the journal [*Bayesian Analysis*]{}, following the papers by Jim Berger [@berger2006] and Michael Goldstein [@goldstein2006] that advocate the objective and subjective points of view, respectively. In particular, Robert Kass [@kass2006] provided a list of nine questions for which answers “may be used to classify various kinds of Bayesians”. The first question is, “Is it important for Bayesian inferences to have good frequentist operating characteristics?” The questions, the explicit answers from Kass, Berger, and Goldstein, and other commentary (from which answers can be gleaned) from other prominent statisticians, are part of my list of “required reading” for physicists. They should go a long way toward broadening the view of any physicists who have swallowed some of the extreme polemics about Bayesian analysis (as found for example in Jaynes [@jaynes2003], which unfortunately seems to have been read uncritically by too many scientists). Analysis of sensitivity to the prior ------------------------------------ Since a Bayesian result depends on the prior probabilities, which are either personalistic or with elements of arbitrariness, it is widely recommended by Bayesian statisticians to study the [*sensitivity*]{} of the result to varying the prior. - “Objective” Bayesian José Bernardo, quoted in Ref. [@bernardodialog]: “Non-subjective Bayesian analysis is just a part—an important part, I believe —of a healthy [*sensitivity analysis*]{} to the prior choice…”. - “Subjective” Bayesian Michael Goldstein, from the Proceedings [@goldsteinphystat]: “…Again, different individuals may react differently, and the sensitivity analysis for the effect of the prior on the posterior is the analysis of the scientific community…” In his transparencies at the conference, he put it simply: “Sensitivity Analysis is at the heart of scientific Bayesianism.” I think that too little emphasis to this important point is given by Bayesian advocates in HEP. Bayesian must-read list for HEP/Astro/Cosmo (including discussion!) ------------------------------------------------------------------- In my experience, some high energy physicists and astrophysicists appear to be overly influenced by the polemical book by E.T. Jaynes [@jaynes2003], (which, for example, argues for the existence priors representing ignorance). I strongly urge anyone diving into Bayesian statistics to read as well the following [ *minimal*]{} set of papers by Bayesian subjectivists and objectivists, and the associated discussion, and rejoinders. (If there is one thing that HEP journals could learn from statisticians, it is to publish such discussion and rejoinder accompanying major papers and reviews!) As I made this list over ten years ago, I would welcome suggestions for more recent additions. Robert E. Kass and Larry Wasserman, “The Selection of Prior Distributions by Formal Rules,” [@kasswasserman] Telba Z. Irony and Nozer D. Singpurwalla, “Non-informative priors do not exist: A dialogue with Jose M. Bernardo,” [@bernardodialog] James Berger, “The Case for Objective Bayesian Analysis,” [@berger2006] Michael Goldstein, “Subjective Bayesian Analysis: Principles and Practice,” [@goldstein2006] J.O. Berger and L.R. Pericchi, “Objective Bayesian Methods for Model Selection: Introduction and Comparison,” [@bergerpericchi2001] Pseudo-Bayesian analyses {#pseudobayes} ------------------------ Jim Berger in 2006 [@berger2006]: “One of the mysteries of modern Bayesianism is the lip service that is often paid to subjective Bayesian analysis as opposed to objective Bayesian analysis, but then the practical analysis actually uses a very adhoc version of objective Bayes, including use of constant priors, vague proper priors, choosing priors to ‘span’ the range of the likelihood, and choosing priors with tuning parameters that are adjusted until the answer ‘looks nice.’ I call such analyses [ *pseudo-Bayes*]{} because, while they utilize Bayesian machinery, they do not carry with them any of the guarantees of good performance that come with either true subjective analysis (with a very extensive elicitation effort) or (well-studied) objective Bayesian analysis…I do not mean to discourage this approach. It simply must be realized that pseudo-Bayes techniques do not carry the guarantees of proper subjective or objective Bayesian analysis, and hence must be validated by some other route.” Berger goes on to give examples of pseudo-Bayes analyses, with the first being (what else?), “Use of the constant prior density”. Pseudo-Bayes analyses pop up from time to time in HEP, for example by those using priors “uniform in the parameter of interest”. Here I mention three examples of “pseudo-Bayes” in HEP that have been criticized by me and others. I previously discussed them at the PhyStat-nu workshop in Tokyo in 2016 [@cousinstokyo] (slides 62–68). The first example was exposed in Luc Demortier’s talk in 2002 at Durham [@demortierdurham]. In a method of Bayesian upper limit calculation that was common at the Tevatron at the time, the use of a uniform prior for a Poisson mean, along with a Gaussian truncated at the origin for a systematic uncertainty in efficiency, led to an integral that Luc showed “by hand” to diverge! The integral was typically evaluated numerically, without first checking that it exists. The answer thus depended on the choice of cutoff that was used in the numerical evaluation. Alternatives to the truncated Gaussian prior are mentioned in Section \[nuisprior\]. The second example comes from Joel Heinrich at the Oxford PhyStat in 2005 [@heinrich2005]. It had been known a long time that a uniform prior for a Poisson mean of a [*signal*]{} yields good frequentist properties for [*upper limits*]{} (but not lower limits). (See Section \[intervalsummary\].) Joel showed the dangers of naively using uniform prior for the means of several background processes. The third example is in the category that I find has some of the worst pseudo-Bayes examples, namely Bayesian model selection (Appendix \[modelselection\]), which is however not attempted as often as Bayesian estimation in HEP. Practitioners are sometimes unaware that: 1. In model selection, unlike estimation, the dependence on some prior pdfs for parameters does [*not*]{} become negligible as the amount of data increases without bound (even for so-called Bayes factors that attempt to separate out the prior probabilities of the hypotheses). 2. Improper priors (such as uniform over a line or half-line) are a disaster, and if they are made proper by adding a cutoff, then the model selection answer is directly proportional to the (often arbitrary) cutoff. Using “1” for the prior just hides the problem. 3. In fact, Jeffreys and followers use priors for model selection that are different from those used for estimation (!). Thus, Bayesian model selection should not be approached naively. For an example in PRL that I criticized, see my Comment in Ref. [@cousins2008]. Harrison Prosper (an early and sustained advocate of Bayesian methods in HEP) has provided an excellent discussion of the care needed in Bayesian analyses in Chapter 12 of [*Data Analysis in High Energy Physics*]{}, edited by O. Behnke et al. [@behnke2013]. Frequentist estimation: confidence intervals {#secconfidence} ============================================ What can be computed without using a prior, with only the frequentist definition of $P$? - [*Not*]{} $P(\textnormal{constant of nature is in some specific interval} \,|\, \textnormal{data}) $ - [*Not*]{} $P(\textnormal{SUSY is true} \,|\, \textnormal{data}) $ - [*Not*]{} $P(\textnormal{SM is false} \,|\, \textnormal{data}) $ Rather: 1. [*Confidence Intervals*]{} for constants of nature or other parameter values, as defined in the 1930’s by Jerzy Neyman. Statements are made about probabilities in [*ensembles*]{} of intervals (fraction containing unknown true value). Confidence intervals have further applications in frequentist hypothesis testing. 2. Likelihoods and thus [*likelihood ratios*]{}, the basis for a large set of techniques for point estimation, interval estimation, and hypothesis testing. Both can be constructed using the [*frequentist*]{} definition of $P$. In this section, we introduce confidence intervals, and in Section \[likelihood\], introduce likelihood ratios for interval estimation. “Confidence intervals”, and this phrase to describe them, were invented by Jerzy Neyman in 1934-37 [@neyman1937]. Statisticians typically mean Neyman’s intervals (or an approximation thereof) when they say “confidence interval”. In HEP the language is a little loose. I highly recommend using “confidence interval” (and “confidence regions” when multi-D) only to describe intervals and regions corresponding to Neyman’s construction, described below, or by recipes of other origin (including Bayesian recipes) only if they yield good approximations thereof. The following subsections use upper/lower limits and closely related central confidence intervals to introduce and illustrate the basic notions, and then discuss Neyman’s more general construction (used e.g. by Feldman and Cousins). Then, after introducing frequentist hypothesis testing (Section \[hypotest\]), we return to make the connection between confidence intervals and hypothesis testing of a particular value of parameter vs other values (Section \[duality\]). Notation -------- $x$ denotes observable(s). More generally, $x$ is any convenient or useful function of the observable(s), and is called a “statistic” or “test statistic”. $\mu$ denotes parameter(s). (Statisticians often use $\theta$.) $p(x| \mu)$ is the probability or pdf (from context) characterizing everything that determines the probabilities/densities of the observations, from laws of physics to experimental setup and protocol. The function $p(x| \mu)$ is called “the statistical model”, or simply “the model”, by statisticians. The basic idea of confidence intervals in two sentences {#twosentences} ------------------------------------------------------- Given the model $p(x| \mu)$ and the observed value $x_0$, we ask: For what values of $\mu$ is $x_0$ an “extreme” value of $x$? Then we include in the confidence interval $[\mu_1, \mu_2]$ those values of $\mu$ for which $x_0$ is [*not*]{} “extreme”. (Note that this basic idea sticks strictly to the frequentist probability of obtaining $x$, and makes no mention of probability (or density) for $\mu$.) ### Ordering principle is required for possible values of $x$ {#ordering} In order to define “extreme”, one needs to choose an [*ordering principle*]{} that ranks the possible values of $x$ applicable to each $\mu$. By convention [*high rank means not extreme*]{}. Some common ordering choices in 1D (when $p(x| \mu)$ is such that higher $\mu$ implies higher average $x$) are: - Order $x$ from largest to smallest: the smallest values of $x$ are the most extreme. Given $x_0$, the confidence interval that contains $\mu$ for which $x_0$ is not extreme will typically not contain the largest values of $\mu$. This leads to confidence intervals known as [*upper limits*]{} on $\mu$. - Order $x$ from smallest to largest. This leads to [*lower limits*]{} on $\mu$. - Order $x$ using [*central*]{} quantiles of $p(x| \mu)$, with the quantiles shorter in $x$ (least integrated probability of $x$) containing higher-ranked $x$, with lower-ranked $x$ added as the central quantile gets longer (contains more integrated probability of $x$). This leads to [*central*]{} confidence intervals for $\mu$. These three orderings apply only when $x$ is 1D. A fourth ordering, using a particular likelihood ratio advocated by Feldman and Cousins, is still to come (Section \[orderFC\]); it is applicable in both 1D and multi-D. ### A “confidence level” must be specified Given model $p(x| \mu)$ and an ordering of $x$, one chooses a particular fraction of highest-ranked values of $x$ that are [*not*]{} considered as “extreme”. This fraction is called the [ *confidence level*]{} ([[C.L.]{.nodecor}]{}), say 68% or 95%. (In this discussion, 68% is more precisely 68.27%; 84% is 84.13%; etc. Also, in the statistics literature there is a fine distinction between confidence level and confidence coefficient, which we ignore here.) We also define $\alpha = 1 - $[[C.L.]{.nodecor}]{}, i.e., the fraction that is the lower-ranked, “extreme” set of values. ### One-sentence summary Then the confidence interval $[\mu_1, \mu_2]$ contains those values of $\mu$ for which $x_0$ is [*not*]{} “extreme” at the chosen [[C.L.]{.nodecor}]{}, [ *given the chosen ordering of $x$*]{}. E.g., at 68% [[C.L.]{.nodecor}]{}, $[\mu_1, \mu_2]$ contains those $\mu$ for which $x_0$ is in the highest-ranked (least extreme) 68% values of $x$ for each respective $\mu$, according probabilities obtained from the model $p(x|\mu)$. Correspondence between upper/lower limits and central confidence intervals -------------------------------------------------------------------------- As illustrated in Fig. \[limits\_central\], the 84% [[C.L.]{.nodecor}]{} upper limit $\mu_2$ excludes $\mu$ for which $x_0$ is in the lowest 16% values of $x$. The 84% [[C.L.]{.nodecor}]{} lower limit $\mu_1$ excludes $\mu$ for which $x_0$ is in the highest 16% values of $x$. Then, the interval $[\mu_1, \mu_2]$ includes $\mu$ for which $x_0$ is in the central 68% quantile of $x$ values. It is a 68% [[C.L.]{.nodecor}]{} [*central*]{} confidence interval (!). For general ${\textnormal{C.L.}}$, the endpoints of central confidence intervals are, by the same reasoning, the same as upper/lower limits with confidence level given by $1-(1-{\textnormal{C.L.}})/2 = (1+{\textnormal{C.L.}})/2$. ![Sketch to illustrate the relationship among 84% C.L. upper limits, 84% C.L. lower limits, and 68% C.L. central confidence interval. The two one-sided tails of 16% compose the two-sided tails totaling 32%.[]{data-label="limits_central"}](figures/limits_central.png){width="40.00000%"} Examples follow for a couple illustrative models, first with continuous $x$, then with discrete $x$. Gaussian pdf $p(x| \mu,\sigma)$ with $\sigma$ a function of $\mu$: $\sigma = 0.2\mu$ {#secgauss02sigma} ------------------------------------------------------------------- It is common in HEP to express uncertainties as a percentage of a mean $\mu$ (even though situations in which this is rigorously motivated are rare). Thus, instead of the most trivial example of Gaussian with unknown mean $\mu$ and [*known*]{} rms $\sigma$, let’s assume that $\sigma$ is 20% of the unknown true value $\mu$: $$\label{eqn-gaussian_prob} p(x| \mu,\sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \textnormal{e}^{-(x-\mu)^2/2\sigma^2} = p(x| \mu) = \frac{1}{\sqrt{2\pi(0.2\mu)^2}} \textnormal{e}^{-(x-\mu)^2/2(0.2\mu)^2},$$ as illustrated in Fig. \[fig-gaussian\_prob\]. ![Plot of $p(x| \mu,\sigma)$, with $\mu=10.0$, $\sigma=0.2\mu$, as in Eqn. \[eqn-gaussian\_prob\]. In the subsequent discussion, we suppose that the observed value is $x_0 = 10.0$.[]{data-label="fig-gaussian_prob"}](figures/gaussian_prob.png){width="49.00000%"} With $\mu$ (and hence $\sigma$) unknown, suppose $x_0 = 10.0$ is observed. What can one say about $\mu$? Regardless of what else is done, it is always useful to plot the likelihood function, in this case obtained by plugging $x_0$ into Eqn. \[eqn-gaussian\_prob\]: $${{\cal L}}(\mu) = \frac{1}{\sqrt{2\pi(0.2\mu)^2}} \textnormal{e}^{-(10.0-\mu)^2/2(0.2\mu)^2},$$ as illustrated in Fig. \[fig-gaussian\_likl\]. The first thing one notices is that the likelihood is asymmetric and obtains it maximum at $\mu$ less than the $x_0$. (There is food for thought here, but we move on.) ![Plot of ${{\cal L}}(\mu)$ for observed $x_0 = 10.$, for the model in Eqn. \[eqn-gaussian\_prob\]. The maximum is at $\mu_{ML}= 9.63$[]{data-label="fig-gaussian_likl"}](figures/gaussian_likl.png){width="49.00000%"} ### What is the [*central confidence interval*]{} for $\mu$? {#secgauss02sigmacentral} First we find $\mu_1$ such that 84% of $p(x|\mu_1,\sigma=0.2\mu_1)$ is [*below*]{} $x_0 = 10.0$; 16% of the probability is above. Solve: $\mu_1 = 8.33$, as in Fig. \[gaussian\_lower\_upper\](left). So $[\mu_1,\infty]$ is an 84% [[C.L.]{.nodecor}]{} confidence interval, and $\mu_1$ is 84% [[C.L.]{.nodecor}]{} [*lower*]{} limit for $\mu$. ![(left) The model in Eqn. \[eqn-gaussian\_prob\], with $\mu$ chosen such that 84% of the probability is [*below*]{} $x_0 = 10.0$. (right) The model in Eqn. \[eqn-gaussian\_prob\], with $\mu$ chosen such that 84% of the probability is [*above*]{} $x_0 = 10.0$.[]{data-label="gaussian_lower_upper"}](figures/gaussian_lower.png "fig:"){width="49.00000%"} ![(left) The model in Eqn. \[eqn-gaussian\_prob\], with $\mu$ chosen such that 84% of the probability is [*below*]{} $x_0 = 10.0$. (right) The model in Eqn. \[eqn-gaussian\_prob\], with $\mu$ chosen such that 84% of the probability is [*above*]{} $x_0 = 10.0$.[]{data-label="gaussian_lower_upper"}](figures/gaussian_upper.png "fig:"){width="49.00000%"} Then we find $\mu_2$ such that 84% of $p(x|\mu_2,\sigma=0.2\mu_2)$ is [*above*]{} $x_0 = 10.0$; 16% of the probability is below. Solve: $\mu_2 = 12.52$, as in Fig. \[gaussian\_lower\_upper\](right). So $[-\infty,\mu_2]$ is an 84% [[C.L.]{.nodecor}]{} confidence interval, and $\mu_2$ is 84% [[C.L.]{.nodecor}]{} [*upper*]{} limit for $\mu$. Then the 68% [[C.L.]{.nodecor}]{} central confidence interval is $[\mu_1,\mu_2] = [8.33,12.52]$. This is “exact”. In fact, the reasoning used here was already laid out by E.B. Wilson in 1927 [@wilson1927] in his discussion of intervals in the Gaussian approximation of the binomial model (Section \[wilson\]). ### Contrast Wilson reasoning with “Wald interval” reasoning {#wald} The Wilson-inspired reasoning is crucial. Note the difference from the superficially attractive reasoning that proceeds as follows, leading to so-called Wald intervals: 1. For $x_0 = 10.0$, the “obvious” point estimate $\hat\mu$ (perhaps thought to be justified by minimum-$\chi^2$) is $\hat\mu = 10.0$. (This makes the mistake of interpreting the sampled value of $x$ as being the “measured value of $\mu$”, and hence as the point estimate of $\mu$, $\hat\mu$.) 2. Then one estimates $\sigma$ by $\hat\sigma = 0.2 \times \hat\mu = 2.0$. (This is potentially dangerous, since it estimates one parameter by plugging the point estimate of another parameter into a relationship between [*true*]{} values of parameters.) 3. Then $\hat\mu\, \pm\, \hat\sigma$ yields the interval $[8.0,12.0]$. (Again this is potentially dangerous, as it evaluates tail probabilities of a model having true parameters via plugging in estimates of the parameters.) Such “Wald intervals” are [*at best*]{} only approximations to exact intervals. For (“exact”) confidence intervals, the reasoning must [*always*]{} involve probabilities for $x$ calculated considering particular possible true values of parameters, as in the Wilson reasoning! Clearly the validity of the approximate Wald-interval reasoning (i.e., how dangerous the “potentially dangerous” steps are) depends on how much $\sigma(\mu)$ changes for $\mu$ relevant to problem at hand. The important point is that the Wald reasoning is not the correct reasoning for confidence intervals. Beware! Confidence intervals for binomial parameter $\rho$ {#sec-binomial} -------------------------------------------------- The binomial model is directly relevant to efficiency calculations in HEP, as well as other contexts. Recall the binomial distribution ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}},{\rho})$ from Eqn. \[eqn-binomial\] for the probability of ${n_\textnormal{\scriptsize on}}$ successes in ${n_\textnormal{\scriptsize tot}}$ trials, each with binomial parameter ${\rho}$. In repeated trials, ${n_\textnormal{\scriptsize on}}$ has mean $$\label{meannon} {n_\textnormal{\scriptsize tot}}\,{\rho}$$ and rms deviation $$\label{rmsnon} \sqrt{{n_\textnormal{\scriptsize tot}}\,{\rho}\,(1-{\rho})}\,.$$ For asymptotically large ${n_\textnormal{\scriptsize tot}}$, ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}},{\rho})$ can be approximated by a normal distribution with this mean and rms deviation. With observed number of successes ${n_\textnormal{\scriptsize on}}$, the likelihood function ${{\cal L}}({\rho})$ follows from reading Eqn. \[eqn-binomial\] as a function of ${\rho}$. The maximum is at $$\hat{\rho}= {n_\textnormal{\scriptsize on}}/{n_\textnormal{\scriptsize tot}}.$$ Suppose one observes ${n_\textnormal{\scriptsize on}}=3$ successes in ${n_\textnormal{\scriptsize tot}}=10$ trials. The likelihood function ${{\cal L}}({\rho})$ is plotted in Fig. \[binomial\_likl\]. ![(left) Likelihood function ${{\cal L}}({\rho})$ for ${n_\textnormal{\scriptsize on}}=3$ successes in ${n_\textnormal{\scriptsize tot}}=10$ trials in the binomial model of Eqn. \[eqn-binomial\]. (right) Looking ahead to Section \[likelihood\], the plot of $-2\ln{{\cal L}}({\rho})$.[]{data-label="binomial_likl"}](figures/binomial_likl.png "fig:"){width="49.00000%"} ![(left) Likelihood function ${{\cal L}}({\rho})$ for ${n_\textnormal{\scriptsize on}}=3$ successes in ${n_\textnormal{\scriptsize tot}}=10$ trials in the binomial model of Eqn. \[eqn-binomial\]. (right) Looking ahead to Section \[likelihood\], the plot of $-2\ln{{\cal L}}({\rho})$.[]{data-label="binomial_likl"}](figures/binomial_2negloglikl.png "fig:"){width="49.00000%"} ### What [*central confidence interval*]{} should we report for $\rho$? We have ${n_\textnormal{\scriptsize on}}= 3$, ${n_\textnormal{\scriptsize tot}}=10$. Let us find the “exact” 68% [[C.L.]{.nodecor}]{} central confidence interval $[\rho_1,{\rho}_2]$. Recall the shortcut in Section \[secgauss02sigmacentral\] for central intervals: Find the [*lower limit*]{} ${\rho}_1$ with [[C.L.]{.nodecor}]{} $= (1+0.68)/2 = 84$%; and find [*upper limit*]{} ${\rho}_2$ with [[C.L.]{.nodecor}]{} $= 84$%. Then $[{\rho}_1,{\rho}_2]$ is a 68% [[C.L.]{.nodecor}]{} central confidence interval. 1. For lower limit at 84% [[C.L.]{.nodecor}]{}, find ${\rho}_1$ such that ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}< 3 | {\rho}_1) = 84$%, i.e., ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}\ge 3 | {\rho}_1) = 16$%. Solve: ${\rho}_1 = 0.142$ as in Fig. \[binomial\_lower\] (left). 2. For upper limit at 84% [[C.L.]{.nodecor}]{}), find ${\rho}_2$ such that ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}> 3 | {\rho}_2) = 84$%, i.e., ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}\le 3 | {\rho}_2) = 16$%. Solve: ${\rho}_2 = 0.508$ as in Fig. \[binomial\_lower\] (right). 3. Then $[{\rho}_1,{\rho}_2] = [0.142, 0.508]$ is a [*central*]{} confidence interval with 68% [[C.L.]{.nodecor}]{}  ![(left) Plot of ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}}=10,{\rho})$, with ${\rho}$ chosen such that 84% of the probability is [*below*]{} ${n_\textnormal{\scriptsize on}}=3$. (right) Plot of ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}}=10,{\rho})$, with ${\rho}$ chosen such that 84% of the probability is [*above*]{} ${n_\textnormal{\scriptsize on}}=3$.[]{data-label="binomial_lower"}](figures/binomial_lower.png "fig:"){width="49.00000%"} ![(left) Plot of ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}}=10,{\rho})$, with ${\rho}$ chosen such that 84% of the probability is [*below*]{} ${n_\textnormal{\scriptsize on}}=3$. (right) Plot of ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}}=10,{\rho})$, with ${\rho}$ chosen such that 84% of the probability is [*above*]{} ${n_\textnormal{\scriptsize on}}=3$.[]{data-label="binomial_lower"}](figures/binomial_upper.png "fig:"){width="49.00000%"} This is the same result as obtained by Clopper and Pearson (C-P) in 1934 [@clopper1934], citing Fisher, Neyman, and Neyman’s students. In HEP, such C-P intervals are the standard for a binomial parameter; they have been in PDG RPP since 2002 [@pdg2018] (although the connection of the given formula with C-P is obscure). The same method was applied to confidence intervals for a Poisson mean by Garwood in his 1934 thesis, and published in 1936. (See Ref. [@cousinsajp1995], and references therein.) They are also the standard in HEP when there is no background. There is controversy when there is background; see PDG RPP (Section 39.4.2.4). Many tables and online calculators for C-P intervals and Garwood intervals exist; I usually use Ref [@confint]. The use of the word “exact” (dating to Fisher) for intervals such as C-P refers to the [*construction*]{} above. But the discreteness of observed $x$ (${n_\textnormal{\scriptsize on}}$ in this case) causes the frequentist coverage (defined and discussed Section \[sec-coverage\] below) not to be exact, but rather generally greater than the chosen [[C.L.]{.nodecor}]{} C-P intervals are thus criticized by some as “wastefully conservative”. For a comprehensive review of both central and non-central confidence intervals for a binomial parameter and for the ratio of Poisson means, see Cousins, Hymes, and Tucker [@cht2010]. Many choices, including C-P, are implemented in ROOT [@tefficiency]. For the closely related construction of upper/lower limits and central confidence intervals for a [*Poisson mean*]{}, see Ref. [@cousinsajp1995]. ### Gaussian approximation for binomial confidence intervals: Wilson score interval of 1927 {#wilson} As mentioned above, ${n_\textnormal{\scriptsize on}}$ has mean and rms deviation given by Eqns. \[meannon\] and \[rmsnon\], and for asymptotically large ${n_\textnormal{\scriptsize tot}}$, ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}},{\rho})$ converges to a Gaussian with mean $\mu({\rho}) = {n_\textnormal{\scriptsize tot}}{\rho}$ and rms $\sigma({\rho}) = \sqrt{{n_\textnormal{\scriptsize tot}}{\rho}(1-{\rho})}$. We can thus compute [*approximate*]{} confidence intervals for $\rho$ while invoking this Gaussian approximation to the binomial distribution. The idea is [*not*]{} to substitute $\hat{\rho}$ for ${\rho}$ in Eqns. \[meannon\] and \[rmsnon\] (potentially big mistake!), but rather to follow the logic already used above from E.B. Wilson in 1927 [@wilson1927]. (This is in fact the example that he was illustrating in that paper!) That is, we use the above recipe for upper and lower limits: 1. Find ${\rho}_1$ such that Gauss$(x\ge3 | \textnormal{mean}\ {\rho}_1, \sigma({\rho}_1) ) = 0.16$. 2. Find ${\rho}_2$ such that Gauss$(x\le3 | \textnormal{mean}\ {\rho}_2, \sigma({\rho}_2) ) = 0.16$. This consistently uses the (different) values of $\sigma$ associated with each ${\rho}$, [*not*]{} $\sigma(\hat{\rho})$. It leads to a quadratic equation that, for our example, has solution $[{\rho}_1,{\rho}_2] = = [0.18, 0.46]$. This is the approximate 68% [[C.L.]{.nodecor}]{} confidence interval known as the [*Wilson score interval*]{}. (See Ref. [@cht2010] and references therein.) Although this Wilson score interval needs only the quadratic formula, it is for some reason relatively unknown, as students are taught the Wald interval (UGH) of the next subsection. ### Gaussian approximation for binomial confidence intervals: potentially disastrous Wald interval to avoid It is tempting instead to follow the so-called “Wald reasoning” mentioned in Section \[wald\], and to substitute $\hat\rho = {n_\textnormal{\scriptsize tot}}{\rho}$ for ${\rho}$ in the expression for the rms deviation in Eqn. \[rmsnon\]. One obtains: $\hat\sigma = \sqrt{{n_\textnormal{\scriptsize tot}}\hat{\rho}(1-\hat{\rho})}$ and it seems a simple step to the potentially disastrous “Wald interval” $\hat{\rho}\pm \hat\sigma$, i.e., $[{\rho}_1,{\rho}_2] = [\hat{\rho}- \hat\sigma, \hat{\rho}+ \hat\sigma]$. The Wald interval does not use the correct logic for frequentist confidence! In fact, $\hat\sigma=0$ when ${n_\textnormal{\scriptsize on}}= 0$ or ${n_\textnormal{\scriptsize on}}= {n_\textnormal{\scriptsize tot}}$. Incredibly, the failure of the Wald interval when ${n_\textnormal{\scriptsize on}}= 0$ (or ${n_\textnormal{\scriptsize on}}= {n_\textnormal{\scriptsize tot}}$) has been used as a [*foundational argument*]{} in favor of Bayesian intervals in at least four public HEP postings (one retracted) and one published astrophysics paper! Of all the misguided things that have been written about statistics (sometimes by me) in nominally scholarly writings, this is among most uninformed that I have seen. (Typically the authors were not informed about Bayesian statistics either, and thought that a prior uniform in ${\rho}$ was obvious, without having read the vast Bayesian literature on priors for a binomial parameter.) Beware! Avoid the Wald interval except for “hallway estimates” — there is no reason to use it. And certainly do not make the silly claim that problems with the Wald interval point to foundational issues with confidence intervals. The Wald interval does not use “confidence” reasoning, and was already obsolete in 1927 as the Gaussian approximation to binomial confidence intervals; and by 1934 the exact intervals using confidence reasoning were published by C-P! They have no problem with the cases “fatal” to Wald intervals, ${n_\textnormal{\scriptsize on}}= 0$ or ${n_\textnormal{\scriptsize on}}= {n_\textnormal{\scriptsize tot}}$, as can be easily verified. ### HEP applications of confidence intervals for binomial parameter As mentioned, the binomial model is directly relevant to efficiency calculations in the usual case where ${n_\textnormal{\scriptsize on}}$ is not fixed by the experiment design but rather sampled. Using a famous math identity, the binomial model is also directly applicable to confidence intervals for the [*ratio of Poisson means*]{} [@cht2010; @jamesroos]. Thus, it is applicable to statistical significance ($Z_\textnormal{Bi}$) of an excess in a signal bin when a sideband is used to estimate the mean background. (See Cousins, Linnemann, and Tucker [@clt2008].) As discussed in our paper, one can even stretch the use of $Z_\textnormal{Bi}$ (using a “rough correspondence”) to the problem of a signal bin when a Gaussian estimate of mean background exists. Perceived problems with upper/lower limits and hence for central confidence intervals {#centralproblems} ------------------------------------------------------------------------------------- For decades, issues with upper limits and central confidence intervals have been discussed in two prototype problems in HEP: - Gaussian measurement resolution near a physical boundary (e.g., neutrino mass-squared is positive, but the sample $x$ is negative); - Poisson signal mean measurement when observed number of events is less than mean expected background (so that the naive “background-subtracted” mean is negative). Many ideas have been put forward, and by 2002 the PDG RPP settled on three, which remains the case [@pdg2018]. I have described some of this interesting history in a “virtual talk” [@cousinsvirtual] and in an arXiv post [@cousins2011]. (See Section \[downward\] in this paper.) In this section, I describe just one of these three ideas, namely using Neyman’s confidence interval construction, going beyond the common orderings of $x$ discussed thus far (Section \[ordering\]), to that advocated by Feldman and Cousins (F-C) [@feldman1998]. (The other two ideas in the RPP are Bayesian upper limits and [$CL_s$]{} (Section \[cls\].) Beyond upper/lower limits and [*central*]{} confidence intervals {#orderFC} ---------------------------------------------------------------- Among the more general choices for ordering $x$ in $p(x| \mu)$, the most common in HEP is that based on a [*likelihood ratio*]{} (LR): For each $\mu$, order $x_0$ using the likelihood ratio ${{\cal L}}(x_0|\mu) / {{\cal L}}(x_0| \mu_\textnormal{best-fit})$, where $\mu_\textnormal{best-fit}$ respects the physical boundaries of $\mu$. This was advocated in HEP by Feldman and Cousins in 1998 [@feldman1998]. In fact, as learned by F-C while their paper was in proof, the dual hypothesis test (Section \[duality\]) appeared in the hypothesis test section of the frequentist classic by “Kendall and Stuart” [@kendall1999] long before and since. Unlike the orderings described above, the LR ordering is applicable in both 1D and multi-D for $x$. Recall from Section \[metricchange\] that likelihood [*ratios*]{} as in F-C are independent of metric in $x$ since the Jacobians cancel, so there is no issue there. In contrast, an alternative that might come to mind, namely ordering $x$ by the probability [*density*]{} $p(x| \mu)$, is [*not*]{} recommended. A change of metric from $x$ to $y(x)$ leads to a Jacobian $|dy/dx|$ in $p(y| \mu) = p(x| \mu) / |dy/dx|$ (as in Section \[metricchange\]). So ordering by $p(y| \mu)$ is different than ordering by $p(x| \mu)$, and so all that would follow from ordering by a pdf would depend on the arbitrary choice of metric. In order to implement this more general ordering by likelihood ratios, we must go beyond finding the confidence interval endpoints $\mu_1$ and $\mu_2$ independently, and perform the construction of the whole interval at the same time, as in the next subsection. Neyman’s Construction of Confidence Intervals {#neymanconstruction} --------------------------------------------- The general method for constructing “confidence intervals”, and the name, were invented by Jerzy Neyman in 1934-37. It takes a bit of time to sink in—given how often confidence intervals are misinterpreted, perhaps the argument is bit too ingenious! In particular, you should understand that the confidence level does not tell you “how confident you are that the unknown true value is in the specific interval you report”—only a [*subjective*]{} Bayesian credible interval has that property! Rather, as stated in Section \[twosentences\], the confidence interval $[\mu_1, \mu_2]$ contains those values of $\mu$ for which the observed $x_0$ is [ *not*]{} “extreme”, according to the ordering principle chosen. In this section, we describe this construction in more detail and more generality. We begin with 1D data and one parameter $\mu$. Given a $p(x|\mu)$: For each value of $\mu$, one draws a horizontal [*acceptance interval*]{} $[x_1,x_2]$ such that $$\label{eqnacc} p(x \in [x_1,x_2] | \mu ) = {\textnormal{C.L.}}= 1 - \alpha,$$ as in Fig. \[neyman\_const\_1.png\](left). The “ordering principle” that was chosen for $x$ is used to well-define which values of $x$ are included. ![Steps in the Neyman construction, as described in the text. From Ref. [@feldman1998].[]{data-label="neyman_const_1.png"}](figures/neyman_const_1.jpg "fig:"){width="49.00000%"} ![Steps in the Neyman construction, as described in the text. From Ref. [@feldman1998].[]{data-label="neyman_const_1.png"}](figures/neyman_const_2.png "fig:"){width="49.00000%"} ![Steps in the Neyman construction, as described in the text. From Ref. [@feldman1998].[]{data-label="neyman_const_1.png"}](figures/neyman_const_3.png "fig:"){width="49.00000%"} Upon observing $x$ and obtaining the value $x_0$, one draws a vertical line through $x_0$, as in Fig. \[neyman\_const\_1.png\](right). The vertical [*confidence interval*]{} $[\mu_1, \mu_2]$ with confidence level ${\textnormal{C.L.}}= 1 - \alpha$ is the union of all values of $\mu$ for which the corresponding acceptance interval is intercepted by the vertical line, as in Fig. \[neyman\_const\_1.png\](bottom). (It need not be simply connected, as in highly nonlinear applications such as neutrino oscillations.) [*Important note:*]{} $x$ and $\mu$ need not have the same range, units, or (in generalization to higher dimensions) dimensionality! In fact, I think it is [*much*]{} easier to avoid confusion if $x$ and $\mu$ are qualitatively different. Louis Lyons gives the example where $x$ is the flux of solar neutrinos and $\mu$ is the temperature at the center of the sun. I like examples where $x$ and $\mu$ have different dimensions: Neyman’s original paper [@neyman1937] has a 2D observation space and 1D parameter space; his figure was crucial for my own understanding of the construction. Famous confusion re Gaussian $p(x| \mu)$ where $\mu$ is mass $\ge0$ {#negativex} ------------------------------------------------------------------- A prototype problem is the Gaussian model $p(x| \mu,\sigma)$ in Eqn. \[eqn-gaussian\] in the case where $\mu$ corresponds to a quantity that is physically non-negative, e.g., a mass, a mass-squared, or an interaction cross section. Note that in this case negative values of $\mu$ [*do not exist in the model*]{}! A particle’s mass-squared might be computed by measuring a particle’s energy-squared ($E^2$) with Gaussian resolution and also (independently) its momentum-squared ($p^2$) with Gaussian resolution. Then the observable $x$ corresponding to mass-squared could be computed from $E^2-p^2$, with Gaussian resolution. If the true $m^2$ is zero or small compared to the resolution, then such a procedure can easily obtain a value of $x$ that is negative. There is [*nothing anomalous*]{} about negative $x$. A [*key*]{} point that was a source of a lot of confusion historically in HEP, and which still trips up people, is the following: It is [ *crucial*]{} to distinguish between the observed data $x$, which [ *can*]{} be negative (no problem), and the model parameter $\mu$, for which negative values [*do not exist in the model*]{}. For parameter $\mu <0$, $p(x|\mu)$ does not exist: You would not know how to simulate the physics of your detector response to negative mass! The constraint $\mu\ge 0$ thus has [*nothing*]{} to do with a Bayesian prior pdf for $\mu$ (!!!) as sometimes mistakenly thought. The constraint is in the [*model*]{}, and hence in the likelihood ${{\cal L}}(\mu)$, not in the prior pdf. The confusion is encouraged since we often refer to $x$ as the “measured value of $\mu$”, and say that $x<0$ is “unphysical”. This is a bad habit! The value $x$ is an observed sample from a Gaussian centered on $\mu$, with $\mu$ being “physical”. Thus a proper Neyman construction graph for the case at hand has $x$ with [*both arithmetic signs*]{} but only non-negative $\mu\ge 0$. This is the case in the plot in Fig. \[neyman\_const\_fc.png\](left), following that in F-C [@feldman1998]. ![(left) The Neyman construction for the nonnegative mean of a Gaussian with acceptance intervals shown in blue, as advocated in Ref. [@feldman1998]. (right) The “confidence belt” obtained from the construction on the left, showing the envelope of the acceptance intervals, which are not drawn.[]{data-label="neyman_const_fc.png"}](figures/neyman_const_fc.png "fig:"){width="48.00000%"} ![(left) The Neyman construction for the nonnegative mean of a Gaussian with acceptance intervals shown in blue, as advocated in Ref. [@feldman1998]. (right) The “confidence belt” obtained from the construction on the left, showing the envelope of the acceptance intervals, which are not drawn.[]{data-label="neyman_const_fc.png"}](figures/neyman_const_belt.png "fig:"){width="49.00000%"} There was a lot of confusion in the early days when the vertical axis on this plot was extended to non-physical values of $\mu$. Since the model does not exist there, it is a conceptual mistake to draw the acceptance intervals there (as some did). For alternative methods for dealing with this situation, see Section \[downward\]. Confidence [*belts*]{} ---------------------- From the earliest days (as in the 1934 Clopper-Pearson paper [@clopper1934]), the horizontal line segments of the acceptance intervals (such as in Fig. \[neyman\_const\_fc.png\](left)) [*have been suppressed*]{}, and only their envelope, as in the black curves in Fig. \[neyman\_const\_fc.png\](right), have been plotted. The black curves (and interior) are called a [*confidence belt*]{}. I added the line segments for demonstrating the construction in the F-C paper after reading Neyman’s 1937 paper [@neyman1937], in which his Fig. 1 has a couple planes with acceptance regions shown, and a 1D vertical line at the observed 2D point intercepting them. This practice in the F-C paper is fortunately spreading. I had found other descriptions of the construction, showing only the belt, to be too obscure when I was trying to learn the construction. There would be statements that I found cryptic, such as “Notice that the confidence belt is [*constructed horizontally*]{} but [*read vertically*]{}.” This made sense to me only [*after*]{} I had understood the construction! Confidence intervals and coverage {#sec-coverage} --------------------------------- Do you recall how a vector is defined in an abstract math class? In math, one defines a [*vector space*]{} as a set with certain properties, and then the definition of a vector is “an element of a vector space”. A vector is not defined in isolation! Similarly, whether constructed in practice by Neyman’s construction or some other technique, a confidence interval is defined to be “an element of a confidence set”, where the confidence set is a set of intervals defined to have the property of frequentist coverage under repeated sampling: Let ${\mu_\textnormal{t}}$ be the unknown true value of $\mu$. In repeated experiments, confidence intervals will have different endpoints $[ \mu_1,\mu_2]$, since the endpoints are functions of the randomly sampled $x$. A little thought will convince you that a fraction [[C.L.]{.nodecor}]{}$ = 1 - \alpha$ of intervals obtained from Neyman’s construction will contain (“cover”) the fixed but unknown ${\mu_\textnormal{t}}$. I.e., $$P({\mu_\textnormal{t}}\in [ \mu_1,\mu_2]) = {\textnormal{C.L.}}= 1 - \alpha. \label{coverage}$$ This equation is the definition of [*frequentist coverage*]{}. In this (frequentist!) equation, ${\mu_\textnormal{t}}$ is [*fixed and unknown*]{}. The endpoints $\mu_1, \mu_2$ are the random variables (!). Coverage is a property of the set of confidence intervals, not of any one interval. \[Here is the “little thought” that explains why Neyman’s construction leads to coverage: For the unknown true value ${\mu_\textnormal{t}}$, the probability that $x_0$ is in its [*acceptance interval*]{} is [[C.L.]{.nodecor}]{}, by construction. When those $x_0$’s are obtained, the vertical line will intercept ${\mu_\textnormal{t}}$’s acceptance region, and so ${\mu_\textnormal{t}}$ be will be put into the confidence interval. Thus the coverage equation is satisfied.\] One of the complaints about confidence intervals is that the consumer often forgets (if he or she ever knew) that the random variables in Eqn. \[coverage\] are $\mu_1$ and $\mu_2$, and not ${\mu_\textnormal{t}}$; and that coverage is a property of the set, not of an individual interval! Please don’t forget! A lot of confusion might have been avoided if Neyman had chosen the names “[*coverage intervals*]{}” and “[ *coverage level*]{}”! (Maybe we can have a summit meeting treaty where frequentists stop saying “confidence” and Bayesians stop saying “noninformative”!) Note: It [*is*]{} true (in precisely the sense defined by the ordering principle used for $x$ in the Neyman construction) that the confidence interval consists of those values of $\mu$ for which the observed $x_0$ is among the [[C.L.]{.nodecor}]{} least extreme values to be observed. ### Over-coverage when observable $x$ is discrete {#overcov} A problem arises in Neyman’s construction when the observable $x$ (or more generally, the test statistic) is discrete. This was already the case in the Clopper-Pearson paper. When constructing an acceptance interval, typically Eqn. \[eqnacc\] cannot be satisfied exactly. The traditional convention (still typically observed in HEP) is to include enough values of $x$ so that the equality becomes “$\ge$”. This means that the coverage equation includes so-called over-coverage: $$\label{overcoverage} P({\mu_\textnormal{t}}\in [ \mu_1,\mu_2]) \ge {\textnormal{C.L.}}= 1 - \alpha.$$ For a discussion of this issue and various opinions about it, see Ref. [@cht2010] and references therein. Frequentist (classical) hypothesis testing {#hypotest} ========================================== At this point, we set aside confidence intervals for the moment and consider from the beginning the nominally different topic of hypothesis testing. In fact, we will soon find that in frequentists statistics, certain hypothesis tests will take us immediately back to confidence intervals. But first we consider the more general framework. Frequentist hypothesis testing, often called “classical” hypothesis testing, was developed by J. Neyman and E. Pearson (N-P) in unfriendly competition with R.A. Fisher. Modern testing has a mix of ideas from both “schools”. Following the N-P approach [@james2006; @neymanpearson1933a], we frame the discussion in terms of a null hypothesis H$_0$ (e.g., the Standard Model), and an alternative H$_1$ (e.g., some Beyond-SM model). As in Section \[notation\], $x$ is a test statistic (function of the observed data) and $\mu$ represents parameters. Then the model $p(x| \mu)$ is different for H$_0$ and H$_1$, either because parameter $\mu$ (often called $\theta$ by statisticians) has a different value, or because $p(x| \mu)$ itself is a different functional form, perhaps with additional parameters. For the null hypothesis H$_0$, we order possible observations $x$ from least extreme to most extreme, using an ordering principle (which can depend on H$_1$ as well). We choose a cutoff $\alpha$ (smallish number). We then “reject” H$_0$ if the observed $x_0$ is in the most extreme fraction $\alpha$ of observations $x$ (generated under H$_0$). By construction: [$\alpha$]{} $=$ probability (with $x$ generated according to H$_0$) of rejecting H$_0$ when it is true, i.e., false discovery claim (Type I error). It is called the [*size*]{} or [ *significance level*]{} of the test. To quantity the performance of this test if H$_1$ is true, we further define: [$\beta$]{} $=$ probability (with $x$ generated according to H$_1$) of not rejecting (“accepting”) H$_0$ when it is false, i.e., not claiming a discovery when there is one (Type II error). The [ *power*]{} of the test is defined as $1-\beta$. Note: If the alternative H$_1$ is not specified, then $\beta$ is not known and optimality cannot be well-defined. The test is then called a [*goodness-of-fit*]{} test of H$_0$, as discussed in Section \[goodness\]. There is tradeoff between Type I and Type II errors. Competing analysis algorithms (resulting in different test statistics) can be compared by looking at graphs of $\beta$ vs $\alpha$ at various $\mu$, and at graphs of $1-\beta$ vs $\mu$ at various $\alpha$ (power function). (See, e.g., Ref. [@james2006], pp. 258, 262.) This is similar to comparing b-tagging efficiency for signal and background. It is equivalent to the ROC curve used on other fields. Appendix \[secjhep\] illustrates the tradeoff between $\alpha$ and $\beta$ with a toy example of spin discrimination of a new resonance. The choice of Type I error probability $\alpha$ {#alphachoice} ----------------------------------------------- The choice of operating point on the $\beta$ vs $\alpha$ curve (or ROC curve) is a long discussion. (It is even longer when considered as the number $N$ of events increases, so that both $\alpha$ and $\beta$ are reduced.) The N-P language of “accept” or “reject” H$_0$ should not be mistaken for a complete theory of decision-making: A decision on whether or not to declare discovery (falsifying H$_0$) requires 2 more inputs: - Prior belief in H$_0$ vs H$_1$. (Can affect choice of $\alpha$) - Cost of Type I error (false discovery claim) vs cost of Type II error (missed discovery). (Can also affect choice of $\alpha$) A one-size-fits-all criterion of $\alpha$ corresponding to 5$\sigma$ is without foundation! For a discussion of this point, see Refs. [@cousinsJL; @lyons2013]. Considerations such as these for the choice of $\alpha$ typically depend on the context. Using the result of an experiment for a single test of a physics hypothesis is a very different context than repeatedly selecting candidate b-jets with a b-tagging algorithm as in Section \[btagsec\]. In the latter case, the tradeoff between $\beta$ vs $\alpha$ is usually determined by considerations downstream in the analysis. Frequentist hypothesis testing: Simple hypotheses {#simple} ------------------------------------------------- In idealized cases that are sometimes reasonably well-approximated in HEP, a hypothesis may have no floating (unfixed) parameters. N-P called such hypotheses [*simple*]{}, in contrast to [*composite*]{} hypotheses that have unfixed parameters. Examples in HEP where both H$_0$ and H$_1$ are truly simple are rare, but we do have a few examples where the quantity of interest is simple in both hypotheses, and the role of unfixed nuisance parameters does not badly spoil the “simplicity”. For example, the hypotheses H$_0$ vs H$_1$ might be: - “jet originated from a quark” vs “jet originated from a gluon”, for a jet reconstructed in the detector - spin-1 vs spin-2 for a new resonance in $\mu^+\mu^-$ - J$^{\rm P}=0^+$ vs J$^{\rm P}=0^-$ for the Higgs-like boson A simplified, detailed illustration of the second test is in Appendix \[secjhep\], while an example of the latter test published by CMS is in Section \[higgsCP\]. Of course, framing these tests in this way makes strong assumptions (in particular that one of the two hypotheses is true) that need to be revisited once data are in hand. ### Testing Simple hypotheses: Neyman–Pearson lemma {#secNP} If the Type I error probability $\alpha$ is specified in a test of simple hypothesis H$_0$ against simple hypothesis H$_1$, then the Type II error probability $\beta$ is minimized by ordering $x$ according to the likelihood ratio [@neymanpearson1933a], $$\lambda = {{\cal L}}(x| {\rm H}_0) /{{\cal L}}(x| {\rm H}_1).$$ One finds the cutoff $\lambda_{{\rm cut,}\alpha}$ for the desired $\alpha$ and rejects H$_0$ if $\lambda\le \lambda_{{\rm cut,}\alpha}$. For a conceptual outline of a proof, see the lecture of Kyle Cranmer [@cranmerNP] as well as Ref. [@kendall1999] (p. 176). As mentioned, Appendix \[secjhep\] has an example. This “Neyman–Pearson lemma” applies only to a very special case: no fitted parameters, not even undetermined parameters of interest! But it has inspired many generalizations, and likelihood ratios are an oft-used component of both frequentist and Bayesian methods. Nested hypothesis testing {#nested} ------------------------- In contrast to two disjoint simple hypotheses, it is common in HEP for H$_0$ to be [*nested*]{} in H$_1$. For example, commonly H$_0$ corresponds to the parameter $\mu$ in H$_1$ being equal to a particular value $\mu_0$. (Typical values of $\mu_0$ are 0 or 1.) So we often consider: [H$_0$:]{} $\mu = \mu_0$ (the “point null”, or “sharp hypothesis”) vs [H$_1$:]{} $\mu \ne \mu_0$ (the “continuous alternative”). Common examples are: - Signal strength $\mu$ of new physics: null $\mu_0 = 0$, alternative $\mu>0$ - H$_0$ $\rightarrow \gamma\gamma$ before discovery of this decay, $\mu =$ signal strength: null $\mu_0 = 0$, alternative $\mu>0$ - H$_0$ $\rightarrow \gamma\gamma$ after discovery of this decay: $\mu$ is the ratio of the signal strength to the SM prediction; null $\mu_0 =1$ (i.e., SM prediction), alternative is any other $\mu \ne \mu_0$. Nested hypothesis testing: Duality with intervals {#duality} ------------------------------------------------- In the classical frequentist formalism (but not the Bayesian formalism), the theory of these tests maps to that of confidence intervals! The argument is as follows. 1. Having observed data $x_0$, suppose the 90% C.L. confidence interval for $\mu$ is $[\mu_1,\mu_2]$. This contains all values of $\mu$ for which the observed $x_0$ is ranked in the [*least*]{} extreme 90% of possible outcomes $x$ according to $p(x|\mu)$ and the ordering principle in use. 2. With the same data $x_0$, suppose that we wish to test H$_0$ vs H$_1$ (as defined in Section \[nested\]) at Type I error probability $\alpha = 10$%. We reject H$_0$ if $x_0$ is ranked in the [*most*]{} extreme 10% of $x$ according to $p(x|\mu)$ and the ordering principle in use. Comparing the two procedures, we see that we reject H$_0$ at $\alpha=10$% if and only if $\mu_0$ is outside the 90% C.L. confidence interval $[\mu_1,\mu_2]$. (In this verbal description, I am implicitly assuming that $x$ is continuous and that $p(x|\mu)$ is a pdf that puts zero probability on a point $x$ with measure zero. Thus I ignore any issues concerning endpoints of intervals.) We conclude: [*Given an ordering:*]{} a test of H$_0$ vs H$_1$ at significance level $\alpha$ is equivalent to asking: Is $\mu_0$ outside the confidence interval for $\mu$ with C.L. $= 1- \alpha$? As Kendall and Stuart put it, “There is thus no need to derive optimum properties separately for tests and for intervals; there is a one-to-one correspondence between the problems as in the dictionary in Table 20.1” [@kendall1999] (p. 175). The table mentioned maps the terminology that historically developed separately for intervals and for testing, e.g., $\alpha$ $\leftrightarrow$ $1 -{\textnormal{C.L.}}$ Most powerful $\leftrightarrow$ Uniformly most accurate Equal-tailed tests $\leftrightarrow$ central confidence intervals Use of this duality is referred to as “inverting a test” to obtain confidence intervals, and vice versa. Feldman-Cousins --------------- As mentioned in Section \[orderFC\], the “new” likelihood-ratio (LR) ordering principle that Gary Feldman and I advocated for confidence intervals in Ref. [@feldman1998] turned out to be one and the same as the time-honored ordering (generalized to include nuisance parameters) described in Ref. [@kendall1999] in the chapter of [*hypothesis testing*]{} using likelihood ratios. We had scoured statistics literature for precedents of “our” intervals without finding anything. But we over-looked the fact that we should also be scouring the literature on hypothesis tests, until Gary realized that, just in time to get a note added in proof. In fact, it was all on $1\frac{1}{4}$ pages of “Kendall and Stuart”, plus nuisance parameters! This led to rapid inclusion of the LR ordering in the PDG RPP, since the proposal suddenly had the weight of real statistics literature behind it. Post-data $p$-values and $Z$-values {#pvalues} ----------------------------------- The above N-P theory is all a [*pre-data*]{} characterization of the hypothesis test. A deep issue is how to apply it after $x_0$ is known, i.e., [*post-data*]{}. In N-P theory, $\alpha$ is [*specified in advance*]{}. Suppose after obtaining data, you notice that with $\alpha=0.05$ previously specified, you reject H$_0$, but with $\alpha=0.01$ previously specified, you accept H$_0$. In fact, you determine that with the data set in hand, H$_0$ would be rejected for $\alpha \ge 0.023$. This interesting value has a name: [*After*]{} data are obtained, the [*p-value*]{} is the smallest value of $\alpha$ for which H$_0$ would be rejected, [*had that value been specified in advance*]{}. This is numerically (if not philosophically) the same as the definition used e.g. by Fisher and often taught: “The $p$-value is the probability under H$_0$ of obtaining $x$ as extreme [*or more extreme*]{} than the observed $x_0$.” [@cowan] (p. 58). See also Ref. [@james2006] (p. 299) and Ref. [@pdg2018] (Section 39.3.2). In HEP, a $p$-value is typically converted to a $Z$-value (unfortunately commonly called “the significance S”), which is the equivalent number of Gaussian standard deviations. E.g., for a one-tailed test in a search for an [*excess*]{}, $p$-value${} = 2.87 \times 10^{-7}$ corresponds to $Z = 5$. Note that Gaussianity of the test statistic is typically [*not*]{} assumed when the $p$-value is computed; this conversion to equivalent Gaussian “number of sigma” is just for perceived ease of communication. This needs to be emphasized when communicating outside HEP, as I hear too often statisticians wondering about Gaussian assumptions, in effect making the conversion counter-productive (!). Although these lectures are not “statistics in practice”, I mention ROOT commands for one-tailed conversions (improved version courtesy of Igor Volobouev): zvalue = -TMath::NormQuantile(pvalue) pvalue = 0.5*TMath::Erfc(zvalue/sqrt(2.0)) In our usual one-tailed test convention, $p$-value $>$ 0.5 corresponds to $Z<0$. ### Interpreting $p$-values and $Z$-values In the example above, it is crucial to realize that that value of $\alpha$ equal to the $p$-value (0.023 in the example) was typically [*not*]{} specified in advance. So $p$-values do [*not*]{} correspond to Type I error probabilities of experiments reporting them. The interpretation of $p$-values (and hence $Z$-values) is a long, contentious story—beware! They are widely bashed; I discuss why in Section \[likelihoodprin\]. I also defend their use in HEP. (See for example my paper on the Jeffreys-Lindley Paradox [@cousinsJL].) Whatever they are, $p$-values are [*not*]{} the probability that H$_0$ is true! This mis-interpretation of $p$-values is unfortunately so common as to be used as an argument against frequentist statistics. Please keep in mind: - That $p$-values are calculated [*assuming that*]{} H$_0$ [*is true*]{}, so they can hardly tell you the probability that H$_0$ is true! - That the calculation of the “probability that H$_0$ is true” requires prior(s) to invert the conditional probabilities, as in Section \[bayesbayes\]. Please help educate press officers and journalists! (and physicists)! ### Early CMS Higgs boson spin-parity test of $0^+$ vs. $0^-$ {#higgsCP} As noted at the beginning of Section \[simple\], the test of H$_0$: J$^{\rm P}=0^+$ vs H$_1$: J$^{\rm P}=0^-$ for the Higgs-like boson is a case where the parameter of interest (parity P) has two discrete values of interest, and the role of the (many) unfixed nuisance parameters does not badly spoil this “simplicity” in practice. Thus the test statistic used is (twice the negative log of) the Neyman–Pearson likelihood ratio $\lambda$ (with nuisance parameters separately optimized for each P). An early result from CMS [@cmshiggsprop2012] is shown in Fig. \[cmsjp\]. The pdf of the test statistic, obtained by simulation, is shown for each hypothesis. ![Figure and caption from the early CMS paper on a test of J$^{\rm P}=0^+$ vs J$^{\rm P}=0^-$ for the Higgs-like boson [@cmshiggsprop2012].[]{data-label="cmsjp"}](figures/cms_parity_higgs.png){width="60.00000%"} The similarity with the spin discrimination example in Appendix \[secjhep\] is evident; the principles are the same, but in the Higgs boson case the likelihood functions are more complicated and there are nuisance parameters. As such examples are fairly rare in HEP (compared to cases of a continuous alternative), there was a fair amount of discussion within CMS about how best to present the results post-data. CMS reported: 1. The observed value $\lambda = -2\ln({{\cal L}}_{0^-} /{{\cal L}}_{0^+}) = 5.5$, favoring $0^+$; 2. For a test of H$_0$: $0^-$, the $p$-value $= 0.0072$; 3. Reversing the role of the two hypotheses and testing H$_0$: $0^+$, the $p$-value $= 0.7$; 4. [$CL_s$]{}${} = (0.0072)/(1-0.7) = 0.024$, “a more conservative value for judging whether the observed data are compatible with $0^-$.” (See Section \[cls\].) Note that for each $p$-value calculation, the relevant tail probability is for the tail in the direction of the other hypothesis. The Bayes factor (Section \[modelselection\]) for this test is similar to the observed value of $\lambda$ reported, as it differs only in the treatment of the nuisance parameters (marginalization rather than separate optimization). Demortier and Lyons [@lyonsp0p1] discuss the two $p$-values in such a simple-vs-simple case, contrasting tail probabilities with the likelihood ratio. See also Fig. 4 of Ref. [@lyonswardle]. ### Post-data choice of the C.L. Section \[duality\] describes how for a hypothesis test dual to intervals, the (pre-data) $\alpha$ corresponds to $1 -{\textnormal{C.L.}}$ Then in Section \[pvalues\], the (post-data) $p$-value is the smallest value of $\alpha$ for which H$_0:\mu = \mu_0$ would be rejected. In light of the duality, the corresponding post-data quantity for intervals is “the largest value of the C.L. (or limiting value thereof) such that $\mu_0$ is not in the confidence interval for $\mu$.” This is clearly equal to one minus the $p$-value, but (remarkably to me) it seems not to have a standard name in the statistics literature. Thus I will call it the “critical C.L.” A more natural way to think about this critical value may be “the smallest value of the C.L. such that $\mu_0$ is [*in*]{} the confidence interval for $\mu$.” For people focused more on the interval aspect of the duality, the critical C.L. may seem a natural way to express a post-data result when a particular value $\mu_0$ is of interest. In fact, shortly after the F-C paper, the CDF collaboration [@cdffc] measured a quantity called $\sin2\beta$ for which a key scientific question was whether $\sin2\beta>0$. They reported the critical value of C.L. (0.93) for which 0 was (just) included in the F-C interval (so that 0 was an endpoint of the interval). I was in a couple email exchanges regarding what this 0.93 number was, and how to interpret it (as people understood that it was not a pre-data coverage probability for the CDF interval). Eventually we all agreed (I think) that it had exactly the same status as a post-data $p$-value. Given the LR ordering used by F-C, a $p$-value associated with that ordering for the test of $\sin2\beta=0$ vs $\sin2\beta>0$ was simply 1 - 0.93 = 0.07. Reporting the critical C.L. was dual to reporting that critical $\alpha$. Classical frequentist goodness of fit (g.o.f.) {#goodness} ---------------------------------------------- If H$_0$ is specified but the alternative H$_1$ is not, then only the Type I error probability $\alpha$ can be calculated, since the Type II error probability $\beta$ depends on a H$_1$. A test with this feature is called a test for [*goodness-of-fit*]{} (to H$_0$). (Fisher called them significance tests. I leave it to others, e.g., Ref. [@lehmann1993], to try to explain his seemingly pathological opposition to explicit formulation of alternative hypotheses.) With no alternative specified, the question “Which g.o.f. test is best?” is thus ill-posed. Despite the popularity of tests with universal maps from test statistics to $\alpha$ (in particular $\chi^2$ and Kolmogorov tests), they may be ill-suited for many problems (i.e., they may have poor power ($1- \beta$) against relevant alternative H$_1$’s). In 1D, the difficulty of the unbinned g.o.f. test question is exemplified by the following simple example: “Given 3 numbers (e.g. neutrino mixing angles) in $(0,1)$, are they consistent with three calls to a random number generator that is uniform on $(0,1)$ ?” Have fun with that! For discussion of g.o.f., see my writeup [@cousinsgoodness] and references therein. As multi-D unbinned ML fits have proliferated in recent decades, there are increasing needs for multi-D unbinned g.o.f. tests. E.g., is it reasonable that 1000 events scattered in a 5D sample space have been drawn from a particular pdf (which may have parameters which were fit using an unbinned M.L. fit to those 1000 events)? Of course this is an ill-posed question, but we are looking for good omnibus tests. Then getting the null distribution of the test statistic from toy MC simulation (Section \[toymc\]) is typically doable, it seems. One can follow an unbinned ML fit with a binned g.o.f. test such as $\chi^2$, but this brings in its own issues. At a loss of power but increase in transparency, one can also perform tests on 1D or 2D distributions of the marginalized densities. For a enlightening study, see Aslan and Zech [@aslanzechdurham] (who propose an omnibus test called the energy test) and others at past PhyStat workshops. 1D issues are well-described in the book by D’Agostino and Stephens [@dagostino] (a must-read for those wanting to invent a new test). A useful review is by Mike Williams [@williamsgoodness]. Likelihood (ratio) intervals for 1 parameter {#likelihood} ============================================ Recall from Part 1: the likelihood ${{\cal L}}(\mu)$ is invariant under reparameterization from $\mu$ to $f(\mu)$: ${{\cal L}}(\mu) = {{\cal L}}(f(\mu))$. So [*likelihood ratios*]{} ${{\cal L}}(\mu_1) /{{\cal L}}(\mu_2)$ and [*log-likelihood differences*]{} $\ln{{\cal L}}(\mu_1) - \ln{{\cal L}}(\mu_2)$ are also invariant. After using the maximum-likelihood method to obtain estimate $\hat\mu$ that maximizes either ${{\cal L}}(\mu)$ or ${{\cal L}}(f(\mu))$, one can obtain a likelihood interval $[ \mu_1,\mu_2]$ as the union of all $\mu$ for which $$\label{deltal} 2\ln{{\cal L}}(\hat\mu) - 2\ln{{\cal L}}(\mu) \le Z^2,$$ for $Z$ real. As sample size increases (under important regularity conditions) this interval approaches a central confidence interval with [[C.L.]{.nodecor}]{} corresponding to $\pm Z$ Gaussian standard deviations. Section 9.3.2 of Ref. [@james2006] has a heuristic argument why this might work; more rigorous derivations are in the literature on “asymptotic” (large sample size) approximations. (See Chapter 6 in Ref. [@cox2006] and the discussions in Severini’s monograph devoted to likelihood methods [@severini2000likelihood].) But! Regularity conditions, in particular the requirement that $\hat\mu$ not be on the boundary (which can also cause practical problems if it is close to a boundary), need to be carefully checked. If $\mu \ge 0$ on physical grounds, then $\hat\mu = 0$ requires care. (See, e.g., Ref. [@ccgv2011] and references therein.) LR interval example: Gaussian pdf $p(x| \mu,\sigma)$ with $\sigma = 0.2\mu$ --------------------------------------------------------------------------- Recall from Section \[secgauss02sigma\] the Gaussian pdf with $\sigma = 0.2\mu$. The likelihood function ${{\cal L}}(\mu)$ for observed $x_0 = 10.0$ is plotted in Fig. \[fig-gaussian\_likl\], with a maximum at $\mu_{\rm ML}= 9.63$. Fig. \[gaussian\_like\_2\] is a plot of $- 2\ln{{\cal L}}(\mu)$, from which the likelihood ratio interval for $\mu$ at approximate 68% [[C.L.]{.nodecor}]{} is $[\mu_1,\mu_2] = [8.10, 11.9]$. Compare with the exact confidence interval, \[8.33,12.5\]. ![Plot of $- 2\ln{{\cal L}}(\mu)$ corresponding to Fig. \[fig-gaussian\_likl\], the case of Gaussian pdf $p(x| \mu,\sigma)$ with $\sigma = 0.2\mu$. Superimposed is the construction using Eqn. \[deltal\] with $Z=1$ to find the approximate 68% C.L. likelihood ratio interval.[]{data-label="gaussian_like_2"}](figures/gaussian_2negloglik.png){width="60.00000%"} Binomial likelihood-ratio interval example {#binomialLR} ------------------------------------------ Recall (Section \[sec-binomial\]) the example of ${n_\textnormal{\scriptsize on}}=3$ successes in ${n_\textnormal{\scriptsize tot}}=10$ trials, with ${{\cal L}}({\rho})$ and $-2\ln{{\cal L}}({\rho})$ plotted in Fig. \[binomial\_likl\]. The minimum value of the latter is 2.64. Solving for solutions to $-2\ln{{\cal L}}(\rho) = 2.64 + 1 = 3.64$, one obtains the likelihood-ratio interval $[\rho_1,\rho_2] = [0.17, 0.45]$. This can be compared to the Clopper-Pearson interval, $[\rho_1,\rho_2] = [0.14, 0.51]$, and the Wilson interval, $[\rho_1,\rho_2] = [0.18, 0.46]$. Poisson likelihood-ratio interval example ----------------------------------------- Recall the plot of ${{\cal L}}(\mu)$ in Fig. \[pois\_likli\_3obs\] for a Poisson process with $n=3$ observed. Fig. \[poisson\_2negloglik\] is a plot of $-2\ln{{\cal L}}(\mu)$, from which the likelihood ratio interval at approximate 68% C.L. can be similarly extracted, yielding $[\mu_1, \mu_2] = [1.58, 5.08]$. The central confidence interval is $[\mu_1, \mu_2] = [1.37, 5.92]$. ![Plot of $- 2\ln{{\cal L}}(\mu)$ corresponding to Fig. \[pois\_likli\_3obs\], the case of Poisson probability in Eqn. \[eqn-poisson\] with $n=3$ observed. Superimposed is the construction using Eqn. \[deltal\] with $Z=1$ to find the approximate 68% C.L. likelihood ratio interval. See also Ref. [@cousinsajp1995] []{data-label="poisson_2negloglik"}](figures/poisson_2negloglik.png){width="49.00000%"} Likelihood principle {#likelihoodprin} ==================== Recall the three methods of interval construction for binomial parameter ${\rho}$ upon observing ${n_\textnormal{\scriptsize on}}=3$ out of ${n_\textnormal{\scriptsize tot}}=10$ trials: Bayesian intervals as briefly outlined in Section \[bayesintro\], confidence intervals in Section \[sec-binomial\], and LR intervals in Section \[binomialLR\]. We can note that: - For constructing Bayesian and likelihood intervals, ${\textnormal{Bi}}({n_\textnormal{\scriptsize on}}|{n_\textnormal{\scriptsize tot}},{\rho})$ is evaluated [*only*]{} at the observed value ${n_\textnormal{\scriptsize on}}=3$. - For constructing confidence intervals we use, in addition, [ *probabilities for values of ${n_\textnormal{\scriptsize on}}$ not observed*]{}. This distinction turns out to be a [*huge*]{} deal! In both Bayesian methods and likelihood-ratio based methods, the probability (density) for obtaining the [*data at hand*]{} is used (via the likelihood function), [*but probabilities for obtaining other data are not used!*]{} In contrast, in typical frequentist calculations (confidence intervals, $p$-values), one also uses probabilities of data that could have been observed but that were [*not observed*]{}. The assertion that only the former is valid is captured by the - [*Likelihood Principle*]{}: If two experiments yield likelihood functions that are proportional, then Your inferences from the two experiments should be identical. There are various versions of the L.P., strong and weak forms, etc. See Ref. [@kendall1999] and the book by Berger and Wolpert [@bergerwolpert]. The L.P. is built into Bayesian inference (except e.g., when Jeffreys prior leads to violation). The L.P. is violated by $p$-values and confidence intervals. Jeffreys [@jeffreys1961] (p. 385) still seems to be unsurpassed in his ironic criticism of tail probabilities, which include probabilities of data [*more extreme*]{} than that observed: “[*What the use of \[the $p$-value\] implies, therefore, is that a hypothesis that may be true may be rejected because it has not predicted observable results that have not occurred.*]{}” Although practical experience indicates that the L.P. may be too restrictive, it is useful to keep in mind. When frequentist results “make no sense” or “are unphysical”, in my experience the underlying reason can be traced to a bad violation of the L.P. Likelihood principle example \#1: the “Karmen problem” ------------------------------------------------------ You expect background events sampled from a Poisson distribution with mean $b=2.8$, assumed known precisely. For signal mean $\mu$, the total number of events $n$ is then sampled from a Poisson distribution with mean $\mu+b$. So $P(n) = (\mu+b)^n \exp(- \mu-b)/n!$. Then suppose you observe no events at all! I.e., $n=0$. (The numbers are taken from an important neutrino experiment [@karmen].) Plugging in, $${{\cal L}}(\mu) = (\mu+b)^0 \exp(- \mu-b)/0! = \exp(- \mu) \exp(-b)$$ Note that changing $b$ from 0 to 2.8 changes ${{\cal L}}(\mu)$ only by the constant factor $\exp(-b)$. This gets renormalized away in any Bayesian calculation, and is irrelevant for likelihood [*ratios*]{}. So for zero events observed, likelihood-based inference about signal mean $\mu$ is [*independent of expected $b$*]{} when zero events are observed. (If the prior depends on $b$, as does the Jeffreys prior for this example in Section \[jeffprior\], then there is potentially an issue. But I do not see this used in HEP.) For essentially all frequentist [*confidence interval*]{} constructions, the fact that $n=0$ is less likely for $b=2.8$ than for $b=0$ results in [*narrower*]{} confidence intervals for $\mu$ as $b$ increases. This is a clear violation of the L.P. Likelihood principle example \#2: binomial stopping rule -------------------------------------------------------- This is famous example among statisticians, translated to HEP. You want to measure the efficiency $\epsilon$ of some trigger selection. You count until reaching ${n_\textnormal{\scriptsize tot}}=100$ zero-bias events, and note that of these, $m=10$ passed the selection. The probability for $m$ is binomial with binomial parameter $\epsilon$: $${\textnormal{Bi}}(m | {n_\textnormal{\scriptsize tot}}, \epsilon) = \frac{{n_\textnormal{\scriptsize tot}}!}{m! ({n_\textnormal{\scriptsize tot}}-m)!} \epsilon^m (1-\epsilon)^{({n_\textnormal{\scriptsize tot}}- m)}$$ The point estimate is $\hat\epsilon = 10/100$, and we can compute the binomial confidence interval (Clopper-Pearson) for $\epsilon$. Also, plugging in the observed data, the likelihood function is $${{\cal L}}(\epsilon) = \frac{100!}{10! 90!} \epsilon^{10} (1-\epsilon)^{90}$$ Suppose that your colleague [*in a different experiment*]{} counts zero-bias events until $m=10$ have passed her trigger selection. She notes that this required ${n_\textnormal{\scriptsize tot}}=100$ minimum-bias events (a coincidence). Intuitively, the fraction 10/100 [*over-estimates*]{} her trigger’s $\epsilon$ because she stopped just upon reaching 10 passed events. Indeed an unbiased estimate of $\epsilon$ and confidence interval will be slightly different from the binomial case. The relevant distribution here is (a version of) the [*negative binomial*]{}: $${\textnormal{NBi}}({n_\textnormal{\scriptsize tot}}|m , \epsilon) = \frac{{n_\textnormal{\scriptsize tot}}- 1!}{(m-1)!} \epsilon^m (1-\epsilon)^{({n_\textnormal{\scriptsize tot}}- m)}$$ Plugging in the observed data, her likelihood function is $${{\cal L}}(\epsilon) = \frac{99!}{9! 90!} \epsilon^{10} (1-\epsilon)^{90}.$$ So both you and your friend observed 10 successes out of 100 trials, but with different [*stopping rules*]{}. Your likelihood function is based on the [*binomial*]{} distribution. Your friend’s is based on the [*negative binomial*]{} distribution. The two likelihoods differ by (only!) a constant factor, so the (strong) LP says that inferences should be [*identical*]{}. In contrast, frequentist inferences that use probabilities of data not obtained result in different confidence intervals and $p$-values for the different stopping rules. Amusing sidebar: The Jeffreys prior is indeed different for the two distributions, so use of Jeffreys prior violates (strong) L.P. Stopping rule principle ----------------------- The two efficiency measurements have different [*stopping rules*]{}: one stops after ${n_\textnormal{\scriptsize tot}}$ events, and the other stops after $m$ events pass the trigger. Frequentist confidence intervals depend on the stopping rule; the likelihood function does not, except for an overall constant. So Bayesians will get the same answer in both cases, unless the [*prior*]{} depends on the stopping rule. The strong L.P. implies, in this example, that the inference is independent of the stopping rule! This [*irrelevance*]{} has been elevated to the “Stopping Rule Principle”. (It is sometimes amusing to ask a recent Bayesian convert if they know that they just bought the Stopping Rule Principle.) Concepts that average/sum over the sample space, such as bias and tail probabilities, do not exist in the pure Bayesian framework. A quote by L.J. (Jimmie) Savage (Ref. [@savagestopping], p. 76), a prominent early subjective Bayesian advocate, is widely mentioned (as seen in Google hits, which can point you to a copy of the original “Savage forum” where you can read his original note): “…I learned the stopping-rule principle from Professor Barnard, in conversation in the summer of 1952. Frankly, I then thought it a scandal that anyone in the profession could advance an idea so patently wrong, even as today I can scarcely believe that some people resist an idea so patently right." Likelihood principle discussion {#lpdisc} ------------------------------- We will not resolve this issue, but we should be aware of it. There is a lot more to the Likelihood Principle than I discuss here. See the book by Berger and Wolpert [@bergerwolpert], but be prepared for the Stopping Rule Principle to set your head spinning. When frequentist confidence intervals from a Neyman construction badly violate the L.P., use great caution! And when Bayesian inferences badly violate frequentist coverage, again use great caution! In these lectures I omitted the important (frequentist) concept of a “sufficient statistic”, due to Fisher. This is a way to describe data reduction without loss of relevant information. E.g., for testing a binomial parameter, one needs only the total numbers of successes and trials, and not the information on exactly which trials had successes. (See Ref. [@kendall1999] for math definitions.) The “Sufficiency Principle” says (paraphrasing—there are strong and weak forms) that if the observed values of the sufficient statistic in two experiments are the same, then they constitute equivalent evidence for use in inference. Birnbaum famously argued (1962) that the Conditionality Principle (Section \[condLP\]) and the Sufficiency Principle imply the Likelihood Principle. Section \[condLP\] has a few more comments on this point. Controversy continues. For a recent discussion and references, see Deborah Mayo’s 2014 detailed article [@mayo2014] and references therein, with comments by six statisticians and rejoinder. Summary of three ways to make intervals {#intervalsummary} ======================================= Table \[threeways\] summarizes properties of the three ways discussed to make intervals. Only frequentist confidence intervals (Neyman’s construction or equivalent) can guarantee coverage. Only Bayesian and likelihood intervals are guaranteed to satisfy the likelihood principle (except when a prior such as Jeffreys’s prior violates it). \[threeways\] ------------------------------------------------------------------------------------------ ---------- --------------- --------------- Bayesian Frequentist Likelihood credible confidence ratio Requires prior pdf? Yes No No Obeys Likelihood Principle? Yes\* No Yes Random variable(s) in $P({\mu_\textnormal{t}}\in [ \mu_1,\mu_2]) = {\textnormal{C.L.}}$ $\mu_t$ $\mu_1,\mu_2$ $\mu_1,\mu_2$ (Over)Coverage guaranteed? No Yes No Provides $P$(parameter$|$data)? Yes No No ------------------------------------------------------------------------------------------ ---------- --------------- --------------- : Summary of three ways to make intervals. The asterisk reminds us that the choice of prior might violate the likelihood principle. Table \[ajpintervals\], inspired by Ref. [@cousinsajp1995], gives illustrative intervals for a specific Poisson case. It is of great historical and practical importance in HEP that the [*right*]{} endpoint of Bayesian central intervals with uniform prior is mathematically identical to that of the frequentist central confidence interval (5.92 in the example of $n=3$, but true for any $n$). By the reasoning of Fig. \[limits\_central\], this identity applies to [ *upper limits*]{}. In contrast, the prior $1/\mu$ is needed to obtain identity of the [*left*]{} endpoint of confidence intervals, and hence of [*lower*]{} limits (1.37 in the example, but true for any $n$). The fact that our field is almost always concerned with upper (rather than lower) limits on Poisson means is responsible for the nearly ubiquitous use of the uniform prior (instead of the Jeffreys prior, for example); as noted in Section \[fivefaces\], the main use of the Bayesian machinery in HEP is as a technical device for generating frequentist inference, i.e., intervals that are (at least approximately) confidence intervals. When known (fixed) background is added to the problem, the Bayesian intervals with uniform prior for $\mu$ become conservative (over-cover), a feature that many are willing to accept in order to obey the likelihood principle (or perhaps for less explicitly stated reasons). (Even without background, frequentist intervals over-cover due to discreteness of $n$, as discussed in Section \[overcov\].) \[ajpintervals\] Method Prior Interval Length Coverage? ------------------------------ ---------------- -------------- -------- ----------- Wald, $n \pm \sqrt{n}$ — (1.27, 4.73) 3.36 no Garwood, Frequentist central — (1.37, 5.92) 4.55 yes Bayesian central 1 (2.09, 5.92) 3.83 no Bayesian central $1/\mu$ (1.37, 4.64) 3.27 no Bayesian central Jeffreys $1/\sqrt{\mu}$ (1.72, 5.27) 3.55 no Likelihood ratio — (1.58, 5.08) 3.50 no : 68 %C.L. confidence intervals for the mean of a Poisson distribution, based on the single observation $n=3$, calculated by various methods. 1D parameter space, 2D observation space ======================================== Until now we have considered 1D parameter space and 1D observation space. Adding a second observation adds surprising subtleties. As before, $\mu$ is a 1D parameter (often called $\theta$ by statisticians). An experiment has two observations, the set $\{x_1, x_2\}$. These could be: - two samples from the same $p(x| \mu)$, or - one sample each of two different quantities sampled from a joint density $p(x_1,x_2 | \mu)$. For frequentist confidence intervals for $\mu$, we proceed with a Neyman construction. Prior to the experiment, for each $\mu$, one uses an ordering principle on the sample space ($x_1,x_2$) to select an acceptance region ${{\cal A}}(\mu)$ in the sample space ($x_1, x_2$) such that $P((x_1,x_2) \in {{\cal A}}(\mu)) = {\textnormal{C.L.}}$ (As mentioned in Section \[neymanconstruction\], this was the illustration in Neyman’s original paper.) Upon performing the experiment and observing the values $\{x_{1,0},x_{2,0}\}$, the confidence interval for $\mu$ at confidence level ${\textnormal{C.L.}}$ is the union of all values of $\mu$ for which the corresponding acceptance region ${{\cal A}}(\mu)$ includes the observed data $\{x_{1,0},x_{2,0}\}$. The problem is thus reduced to choosing an ordering of the points $(x_1,x_2)$ in the sample space, in order to well-define ${{\cal A}}(\mu)$, given a [[C.L.]{.nodecor}]{} This turns out to be surprisingly subtle, exposing a further foundational issue. Conditioning: Restricting the sample space used\ by frequentists {#conditioning} ------------------------------------------------ We now return to the point mentioned in Section \[aside\] regarding the “whole space” of possibilities that is considered when computing probabilities. In Neyman’s construction in the 2D sample space ($x_1,x_2$), the probabilities $P((x_1,x_2) \in {{\cal A}}(\mu))$ associated with each acceptance region ${{\cal A}}(\mu)$ are [*unconditional*]{} probabilities with respect to the “whole” sample space of all possible values of ($x_1,x_2$). In contrast, Bayesian inference is based on a single point in this sample space, the observed ($x_{1,0},x_{2,0}$), per the Likelihood Principle. There can be a middle ground in frequentist inference, in which the probabilities $P((x_1,x_2) \in {{\cal A}}(\mu))$ are [*conditional*]{} probabilities conditioned on a function of ($x_1,x_2$), in effect restricting the sample space to a “recognizable subset” depending on the observed data. Restricting the sample space in this way is known as [ *conditioning*]{}. Here I discuss two famous examples: - A somewhat artificial example of Welch [@welch1939] where the conditioning arises from the mathematical structure; - A more physical example of Cox [@cox1958] where the argument for conditioning seems “obvious”. ### Example of B.L. Welch (1939) In this example, $x_1$ and $x_2$ are two samples from same $p(x| \mu)$, a rectangular pdf given by (Fig. \[welchpdf\]) $$\label{rectangle} p(x|\mu) = \begin{cases} 1, & \text{if~} \mu- \frac{1}{2} \le x \le \mu + \frac{1}{2}\\ 0, & \text{otherwise}. \end{cases}$$ The observed data is a set of two values $\{x_1, x_2\}$ sampled from this pdf. From these data, the point estimate for $\mu$ is the sample mean, $\hat\mu = \bar x = (x_1 + x_2)/2$. (Aside: if more than two samples are observed, the point estimate is the mean of the outermost two, not the whole sample mean; see Section \[point-est\].) ![The rectangular pdf $p(x|\mu)$ of the Welch example, centered on $\mu$.[]{data-label="welchpdf"}](figures/welch_rect_prob.png){width="49.00000%"} What is a 68% [[C.L.]{.nodecor}]{} central confidence interval for $\mu$? To perform a Neyman construction, for each $\mu$ we must define an acceptance region ${{\cal A}}(\mu)$ containing 68% of the unit square $(x_1, x_2)$ centered on $\mu$, as in Fig. \[welchA\]. Which 68% should one use? Centrality implies symmetry, but we need something else to rank points in the plane. The N-P Lemma suggests a likelihood ratio, but first let’s think about some examples of possible pairs $\{x_1, x_2\}$. ![The starting point for constructing confidence intervals in the Welch example: in the 2D data space centered on ($x_1=\mu, x_2=\mu$) what should be the acceptance region ${{\cal A}}(\mu)$ containing 68% of the unit square? One should imagine a $\mu$ axis perpendicular to the plane of the square, with such a square at each $\mu$.[]{data-label="welchA"}](figures/welch2by2_a.png){width="49.00000%"} A “lucky” sample with $|x_1 - x_2|$ close to 1 is shown in Fig. \[lucky\](left): ${{\cal L}}(\mu) = {{\cal L}}_1(\mu) \times {{\cal L}}_2(\mu)$ is very narrow. Is it thus reasonable to expect small uncertainty in $\hat\mu$? An “unlucky” sample with $|x_1 - x_2|$ close to 0 is shown in Fig. \[lucky\](right): ${{\cal L}}(\mu)$ has full width close to 1, as the second observation adds almost no useful information. Should we expect a 68% C.L. confidence interval for $\mu$ that is the same as for only one observation, i.e. with length 0.68? ![(left) A “lucky” set of observations $\{x_1, x_2\}$ having small overlap of the likelihood functions, so that possible true $\mu$ is localized. (right) an “unlucky” set of observations $\{x_1, x_2\}$, for which the second observation adds little additional information.[]{data-label="lucky"}](figures/welch_two_rect.png "fig:"){width="49.00000%"} ![(left) A “lucky” set of observations $\{x_1, x_2\}$ having small overlap of the likelihood functions, so that possible true $\mu$ is localized. (right) an “unlucky” set of observations $\{x_1, x_2\}$, for which the second observation adds little additional information.[]{data-label="lucky"}](figures/welch_two_rect_unlucky.png "fig:"){width="39.00000%"} Intuition says that a reasonable answer might be a confidence interval centered on $\hat\mu$, with a length that is 68% of the width of ${{\cal L}}$, i.e., the interval $\hat\mu \pm 0.34(1-|x_1 - x_2|)$. From this argument, it seems reasonable for the [*post-data*]{} uncertainty to depend on $|x_1 - x_2|$, which of course cannot be known in advance. This quantity $|x_1 - x_2|$ is a classic example of an [*ancillary statistic*]{} $A$: it has information on the [ *uncertainty*]{} on the estimate of $\mu$, but no information on $\mu$ itself, because the distribution of $A$ does not depend on $\mu$. An idea dating to Fisher and before is to divide the full “unconditional” sample space into “recognizable subsets” (in this case having same or similar values of $A$), and calculate probabilities using the “relevant” subset rather than the whole space! The (representative) diagonal lines in Fig. \[neymanA\](left) show a partition of the full sample space via the ancillary statistic $ A= |x_1 - x_2|$. Within each partition, in Fig. \[neymanA\](right), the shading shows a central 68% probability acceptance region (red fill). We are thus using [*conditional probabilities*]{} (still frequentist!) $p(x|A,\mu)$ in this Neyman construction, with desired probability 68% within each partition. (Aside: A set of measure zero has zero probability even if non-zero pdf, so in general, care is needed in conditioning on exact value of continuous $A$ in $p(x|A,\mu)$. This is not an issue here in this example.) ![(left) Dividing the observation space into subsets based on the ancillary statistic $|x_1 - x_2|$. (right), within each subset, acceptance regions containing the 68% central values of $|x_1 + x_2|$.[]{data-label="neymanA"}](figures/welch_rect_prob_partition.png "fig:"){width="49.00000%"} ![(left) Dividing the observation space into subsets based on the ancillary statistic $|x_1 - x_2|$. (right), within each subset, acceptance regions containing the 68% central values of $|x_1 + x_2|$.[]{data-label="neymanA"}](figures/welch_rect_prob_partition_central.png "fig:"){width="49.00000%"} The resulting ${{\cal A}}(\mu)$ for the whole square fills 68% of the square, so there is correct unconditional probability as well. Imagine a plane such as this for every $\mu$, and then obtaining the data $\{x_1, x_2\}$. The confidence interval for $\mu$ is then the union of all values of $\mu$ for which the observed data are in ${{\cal A}}(\mu)$. A moment’s thought will confirm that this results in confidence intervals centered on $(x_1 + x_2)/2$, with a length that is 68% of $|x_1 - x_2|$, i.e., $\hat\mu \pm 0.34(1-|x_1 - x_2|)$, as intuitively thought reasonable! This construction is known as “conditioning” on the ancillary statistic $A$. In fact, it can be more simply stated: [ *post*]{}-data, ignore the construction in the whole sample space for values of $A$ other than that observed, and [*proceed as if $A$ had been fixed, rather than randomly sampled!*]{} Now the catch: one can find acceptance regions ${{\cal A}}(\mu)$ that correspond to hypothesis tests with [*more power*]{} (lower Type 2 error probability $\beta$) in the unconditional sample space! A couple examples from the literature are shown in Fig. \[welchpower\]. The construction on the left results in 68% C.L. intervals with length independent of $|x_1 - x_2|$, namely $\hat\mu \pm 0.22$ at 68% C.L. They obtain 68% coverage in the unconditional sample space by having 100% coverage in the subspace where $|x_1 - x_2| \approx 1$ (narrow likelihood), while badly undercovering when $|x_1 \approx x_2|$. .[]{data-label="welchpower"}](figures/welch_rect_prob_partition_power1.png "fig:"){width="49.00000%"} .[]{data-label="welchpower"}](figures/welch_rect_prob_partition_power2.png "fig:"){width="49.00000%"} Welch’s 1939 paper argued [*against*]{} conditioning because it is less powerful in the unconditional sample space! Neyman’s position is not completely clear but he also seems to have been against conditioning on ancillaries (which was Fisher’s idea) when it meant an overall loss of power [@lehmann1993; @berger2003]. Most modern writers use Welch’s example as an “obvious” argument [*in favor*]{} of conditioning, unless one is in an “industrial” setting where the unconditional ensemble is sampled repeatedly and the result for an individual sample is not of much interest. ### Efron example that is structurally similar to Welch example Brad Efron’s talk at PhyStat-2003 [@efronslac] included a similar example using the Cauchy distribution, where the ancillary statistic is the curvature of ${{\cal L}}(\mu)$. He called the conditional answer “correct”. ### Cox two-measuring device example (1957) For measuring a mean $\mu$ with Gaussian resolution, one of two devices is selected randomly with equal probability: Device \#1 with $\sigma_1$ Device \#2 with $\sigma_2 \ll \sigma_1$. Then a single measurement sample is made with the chosen device. In my notation, $x_1$ is the index (1 or 2) chosen randomly and specifying the device, and $x_2$ is the single sample from the selected Gaussian measurement. The total observed data is then, as before, $\{x_1, x_2\}$. But in this example, $x_1$ and $x_2$ are samples of two different quantities from the joint density $p(x_1,x_2 | \mu)$. In Ref. [@behnke2013] (p. 112) Luc Demortier gives a nice example in HEP: $\mu$ is the mass of a decaying particle with probability $p_h$ to decay hadronically (mass resolution $\sigma_1$) and probability $1-p_h$ to decay leptonically with mass resolution $\sigma_2$). Thus the “measuring machine” chosen randomly is the detector used to measure the decay mode that is randomly chosen by quantum mechanics. So $\hat\mu = x_2$. What is the confidence interval? The index $x_1$ is an ancillary statistic, and it is reasonable (obvious?) to condition on it. I.e., we report a confidence interval giving correct coverage in the [*subspace of measurements that used the same measuring device that we used*]{}. So the 68% [[C.L.]{.nodecor}]{} confidence interval is: $\hat\mu \pm \sigma_1$ if Device \#1 was randomly selected $\hat\mu \pm \sigma_2$ if Device \#2 was randomly selected Again it turns out that more powerful tests can be found. Demortier gives details on how the average length of intervals optimized in the unconditional sample space is shorter in the HEP example. Here I give Cox’s discussion. If one is testing $\mu=0$ vs $\mu= \mu_1$, with $\mu_1$ roughly the size of $\sigma_1$ (the larger $\sigma$), consider the following intervals: $\hat\mu \pm (0.48)\sigma_1$ if Device \#1 used (covers true $\mu$ in 37% of uses), and $\hat\mu \pm 5\sigma_2$ if Device \#2 used (covers true $\mu$ nearly 100% of uses). Then the true $\mu$ is covered in (37/2 + 100/2)% = 68% of all intervals! The unconditional (full sample space) coverage is correct, but conditional coverage is not. Due to the smallness of $\sigma_2$, the average length of all intervals [*when averaging over the entire unconditional sample space*]{} is smaller than for conditional intervals with independent coverage. One gives up power with Device \#1 and uses it in Device \#2. Cox asserts: “If, however, our object is to say ‘what can we learn from the data that we have’, the unconditional test is surely no good.” [@cox1958] (p. 361) (See also Ref. [@cox2006] (pp. 47-48).) Conditioning in HEP ------------------- A classic example is a [*measurement of the branching fraction of a particular decay mode*]{} when the [*total*]{} number of decays $N$ can fluctuate because the experimental design is to run for a fixed length of time. Then $N$ is an ancillary statistic. You perform an experiment and obtain $N$ total decays, and then do a toy MC simulation (Section \[toymc\]) of repetitions of the experiment. Do you let $N$ fluctuate, or do you fix it to the value observed? It may seem that the toy MC should include your complete procedure, including fluctuations in $N$. But the above arguments would point toward [*conditioning on the value of the ancillary statistic actually obtained*]{}. So your branching fraction measurement is binomial with trials $N$. This was originally discussed in HEP by F. James and M. Roos [@jamesroos]. For more complete discussion, see Ref. [@cht2010]. Conditioning and the likelihood principle {#condLP} ----------------------------------------- To summarize: conditioning on an ancillary statistic $A$ means: Even though $A$ was randomly sampled in the experimental procedure, after data are obtained, proceed as if $A$ had been fixed to the value observed. Ignore the rest of the sample space with all those other values of $A$ that you could have obtained, but did not. [*The Welch and Cox (and Efron) examples reveal a real conflict between N-P optimization for power and conditioning to optimize relevance*]{}. The assertion that inference should be conditioned on an ancillary in the Welch example (where it comes out of the math) is often called the “Conditionality Principle” (CP). Conditioning in the Cox example (a “mixture experiment” where the ancillary has physical meaning about which experiment was performed) is then called the “Weak Conditionality Principle” (WCP). But note: in sufficiently complicated cases (for example if there is more than one ancillary statistic), the procedure is less clear. In many situations, ancillary statistics do not exist, and it is not at all clear how to restrict the “whole space” to the relevant part for frequentist coverage. The pure Bayesian answer is to collapse the whole sample space to the data observed, and refer only to the probability of the data observed, i.e., the likelihood principle discussed in Section \[likelihoodprin\]. This is literally the ultimate extreme in conditioning, [*conditioning (in the continuous case) on a point of measure zero!*]{} (You can’t get any more “relevant”.) But the price is giving up coverage. When there are “recognizable subsets” with varying coverage, Buehler [@buehler1959] has discussed how a “conditional frequentist” can win bets against an “unconditional frequentist”. (See Refs. [@cousinsvirtual; @cousins2011].) I emphasize conditioning not only for the practical issues, but also to explain that there are intermediate positions between the full unconditional frequentist philosophy and the Likelihood Principle of Section \[likelihoodprin\]. A key point is that unconditional frequentist coverage is a [*pre-data*]{} assessment: the entire confidence belt is constructed independent of where the observation lies. Thus a big argument is whether [*unconditional*]{} coverage remains appropriate [*post-data*]{}, after one knows where one’s observed data lies in the sample space. When the “measurement uncertainty” depends strongly on where one’s data lies, then the arguments for conditioning seem strong. Whether or not one takes conditioning to the extreme and considers only the (measure-zero) subset of the sample space corresponding to the data observed is the issue of the Likelihood Principle [@bergerwolpert]. It is not surprising that pure Bayesians argue for the importance of [*relevance*]{} of the inference, and criticize frequentists for the danger of irrelevance (and the difficulty of diagnostic of irrelevance). And it is not surprising that pure frequentists argue for the importance of a useful measure of “error rates”, in the sense of Type 1 and Type 2 errors, coverage, etc., which may at best be estimates if the L.P. is observed. 2D parameter space, multi-D observation space ============================================= We (finally) generalize to two parameters $\mu_1$ and $\mu_2$, with both true values unknown. (I hope that from context, there is no confusion from using the subscripts 1 and 2 to indicate different parameters, whereas in the 1D case above, they indicate the endpoints of a confidence interval on the single parameter $\mu$.) Let data $x$ be a multi-D vector, so the model is $p(x| \mu_1, \mu_2)$. The observed vector value is $x_0$. First consider the desire to obtain a 2D confidence/credible [ *region*]{} in the parameter space $(\mu_1, \mu_2)$. All three methods discussed for intervals handle this in a straightforward (in principle) generalization. We mention the first two briefly and devote a subsection to the third: [*Bayesian:*]{} Put the observed data vector $x_0$ into $p(x| \mu_1, \mu_2$) to obtain the likelihood function ${{\cal L}}(\mu_1, \mu_2$). Multiply by the prior pdf $p(\mu_1, \mu_2$) to obtain the 2D posterior pdf $p(\mu_1, \mu_2|x_0)$. Use the posterior pdf to obtain credible regions, etc., in $(\mu_1, \mu_2)$. [*Confidence intervals:*]{} Perform a Neyman construction: Find acceptance [*regions*]{} ${{\cal A}}(\mu_1, \mu_2)$ for $x$ as a function of $(\mu_1, \mu_2)$. The 2D confidence region is the union of all $(\mu_1, \mu_2)$ for which $x_0$ is in ${{\cal A}}(\mu_1, \mu_2)$. Likelihood [*regions*]{} in $\ge 2D$ parameter space {#rppregion} ---------------------------------------------------- Recall the method for 1D confidence [*intervals*]{}, Eqn. \[deltal\]. For a joint 2D likelihood [*region*]{}, first find the global maximum of ${{\cal L}}(\mu_1, \mu_2$), yielding point estimates $\hat\mu_1, \hat\mu_2$. Then find the 2D contour bounded by $$\label{wilks} 2\Delta \ln {{\cal L}}= 2\ln{{\cal L}}(\hat\mu_1, \hat\mu_2) - 2\ln{{\cal L}}(\mu_1, \mu_2) \le C,$$ where C comes from Wilks’s Theorem, tabulated in the PDG RPP [@pdg2018] (Table 39.2) for various C.L. and for various values of $m$, the dimensionality of the confidence region. Here we have the case $m=2$, for which $C= 2.3$ for 68% C.L. ![Sketch of a 2D joint 68% C.L. likelihood region for $(\mu_1, \mu_2)$, obtained via Eqn. \[wilks\].[]{data-label="wilksfig"}](figures/likl_region_2d.png){width="49.00000%"} This region is an approximate confidence region, as sketched in Fig. \[wilksfig\]. As in 1D, Wilks’s Theorem is an asymptotic (large $N$) result, with various “regularity conditions” to be satisfied. (Again see, e.g., Ref. [@ccgv2011] and references therein.) Nuisance parameters ------------------- Frequently one is interested in considering one parameter at a time, irrespective of the value of other parameter(s). The parameter under consideration at the moment is called the “parameter of interest” and the other parameters (at that moment) are called “nuisance parameters”. E.g., if $\mu_1$ is of interest and $\mu_2$ is a nuisance parameter, then ideally one seeks a 2D confidence region that is a vertical “stripe” in the ($\mu_1, \mu_2$) plane as in Fig. \[stripe\](left); this allows the same 1D interval to be quoted for $\mu_1$, independent of $\mu_2$. Or, in different moment, $\mu_2$ may be of interest and $\mu_1$ is a nuisance parameter; then one seeks a horizontal stripe, as in Fig. \[stripe\](right). How can one construct those stripes? ![The dashed curve is the boundary of the 2D confidence region when both parameters are of interest. That shaded stripes are (left) 2D region to get 1D 68% C.L. interval for $\mu_1$ ($\mu_2$ is nuisance). (right) 2D region to get 1D 68% C.L. interval for $\mu_2$ ($\mu_1$ is nuisance) []{data-label="stripe"}](figures/2d_vertical_stripe.png "fig:"){width="49.00000%"} ![The dashed curve is the boundary of the 2D confidence region when both parameters are of interest. That shaded stripes are (left) 2D region to get 1D 68% C.L. interval for $\mu_1$ ($\mu_2$ is nuisance). (right) 2D region to get 1D 68% C.L. interval for $\mu_2$ ($\mu_1$ is nuisance) []{data-label="stripe"}](figures/2d_horizontal_stripe.png "fig:"){width="49.00000%"} Each of the three main classes of constructing intervals (Bayesian, Neyman confidence, likelihood ratio) has a “native” way to incorporate the uncertainty on the nuisance parameters, described in Sections \[nuisbayes\]–\[nuislikl\]. [*But this remains a topic of frontier statistics research*]{}. ### Systematic uncertainties as nuisance parameters Systematic uncertainties provide prototype examples of parameters that are frequently nuisance parameters, so I mention them briefly here. A typical measurement in HEP has many subsidiary measurements of quantities not of direct physics interest, but which enter into the calculation of the physics quantity of particular interest. E.g., if an absolute cross section is measured, one will have uncertainty in the integrated luminosity $L$, in the background level $b$, the efficiency $e$ of detecting the signal, etc. In HEP, we call these [*systematic uncertainties*]{}, but statisticians (for the obvious reason) refer to $L$, $b$, and $e$ as [*nuisance parameters*]{}. For discussion of many of the issues with systematic uncertainties in HEP, see Refs. [@sinervoslac; @lyonshein2007; @barlowsyst]. However, it is important to keep in mind is that whether or not a parameter is considered to be a nuisance parameter depends on context. For example, in measurements of Higgs boson couplings, the mass of the Higgs boson is typically regarded as a nuisance parameter. But clearly the mass of the Higgs boson can itself be the primary object of a measurement, in which case the couplings are the nuisance parameters. Nuisance parameters I: Bayesian credible intervals {#nuisbayes} -------------------------------------------------- Construct a multi-D prior pdf $p(\textnormal{parameters})$ for the space spanned by all parameters. Multiply by the likelihood function ${{\cal L}}({\rm data}|{\rm parameters})$ for the data obtained to obtained the multi-D posterior pdf. Integrate over the full subspace of all nuisance parameters (marginalization). Thus obtain the 1D posterior pdf for the parameter of interest. Further use of the posterior pdf is thus reduced to the case of no nuisance parameters. [*Problems*]{}: The multi-D prior pdf is a problem for both subjective and non-subjective priors. In HEP there has been little use of the favored non-subjective priors (reference priors of Bernardo and Berger). The high-D integral can be a technical problem, more and more overcome by Markov Chain Monte Carlo. As with all Bayesian analyses, how does one interpret probability if “default” priors are used, so that coherent subjective probability is not applicable? ### Priors for nuisance parameters {#nuisprior} It used to be (unfortunately) common practice to express, say, a 50% systematic uncertainty on a positive quantity as a Gaussian with 50% rms. Then one “truncated” the Gaussian by not using non-positive values. As mentioned in Section \[pseudobayes\] but worth repeating, in Bayesian calculations, the interaction of a uniform prior for a Poisson mean and a “truncated Gaussian” for systematic uncertainty in efficiency leads to an integral that diverges if the truncation is at origin [@demortierdurham]. In evaluating the integral numerically, some people did not even notice! [*Recommendation*]{}: Use lognormal or (certain) Gamma distributions instead of truncated Gaussian. Recipes are in my note [@cousinslognormal]. Nuisance parameters II: Neyman construction ------------------------------------------- For each point in the subspace of nuisance parameters, treat them as fixed true values and perform a Neyman construction for multi-D confidence regions in the full space of all parameters. Project these regions onto the subspace of the parameter of interest. [*Problems*]{}: Typically intractable and causes overcoverage, and therefore rarely attempted. Tractability can sometimes be recovered by doing the construction in the lower dimensional space of the profile likelihood function, obtaining approximate coverage. (This is one way to interpret the Kendall and Stuart pages on the likelihood ratio test with nuisance parameters [@kendall1999].) [*Problem*]{}: Not well-studied. Typically “elimination” is done in a way technically feasible, and the coverage studied with toy MC simulation (Section \[toymc\]). Nuisance parameters III: Likelihood ratio intervals {#nuislikl} --------------------------------------------------- Many of us raised on MINUIT MINOS read the article by F. James, “Interpretation of the Shape of the Likelihood Function around Its Minimum,” [@james1980]. Whereas the 2D region in Section \[rppregion\] has $m=2$ and hence $2\Delta\ln{{\cal L}}\le 2.3$, for 1D intervals on $\mu_1$, we first a make 2D [*contour*]{} with the $m=1$ value, $2\Delta \ln{{\cal L}}= 1$, as shown by the black dashed curve in Fig. \[mlstripe\](right). Then the [*extrema*]{} in $\mu_1$ of this curve correspond to the endpoints of the approximate confidence interval for $\mu_1$. ![Sketches for likelihood ratio regions. (left) The 2D confidence region from Fig. \[wilksfig\], when both parameters are of interest. (right) On the same scale, the 2D confidence region if $\mu_1$ is of interest and $\mu_2$ is a nuisance parameter, so that effectively one obtains a confidence interval for $\mu_1$ that is independent of $\mu_2$. The width of the stripe is smaller than the width of the extrema of the region on the left, since a smaller value of $C$ is used in Eqn. \[wilks\]: the dashed contour on the right is inside the solid contour on the left. Both the left and the right shaded regions correspond to the same C.L.; the left region is relevant for joint inference on the pair of parameters, while the right region is relevant when only $\mu_1$ is of interest.[]{data-label="mlstripe"}](figures/2d_likl_region.png "fig:"){width="49.00000%"} ![Sketches for likelihood ratio regions. (left) The 2D confidence region from Fig. \[wilksfig\], when both parameters are of interest. (right) On the same scale, the 2D confidence region if $\mu_1$ is of interest and $\mu_2$ is a nuisance parameter, so that effectively one obtains a confidence interval for $\mu_1$ that is independent of $\mu_2$. The width of the stripe is smaller than the width of the extrema of the region on the left, since a smaller value of $C$ is used in Eqn. \[wilks\]: the dashed contour on the right is inside the solid contour on the left. Both the left and the right shaded regions correspond to the same C.L.; the left region is relevant for joint inference on the pair of parameters, while the right region is relevant when only $\mu_1$ is of interest.[]{data-label="mlstripe"}](figures/2d_likl_region_vertical_stripe.png "fig:"){width="49.00000%"} ### [*Profile*]{} likelihood function {#profilel} At the Fermilab Confidence Limits Workshop in 2000, statistician Wolfgang Rolke expressed the construction in a different (but [ *exactly equivalent*]{}) way [@rolkeclk; @rolke2005], as illustrated in Fig. \[profile\] and paraphrased as follows: - For each $\mu_1$, find the value $\hat{\hat\mu}_2$ that minimizes $-2\ln{{\cal L}}(\mu_1,\hat{\hat\mu}_2)$. Make a 1D plot vs $\mu_1$ of (twice the negative log of) this “profile likelihood function” $-2\ln{{\cal L}}_{\rm profile}(\mu_1)$. Use the $m=1$ threshold on $-2\ln{{\cal L}}_{\rm profile}(\mu_1)$, i.e., Eqn. \[deltal\], to obtain intervals at the desired C.L. The interval one obtains in Fig. \[profile\] is the exact [*same*]{} interval as obtained by “MINOS” in Fig. \[mlstripe\](right). Can you see why? Since 2000, the “profile” statistical terminology has permeated HEP. The “hat-hat” notation (stacked circumflex accents) is also used by “Kendall and Stuart” [@kendall1999] in the generalized hypothesis test that is dual to the intervals of Feldman and Cousins. See also Ref. [@ccgv2011]. [*Warning:*]{} Combining profile likelihoods from two experiments is unreliable. Apply profiling after combining the full likelihoods [@lyonscombine]. ![Red curve is path $(\mu_1,\hat{\hat\mu}_2)$ along which profile ${{\cal L}}$ is evaluated[]{data-label="profile"}](figures/2d_likl_region_profile.png){width="49.00000%"} [*Problems with profile likelihood*]{}: Coverage is not guaranteed, particularly with small sample size. By using the best-fit value of the nuisance parameters corresponding to each value of the parameter of interest, this has an (undeserved?) reputation for underestimating the true uncertainties. In Poisson problems, the profile likelihood (MINUIT MINOS) gives surprisingly good performance in many problems. See Rolke, et al. [@rolke2005]. In some cases (for example when there are spikes in ${{\cal L}}$), marginalization may give better frequentist performance, I have heard. For small sample sizes, there is no theorem to tell us whether profiling or marginalization of nuisance parameters will give better frequentist coverage for the parameter of interest. [*Not*]{} eliminating a nuisance parameter: the “raster scan” {#raster} ------------------------------------------------------------- In the 2D example of Figure \[mlstripe\], the right side is an attempt to “eliminate” the nuisance parameter $\mu_2$ by obtaining an interval estimate for $\mu_1$ that is independent of $\mu_2$. However, it may be that it is preferable simply to quote a 1D interval estimate for $\mu_1$ [*as a function of assumed true values for $\mu_2$.*]{} In a toy neutrino oscillation example, Feldman and Cousins [@feldman1998] contrasted their unified approach with this method, which they referred to as a “raster scan”, in analogy with the way old televisions drew lines across the screen. As an example, I am reminded that before the top quark mass was known, other measurements (and theoretical predictions) were typically given as a function of the unknown top quark mass, with no attempt to “eliminate” it (for example by putting a Bayesian prior pdf on it and integrating it out). In the search for the Higgs boson and ultimately the discovery, its unknown mass was a nuisance parameter that was also treated by raster scan: all the plots of upper limits on cross section, as well as $p$-values testing the background-only hypothesis, are given as a function of mass. When to use a raster scan is a matter of judgment; for some useful considerations and detailed explanations, see Ref. [@lyonsraster]. Hybrid techniques: Introduction to pragmatism {#pragmatism} --------------------------------------------- Given the difficulties with all three classes of interval estimation, especially when incorporating nuisance parameters, it is common in HEP to relax foundational rigor and: - Treat nuisance parameters in a Bayesian way (marginalization) while treating the parameter of interest in a frequentist way. Virgil Highland and I were early advocates of this for the luminosity uncertainty in upper limit calculation [@cousinshighland1992]. At PhyStat 2005 at Oxford, Kyle Cranmer revealed problems when used for background mean in a 5$\sigma$ discovery context [@cranmeroxford2005]. For a review of the background case and connection to George Box’s semi-Bayesian “prior predictive $p$-value”, see Cousins, Linnemann, and Tucker [@clt2008]. - Or, treat nuisance parameters by profile likelihood while treating parameter of interest another way, - Or, use the Bayesian framework (even without the priors recommended by statisticians), but evaluate the frequentist performance [@bergerclk]. In effect (as in profile likelihood) one gets approximate coverage while respecting the L.P. In fact, the statistics literature has attempts to find prior pdfs that lead to posterior pdfs with good frequentist coverage: [*probability matching priors. At lowest order in 1D, the matching prior is the Jeffreys prior!*]{} [@welchpeers1963]. A couple looks at the literature on nuisance parameters ------------------------------------------------------- In the mid-2000s, Luc Demortier and I both looked in the statistics literature regarding nuisance parameters. I thought that my note was fairly thorough until I read his! Our writeups: - R.D. Cousins, “Treatment of Nuisance Parameters in High Energy Physics, and Possible Justifications and Improvements in the Statistics Literature,” presented at the PhyStat 2005 at Oxford [@cousinsoxford2005] with response by statistician Nancy Reid [@reidoxford2005]. <!-- --> - Luc Demortier, “P Values: What They Are and How to Use Them,” [@lucpvalue]. See also Luc’s scholarly Chapter 4 on interval estimation in Ref. [@behnke2013]. HEP “State of the art” for dealing with nuisance parameters ----------------------------------------------------------- All three main classes of methods are commonly used on the parameter of interest. In addition: - Both marginalization and profiling are commonly used to treat nuisance parameters. - Many people have the good practice of checking coverage. - Too little attention is given to priors, in my opinion. But the flat prior for Poisson mean is “safe” (from frequentist point of view) for [*upper*]{} limits (only!). A serious analysis using any of the main methods requires coding up the model $p(x|\mu)$. (It is needed at $x=x_0$ to obtain the likelihood function, and at other $x$ as well for confidence intervals.) Doing this (once!) with the RooFit [@roofit] modeling language gives access to RooStats [@roostats] techniques for all three classes of calculations, and one can mix/match nuisance parameter treatments. Evaluation of coverage with toy MC simulation {#toymc} ============================================= For a single parameter of interest $\mu$, one typically reports the confidence interval $[\mu_1,\mu_2]$ at some [[C.L.]{.nodecor}]{} after elimination of nuisance parameters by some approximate method and construction of intervals perhaps involving more approximations. It is important to check that the approximations in the whole procedure have not materially altered the claimed coverage, defined in Eqn. \[coverage\]. Typically the performance is evaluated with a simplified MC simulation, referred to as “toy Monte Carlo simulation”. First I describe the most thorough evaluation (very CPU intensive), and then some approximations. In frequentist statistics, the true values of all parameters are typically [*fixed but unknown*]{}. A complete, rigorous check of coverage considers a fine multi-D grid of [*all*]{} parameters, and [*for each multi-D point in the grid*]{}, generates an ensemble of toy MC pseudo-experiments, runs the full analysis procedure, and finds the fraction of intervals covering the ${\mu_\textnormal{t}}$ of interest that was used for that ensemble. I.e., one calculates $P({\mu_\textnormal{t}}\in [ \mu_1,\mu_2])$, and compares to [[C.L.]{.nodecor}]{} Thus a thorough check of frequentist coverage includes: 1. Fix all parameters (of interest and nuisance) to a single set of true values. For this set, 1. Loop over “pseudo-experiments” 2. For each pseudo-experiment, loop over events, generating each event with toy data generated from the statistical model with parameters set equal to the fixed set. 3. Perform the same analysis on the toy events in the pseudo-experiment as was done for the real data. 4. Find that fraction of the pseudo-experiments for which parameter(s) of interest are included in stated confidence intervals or regions. 2. [*Repeat for various other fixed sets of all parameters*]{}, ideally a fine grid. [*But*]{}…the ideal of a fine grid is usually impractical. So the issue is what selection of “various other fixed sets” is adequate. Obviously one should check coverage for the set of true values set equal to the global best-fit values. Just as obviously, this may not be adequate. Some exploration is needed, particularly in directions where the uncertainty on a parameter depends strongly on the parameter value. One can start by varying a few critical parameters by one or two standard deviations, trying parameters near boundary/ies, and seeing how stable coverage is. A Bayesian-inspired approach is to calculate a weighted average of coverage over a neighborhood of parameter sets for the nuisance parameters. This requires a choice of multi-D prior. Instead of fixing the true values of nuisance parameters during the toy MC simulation, one samples the true parameters from the posterior pdf of the nuisance parameters. Numerous studies have been done for elimination of nuisance parameters in the test statistic (typically a likelihood ratio), many concluding that results are relatively insensitive to profiling vs marginalization, so that choice can be made based on CPU time. See for example John Conway’s talk and writeup at PhyStat-2011 [@conwayphystat]. It seems that the method for treating nuisance parameters in the toy MC generation of events may be more important than the treatment choice in test statistic: with poor treatment in the test statistic, one may lose statistical power but still calculate coverage correctly, while poor treatment in the toy MC generation may lead to incorrect coverage calculation. Downward fluctuations in searches for excesses {#downward} ============================================== As mentioned in Section \[centralproblems\] and discussed in more detail in Section \[negativex\], a key problem that has been a driver of the development of special methods for upper limits in HEP is the situation where there is Gaussian measurement resolution near a physical boundary. Specifically, in the Gaussian model $p(x| \mu,\sigma)$ in Eqn. \[eqn-gaussian\], $\mu$ may be a quantity that is physically non-negative, e.g., a mass, a mass-squared, or an interaction cross section. Recall (Section \[negativex\]) that in this case negative values of the parameter $\mu$ [*do not exist in the model*]{}, but that negative values of the observation $x$ [*do exist*]{} in the sample space. The traditional Neyman construction of frequentist one-sided 95% C.L. upper limits, for $\alpha = 1 - $C.L. $= 5$%, is shown in Fig. \[ULgauss\]. As the observation $x_0$ becomes increasingly negative, the standard frequentist upper limit (obtained by drawing a vertical line at observed $x_0$) becomes small, and then for $x_0 < -1.64 \sigma$, the upper limit is the [*null*]{} set! Some people prefer to extend the construction to negative $\mu$, which is a bad idea in my opinion, since the model does not exist there (!); this leads to a different description of the issue, so-called “unphysical” upper limits. In any case, one should report enough information so that the consumer can make interval estimates by any desired method. This would include the observed $x$ (not constrained to positive values) and the model $p(x|\mu)$. Such information is also essential for combining results of different experiments. ![The traditional frequentist construction for one-sided upper limits at 95% C.L., for a Gaussian measurement with unit rms. A vertical line drawn through observed $x_0$, for $x_0 < -1.64 \sigma$, intersects no acceptance intervals, resulting in an empty-set confidence interval.[]{data-label="ULgauss"}](figures/belt_UL_gauss.png){width="60.00000%"} This was an acute issue 20–30 years ago in experiments to measure the $\bar\nu_e$ mass (actually the mass-squared) in tritium $\beta$ decay: several observed $x_0 < 0$. (With neutrino mixing, $\bar\nu_e$ is presumably not a mass eigenstate, but one still speaks loosely of its mass, which is actually an expectation value.) At the time (and still today, unfortunately), $x_0$ was referred to as the “measured value”, or point estimate $\hat m_\nu^2$. The resulting confusion and resolutions make a very long story; see my “virtual talk”, “Bayes, Fisher, Neyman, Neutrino Masses, and the LHC” [@cousinsvirtual] and arXiv post [@cousins2011]. (These also contain an introduction to “Buehler’s betting game”, related to conditioning.) The confidence intervals proposed by F-C and described in Section \[negativex\] are just one of the three proposed ways to deal with this issue that have been widely adopted. The other two are Bayesian with uniform prior (for the quantity with Gaussian resolution), and the [$CL_s$]{} criterion of the next subsection. [$CL_s$]{} {#cls} ---------- The unfortunately named [$CL_s$]{} is the traditional frequentist one-tailed $p$-value for upper limits divided by another tail probability (associated with the alternative hypothesis), i.e., by a number less than 1. The limits are thus (intentionally) conservative. A brief definition in good notation is in the PDG RPP [@pdg2018] (Eqn. 39.78, Section 39.4.2.4). [$CL_s$]{} is a generalization of an earlier result by Günter Zech [@zech1988] that considered the case of Poisson distribution with background and applied a non-standard form of conditioning on an inequality. The rationale is further described by Alex Read in the early papers [@readclw; @readjphysg]; an implementation with further commentary is described by Tom Junk [@junkcls]. What is new (non-standard statistics) in [$CL_s$]{} is combining two $p$-values into one quantity. (This is referred to as a “modification” to the usual frequentist $p$-value.) The combination (equivalent to a non-standard form of conditioning) is designed to avoid rejecting H$_0$ when the data are similarly incompatible with H$_1$. This situation can arise when an experiment has little sensitivity for distinguishing H$_0$ from H$_1$. There is no established foundation in the statistics literature for this as far as I know. In fact, the two [ *un*]{}combined $p$-values are considered to be the (irreducible) post-data “evidence” for simple-vs-simple hypothesis testing in a philosophical monograph by Bill Thompson [@thompson2007] (p. 108). Clearly, with both $p$-values at hand, as with the Higgs spin-parity example in Section \[higgsCP\], the consumer has the complete information regarding post-data tail probabilities. (Ofer Vitells has unearthed a suggestion in a 1961 paper by Birnbaum [@birnbaum1961] (p. 434) that combines the pre-data Type I and Type II errors rates $\alpha$ and $\beta$ into one quantity in the identical manner, i.e., $\alpha/(1-\beta)$; Birnbaum’s motivation was that this quantity equals the likelihood ratio in a very special case where the test statistic is the single binary digit “reject H$_0$” or “do not reject H$_0$” (rather than the full likelihood ratio). But this seems to be obscure to modern statisticians, and Birnbaum’s last paper [@birnbaum1977] examined both $\alpha$ and $1-\beta$ (the usual “power”), not just the combination $\alpha/(1-\beta)$, in defining his “confidence concept”.) In any case, the [$CL_s$]{} quantity does have properties that many find to be attractive. In particular, for the two simple prototype problems (Poisson with known mean background, and the bounded Gaussian problem), the results are numerically identical to the Bayesian answers with uniform prior, and hence the likelihood principle is respected. These Bayesian interval estimates over-cover from a frequentist point, which is not considered to be as bad as under-coverage. (Regarding the uniform priors, for both these models negative $\mu$ does not exist in the model, so it is incorrect to speak of the prior being zero for negative $\mu$, as in Section \[negativex\]) The step of combining the two $p$-values into one quantity is called “the [$CL_s$]{} criterion” in (most) CMS papers. Unfortunately, the calculation of the $p$-values themselves, typically using a likelihood-ratio test statistic, is sometimes called the “[$CL_s$]{} method” in HEP. But these $p$-values themselves have long existed in the statistics literature and should be designated that way (and not denoted with equally unfortunate names that the original papers of [$CL_s$]{} use for them). The many issues of $p$-values are of course inherited by [$CL_s$]{}, namely: - What specific likelihood ratio is used in the test statistic; - How nuisance parameters are treated (marginalization, profiling); - What ensembles are used for “toy MC” simulation used to get the distribution of the test statistic under H$_0$ (e.g. no Higgs) and H$_1$ (e.g. SM with Higgs). LEP, Tevatron, and LHC Higgs experimenters differed in the choices made (!). ATLAS and CMS Conventions ========================= For many years, ATLAS and CMS physicists have collaborated on statistics tools (the RooStats [@roostats] software), and attempted to have some coherence in methods, so that results could be compared, and (when worth the effort) combined. A key development was the paper by Cowan, Cranmer, Gross and Vitells (CCGV) [@ccgv2011] that extends asymptotic formulas to various cases where Wilks’s theorem is not valid. As the CCGV asymptotic formulas correspond to the “fully frequentist” treatment of nuisance parameters, for consistency we tended to use that treatment in many cases at small $N$ as well. Toy MC simulation is thus performed in an approximate frequentist manner in which the underlying parametric distribution of the data is known up to one or more parameters, and sampling is performed from that distribution after substituting estimates for the parameter(s). (This is known as the [*parametric bootstrap*]{} [@luc2012slac].) For upper limits, there was a lot of discussion among CMS and ATLAS physicists in the early LHC days without convergence, with the result that the two experiments’ physics coordinators in 2010 decreed that [$CL_s$]{} (Section \[cls\]) be used in most cases. The ATLAS/CMS Higgs boson results followed these trends. A jointly written description is, “Procedure for the LHC Higgs boson search combination in Summer 2011” [@LHC-HCG]. Many issues were further discussed and described in the ATLAS-CMS combination papers for mass [@atlascmsmass] and couplings [@atlascmscouplings]. In particular, a lot of attention was paid to correlations. In the last couple years, Feldman-Cousins [@feldman1998] starts to be used in some analyses, without my pushing. (Initially some at the LHC were very opposed, evidently because it could return a two-sided interval not including zero when they insisted on a strict upper limit.) My advocacy for $>$10 years =========================== Have in place tools to allow computation of results using a variety of recipes, for problems up to intermediate complexity: - Bayesian with analysis of sensitivity to prior - Profile likelihood ratio (MINUIT MINOS) - Frequentist construction with approximate treatment of nuisance parameters - Other “favorites” such as [$CL_s$]{}  (an HEP invention). The community can (and should) then demand that a result shown with one’s preferred method also be shown with the other methods, [*and sampling properties studied*]{}. When the methods all agree, we are in asymptotic Nirvana (idyllic state). When methods disagree, we are reminded that the results are answers to different questions, and we learn something! E.g.: - Bayesian methods can have poor frequentist properties - Frequentist methods can badly violate Likelihood Principle. In fact, the community reached the point of having the tools in place (RooStats [@roostats]) by the time of the Higgs boson discovery, and they have continued to be improved. What is not as far along is the “demands” of the community, in my opinion. I would prefer that it be more common for papers to compare explicitly the chosen method with other methods, as is sometimes done. Unsound statements you can now avoid ==================================== - “It makes no sense to talk about the probability density of a constant of nature.” - “Frequentist confidence intervals for efficiency measurements don’t work when all trials give successes.” - “We used a uniform prior because this introduces the least bias.” - “We used a uniform positive prior as a function of the parameter of interest.” - “A noninformative prior probability density does not contain any information.” - “The total number of events could fluctuate in our experiment, so [*obviously*]{} our toy Monte Carlo simulation should let the number of events fluctuate.” - “We used Delta-likelihood contours so there was no Gaussian approximation.” - “A five-sigma departure from the SM constitutes a discovery.” - “The confidence level tells you how much confidence one has that the true value is in the confidence interval.” - “We used the tail area under the likelihood function to measure the significance.” - “Statistics is obvious, so I prefer not to read the literature and just figure it out for myself!” Details of spin discrimination example of simple-vs-simple hypothesis test {#secjhep} ========================================================================== Ref. [@cousinsspin] describes an enlightening toy example with simple hypotheses. The setup is the observation of a new resonance with a mass of 1.5 TeV in the dilepton final state at the LHC (unfortunately optimistic thus far). We consider here the case of just two simple hypotheses for the spin of the new resonance, either a spin-1 vector boson or a spin-2 graviton. In order to avoid confusion between subscripts and spin values, the hypotheses are called ${{\rm H}_{\rm A}}$ and ${{\rm H}_{\rm B}}$. The distinguishing observable is the so-called Collins-Soper angle ${\theta^{\ast}_{\rm CS}}$, which is a useful approximation to the angle in the CM frame between the incoming quark and the outgoing $\mu^-$. Figure \[fig:angdists\] shows the simulated distributions of ${\cos{\theta^{\ast}_{\rm CS}}}$ for the two spin hypotheses. The events in the detector’s geometrical acceptance (solid histograms) are used in the analysis. The histograms numerically define the models $p({\cos{\theta^{\ast}_{\rm CS}}}|{{\rm H}_{\rm A}})$ and $p({\cos{\theta^{\ast}_{\rm CS}}}|{{\rm H}_{\rm B}})$. ![Histograms of ${\cos{\theta^{\ast}_{\rm CS}}}$ of individual events from the generated (dashed) and accepted (solid) samples of spin-1 ${\rm Z^{\prime}}$ (left) and spin-2 ${\rm G^*}$ (right) for 1.5 TeV mass. From Ref. [@cousinsspin].[]{data-label="fig:angdists"}](figures/angdists.pdf){width="90.00000%"} As dictated by the Neyman–Pearson lemma (Section \[secNP\]), the optimal test statistic for distinguishing the two hypotheses is the likelihood ratio $$\lambda = {{\cal L}}({{\rm H}_{\rm A}}) / {{\cal L}}({{\rm H}_{\rm B}}),$$ from which one usually considers the monotonic function $-2\ln\lambda$. For a data set with $N$ events indexed by $i$, $\lambda$ is the product of event likelihood ratios, so that $-2\ln\lambda$ is the sum of the individual event quantities, $$\label{sumlambda} -2\ln\lambda =\sum_i^N -2\ln\left (\frac{p({\cos{\theta^{\ast}_{{\rm CS},i}}}|{{\rm H}_{\rm A}})}{ p({\cos{\theta^{\ast}_{{\rm CS},i}}}|{{\rm H}_{\rm B}})}\right).$$ Figure \[fig:lrdists\](left) and right are histograms of the individual terms in the sum, with events on the left simulated according to ${{\rm H}_{\rm A}}$ as in Figure \[fig:angdists\](left), and events on the right simulated according to ${{\rm H}_{\rm B}}$, as in Figure \[fig:angdists\](right). ![Histograms of $-2\ln{\lambda}= 2\ln{{\cal L}}({{\rm H}_{\rm A}}) - 2\ln{{\cal L}}({{\rm H}_{\rm B}})$ for individual events from the accepted samples generated according to ${{\rm H}_{\rm A}}$ (spin-1) and ${{\rm H}_{\rm B}}$ (spin-2) for a 1.5 TeV resonance. From Ref. [@cousinsspin].[]{data-label="fig:lrdists"}](figures/lrs2s1.pdf){width="90.00000%"} We then consider data sets (“experiments”) that each contain a sample of $N$ events from the distributions in Figure \[fig:lrdists\], with the values summed as in Eqn. \[sumlambda\]. Figure \[fig:lrdists\_n50\] shows on the left the distribution for 10,000 simulated experiments, each with $N=50$, and on the right for $N=200$. We see the dramatic effect of the central limit theorem, which says that each histogram of sums tends to Gaussian in spite of the highly non-Gaussian distribution of the addends from Fig. \[fig:lrdists\]; and that furthermore the mean and rms of each are correctly related to those in Fig. \[fig:lrdists\]. ![Histograms of $-2\ln{\lambda}$ for simulated experiments with 1.5 TeV dilepton mass, each of which is a sum of $N$ values of $-2\ln{\lambda}$ from spin-1 and spin-2 events in Fig. \[fig:lrdists\], for $N=50$ (left) and $N=200$ (right) events per experiment. The superimposed Gaussian curves have means fixed at a factor of $N$ greater than that of the events in Fig. \[fig:lrdists\], and rms deviations a factor of $\sqrt{N}$ greater. From Ref. [@cousinsspin].[]{data-label="fig:lrdists_n50"}](figures/g1500zp1500_50_200.pdf){width="90.00000%"} Recalling the prescriptions in Sections \[hypotest\] and \[secNP\], to perform a N-P hypothesis test of ${{\rm H}_{\rm A}}$, one finds the cutoff $\lambda_{{\rm cut,}\alpha}$ for the desired Type I error probability $\alpha$ and rejects ${{\rm H}_{\rm A}}$ if $\lambda \le \lambda_{{\rm cut,}\alpha}$. The Type II error probability $\beta$ then follows. Both of these error probabilities are easily obtained from the histograms in Fig. \[fig:lrdists\_n50\] and plotted as a function of $-2\ln \lambda_{{\rm cut,}\alpha}$, in Fig.\[alphabeta\](left), for $N=50$. One can also plot $\beta$ vs $\alpha$ as in Fig. \[alphabeta\](right) for $N=50$. As noted in Section \[alphachoice\], the choice of operating point on the $\beta$ vs $\alpha$ curve (or equivalent ROC curve) requires multiple considerations. . On the right is a plot of $\beta$ vs $\alpha$ derived from the curves on the left.[]{data-label="alphabeta"}](figures/alphabeta "fig:"){width="57.00000%"} . On the right is a plot of $\beta$ vs $\alpha$ derived from the curves on the left.[]{data-label="alphabeta"}](figures/betavsalpha "fig:"){width="42.00000%"} Look-elsewhere effect ===================== In these lectures, I did not have time for the important (and increasingly mandatory) calculation of the LEE. A starting point for self-study is the discussion by Louis Lyons, “Comments on ‘Look Elsewhere Effect’ ” [@lyonslee2010]. See also Section 9.2 of my paper on the Jeffreys-Lindley Paradox [@cousinsJL]. An important paper for practical calculations, and also for qualitative insight, is by Eilam Gross and Ofer Vitells, “Trial factors for the look elsewhere effect in high energy physics,” [@grossvitells2010]. Bayesian hypothesis testing (model selection) {#modelselection} ============================================= As mentioned very briefly in Sections \[bayesintro\], \[pseudobayes\], and \[intervalsummary\], the duality used in frequentist hypothesis testing (Section \[duality\]) is not used in Bayesian statistics. The usual methods follow Chapter 5 of Harold Jeffreys’s book [@jeffreys1961]: Bayes’s Theorem is applied to the models themselves after integrating out all parameters, including the parameter of interest! This is typically presented by Bayesian advocates as “logical” and therefore simple to use, with great benefits such as automatic “Occam’s razor” penalizing less predictive models, etc. In fact, Bayesian model selection is full of subtleties, and even for the experts, it can be a “can of worms” (James Berger [@berger2006], Rejoinder, p. 459). As just one indication, Jeffreys and followers use [*different priors*]{} for integrating out parameter(s) in model selection than for the same parameter(s) in parameter estimation. Here I mainly just say: Beware! There are posted/published applications in HEP that are silly (by Bayesian standards). As mentioned in Section \[pseudobayes\], a pseudo-Bayes example in PRL provoked me to write a Comment [@cousins2008] that has some references to useful Bayesian literature. The dependence on prior probabilities for the [*models themselves*]{} can be factored out, leading to a “Bayes factor” that is the ratio of posterior odds to prior odds. However, the Bayes factor still depends on prior pdfs for [*parameters*]{} in the models, and this leads to direct sensitivity to the prior pdf for a parameter that is in one model but not in the other. For testing H$_0$:$\mu=\mu_0$ vs H$_1$:$\mu\ne\mu_0$, improper priors for $\mu$ that work fine for estimation become a disaster. Adding a cut-off to make them proper just gives (typically arbitrary) cutoff dependence. In the asymptotic limit of lots of data, your answers in a test of a point null vs a continuous alternative (either the probability H$_0$ is true, or also the Bayes factor) [*remain directly proportional to the prior pdf for the parameter of interest*]{}. This is [ *totally different*]{} behavior compared to Bayesian interval estimation, where the effect of a prior typically becomes negligible as the likelihood function becomes narrowly peaked at large $N$. For a review and comparison to $p$-values in the discovery of Higgs boson, see my paper, “The Jeffreys-Lindley Paradox and Discovery Criteria in High Energy Physics” [@cousinsJL]. As mentioned in Section \[pseudobayes\], see also Chapter 12 by Harrison Prosper in Ref. [@behnke2013]. Point estimation {#pointest} ================ Most of this paper is about intervals; I have do not say much about what to quote as the “measured value” (typically using the MLE by default). Statisticians call this the “point estimate”. There is a huge literature on point estimation; see e.g. Chapters 7 and 8 in Ref. [@james2006]. If you are well-grounded in interval estimation, one approach is to use that machinery to get a point estimate. E.g., one might take the mid-point of (say) your 68% [[C.L.]{.nodecor}]{} central interval. But a better approach is probably to let the [[C.L.]{.nodecor}]{} go to 0, so that your interval gets shorter and shorter, and use the limiting point. For both the LR ordering for confidence intervals (F-C), and for likelihood ratio intervals, this results in the maximum likelihood estimate. To give an idea of how rich the subject is, I show a few interesting things from Ref. [@james2006]. ### Point estimation: Traditional desiderata {#point-est} - Consistency: Estimate converges toward true value as number of observations $N$ increases - Unbiasedness: Expectation value of estimate is equal to the true value. (Bias and consistency are independent properties; see Fig. 7.2 in Ref. [@james2006].) - Efficiency: Estimate has minimum variance - Minimum loss of information: (technical definition) - Robustness: Insensitivity to departures from the assumed distribution One can add: - Simplicity: transparent and understandable - Minimum computer time: still relevant in online applications, less relevant otherwise - Minimum loss of physicist’s time (how much weight to put on this?) These desired properties can be impossible to achieve simultaneously. How to choose? A thorough analysis requires further input: what are the costs of not incorporating various desiderata? Then formal [ *decision theory*]{} can be used to choose estimator. In practice in HEP, maximum likelihood estimates are often used (even though they are typically not unbiased). Typically the MLE is consistent and has other excellent asymptotic properties (e.g., estimate is asymptotically normal). For finite $N$, the MLE works well in the so-called exponential family that includes binomial, Poisson, and Gaussian. As noted in Section \[invariantl\], the MLE is invariant under reparameterization. (This means that if it is unbiased in one metric, it is typically biased in other metrics, as discussed in Ref. [@barlow], Section 5.3.1.) ### Alternatives to the arithmetic mean when model is non-Gaussian Fred James [@james2006] (pp. 209 ff) has a very illuminating discussion of the MLEs for a diverse set of pdfs, representing long tails, short tails, and in between. If $p(x| \mu) = f(x- \mu)$, where $f$ is some pdf, then $\mu$ is called a [*location parameter*]{}. Some common examples of pdfs were $\mu$ is a location parameter are: [Normal:]{} $p \sim \exp(-(x- \mu)^2/2\sigma^2)$ [Uniform:]{} p = constant for $|x- \mu|<a$; $p=0$ otherwise [Cauchy:]{} $p \sim 1/(a^2 + (x- \mu)^2)$ [Double exponential:]{} $p \sim \exp( -a |x- \mu| )$ These examples are all symmetric about $\mu$: $p(\mu+y) = p(\mu-y)$ Suppose you are given N=11 values of $x$ randomly sampled from $p(x| \mu)$. What estimator (function of the 11 values) gives you the “best” estimate of $\mu$ ? If by “best” you mean minimum variance, it is the M.L. estimate, resulting in a different formula for each of the above cases! Three of the four are special cases of $L^p$, the estimator that minimizes the sum over the observations of $|x_i - \mu|^p$. Different values of $p$ put different emphasis on observations in the tails. [*Only for the normal*]{} pdf is the MLE equal to the arithmetic mean, obtained by the familiar least-squares technique ($p=2$). Can you guess what the MLE is for the uniform and the exponential cases? (For the Cauchy pdf, with its long tails, the arithmetic mean is particularly useless; see also discussion by Efron [@efronslac]). [*If the true distribution departs from that assumed, then the estimate of location is no longer optimal. Sensitivity is in the tails!*]{} See the nice discussion of asymptotic variance and robustness in Ref. [@james2006] (pp. 211 ff). Acknowledgments {#acknowledgments .unnumbered} =============== Thanks to many in HEP (Frederick James, Gary Feldman, Louis Lyons, Luc Demortier, and numerous others) from whom I learned…and to many statisticians that Louis invited to PhyStat meetings. For Bayesian statistics that was especially Jim Berger (multiple times) and Michael Goldstein, and more recently, David van Dyk (multiple times). Thanks also to CMS Statistics Committee (Olaf Behnke et al.) for many discussions and comments on earlier versions of the slides and writeup…and to the authors of numerous papers from which I learned, including early (1980s) Bayesian papers by Harrison Prosper…. Thanks also to Diego Tonelli of the LHCb experiment for encouragement to update the slides in 2017 and for comments on an earlier version. This work was partially supported by the U.S. Department of Energy under Award Number [DE]{}–[SC]{}0009937. [^1]: [email protected]

--- abstract: 'We derive an empirical formula for the width of quantum Hall effect plateaus which is free of adjustable parameters. It describes the integer, fractional and $\nu = 0$ (Wigner insulator) quantum Hall effect in single heterojunctions. The temperature scale of the existence of these three phenomena is the same as the melting temperature of a classical Wigner crystal. We conclude that the basic assumption of the current theory of QHE that the plateau width is determined by the disorder is highly improbable.' address: | $^a)$ Institut für Theoretische Festkörperphysik, Universität Karlsruhe,\ 76128 Karlsruhe, Federal Republic of Germany\ $^b)$ Department of Theoretical Physics, University of Sofia, 5 J. Boucher Blvd.,\ 1126 Sofia, Bulgaria. author: - 'Atanas Groshev$^{a,b}$' title: ' Formula for the widths of the plateaus of the quantum Hall effect\' --- Three phenomena of completely different nature are believed to exist in a 2-dimensional electron system (2DES) in high magnetic fields at low temperatures [@pra1]. The integer quantum Hall effect is thought to be describable as single particle localization (by the disorder) of a noninteracting electron gas [@pra1]. The fractional Hall effect is taught to be due to quasiparticle localization (by the disorder) on top of a many-particle liquid state [@pra1]. A Wigner crystal (many-body effect) exists at low filling factors [@sha1]. The essence of the quantum Hall effect (QHE) is the existence of plateaus of [*finite*]{} width in the Hall resistance [@kli1]. In spite of substantial theoretical efforts devoted to this subject no successful formula has yet been derived for the width of these plateaus. However, since it is believed that the plateau width is determined by localization in the disorder potential, one would expect it to be sample dependent and moreover to decrease and disappear in the low disorder limit. Here we discuss experimental facts which contradict this assumption. On the basis of the symmetries of the experimental data we derive an empirical formula for the width of the plateaus of the quantum Hall effect (integer and fractional) as well as for the width of the filling factor region of the Wigner solid.We also show that the temperature scales of the integer QHE, fractional QHE and the Wigner solid are the same and close to the classical melting temperature of the Wigner crystal. The Landau level filling factor of the 2DES of density $n_s$ in a magnetic field $B$ is given by $\nu = n_s h c/eB$. In the absence of the quantum Hall effect (for example at high temperatures) the Hall conductivity is given by the classical value $\sigma_{xy} = e n c/B = \nu e^2/h$. The quantum Hall effect plateaus appear at integer and fractional values of $\nu$. The most important factors influencing the width of plateaus are the temperature $T$ and the disorder [@rem1]. We will be interested in the limit of very low temperatures and disorder. We will quantify the meaning of “very low” below. We start with several observations concerning the shape of the dependence $\sigma_{xy}(\nu)$ in this limit. They are based on visual examination of the available data and should be considered as empirical rules.\ 1. The transitions between adjacent plateaus are sharp steps.\ 2. The plateaus extend symmetrically on both sides of the classical line $\sigma_{xy}(\nu) = \nu e^2/h$ (see Fig. \[platofg1\]). We denote the half-width of the plateau at the filling factor $\nu$ by $\Delta\nu_{\nu}$.\ 3. The following symmetry relations are fulfilled $$\Delta\nu_{1\pm\nu} = \Delta\nu_{\nu} \label{s}$$ 4. The plateaus are grouped in sequences as follows.\ A main sequence converging towards 1/2: $$\nu = p/(2p+1),\ \ \ p = 0,1,2\ldots$$ and its symmetry partners according to (\[s\]). This is the only sequence in 2D hole systems.\ Two additional secondary sequences converging towards 1/4 exist in 2D electron systems: $$\nu = p/(4p\pm 1),\ \ \ p = 0,1,2\ldots$$ and their symmetry partners according to (\[s\]).\ The positions of these plateaus is given by the single formula $\nu_{m,p} = p/(2 m p\pm 1)$ where $m = 1$ corresponds to the main sequence and $m = 2$ to the secondary sequences.\ We would like to stress that the property 4. has been discussed earlier in the literature [@cha1; @wil1; @du1; @jai1; @kiv1; @hal1]. The symmetry $\nu \rightarrow 1\pm \nu$ has also been discussed earlier [@kiv1; @hal1] although in the present form (\[s\]) we give it for the first time. To our knowledge the properties 1. and 2. have not been discussed earlier, although we consider them apparent from the experimental data (see for example [@dav1; @wil1; @saj1]). Using these facts we find a formula for the widths of the plateaus. Indeed, let $\nu_{m,p}$ and $\nu_{m,p+1}$ be two adjacent plateaus within a particular sequence. As a consequence of 1. and 2., the following recursive equation holds for the plateau widths (see also Fig. \[platofg1\]): $$\Delta\nu_{\nu_{m,p}} + \Delta\nu_{\nu_{m,p+1}} = |\nu_{m,p+1} - \nu_{m,p}| \label{equ}$$ The solution of this recursive equation is unique for each of the sequences and is given by the formula: $$\Delta\nu_{p/(2mp \pm 1)} = \frac{1}{8m^2} \left[ 2 \psi\left({p\over 2} + {1\over 2} \pm {1\over 4m}\right) - \psi\left({p\over 2} + 1 \pm {1\over 4m}\right) - \psi\left({p\over 2} \pm {1\over 4m}\right) \right] %\sum\limits_{k=0}^{\infty} %{(-1)^k \over [2m(p+k)+ 2m \pm 1][2m(p+k) \pm 1]} \label{sol}$$ where $\psi(z)$ is the digamma function [@abr1]. This is our main result. The staircases obtained from formula (\[sol\]) are presented in Fig. \[platofg2\]. The main staircase spans the space $0 < \nu < 1/2$ while the secondary staircase spans the space $0 < \nu < 1/3+\Delta\nu_{1/3} = 0.37187$. The plateau widths of the most prominent plateaus are presented in Tables I-III. Now we compare with the experiment. The experimental data show that in 2-dimensional hole systems $\Delta\nu_0$ and $\Delta\nu_{1/3}$ belong to the main sequence. Indeed $\Delta\nu_0 = 0.2854$ agrees perfectly well with the experimental value $\sim 0.28$ of the critical filling of the existence of insulating phase $\sigma_{xy} = 0$ [@rod1]. It is also in perfect agreement with the half-width of the $\nu = 1$ plateau $\Delta\nu_1 = \Delta\nu_0 = 0.2854$ [@dav1]. In general the main sequence plateaus (which is the only sequence present in hole systems) are very well described by the formula (\[sol\]). In 2-dimensional electron systems also the secondary sequences are observed [@wil1; @saj1; @du1]. In fact the experimental data shows that the range $0 < \nu < 1/3+\Delta\nu_{1/3} = 0.37187$ belongs to the secondary staircase while the range $ 2/5-\Delta\nu_{2/5} = 0.38127 < \nu < 1/2$ belongs to the main staircase. For example the width of the Wigner solid region [@sha1] is equal to the half-width of the $\nu = 1$ plateau $\sim 0.19$ which is again very close to our value $\Delta\nu_0 = 0.1835$ taken from the secondary sequence. The Hall conductance in the gap between the two staircases $0.37187 < \nu < 0.38127$ shows a peculiar behavior [@rem2]. For the sake of easy visual comparison with the experimental data we present the Hall conductivity $\sigma_{xy}(\nu)$ and resistivity $\rho_{xy}(1/\nu)$ for electron systems in Fig. \[platofg3\] and Fig. \[platofg4\]. We would like to stress that although we can not say which sequences will be realized for a particular material we can predict the plateau width [*if*]{} we know which sequences are realized. Further we would like to discuss the conditions under which the real system data is close to the ideal one described above. First we discuss the effect of disorder. In Fig. \[platofg5\] we present a schematic dependence of the half width of the $\nu = 1$ plateau as a function of the sample mobility $\mu$ at low temperatures and zero magnetic field. It is a summary of our investigation of the available data. The mobility is a measure (although indirect) of the disorder in the sample. At very low mobilities there is no quantum Hall effect. At mobilities in the range of 1-2$\times 10^{5}$ cm$^{2}$/V.s the plateau width has a maximum. It is due to the broadening of the plateaus by disorder-induced local electron reservoirs spread in the sample. This broadening is reduced when the sample quality is improved. Above $\mu$ = 1-2$\times 10^{6}$ cm$^{2}$/V.s the plateau width saturates to the value $\Delta\nu_1 \approx 0.19$ ( for the best published sample $\mu=1.2\times 10^{7}$ cm$^{2}$/V.s [@du1]). If the disorder would be relevant to the width of the plateaus one would expect a gradual decrease to zero of $\Delta\nu_1$ when $\mu$ is increased. A measure of the disorder is the width of the smallest resolvable plateau. For the data published in Ref. [@du1] it is for example $\Delta\nu_{5/11} \approx 0.004$. In low density systems the effect of disorder is more pronounced. We can estimate it from the difference between the experimental width of the Winger insulator region $\approx 0.19$ and our theoretical value 0.1835 to be of order of 0.007. Next we discuss the effect of temperature. In Fig. \[platofg6\] we present a combined plot of quantum Hall effect data and Wigner insulator data. On the horizontal axis the filling factor $\nu$ in the case of Wigner insulator and and $1-\nu$ in the case of QHE is given. The black points are the melting temperature $T$ of the Wigner insulator normalized to the melting temperature of a classical Wigner crystal $T_{cl} = \sqrt{n_s} e^2/127\varepsilon \sqrt{\pi}$. The data is from the review of Williams [*et al.*]{} [@sha1] and is obtained with different techniques in different groups on samples with densities ranging from 3.4$\times 10^{10}$ cm$^{-2}$ to 10.2$\times 10^{10}$ cm$^{-2}$. No systematic trend of the density dependence of the melting temperature exists. On the same plot we give experimental $\sigma_{xy}(1-\nu)h/e^2$ taken from Sajoto [*et al.*]{} (solid lines) [@rem3] and the half width of the $\nu = 1$ plato from Clark [*et al.*]{} [@cla1] (circles). The $\sigma_{xy}(1-\nu)h/e^2$ dependencies has been offset vertically to the corresponding normalized temperature. The density of the sample is 5.5$\times 10^{10}$ cm$^{-2}$ ($T_{cl} = 380$ mK). The density of the sample of Clark [*et al.*]{} is 1.9$\times 10^{11}$ cm$^{-2}$ corresponding to almost twice higher $T_{cl}$. As it is obvious from the figure the characteristic temperature scale of the existence of both IQHE and Wigner insulator is the melting temperature of the classical Wigner crystal. Moreover, visual inspection of the data of Sajoto [*et al.*]{} and Goldman [*et al.*]{} Ref. [@saj1] shows that also the [*fractional*]{} QHE exists on the same temperature scale. Typical lowest dilution refrigerator temperatures are $20-30$ mK $\ll T_{cl}$ which shows that the low temperature limit is reached experimentally. The connection of the melting of a Wigner crystal with the width of the plateaus in the integer QHE was discussed in Ref. [@gro1]. We would like to mention that here we have neglected the effect of dissipation ($\sigma_{xx} = 0$) and related to it reentrant phases [@saj1; @san1]. At the end we would like to discuss the consequences of the existence of formula (\[sol\]).\ 1. Integer and fractional Hall effect seem to have the same origin as well as the low-filling factor insulating phase (Wigner crystal) which can be viewed as $\nu = 0$ quantum Hall effect.\ 2. Disorder is highly improbable to be the reason for the finite width of the plateaus.\ These two conclusions cast serious doubts on the validity of the standard picture[@pra1] of the behavior of 2DEG in magnetic field. The symmetry relations $\nu \rightarrow 1\pm \nu$ as well as recursive relations of the type of (\[equ\])are natural for the properties of electrons moving in periodic potential [@hof1]. It has been shown, however, [@tho1] that if one uses a Kubo formula for the Hall conductivity of independent spinless electrons in external periodic potential, one can not obtain the fractional QHE. The problem we see with the Kubo formula is that it is a rewritten fluctuation-dissipation theorem and it is not clear if it could be applied for description of a dissipationless phenomenon like QHE. For example one can not obtain superconductivity by applying the Kubo formula to a noninteracting electron gas. I would like to thank G. Schön and A. MacDonald for stimulating discussions, H.L. Störmer for sending me preprints, C. Bruder and G. Falci for useful suggestions. I would like to acknowledge the support of the A. von Humboldt Foundation. The work is supported by the Sonderforschungsbereich 195 of the DFG. See: [*The Quantum Hall Effect*]{}, eds. R.E. Prange and S.M. Girvin (Springer-Verlag 1990); A.H. MacDonald in: [*Quantum Coherence in Mesoscopic Systems*]{}, ed. B. Kramer (Plenum, New York 1991). For recent reviews see: M. Shayegan, in [*Low-Dimensional Electronic Systems, New Concepts*]{}, eds. G. Bauer, F. Kuchar, and H. Heinrich (Springer, Berlin 1992); F.I.B. Williams [*et al.*]{}, Surf. Sci. [**263**]{}, 23 (1992). K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. [**45**]{}, 494 (1980). Other factors are the measuring current and the shape of the confining potential. The measuring current can be made low enough so that the plateaus’ width does not change any more. The shape of the confining potential depends on the particular structure. We will consider here single heterojunction systems. A.M. Chang, P. Berglind, D.C. Tsui, H.L. Störmer, and J.C.M. Hwang, Phys. Rev. Lett. [**53**]{}, 997 (1984). R. Willet, J.P. Eisenstein, H.L. Störmer, D.C. Tsui, A.C. Gossard, and J.H. English, Phys. Rev. Lett. [**59**]{}, 1776 (1987). R.R. Du, H.L. Störmer, D.C. Tsui, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. [**70**]{}, 2944 (1993). J.K. Jain, Phys. Rev. Lett. [**63**]{}, 199 (1989). S. Kivelson, D.-H. Lee, S.-C. Zhang, Phys. Rev. B [**46**]{}, 2223 (1992). B.I. Halperin, P.A. Lee, N. Read, Phys. Rev. B [**47**]{}, 7312 (1993). A.G. Davies, R. Newbury, M. Pepper, J.E.F. Frost, D.A. Ritchie, and G.A.C. Jones, Phys. Rev. B [**44**]{}, 13128 (1991). T. Sajoto, Y.P. Li, L.W. Engel, D.C. Tsui, and M. Shayegan, Phys. Rev. Lett. [**70**]{}, 2321 (1993); V.J. Goldmann, J.K. Wang, B. Su, and M. Shayegan, Phys. Rev. Lett. [**70**]{}, 647 (1993). , eds. M. Abramovitz and I. Stegun, (Dover Publications, New York 1965). P.J. Rodgers, C.J.G.M. Langerak, B.L. Gallagher, R.J. Barraclough, M. Heini, T.J. Foster, G. Hill, S.A.J. Wieger and J.A.A.J Perenboom, Physica B [**184**]{}, 95 (1993). More correctly saying the peculiar behavior is observed in the regions of the symmetry counterparts of this gap between 2/3 and 3/5 and between 5/3 and 8/5. See V.J. Goldman, M. Shayegan, Surf. Sci. [**229**]{}, 10 (1990); J.P. Eisenstein, H.L. Störmer, L.N. Pfeiffer and K.W. West, Phys. Rev. B [**41**]{}, 7910 (1990); L.W. Engel, S.W. Hwang, T. Sajoto, D.C. Tsui, and M. Shayegan, Phys. Rev. B [**45**]{}, 3418 (1992). J.P. Eisenstein, H.L. Störmer, L.N. Pfeiffer, and K.W. West, Surf. Sci. [**229**]{}, 21 (1990). We have magnified and digitalized the data of $\rho_{xy}(1/\nu)$ from Sajoto [*et al.*]{} [@saj1] and than inverted it to get $\sigma_{xy}(1-\nu)$. R.G. Clark, R.J. Nicholas, A. Usher, C.T. Foxon and J.J. Harris, Surf. Sci. [**170**]{}, 141 (1986). The circles in Fig. \[platofg6\] are obtained from the maximum in $\rho_{xx}$. M.B. Santos, Y.W. Suen, M. Shayegan, Y.P. Li, L.W. Engel, and D.C. Tsui, Phys. Rev. Lett. [**68**]{}, 1188 (1992). A. Groshev and G. Schön, to be published in Physica B. D.R. Hofstadter, Phys. Rev. B [**14**]{}, 2239 (1976); F.H. Carlo and G.H. Wannier, Phys. Rev. B [**19**]{}, 6068 (1979). D.J. Thouless, in Ref. [@pra1]. [c|cccccc]{} $\nu$ & 0 & 1/3 & 2/5 & 3/7 & 4/9 & 5/11 $\Delta\nu_{\nu}$ & 0.2854 & 0.0479 & 0.0187 & 0.0098 & 0.0060 & 0.0041 [c|cccc]{} $\nu$ &0 & 1/5 & 2/9 & 3/13 $\Delta\nu_{\nu}$ &0.1835 & 0.0165 & 0.0057 & 0.0028 [c|cccc]{} $\nu$ &1/3 & 2/7 & 3/11 & 4/15 $\Delta\nu_{\nu}$ &0.0385 & 0.0091 & 0.0039 & 0.002

--- author: - | \ St.Petersburg State University\ E-mail: - | Igor Altsybeev\ St.Petersburg State University\ - | Olga Kochebina\ Laboratoire de l’Accelerateur Lineaire (LAL), CNRS : UMR8607 - IN2P3 - Universite Paris XI - Paris Sud\ title: 'Constraints on string percolation model from anomalous centrality evolution data in Au-Au collisions at $\mathbf{\sqrt{s_{NN}}=}$ 62 and 200 GeV' --- Introduction {#intro} ============ The first predictions of the azimuthal asymmetry of multiple-production of secondary hadrons in high energy nucleus-nucleus collisions were done [@Abramovsky-1980; @Abramovsky-1988] using the concept of interacting color flux tubes (strings). The color strings may be viewed as tubes of the color field created by the colliding partons [@Capella; @Kaidalov; @Armesto2000]. Production of particles goes via spontaneous emission of quark-antiquark pairs in this color field. These strings are the phenomenological objects extended in rapidity and are related to the cut Pomerons. Their cross-section in the transverse plane is considered as small discs of $\pi r_{0}^2$ area, where $r_{0}$ is the string radius usually taken to be about 0.2 fm. With growing energy and/or atomic number of colliding particles the number of strings grows, therefore they start to overlap and may interact. In case of existence of string-string interaction, the event-by-event fluctuations of the initial geometry of collisions should manifest themselves as the azimuthal ($\phi$) anisotropy in two-particle correlations functions. The second important outcome of this hypothesis [@Abramovsky-1988] is that this spatial $\phi$ asymmetry will be also manifested as the long-range (extended in pseudorapidity $\eta$) correlations. Experimental evidences of the long-range azimuthal anisotropy in two-particle correlations in heavy-ion collisions at RHIC and LHC are well-known. The ridge was defined as a two-particle correlation structure relatively narrow in azimuthal angle and extended over several units in pseudo-rapidity [@STAR; @2006-05; @Putschke; @STAR; @2006]. These structures were also observed both in $\pt$-integrated and in special $\pt$-selected analyses of the 62 and 200 GeV Au-Au and Cu-Cu data (one may see a detailed overview of STAR ridges in a recent work [@star-ridges]). Recently the experimental ridge landscape was considerably broadened by the observation of the CMS Collaboration at the LHC when the unexpected long-range azimuthal two-particle correlations where found in pp collisions [@CMS-ridge-pp]. Ridge structures were also reported in Pb-Pb and in p-Pb collisions at LHC [@CMS-ridge-PbPb; @CMS-ridge-pPb; @ALICE-ridge-PbPb-harmonics]. The onset of the ridge and the role of initial conditions in the ridge formation were considered by a rather large number of theoretical models that were motivated by the experimental discoveries. Several models were proposed to explain qualitatively an origin of the ridge using various concepts like an interaction of high-$\pt$ partons or jets with medium, formation of jets in small-$\pt$, parton-jets collisions, glasma flux tubes with radial flow etc. (see references in [@Putschke; @star-ridges; @CMS-ridge-PbPb]). The Color Glass Condensate (CGC) model [@ridge-cgc] in addition to the long-range rapidity correlation points to the possibility of intrinsic angular correlation which is assumed to come from the particle production process due to glasma tubes formation on transverse distance scales 1/$Q_s$ much smaller than the proton size (here $Q_s$ is the saturation scale of the colliding nuclei). In [@Perc_ridge] the string percolation phenomenology was compared to CGC results on effective string or glasma flux tube intrinsic correlations, including the ridge phenomena and long-range forward-backward azimuthal correlations. Color string percolation model and its similarities with the CGC are discussed in [@Paj-2011]. Fourier harmonics decomposition of the two-particle azimuthal correlations in nucleus-nucleus collisions was found to describe various ridge structures observed in the experiment [@ALICE-ridge-PbPb-harmonics; @ALICE-H]. However, the dynamical origin of the harmonics and of the onset of these collective phenomena are still not clear enough. In the present work we use as the main working hypothesis the interaction among the quark-gluon strings formed in the nucleus-nucleus collisions. In Section \[onset\], estimations of string density that might be reached in nucleus-nucleus collisions are done. Following [@OK-2010], we assume here that the intriguing phenomena of sharp change in two-particle correlation function, observed for the first time in Au-Au collisions at certain collision centralities at $\sqrt{s_{NN}}=62$ and 200 GeV [@Daugherity], is related to the critical string density reached in the interaction region. In Section \[MC-toy\], a toy-model with interacting strings in nucleus-nucleus collisions [@TOY] is applied for the analysis of the topology of two-particle correlations to study the phenomena of the onset of the azimuthal peculiarities. In this model, a string repulsion is considered as the collective effect of a large number of interacting strings. The Monte Carlo model allows to understand in a qualitative way the formation of the initial conditions representing the dynamic origin for the elliptic flow and for the higher-order components of the two-particle angular correlations observed in nucleus-nucleus collisions. String density in nucleus-nucleus collisions {#onset} ============================================ In this section we use the string percolation model for the analysis of the onset of the long-range correlations in Au-Au collisions at RHIC and estimate string densities of nucleus-nucleus collisions at different collision energies and centralities. Onset of ridge phenomena in Au-Au collisions at RHIC ---------------------------------------------------- The very first experimental data on the ridge onset were obtained in detailed study of centrality dependence of two-particle correlations done by STAR in Au-Au collisions at 62 and 200 GeV at RHIC for all charged hadrons with rather low-$\pt$ ($\pt > 0.15$ GeV/[*c*]{}) [@Daugherity]. These preliminary results revealed that the “soft ridge” structure appears in Au-Au collisions after reaching certain collision centrality that might be characterized by definite (“critical”) number of participating nucleons ($N_{part}^{crit}(\sqrt{s})$). These “critical centralities” were found to be different for two collision energies: the onset of the ridge was observed in Au-Au at approximately 55% centrality for collision energy $\sqrt{s_{NN}}$ 200 GeV and at about 40% for $\sqrt{s_{NN}}=62$ GeV [@Daugherity]. One may see from the data [@Daugherity] that at $\sqrt{s_{NN}}=62$ GeV this phenomenon starts at $N_{part}^{crit}\approx90$, while at $\sqrt{s_{NN}}=200$ GeV the relevant threshold is marked by $N_{part}^{crit}\approx40$. The uncertainties of these values of $N_{part}^{crit}$, extracted from the RHIC data, produce some systematic error that is taken in account in our calculations. Moreover, it was also found in [@Daugherity] that transverse particle density $$\label{rho} \tilde{\rho}=\frac{3}{2}~{\frac{dN_{ch}}{dy}}/{\av{S}}$$ brings the transition points for these two energies to the same value $2.6\pm0.2$ fm$^{-2}$. Here $\frac{dN_{ch}}{dy}$ is the total charge particle multiplicity per rapidity unit at a given centrality, $\av{S}$ is the collision overlap area, the factor $\frac{3}{2}$ appears because both charge and neutral particles are taken into account. It is important to note that both $\frac{dN_{ch}}{dy}$ and $\av{S}$ in Eq. \[rho\] depend on centrality of collisions defined by the number of nucleons-participants $N_{part}$. ![Example of changes in 2D two-particle angular correlations with centrality of Au-Au collisions at $\sqrt{s}$ = 200 GeV: from peripheral (84$\%$ - 93$\%$) to semi-peripheral (55$\%$ - 64$\%$) collisions (see more detailed plots in [@STAR-2012]). []{data-label="ris:onset"}](fig/STAR_200GeV_twoCentralities.eps){width="75.00000%"} The detailed analysis of anomalous evolution of two-particle correlations with energy and centrality of Au-Au collisions was followed in [@STAR-2012]. The sudden change in 2D angular correlations, observed by STAR at some critical centrality (example is shown in Figure \[ris:onset\]) motivated our application of the string percolation model to describe this phenomenon. String density in Au-Au collisions in string percolation model {#density} -------------------------------------------------------------- In the present study we assume that the onset of the low-$\pt$ manifestation of a near-side ridge phenomenon in Au-Au collisions discussed above, is related to the critical quark-gluon string density reached at certain centrality. Under these conditions, a “macroscopic” cluster could appear, which would be composed of a large number of overlapped strings extended in rapidity. Such a cluster might be considered as a new kind of source emitting correlated particles. Cluster formation and the azimuthal effects in correlation functions might be due to some process that starts to be visible above the percolation threshold. To characterize mathematically the string density, a dimensionless percolation parameter $\tilde{\eta}$ is introduced [@Armesto1996; @Nardi; @Braun2000]: $$\label{perc} \tilde{\eta}=\frac{\pi r_{0}^2 N_{str}}{\av{S}}\ .$$ Here $\av{S}$ is the transverse area of the overlap of colliding nuclei, $N_{str}$ is a number of strings. The critical value of the parameter $\tilde{\eta}$ marking the percolation transition ($\tilde{\eta}^{crit}$) could be calculated from the geometrical considerations and is estimated to be $\tilde{\eta}^{crit}\approx~1.12-1.175$ [@BraunPRL], string radius is usually taken as $r_{0}=0.2-0.3$ fm [@Armesto2000; @Dias; @Pajares]. In our calculations we use $\tilde{\eta}^{crit}~\approx~1.15\pm0.03$ and $r_{0}~=~0.25$ fm. (We have to note here that only the product of ${r_{0}^2 N_{str}}$ could be constrained using Eq. \[perc\]. So that one will get different value of $N_{str}$ in case of using the different value of $r_{0}$). The number of particle emitting strings $N_{str}$ generally depends on the centrality of nuclus-nucleus collision, on the type of colliding system and on the collision energy $\sqrt{s}$. In our approach, $N_{str}$ and the overlap area $\av{S}$ depend on $N_{part}$. However, these variables could not be measured directly. In this work we exclude $\av{S}$ from the estimations by considering the ratio $\tilde{\rho}^{crit}/\tilde{\eta}^{crit}$ at the “critical” point, that marks the onset of the low-$\pt$ ridge manifestation mentioned above. Thus at the critical point one may obtain the following value: $$\label{ratio} \frac{\tilde{\rho}^{crit}(N_{part})}{\tilde{\eta}^{crit}(N_{part})}=\frac{3}{2} \frac{1}{\pi r_{0}^2} \frac{dN_{ch}}{dy} \frac{1}{N_{str}} = 2.3 \pm 0.2 fm^{-2},$$ here the error is coming mainly from the systematic uncertainties of $\tilde{\rho}^{crit}$ and $\tilde{\eta}^{crit}$. The total number of strings $N_{str}$ at the “critical” points in Au-Au collisions at $\sqrt{s_{NN}}=62$ and 200 GeV could be easily found from the Eq. \[ratio\]. The results of the calculations are presented in Table \[tab:results-AA\]. In order to make rough estimates for the dependence of the mean number of strings formed in nucleus-nucleus collisions vs. energy and centrality, we use the concept of valence and sea strings. A number of the valence strings $N_{V}$ is defined by $N_{part}$ and a number of the sea strings $N_{S}$ is proportional to $N_{coll}$, with a coefficient $a$. For the total number of the strings $N_{str}$, formed in nucleus-nucleus collisions at some given energy, we use the following parametrization: $$\label{a-prop} N_{str}=N_{V}+N_{S}=N_{part}+a\cdot N_{coll}.$$ The number $N_{str}$ can be estimated using Eq. \[ratio\] at the “critical” points, characterized by certain values of $N_{part}$ and estimated $N_{coll}$, after that the coefficient $a$ can calculated from Eq. \[a-prop\]. Results of the calculations of the parameter $a$ for $\sqrt{s_{NN}}$ = 62 and 200 GeV are presented in Table \[tab:results-AA\]. In the third line of the Table we also added results of our previous similar estimations [@j-psi] in the framework of string percolation model based on the observed threshold of anomalous suppression of $J/\psi$ in Pb-Pb collisions at $\sqrt{s_{NN}}= 17.3$ GeV at SPS. [lllllll]{} $\sqrt{s},~GeV~~~~~~~$ & $N_{part}$  & $(dN_{ch}/d\eta)$/$(0.5*N_{part})$  & $N_{str}$       & $N_{s}$        & $N_{coll}$       & $a$        \ 200(Au-Au) & 40 & $2.97\pm0.30$ [@PHOBOS130GeV-200GeV] & $194\pm25$ & $155\pm23$& $59\pm4$ & $2.6\pm0.4$\ 62 (Au-Au)& 90 & $2.30\pm0.23$ [@PHOBOS62] & $352\pm28$ & $262\pm23$ & $167\pm4$ & $1.6\pm0.2$\ 17.3 (Pb-Pb)& 110 & $1.62\pm0.21$ [@j-psi-pb] & $302\pm45$ & $192\pm30$ & $158\pm5$& $1.2\pm0.2$\ $\begin{array}{ccc} \begin{overpic}[width=0.62\textwidth, clip=true, trim=10 0 10 10] {fig/eta-percol-npart} \end{overpic} \end{array}$ It is possible to extrapolate the parameter $a$ to other energies and centralities of collisions with definite uncertainties (see details in [@OK-2010]). In order to get the centrality dependence of $\tilde{\eta}$ in nucleus-nucleus collisions, the calculations of the interaction area $\av{S}$ are performed by applying the relation $\av{S}\sim~N_{part}^{2/3}$ [@S_Npart]. The coefficient of proportionality is derived from the information obtained at the “critical” point of the transverse particle density $\tilde{\rho}$ as it is mentioned above in Eq. \[rho\]. The Modified Glauber model [@Ivanov] is used here for calculations of $N_{part}$ and $N_{coll}$. One may see on Figure \[fig:aAndEta\] that rather large values of average string density $\tilde{\eta}$ exceeding considerably the “critical” density value are obtained. The observation of the ridge at SPS energies, reported in [@na49-ridge], could be the first experimental hint, confirming that the critical string density is reached in central Pb-Pb collisions at $\sqrt{s} = 17.3$ GeV. At the same time string density acquired in Pb-Pb collisions at the LHC energies exceeds the percolation threshold in all centrality classes. String-string interaction in Monte Carlo toy model {#MC-toy} =================================================== Monte Carlo toy model --------------------- Interaction between color strings formed in nucleus-nucleus collisions might produce clear experimental manifestations in two-particle angular correlations. An exact form of this string-string interactions is not known. As it was proposed in [@Abramovsky-1988], it could be attractive or repulsive depending on the directions of the chromo-electric field inside the string. The issue of the string-string interaction has not yet been systematically addressed till recently. One can find the overview of the problem in [@strings-interaction]. The magnitude of this interaction in string tension units was found to be small ($\sim 10^{-1}-10^{-2}$ [@strings-interaction]). It is natural that deeper basic fundamental understanding of string-string interaction is required. “Mesonic clouds” around the color flux tubes and exchange of the scalar $\sigma$-meson were proposed in [@strings-interaction] and could be considered as the origin of interaction. $\begin{array}{ccc} \begin{overpic}[width=0.38\textwidth, clip=true, trim=0 0 20 0] {fig/tubes-2.eps} \end{overpic} & \begin{overpic}[width=0.38\textwidth, clip=true, trim=0 0 0 0] {fig/eventView_99.eps} \end{overpic} \end{array}$ In our study we consider a simplified approach to string-string interaction mechanism for the case of repulsion. A Monte Carlo (MC) toy model [@TOY] is used. It is assumed that quark-gluon strings, formed at early stage of hadron-hadron collision, may overlap in case of sufficiently high density and interact. An efficient string-string interaction radius $R_{\rm int}$ is introduced. We consider this free parameter differently from the string radius $r_{0}$. Doubled string-string interaction radius $R_{\rm int}$ can be interpreted as the effective distance of interaction between strings in the transverse plane. In this MC model we consider the case of [*repulsing*]{} strings. We do not take in account neither string attraction nor fusion. The repulsion mechanism between two strings is considered to be similar to [@Abramovsky-1988]: two completely overlapped strings have the energy density of $2E +2E_1$, while the density of partial overlapping is $2E +2E_1 \cdot S/S_{\rm 0}$. Here $E$ is the energy density of a single free string and $E_1$ is the energy density excess due to overlapping. $S$ is the area of the overlap (i.e. it is assumed that effectively interacting strings are “the black discs” with the area $S_{\rm 0}$ in the transverse plane). Thus, the total energy of the cluster formed by highly overlapped strings, reached in high-energy A-A collision, is larger then the sum of energies of individual separated non-interacting strings. This energy excess is responsible for the string repulsion [@Abramovsky-1988]. In this simplified approach, for any interacting string we consider a coherent sum of interactions of this string with all strings within the efficient interaction radius. Thus each string acquires transverse momentum $p_{\rm T}^{\rm string}$ [@Abramovsky-1988; @TOY], and all the particles produced during hadronization of this string gain in all region of rapidity a transverse Lorentz boost. In such a way, due to the string-string interaction, the initial asymmetry of azimuthal configuration of quark-gluon strings could be transferred into the final state with different harmonics of the azimuthal flow. Schematic view of two color flux tubes (strings) boosted apart and generating azimuthally asymmetric flow in the transverse plane due to repulsion is shown in Figure \[fig:tubes\] (left). An example of the toy model simulation of high string density in the transverse plane in semi-central nucleus-nucleus collision is shown in Figure \[fig:tubes\] (right). One may see that the flow appears as a result of multiple string interactions. Anisotropic flow in the Monte Carlo toy model ---------------------------------------------- We applied the MC model of efficient string repulsion [@TOY] for the analysis of two-particle correlation topology in order to study the origin of the elliptic flow and the higher harmonics observed in nucleus-nucleus collisions at RHIC and LHC. Two-particle correlation functions are obtained in the MC model for various centralities of high energy nucleus-nucleus collisions [@QC-2014-TOY]. They illustrate the onset of collectivity when passing from peripheral to central nucleus-nucleus collisions in a qualitative agreement with RHIC data. Different values of $R_{\rm int} =1$ and 2 fm were also tested for evaluation of the contribution of different harmonics and their centrality dependence, see [@QC-2014-TOY]. This qualitative approach demonstrates also the onset of the elliptic flow and the higher harmonics in heavy-ion collisions. ![The two-particle azimuthal correlation, measured in $0 < \Delta\phi <\pi$ and shown symmetrized over 2$\pi$, between a trigger particle with $2<\pt< 3$ GeV/$c$ and an associated particle with $1<\pt < 2$ GeV/$c$ for the 0-1% centrality class. The solid red line shows the sum of the measured anisotropic flow Fourier coefficients v$_2$, v$_3$, v$_4$, and v$_5$ (dashed lines) [@ALICE-H].[]{data-label="ris:double"}](fig/Double-R_inVectorFormat.eps){width="50.00000%"} The measurement of the triangular v${_3}$, quadrangular v${_4}$, and pentagonal v${_5}$ charged particle flow in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV was recently reported in [@ALICE-H]. In particular, it was shown that one of the remarkable observations, so-called double-ridge structure in very central Pb-Pb collisions (see Figure \[ris:double\]) is related to the triangular flow and can be understood from the initial spatial anisotropy. Figure \[fig:harmonics\_dihadron\_mostcentral\] (left) shows two-particle azimuthal correlation function obtained in the MC model simulations of the most central Pb-Pb events, for charged particles with $\pt\in[3,5]$ GeV/[*c*]{}. We use the correlation measure $\Delta\rho/\sqrt{\rho_{\rm ref}}$, which is described in detail, in particular, in [@STAR-2012]. The harmonic decomposition of the azimuthal profile of this function is presented in the right pad of Figure \[fig:harmonics\_dihadron\_mostcentral\]. These data are in a nice correspondence with the experimental picture [@ALICE-H] shown in Figure \[ris:double\] (up to a numerical factor between the two different observables). $ \begin{array}{ccc} \begin{overpic}[width=0.5\textwidth, clip=true, trim=10 0 61 0]{fig/hist2D_with_large_v3.eps} \put(0,75){ central events} \put(0,70){ $\pt$ 3-5 GeV/c } \end{overpic} & \begin{overpic}[width=0.5\textwidth]{fig/harmonics_central_large_v3.eps} \put(50,66){ $\pt$ 3-5 GeV/{\it c} } \end{overpic} \end{array}$ Discussion ---------- All this indicates that string percolation with introduced repulsion mechanism between interacting quark-gluon strings, both valence and sea-quark, may lead to some collective phenomena in nucleus-nucleus collisions. The model gives adequate description of the transition from peripheral to central collisions as well as rise and development of the contribution of the elliptic flow and interplay with higher harmonics. In these first calculations we neglect the finite string length in rapidity and concentrate on the azimuthal asymmetry of correlation functions. For this case a phenomenological approach of repulsive string-string interaction is shown to be a possible dynamic origin of the observed azimuthal asymmetries of two-particle correlation functions. Our results show also that the increase of the number of strings and correspondingly the density of the overlapped strings with centrality and collision energy is related mainly to increase of the number of sea-strings. Therefore, for example, the observed rise with centrality of the amplitude and a pseudorapidity width of the so-called same-side 2D Gaussian obtained in [@Daugherity; @STAR-2012] may require an accurate consideration of sea-quark strings formation and their interaction at midrapidity. The onset of the ridge structure in AA, pA and pp collisions was also considered in the frame of string percolation in a recently published paper [@Pajares-2014]. The increase of the rapidity length of the effective cluster formed by overlapping sea-strings is discussed. The total energy-momentum of the string cluster is taken here to be the sum of the energy-momenta of the individual strings. In our approach, following [@Abramovsky-1980; @Abramovsky-1988], the energy of the cluster of the overlapped strings is higher than the sum of individual non-interacting strings, therefore, the effective cluster may be more extended in rapidity. Another kind of string interaction, string fusion [@Amelin1994; @Braun], could explain other observed effects like increased production of strange and multistrange particles with centrality of nucleus-nucleus collisions. String fusion could be considered as an initial stage leading to glasma or QGP formation. New constraints on the mentioned models could be obtained from experiment. Conclusion ========== The hypothesis of string-string interaction and percolation string transition looks reasonable in the quantification of the onset of the low-$\pt$ near side ridge phenomena in Au-Au collisions at RHIC and in Pb-Pb collisions at LHC. One may assume that the onset of string percolation at sufficiently high string densities leads to the formation of rather large clusters composed of overlapped strings extended in rapidity. Collective effects of interactions between strings inside this cluster could be one of the possible processes leading to repulsion of strings thus shaping topology of two-particle correlation functions. The Monte Carlo toy-model with the efficient account of string repulsion of color flux tubes describes for the first time in a qualitative way the dynamics of the initial conditions of high energy nucleus-nucleus collisions. The value of the efficient string-string interaction radius $R_{\rm int}\sim 2$ fm provides qualitative description of elliptic and triangular flows in nucleus-nucleus collisions at RHIC and LHC energies. This radius is found to be larger than the usual string radius, $r_{0}=0.25$ fm. More detailed quantitative estimates including the case of proton-proton and proton-nucleus collisions will follow. Authors would like to thank V. Vechernin for fruitful discussions and for permanent interest to this work. The work was supported for G. F. and I. A. by the St.Petersburg State University grant 11.38.242.2015. [99]{} V.A. Abramovsky, O.V. Kanchely, JETP letters **31**, (1980) 566. Abramovskii V. A., Gedalin E. V., Gurvich E. G., Kancheli O. V., JETP Lett., vol.47, 337-339, 1988. A.Capella, U.P.Sukhatme, C.I.Tan and J.Tran Thanh Van, Phys. Lett. B **81**, (1979) 68; Phys. Rep. **236**, (1994) 225. A.B.Kaidalov, Phys. Lett., **116B**, (1982) 459; A.B.Kaidalov K.A.Ter-Martirosyan, Phys. Lett., **117B**, (1982) 247. N.Armesto and C.Pajares, Int. J. Mod. Phys. A **15**, (2000) 2019. J. Adams et al. (STAR Collaboration), J. Phys. G 32, L37 (2006); J. Adams et al. (STAR Collaboration), Phys. Rev. C 72, 044902 (2005). J. Putschke et al. (STAR Collaboration), J. Phys. G **34**, (2007) S679. Adams J et al (STAR Collaboration) Phys. Rev. C **73**, (2006) 064907. L.Ray (for the STAR Collaboration), Nuclear Physics A 854 (2011) 89-96. CMS Collaboration, JHEP 1009 (2010) 091; arXiv:1009.4122v1. CMS Collaboration, CMS-HIN-11-001, CERN-PH-EP-2011-056. CMS Collaboration, Phys. Lett. B 718 (2013) 795, DOI:10.1016/j.physletb.2012.11.025, arXiv:1210.5482. Phys.Lett. B708 (2012) 249-264 ISSN 0370-2693, 10.1016/j.physletb.2012.01.060. A. Dumitru et al.,Nucl. Phys. A810 (2008) 91, arXiv:0804.3858. J. Dias de Deus and C. Pajares, Phys.Lett B**695** (2011) 211-213. C. Pajares, Nucl. Phys, A 854 (2011) 125-130, arXiv:1007.3610. O.Kochebina, G.Feofilov , Onset of ”ridge phenomenon” in AA and pp collisions and percolation string model, arXiv.org(hep-ph)arXiv:1012.0173, 1 Dec 2010. M.Daugherity (STAR Collaboration), J. Phys. G**35**, (2008) 104090, arXiv:0806.2121v2 (2008); M. Daugherity (for STAR Collaboration), report on QM2008. I. Altsybeev, in the materials of Proceedings of the XI Quark Confinement and the Hadron Spectrum Conference, Saint-Petersburg, 2014. G.Agakishev et al. (STAR Collaboration), Phys.Rev.C 86, 064902(2012). N. Armesto, M.A. Braun, E.G. Ferreiro and C. Pajares, Phys. Rev. Lett. **77**, (1996) 3736. M.Nardi and H.Satz, Phys. Lett. B442 (1998) 14;H.Satz, Nucl. Phys. A**661**, (2000) 104c. M.A.Braun and C.Pajares, Eur. Phys. J. C**16**, (2000) 349. M.A.Braun and C.Pajares, Phys. Rev. Lett. **85**, (2000) 4864. J.Dias de Deus and A. Rodrigues, Phys. Rev. C **67**, (2003) 064903. C. Pajares, hep-ph/0501125v1 (2005). G.Feofilov, O.Kochebina, Proceedings of the XIX International Baldin Seminar on High Energy Physics Problems, **Vol.I**, (2008) 120-126, arXiv:0905.2494v2. B. B. Back, et al.(PHOBOS Collaboration), Phys. Rev. C **65**, (2002) 061901(R). B. B. Back, et al.(PHOBOS Collaboration), Phys. Rev. C **74**, (2006) 021901(R). B. Alessandro et al. (NA50 Collaboration), Eur. Phys. J. C39, 335 (2005) ; arXiv:hep-ex/0412036v1. Elena G. Ferreiro, ACTA PHYSICA POLONICA B, No 2, Vol. **36**, (2005) 543-550. G.Feofilov, A.Ivanov, Journal of Physics: Conference Series **5**, (2005) 230-237. M.Szuba (for the NA49 Collaboration), arXiv: 0908.2155v1 \[nucl.ex\], 15 Aug. 2009. Terence J. Tarnowsky, Brijesh K. Srivastava, Rolf P. Scharenber (for the STAR Collaboration), Nukleonika 51S3:S109-S112,2006; arXiv: nucl-ex/ 0606019v2, 2007. T. Kalaydzhyan and E. Shuryak, arXiv:1402.7363 \[hep-ph\]. I.Altsybeev, G.Feofilov, O.Kochebina, in the materials of Proceedings of the XI Quark Confinement and the Hadron Spectrum Conference, Saint-Petersburg, 2014. K. Aamodt et al.(ALICE Collaboration), PRL, 107, 032301(2011) C. Andres, A. Moscoso, and C. Pajares, arXiv:1405.3632v3 \[hep-ph\], 12 Dec 2014. N.S. Amelin, N. Armesto, M.A. Braun, E.G. Ferreiro and C. Pajares,Phys. Rev. Lett. 73, 2813 (1994). M.A.Braun, C.Pajares and J. Ranft. Int. J. of Mod. Phys. A**14**, (1999) 2689; hep-ph/9707363.

--- abstract: | Given a Banach algebra $ \mathcal{A} $ and a continuous homomorphism $\sigma$ on it, the notion of $\sigma$-biflatness for $ \mathcal{A} $ is introduced. This is a generalization of biflatness and it is shown that they are distinct. The relations between $\sigma$-biflatness and some other close concepts such as $\sigma$-biprojectivity and $\sigma$-amenability are studied. The $\sigma$-biflatness of tensor product of Banach algebras are also discussed. address: - 'Department of Mathematics, Central Tehran Branch, Islamic Azad University,Tehran, Iran, e-mail: [[email protected]]{}' - 'Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: [a\[email protected]]{}' author: - Sanaz Haddad sabzevar - Amin Mahmoodi title: 'Biflat-like Banach algebras ' --- Keywords: $\sigma$-amenable, $\sigma$-biflat, $\sigma$-virtual diagonal, $\sigma$-biprojective, $\sigma$-derivation. MSC 2010: Primary: 46H25; Secondary: 16E40, 43A20. Introduction ============= Biprojectivity and biflatness for Banach algebras, as introduced and studied in the works of Helemskii (see for instance \[5\]), have proved to be important and fertile notions. There are close relationship between these notions and some other concepts of Banach algebras such as amenability. It is known that every biprojective Banach algebra with a bounded approximate identity is amenable and in the presence of a bounded approximate identity, biflatness and amenability are the same notions [@H]. Before preceding further, we recall some preliminaries. Let $ \mathcal{A} $ be a Banach algebra. Then its projective tensor product $ \mathcal{A}\widehat{\otimes}{\mathcal{A}}$ is a Banach $ \mathcal{A} $-bimodule through $$a\cdot(b\otimes c)=ab\otimes c \ \ \ \text{and} \ \ \ (b\otimes c)\cdot a=b\otimes ca \ \ \ (a,b,c\in \mathcal{A}).$$ For a Banach $ \mathcal{A} $-bimodule $X$, the dual space $X^*$ becomes a Banach $ \mathcal{A} $-bimodule in a natural manner. Let $X$ and $Y$ be Banach $ \mathcal{A} $-bimodules. A bounded linear map $T : X \longrightarrow Y$ is an $ \mathcal{A} $-*bimodule homomorphism* if $ T(a\cdot x)=a \cdot T(x) $ and $ T(x\cdot a)= T(x)\cdot a $ , for $a \in {\mathcal A}$, $x \in X$. It is obvious that the *diagonal operator* $ \pi_{\mathcal{A}}:\mathcal{A}\widehat{\otimes}\mathcal{A} \rightarrow \mathcal{A}$ given by $ \pi( a\otimes b) = ab $, is an $ \mathcal{A} $-bimodule homomorphism. If it is clear to which algebra $ \mathcal{A} $ we refer, we simply write $\pi$. A Banach algebra $ \mathcal{A} $ is *biprojective* if $ \pi$ has a right inverse which is an $\mathcal{A}$-bimodule homomorphism. If there is an $\mathcal{A}$-bimodule homomorphism which is a left inverse of $\pi^*$, then we say that $ \mathcal{A} $ is *biflat*. Let $\mathcal{A}$ be a Banach algebra. We write $Hom(\mathcal{A})$ for the set of all continuous homomorphisms on $ \mathcal{A} $. Let $X$ and $Y$ be Banach $\mathcal{A}$-bimodule, and let $\sigma\in Hom(\mathcal{A})$. A bounded linear map $ T:X \rightarrow Y $ is a $\sigma$-$ \mathcal{A} $-*bimodule homomorphism* if $ T(a\cdot x)=\sigma (a) \cdot T(x) $ and $ T(x\cdot a)= T(x)\cdot\sigma (a) $ where $ x\in X$ and $a\in\mathcal{A} $. A Banach algebra $ \mathcal{A} $ is $\sigma$-*biprojective* if there exists a $ \sigma $-$ \mathcal{A} $-bimodule homomorphism $ \rho : \mathcal{A} \longrightarrow \mathcal{A}\widehat{\otimes}\mathcal{A}$ such that $ \pi \circ \rho = \sigma$ \[13\]. The purpose of this paper is to study the concept of $\sigma$-biflatness for Banach algebras. We have to stress that our definition is completely different from what have introduced in \[4, Definition 2.11\]. For comparison, unlike our definition, $\sigma$-biflatness in \[4\] is not a generalization of the notion of biflatness \[4, Remark 2.13\]. The organization of the paper is as follows. Firstly, in section 2 we investigate some basic properties of $\sigma$-biflat Banach algebras. We find an equivalent condition to $\sigma$-biflatness (Theorem 2.3). We prove that every $\sigma$-biprojective Banach algebra is $\sigma$-biflat (Proposition 2.4). However biflat Banach algebras are $\sigma$-biflat (Proposition 2.5), we give an example to show that the class of $\sigma$-biflat Banach algebras is larger than that for biflat Banach algebras (Example 2.1). In section 3, we find the relations between $\sigma$-biflatness and both $\sigma$-amenability and the existence of some certain $\sigma$-diagonals. There are examples of $\sigma$-biflat Banach algebras which are not $\sigma$-amenable (Examples 3.1 and 3.2). In section 4, we deal with the short exact sequence $ \Sigma : 0 \longrightarrow ker \pi \stackrel \imath \longrightarrow \mathcal{A} \widehat{\otimes}\mathcal{A} \stackrel \pi \longrightarrow \mathcal{A} \longrightarrow 0 $, where $ \mathcal{A}$ is a Banach algebra. We prove that if $ \mathcal{A}$ is $\sigma$-amenable, then $ \Sigma ^*$ and $ \Sigma ^{**}$ behave like splitting sequences (Proposition 4.2). Finally in section 5, we generalize Ramsden’s theorem \[11, Proposition 2.5\] related to biflatness of tensor product of Banach algebras to the $\sigma$-case (Theorem 5.1). Basic properties ================ \[2.3\] Let $ \mathcal{A} $ be a Banach algebra and let $ \sigma\in Hom(\mathcal{A}) $. Then $ \mathcal{A} $ is *$ \sigma $-biflat* if there exists a bounded linear map $ \rho : ( \mathcal{A}\widehat{\otimes}\mathcal{A})^* \longrightarrow \mathcal{A}^* $ satisfying $$\rho(\sigma(a)\cdot\lambda)=a\cdot\rho(\lambda) \ \ \ \text{and} \ \ \ \rho(\lambda\cdot\sigma(a))=\rho(\lambda)\cdot a \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ for $ a\in \mathcal{A} , \lambda\in\mathcal{A}^{\ast}$ such that $ \rho \circ \pi^* = \sigma^*$. \[2.2\] Let $ \mathcal{A} $ be a Banach algebra, let $ \sigma\in Hom(\mathcal{A}) $ and let $ X $ and $ Y $ be Banach $ \mathcal{A} $-bimodules. If $ T:X\rightarrow Y $ is a $ \sigma $-$ \mathcal{A} $-bimodule homomorphism, then $ T^{\ast}$ satisfies $(1) $, that is, $ T^{\ast}(\sigma(a)\cdot\lambda)=a\cdot T^{\ast}(\lambda) $ and $T^{\ast}(\lambda\cdot\sigma(a))=T^{\ast}(\lambda)\cdot a$, for $a\in\mathcal{A},\lambda\in Y^{\ast}$. For every $ x\in X $, $ a\in A $ and $ \lambda\in Y^{\ast} $, we have $$\begin{aligned} \langle T^{\ast}(\sigma(a)\cdot\lambda),x\rangle &=\langle\sigma(a)\cdot\lambda,T(x)\rangle \\&=\langle\lambda,T(x)\cdot\sigma(a)\rangle= \langle\lambda,T(x\cdot a)\rangle\\&=\langle T^{\ast}(\lambda),x\cdot a\rangle=\langle a\cdot T^{\ast}(\lambda),x\rangle,\end{aligned}$$ so $ T^{\ast}(\sigma(a)\cdot\lambda)=a\cdot T^{\ast}(\lambda) $. Similarly $ T^{\ast}(\lambda\cdot\sigma(a))=T^{\ast}(\lambda)\cdot a $. The following characterization is useful. \[2.5\] Let $ \mathcal{A} $ be a Banach algebra and let $ \sigma\in Hom(\mathcal{A}) $. Then, the following are equivalent: 1. $ \mathcal{A} $ is $ \sigma $-biflat; 2. There is a $ \sigma $-$ \mathcal{A} $-bimodule homomorphism $ \rho:\mathcal{A}\rightarrow(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast\ast} $ such that $ \pi^{\ast\ast}\circ\rho=\kappa_{\mathcal{A}}\circ\sigma $, where $\kappa_{\mathcal{A}}$ is the canonical embedding of $ \mathcal{A} $ into $\mathcal{A}^{\ast\ast}$. (i)$ \Longrightarrow $(ii) Since $ \mathcal{A} $ is $ \sigma $-biflat, there exists a bounded linear map $ \theta:(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast}\rightarrow\mathcal{A}^{\ast} $ satisfying ($ 1 $) and $ \theta\circ\pi^{\ast}=\sigma^{\ast} $. We set $ \rho $ to be the restriction of $ \theta^{\ast} $ to $ \mathcal{A} $. Then, for every $ a\in\mathcal{A} $ and $ \lambda\in\mathcal{A}^{\ast} $ we have $$\begin{aligned} \langle\pi^{\ast\ast}\circ\rho(a),\lambda\rangle &=\langle\rho(a),\pi^{\ast}(\lambda)\rangle=\langle\theta^{\ast}(a),\pi^{\ast}(\lambda)\rangle \\ &=\langle a,\theta\circ\pi^{\ast}(\lambda)\rangle=\langle a,\sigma^{\ast}(\lambda)\rangle \\ &=\langle\sigma(a),\lambda\rangle=\langle\kappa_{\mathcal{A}}\circ\sigma(a),\lambda\rangle. \end{aligned}$$ Next, for every $ a,b\in\mathcal{A} $ and $ \xi\in(\mathcal{A}\otimes\mathcal{A})^{\ast} $ $$\begin{aligned} \langle \rho(ab),\xi\rangle&=\langle ab,\theta(\xi)\rangle=\langle b,\theta(\xi)\cdot a\rangle\\&=\langle b,\theta(\xi\cdot\sigma(a))\rangle=\langle\rho(b),\xi\cdot\sigma(a)\rangle=\langle\sigma(a)\cdot\rho(b),\xi\rangle, \end{aligned}$$ so $ \rho(ab)=\sigma(a)\cdot\rho(b) $. A Similar argument shows that $ \rho(ba)=\rho(b)\cdot\sigma(a) $.\ (ii)$ \Longrightarrow $(i) Let $ \rho $ be as specified in the clause (ii). Suppose that $ \tilde{\rho}:(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast}\rightarrow\mathcal{A}^{\ast} $ is the restriction of $ \rho^{\ast} $ into $ (\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast} $. Clearly $ \tilde{\rho} $ is a bounded linear map and satisfies $ (1) $, by Lemma \[2.2\]. For every $ a\in\mathcal{A} $ and $ \lambda\in\mathcal{A^{\ast}} $ $$\begin{aligned} \langle\tilde{\rho}\circ\pi^{\ast}(\lambda),a\rangle &=\langle\pi^{\ast}(\lambda),\rho(a)\rangle=\langle\lambda,\pi^{\ast\ast}\circ\rho(a)\rangle \\ &=\langle\lambda,\kappa_{\mathcal{A}}\circ\sigma(a)\rangle=\langle\lambda,\sigma(a)\rangle=\langle\sigma^{\ast}(\lambda),a\rangle.\end{aligned}$$ showing that $ \tilde{\rho}\circ\pi^{\ast}=\sigma^{\ast} $. It is well-known that every biprojective Banach algebra is biflat. The next proposition gives a generalization of this fact. \[2.6\] Let $ \mathcal{A} $ be a Banach algebra and let $ \sigma\in Hom(\mathcal{A}) $. If $ \mathcal{A} $ is $ \sigma $-biprojective, then $ \mathcal{A} $ is $ \sigma $-biflat. If $ \mathcal{A} $ is a $ \sigma $-biprojective Banach algebra, then there exists a bounded $ \sigma $-$ \mathcal{A} $-bimodule homomorphism $ \rho:\mathcal{A}\rightarrow\mathcal{A}\widehat{\otimes}\mathcal{A} $ such that $ \pi\circ\rho=\sigma $. For every $ \lambda\in\mathcal{A}^{\ast} $ and $ a\in\mathcal{A} $ we have $$\langle\rho^{\ast}\circ\pi^{\ast}(\lambda),a\rangle=\langle\pi^{\ast}(\lambda),\rho(a)\rangle=\langle\lambda,\pi\circ\rho(a)\rangle= \langle\lambda,\sigma(a)\rangle=\langle\sigma^{\ast}(\lambda),a\rangle.$$ Since $ \rho $ is a $ \sigma $-$ \mathcal{A} $-bimodule homomorphism, $ \rho^{\ast} $ satisfies $ (1) $ by Lemma \[2.2\], so $ \mathcal{A} $ is $ \sigma $-biflat. The relation between biflatness and $ \sigma$-biflatness appears as follows. \[2.11\] Let $ \mathcal{A} $ be a Banach algebra and let $ \sigma\in Hom(\mathcal{A}) $. Then: 1. If $ \mathcal{A} $ is biflat, then $ \mathcal{A} $ is $ \sigma $-biflat. 2. If $ \mathcal{A} $ is $ \sigma $-biflat and has a bounded approximate identity, and if $ \sigma $ has a dense range, then $ \mathcal{A} $ is biflat. \(i) There exists a bounded $ \mathcal{A} $-bimodule homomorphism $ \rho:(\mathcal{A}\hat{\otimes}\mathcal{A})^{*}\longrightarrow\mathcal{A}^{\ast} $ such that $ \rho\pi^{\ast}=i_{\mathcal{A}^{\ast}} $. Define $ \acute{\rho}=\sigma^{\ast}\rho:(\mathcal{A}\hat{\otimes}\mathcal{A})^{\ast}\longrightarrow\mathcal{A}^{\ast} $. Then, for every $ \lambda\in (\mathcal{A}\hat{\otimes}\mathcal{A})^{\ast} $ and $ a,b\in\mathcal{A} $, we have $$\begin{aligned} \langle b,\acute{\rho}(\sigma(a)\cdot\lambda)\rangle&=\langle b,\sigma^{\ast}\rho(\sigma(a)\cdot\lambda)\rangle=\langle b,\sigma^{\ast}(\sigma(a)\cdot\rho(\lambda))\rangle\\&=\langle\sigma(b),\sigma(a)\cdot\rho(\lambda)\rangle=\langle\sigma(ba),\rho(\lambda)\rangle\\&=\langle ba,\sigma^{\ast}\rho(\lambda)\rangle=\langle b,a\cdot\acute{\rho}(\lambda)\rangle\end{aligned}$$ and analogously, $ \acute{\rho}(\lambda\cdot\sigma(a))=\acute{\rho}(\lambda)\cdot a $. It is clear that $ \acute{\rho}\pi^{\ast}=\sigma^{\ast} $, and hence $ \mathcal{A} $ is $ \sigma $-biflat.\ (ii) By Proposition \[2.9\] below, $ \mathcal{A} $ is amenable and then by [@D Theorem 2.9.65], $ \mathcal{A} $ is biflat. Now, we give a $ \sigma $-biflat Banach algebra which is not biflat. It is known that $ \mathcal{A}=l^{1}(\mathbb{Z}^{+}) $ is not amenable, and so $ \mathcal{A}^{\sharp} $ is not amenable by [@R Corollary 2.3.11]. Therefore, $ \mathcal{A}^{\sharp} $ is not biflat by [@H]. We consider the homomorphism $ \sigma:\mathcal{A}^{\sharp}\rightarrow\mathcal{A}^{\sharp} $ defined by $ \sigma(a+\lambda e)=\lambda $, $ (a\in \mathcal{A},\lambda\in \mathbb{C}) $. An argument similar to [@G Example 2.7], shows that $ \mathcal{A}^{\sharp} $ is $ \sigma $-amenable. Then by Remark \[2.10\] below, $ \mathcal{A}^{\sharp} $ is $ \sigma $-biflat. Relation to $ \sigma $-amenability ================================== Let $ \mathcal{A} $ be a Banach algebra, let $ \sigma \in Hom(\mathcal{A})$, and let $X$ be a Banach $ \mathcal{A} $-bimodule. A bounded linear map $ {D}:\mathcal{A}\rightarrow X $ is a *$\sigma$-derivation* if $$D(ab)=\sigma(a)\cdot D(b)+D(a)\cdot \sigma(b) \qquad (a,b\in\mathcal{A}) ,$$ and it is *$\sigma$-inner derivation* if there is an element $x\in X$ such that $ D(a)=\sigma(a)\cdot x-x\cdot \sigma(a) $ for all $ a\in\mathcal{A} $. A Banach algebra $ \mathcal{A} $ is *$\sigma$-amenable* if for every Banach $ \mathcal{A} $-bimodule $X$, every $\sigma$-derivation $ {D}:\mathcal{A}\rightarrow X^* $ is $\sigma$-inner [@M1; @M2]. An element $ M\in(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast\ast} $ is said to be a *$ \sigma $-virtual diagonal* for $ \mathcal{A} $, if $ \sigma(a)\cdot M=M\cdot\sigma(a) $ and $ \pi^{**}(\it M)\cdot\sigma(a)=\sigma(a) $ for all $ a\in\mathcal{A} $. A bounded net $ (m_{\alpha})\subseteq\mathcal{A}\widehat{\otimes}\mathcal{A} $ is said to be a *$ \sigma $-approximate diagonal* for $ \mathcal{A} $ if $ \lim _{\alpha}m_{\alpha}\cdot\sigma(a)-\sigma(a)\cdot m_{\alpha}=0 $ and $ \lim_{\alpha}\pi(m_{\alpha})\cdot\sigma(a)=\sigma(a) $ for all $ a\in\mathcal{A} $ [@M]. In [@G Proposition 2.4] it is shown that if $ \mathcal{A} $ has a $ \sigma $-virtual diagonal, then it has a $ \sigma $-approximate diagonal. An easy verification shows that the converse is also true. In the following two propositions, we establish a connection between $ \sigma $-biflatness and existence of $ \sigma $-virtual diagonals. \[2.7\] Let $ \mathcal{A} $ be a Banach algebra with a bounded approximate identity and $ \sigma\in Hom(\mathcal{A}) $. If $ \mathcal{A} $ is $ \sigma $-biflat, then $ \mathcal{A} $ has a $ \sigma $-virtual diagonal. Let $ (e_{\alpha}) $ be a bounded approximate identity for $ \mathcal{A} $. Since $ \mathcal{A} $ is $ \sigma $-biflat there exists a bounded linear map $ \rho:(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast}\rightarrow\mathcal{A}^{\ast} $ satisfying $ (1) $ such that $ \rho\circ\pi^{\ast}=\sigma^{\ast} $. We may suppose that $ \rho^{\ast}(e_{\alpha}) $ converges in the weak\* topology to an element of $(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast\ast}$, say $ M $. Then for each $ a\in\mathcal{A} $ and $ \lambda\in(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast} $, we have $$\begin{aligned} \langle \sigma(a)\cdot M,\lambda\rangle &=\langle M,\lambda\cdot\sigma(a)\rangle= w^{\ast}-\lim _{\alpha}\langle \rho^{\ast} (e_{\alpha}),\lambda\cdot\sigma(a)\rangle\\& =\lim _{\alpha}\langle e_{\alpha},\rho(\lambda\cdot\sigma(a))\rangle=\lim _{\alpha}\langle e_{\alpha},\rho(\lambda)\cdot a\rangle\\&= \lim_{\alpha}\langle ae_{\alpha},\rho(\lambda)\rangle=\langle a,\rho(\lambda)\rangle \end{aligned}$$ and similarly $ \langle M\cdot\sigma(a),\lambda\rangle=\langle a,\rho(\lambda)\rangle $. Thus $ \sigma(a)\cdot M=M\cdot\sigma(a) $. An application of Theorem \[2.5\] shows that $$\begin{aligned} \pi^{\ast\ast}_{\mathcal{A}}(M)\cdot\sigma(a) &=w^{\ast}-\lim_{\alpha}\pi^{\ast\ast}_{\mathcal{A}}(\rho^{\ast}(e_{\alpha}))\cdot\sigma(a)=w^{\ast}- \lim_{\alpha}\kappa_{\mathcal{A}}\circ\sigma(e_{\alpha})\cdot\sigma(a)=\sigma(a), \end{aligned}$$ hence $ M $ is a $ \sigma $-virtual diagonal for $ \mathcal{A} $. \[2.8\] Let $ \mathcal{A} $ be a Banach algebra and $ \sigma\in Hom(\mathcal{A}) $. If $ \mathcal{A} $ has a $ \sigma $-virtual diagonal, then $ \mathcal{A} $ is $ \sigma $-biflat. Let $ M\in(\mathcal{A}\hat{\otimes}\mathcal{A})^{\ast\ast} $ be a $ \sigma $-virtual diagonal. Define $ \rho:\mathcal{A}\rightarrow(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast\ast} $ by $ \rho(a)=\sigma(a)\cdot M $ $ (a\in\mathcal{A}) $. Clearly $ \rho $ is bounded, linear and $ \sigma $-$ \mathcal{A} $-bimodule homomorphism. Also $$\pi^{\ast\ast}\circ\rho(a)=\pi^{\ast\ast}(\sigma(a)\cdot M)=\pi^{\ast\ast}(M)\cdot\sigma(a)=\sigma(a)=\kappa_{\mathcal{A}}(\sigma(a)).$$ Thus $ \mathcal{A} $ is $ \sigma $-biflat. Now, we describe the relation between $ \sigma $-biflatness and $ \sigma $-amenability. \[2.9\] Let $ \mathcal{A} $ be a $ \sigma $-biflat Banach algebra with a bounded approximate identity. If $ \sigma\in Hom(\mathcal{A}) $ has a dense range, then $ \mathcal{A} $ is amenable so is $ \sigma $-amenable. Let $ \mathcal{A} $ be a $ \sigma $-biflat Banach algebra. By Proposition \[2.7\] $ \mathcal{A} $ has a $ \sigma $-virtual diagonal, equivalently, $ \mathcal{A} $ has a $ \sigma $-approximate diagonal $ m_{\alpha}\subseteq(\mathcal{A}\hat{\otimes}\mathcal{A}) $. We show that $ \mathcal{A} $ has an approximate diagonal, so it is amenable by [@D Theorem 2.9.65] and thus by [@M Corollay 2.2] it is $ \sigma $-amenable, as required. Suppose that $ a\in\mathcal{A} $ and $ \varepsilon>0 $. There exists $ N $ such that for every $ n\geq N $, $ \lVert a-\sigma(a_{n})\rVert<\varepsilon $ then $ \lVert a-\sigma(a_{N})\rVert<\varepsilon $. There is $ \alpha_{0} $ such that for all $ \alpha\geq\alpha_{0} $, $ \lVert\sigma(a_{N})\cdot m_{\alpha}-m_{\alpha}\cdot\sigma(a_{N})\rVert<\varepsilon $. Hence for $ \alpha\geq\alpha_{0} $, we have $ \lVert a\cdot m_{\alpha}-m_{\alpha}\cdot a\rVert\leq\lVert a\cdot m_{\alpha}-\sigma(a_{N})\cdot m_{\alpha}\rVert+\lVert\sigma(a_{N}) \cdot m_{\alpha}-m_{\alpha}\cdot\sigma(a_{N})\rVert+\lVert m_{\alpha}\cdot\sigma(a_{N})-m_{\alpha}\cdot a\rVert\leq(2M+1)\varepsilon $, where $ M $ is a bounded of $ (m_{\alpha}) $. This shows that $ a\cdot m_{\alpha}-m_{\alpha}\cdot a\rightarrow 0 $ and similarly, $ a\pi(m_{\alpha})\rightarrow a $. \[2.10\] There is a converse for Proposition \[2.9\]. Indeed if $ \mathcal{A} $ is $ \sigma $-amenable with a bounded approximate identity, then it has $ \sigma $-virtual diagonal [@G Theorem 2.2], and whence by Proposition \[2.8\], $ \mathcal{A} $ is $ \sigma $-biflat. We conclude the current section with two examples of $ \sigma $-biflat Banach algebras which are not $ \sigma $-amenable. We refer the reader to [@R Definition 3.1.8 and Definition C.1.1] for the definitions of *property* ($ \mathbb{A} $) and *approximation property* for Banach spaces. We also write $ \mathfrak{A}(E) $ for the space of *approximable operators* on a Banach space $E$. let $ E $ be a Banach space with property ($ \mathbb{A} $) such that $ E^{**} $ does not have the bounded approximation property. Then $ \mathfrak{A}(E^{*}) $ is biflat while it is not amenable [@R Theorem 4.3.24]. Therefore, $ \mathfrak{A}(E^{*}) $ is $ \sigma $-biflat for each $ \sigma\in Hom(\mathfrak{A}(E^{*})) $. On the other hand, One may check that every $ \sigma $-amenable Banach algebra for which $ \sigma $ has a dense range, is amenable. Hence, choosing a homomorphism $ \sigma\in Hom(\mathfrak{A}(E^{*})) $ with a dense range, it is readily seen that $ \mathfrak{A}(E^{*}) $ is not $ \sigma $-amenable. let $ H $ be an infinite dimensional Hilbert space. It was shown in [@R Example 4.3.25] that $ \mathfrak{A}( H \widehat{\otimes} H)$ is biflat but not amenable. Then, an argument similar to Example 3.1 shows that $ \mathfrak{A}( H \widehat{\otimes} H)$ is $ \sigma $-biflat which is not $ \sigma $-amenable, whereas $ \sigma$ is a homomorphism in $ Hom(\mathfrak{A}( H \widehat{\otimes} H)) $ with a dense range. The role of sequences ======================= We start with the following which is similar to Lemma 2.2. Let $ \mathcal{A} $ be a Banach algebra, $X$ and $Y$ be Banach $ \mathcal{A} $-bimodule and let $ \sigma\in Hom(\mathcal{A}) $. If $T : X \longrightarrow Y$ is a bounded linear map satisfying $T(\sigma(a)\cdot x) = a\cdot Tx$ and $T( x\cdot \sigma(a)) = Tx\cdot a$ ($a \in \mathcal{A}, x \in X$), then $T^*$ is a $\sigma$-$ \mathcal{A} $-bimodule homomorphism. For a Banach algebra $ \mathcal{A} $, we consider the short exact sequence $$\Sigma : 0 \longrightarrow ker \pi \stackrel \imath \longrightarrow \mathcal{A} \widehat{\otimes}\mathcal{A} \stackrel \pi \longrightarrow \mathcal{A} \longrightarrow 0 \ ,$$ and its duals $ \Sigma^*$ and $ \Sigma^{**}$, where $\imath$ is the natural injection. It is known that a Banach algebra $ \mathcal{A}$ with a bounded approximate identity is amenable if and only if $ \Sigma^*$ (and so $ \Sigma^{**}$) split [@C-L]. Let $ \mathcal{A} $ be a Banach algebra with a bounded approximate identity and $ \sigma\in Hom(\mathcal{A}) $. Suppose that $ \mathcal{A} $ is $ \sigma $-amenable. Then we have the following statements: $(i)$ Regarding the sequence $ \Sigma^*$, there is a bounded linear map $ \theta : (\mathcal{A} \widehat{\otimes}\mathcal{A})^* \longrightarrow \mathcal{A}^*$ such that $ \theta \pi^* = \sigma^*$, and $$\theta( \sigma(a)\cdot f ) = a\cdot \theta(f) \ \ \ \text{and} \ \ \ \theta( f\cdot \sigma(a)) = \theta(f)\cdot a \ \ \ ( a \in \mathcal{A}, f \in (\mathcal{A} \widehat{\otimes}\mathcal{A})^*) \ .$$ $(ii)$ Regarding the sequence $ \Sigma^{**}$, there is a $ \sigma $-$ \mathcal{A} $-bimodule homomorphism $ \gamma : \mathcal{A}^{**} \longrightarrow (\mathcal{A} \widehat{\otimes}\mathcal{A})^{**}$ such that $ \pi^{**} \gamma = \sigma^{**}$. $(i)$ Let $M \in (\mathcal{A} \widehat{\otimes}\mathcal{A})^{**}$ be a $ \sigma $-virtual diagonal for $ \mathcal{A} $. We define the map $ \theta : (\mathcal{A} \widehat{\otimes}\mathcal{A})^* \longrightarrow \mathcal{A}^*$ via $$\langle a , \theta(f) \rangle := \langle f\cdot \sigma(a) , M \rangle \ \ \ (a \in \mathcal{A}, f \in (\mathcal{A} \widehat{\otimes}\mathcal{A})^*) \ .$$ Then, for $a \in \mathcal{A}$ and $\lambda \in \mathcal{A}^*$ $$\begin{aligned} \langle a , \theta \pi^* \lambda \rangle &= \langle (\pi^* \lambda)\cdot \sigma(a), M \rangle = \langle \pi^*( \lambda\cdot \sigma(a)) , M \rangle = \langle \lambda\cdot \sigma(a) , \pi^{**} M \rangle \\&= \langle \lambda , \sigma(a) \pi^{**} M \rangle = \langle \lambda , \sigma(a) \rangle = \langle a , \sigma^* \lambda \rangle\end{aligned}$$ so that $ \theta \pi^* = \sigma^*$. Next for $a , b \in \mathcal{A}$ and $f \in (\mathcal{A} \widehat{\otimes}\mathcal{A})^*$, we have $$\begin{aligned} \langle b , \theta(\sigma(a) \ . \ f) \rangle &= \langle (\sigma(a) \ . \ f) \ . \ \sigma(b) , M \rangle = \langle f \ . \ \sigma(b) , M \ . \ \sigma(a) \rangle \\&= \langle f \ . \ \sigma(b) , \sigma(a) \ . \ M \rangle = \langle f \ . \ \sigma(ba) , M \rangle \\&= \langle ba , \theta(f) \rangle = \langle b , a \ . \ \theta(f) \rangle\end{aligned}$$ whence $ \theta( \sigma(a) \ . \ f ) = a \ . \ \theta(f) $. The equality $ \theta( f \ . \ \sigma(a)) = \theta(f) \ . \ a$ is even easier. $(ii)$ Take $ \gamma := \theta^*$, where $\theta$ is given by the clause $(i)$. Then, it is immediate by Lemma 4.1. Finally, we generalize [@R Proposition 4.3.23] where the proof reads somehow the same lines. Let $ \mathcal{A} $ be Banach algebra, let $ \mathcal{B} $ be a closed subalgebra of $ \mathcal{A} $ and $ \sigma\in Hom(\mathcal{A}) $ for which $ \sigma(\mathcal{B}) \subseteq \mathcal{B} $ with the followings properties: 1. $ \mathcal{B} $ is $ \sigma $-amenable; 2. $ \mathcal{B} $ is a left ideal of $ \mathcal{A} $; 3. $ \mathcal{B} $ has a bounded approximate identity which is also a bounded left approximate identity for $ \mathcal{A} $. Then $ \mathcal{A} $ is $ \sigma $-biflat. Since $ \mathcal{B} $ is $ \sigma $-amenable, Proposition 4.2(ii) yields the existence of a $ \sigma $-$ \mathcal{B} $-bimodule homomorphism $ \rho:\mathcal{B}^{**}\longrightarrow(\mathcal{B}\hat{\otimes}\mathcal{B})^{**} $ such that $ \pi^{**}_{\mathcal{B}} \circ \rho=(\sigma\mid_{\mathcal{B}})^{**}$. Set $ \tilde{\rho}=(\iota\hat{\otimes}\iota)^{**} \circ \rho $, where $ \iota:\mathcal{B}\hookrightarrow\mathcal{A} $ is the canonical embedding. Let $ (e_{\alpha})_{\alpha} $ be a bounded approximate identity for $ \mathcal{B} $ which is a left bounded approximate identity for $ \mathcal{A} $. For each $a \in \mathcal{A}$, $ ( \tilde{\rho}(e_{\alpha})\cdot\sigma(a))_\alpha$ is a bounded net in $(\mathcal{A}\hat{\otimes}\mathcal{A})^{**} $, and so without loss of generality we may suppose that it is convergence. Therefore we obtain a bounded linear map $ \bar{\rho}:\mathcal{A}\longrightarrow(\mathcal{A}\hat{\otimes}\mathcal{A})^{**} $, defined by $ \bar{\rho}(a)=w^{*}-\lim_\alpha \tilde{\rho}(e_{\alpha})\cdot\sigma(a) $. It is immediate that $\bar{\rho}$ is a right $ \sigma $-$\mathcal{A}$-module homomorphism. To check that $\bar{\rho}$ is a left $ \sigma $-$\mathcal{A}$-module homomorphism, we first notice that $$\begin{aligned} \bar{\rho}(axb)&=w^{*}-\lim_\alpha\tilde{\rho}(e_{\alpha})\cdot\sigma(axb)\\ &=w^{*}-\lim_\alpha\sigma(axb)\cdot\tilde{\rho}(e_{\alpha}) \ \ \ \ (\text{since} \ axb \in\mathcal{B}) \\ &=w^{*}-\lim_\alpha \sigma(a) \sigma(xb)\cdot\tilde{\rho}(e_{\alpha}) \\ &=w^{*}-\lim_\alpha\sigma(a)\cdot\tilde{\rho}(e_{\alpha})\cdot\sigma(xb) \ \ \ \ (\text{since} \ xb \in\mathcal{B}) \\ &=\sigma(a)\cdot\bar{\rho}(x)\cdot\sigma(b) \ \ \ \ (a,x\in\mathcal{A}, \ b\in\mathcal{B} ). \end{aligned}$$ Let $a,x,b\in\mathcal{A}$. By Cohen’s factorization theorem, $ b=cd $ for some $ c\in\mathcal{B}$ and $ d\in\mathcal{A} $. Then, it follows from the above observation that $$\begin{aligned} \bar{\rho}(axb)&=\bar{\rho}(axcd)=\bar{\rho}(axc)\cdot\sigma(d)=\sigma(a)\cdot\bar{\rho}(x)\cdot\sigma(c)\cdot\sigma(d)\\ &=\sigma(a)\cdot\bar{\rho}(x)\cdot\sigma(cd)=\sigma(a)\cdot\bar{\rho}(x)\cdot\sigma(b). \end{aligned}$$ Now, let $ a,x\in\mathcal{A} $. Again, by Cohen’s factorization theorem, there are $ y,z \in\mathcal{A} $ such that $ b=yz $. Then $$\bar{\rho}(ax)=\bar{\rho}(ayz)=\sigma(a)\cdot\bar{\rho}(y)\cdot\sigma(z)=\sigma(a)\cdot\bar{\rho}(yz)=\sigma(a)\cdot\bar{\rho}(x).$$ So $ \bar{\rho} $ is $ \sigma $-$ \mathcal{A} $-bimodule homomorphism. It remains to prove that $ \pi^{**}_{\mathcal{A}}\circ\bar{\rho}=\kappa_{\mathcal{A}}\circ\sigma $. For each $ a\in\mathcal{A} $ we have $$\begin{aligned} \pi^{**}_{\mathcal{A}} (\bar{\rho}(a))&=w^{*}-\lim_\alpha \pi^{**}_{\mathcal{A}}(\tilde{\rho}(e_{\alpha})\cdot\sigma(a)) =w^{*}-\lim_\alpha \pi^{**}_{\mathcal{A}}(\tilde{\rho}(e_{\alpha})) \ . \ \sigma(a) \\ &=w^{*}-\lim_\alpha \pi^{**}_{\mathcal{B}}(\rho(e_{\alpha})) \ . \ \sigma(a) =\lim_\alpha \sigma^{**}(e_{\alpha}) \ . \ \sigma(a) = \lim_\alpha \kappa_{\mathcal{A}} ( \sigma(e_{\alpha} a)) \\&= \kappa_{\mathcal{A}} (\sigma(a)) . \end{aligned}$$ So by Theorem \[2.5\], $ \mathcal{A} $ is $ \sigma $-biflat. Application for Tensor products ================================ Let $ \mathcal{A} $ be a Banach algebra, let $ \sigma\in Hom(\mathcal{A}) $ and let $ C>0 $. We say that $ \mathcal{A} $ is $ C $-$ \sigma $-biflat, If there exists a map $ \rho:(\mathcal{A}\otimes\mathcal{A})^{\ast}\longrightarrow\mathcal{A}^{\ast} $, satisfying $ (1) $ such that $ \rho\circ\pi^{\ast}=\sigma^{\ast} $ and $ \lVert \rho\rVert< C $. Let $ \mathcal{A} $ and $ \mathcal{B} $ be Banach algebras and $ \sigma_{\mathcal{A}}\in Hom(\mathcal{A}) $ and $ \sigma_{\mathcal{B}}\in Hom(\mathcal{B}) $. Let $ E $ be a Banach $ \mathcal{A} $-bimodule, and let $ F $ be a Banach $ \mathcal{B} $-bimodule. We regard $ E\widehat{\otimes}F $ as a Banach $ \mathcal{A}\widehat{\otimes}\mathcal{B} $-bimodule with the actions $$\begin{array}{rl} (a\otimes b)\cdot(x\otimes y)=(a\cdot x)\otimes(b\cdot y)\\ (x\otimes y)\cdot(a\otimes b)=(x\cdot a)\otimes(y\cdot b), \end{array}$$ for every $ a\in\mathcal{A},b\in\mathcal{B},x\in E $ and $ y\in F $. Using Ramsden’s notation [@P], we construct $ \tilde{\mathcal{B}}(E,F) $ and $ \hat{\mathcal{B}}(F,E) $ as follows. Let $ \tilde{\mathcal{B}}(E,F) $ be the space $ \mathcal{B}(E,F) $ with the following module actions: $$\begin{array}{rl} ((a\otimes b)\cdot T)(x)=\sigma_{\mathcal{B}}(b)\cdot T(x\cdot\sigma_{\mathcal{A}}(a))\\ (T\cdot(a\otimes b))(x)=T(\sigma_{\mathcal{A}}(a)\cdot x)\cdot\sigma_{\mathcal{B}}(b), \end{array}$$ for every $ T\in\mathcal{B}(E,F)$, $a\in\mathcal{A}$, $b\in\mathcal{B}$, $x\in E$. Consider the map $ \tilde{T}: (E\widehat{\otimes}F)^{\ast}\rightarrow{\tilde{\mathcal{B}}}(E,F^{*}); \lambda\rightarrow\tilde{T}(\lambda) $ defined by $ \tilde{T}(\lambda)(x)(y)=\lambda(x\otimes y) $. By [@B 42. Proposition 13], $ \tilde{T} $ is an isometric isomorphism of Banach spaces. We claim that $ \tilde{T} $ satisfies (1). Indeed $$\begin{aligned} \tilde{T}(\sigma_{\mathcal{A}\otimes\mathcal{B}}(a\otimes b)\cdot\lambda)(x)(y)&=\tilde{T}((\sigma_{\mathcal{A}}(a)\otimes\sigma_{\mathcal{B}}(b)) \cdot\lambda)(x)(y)\\&=((\sigma_{\mathcal{A}}(a)\otimes\sigma_{\mathcal{B}}(b))\cdot\lambda)(x\otimes y)\\&=\lambda((x\otimes y) \cdot(\sigma_{\mathcal{A}}(a)\otimes\sigma_{\mathcal{B}}(b)))\\&=\lambda(x\cdot\sigma_{\mathcal{A}}(a)\otimes y\cdot\sigma_{\mathcal{B}}(b)),\end{aligned}$$ and on the other hand $$\begin{aligned} (a\otimes b)\cdot\tilde{T}(\lambda)(x)(y)&=\sigma_{\mathcal{B}}(b)\cdot\tilde{T}(\lambda)(x\cdot \sigma_{\mathcal{A}}(a))(y)\\&=\sigma_{\mathcal{B}}(b)\cdot\lambda(x\cdot\sigma_{\mathcal{A}}(a)\otimes y)\\&= \lambda(x\cdot\sigma_{\mathcal{A}}(a)\otimes y)\cdot\sigma_{B}(b)\\&=\lambda(x\cdot\sigma_{\mathcal{A}}(a)\otimes y\cdot\sigma_{\mathcal{B}}(b)).\end{aligned}$$ Let $ \widehat{\mathcal{B}}(F,E) $ be $ \mathcal{B}(F,E) $ with the following module actions $$\begin{array}{rl} ((a\otimes b)\cdot T)(y)=\sigma_{\mathcal{A}}(a)\cdot T(y\cdot\sigma_{\mathcal{B}}(b))\\ (T\cdot(a\otimes b))(y)=T(\sigma_{\mathcal{B}}(b)\cdot y)\cdot\sigma_{\mathcal{A}}(a), \end{array}$$ for all $ T\in\mathcal{B}(F,E)$, $a\in A$, $b\in B$, $y\in F$. Consider the map $ \widehat{T} :(E\widehat{\otimes}F)^{\ast}\rightarrow{\hat{\mathcal{B}}}(F,E^{\ast});\lambda\mapsto\widehat{T}(\lambda) $ defined by $ \widehat{T}(\lambda)(y)(x)=\lambda(x\otimes y) $. A similar argument as we use for $ \tilde{T} $, shows that $ \hat{T} $ is an isometric isomorphism of Banach spaces satisfying (1) Now, we are ready to prove the main goal of the current section. Let $ \mathcal{A} $ be $ C_{1} $-$ \sigma_{\mathcal{A}} $-biflat Banach algebra and let $ \mathcal{B} $ be $ C_{2} $-$ \sigma_{\mathcal{B}} $-biflat Banach algebra and if both $ \sigma_{\mathcal{A}} $ and $ \sigma_{\mathcal{B}} $ are idempotents. Then $ (\mathcal{A}\widehat{\otimes}\mathcal{B}) $ is $ C_{1} C_{2} $-$ \sigma_{\mathcal{A}}\otimes\sigma_{\mathcal{B}} $-biflat. There exists a bounded linear map $ \rho_{\mathcal{A}}:(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast}\rightarrow\mathcal{A}^{\ast} $ such that $ \rho_{\mathcal{A}}\circ\pi^{\ast}_{\mathcal{A}}=\sigma^{\ast}_{\mathcal{A}} $, satisfying (1) and $ \Vert\rho_{\mathcal{A}}\Vert\leq C_{1} $ and also a bounded linear map $ \rho_{\mathcal{B}}:(\mathcal{B}\widehat{\otimes}\mathcal{B})^{\ast}\rightarrow\mathcal{B}^{\ast} $ such that $ \rho_{\mathcal{B}}\circ\pi^{\ast}_{\mathcal{B}}=\sigma^{\ast}_{\mathcal{B}} $, satisfying (1) and $ \Vert\rho_{\mathcal{B}}\Vert\leq C_{2} $. Consider the composition $$\begin{aligned} \rho_{\mathcal{A}\widehat{\otimes}\mathcal{B}}: ((\mathcal{A}\widehat{\otimes}\mathcal{B})\widehat{\otimes}(\mathcal{A}\widehat{\otimes}\mathcal{B})) ^{\ast}&\cong\tilde{\mathcal{B}}(\mathcal{A}\widehat{\otimes}\mathcal{A},(\mathcal{B}\widehat{\otimes}\mathcal{B})^{\ast})\\ &\xrightarrow{T\mapsto\rho_\mathcal{B}\circ T}\tilde{\mathcal{B}}(\mathcal{A} \hat{ \otimes} \mathcal{A}, \mathcal{B}^\ast))\cong\widehat{\mathcal{B}} (\mathcal{B},(\mathcal{A}\widehat{\otimes}\mathcal{A})^{\ast})\\ &\xrightarrow{T\mapsto\rho_\mathcal{A}\circ T}\widehat{\mathcal{B}}(\mathcal{B},\mathcal{A}^{*})\cong(\mathcal{A}\widehat{\otimes}\mathcal{B})^{\ast}. \end{aligned}$$ All the maps satisfy the equation (1), and $ \Vert\rho_{\mathcal{A}\widehat{\otimes}\mathcal{B}}\Vert\leq C_{1}C_{2} $. For $ \lambda\in(\mathcal{A}\widehat{\otimes}\mathcal{B})^{\ast} $, we follow $ \pi^{\ast}_{\mathcal{A}\widehat{\otimes}\mathcal{B}}(\lambda) $ under the sequence of composition $ \rho_{\mathcal{A}\widehat{\otimes}\mathcal{B}} $. We have $$\rho_{\mathcal{B}}\circ\pi^{\ast}_{\mathcal{A}\widehat{\otimes}\mathcal{B}}(\lambda)\mapsto \rho_{\mathcal{B}}\circ\pi^{\ast}_{\mathcal{B}}\circ\tilde{T} _{\lambda}\circ\pi_{\mathcal{A}}\mapsto\sigma^{\ast}_{\mathcal{B}}\circ\tilde{T}_{\lambda}\circ\pi_{\mathcal{A}}\mapsto\pi^{\ast}_{\mathcal{A}}\circ\widehat{T} _{\lambda}\circ\sigma_{\mathcal{B}}.$$ So $$\begin{aligned} \rho_{\mathcal{A}}\circ\rho_{\mathcal{B}}\circ\pi^{\ast}_{\mathcal{A}\widehat{\otimes}\mathcal{B}}(\lambda)\mapsto\rho_{\mathcal{A}} \circ\pi^{\ast}_{\mathcal{A}}\circ\widehat{T}_{\lambda}\circ\sigma_{\mathcal{B}} \mapsto\sigma^{\ast}_{\mathcal{A}}\circ\widehat{T}_{\lambda}\circ\sigma_{\mathcal{B}} \mapsto\sigma^{\ast}_{\mathcal{A}\widehat{\otimes}\mathcal{B}}(\lambda),\end{aligned}$$ hence $\rho_{\mathcal{A}\widehat{\otimes}\mathcal{B}}\circ\pi^{\ast}_{\mathcal{A}\widehat{\otimes}\mathcal{B}}= \sigma^{\ast}_{\mathcal{A}\widehat{\otimes}\mathcal{B}} $. Note that the last equality comes from the following isomorphisms $$(\mathcal{A}\widehat{\otimes}\mathcal{B})^{\ast}\cong\widetilde{\mathcal{B}}(\mathcal{A},\mathcal{B}^{\ast}) \cong\widehat{\mathcal{B}}(\mathcal{B},\mathcal{A}^{\ast}).$$ [99]{} F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Varlag, Berlin Heidelberg New York, 1973. P. C. Curtis, R. J. Loy, The structure of amenable Banach algebras, *J. London Math. Soc.*, (2) **40** (1989), 89-104. H. G. Dales, Banach Algebras And Automatic continuity, London Mathematical Society Monographs 24, Clarendon Press, Oxford, 2000. Z. Ghorbani and M. L. Bami, $\varphi$-amenable and $\varphi$-biflat Banach algebras, [*Bull. Iranian Math. Soc.*]{} **39** (3) (2013), 507-515. A. Y. Helemskii, The Homology of Banach and Topological Algebras, Dordrecht, Netherlands, Kluwer, 1989. B. E. Johnson, Cohomology in Banach algebras, *Mem. Amer. Math. Soc.* **127** (1972). B. E. Johnson, Approximate diagonals and Cohomology of certain annihilator Banach algebras, [*Amer. J. Math.*]{} **94** (1972), 685-698. M. Mirzavaziri and M. S. Moslehian, $\sigma$-derivation in Banach algebras, [*Bull. Iranian Math. Soc.*]{} **32** (1) (2006), 65-78. M. Mirzavaziri and M. S. Moslehian, $\sigma$-amenability of Banach algebras, [*Southeast Asian Bull. Math.*]{} **33** (2009), 89-99. M. S. Moslehian and A. N. Motlagh, Some notes on ($\sigma,\tau$)-amenable of Banach algebras, [*Stud. Univ. Babeş-Bolyai Math.*]{} **53** (3) (2008), 57-68. P. Ramsden, Biflatness of semigroup algebras, *Semigroup Forum*, **79** (2009), 515-530. V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin, 2002. T. Yazdanpanah and H. Najafi, $\sigma$-contractible and $\sigma$-biprojective Banach algebras, [*Quaestiones Math.*]{} **33** (2010), 485-495.

--- author: - Andrea Pinna - Roberto Tonelli - Matteo Orrú - Michele Marchesi title: A Petri Nets Model for Blockchain Analysis --- @todonotes@disabled Introduction {#sec:Intro} ============ Related works {#sec:Related} ============= The Bitcoin Cash System: an overview. {#sec:Overview} ====================================== The model: the Blockchain Petri Net {#sec:PTNets} =================================== Results {#sec:Results} ======= Discussion {#sec:Discussion} ========== Conclusions {#sec:Conlcusions} =========== Chains of disposable addresses {#apx:chains} ==============================

--- abstract: 'It is discussed a physical vacuum excitation as a mechanism of sonoluminescence of gas bubble. This Schwinger’s theory was based on the assumption that the sudden change of the collapse rate of bubble in the water leads to the jump of the dielectric constant of gas. It is shown that the dependence of the dielectric constant on the gas density really leads to the jump of the dielectric constant at the shock-wave propagation in a collapsing gas bubble.' author: - 'Yu.P.Stepanovsky and G.G.Sergeeva' title: Sonoluminescence as a physical vacuum excitation --- 160 true mm 250 true mm -10.0 true mm 0 true mm 0 true mm Sonoluminescence is a transformation of sound into the light. The liquid (it can be the distillate water) is incurred to the acoustic waves, and photons are radiated at that. The radiated light energy is comparable with introducing sound energy. The sound energy density is of the order $10^{-11}$ eV/atom, and as the energy of radiation photons is of the order 10 eV, one can say that the density of the energy is increases in $10^{12}$ times. Sonoluminescence is well-known and studied for 60 years. The cavitation nature of the radiation is determined [@Lev]. The dependence of sonoluminescence intensity from chemical structure of liquid and dissolved gas in this liquid, their thermal conductivity coefficients, from temperature, from sound frequency and amplitude is studied. In spite of, that after stable luminescence observation of single gas bubble [@Gai; @Bar], the sonoluminescence attracted attention of many investigators \[4-12\], the conclusive explanation of physical mechanism of this effect is not find out now. The sonoluminescence in Schwinger’s works is considered as a manifestation of nonstationary Casimir effect [@Sch], which is a result of the change of vacuum properties under different external influences. At the studying of sonoluminescence the gas bubbles (cavities in the water with radius $r$, which are filled up by the gas) are putting to compression and expansion in the response to the positive and negative changes of pressure under the acoustic field influence. Schwinger’s theory is based on the assumption that the sudden change of the collapse rate of bubble in the water leads to the jump of its dielectric constant. This is accompanied by the excitation of electromagnetic vacuum and by the photons radiation. In [@Sch] the probability of the photons creation is calculated and the numeral accounts which are qualitatively explain the part of experiment datas, are doing. The task of this paper is to lead some reasons for the sustention of Schwinger’s idea and to show that the dielectric constant of gas is really jump function of the time at nonadiabatic stage of the bubble collapse. Well-known Schwinger’s expression for the total number of photons created in the volume V [@Sch]: $$\label{1} N=\int\limits_{}^{}\frac{(\sqrt[]{\varepsilon}-1)^2}{4~\sqrt[]{\varepsilon}} \frac{d^3k}{(2\pi)^3}V.$$ can be obtained as solution of an equation, which is described the scalar field of photons [@Step]. This fact is evidenced about the validity of Schwinger’s idea. For removing the integration divergency in the formula (1) we must to make cut-off wave-number $k_{max}$, which corresponds to the minimum wave length $\lambda_{min}<r_m$ ( $r_m$ is a bubble radius at the moment of photons radiation). Disregarding the dispersion, Schwinger obtained the following expression for the energy $E$ emitted in unite volume: $$\label{2} \frac{E}{V}=\frac{(\sqrt[]{\varepsilon}-1)^2}{4~\sqrt[]{\varepsilon}} \int\limits_{}^{k_{max}}\hbar\omega\frac{d^3k}{(2\pi)^3}= 2{\pi}^2\frac{(\sqrt[]{\varepsilon}-1)^2}{4~\sqrt[]{\varepsilon}} \frac{\hbar c}{\lambda^4_{min}}.$$ The expression (2) is in qualitative accordance with the result of Ref [@Milt], which was obtained at the divergency removal by taking into account the contribution of the surface energy. This is the serious substantiation of the validity for conducted by Shwinger removal of the divergency. The value $\varepsilon\approx1$ for the many gases in normal conditions at adiabatic compression. From formula (2) one can see that the number of photons and the emitting energy are little as $(\sqrt[]{\varepsilon}-1)^2$. But the value of gas temperature within the bubble ( $10^5$ K) which observed in the experiment, compelled to remember the Jarman’s idea [@Jar] about excitation of gathering shock wave at the nonadiabatic stage of bubble collapse [@Green; @Wu]. The numerical solution of Rayleigh-Plesset equation of a bubble surface, of van der Waals equations for the gas inside bubble and of tasks about generation and motion of the shock wave was carried out in a work [@Wu]. At a first stage of a bubble growth it was used an adiabatic approximation: by the time $\sim16,65 \mu s$, the bubble radius increased from the initial value $r_0=4,5 \mu m$ to the maximum value $\sim 37,09 \mu m$ and the gas density decreased to the value $0,0023~kg~m^{-3}$. At the second stage of a bubble collapse calculations were done in a nonadiabatic approximation, which leads to the Guderley’s decision for an ideal gas [@Gud]. As it was shown in \[10\], at nonadiabatic stage of collapse at the focusing shock wave in the centre of bubble the density of nonideal gas increases in $10^{5}$ times by the time $\sim 3,84 \mu s$, and achieving the maximum value $\rho \sim \rho_m \sim 794 kg~m^{-3}$. This conducts by the change of the rate of bubble collapse to $\sim 2\cdot 10^{4}m~ s^{-1}$. Further it will be shown that it leads to the jump of the dielectric constant of gas. The value $\varepsilon$ for gas is determined by Clausis-Mossotti formula  [@Fr] $$\label{3} \varepsilon-1=\frac{4\pi p\rho}{w(1-\frac{4\pi p\rho}{3w})},$$ where $p$ is the molar polarizability of gas, $\rho$ is the gas density and $w$ is his molecular weight. The liquid which surrounds the gas bubble (in the experiment this is the water with small addition of glycerin) leads to the additional polarization and to the molar polarizability of gas dependence on the dielectric constant of liquid $\varepsilon_1$: $$\label{4} p=\frac{p_0}{3(1+\frac{\varepsilon}{2\varepsilon_1})}.$$ where $p_0$ is the molar polarizability of gas in normal conditions. Even taking into account that $\rho_m\sim \rho$ at focusing shock wave, the ratio $\varepsilon/\varepsilon_1 << 1$, so $$\label{5} \varepsilon(t)-1\approx\frac{4\pi p_0\rho(t)}{3w}.$$ Such dependence of $\varepsilon(t)$ on $\rho(t)$ leads to the jump of the dielectric constant at the jump of the gas density as shock wave focuses at the centre of bubble. As we can see from (2) this jump accompanies by the vacuum excitation and to the photons radiation, the intensity of which depends from the characteristic time of the jump of the gas density. // Dokladi Akademii Nauk SSSR (russian)1937,v.•VI N8, 407. //J.Acoust.Soc.Am.Suppl. 1990,1,87,S141. // Nature (London)1991, 352, 318. //Phys. Rev. Lett. 1992. V. 69. P. 1182. //Phys. Rev. Lett. 1992. V. 69. P. 3839. //Phys. Today. 1994. V. 47, N9. P. 22. //Proc. Nat. Acad. Sci. USA. 1992 — V.89. — PP. 4091, 11118. — 1993 — V.90. — PP. 958, 2105, 4505, 7285. — 1994 — V.91. — P. 6473. // Phys.Fluids A 5(4),1065.1993. // Phys.Rev.Lett. 70.3423,1993 //Scientific American. 1995. February. P. 47. //Phys. Rev. Lett. 1996. V. 76. P. 3842. //in “Supersymmetry and Quantum Field Theory”, J.Wess, V.Akulov, Springer-Verlag Berlin Heidelberg, 1998, (Lecture notes in physics, Vol.509, p.246-251). //Phys. Rev. E55, p.4209, 1997. // J.Acoust.Soc.Am.69,p.1459, 1960. // Luftfahnrtforsch, 19, p.302, 1942. //Theory of Dielectrics. Oxford: Clarendon Press, 1958.

--- abstract: | The Gamma kernel is a projection kernel of the form $(A(x)B(y)-B(x)A(y))/(x-y)$, where $A$ and $B$ are certain functions on the one-dimensional lattice expressed through Euler’s ${\Gamma}$-function. The Gamma kernel depends on two continuous parameters; its principal minors serve as the correlation functions of a determinantal probability measure $P$ defined on the space of infinite point configurations on the lattice. As was shown earlier (Borodin and Olshanski, Advances in Math. 194 (2005), 141-202; arXiv:math-ph/0305043), $P$ describes the asymptotics of certain ensembles of random partitions in a limit regime. Theorem: The determinantal measure $P$ is quasi-invariant with respect to finitary permutations of the nodes of the lattice. This result is motivated by an application to a model of infinite particle stochastic dynamics. address: 'Institute for Information Transmission Problems, Bolshoy Karetny 19, Moscow 127994, Russia; Independent University of Moscow, Russia' author: - Grigori Olshanski date: 'November 19, 2009' title: 'The quasi-invariance property for the Gamma kernel determinantal measure' --- [^1] Introduction {#introduction .unnumbered} ============ Preliminaries: a general problem {#0-1} -------------------------------- Recall a few well-known notions from measure theory. Let $\mathfrak A$ be a Borel space (that is, a set with a distinguished sigma-algebra of subsets). Two Borel measures $P_1, P_2$ on $\mathfrak A$ are said to be [*equivalent*]{} if $P_1$ has a density with respect to $P_2$ and vice versa. They are said to [*disjoint*]{} or [*mutually singular*]{} if there exist disjoint Borel subsets $B_1$ and $B_2$ such that $P_1$ is supported by $B_1$ and $P_2$ is supported by $B_2$ (that is, $P_1(\mathfrak A\setminus B_1)=P_2(\mathfrak A\setminus B_2)=0$). Assume $G$ is a group acting on $\mathfrak A$ by Borel transformations; then a Borel measure $P$ is said to be $G$-[*quasi-invariant*]{} if $P$ is equivalent to its transform by any element $g\in G$. In practice, especially for measures living on “large” spaces, verifying the property of equivalence, disjointness or quasi-invariance, and explicit computation of densities (Radon–Nikodým derivatives) for equivalent measures can be a nontrivial task. There exist nice general results for particular classes of measures: infinite product measures (Kakutani’s theorem [@Ka]), Gaussian measures on infinite-dimensional spaces (Feldman–Hajek’s theorem and related results, see [@Kuo Ch. II]), Poisson measures (see [@Bro]). Assume that ${\mathfrak X}$ is a locally compact space, take as the “large” space $\mathfrak A$ the space ${\operatorname{Conf}}({\mathfrak X})$ of locally finite point configurations on ${\mathfrak X}$, and assume that the measures under consideration are probability measures on $\mathfrak A={\operatorname{Conf}}({\mathfrak X})$; they are also called [*point processes*]{} on ${\mathfrak X}$ (for fundamentals of point processes, see, e.g., [@Le]). Poisson measures are just the simplest yet important example of point processes. The next by complexity example is the class of [*determinantal*]{} measures (processes). Determinantal measures are specified by their correlation kernels which are functions $K(x,y)$ on ${\mathfrak X}\times{\mathfrak X}$. Note an analogy with covariation kernels of Gaussian measures which are also functions in two variables. Note also that, informally, Poisson measures can be viewed as a degenerate case of determinantal measures corresponding to kernels $K(x,y)$ concentrated on the diagonal $x=y$. Many concrete examples of determinantal measures are furnished by random matrix theory and other sources, see, e.g., the surveys [@So] and [@Bor]. The interest to determinantal measures especially increased in the last years. However, to the best of my knowledge, the following problem was never discussed in the literature: \[P1\] [*Assume we are given two determinantal measures, $P_1$ and $P_2$ on a common space ${\operatorname{Conf}}({\mathfrak X})$. How to test their equivalence (or, on the contrary, disjointness)? Is it possible to decide this by inspection of the respective correlation kernels $K_1(x,y)$ and $K_2(x,y)$?*]{} One could imagine that equivalence $P_1\sim P_2$ holds if the kernels are close to each other in an appropriate sense. However, there is a subtlety here, see Subsection \[1-6\] below. Let $G$ be a group of homeomorphisms $g:{\mathfrak X}\to{\mathfrak X}$. Then $G$ also acts, in a natural way, on the space ${\operatorname{Conf}}({\mathfrak X})$ and hence on the space of probability measures on ${\operatorname{Conf}}({\mathfrak X})$. Observe that the latter action preserves the determinantal property: If $P$ is a determinantal measure on ${\operatorname{Conf}}({\mathfrak X})$ with correlation kernel $K(x,y)$, then the transformed measure $g(P)$ is determinantal, too, and $K(g^{-1}x, g^{-1}y)$ serves as its correlation kernel. Thus, the question of $G$-quasi-invariance of $P$ becomes a special instance of Problem \[P1\]: \[P2\] [*Let $P$ and $G$ be as above. How to test whether $P$ is $G$-quasi-invariant? Is it possible to decide this by comparing the correlation kernels $K(x,y)$ and $K(g^{-1}x, g^{-1}y)$ for $g\in G$?*]{} I think it would be interesting to develop general methods for solving Problem \[P2\] and more general Problem \[P1\]. They seem to be nontrivial even in the case when ${\mathfrak X}$ is a countably infinite set with discrete topology. The Gamma kernel measure ------------------------ In the present paper we are dealing with a concrete model of determinantal measures, introduced in [@BO2]. The space ${\mathfrak X}$ is assumed to be discrete and countable; it is convenient to identify it with the lattice ${\mathbb Z}':={\mathbb Z}+\frac12$ of half–integers. Then the space ${\operatorname{Conf}}({\mathfrak X})={\operatorname{Conf}}({\mathbb Z}')$ is simply the space of all subsets of ${\mathbb Z}'$. We consider a two-parameter family of kernels on ${\mathbb Z}'\times{\mathbb Z}'$. Following [@BO2], we denote them as ${\underline K\,}_{z,z'}(x,y)$; here $z$ and $z'$ are some continuous parameters, and $x,y$ are the arguments, which range over ${\mathbb Z}'$. Each kernel is real-valued and symmetric. Moreover, it is a projection kernel meaning that it corresponds to a projection operator in the Hilbert space $\ell^2({\mathbb Z}')$. Like many examples of kernels from random matrix theory, our kernels can be written in the so-called integrable form [@IIKS], [@De] $$\frac{\mathcal A(x)\mathcal B(y)-\mathcal B(x)\mathcal A(y)}{x-y},$$ resembling Christoffel–Darboux kernels associated to orthogonal polynomials. In our situation $\mathcal A$ and $\mathcal B$ are certain functions on the lattice ${\mathbb Z}'$, which are expressed through Euler’s ${\Gamma}$-function. For this reason we call ${\underline K\,}_{z,z'}(x,y)$ the [*Gamma kernel*]{}. In [@BO2] we conjectured that the Gamma kernel might be a universal microscopic limit of the Christoffel–Darboux kernels for generic discrete orthogonal polynomials, in an appropriate asymptotic regime. The Gamma kernel serves as the correlation kernel for a determinantal measure on ${\operatorname{Conf}}({\mathbb Z}')$, called the [*Gamma kernel measure*]{} and denoted as ${\underline P\,}_{z,z'}$. According to the general definition of determinantal measures (see [@So], [@Bor]), the measure ${\underline P\,}_{z,z'}$ is characterized by its correlation functions $$\rho_n(x_1,\dots,x_n):={\underline P\,}_{z,z'}\{X\in{\operatorname{Conf}}({\mathbb Z}')\mid X\ni x_1,\dots,x_n\}$$ which in turn are equal to principal $n\times n$ minors of the kernel: $$\rho(x_1,\dots,x_n):=\det[{\underline K\,}_{z,z'}(x_i,x_j)]_{i,j=1}^n.$$ Here $n=1,2,\dots$ and $x_1,\dots,x_n$ is an arbitrary $n$-tuple $x_1,\dots,x_n$ of pairwise distinct points from ${\mathbb Z}'$. As shown in [@BO2], the Gamma kernel measure arises from several models of representation–theoretic origin, through certain limit transitions. A more detailed information about ${\underline K\,}_{z,z'}(x,y)$ and ${\underline P\,}_{z,z'}$ is given in Section \[1\] below, see also [@BO2], [@Ol2]. The main result --------------- We take as $G$ the group $\mathfrak S$ of permutations of the set ${\mathbb Z}'$ fixing all but finitely many points. Such permutations are said to be [*finitary*]{}. Clearly, $\mathfrak S$ is a countable group. It is generated by the elementary transpositions ${\sigma}_n$ of the lattice ${\mathbb Z}'$: Here $n\in{\mathbb Z}$ and ${\sigma}_n$ transposes the points $n-\frac12$ and $n+\frac12$ of ${\mathbb Z}'$. Each permutation ${\sigma}\in\mathfrak S$ induces, in a natural fashion, a transformation of the space ${\operatorname{Conf}}({\mathbb Z}')$, which in turn results in a transformation $P\mapsto {\sigma}(P)$ of probability measures on ${\operatorname{Conf}}({\mathbb Z}')$. The main result of the paper says that the Gamma kernel measure is quasi-invariant with respect to the action of the group $\mathfrak S$: [**Main Theorem.**]{} [*For any $\sigma\in\mathfrak S$, the measures ${\underline P\,}_{z,z'}$ and ${\sigma}({\underline P\,}_{z,z'})$ are equivalent. Moreover, the Radon–Nikodým derivative ${\sigma}({\underline P\,}_{z,z'})/{\underline P\,}_{z,z'}$ can be explicitly computed*]{}. This result gives a solution to the first question of Problem \[P2\] in a concrete situation. As will be shown in another paper, the quasi-invariance property established in the theorem makes it possible to construct an equilibrium Markov process on ${\operatorname{Conf}}({\mathbb Z}')$ with determinantal dynamical correlation functions and equilibrium distribution ${\underline P\,}_{z,z'}$. This application is one of the motivations of the present work. [^2] It seems plausible that ${\underline P\,}_{z,z'}$ is [*not*]{} quasi-invariant with respect to the transformations of ${\operatorname{Conf}}({\mathbb Z}')$ generated by the translations of the lattice. Note that the translation $x\mapsto x+k$ with $k\in{\mathbb Z}$ amounts to the shift $(z,z')\to(z-k,z'-k)$ of the parameters (see Theorem \[1.4\]). One can ask, more generally, whether any two Gamma kernel measures with distinct parameters are disjoint. [^3] Scheme of proof of Main Theorem {#0-4} ------------------------------- The proof relies on the fact that for fixed $z,z'$, the measure ${\underline P\,}_{z,z'}$ can be approximated by simpler measures which are $\mathfrak S$–quasiinvariant and whose Radon–Nikodým derivatives (with respect to the action of the group $\mathfrak S$) are readily computable. The approximating measures depend on an additional parameter $\xi\in(0,1)$ and are denoted as ${\underline P\,}_{z,z',\xi}$. These are purely atomic probability measures supported by a single $\mathfrak S$–orbit. They come from certain probability distributions on Young diagrams, and are called the [*z–measures*]{} (Kerov–Olshanski–Vershik [@KOV], Borodin–Olshanski [@BO1]). As $\xi$ goes to $1$, the measures ${\underline P\,}_{z,z',\xi}$ weakly converge to ${\underline P\,}_{z,z'}$: this is simply the initial definition of ${\underline P\,}_{z,z'}$ given in [@BO2]. What we actually need to prove is that the convergence of the measures holds not only in the weak topology (that is, on bounded continuous test functions) but also in a much stronger sense: Namely, $$\langle F, {\underline P\,}_{z,z',\xi}\rangle\, \underset{\xi\to1}{\longrightarrow}\, \langle F, {\underline P\,}_{z,z'}\rangle$$ for certain test functions $F$ which, like the Radon–Nikodým derivatives, may be unbounded and not everywhere defined. Here and in the sequel the angular brackets denote the pairing between functions and measures. To explain this point more precisely we need some preparation. First of all, it is convenient to transform all the measures in question by means of an involutive homeomorphism of the compact space ${\operatorname{Conf}}(\mathfrak {\mathbb Z}')$. This homeomorphism, denoted as “${\operatorname{inv}}$”, assigns to a configuration $X\in{\operatorname{Conf}}({\mathbb Z}')$ its symmetric difference with the set ${\mathbb Z}'_-=\{\dots,-\frac32,-\frac12\}$. An equivalent description is the following. Regard $X$ as a configuration of charged particles occupying some of the sites of the lattice ${\mathbb Z}'$, while the holes (that is, the unoccupied sites of ${\mathbb Z}'$) are interpreted as anti–particles with opposite charge. Now, the new configuration ${\operatorname{inv}}(X)$ is formed by the particles sitting to the right of 0 and the anti–particles to the left of 0. We call “${\operatorname{inv}}$” the [*particle/hole involution*]{} on ${\mathbb Z}'_-$. For instance, if $X={\mathbb Z}'_-$ then ${\operatorname{inv}}(X)=\varnothing$, the empty configuration. The configuration $X={\mathbb Z}'_-$ plays a distinguished role because the $\mathfrak S$–orbit of this configuration is the support of the pre–limit measures ${\underline P\,}_{z,z',\xi}$. The map “${\operatorname{inv}}$” transforms this distinguished orbit into the set of all finite [*balanced*]{} configurations, that is, finite configurations with equally many points to the right and to the left of $0$. Note that the transform by “${\operatorname{inv}}$” leaves intact the action of all the elementary transpositions ${\sigma}_n$ with $n\ne0$, only the action of ${\sigma}_0$ is perturbed. Note also that if ${\underline P\,}$ is a determinantal measure on ${\operatorname{Conf}}({\mathbb Z}')$ then so is its push–forward $P:={\operatorname{inv}}({\underline P\,})$, and there is a simple relation between the correlation kernels of ${\underline P\,}$ and $P$ ([@BOO Appendix]). If the kernel of ${\underline P\,}$ is symmetric then that of $P$ has a different kind of symmetry: it is symmetric with respect to an indefinite inner product (see [@BO1 Proposition 2.3 and Remark 2.4]), which reflects the presence of two kinds of particles. Instead of the measures ${\underline P\,}_{z,z',\xi}$ and ${\underline P\,}_{z,z'}$ we will deal with their transforms by “${\operatorname{inv}}$”, denoted as $P_{z,z',\xi}:={\operatorname{inv}}({\underline P\,}_{z,z',\xi})$ and $P_{z,z'}:={\operatorname{inv}}({\underline P\,}_{z,z'})$. Clearly, the transform does not affect the formulation of the theorem, only the initial action of the group $\mathfrak S$ on ${\operatorname{Conf}}({\mathbb Z}')$ has to be conjugated by the involution: an element ${\sigma}\in\mathfrak S$ now acts as the transformation $$\label{f0.1} {\widetilde{\sigma}}:={\operatorname{inv}}\circ\,{\sigma}\circ{\operatorname{inv}}.$$ An advantage of the transformed measures as compared to the initial ones is that the pre–limit measures $P_{z,z',\xi}$ live on finite configurations. In a weaker form, this property is inherited by the limit measures. Namely, let us say that a configuration $X\in{\operatorname{Conf}}({\mathbb Z}')$ is [*sparse*]{} if $$\sum_{x\in X}|x|^{-1}<\infty.$$ Denote the set of all sparse configurations as ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$. There is a natural embedding ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')\hookrightarrow\ell^1({\mathbb Z}')$ assigning to a sparse configuration $X$ its characteristic function multiplied by the function $|x|^{-1}$, and we equip ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ with the “$\ell^1$–topology”, that is, the one induced by the norm of the Banach space $\ell^1({\mathbb Z}')$. The $\ell^1$–topology is finer than the topology induced from the ambient space ${\operatorname{Conf}}({\mathbb Z}')$. Now we are in a position to describe the scheme of proof. \[C1\] [*The limit measures $P_{z,z'}$ are concentrated on the set of sparse configurations.*]{} The claim makes sense because the set ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ is a Borel subset in ${\operatorname{Conf}}({\mathbb Z}')$. Given a function $f$ on the lattice ${\mathbb Z}'$ such that $|f(x)|=O(|x|^{-1})$, we define a function $\Phi_f(X)$ on the set of sparse configurations by the formula $$\label{f0.mult} \Phi_f(X)=\prod_{x\in X}(1+f(x))$$ (the product is convergent). Such functions $\Phi_f$ will be called [*multiplicative functionals*]{} on configurations. Any multiplicative functional $\Phi_f$ is continuous in the $\ell^1$–topology. Given a permutation ${\sigma}\in\mathfrak S$ and a measure $P$ on ${\operatorname{Conf}}({\mathbb Z}')$, we denote by ${\widetilde{\sigma}}(P)$ the push–forward of $P$ under the transformation ${\widetilde{\sigma}}$, see . Let $z,z'$ be fixed and $\xi$ range over $(0,1)$. For any ${\sigma}\in\mathfrak S$, let $\mu_{z,z',\xi}({\sigma},X)$ be the Radon–Nikodým derivative of the measure ${\widetilde{\sigma}}(P_{z,z',\xi})$ with respect to the measure $P_{z,z',\xi}$. That is, $$\mu_{z,z',\xi}({\sigma},X)=\frac{{\widetilde{\sigma}}(P_{z,z',\xi})(X)}{P_{z,z',\xi}(X)} =\frac{P_{z,z',\xi}({\widetilde{\sigma}}^{-1}(X))}{P_{z,z',\xi}(X)};$$ here $X$ belongs to the countable set of finite balanced configurations. \[C2\] *Fix an arbitrary ${\sigma}\in\mathfrak S$.* [(i)]{} The function $\mu_{z,z',\xi}({\sigma},X)$ has a unique extension to a continuous function on ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$. [(ii)]{} As $\xi\to1$, the extended functions obtained in this way converge pointwise to a continuous function $\mu_{z,z'}({\sigma},X)$ on ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$. [(iii)]{} The limit function $\mu_{z,z'}({\sigma},X)$ can be written as a finite linear combination of multiplicative functionals of the form . Here continuity is assumed with respect to the $\ell^1$–topology. Actually, a somewhat stronger claim holds, see Proposition \[3.1\] and the subsequent discussion. Claim \[C2\] suggests that the limit function $\mu_{z,z'}({\sigma},X)$ might serve as the Radon–Nikodým derivative for the limit measure, that is, $${\widetilde{\sigma}}(P_{z,z'})=\mu_{z,z'}({\sigma},\,\cdot\,)P_{z,z'}.$$ This relation is indeed true. We reduce it to the following claim. \[C3\] [*Let $f$ be an arbitrary function on ${\mathbb Z}'$ such that $|f(x)|=O(|x|^{-1})$. Then the multiplicative functional $\Phi_f$ given by is absolutely integrable with respect to both the pre–limit and limit measures, and we have*]{} $$\lim_{\xi\to1}\langle\Phi_f,P_{z,z',\xi}\rangle=\langle\Phi_f,P_{z,z'}\rangle.$$ This claim is stronger than the assertion about the weak convergence of measures $P_{z,z',\xi}\to P_{z,z'}$ which was known previously. Indeed, weak convergence of measures on ${\operatorname{Conf}}({\mathbb Z}')$ means convergence on continuous test functions, while multiplicative functionals are, generally speaking, unbounded functions on ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ and thus cannot be extended to continuous functions on the compact space ${\operatorname{Conf}}({\mathbb Z}')$. To prove Claim \[C3\] we use the well–known fact that the expectation of a multiplicative functional with respect to a determinantal measure can be expressed as a Fredholm determinant involving the correlation kernel. This makes it possible to reformulate the claim in terms of the correlation operators $K_{z,z', \xi}$ and $K_{z,z'}$ (these are operators in the Hilbert space $\ell^2({\mathbb Z}')$ whose matrices are the correlation kernels of the measures $P_{z,z',\xi}$ and $P_{z,z'}$, respectively). The reformulation is given in Claim \[C4\] below. Represent the Hilbert space $H:=\ell^2({\mathbb Z}')$ as the direct sum of two subspaces $H_{\pm}=\ell^2({\mathbb Z}'_\pm)$ according to the splitting ${\mathbb Z}'={\mathbb Z}'_+\sqcup{\mathbb Z}'_-$ (positive and negative half–integers). Then any bounded operator in $H$ can be written as a $2\times2$ matrix with operator entries (or “blocks”). Let ${\mathcal L_{1|2}}(H)$ denote the set (actually, algebra) of bounded operators in $H$ whose two diagonal blocks are trace class operators and two off–diagonal blocks are Hilbert–Schmidt operators. If $\mathcal K\in{\mathcal L_{1|2}}(H)$ then the Fredholm determinant $\det(1+\mathcal K)$ makes sense ([@BOO Appendix]). We equip ${\mathcal L_{1|2}}(H)$ with the [*combined topology*]{} determined by the trace class norm on the diagonal blocks and the Hilbert–Schmidt norm on the off–diagonal ones. \[C4\] [*Let $A$ stand for the operator of pointwise multiplication by the function $x\mapsto|x|^{-1/2}$ in the space $H$. The operator $AK_{z,z'}A$ lies in ${\mathcal L_{1|2}}(H)$, and, as $\xi$ goes to $1$, the operators $AK_{z,z',\xi}A$ approach the operator $AK_{z,z'}A$ in the combined topology of ${\mathcal L_{1|2}}(H)$.*]{} Note that the operator $A^2$ [*is not*]{} in ${\mathcal L_{1|2}}(H)$, because the series $\sum |x|^{-1}$ taken over all $x\in{\mathbb Z}'$ is divergent. This is the source of difficulties. For instance, the assertion that $AK_{z,z'}A$ belongs to ${\mathcal L_{1|2}}(H)$ is not a formal consequence of the boundedness of $K_{z,z'}$. Claim \[C4\] is the key technical result of the paper. The proof relies on explicit expressions for the correlation kernels in terms of contour integrals and requires a considerable computational work. I do not know whether the operators $AK_{z,z',\xi}A$ approach $AK_{z,z'}A$ simply in the trace class norm. The point is that the diagonal blocks are Hermitian nonnegative operators while the operators themselves are not. For nonnegative operators, one can use the fact that the convergence in the trace class norm is equivalent to the weak convergence together with the convergence of traces. For non–Hermitian operators, dealing with the trace class norm is difficult, while the Hilbert–Schmidt norm turns out to be much easier to handle. Fortunately, for the off–diagonal blocks, the convergence in the Hilbert–Schmidt norm already suffices. Organization of the paper ------------------------- Section \[1\] contains the basic notation and definitions related to the measures under consideration and their correlation kernels. Section \[2\] starts with basic facts related to multiplicative functionals and their connection to Fredholm determinants; then the proof of Claim \[C1\] follows; it is readily derived from the explicit expression for the first correlation function of the measure ${\underline P\,}_{z,z'}$. Section \[3\] is devoted to the proof of Claim \[C2\]. In Section \[4\], we formulate the main result (Theorem \[4.1\]). Then we reduce to it to Theorem \[4.2\] and next to Theorem \[4.3\]; they correspond to Claims \[C3\] and \[C4\], respectively. The main technical work is done in Sections \[5\] and \[6\], where we prove Theorem \[4.3\] (or Claim \[C4\]) separately for diagonal and off–diagonal blocks. Acknowledgment -------------- I am very much indebted to Alexei Borodin for a number of important suggestions which helped me in dealing with asymptotics of contour integral representations. I am also grateful to Leonid Petrov and Sergey Pirogov for valuable remarks. Z–measures and related objects {#1} ============================== Partitions and lattice point configurations {#1-1} ------------------------------------------- A [*partition*]{} is an infinite sequence ${\lambda}=({\lambda}_1,{\lambda}_2,\dots)$ of nonnegative integers ${\lambda}_i$ such that ${\lambda}_i\ge{\lambda}_{i+1}$ and only finitely many ${\lambda}_i$’s are nonzero. We set $|{\lambda}|=\sum{\lambda}_i$. Let ${\mathbb Y}$ denote the set of all partitions; it is a countable set. Following [@Ma], we identify partitions and Young diagrams. Let ${\mathbb Z}'$ denote the set of all half–integers; that is, ${\mathbb Z}'={\mathbb Z}+\frac12$. By ${\mathbb Z}'_+$ and ${\mathbb Z}'_-$ we denote the subsets of positive and negative half–integers, so that ${\mathbb Z}'$ is the disjoint union of ${\mathbb Z}'_+$ and ${\mathbb Z}'_-$. Subsets of ${\mathbb Z}'$ are viewed as [*configurations of particles*]{} occupying the nodes of the lattice ${\mathbb Z}'$. The unoccupied nodes are called [*holes*]{}. Let ${\operatorname{Conf}}({\mathbb Z}')$ denote the space of all particle configurations on ${\mathbb Z}'$. The space ${\operatorname{Conf}}({\mathbb Z}')$ can be identified with the infinite product space $\{0,1\}^{{\mathbb Z}'}$ and we equip it with the product topology. In this topology, ${\operatorname{Conf}}({\mathbb Z}')$ is a totally disconnected compact space. Recall (see Subsection \[0-4\]) that the [*particle/hole involution*]{} on ${\mathbb Z}'_-$ is the involutive map ${\operatorname{Conf}}({\mathbb Z}')\to{\operatorname{Conf}}({\mathbb Z}')$ keeping intact particles and holes on ${\mathbb Z}'_+\subset{\mathbb Z}'$ and changing particles by holes and vice versa on ${\mathbb Z}'_-\subset{\mathbb Z}'$. We denote the particle/hole involution by the symbol “${\operatorname{inv}}$”. In a more formal description, “${\operatorname{inv}}$” assigns to a configuration its symmetric difference with ${\mathbb Z}'_-$. In particular, ${\operatorname{inv}}({\mathbb Z}'_-)=\varnothing$. To a partition ${\lambda}\in{\mathbb Y}$ we assign the semi–infinite point configuration $${\underline X}({\lambda})=\{{\lambda}_i-i+\tfrac12\}_{i=1,2,\dots}\in{\operatorname{Conf}}({\mathbb Z}').$$ Note that among ${\lambda}_i$’s some terms may repeat while the numbers ${\lambda}_i-i+\frac12$ are all pairwise distinct. Clearly, the correspondence ${\lambda}\mapsto{\underline X}({\lambda})$ is one–to–one. The configuration ${\underline X}({\lambda})$ is sometimes called the [*Maya diagram*]{} of ${\lambda}$, see Miwa–Jimbo–Date [@MJD]. For instance, the Maya diagram of the zero partition ${\lambda}=(0,0,\dots)$ is ${\mathbb Z}'_-$. Any Maya diagram can be obtained from this one by finitely many elementary moves consisting in shifting one particle to the neighboring position on the right provided that it is unoccupied. A [*finite*]{} configuration $X\subset{\mathbb Z}'$ is called [*balanced*]{} if $|X\cap{\mathbb Z}'_+|=|X\cap{\mathbb Z}'_-|$. An important fact is that “${\operatorname{inv}}$” establishes a bijective correspondence between the Maya diagrams ${\underline X}({\lambda})$ and the balanced configurations. We set $$X({\lambda})={\operatorname{inv}}({\underline X}({\lambda})), \qquad X_{\pm}({\lambda})=X({\lambda})\cap{\mathbb Z}'_\pm.$$ An alternative interpretation of the balanced configuration $X({\lambda})$ is as follows: $X({\lambda})=X_+({\lambda})\cup X_-({\lambda})$ with $$X_+({\lambda})=\{p_1<\dots<p_d\}, \quad X_-({\lambda})=\{-q_d<\dots<-q_1\}$$ (recall that $|X_+({\lambda})|=|X_-({\lambda})|$), where the positive half–integers $p_i$ and $q_i$ are the [*modified Frobenius coordinates*]{} (Vershik–Kerov [@VK]) of the Young diagram ${\lambda}$. They differ from the conventional Frobenius coordinates [@Ma] by the additional summand $\frac12$. A slight divergence with the conventional notation is that we arrange the coordinates in the ascending order. A direct explanation: $d$ is the number of diagonal boxes in ${\lambda}$ and $$\aligned ({\lambda}_1-1+\tfrac12,{\lambda}_2-2+\tfrac12,\dots,{\lambda}_d-d+\tfrac12) &=(p_d,p_{d-1}\dots,p_1)\\ ({\lambda}'_1-1+\tfrac12,{\lambda}'_2-2+\tfrac12,\dots,{\lambda}'_d-d+\tfrac12)&=(q_d,q_{d-1},\dots,q_1), \endaligned$$ where ${\lambda}'$ is the transposed diagram. Thus, we have defined two embeddings of the countable set ${\mathbb Y}$ into the space ${\operatorname{Conf}}({\mathbb Z}')$, namely, ${\lambda}\mapsto{\underline X}({\lambda})$ and ${\lambda}\mapsto X({\lambda})$. These two embeddings are related to each other by the particle/hole involution on ${\mathbb Z}'_-$. Note that each of the two embeddings maps ${\mathbb Y}$ onto a dense subset in ${\operatorname{Conf}}({\mathbb Z}')$. Z-measures on partitions {#1-2} ------------------------ Here we introduce a family $\{M_{z,z',\xi}\}$ of probability measures on ${\mathbb Y}$, called the [*z–measures*]{}. The subscripts $z$, $z'$, and $\xi$ are continuous parameters. Their range is as follows: parameter $\xi$ belongs to the open unit interval $(0,1)$, and parameters $z$ and $z'$ should be such that $(z+k)(z'+k)>0$ for any integer $k$. Detailed examination of this condition shows that either $z\in{\mathbb C}\setminus{\mathbb R}$ and $z'=\bar z$ (the [*principal series*]{} of values), or both $z$ and $z'$ are real numbers contained in an open interval $(N,N+1)$ with $N\in{\mathbb Z}$ (the [*complementary series*]{} of values). We shall need the [*generalized Pochhammer symbol*]{} $(x)_{\lambda}$: $$(x)_{\lambda}=\prod_{i=1}^{\ell({\lambda})}(x-i+1)_{{\lambda}_i}\,, \qquad x\in{\mathbb C}, \quad {\lambda}\in{\mathbb Y},$$ where $\ell({\lambda})$ is the number of nonzero coordinates ${\lambda}_i$ and $$(x)_k=x(x+1)\dots(x+k-1)=\frac{{\Gamma}(x+k)}{{\Gamma}(x)}$$ is the conventional Pochhammer symbol. Note that $$(x)_{\lambda}=\prod_{(i,j)\in{\lambda}}(x+j-i),$$ where the product is taken over the boxes $(i,j)$ of the Young diagram ${\lambda}$, and $i$ and $j$ stand for the row and column numbers of a box. In this notation, the weight of ${\lambda}\in{\mathbb Y}$ assigned by the z–measure $M_{z,z',\xi}$ is written as $$\label{f1.1} M_{z,z',\xi}({\lambda})=(1-\xi)^{zz'}\,\xi^{|{\lambda}|}\,(z)_{\lambda}(z')_{\lambda}\, \left(\frac{\operatorname{dim}{\lambda}}{|{\lambda}|!}\right)^2\,,$$ where $\operatorname{dim}{\lambda}$ is the dimension of the irreducible representation of the symmetric group of degree $|{\lambda}|$ indexed by ${\lambda}$. Note that $z$ and $z'$ enter the formula symmetrically, so that their interchange does not affect the z-measure. For the origin of formula and the proof that $M_{z,z',\xi}$ is indeed a probability measure, see Borodin–Olshanski [@BO1], [@BO2], [@BO3] and references therein. Note that all the weights are strictly positive: this follows from the conditions imposed on parameters $z$ and $z'$. The z–measures form a deformation of the poissonized Plancherel measure and are a special case of Schur measures (see Okounkov [@Ok]). Limit measures {#1-3} -------------- Throughout the paper the parameters $z$ and $z'$ are assumed to be fixed. If the third parameter $\xi$ approaches 0, then the z–measures $M_{z,z',\xi}$ converge to the Dirac measure at the zero partition: this is caused by the factor $\xi^{|{\lambda}|}$. A much more interesting picture arises as $\xi$ approaches 1. Then the factor $(1-\xi)^{zz'}$ forces each of the weights $M_{z,z',\xi}({\lambda})$ to tend to 0 (note that $zz'>0$). This means that the z–measures on the discrete space ${\mathbb Y}$ escape to infinity. However, the situation changes when we embed ${\mathbb Y}$ into ${\operatorname{Conf}}({\mathbb Z}')$. Recall that we have two embeddings, one producing semi–infinite configurations ${\underline X}({\lambda})$ and the other producing finite balanced configurations $X({\lambda})$. Denote by ${\underline P\,}_{z,z',\xi}$ and $P_{z,z',\xi}$ the push–forwards of the z–measure $M_{z,z',\xi}$ under these two embeddings. Then the following result holds, see [@BO2]: \[1.1\] In the space of probability measures on the compact space ${\operatorname{Conf}}({\mathbb Z}')$, there exist weak limits $${\underline P\,}_{z,z'}=\lim_{\xi\to1}{\underline P\,}_{z,z',\xi}, \quad P_{z,z'}=\lim_{\xi\to1}P_{z,z',\xi}.$$ Of course, ${\underline P\,}_{z,z',\xi}$ and $P_{z,z',\xi}$ are transformed to each other under the particle/hole involution on ${\mathbb Z}'_-$, and the same holds for the limit measures. Projection correlation kernels {#1-4} ------------------------------ All the measures appearing in Theorem \[1.1\] are determinantal measures. Here we explain the structure of their correlation kernels (for a detailed exposition, see [@BO2], [@BO3], [@BO4], and [@Ol2]). The key object is a second order difference operator $D_{z,z',\xi}$ on the lattice ${\mathbb Z}'$. This operator acts on a test function $f(x)$, $x\in{\mathbb Z}'$, according to $$\aligned D_{z,z',\xi}f(x)&=\sqrt{\xi(z+x+\tfrac12)(z'+x+\tfrac12)}\,f(x+1)\\ &+\sqrt{\xi(z+x-\tfrac12)(z'+x-\tfrac12)}\,f(x-1) \\ &-[x+\xi(z+z'+x)]\,f(x). \endaligned$$ Since $x\pm\frac12$ is an integer for $x\in{\mathbb Z}'$, the expressions under the square root are strictly positive, due to the conditions imposed on the parameters $z$ and $z'$. As shown in [@BO4], $D_{z,z',\xi}$ determines an unbounded selfadjoint operator in the Hilbert space $H=\ell^2({\mathbb Z}')$. This operator has simple, purely discrete spectrum filling the subset $(1-\xi){\mathbb Z}'\subset{\mathbb R}$. In the sequel we will freely pass from bounded operators in $H$ to their kernels and vice versa using the natural orthonormal basis $\{e_x\}$ in $H$ indexed by points $x\in{\mathbb Z}'$: If $A$ is an operator in $H$ then its kernel (or simply matrix) is defined as $A(x,y)=(Ae_y,e_x)$. Let ${\underline K\,}_{z,z',\xi}$ denote the projection in $H$ onto the positive part of the spectrum of $D_{z,z',\xi}$, and let ${\underline K\,}_{z,z',\xi}(x,y)$ denote the corresponding kernel. (Here and below all projection operators are assumed to be orthogonal projections.) \[1.2\] ${\underline K\,}_{z,z',\xi}(x,y)$ is the correlation kernel of the measure ${\underline P\,}_{z,z',\xi}$. The operator corresponding to a correlation kernel of a determinantal measure will be called its [*correlation operator*]{}. Thus, the projection ${\underline K\,}_{z,z',\xi}$ is the correlation operator of ${\underline P\,}_{z,z',\xi}$. Let $D_{z,z'}$ denote the difference operator on ${\mathbb Z}'$ which is obtained by setting $\xi=1$ in the above formula defining $D_{z,z',\xi}$. One can show that $D_{z,z'}$ still determines a selfadjoint operator in $\ell^2({\mathbb Z}')$. Its spectrum is simple, purely continuous, filling the whole real line. Let ${\underline K\,}_{z,z'}$ denote the projection onto the positive part of the spectrum. \[1.3\] [(i)]{} As $\xi$ goes to $1$, the projection operators ${\underline K\,}_{z,z',\xi}$ weakly converge to a projection operator ${\underline K\,}_{z,z'}$. [(ii)]{} ${\underline K\,}_{z,z'}$ serves as the correlation operator of the limit measure ${\underline P\,}_{z,z'}$, that is, the kernel ${\underline K\,}_{z,z'}(x,y)$ is the correlation kernel of ${\underline P\,}_{z,z'}$. Note that the weak convergence of operators in $\ell^2({\mathbb Z}')$ whose norms are uniformly bounded is the same as the pointwise convergence of the corresponding kernels. Note also that on the set of projections, the weak operator topology coincides with the strong operator topology. The above definition of the operators ${\underline K\,}_{z,z',\xi}$ and ${\underline K\,}_{z,z'}$ through the difference operators $D_{z,z',\xi}$ and $D_{z,z'}$ is nice and useful but one often needs explicit expressions for the correlation kernels. Various such expressions are available: - Presentation in the integrable form [@BO1], [@BO2] $$\frac{\mathcal A(x)\mathcal B(y)-\mathcal B(x)\mathcal A(y)}{x-y}\,.$$ - Series expansion (or integral representation) involving eigenfunctions of the difference operators [@BO3], [@BO4]. - Double contour integral representation [@BO3], [@BO4]. In Sections \[5\] and \[6\] we will work with contour integrals. Theorems \[1.4\] and \[1.5\] below describe the integrable form for the limit kernel ${\underline K\,}_{z,z'}(x,y)$. This presentation will be used in Section \[2\]. \[1.4\] Assume $z\ne z'$. For $x,y\in{\mathbb Z}'$ and outside the diagonal $x=y$, $${\underline K\,}_{z,z'}(x,y)=\frac{\sin(\pi z)\sin(\pi z')}{\pi\sin(\pi(z-z'))} \cdot\frac{\mathcal P(x)\mathcal Q(y)-\mathcal Q(x)\mathcal P(y)}{x-y}\,,$$ where $$\begin{split} \mathcal P(x) =\frac{{\Gamma}(z+x+\tfrac12)} {\sqrt{{\Gamma}(z+x+\tfrac12){\Gamma}(z'+x+\tfrac12)}}\,, \\ \mathcal Q(x) =\frac{{\Gamma}(z'+x+\tfrac12)} {\sqrt{{\Gamma}(z+x+\tfrac12){\Gamma}(z'+x+\tfrac12)}} \end{split}$$ and ${\Gamma}(\cdot)$ is Euler’s ${\Gamma}$–function. On the diagonal $x=y$, $${\underline K\,}_{z,z'}(x,x)=\frac{\sin(\pi z)\sin(\pi z')} {\pi\sin(\pi(z-z'))}\, (\psi(z+x+\tfrac12)-\psi(z'+x+\tfrac12)),$$ where $\psi(x)={\Gamma}'(x)/{\Gamma}(x)$ is the logarithmic derivative of the ${\Gamma}$–function. See [@BO2] for a proof. In that paper, we called the kernel ${\underline K\,}_{z,z'}(x,y)$ the [*Gamma kernel*]{}. In the case $z=z'$ (then necessarily $z\in{\mathbb R}\setminus{\mathbb Z}$) an explicit expression can be obtained by taking the limit $z'\to z$ (see [@BO2]), and the result is expressed through the $\psi$ function (outside the diagonal) or its derivative $\psi'$ (on the diagonal): \[1.5\] Assume $z=z'\in{\mathbb R}\setminus{\mathbb Z}$. For $x,y\in{\mathbb Z}'$ and outside the diagonal $x=y$, $${\underline K\,}_{z,z'}(x,y)=\left(\frac{\sin(\pi z)}{\pi}\right)^2\, \frac{\psi(z+x+\tfrac12)-\psi(z+y+\tfrac12)}{x-y}\,.$$ On the diagonal $x=y$, $${\underline K\,}_{z,z'}(x,x)=\left(\frac{\sin(\pi z)}{\pi}\right)^2\, \psi'(z+x+\tfrac12).$$ Here is a simple corollary of the above formulas, which we will need later on: \[1.6\] Let $\rho_1^{(z,z')}(x)$ denote the density function of $P_{z,z'}$. We have $$\rho_1^{(z,z')}(x)\sim \frac{C(z,z')}{|x|}\,, \qquad |x|\to\infty,$$ where $$C(z,z')=\begin{cases} \dfrac{\sin(\pi z)\sin(\pi z')(z-z')} {\pi\sin(\pi(z-z'))}, & z\ne z'\\ \left(\dfrac{\sin(\pi z)}{\pi}\right)^2, & z=z'\in{\mathbb R}\setminus{\mathbb Z}. \end{cases}$$ Recall that $P_{z,z'}$ is related to ${\underline P\,}_{z,z'}$ by the particle/hole involution transformation on ${\mathbb Z}'_-$. It follows that the density functions of the both measures coincide on ${\mathbb Z}'_+$. By the very definition of determinantal measures, the density function of ${\underline P\,}_{z,z'}$ is given by the values of the correlation kernel on the diagonal $x=y$. The formulas of Theorem \[1.4\] and Theorem \[1.5\] express ${\underline K\,}_{z,z'}(x,x)$ through the psi–function and its derivative. The asymptotic expansion of $\psi(y)$ as $y\to+\infty$ is given by formula 1.18(7) in Erdelyi [@Er], which implies $$\psi(y)=\log y-(2y)^{-1}+O(y^{-2}), \quad \psi'(y)=y^{-1}+O(y^{-2}) \qquad (y\to+\infty).$$ Using this we readily get $$\rho_1^{(z,z')}(x)=\frac{C(z,z')}{x}+O(x^{-2}), \qquad x\to+\infty,$$ To handle the case $x\to-\infty$ one can use the relation (see ) $$K_{z,z'}(x,x)=1-{\underline K\,}_{z,z'}(x,x), \qquad x\in{\mathbb Z}'_-\,,$$ and then employ the identity ([@Er 1.7.1]) $$\psi(y+\tfrac12)-\psi(-y+\tfrac12)=\pi\tan(\pi y).$$ A simpler way is to use the symmetry property of $P_{z,z'}$ discussed in Subsection \[1-7\] below. It immediately gives $$\rho_1^{(z,z')}(-x)=\rho_1^{(-z,-z')}(x), \qquad x\in{\mathbb Z}'_+\,.$$ Since $C(-z,-z')=C(z,z')$, we get the desired formula. \[1.7\] As is seen from Theorems \[1.4\] and \[1.5\], the limit kernel ${\underline K\,}_{z,z'}(x,y)$ is real-valued. The same is true for the pre–limit kernels ${\underline K\,}_{z,z',\xi}(x,y)$: this can be seen from their integrable form presentation or from the series expansion. The fact that the kernels are real-valued will be employed in Section \[6\]. $J$–Symmetric kernels and block decomposition {#1-5} --------------------------------------------- For technical reasons, it will be more convenient for us to deal, instead of ${\underline K\,}_{z,z',\xi}$ and ${\underline K\,}_{z,z'}$, with the correlation kernels for the measures $P_{z,z',\xi}$ and $P_{z,z'}$. The latter kernels will be denoted as $K_{z,z',\xi}(x,y)$ and $K_{z,z'}(x,y)$, respectively. The link between two kinds of kernels, the “${\underline K\,}$ kernels” and the “$K$ kernels”, is given by the following relation (see [@BOO Appendix] for a proof): $$\label{f1.2} {\varepsilon}(x)K(x,y){\varepsilon}(y)=\begin{cases} {\underline K\,}(x,y), & x\in {\mathbb Z}'_+\\ {\delta}_{xy}-{\underline K\,}(x,y), & x\in{\mathbb Z}'_-\end{cases}\,,$$ where $${\varepsilon}(x)=\begin{cases} 1, & x\in{\mathbb Z}'_+\\ (-1)^{|x|-\frac12}, & x\in{\mathbb Z}'_- \end{cases}.$$ Note that the factor ${\varepsilon}(x)=\pm1$ does not affect the correlation functions (see Subsection \[1-6\]). This factor becomes important in the limit regime considered in [@BO1] and [@BO3 §8], but for the purpose of the present paper, it is inessential and could be omitted; I wrote it only to keep the notation consistent with that of the previous papers [@BO1], [@BO2], [@BO3]. Decompose the Hilbert space $H=\ell^2({\mathbb Z}')$ into the direct sum $H=H_+\oplus H_-$, where $H_\pm=\ell^2({\mathbb Z}'_\pm)$. Then every operator $A$ in $H$ can be written in a block form, $$A=\bmatrix A_{++} & A_{+-}\\ A_{-+} & A_{--} \endbmatrix\,,$$ where $A_{++}$ acts from $H_+$ to $H_+$, $A_{+-}$ acts from $H_-$ to $H_+$, etc. In terms of the block form, can be rewritten as follows (below $A_{\varepsilon}$ denotes the operator of multiplication by ${\varepsilon}(x)$): $$\begin{aligned} K_{++}&={\underline K\,}_{++} &K_{+-}&={\underline K\,}_{+-}A_{\varepsilon}\\ K_{-+}&=-A_{\varepsilon}{\underline K\,}_{-+} &K_{--}&=1-A_{\varepsilon}{\underline K\,}_{--}A_{\varepsilon}.\end{aligned}$$ It follows that if ${\underline K\,}$ is an Hermitian operator in $H$ then $K$ is also Hermitian, but with respect to an [*indefinite*]{} inner product in $H$: $$[f,g]:=(Jf,g), \qquad f,g\in H, \quad J=\bmatrix 1 & 0\\0 &-1\endbmatrix.$$ Such operators are called [*$J$–Hermitian*]{} or [*$J$–symmetric*]{} operators. Thus, the operators $K_{z,z',\xi}$ and $K_{z,z'}$ are $J$–symmetric. \[1.8\] The pre–limit operators $K_{z,z',\xi}$ belong to the trace class. This claim is not obvious from the definition of the operators nor from the explicit expressions for the kernels, but can be easily derived from the results of [@BO1] (it is immediately seen that the “$L$–operator” related to $K:=K_{z,z',\xi}$ through the formula $K=L(1+L)^{-1}$ is of trace class). The trace class property of $K_{z,z',\xi}$ is related to the fact that the measure $P_{z,z',\xi}$ lives on finite configurations (note that the trace of a correlation operator equals the expected total number of particles). As for the limit measure $P_{z,z'}$, it lives on infinite configurations, and the limit operator $K_{z,z'}$ is not of trace class. Gauge transformation of correlation kernels {#1-6} ------------------------------------------- An arbitrary transformation of correlation kernels of the form $$\mathcal K(x,y)\mapsto \phi(x)\mathcal K(x,y)\phi(y)^{-1}$$ with a nonvanishing function $\phi(x)$ does not affect the minors giving the values of the correlation functions. We call this a [*gauge transformation*]{}. Thus, the correlation kernel [*is not*]{} a canonical object attached to a determinantal measure. This circumstance must be taken into account in attempting to solve Problem \[P1\]. Symmetry {#1-7} -------- Recall that by ${\lambda}\mapsto{\lambda}'$ we denote transposition of Young diagrams. Return to formula for the z–measure weights and observe that $\dim{\lambda}'=\dim{\lambda}$ and $(z)_{\lambda}=(-1)^{|{\lambda}|}(-z)_{{\lambda}'}$. This implies the important symmetry relation $$\label{f1.3} M_{z,z',\xi}({\lambda}')=M_{-z,-z',\xi}({\lambda}), \qquad {\lambda}\in{\mathbb Y}.$$ Next, observe that under transposition ${\lambda}\mapsto{\lambda}'$, the modified Frobenius coordinates interchange: $p_i\leftrightarrow q_i$. Together with the above symmetry relation this implies that the transformation of the measure $P_{z,z',\xi}$ induced by the reflection symmetry $x\mapsto-x$ of the lattice ${\mathbb Z}'$ amounts to the index transformation $(z,z')\to(-z,-z')$. The same holds for the limit measures $P_{z,z'}$. It is worth noting that the behavior of the measures ${\underline P\,}_{z,z',\xi}$ and their limits under the reflection symmetry of ${\mathbb Z}'$ is more complex: besides the change of sign of $z$ and $z'$ one has to apply the particle/hole involution [*on the whole lattice*]{}. The symmetry is reflected in the following symmetry property for the kernels $K_{z,z',\xi}$: \[1.9\] We have $$K_{z,z',\xi}(x,y)=(-1)^{\operatorname{sgn}(x)\operatorname{sgn}(y)}K_{-z,-z',\xi}(-x,-y).$$ This follows from [@BO3 Theorem 7.2], see the comments to this theorem. Passing to the limit as $\xi\to1$, we get the same property for the limit kernel $K_{z,z'}(x,y)$. Multiplicative functionals and Fredholm determinants {#2} ==================================================== Generalities {#2-1} ------------ Let ${\mathfrak X}$ be a countable set. Below we will return to ${\mathfrak X}={\mathbb Z}'$ but at this moment we do not need any structure on ${\mathfrak X}$. As in Subsection \[1-1\], we mean by a [*configuration*]{} in ${\mathfrak X}$ an arbitrary subset $X\subseteq{\mathfrak X}$ and we denote by ${\operatorname{Conf}}({\mathfrak X})$ the space of all configurations. Again, we equip ${\operatorname{Conf}}({\mathfrak X})$ with the topology determined by the identification ${\operatorname{Conf}}({\mathfrak X})$ with the infinite product space $\{0,1\}^{{\mathfrak X}}$; then ${\operatorname{Conf}}({\mathfrak X})$ becomes a metrizable compact topological space. We also endow ${\operatorname{Conf}}({\mathfrak X})$ with the corresponding Borel structure. Given a configuration $X\in{\operatorname{Conf}}({\mathfrak X})$, let $1_X$ denote its indicator function: for $x\in\mathfrak X$, the value $1_X(x)$ equals 1 or 0 depending on whether $x$ belongs or not to $X$. Viewing $1_X$ as a collection of $0$’s and $1$’s indexed by points $x\in{\mathfrak X}$ we just get the identification of ${\operatorname{Conf}}({\mathfrak X})$ with $\{0,1\}^{{\mathfrak X}}$. A function $F(X)$ on ${\operatorname{Conf}}({\mathfrak X})$ is said to be a [*cylinder function*]{} if it depends only on the intersection $X\cap Y$ with some finite subset $Y\subset{\mathfrak X}$. Cylinder functions are continuous and form a dense subalgebra in $C({\operatorname{Conf}}({\mathfrak X}))$, the Banach algebra of continuous functions on the compact space ${\operatorname{Conf}}({\mathfrak X})$. We will need cylinder functions in Section \[4\]. Multiplicative functionals $\Phi_f$ {#2-2} ----------------------------------- To a function $f(x)$ on $\mathfrak X$, we would like to assign a [*multiplicative functional*]{} on ${\operatorname{Conf}}(\mathfrak X)$ by means of the formula $$\Phi_f(X)=\prod_{x\in X}(1+f(x))=\prod_{x\in{\mathfrak X}}(1+1_X(x)f(x)), \quad X\in{\operatorname{Conf}}(\mathfrak X).$$ Let us say that $\Phi_f(X)$ is [*defined at $X$*]{} if the above product is absolutely convergent, which is equivalent to saying that the sum $$\label{f2.1} \sum_{x\in X}|f(x)|=\sum_{x\in{\mathfrak X}}1_X(x)|f(x)|$$ is finite. Thus, the domain of definition for $\Phi_f$ is the set of all configurations $X$ for which is finite. Obviously, this set coincides with the whole space ${\operatorname{Conf}}({\mathfrak X})$ if and only if $f$ belongs to $\ell^1(\mathfrak X)$. In particular, this happens if $f$ vanishes outside a finite subset $Y\subset{\mathfrak X}$, and then $\Phi_f$ is simply a cylinder function. However, we will need to deal with multiplicative functionals which are defined on a proper subset of ${\operatorname{Conf}}({\mathfrak X})$ only. Observe that for any function $f$, the domain of definition of $\Phi_f$ is a Borel subset in ${\operatorname{Conf}}(\mathfrak X)$ (more precisely, a subset of type $F_\sigma$), and $\Phi_f$ is a Borel function on this subset, because $\Phi_f$ is a pointwise limit of cylinder functions. Given a probability measure $P$ on ${\operatorname{Conf}}(\mathfrak X)$, it is important for us to see if the domain of definition of $\Phi_f$ is of full $P$–measure. Here is a simple sufficient condition for this, expressed in terms of the density function $\rho_1(x)$. Recall its meaning: $\rho_1(x)$ is the probability that the random (with respect to $P$) configuration $X$ contains $x$. \[2.1\] Let $P$ be a probability measure on ${\operatorname{Conf}}(\mathfrak X)$, $\rho_1(x)$ be its density function, and $f(x)$ be a function on $\mathfrak X$. If $$\sum_{x\in\mathfrak X}\rho_1(x)|f(x)|<\infty$$ then the multiplicative functional $\Phi_f(X)$ is defined $P$–almost everywhere on ${\operatorname{Conf}}(\mathfrak X)$. Regard the quantity as a function in $X$, with values in $[0,+\infty]$. This function is almost everywhere finite if its expectation is finite. Now, take the expectation of the right–hand side of . Since the expectation of $1_X(x)$ is $\rho_1(x)$, the result is $\sum_{x\in\mathfrak X}\rho_1(x)|f(x)|$, which is finite by the assumption. Fix a function $r(x)>0$ on ${\mathfrak X}$. Let us say that a configuration $X$ is [*$r$–sparse*]{} (more precisely, sparse with respect to weight $r^{-1}$) if the series $$\sum_{x\in X}r^{-1}(x)=\sum_{x\in\mathfrak X}1_X(x)r^{-1}(x)$$ converges. Let ${\operatorname{Conf}}_r({\mathfrak X})$ denote the subset of all $r$–sparse configurations. If the series $\sum_{x\in\mathfrak X}r^{-1}(x)$ converges then obviously ${\operatorname{Conf}}_r({\mathfrak X})={\operatorname{Conf}}({\mathfrak X})$; otherwise ${\operatorname{Conf}}_r({\mathfrak X})$ is a proper subset of ${\operatorname{Conf}}({\mathfrak X})$. Note that it is a Borel subset (more precisely, a subset of type $F_\sigma$). \[2.2\] Let $P$ be a probability measure on ${\operatorname{Conf}}({\mathfrak X})$ and $\rho_1(x)$ be its density function. If $$\sum_{x\in{\mathfrak X}}\rho_1(x)r^{-1}(x)<\infty$$ then $P$ is concentrated on the Borel subset ${\operatorname{Conf}}_r({\mathfrak X})$. The argument is the same as in the proof of Proposition \[2.1\]. Consider the function $$\varphi(X)=\sum_{x\in{\mathfrak X}}1_X(x)r^{-1}(x), \qquad X\in{\operatorname{Conf}}({\mathfrak X}),$$ which is allowed to take the value $+\infty$. We have to prove that $\varphi(X)$ is finite almost surely with respect to the measure $P$. This is obvious, because the expectation of $\varphi$ equals $$\sum_{x\in{\mathfrak X}}\rho_1(x)r^{-1}(x),$$ which is finite by the assumption. Consider the correspondence $X\mapsto m_X$ that assigns to a configuration $X$ the function $m_X(x)=1_X(x)r^{-1}(x)$ on ${\mathfrak X}$. This correspondence determines an embedding of the set ${\operatorname{Conf}}_r({\mathfrak X})$ into the Banach space $\ell^1({\mathfrak X})$. Using this embedding we equip ${\operatorname{Conf}}_r({\mathfrak X})$ with the topology induced by the norm topology of $\ell^1({\mathfrak X})$. Let us call this topology the [*$\ell^1$–topology*]{}; of course, the definition depends on the choice of the function $r(x)$. If the series $\sum_{x\in\mathfrak X}r^{-1}(x)$ diverges then the $\ell^1$–topology on ${\operatorname{Conf}}_r({\mathfrak X})$ is stronger than that induced by the canonical topology of the space ${\operatorname{Conf}}({\mathfrak X})$. \[2.3\] Let $f$ be a function on ${\mathfrak X}$ such that the function $|f(x)|r(x)$ is bounded. Then the multiplicative functional $\Phi_f$ is well defined on ${\operatorname{Conf}}_r({\mathfrak X})$. Moreover, $\Phi_f$ is continuous in the $\ell^1$–topology of ${\operatorname{Conf}}_r({\mathfrak X})$ defined above. The first claim is trivial. Indeed, set $g(x)=f(x)r(x)$. This function is in $\ell^\infty({\mathfrak X})$ while the function $m_X(x)$ is in $\ell^1({\mathfrak X})$. Since the quantity can be represented as the pairing between $|g|$ and $m_X$, we conclude that is finite. To prove the second claim we observe that $$\Phi_f(X)=\prod_{x\in{\mathfrak X}}(1+m_X(x)g(x)).$$ Now the claim readily follows from the fact that the pairing between $g\in\ell^\infty({\mathfrak X})$ and $m_X\in\ell^1({\mathfrak X})$ is continuous in the second argument. \[2.4\] Let ${\mathfrak X}={\mathbb Z}'$ and $P=P_{z,z'}$. We know from Corollary \[1.6\] that in this concrete case the density function $\rho_1(x)$ on ${\mathbb Z}'$ decays as $|x|^{-1}$ as $x\to\pm\infty$. Then Proposition \[2.2\] says that $P_{z,z'}$ is concentrated on ${\operatorname{Conf}}_r({\mathbb Z}')$ provided that the function $r(x)>0$ on ${\mathbb Z}'$ is such that the series $\sum_{x\in{\mathbb Z}'}r^{-1}(x)|x|^{-1}$ converges. For instance, one may take $r(x)=|x|^{\delta}$ with any ${\delta}>0$. (Later on we will choose $r(x)=|x|$.) Proposition \[2.3\] says that a multiplicative functional $\Phi_f$ is well defined on ${\operatorname{Conf}}_r({\mathbb Z}')$ if $f(x)=O(r^{-1}(x))$ as $x\to\pm\infty$. In particular, $\Phi_f(X)$ is well defined for $P_{z,z'}$–almost all configurations $X$ provided that $f$ satisfies the above condition for a positive function $r$ such that $\sum_{x\in{\mathbb Z}'}r^{-1}(x)|x|^{-1}<\infty$. This condition on $f$ essentially coincides with the condition of Proposition \[2.1\] (only that proposition avoids the intermediation of $r$), which is not surprising because the both propositions exploit the same idea. Condition of integrability for $\Phi_f$ {#2-3} --------------------------------------- Let again $\mathfrak X$ be a countable set and $P$ be a probability Borel measure on ${\operatorname{Conf}}(\mathfrak X)$. Here we give a condition for $\Phi_f$ to be not only defined $P$–almost everywhere but also to have finite expectation. The condition involves the correlation functions of all orders. It is convenient to combine them into a single function $\rho(X')$ defined on arbitrary finite subsets $X'\subset\mathfrak X$: By definition, $\rho(X')$ equals the probability of the event that the random configuration $X$ contains $X'$. Let $X'\Subset X$ mean that $X'$ is a finite subset of $X$. By $\mathbb E_P(\,\cdot\,)$ we denote expectation with respect to $P$. \[2.5\] Let $f(x)$ be a function on $\mathfrak X$ such that $$\sum_{X'\Subset\mathfrak X}\rho(X')\prod_{x\in X'}|f(x)|<\infty.$$ Then the multiplicative functional $\Phi_f$ is defined almost everywhere with respect to $P$, is absolutely integrable, and its expectation equals $$\label{f2.2} \mathbb E_P(\Phi_f)=\sum_{X'\Subset\mathfrak X}\rho(X')\prod_{x\in X'}f(x).$$ The above condition on $f$ is stronger than the condition of Proposition \[2.1\], so that the first claim follows from Proposition \[2.1\]. Checking the second and third claims uses the same argument as in that proposition. Observe that $$\prod_{x\in X}(1+|f(x)|)=\sum_{X'\Subset X}\prod_{x\in X'}|f(x)|$$ in the sense that the both sides are simultaneously either finite or infinite, and if they are finite then they are equal. Denote by $\eta_{X'}(X)$ the function on ${\operatorname{Conf}}(\mathfrak X)$ equal to 1 or 0 depending on whether $X$ contains $X'$ or not. The above equality can be rewritten as $$\prod_{x\in X}(1+|f(x)|)=\sum_{X'\Subset \mathfrak X}\eta_{X'}(X)\prod_{x\in X'}|f(x)|.$$ Take the expectation of the both sides. Since $\mathbb E_P(\eta_{X'})=\rho(X')$, we get $$\mathbb E_P(\Phi_{|f|})=\sum_{X'\Subset\mathfrak X}\rho(X')\prod_{x\in X'}|f(x)|<\infty.$$ Thus, we have checked the second and third claims for the function $|f|$. Since $|\Phi_f(X)|\le\Phi_{|f|}(X)$, it follows that $\Phi_f$ is absolutely integrable. Now we can repeat the above argument with $f$ instead of $|f|$. The above computation with $|f|$ provides the necessary justification for manipulations with infinite sums. Fredholm determinants {#2-4} --------------------- Let $\mathfrak X$ and $P$ be as in Subsection \[2-3\], and assume additionally that $P$ is determinantal with a correlation kernel $K(x,y)$ corresponding to a bounded operator $K$ in the Hilbert space $H:=\ell^2(\mathfrak X)$ (so $K$ is the correlation operator of $P$). For a bounded function $f(x)$ on $\mathfrak X$, we denote by $A_f$ the operator in $H$ given by multiplication by $f$. \[2.6\] If $f$ is finitely supported then $$\mathbb E_P(\Phi_f)=\det(1+A_fK).$$ Note that the determinant is well defined because, due to the assumption on $f$, the operator $A_fK$ has finite–dimensional range. This directly follows from . Indeed, according to the definition of determinantal measures, $\rho(X')=\det\left([K(x,y)]_{x,y\in X'}\right)$. This implies $$\rho(X')\prod_{x\in X'}f(x)=\det\left([f(x)K(x,y)]_{x,y\in X'}\right).$$ Then we employ a well–known identity from linear algebra: If $B=[B(x,y)]$ is a matrix then $\det(1+B)$ equals the sum of the principal minors of $B$. The identity holds for matrices of finite size but we can apply it to $B=A_fK$ because the matrix $[f(x)K(x,y)]$ has only finitely many nonzero rows. This gives us the equality $$\sum_{X'\Subset\mathfrak X}\det\left([f(x)K(x,y)]_{x,y\in X'}\right)=\det(1+A_fK),$$ which concludes the proof. The hypothesis of the lemma is, of course, too restrictive: the above argument can be easily extended to the case when the operator $A_fK$ is of trace class. A slightly more general fact is established below in Proposition \[2.7\], which is specially adapted to the application we need. First, state a few general results from [@BOO Appendix]. Let $H=H_+\oplus H_-$ be a ${\mathbb Z}_2$–graded Hilbert space. Any operator $A$ in $H$ can be written in block form, $$A=\begin{bmatrix} A_{++} & A_{+-}\\ A_{-+} & A_{--} \end{bmatrix}$$ where $A_{++}$ acts from $H_+$ to $H_+$, $A_{+-}$ acts from $H_-$ to $H_+$, etc. Let ${\mathcal L_{1|2}}(H)$ be the set of bounded operators $A$ whose diagonal blocks $A_{++}$ and $A_{--}$ are trace class operators while the off–diagonal blocks $A_{+-}$ and $A_{-+}$ are Hilbert–Schmidt operators. The set ${\mathcal L_{1|2}}(H)$ is an algebra. We equip it with the corresponding combined topology: the topology of the trace class norm $\Vert\cdot\Vert_1$ for the diagonal blocks and the topology of the Hilbert–Schmidt norm $\Vert\cdot\Vert_2$ for the off–diagonal blocks. There exists a unique continuous function on ${\mathcal L_{1|2}}(H)$, $$A\to\det(1+A),$$ coinciding with the conventional determinant when $A$ is a finite rank operator. This function can be defined as $$\det(1+A)=\det((1+A)e^{-A})e^{{\operatorname{tr}}(A_{++})+{\operatorname{tr}}(A_{--})},$$ where the determinant in the right–hand side is the conventional one: the point is that $A\mapsto(1+A)e^{-A}-1$ is a continuous map from ${\mathcal L_{1|2}}(H)$ to the set of trace class operators. If $\{E_N\}_{N=1,2,\dots}$ is an ascending chain of projection operators in $H$ strongly convergent to 1 then $$\label{f2.3} \det(1+A)=\lim_{N\to\infty}(1+E_NAE_N).$$ From now on we assume $\mathfrak X={\mathbb Z}'$ and we set $$H=\ell^2({\mathbb Z}'), \quad H_+=\ell^2({\mathbb Z}'_+), \quad H_-=\ell^2({\mathbb Z}'_-).$$ As before, $\{e_x\}_{x\in{\mathbb Z}'}$ denotes the natural basis in $H$. \[2.7\] Let $P$ be a determinantal probability measure on ${\operatorname{Conf}}({\mathbb Z}')$, $K(x,y)$ be its correlation kernel and $K$ denote the corresponding correlation operator in $H=\ell^2({\mathbb Z}')$. Further, assume that $f(x)$ is a function on ${\mathbb Z}'$ which can be written in the form $f(x)=g(x)h^2(x)$, where $g(x)$ is bounded and $h$ is nonnegative and such that $A_h K A_h\in{\mathcal L_{1|2}}(H)$. Then the functional $\Phi_f$ is defined almost everywhere with respect to $P$, is absolutely integrable with respect to $P$, and $$\mathbb E_P(\Phi_f)=\det(1+A_gA_h K A_h).$$ For $N=1,2,\dots$, let $E_N$ be the projection in $H$ onto the finite–dimensional subspace of functions concentrated on $[-N,N]\cap{\mathbb Z}'$. As $N\to\infty$, the projections $E_N$ converge to 1. Assume first that $g\equiv1$. Then $f(x)$ is nonnegative and the same argument as in Lemma \[2.6\] shows that $$\begin{gathered} \det(1+E_NA_h K A_hE_N)=\det(1+A_fE_NKE_N)\\ =\sum_{X'\subset[-N,N]\cap{\mathbb Z}'}\det\left([K(x,y)]_{x,y\in X'}\right)\prod_{x\in X'}f(x)\\ = \sum_{X'\subset[-N,N]\cap{\mathbb Z}'}\rho(X')\prod_{x\in X'}f(x),\end{gathered}$$ where we used the fact that $E_N$ and $A_h$ commute. As $N\to\infty$, the resulting quantity converges to the infinite sum $$\sum_{X'\Subset{\mathbb Z}'}\rho(X')\prod_{x\in X'}f(x).$$ On the other hand, by virtue of , $$\det(1+E_NA_gA_h K A_hE_N)\to\det(1+A_h K A_h).$$ Consequently, $$\det(1+A_hKA_h)=\sum_{X'\Subset{\mathbb Z}'}\rho(X')\prod_{x\in X'}f(x)=\mathbb E_P(\Phi_f),$$ where the last equality follows from Proposition \[2.5\]. For an arbitrary bounded $g$ the computation is the same, and the above argument with $f\ge0$ is used to check the absolute convergence of the arising infinite sum and to guarantee applicability of Proposition \[2.5\]. Radon–Nikodým derivatives {#3} ========================= Denote by $\mathfrak S$ the group of finitary permutations of the set ${\mathbb Z}'$. This is a countable group generated by the elementary transpositions $$\dots,{\sigma}_{-1},{\sigma}_0,{\sigma}_1,\dots,$$ where ${\sigma}_n$ transposes $n-\frac12$ and $n+\frac12$. The action of the group $\mathfrak S$ on ${\mathbb Z}'$ induces its action on ${\operatorname{Conf}}({\mathbb Z}')$. For ${\sigma}\in\mathfrak S$, we denote the corresponding transformation of ${\operatorname{Conf}}({\mathbb Z}')$ by the same symbol ${\sigma}$. Let us represent configurations $X\in{\operatorname{Conf}}({\mathbb Z}')$ as two–sided infinite sequences of black and white circles separated by vertical bars, like this: $$\cdots\bullet\mathop{|}_{n-2}\bullet \mathop{|}_{n-1}\circ\mathop{|}_{n}\bullet\mathop{|}_{n+1}\circ \mathop{|}_{n+2}\circ\cdots$$ Here black and white circles represent particles and holes, respectively, and the subscripts under the bars are used to mark the positions of integers interlacing with half–integers. In this picture, the action of ${\sigma}_n$ affects only the two–circle fragment around the $n$th vertical bar and amounts to replacing “$\circ\mathop{|}\limits_{n}\bullet$” by “$\bullet\mathop{|}\limits_{n}\circ$” and vice versa (the fragments “$\circ\mathop{|}\limits_{n}\circ$” and “$\bullet\mathop{|}\limits_{n}\bullet$” remain intact). This action preserves the set of Maya diagrams ${\underline X}({\lambda})$, so that we get an action of $\mathfrak S$ on ${\mathbb Y}$, which can be directly described as follows: application of the elementary transposition ${\sigma}_n$ to a Young diagram ${\lambda}$ amounts to adding or removing a box $(i,j)$ with $j-i=n$, if this operation is possible. The transformation “$\bullet\mathop{|}\limits_{n}\circ\to \circ\mathop{|}\limits_{n}\bullet$” corresponds to adding a box, and the inverse transformation corresponds to removing a box. In particular, ${\sigma}_0$ adds/removes boxes on the main diagonal of ${\lambda}$. Our aim is to study the transformation of the measures ${\underline P\,}_{z,z',\xi}$ and their limits ${\underline P\,}_{z,z'}$ under the action of $\mathfrak S$. However, for technical reasons, it is more convenient to deal with the measures $P_{z,z',\xi}$ and $P_{z,z'}$ which are related to the former measures by the particle/hole involution on ${\mathbb Z}'_-$. To this end we introduce the [*modified action*]{} of $\mathfrak S$ on ${\operatorname{Conf}}({\mathbb Z}')$: it differs from the natural one by conjugation with the particle/hole involution. Given ${\sigma}\in\mathfrak S$, we denote the modified action of ${\sigma}$ on ${\operatorname{Conf}}({\mathbb Z}')$ by the symbol ${\widetilde{\sigma}}$. The relation between ${\sigma}$ and ${\widetilde{\sigma}}$ is $${\widetilde{\sigma}}(X)={\operatorname{inv}}({\sigma}({\operatorname{inv}}(X))), \qquad {\sigma}\in\mathfrak S, \quad X\in{\operatorname{Conf}}({\mathbb Z}').$$ The modified transformations ${\widetilde{\sigma}}:{\operatorname{Conf}}({\mathbb Z}')\to{\operatorname{Conf}}({\mathbb Z}')$ induce transformations of measures denoted as $P\mapsto {\widetilde{\sigma}}(P)$. Let us emphasize that the modified action is defined only on ${\operatorname{Conf}}({\mathbb Z}')$, not on ${\mathbb Z}'$ itself. In the case of elementary transpositions ${\sigma}={\sigma}_n$, the modified action differs from the natural one for $n=0$ only. Namely, the modified action of ${\sigma}_0$ amounts to switching “$\circ\mathop{|}\limits_{0}\circ\leftrightarrow \bullet\mathop{|}\limits_{0}\bullet$”, while the fragments “$\bullet\mathop{|}\limits_{0}\circ$” and “$\circ\mathop{|}\limits_{0}\bullet$” remain intact. Recall that a finite configuration $X\Subset{\mathbb Z}'$ has the form ${\operatorname{inv}}({\underline X}({\lambda}))$ with ${\lambda}\in{\mathbb Y}$ if and only if $X$ is [*balanced*]{} in the sense that $|X\cap{\mathbb Z}'_+|=|X\cap{\mathbb Z}'_-|$. Since the initial action of $\mathfrak S$ preserves the set of the semi–infinite configurations of the form ${\underline X}({\lambda})$, the modified action preserves the set of the finite balanced configurations. Obviously, if ${\sigma}\in\mathfrak S$, ${\underline X}={\underline X}({\lambda})$, and $X=X({\lambda})={\operatorname{inv}}({\underline X})$, then we have $$\frac{{\sigma}({\underline P\,}_{z,z',\xi})({\underline X})}{{\underline P\,}_{z,z',\xi}({\underline X})} =\frac{{\widetilde{\sigma}}(P_{z,z',\xi})(X)}{P_{z,z',\xi}(X)}\,.$$ We introduce a special notation for this Radon–Nikodým derivative: $$\label{f3.1} \mu_{z,z',\xi}({\sigma},X):=\frac{{\widetilde{\sigma}}(P_{z,z',\xi})(X)}{P_{z,z',\xi}(X)} =\frac{P_{z,z',\xi}({\widetilde{\sigma}}^{-1}(X))}{P_{z,z',\xi}(X)}, \qquad {\sigma}\in\mathfrak S,$$ where $X$ is a finite balanced configuration. In the remaining part of the section we prove the following result. \[3.1\] Fix an arbitrary couple $(z,z')$ of parameters belonging to the principal or complementary series. Let $\xi$ range over $(0,1)$ and $X$ range over the set of finite balanced configurations on ${\mathbb Z}'$. For any fixed ${\sigma}\in\mathfrak S$, the Radon–Nikodým derivative can be written as a finite linear combination of multiplicative functionals of the form $\Phi_f$ multiplied by factors $\xi^k$ with $k\in{\mathbb Z}$, where each function $f(x)$ decays at infinity at least as $|x|^{-1}$[:]{} $$\label{f3.2} \begin{split} \mu_{z,z',\xi}({\sigma},X)=\sum_{i=1}^m a_i\xi^{k_i}\Phi_{f_i}(X), \\ a_i\in{\mathbb R}, \quad k_i\in{\mathbb Z}, \quad f_i(x)=O(|x|^{-1}). \end{split}$$ Let us emphasize that the right–hand side depends on $\xi$ through the factors $\xi^k$ only. Proposition \[3.1\] provides a refinement of Claim \[C2\] of the Introduction, as explained after the end of the proof of the proposition. [*Step*]{} 1. Given a subset $X'\subseteq{\mathbb Z}'\cap[-N,N]$, where $N\in\{1,2,\dots\}$, and a function $f^{{\operatorname{\,out}}}$ on ${\mathbb Z}'\setminus[-N,N]$, we set $$\begin{split} \eta_{N,X'}(X)=\begin{cases} 1, & X\cap[-N,N]=X'\\ 0, &\text{\rm otherwise}\end{cases}, \\ \Phi_{N,f^{{\operatorname{\,out}}}}(X)=\prod_{x\in X\setminus[-N,N]}(1+f^{{\operatorname{\,out}}}(x)). \end{split}$$ We will prove that for any ${\sigma}\in\mathfrak S$ and all $N$ large enough there exists a representation of the form $$\label{f3.3} \begin{gathered} \mu_{z,z',\xi}({\sigma},X)=\sum_{i=1}^m a_i\xi^{k_i}\eta_{N,X'_i}(X)\Phi_{N,f^{{\operatorname{\,out}}}_i}(X), \\ a_i\in{\mathbb R}, \quad k_i\in{\mathbb Z}, \quad X'_i\subseteq{\mathbb Z}'\cap[-N,N], \quad f^{{\operatorname{\,out}}}_i(x)=O(|x|^{-1}). \end{gathered}$$ Observe that is equivalent to , because each function of the form $\eta_{N,X'}(X)$, being a cylinder functional depending on $X\cap[-N,N]$ only, can be written as a linear combination of functionals of the form $$X\mapsto\prod_{x\in X\cap[-N,N]}(1+f^{{\operatorname{\,in}}}(x))$$ with appropriate functions $f^{{\operatorname{\,in}}}$ on ${\mathbb Z}'\cap[-N,N]$. [*Step*]{} 2. Next, we want to reduce the problem to the particular case when ${\sigma}$ is an elementary transposition. Since the elementary transpositions generate the whole group $\mathfrak S$, to perform the desired reduction, it suffices to prove that if the presentation exists for two elements ${\sigma},\tau\in\mathfrak S$ then it also exists for the product ${\sigma}\tau$. It follows from the definition that $$\mu_{z,z',\xi}({\sigma}\tau,X) =\mu_{z,z',\xi}({\sigma},X)\cdot \mu_{z,z',\tau}(\tau,{\widetilde}{\sigma}^{-1}(X)).$$ We may assume that $N$ is so large that the permutation ${\sigma}:{\mathbb Z}'\to{\mathbb Z}'$ does not move points outside $[-N,N]$. Then it is clear that if the function $X\mapsto \mu_{z,z',\xi}(\tau,X)$ admits a presentation of the form then the same holds for the function $X\mapsto \mu_{z,z',\xi}(\tau,{\widetilde}{\sigma}^{-1}(X))$ as well. Thus, it remains to check that the set of functions admitting a representation of the form (with $N$ fixed) is closed under multiplication. This is obvious, because the product of two functionals of the form $\Phi_{N,f^{{\operatorname{\,out}}}}$ is a functional of the same kind: $$\Phi_{N,f^{{\operatorname{\,out}}}}\Phi_{N,g^{{\operatorname{\,out}}}}=\Phi_{N,h^{{\operatorname{\,out}}}}$$ with $$h^{{\operatorname{\,out}}}(x):=f^{{\operatorname{\,out}}}(x)g^{{\operatorname{\,out}}}(x)+f^{{\operatorname{\,out}}}(x)+g^{{\operatorname{\,out}}}(x),$$ and, moreover, if $f^{{\operatorname{\,out}}}(x)=O(|x|^{-1})$ and $g^{{\operatorname{\,out}}}(x)=O(|x|^{-1})$ then $h^{{\operatorname{\,out}}}(x)=O(|x|^{-1})$. [*Step*]{} 3. Thus, we have to analyze the ratio $$\label{f3.4} \mu_{z,z',\xi}({\sigma}_n,X) =\frac{P_{z,z',\xi}({\widetilde{\sigma}}_n^{-1}(X))}{P_{z,z',\xi}(X)} =\frac{P_{z,z',\xi}({\widetilde{\sigma}}_n(X))}{P_{z,z',\xi}(X)}\,,$$ where the second equality holds because ${\widetilde{\sigma}}_n^{-1}={\widetilde{\sigma}}_n$. We aim to prove that for any $N>|n|$, there exists a single term representation $$\label{f3.5} \begin{split} \frac{P_{z,z',\xi}({\widetilde{\sigma}}_n(X))}{P_{z,z',\xi}(X)}=a\xi^k\Phi_{N,f^{{\operatorname{\,out}}}}(X),\\ k=0,\pm1, \quad f^{{\operatorname{\,out}}}(x)=O(|x|^{-1}), \end{split}$$ where, in contrast to , $a$, $k$, and $f^{{\operatorname{\,out}}}$ may depend on the intersection $X':=X\cap[-N,N]$. Observe that once is established, we can combine various variants of (which depend on $X'$) into a single representation by making use of the factors $\eta_{N,X'}(X)$. Thus, it suffices to prove . [*Step*]{} 4. Here we exhibit a convenient explicit expression for $P_{z,z',\xi}(X)$, where $X$ is an arbitrary balanced configuration. Employing the notation introduced in Subsection \[1-1\] we write $$X=\{-q_d,\dots,-q_1,p_1,\dots,p_d\}.$$ By definition, $P_{z,z',\xi}(X)=M_{z,z',\xi}({\lambda})$, where ${\lambda}$ is such that $X=X({\lambda})$. We have to rewrite the expression for $M_{z,z',\xi}({\lambda})$ given in Subsection \[1-2\] in terms of the $p_i$’s and $q_i$’s. For the terms $(z)_{\lambda}$ and $(z')_{\lambda}$ this is easy, and for $\dim{\lambda}/|{\lambda}|!$ we employ the formula $$\frac{\dim{\lambda}}{|{\lambda}|!}=\frac{\prod\limits_{1\le i<j\le d}(p_j-p_i)(q_j-q_i)}{\prod\limits_{i=1}^d(p_i-\frac12)!(q_i-\frac12)!\cdot \prod\limits_{i,j=1,\dots,d}(p_i+q_j)}$$ given, e.g., in [@Ol1 (2.7)]. The result is $$\label{f3.6} \begin{split} &P_{z,z',\xi}(X)=(1-\xi)^{zz'}\xi^{\sum_{i=1}^d(p_i+q_i)}(zz')^d \\ &\quad\times\prod_{i=1}^d \frac{(z+1)_{p_i-\frac12}(z'+1)_{p_i-\frac12} (-z+1)_{q_i-\frac12}(-z'+1)_{q_i-\frac12}}{((p_i-\frac12)!)^2((q_i-\frac12)!)^2}\\ &\quad\times\frac{\prod\limits_{1\le i<j\le d}(p_j-p_i)^2(q_j-q_i)^2}{\prod\limits_{i,j=1,\dots,d}(p_i+q_j)^2}. \end{split}$$ At first glance formula might appear cumbersome but actually it is well suited for our purpose, because it already has multiplicative form and after substitution into many factors are cancelled out. Below we examine separately the three cases: $n=1,2,\dots$, $n=-1,-2,\dots$, and $n=0$. [*Step*]{} 5. Consider the case $n=1,2,\dots$. Then the transformed configuration ${\widetilde{\sigma}}_n(X)$ is the same as the configuration ${\sigma}_n(X)$, which in turn either coincides with the initial configuration $X$ or differs from it by shifting a single coordinate $p_i$ by $\pm1$. The shift $p_i\to p_i+1$ arises if there exists $i$ such that $p_i=n-\frac12$ and either $i=d$ or $p_i+1\ne p_{i+1}$, which means that $X$ contains the fragment $\bullet\underset{n}{|}\circ$. The shift $p_i\to p_i-1$ arises if there exists $i$ such that $p_i=n+\frac12$ and either $i=1$ or $p_i-1\ne p_{i-1}$, which means that $X$ contains the fragment $\circ\underset{n}{|}\bullet$. In all other cases ${\sigma}_n(X)=X$. Clearly, what of these possible variants takes place is uniquely determined by the intersection $X':=X\cap[-N,N]$ (recall that, by assumption, $N>|n|$). If ${\sigma}_n(X)=X$ then the ratio simply equals 1. If ${\sigma}_n$ transforms $p_i$ to $p_i\pm1$ then, as directly follows from , the ratio equals $$\xi^{\pm1}\, \left(\dfrac{(z+p_i\pm\tfrac12)(z'+p_i\pm\tfrac12)}{(p_i\pm\tfrac12)^2}\right)^{\pm1} \dfrac{\prod\limits_{j:\, j\ne i}\left(\dfrac{p_j-p_i\mp1}{p_j-p_i}\right)^2} {\prod\limits_j\left(\dfrac{q_j+p_i\pm1}{q_j+p_i}\right)^2}\,.$$ This has the desired form with $k=\pm1$ and $$1+f^{{\operatorname{\,out}}}(x)=\begin{cases} \left(1\mp\dfrac1{x-p_i}\right)^2, & x>N\\ \left(1\pm\dfrac1{|x|+p_i}\right)^{-2}, & x<-N. \end{cases}$$ [*Step*]{} 6. In the case $n=-1,-2,\dots$ one can repeat the argument of step 5. Alternatively, one can use the symmetry $p_i\leftrightarrow q_i$ (Subsection \[1-7\]). [*Step*]{} 7. Finally, consider the case $n=0$. Then either ${\widetilde{\sigma}}_0(X)=X$ or ${\widetilde{\sigma}}_0(X)$ differs from $X$ by adding or removing the couple of coordinates $p_1=\frac12$, $q_1=\frac12$. Therefore, ratio either equals 1 or has the form $$(zz')^{\pm1}\xi^{\pm1}\left(\prod_j\dfrac{(p_j-\tfrac12)(q_j-\tfrac12)} {(p_j+\tfrac12)(q_j+\tfrac12)}\right)^{\pm2}.$$ This has the desired form with $k$ equal to $0$ or $\pm1$ and $f^{{\operatorname{\,out}}}$ equal to 0 or $$1+f^{{\operatorname{\,out}}}(x)=\left(1-\frac1{2|x|}\right)^{\pm2}\left(1+\frac1{2|x|}\right)^{\mp2}.$$ In the discussion below we use the notions introduced in Subsection \[2-2\]. \[3.2\] Take the function $r(x)=|x|$ on ${\mathbb Z}'$. The corresponding subset ${\operatorname{Conf}}_r({\mathbb Z}')\subset{\operatorname{Conf}}({\mathbb Z}')$ of $r$–sparse configurations will be denoted as ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$. We equip ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ with the $\ell^1$–topology. By virtue of Proposition \[2.3\], any function of the form is well defined and continuous on ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$. Thus, for any ${\sigma}\in\mathfrak S$, the function $\mu_{z,z',\xi}({\sigma},X)$, initially defined on finite balanced configurations $X$, admits a continuous extension to the larger set ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$. Although the presentation is not unique, the result of the continuous extension provided by formula does not depend on the specific presentation. Indeed, this follows from the fact that the finite balanced configurations form a dense subset in ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ (which is readily checked). \[3.3\] For any ${\sigma}\in\mathfrak S$, let $\mu_{z,z'}({\sigma},X)$ stand for the function on ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ obtained by specializing $\xi=1$ in the right–hand side of formula . Obviously, $\mu_{z,z'}({\sigma},X)$ coincides with the pointwise limit, as $\xi\to1$, of the continuous extensions of the functions $\mu_{z,z',\xi}({\sigma},X)$. This shows that $\mu_{z,z'}({\sigma},X)$ does not depend on a specific presentation . Moreover, Proposition \[2.3\] ensures that the limit function is continuous in the $\ell^1$–topology. Thus, we have shown that Proposition \[3.1\] implies Claim \[C2\] (Subsection \[0-4\]) in a refined form. Main result: Formulation and beginning of proof {#4} =============================================== The following theorem is the main result of the paper. \[4.1\] [(i)]{} The measures $P_{z,z'}$ are quasiinvariant with respect to the modified action of the group $\mathfrak S$ on probability measures on the space ${\operatorname{Conf}}({\mathbb Z}')$, as defined in Section \[3\]. [(ii)]{} For any permutation ${\sigma}\in\mathfrak S$, the Radon–Nikodým derivative ${\widetilde{\sigma}}(P_{z,z'})/P_{z,z'}$ coincides with the limit expression $\mu_{z,z'}({\sigma},X)$ introduced in Definition \[3.3\], within a $P_{z,z'}$–null set. Recall that each of the measures $P_{z,z'}$ is concentrated on the Borel subset ${\operatorname{Conf}}_{{\text{\rm sparse}}}({\mathbb Z}')$ (see Example \[2.4\]) and each of the functions $\mu_{z,z'}({\sigma},X)$ is well defined on the same subset and is a Borel function. In this section, we will reduce Theorem \[4.1\] to Theorem \[4.3\] through an intermediate claim, Theorem \[4.2\], which is of independent interest. The proof of Theorem \[4.3\] occupies Sections \[5\] and \[6\]. As before, we use the angular brackets to denote the pairing between functions and measures. \[4.2\] Let $f(x)$ be an arbitrary function on ${\mathbb Z}'$ such that $f(x)=O(|x|^{-1})$ as $x\to\pm\infty$. Then the multiplicative functional $\Phi_f$ is absolutely integrable with respect to the measures $P_{z,z',\xi}$ and $P_{z,z'}$ and $$\lim_{\xi\to1}\langle\Phi_f,\, P_{z,z',\xi}\rangle=\langle\Phi_f,\, P_{z,z'}\rangle.$$ Recall that the function $\mu_{z,z'}({\sigma},X)$ is a finite linear combination of the multiplicative functionals $\Phi_f$ with $f(x)=O(|x|^{-1})$, see Definition \[3.3\]. Theorem \[4.2\] says that such functionals are absolutely integrable with respect to $P_{z,z'}$, which implies that so is $\mu_{z,z'}({\sigma},X)$. Thus, $\mu_{z,z'}({\sigma},\,\cdot\,)P_{z,z'}$ is a finite Borel measure. The claim of Theorem \[4.1\] is equivalent to the following one: For any ${\sigma}\in\mathfrak S$, $${\widetilde{\sigma}}(P_{z,z'})=\mu_{z,z'}({\sigma},\,\cdot\,)P_{z,z'}.$$ Recall that the cylinder functions on ${\operatorname{Conf}}({\mathbb Z}')$ are dense in $C({\operatorname{Conf}}({\mathbb Z}'))$, see Subsection \[2-1\]. Therefore, it suffices to prove that for any cylinder function $F$ $$\langle F, \,{\widetilde{\sigma}}(P_{z,z'})\rangle=\langle \mu_{z,z'}({\sigma},\,\cdot\,)F,\, P_{z,z'}\rangle.$$ Set $$F^{{\sigma}}(X)=F({\widetilde{\sigma}}(X))$$ and observe that $F^{\sigma}$ is a cylinder function, too. We may rewrite the desired equality in the form $$\label{f4.1} \langle F^{\sigma},\, P_{z,z'}\rangle=\langle \mu_{z,z'}({\sigma},\,\cdot\,)F,\, P_{z,z'}\rangle.$$ By the very definition of $\mu_{z,z',\xi}$, we have $${\widetilde{\sigma}}(P_{z,z',\xi})=\mu_{z,z',\xi}({\sigma},\,\cdot\,)P_{z,z',\xi},$$ so that $$\label{f4.2} \langle F^{\sigma},\, P_{z,z',\xi}\rangle=\langle \mu_{z,z',\xi}({\sigma},\,\cdot\,)F,\, P_{z,z',\xi}\rangle.$$ A natural idea is to derive from by passing to the limit as $\xi$ goes to 1. We know that the measures $P_{z,z',\xi}$ weakly converge to the measure $P_{z,z'}$ (Theorem \[1.1\] above). Therefore, the left–hand side of converges to the left–hand side of . Consequently, to establish it remains to prove that the similar limit relation holds for the right–hand sides, namely $$\label{f4.3} \lim_{\xi\to1}\langle \mu_{z,z',\xi}({\sigma},\,\cdot\,)F,\, P_{z,z',\xi}\rangle = \langle \mu_{z,z'}({\sigma},\,\cdot\,)F,\, P_{z,z'}\rangle.$$ According to the definition of the function $\mu_{z,z'}({\sigma},X)$ (see Theorem \[3.1\] and Definition \[3.3\]), the limit relation can be reduced to the following one: $$\label{f4.4} \lim_{\xi\to1}\langle \xi^k\Phi_f F,\, P_{z,z',\xi}\rangle =\langle \Phi_f F,\, P_{z,z'}\rangle,$$ where $k\in{\mathbb Z}$, $F$ is a cylinder function, and $f(x)=O(|x|^{-1})$. Obviously, the factor $\xi^k$, which tends to $1$, is inessential and can be neglected, so that can be simplified: $$\label{f4.5} \lim_{\xi\to1}\langle \Phi_f F,\, P_{z,z',\xi}\rangle =\langle \Phi_f F,\, P_{z,z'}\rangle.$$ Next, fix a finite subset $Y\subset{\mathfrak X}$, so large that $F(X)$ depends on the intersection $X\cap Y$ only. It is readily verified that $F$ can be written as a finite linear combination of multiplicative functionals $\Phi_{g_i}$, where each $g_i$ vanishes outside $Y$ (this claim actually concerns functions on the finite set $\{0,1\}^Y$). Observe that $$\Phi_f\Phi_{g_i}=\Phi_{f_i}\,, \qquad f_i:=f+g_i+fg_i$$ (we have already used such an equality in the proof of Proposition \[3.1\], step 2). It follows that the product $\Phi_f F$ can be written as a finite linear combination of multiplicative functionals $\Phi_{f_i}$, where each $f_i$ coincides with $f$ outside $Y$ and hence obeys the same decay condition, $f_i(x)=O(|x|^{-1})$. Thus, we have reduced to the claim of Theorem \[4.2\]. The essence of difficulty in proving Theorem \[4.2\] is that, for generic $f$ decaying as $|x|^{-1}$, the multiplicative functional $\Phi_f$ is unbounded and so cannot be extended to a continuous function on the whole space ${\operatorname{Conf}}({\mathbb Z}')$. Thus, for our purpose, the fact of the weak convergence $P_{z,z',\xi}\to P_{z,z'}$, that is, convergence on continuous test functions, is insufficient: we have to enlarge the set of admissible test functions to include the functions like $\Phi_f$. [^4] The idea is to relate the required stronger convergence of the measures to an appropriate convergence of their correlation operators. Set $h(x)=|x|^{-1/2}$, where $x\in{\mathbb Z}'$, and recall that $A_h$ denotes the operator of multiplication by $h$ in the Hilbert space $H=\ell^2({\mathbb Z}')$. Below we use the notions introduced in Subsection \[2-4\]. \[4.3\] [(i)]{} The operator $A_hK_{z,z'}A_h$ lies in ${\mathcal L_{1|2}}(H)$. [(ii)]{} As $\xi$ goes to $1$, the operators $A_hK_{z,z',\xi}A_h$ converge to the operator $A_hK_{z,z'}A_h$ in the topology of the space ${\mathcal L_{1|2}}(H)$. [*Comments.*]{} Note that the pre–limit operators $A_hK_{z,z',\xi}A_h$ also lie in ${\mathcal L_{1|2}}(H)$, because the operators $K_{z,z',\xi}$ are of trace class, see Proposition \[1.8\]. According to the definition of ${\mathcal L_{1|2}}(H)$ (see Subsection \[2-4\]), the claim of the theorem means that the diagonal blocks converge in the trace class norm while the off–diagonal blocks converge in the Hilbert–Schmidt norm. I do not know whether, in the case of the off–diagonal blocks, the Hilbert–Schmidt norm can be replaced by the trace class norm. The assumption on the function $f$ allows one to write it as $f=gh^2$, where $|g(x)|$ is bounded (recall that $h(x)=|x|^{-1/2}$). By virtue of Proposition \[2.7\] and claim (i) of Theorem \[4.3\], $\Phi_f$ is absolutely integrable with respect to the limit measure $P_{z,z'}$, and the same also holds for the pre–limit measures $P_{z,z',\xi}$ (see the comments above). Moreover, $$\langle \Phi_f,\, P_{z,z',\xi}\rangle=\det(1+A_gA_hK_{z,z',\xi}A_h), \qquad \langle \Phi_f,\, P_{z,z'}\rangle=\det(1+A_gA_hK_{z,z'}A_h).$$ Finally, claim (ii) of Theorem \[4.2\] implies that the operators $A_gA_hK_{z,z',\xi}A_h$ converge to the operator $A_gA_hK_{z,z'}A_h$ in the topology of ${\mathcal L_{1|2}}(H)$. Therefore, the corresponding determinants converge, too. Here we use the fact that the function $A\mapsto\det(1+A)$ is continuous on ${\mathcal L_{1|2}}(H)$, see Subsection \[2-4\]. Thus, we have reduced Theorem \[4.2\], and hence Theorem \[4.1\], to Theorem \[4.3\]. The latter theorem is proved separately for the diagonal and off–diagonal blocks in Sections \[5\] and \[6\], respectively. Convergence of diagonal blocks in the topology of the trace class norm {#5} ====================================================================== Here we prove the claims of Theorem \[4.3\] for the diagonal blocks $(\cdot)_{++}$ and $(\cdot)_{--}$ of the operators in question. Due to the symmetry relation of Proposition \[1.9\], the latter block is obtained from the former one by a simple change of the basic parameters, $(z,z')\to(-z,-z')$. Thus, it suffices to focus on the limit behavior of the block $(\cdot)_{++}$. That is, we have to prove that the operator $(A_hK_{z,z'}A_h)_{++}$ in the Hilbert space $\ell^2({\mathbb Z}'_+)$ is of trace class and $$\lim_{\xi\to1}\Vert (A_hK_{z,z',\xi}A_h)_{++}-(A_hK_{z,z'}A_h)_{++}\Vert_1=0,$$ where $\Vert\cdot\Vert_1$ is the trace class norm. Since the proof is long, let us describe its scheme. First of all, observe that, as long as we are dealing with the $++$ block, there is no difference between $K_{z,z',\xi}$ and ${\underline K\,}_{z,z',\xi}$ (and the same for the limit operators). Indeed, this follows from the relation between the both kind of operators, see Subsection \[1-5\]. In Proposition \[5.1\] we rederive the weak convergence ${\underline K\,}_{z,z',\xi}\to {\underline K\,}_{z,z'}$, which implies the weak convergence $(A_hK_{z,z',\xi}A_h)_{++}\to(A_hK_{z,z'}A_h)_{++}$. Recall that ${\underline K\,}_{z,z',\xi}$ is a projection operator (see Subsection \[1-4\]). This implies that its $++$ block is a nonnegative operator, so that $(A_hK_{z,z',\xi}A_h)_{++}$ is also a nonnegative operator. Therefore, the limit operator $(A_hK_{z,z'}A_h)_{++}$ is nonnegative, too. Consequently, to prove that the operator $(A_hK_{z,z'}A_h)_{++}$ is of trace class it suffices to prove that its trace is finite. This is done in Proposition \[5.2\]. The similar fact for the pre–limit operator $(A_hK_{z,z',\xi}A_h)_{++}$ is a trivial consequence of Proposition \[1.8\]. In Proposition \[5.4\] we establish the convergence of traces, $$\lim_{\xi\to1}{\operatorname{tr}}\left((A_hK_{z,z',\xi}A_h)_{++}\right) ={\operatorname{tr}}\left((A_hK_{z,z'}A_h)_{++}\right).$$ This concludes the proof, because, for nonnegative operators, weak convergence together with convergence of traces is equivalent to convergence in the trace class norm (see, e.g., [@BOO Proposition A.9]). Let us proceed to the detailed proof. Our starting point is the double contour integral representation (see below) for the kernel ${\underline K\,}_{z,z',\xi}(x,y)$ on the lattice ${\mathbb Z}'$. Formula is a particular case of a more general formula obtained in [@BO3 §9, p. 148]. [^5] $$\begin{gathered} \label{f5.1} {\underline K\,}_{z,z',\xi}(x;y)\\ =\frac{\Gamma(-z'-x+\frac12)\Gamma(-z-y+\frac12)} {\bigl(\Gamma(-z-x+\frac12)\Gamma(-z'-x+\frac12) \Gamma(-z-y+\frac12)\Gamma(-z'-y+\frac12)\bigr)^{\frac12}} \\ \times\frac{1-\xi}{(2\pi i)^2}\oint\limits_{\{{\omega}_1\}}\oint\limits_{\{{\omega}_2\}} \left(1-\sqrt{\xi}{\omega}_1\right)^{z'+x-\tfrac12} \left(1-\frac{\sqrt{\xi}}{{\omega}_1}\right)^{-z-x-\tfrac12}\\ \times\left(1-\sqrt{\xi}{\omega}_2\right)^{z+y-\tfrac12} \left(1-\frac{\sqrt{\xi}}{{\omega}_2}\right)^{-z'-y-\tfrac12} \frac{{\omega}_1^{-x-\frac 12}{\omega}_2^{-y-\frac 12}}{\omega_1\omega_2-1} \,{d{\omega}_1}{d{\omega}_2}.\end{gathered}$$ Let us explain the notation. Here $\{{\omega}_1\}$ and $\{{\omega}_2\}$ are arbitrary simple, positively oriented loops in ${\mathbb C}$ with the following properties: $\bullet$ Each of the contours surrounds the finite interval $[0,\sqrt\xi]$ and leaves outside the semi–infinite interval $[1/\sqrt\xi,+\infty)\subset{\mathbb R}$. $\bullet$ On the direct product of the contours, ${\omega}_1{\omega}_2\ne1$, so that the denominator ${\omega}_1{\omega}_2-1$ in does not vanish. The simplest contours satisfying these conditions are the circles centered at 0, with radii slightly greater than 1. However, to pass to the limit as $\xi\to1$, we will deform these contours to a more sophisticated form, as explained below. To make the integrand meaningful, we have to specify the branches of the power functions entering , and this is done in the following way. For the terms $(1-\sqrt{\xi}{\omega})^{\alpha}$ (where ${\omega}$ stands for ${\omega}_1$ or ${\omega}_2$ and ${\alpha}$ equals $z'+x-\frac12$ or $z+y-\frac12$), we use the fact that $\{{\omega}\}$ is contained in the simply connected region ${\mathbb C}\setminus[1/\sqrt\xi,+\infty)$ and specify the branch by setting $\arg(1-\sqrt{\xi}{\omega})=0$ for real negative values of ${\omega}$. Likewise, the terms ${\omega}\mapsto (1-\sqrt{\xi}/{\omega})^{\alpha}$ are well defined in the simply connected region $({\mathbb C}\cup\{\infty\})\setminus[0,\sqrt\xi]$, with the convention that $\arg(1-\sqrt{\xi}/{\omega})=0$ for real ${\omega}$ greater than $\sqrt\xi$. Finally, note that the ${\Gamma}$–factors in the numerator are not singular because their arguments are not integers. Indeed, parameters $z$ and $z'$ are forbidden to take integral values while $x-\frac12$ and $y-\frac12$ are integers. As for the ${\Gamma}$–factors in the denominator, their product is strictly positive, again by virtue of the basic conditions on the parameters. Thus, we may and do assume that the square root extracted from this product is positive, too. To perform the limit transition as $\xi\to1$ we will need a special contour $C(R,r,\xi)$ in the complex ${\omega}$–plane. This contour depends on the parameters $R>0$, $r>0$, and $\xi$, and looks as follows (see Fig. 1): Figure 1. The contour $C(R,r,\xi)$: $A=Re^{i\theta}$, $B=1/\sqrt\xi$, $C=\sqrt\xi$. Here we assume that the parameter $r>0$ is small enough while the parameter $R>0$ is big enough. The integration path starts at the point $A=R e^{i\theta}$ with small $\theta>0$ such that $\Im{\omega}=R\sin\theta=r$, first goes along the circle $|\omega|=R$ in the positive direction till the point $Re^{-i\theta}$, then goes in parallel to the real line until the point $1/\sqrt{\xi}-ir$, further goes to the point $1/\sqrt{\xi}+ir$ along the left semicircle $|{\omega}-1/\sqrt{\xi}|=r$ (so that 0 and $\sqrt\xi$ are left on the left), and finally returns to the initial point $Re^{i\theta}$ in parallel to the real line. [^6] Note that $|{\omega}|>1$ along the whole contour provided that $r$ is so small that $1/\sqrt\xi-r>1$. This condition also implies that $\sqrt\xi$ lies inside the contour, as required. Next, consider the following contour in the complex $u$–plane (see Fig. 2). Here $\rho>0$ is the parameter, the integration path starts at infinity, goes towards 0 in the right half–plane, along the line $\Im u=-\rho$, then turns around 0 in the negative direction along the semicircle $|u|=\rho$, and finally returns to infinity in the right half–plane, along the line $\Im u=\rho$. Let us denote this contour as $[+\infty-i\rho,0-,+\infty+i\rho]$. Figure 2. The contour $[+\infty-i\rho,0-,+\infty+i\rho]$. \[5.1\] Assume $x,y\in{\mathbb Z}'$ are fixed. There exists the limit $$\lim_{\xi\to1}{\underline K\,}_{z,z',\xi}(x,y)={\underline K\,}_{z,z'}(x,y)$$ with $$\begin{gathered} \label{f5.2} {\underline K\,}_{z,z'}(x,y)= \frac{{\Gamma}(-z'-x+\frac12){\Gamma}(-z-y+\frac12)} {\bigl({\Gamma}(-z-x+\frac12){\Gamma}(-z'-x+\frac12) {\Gamma}(-z-y+\frac12){\Gamma}(-z'-y+\frac12)\bigr)^{\frac12}}\\ \times\frac1{(2\pi i)^2}\oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}} \frac{(-u_1)^{z'+x-\frac12}(1+u_1)^{-z-x-\frac12} (-u_2)^{z+y-\frac12}(1+u_2)^{-z'-y-\frac12}\,du_1 du_2}{u_1+u_2+1},\end{gathered}$$ where both contours $\{u_{1,2}\}$ are of the form $[+\infty-i\rho,0-,+\infty+i\rho]$ as defined above, with $\rho<\frac12$. [*Comments*]{}. To give a sense to the function $(-u)^{\alpha}$ with ${\alpha}\in{\mathbb C}$ we cut the complex $u$–plane along $[0,+\infty)$ and agree that the argument of $-u$ equals 0 when $u$ intersects the negative real axis $(0,-\infty)$. This is equivalent to say that the argument of $-u$ equals $+\pi$ just below the cut and $-\pi$ just above the cut. Thus, one could remove the minus sign from $(-u_1)^{z'+x-\frac12}$ and $(-u_2)^{z+y-\frac12}$ and put instead in front of the integral the extra factor $e^{i\pi(z'+z+x+y-1)}$, with the understanding that the argument of $u$ is equal to 0 just below the cut and to $-2\pi$ just above it. We also assume that the branch of $(1+u)^C$ (where $u$ is $u_1$ or $u_2$ and $C$ equals $-z-x-\frac12$ or $z+y-\frac12$) is defined with the understanding that the argument of $1+u$ is in $(-\pi/2,\pi/2)$. The existence of the limit kernel was first established in [@BO2]. In that paper we worked with the integrable form of the kernels ${\underline K\,}_{z,z',\xi}(x,y)$ and ${\underline K\,}_{z,z'}(x,y)$, as written down above in Subsection \[1-4\]. In the present form, the claim of the proposition is a particular case of a more general result obtained in [@BO3 §9]. I will reproduce, with minor variations and with more details, the argument of [@BO3] because all its steps will be employed in the sequel. [^7] [*Step*]{} 1. Let us check that the double integral in is absolutely convergent: $$\label{f5.3} \oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}} \left|\frac{(-u_1)^{z'+x-\frac12}(1+u_1)^{-z-x-\frac12} (-u_2)^{z+y-\frac12}(1+u_2)^{-z'-y-\frac12}\,du_1 du_2}{u_1+u_2+1}\right|<+\infty$$ First of all, the restriction $\rho<\frac12$ guarantees that the denominator $u_1+u_2+1$ remains separated from 0 as $u_1$ and $u_2$ range over the contours. Set $\mu=\Re(z'-z)$. For the principal series $\mu=0$, and for the complementary series $-1<\mu<1$. Note that the modulus of the integrand is bounded from above, so that we have to check the convergence only in the case when at least one of the variables goes to infinity. Fix a constant $C>0$ large enough. In the region where $|u_1|\le C$ and $|u_2|\le C$, as was already pointed out, there is no problem of convergence. Assume $|u_2|\le C$ while $|u_1|\ge C$. Then we may exclude $u_2$. That is, we replace the quantity $\left|(-u_2)^{z+y-\frac12}(1+u_2)^{-z'-y-\frac12}\right|$ by an appropriate constant and also use the bound $$\frac1{|u_1+u_2+1|}\le{\operatorname{const}}\frac1{|u_1|}.$$ This allows one to discard integration over $u_2$. Further, for large $|u_1|$, on our contour, the arguments of $|u_1|$ and $|1+u_1|$ are small, which makes it possible to replace both $u_1$ and $1+u_1$ by the real variable $u=\Re u_1$. Then we are lead to the one–dimensional integral $$\int_C^{+\infty}u^{\mu-2}du$$ whose convergence is obvious because $\mu<1$. Interchanging $u_1\leftrightarrow u_2$ gives the same effect, due to the symmetry $z\leftrightarrow z'$. Assume now that both variables are large, $|u_1|\ge C$ and $|u_2|\ge C$. Then the same argument as above leads to the real integral $$\iint\limits_{u_1\ge C, u_2\ge C}\frac{u_1^{\mu-1}u_2^{-\mu-1}}{u_1+u_2}du_1du_2.$$ To handle it we use the bound $$\frac1{u_1+u_2}\le\frac{1}{u_1^\nu u_2^{1-\nu}}\,,$$ which holds for any $\nu\in(0,1)$. Let us choose $\nu=\frac12+\frac12\mu$; then $1-\nu=\frac12-\frac12\mu$. Since $\mu\in(-1,1)$, the requirement $\nu\in(0,1)$ is satisfied. This bound reduces our double integral to the product of two simple integrals of the form $$\int_{u\ge C}u^{\pm\frac12\mu-\frac32}du,$$ which are convergent. [*Step*]{} 2. Let us turn to the kernel . Consider the contour $\{{\omega}\}=C(R, r,\xi)$ where $r=r(\xi)=(1-\xi)\rho$ with $\rho<\frac12$, as above, and $R>0$ is large enough and fixed. It is readily verified that $1/\sqrt\xi-r(\xi)>1$. As mentioned above, this inequality guarantees that $|{\omega}|>1$ on the whole contour, so that the both contours in can be deformed to the form $C(R,(1-\xi)\rho,\xi)$ without changing the value of the double integral. Let us split each contour on two parts, the big arc on the circle $|{\omega}|=R$, which we will denote as $C^-(R, (1-\xi)\rho,\xi)$, and the rest (inside the circle), denoted as $C^+(R, (1-\xi)\rho,\xi)$. The ${\Gamma}$–factors in and in are the same, so that we may ignore them. On the contrary, the prefactor $1-\xi$ in will play the key role. Our plan for the remainder of the proof is as follows: First, we show that when we restrict the double integral in (together with the prefactor $1-\xi$) on the product of two copies of $C^+(R, (1-\xi)\rho,\xi)$ and pass to the limit as $\xi\to1$, we get the integral in . Next, we check that the contribution from the rest of the double integral is asymptotically negligible. [*Step*]{} 3. Here we assume that both ${\omega}_1$ and ${\omega}_2$ range over the contour $C^+(R, (1-\xi)\rho,\xi)$. Make the change of variables ${\omega}_1\to u_1$, ${\omega}_2\to u_2$ according to the relation $$\label{f5.4} {\omega}={\omega}_\xi(u)=\frac1{\sqrt\xi}+(1-\xi)u.$$ After this transformation, the contour $C^+(R, (1-\xi)\rho,\xi)$ turns into a truncation of the contour $[+\infty-i\rho,0-,+\infty+i\rho]$ (we have to impose the constraint $\Re u\le (1-\xi)^{-1}{\widetilde}R$, where ${\widetilde}R=R-(1/\sqrt\xi)\approx R-1$). As $\xi$ goes to 1, the threshold of the truncation shifts to the right, and in the limit we get the whole contour $[+\infty-i\rho,0-,+\infty+i\rho]$. Substituting $$\begin{gathered} 1-\sqrt\xi{\omega}=-(1-\xi)\sqrt\xi\cdot u\\ {\omega}-\sqrt\xi=(1-\xi)\left(u+\frac1{\sqrt\xi}\right)\\ {\omega}_1{\omega}_2-1=\frac{1-\xi}\xi\,\left(\sqrt\xi(u_1+u_2)+1+\xi(1-\xi)u_1u_2\right)\\ d{\omega}_1 d{\omega}_2=(1-\xi)^2du_1du_2\end{gathered}$$ into the integrand of , we see that all the terms consisting of various powers of $1-\xi$, including the prefactor $1-\xi$, cancel out. The integrand that we get can be written as the product of the integrand of with the following expression depending on $\xi$: $$\label{f5.5} \begin{gathered} \xi^{\frac12(z+z'+x+y-1)}\left(\frac{1+u_1}{\frac1{\sqrt\xi}+u_1}\right)^{z+x+\frac12} \left(\frac{1+u_2}{\frac1{\sqrt\xi}+u_2}\right)^{z'+y+\frac12}\\ \times\frac{u_1+u_2+1}{\sqrt\xi(u_1+u_2)+1+\xi(1-\xi)u_1u_2}\, ({\omega}_\xi(u_1))^z({\omega}_\xi(u_2))^{z'} \end{gathered}$$ As $\xi$ goes to 1, this expression converges to 1 pointwise (note that ${\omega}_\xi(u)\to1$). On the other hand, as $u_1$ and $u_2$ range over the contour $[+\infty-i\rho, 0-,+\infty+i\rho]$, the modulus of this expression remains bounded, uniformly on $\xi$. Indeed, this is obvious for the first four factors. As for the last term, $|({\omega}_\xi(u_1))^z({\omega}_\xi(u_2))^{z'}|$, it is bounded because $|{\omega}_\xi(u)|\le R$ and the argument of ${\omega}_\xi(u)$ is close to 0 for $\xi$ close to 1 (recall that ${\omega}_\xi(u)$ is close to $[1,+\infty)$). Consequently, by virtue of step 1, our integral converges absolutely and uniformly on $\xi$, so that we may pass to the limit under the sign of the integral, which results in the desired integral . [*Step*]{} 4. On this last step, we will check that the contribution from the remaining parts of the contours, together with the prefactor $1-\xi$, is asymptotically negligible. To do this, let us evaluate the modulus of the integrand in . The crucial observation is that the factor ${\omega}_1{\omega}_2-1$ in the denominator remains separated from 0. Indeed, when at least one of the variables ranges over the big arc $C^-(R,(1-\xi)\rho,\xi)$ (which is just our case), we have $|{\omega}_1{\omega}_2|\ge R>1$. Consequently, we may ignore this factor, and then our double integral factorizes into the product of two one–dimensional integrals: one is $$\oint\limits_{\{{\omega}_1\}} \left|\left(1-\sqrt{\xi}{\omega}_1\right)^{z'+x-\tfrac12} \left(1-\frac{\sqrt{\xi}}{{\omega}_1}\right)^{-z-x-\tfrac12} {\omega}_1^{-x-\frac12}\,d{\omega}_1\right|$$ and the other has the same form, only $x$ is replaced by $y$ and $z$ is interchanged with $z'$: $$\oint\limits_{\{{\omega}_2\}} \left|\left(1-\sqrt{\xi}{\omega}_2\right)^{z+y-\tfrac12} \left(1-\frac{\sqrt{\xi}}{{\omega}_2}\right)^{-z'-y-\tfrac12} {\omega}_2^{-y-\frac12}\,d{\omega}_2\right|.$$ If one of the variables ${\omega}_1, {\omega}_2$ ranges over the big arc $C^-(R,(1-\xi)\rho,\xi)$, then the integrand of the corresponding integral is bounded uniformly on $\xi$, so that this integral remains bounded. Thus, the case when both ${\omega}_1$ and ${\omega}_2$ range over the big arc is trivial: the prefactor $1-\xi$ forces the whole expression to go to 0. Examine now the case when one of the variables (say, ${\omega}_1$) ranges over $C^+(R,(1-\xi)\rho,\xi)$, while the other variable (hence, ${\omega}_2$) ranges over the big arc. Then we are left with the first integral together with the prefactor $1-\xi$. Let us make the same change of a variable as above: ${\omega}_1={\omega}_\xi(u)$. Then the integral reduces to $$(1-\xi)^\mu(\sqrt\xi)^{z'+x-\frac12}\oint\limits_{\{u\}} \left|(-u)^{z'+x-\frac12}\left(u+\frac1{\sqrt\xi}\right)^{-z-x-\frac12} ({\omega}_\xi(u))^zdu\right|.$$ The quantity $|({\omega}_\xi(u))^z|$ is bounded uniformly on $\xi$ and hence may be ignored. The factor $(\sqrt\xi)^{z'+x-\frac12}$ is inessential, too. Further, using the same argument as on step 1, we reduce our expression to the real integral $$(1-\xi)^\mu\int_{C}^{C_1/(1-\xi)}u^{\mu-1}du$$ where the upper limit arises due to the fact that $\Re({\omega}_\xi(u))\le R$. Recall that $\mu=\Re(z'-z)\in(-1,1)$. If $\mu\in(-1,0)$ then the integral is uniformly convergent, so that the whole expression grows as $(1-\xi)^\mu$. If $\mu=0$ then the integral grows as $\log\left((1-\xi)^{-1}\right)$, and so is the growth of the whole expression. If $\mu\in(0,1)$ then the integral grows as $(1-\xi)^{-\mu}$ and the whole expression remains bounded. In all these cases the small prefactor $(1-\xi)$ in dominates and makes the result asymptotically negligible. \[5.2\] The matrix $$\left[(m+\tfrac12)^{-1/2} (K_{z,z'})_{++}(m+\tfrac12,n+\tfrac12)(n+\tfrac12)^{-1/2}\right]_{m,n=0,1,2,\dots}$$ is of trace class. We have to prove that $$\sum_{m=0}^\infty (m+\tfrac12)^{-1}(K_{z,z'})_{++}(m+\tfrac12,m+\tfrac12)<+\infty.$$ Since the $++$ blocks of $K_{z,z'}$ and ${\underline K\,}_{z,z'}$ are the same, we may use formula . On the diagonal $x=y=m+\frac12$, the ratio in formed by the ${\Gamma}$–factors equals 1, so that we may omit them. Therefore, $$\begin{gathered} (K_{z,z'})_{++}(m+\tfrac12,m+\tfrac12)\\=\frac1{(2\pi i)^2}\oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}}\frac{(-u_1)^{z'+m}(1+u_1)^{-z-m-1} (-u_2)^{z+m}(1+u_2)^{-z'-m-1}\,du_1 du_2}{u_1+u_2+1}\end{gathered}$$ where the contours are the same as in . We have $$\begin{gathered} (K_{z,z'})_{++}(m+\tfrac12,m+\tfrac12)\\ \le \frac1{4\pi^2}\oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}} \left| \frac{(-u_1)^{z'+m}(1+u_1)^{-z-m-1} (-u_2)^{z+m}(1+u_2)^{-z'-m-1}\,du_1 du_2}{u_1+u_2+1}\right|\\ =\frac1{4\pi^2}\oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}}\left|\frac{u_1u_2}{(1+u_1)(1+u_2)}\right|^m\cdot\left| \frac{(-u_1)^{z'}(1+u_1)^{-z-1}(-u_2)^{z}(1+u_2)^{-z'-1}du_1 du_2}{u_1+u_2+1}\right|\,.\end{gathered}$$ Introduce the factor $(m+\tfrac12)^{-1}$ inside the integral and sum over $m=0,1,\dots$. Observe that $$\gathered \sum_{m=0}^\infty (m+\tfrac12)^{-1}\left|\frac{u_1u_2}{(1+u_1)(1+u_2)}\right|^m\\ \le 2+2\sum_{m=1}^\infty m^{-1}\left|\frac{u_1u_2}{(1+u_1)(1+u_2)}\right|^m\\ =2+2\log\left(\dfrac1{1-\left|\dfrac{u_1u_2}{(1+u_1)(1+u_2)}\right|}\right)=:F(u_1,u_2). \endgathered$$ Thus, we have to prove that $$\oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}}F(u_1,u_2)\cdot\left| \frac{(-u_1)^{z'}(1+u_1)^{-z-1}(-u_2)^{z}(1+u_2)^{-z'-1}du_1 du_2}{u_1+u_2+1}\right|<+\infty.$$ Without the factor $F(u_1,u_2)$, this integral coincides with the integral with $x=y=\frac12$, whose convergence has already been verified (see step 1 in the proof of Proposition \[5.1\]). Let us show that the extra factor $F(u_1,u_2)$ does not add very much. Indeed, it grows only as both $u_1$ and $u_2$ go to infinity, so that we may assume that both $u_1$ and $u_2$ are far from the origin. If $u$ is a point on one of the contours, far from the origin, then writing $u=|u|e^{i\theta}$ we have $|u|$ large and $\theta$ small. Then $$\left|\frac{u}{1+u}\right|^2 =\frac{|u|^2}{|u|^2+2|u|\cos\theta+1}=1-2|u|^{-1}\cos\theta+O(|u|^{-2})$$ and consequently $$\left|\frac{u}{1+u}\right|=1-|u|^{-1}\cos\theta+O(|u|^{-2})$$ with $\cos\theta\to1$ as $u\to\infty$. This allows one to estimate the growth of the logarithm in $F(u_1,u_2)$. Omitting unessential details we get that it behaves roughly as $$\log\left(\frac1{|u_1|^{-1}+|u_2|^{-1}}\right)\le{\operatorname{const}}|u_1|^{\delta}|u_2|^{\delta},$$ where ${\delta}>0$ can be chosen arbitrarily small. Using this bound and examining the argument of step 1 in Proposition \[5.1\] we see that the same arguments work equally well with the extra factor $F(u_1,u_2)$. In the sequel we use the notation $${\varepsilon}=1-\xi.$$ \[5.3\] Fix an arbitrary $c\in(0,\frac12)$. If $R$ is large enough and $\rho<\frac12$ then for all sufficiently small ${\varepsilon}$ the following estimate holds $$\left|\frac{1-\sqrt{\xi}{\omega}}{{\omega}-\sqrt\xi}\right|<1-c{\varepsilon}, \qquad {\omega}\in C(R,{\varepsilon}\rho,\xi).$$ Consider the transform ${\omega}\mapsto {\omega}'=\frac{1-\sqrt{\xi}{\omega}}{{\omega}-\sqrt\xi}$. Its inverse has the form ${\omega}=\frac{1+\sqrt{\xi}{\omega}'}{{\omega}'+\sqrt\xi}$ and sends the interior part of the circle $|{\omega}'|=1-c{\varepsilon}$ to the exterior part $S^{\operatorname{ext}}$ of a circle $S$. Thus, we have to check that $C(R,{\varepsilon}\rho,\xi)\subset S^{\operatorname{ext}}$. The circle $S$ is symmetric relative the real axis and intersects it at the points ${\omega}_\mp$ corresponding to ${\omega}'_\mp=\mp(1-c{\varepsilon})$. Let us prove that both ${\omega}_-$ and ${\omega}_+$ are on the left of $\frac1{\sqrt\xi}-{\varepsilon}\rho$, the leftmost point of the contour $C^+(R,{\varepsilon}\rho,\xi)$. We have $${\omega}_-=\frac{1-\sqrt{\xi}(1-c{\varepsilon})}{-(1-c{\varepsilon})+\sqrt\xi} =\frac{1-(1-\tfrac12{\varepsilon}+O({\varepsilon}^2))(1-c{\varepsilon})}{-(1-c{\varepsilon})+(1-\tfrac12{\varepsilon}+O({\varepsilon}^2))} =\frac{c+\tfrac12}{c-\tfrac12}+O({\varepsilon})$$ and $${\omega}_+=\frac{1+\sqrt{\xi}(1-c{\varepsilon})}{(1-c{\varepsilon})+\sqrt\xi} =1+\frac{(1-\sqrt{\xi})c{\varepsilon}}{(1-c{\varepsilon})+\sqrt\xi} =1+O({\varepsilon}^2).$$ Since $c<\frac12$, ${\omega}_-$ lies on the left of 0. As for ${\omega}_+$, it lies on the left of $$\frac1{\sqrt\xi}-{\varepsilon}\rho\approx 1+(\tfrac12-\rho){\varepsilon},$$ because $\rho<\frac12$ by the assumption. This shows that the interior part of the contour, $C^+(R,{\varepsilon}\rho,\xi)$ lies in $S^{\operatorname{ext}}$. The same also holds for the exterior part, $C^-(R,{\varepsilon}\rho,\xi)$, because $R$ can be made arbitrarily large. Indeed, as seen from the above expressions for the points ${\omega}_-$ and ${\omega}_+$, they do not run to infinity as $\xi\to1$, so that if $R$ is chosen large enough, the circle of radius $R$ will enclose the circle $S$ for any $\xi$. \[5.4\] $$\begin{gathered} \lim_{\xi\to1}\left(\sum_{m\ge0}(m+\tfrac12)^{-1} (K_{z,z',\xi})_{++}(m+\tfrac12,m+\tfrac12)\right)\\ =\sum_{m\ge0}(m+\tfrac12)^{-1} (K_{z,z'})_{++}(m+\tfrac12,m+\tfrac12).\end{gathered}$$ We will argue as in the proof of Proposition \[5.1\], steps 2–4. The role of step 1 in that proof will be played by Proposition \[5.2\]. First of all, observe that on the diagonal $x=y$, the expression formed by the ${\Gamma}$–factors in front of the integrals in and equals 1. Consequently, we may ignore these factors. Let $F_{m;\xi}({\omega}_1,{\omega}_2)$ denote the integrand in the integral corresponding to $x=y=m+\frac12$. Likewise, let $F_m(u_1,u_2)$ denote the integrand in the integral with $x=y=m+\frac12$. We have to prove that $$\begin{gathered} \label{f5.6} \lim_{\xi\to1}\left((1-\xi)\sum_{m=0}(m+\tfrac12)^{-1} \oint\limits_{\{{\omega}_1\}}\oint\limits_{\{{\omega}_2\}}F_{m;\xi}({\omega}_1,{\omega}_2)d{\omega}_1d{\omega}_2\right)\\ =\sum_{m=0}(m+\tfrac12)^{-1} \oint\limits_{\{u_1\}}\oint\limits_{\{u_2\}}F_m(u_1,u_2)du_1du_2\,.\end{gathered}$$ We already dispose of all the necessary information to conclude that holds provided that we truncate the both contours in the left–hand side to $C^+(R,{\varepsilon}\rho,\xi)$. Indeed, the ratio $$\frac{(1-\xi)F_{m;\xi}({\omega}_\xi(u_1),{\omega}_\xi(u_2))}{F_m(u_1,u_2)}$$ is the particular case of corresponding to $x=y=m+\frac12$. The argument of step 3 in Proposition \[5.1\] shows that this expression goes to 1 pointwise. Moreover, its modulus is bounded uniformly on both $\xi$ and $m$: To see this observe that $$\left|\frac{1+u}{\frac1{\sqrt\xi}+u}\right|\le 1, \qquad u=u_1,u_2,$$ and recall that $|{\omega}_\xi(u)|\le R_1$, $u=u_1,u_2$. Together with the result of Proposition \[5.2\] this implies the claim. Now we have to check that the contribution of the remaining parts of the contours to the left–hand side of is asymptotically negligible. This can be done by slightly modifying the argument of step 4 in Proposition \[5.1\]. Indeed, the integrand $F_{m,\xi}({\omega}_1,{\omega}_2)$ can be written in the form $$F_{m;\xi}({\omega}_1,{\omega}_2)=F_{0;\xi}({\omega}_1,{\omega}_2) \left(\frac{1-\sqrt{\xi}{\omega}_1}{{\omega}_1-\sqrt\xi}\right)^m \left(\frac{1-\sqrt{\xi}{\omega}_2}{{\omega}_2-\sqrt\xi}\right)^m .$$ Compare our task with the one we had on step 4 in Proposition \[5.1\] (where we set $x=y=\frac12$). The only difference is that now we have in the integral the extra factor equal to $$\label{f5.7} \sum_{m=0}^\infty(m+\tfrac12)^{-1}\left|\left(\frac{1-\sqrt{\xi}{\omega}_1}{{\omega}_1-\sqrt\xi}\right) \left(\frac{1-\sqrt{\xi}{\omega}_2}{{\omega}_2-\sqrt\xi}\right)\right|^m.$$ By Lemma \[5.3\], the quantity is majorated by $$\sum_{m=0}^\infty(m+\tfrac12)^{-1}(1-c\xi)^{2m},$$ which grows as $\log({\varepsilon}^{-1})$. Such an extra factor does not affect the estimates on step 4 of Proposition \[5.1\]. Convergence of off–diagonal blocks in Hilbert–Schmidt norm {#6} ========================================================== Here we prove the claims of Theorem \[4.3\] for the off–diagonal blocks. As in Section \[5\], we apply the symmetry relations of Proposition \[1.9\] to reduce the case of the $-+$ block to that of the $+-$ block, and due to the relations of Subsection \[1-5\], we may freely switch from $K_{z,z',\xi}$ to ${\underline K\,}_{z,z',\xi}$. However, to compute the Hilbert–Schmidt norm of the $+-$ block we cannot use anymore the basic contour integral representation as we did in Section \[5\], because this would lead to divergent series in the integrand. The reason of this is that the variable $y$, which previously ranged over ${\mathbb Z}'_+$, will now range over ${\mathbb Z}'_-$. It turns out that we still may employ essentially the same estimates and arguments as in Section \[5\] but beforehand we have to change the integral representation . Return to the initial version of the contours $\{{\omega}_1\}$ and $\{{\omega}_2\}$ in , when they were circles slightly greater than the unit circle, and make the change of a variable ${\omega}_2\to{\omega}_2^{-1}$. Then the former condition ${\omega}_1{\omega}_2\ne1$ will turn into the requirement that the second contour must lie inside the first contour. Further, we deform the contours so that they take the form $C(R_1,{\varepsilon}\rho_1,\xi)$ and $C(R_2,{\varepsilon}\rho_2,\xi)$, respectively, where $R_1>R_2$ and $\rho_1<\rho_2$: these inequalities just guarantee that the contour $C(R_2,{\varepsilon}\rho_2,\xi)$ lies inside $C(R_1,{\varepsilon}\rho_1,\xi)$. After the transform ${\omega}_2\to{\omega}_2^{-1}$ the formula turns into $$\label{f6.1} \begin{aligned} {\underline K\,}_{z,z',\xi}&(x;y)\\ &=\frac{\Gamma(-z'-x+\frac12)\Gamma(-z-y+\frac12)} {\bigl(\Gamma(-z-x+\frac12)\Gamma(-z'-x+\frac12) \Gamma(-z-y+\frac12)\Gamma(-z'-y+\frac12)\bigr)^{\frac12}} \\ &\times\frac{1-\xi}{(2\pi i)^2}\oint\limits_{\{{\omega}_1\}}\oint\limits_{\{{\omega}_2\}} \left(1-\sqrt{\xi}{\omega}_1\right)^{z'+x-\tfrac12} \left(1-\frac{\sqrt{\xi}}{{\omega}_1}\right)^{-z-x-\tfrac12}\\ &\times\left(1-\sqrt{\xi}{\omega}_2\right)^{-z'-y-\tfrac12} \left(1-\frac{\sqrt{\xi}}{{\omega}_2}\right)^{z+y-\tfrac12} \frac{{\omega}_1^{-x-\frac 12}{\omega}_2^{y-\frac 12}}{\omega_1-\omega_2} \,{d{\omega}_1}{d{\omega}_2}. \end{aligned}$$ According to Proposition \[5.1\], for any fixed $x,y\in{\mathbb Z}'$, there exists a limit $$\lim_{\xi\to1}{\underline K\,}_{z,z',\xi}(x,y)={\underline K\,}_{z,z'}(x,y),$$ for which we dispose of the double contour integral representation . However, now we will need a different integral representation, which is consistent with : \[6.1\] The following formula holds $$\begin{gathered} \label{f6.2} {\underline K\,}_{z,z'}(x,y)= \frac{{\Gamma}(-z'-x+\frac12){\Gamma}(-z-y+\frac12)} {\bigl({\Gamma}(-z-x+\frac12){\Gamma}(-z'-x+\frac12) {\Gamma}(-z-y+\frac12){\Gamma}(-z'-y+\frac12)\bigr)^{\frac12}}\\ \times\frac1{(2\pi i)^2}\oint\limits_{\{u_1\}} \oint\limits_{\{u_2\}} \frac{(-u_1)^{z'+x-\frac12}(1+u_1)^{-z-x-\frac12} (-u_2)^{-z'-y-\frac12}(1+u_2)^{z+y-\frac12}\,du_1 du_2}{u_1-u_2}.\end{gathered}$$ where the contours have the form $$\{u_1\}=[+\infty-i\rho_1,0-,+\infty+i\rho_1],\qquad \{u_2\}=[+\infty-i\rho_2,0-,+\infty+i\rho_2]$$ and $\rho_1<\rho_2$. The ${\Gamma}$–factors in and are the same, so that it suffices to prove that the integral in together with the prefactor $(1-\xi)$ converges to the integral in . We will follow the scheme of the proof of Proposition \[5.1\]. [*Step*]{} 1. Let us check that that the integral is absolutely convergent. Arguing as on step 1 of Proposition \[5.1\] we reduce this claim to finiteness of the real integral $$\label{f6.3} \iint\limits_{\substack{u_1,u_2\ge C\\ |u_1-u_2|\ge \rho}} \frac{u_1^{\mu-1}u_2^{-\mu-1}} {|u_1-u_2|}du_1 du_2,$$ where $\mu=\Re(z'-z)\in(-1,1)$, $\rho=\rho_2-\rho_1>0$, and $C$ is a positive constant. By symmetry we may assume $u_1\ge u_2$. Making the change of variables $u_1=u+a$, $u_2=u$, where $u\ge C$, $a\ge \rho$, we get the integral $$\begin{gathered} \int_{a\ge \rho}a^{-1}da\int_{u\ge C} (u+a)^{\mu-1}u^{-\mu-1}du\\ =\int_{a\ge \rho}a^{-2}da\int_{u\ge C/a} (u+1)^{\mu-1}u^{-\mu-1}du.\end{gathered}$$ Consider the interior integral in the second line: For large $u$, the integrand decays as $u^{-2}$, which is integrable near infinity, while for $u$ near 0, the integrand behaves as $u^{-\mu-1}$. It follows that if $\mu<0$ then the integral over $u$ converges uniformly on $a$; if $\mu=0$ then it grows as $\log a$ for large $a$; and if $\mu>0$ then it grows as $a^{\mu}$. Consequently, the double integral can be estimated by one of the three convergent integrals $$\int_{a\ge \rho}a^{-2}da, \qquad \int_{a\ge \rho}a^{-2}\log a\, da, \qquad \int_{a\ge \rho}a^{\mu-2}da \quad (0<\mu<1).$$ [*Step*]{} 2. Relying on the result of step 1, we can verify the required convergence of the integrals provided that we restrict the integration in to interior parts of the contours. This is done exactly as on step 3 of Proposition \[5.1\]. A minor simplification is that under the change of a variable , the quantity ${\omega}_1-{\omega}_2$ is transformed simpler than ${\omega}_1{\omega}_2-1$. [*Step*]{} 3. Finally, we have to prove that the contributions from the remaining parts of the contours is asymptotically negligible due to the prefactor $1-\xi$. Here we argue exactly as on step 4 of Proposition \[5.1\], with the only exception: In the case when ${\omega}_1$ ranges over the interior part of the contour, $C^+(R_1,{\varepsilon}\rho_1,\xi)$, while ${\omega}_2$ ranges over the exterior part, $C^-(R_2,{\varepsilon}\rho_2,\xi)$, we cannot automatically discard the denominator ${\omega}_1-{\omega}_2$. The reason is that in this special case, the two parts of the contours come close at the distance of order ${\varepsilon}$, which happens near the point $R_2$. This difficulty can be resolved in the following way. Dissect $C^+(R_1,{\varepsilon}\rho_1,\xi)$ into two parts by a vertical line $\Re {\omega}_1={\operatorname{const}}$ so that the points on the left be separated from $R_2$ while the points on the right be separated from $1/\sqrt\xi$. Now we have two cases: When ${\omega}_1$ ranges over the left part, we may discard the denominator ${\omega}_1-{\omega}_2$ and argue as on step 4 of Proposition \[5.1\]. When ${\omega}_1$ ranges over the right part, the argument is different: Observe that the modulus of the whole integrand in , except the denominator ${\omega}_1-{\omega}_2$, is bounded uniformly on $\xi$, while the quantity $|{\omega}_1-{\omega}_2|^{-1}$ is integrable, so that the prefactor $1-\xi$ makes the contribution negligible. Checking the integrability of $|{\omega}_1-{\omega}_2|^{-1}$ is easy: Indeed, here it is even unessential that $C^-(R_2,{\varepsilon}\rho_2,\xi)$ does not contain a small arc around the point $R_2$. What is important is that the singularity $|{\omega}_1-{\omega}_2|^{-1}$ arising near the point $R_2$ is of the same kind as the singularity $(a^2+b^2)^{-1/2}$ in the real $(a,b)$–plane near the origin. Although formulas and are valid for any $x,y\in{\mathbb Z}'$, we will deal exclusively with $x\in{\mathbb Z}'_+$ and $y\in{\mathbb Z}'_-$. Below we set $$x=m+\tfrac12, \quad y=-n-\tfrac12, \qquad m,n\in{\mathbb Z}_+$$ and use the notation $$\begin{gathered} (K_{z,z',\xi})_{+-}(m,n)=(-1)^n{\underline K\,}_{z,z',\xi}(m+\tfrac12,-n-\tfrac12)\\ (K_{z,z'})_{+-}(m,n)=(-1)^n{\underline K\,}_{z,z'}(m+\tfrac12,-n-\tfrac12).\end{gathered}$$ Here the factor $(-1)^n$ comes from the factor ${\varepsilon}(y)$ in . The next result is similar to Proposition \[5.2\]: \[6.2\] The matrix $$\left[(m+\tfrac12)^{-1/2}(K_{z,z'})_{+-} (m,n)(n+\tfrac12)^{-1/2}\right]_{m,n\in{\mathbb Z}'_+}$$ belongs to the Hilbert–Schmidt class. Since all the matrix entries are real (Remark \[1.7\]), the claim means that the series $$\label{f6.4} \sum_{m,n=0}^\infty(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}((K_{z,z'})_{+-} (m,n))^2$$ is finite. To get the squared matrix entry we multiply out two copies of the double integral representation , where in the second copy we swap $z$ and $z'$. This operation does not change the kernel, as it follows from a series expansion for the kernel (see [@BO4 (3.3)]) and the fact that the functions entering this expansion depend symmetrically on $z$ and $z'$. On the other hand, after this operation the ${\Gamma}$–factors in front of the integral will cancel out (this trick is borrowed from [@BO3] and [@BO4], see, e.g., the proof of Proposition 2.3 in [@BO4]). The resulting expression for the sum can be written in the form $$\begin{gathered} \label{f6.5} \sum_{m,n\in{\mathbb Z}'_+}(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}\\ \times\frac1{16\pi^4}\oint\limits_{\{u_1\}}\oint\limits_{\{u_2\}} \oint\limits_{\{u'_1\}}\oint\limits_{\{u'_2\}} F_{mn}(u_1,u_2)F'_{mn}(u'_1,u'_2) du_1d u_2 du'_1 du'_2,\end{gathered}$$ where $$\begin{gathered} \{u_1\}=\{u'_1\}=[+\infty-i\rho_1,0-,+\infty+i\rho_1], \\ \{u_2\}=\{u'_2\}=[+\infty-i\rho_2,0-,+\infty+i\rho_2]\end{gathered}$$ and $$\begin{gathered} F_{mn}(u_1,u_2)= \frac{(-u_1)^{z'+m}(1+u_1)^{-z-m-1} (-u_2)^{-z'+n}(1+u_2)^{z-n-1}}{u_1-u_2}\,,\\ F'_{mn}(u'_1,u'_2)= \frac{(-u'_1)^{z+m}(1+u'_1)^{-z'-m-1} (-u'_2)^{-z+n}(1+u'_2)^{z'-n-1}}{u'_1-u'_2}\end{gathered}$$ (in the second line $z$ and $z'$ are interchanged). To show that the sum is finite we replace the integrand by its modulus and then interchange summation and integration. Then we get the integral $$\begin{gathered} \label{f6.6} \frac{1}{16\pi^4}\oint\limits_{\{u_1\}}\oint\limits_{\{u_2\}} \oint\limits_{\{u'_1\}}\oint\limits_{\{u'_2\}} \sum_{m,n\in{\mathbb Z}'_+}(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}\\ \times\left|F_{mn}(u_1,u_2)F'_{mn}(u'_1,u'_2) du_1d u_2 du'_1 du'_2\right|.\end{gathered}$$ It suffices to check that it is finite. We have $$\begin{gathered} \sum_{m,n=0}^\infty(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}\left|F_{mn}(u_1,u_2)F'_{mn}(u'_1,u'_2)\right| =\left|F_{00}(u_1,u_2)F'_{00}(u'_1,u'_2)\right|\\ \times\sum_{m,n=0}^\infty(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}\left|\frac{u_1u'_1}{u_1+u'_1}\right|^m \left|\frac{u_2u'_2}{u_2+u'_2}\right|^n.\end{gathered}$$ The same argument as in the proof of Proposition \[5.2\] shows that for the latter sum there exists the upper bound of the form $$\sum_{m,n=0}^\infty(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}\left|\frac{u_1u'_1}{u_1+u'_1}\right|^m \left|\frac{u_2u'_2}{u_2+u'_2}\right|^n \le{\operatorname{const}}|u_1|^{\delta}|u'_1|^{\delta}|u_2|^{\delta}|u'_2|^{\delta},$$ where ${\delta}>0$ can be chosen as small as is needed. Substituting this estimate into the 4–fold integral leads to its splitting into the product of two double integrals, one of which is $$\label{f6.7} \oint\limits_{\{u_1\}}\oint\limits_{\{u_2\}} |u_1|^{\delta}|u_2|^{\delta}\left|\frac{(-u_1)^{z'}(1+u_1)^{-z-1} (-u_2)^{-z'}(1+u_2)^{z-1}}{u_1-u_2} du_1d u_2 \right|$$ and the other has the similar form, with $z$ and $z'$ interchanged. Therefore, it suffices to prove the finiteness of the integral . Arguing as on step 1 of Proposition \[5.1\] we reduce this integral to the real integral $$\label{f6.8} \iint\limits_{\substack{u_1,u_2\ge C\\ |u_1-u_2|\ge \rho}} \frac{u_1^{\mu+{\delta}-1}u_2^{-\mu+{\delta}-1}} {|u_1-u_2|}du_1 du_2.$$ This integral only slightly differs from the integral examined on step 1 of Proposition \[6.1\], and the same argument as in Proposition \[6.1\] shows that is finite provided that ${\delta}$ is chosen small enough. \[6.3\] As $\xi$ goes to $1$, the matrices $$\left[(m+\tfrac12)^{-1/2}(K_{z,z',\xi})_{+-} (m+\tfrac12,-n-\tfrac12)(n+\tfrac12)^{-1/2}\right]_{m,n\in{\mathbb Z}'_+}$$ converge to the matrix $$\left[(m+\tfrac12)^{-1/2}(K_{z,z'})_{+-} (m+\tfrac12,-n-\tfrac12)(n+\tfrac12)^{-1/2}\right]_{m,n\in{\mathbb Z}'_+}$$ in the topology of the Hilbert–Schmidt norm. We know that the convergence takes place in the weak operator topology, that is, for the matrix entries (this follows from Proposition \[5.1\]). Consequently, it suffices to prove the convergence of the squared Hilbert–Schmidt norms: $$\begin{gathered} \sum_{m,n=0}^\infty(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}((K_{z,z',\xi})_{+-} (m,n))^2\\ \to \sum_{m,n=0}^\infty(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}((K_{z,z'})_{+-} (m,n))^2\end{gathered}$$ (recall that our matrices are real). We will follow the arguments of Propositions \[5.4\], \[6.1\], and \[5.1\]. Write the left–hand side in the form similar to . Namely, let $F_{mn;\xi}({\omega}_1,{\omega}_2)$ denote the integrand in , where we substitute $x=m+\frac12$ and $y=-n-\frac12$: $$\begin{gathered} F_{mn;\xi}({\omega}_1,{\omega}_2)=\left(1-\sqrt{\xi}{\omega}_1\right)^{z'+m} \left(1-\frac{\sqrt{\xi}}{{\omega}_1}\right)^{-z-m-1}\\ \times\left(1-\sqrt{\xi}{\omega}_2\right)^{-z'+n} \left(1-\frac{\sqrt{\xi}}{{\omega}_2}\right)^{z-n-1} \frac{{\omega}_1^{-m-1}{\omega}_2^{-n-1}}{\omega_1-\omega_2}\,,\end{gathered}$$ and let $F'_{mn;\xi}({\omega}'_1,{\omega}'_2)$ be the similar quantity with $z$ and $z'$ interchanged: $$\begin{gathered} F'_{mn;\xi}({\omega}'_1,{\omega}'_2)=\left(1-\sqrt{\xi}{\omega}_1\right)^{z+m} \left(1-\frac{\sqrt{\xi}}{{\omega}_1}\right)^{-z'-m-1}\\ \times\left(1-\sqrt{\xi}{\omega}_2\right)^{-z+n} \left(1-\frac{\sqrt{\xi}}{{\omega}_2}\right)^{z'-n-1} \frac{({\omega}'_1)^{-m-1}({\omega}'_2)^{-n-1}}{\omega'_1-\omega'_2}\,.\end{gathered}$$ As above, swapping $z$ and $z'$ kills the ${\Gamma}$–factors. In this notation, the series in question takes the form $$\begin{gathered} \label{f6.9} \frac{1}{16\pi^4}\sum_{m,n\in{\mathbb Z}'_+}(m+\tfrac12)^{-1}(n+\tfrac12)^{-1}\\ \times\oint\limits_{\{{\omega}_1\}}\oint\limits_{\{{\omega}_2\}} \oint\limits_{\{{\omega}'_1\}}\oint\limits_{\{{\omega}'_2\}} (1-\xi)^2 F_{mn;\xi}({\omega}_1,{\omega}_2)F'_{mn;\xi}({\omega}'_1,{\omega}'_2) d{\omega}_1d{\omega}_2d{\omega}'_1d{\omega}'_2\,,\end{gathered}$$ where $$\{{\omega}_1\}=\{{\omega}'_1\}=C(R_1,{\varepsilon}\rho_1,\xi), \qquad \{{\omega}_2\}=\{{\omega}'_2\}=C(R_2,{\varepsilon}\rho_2,\xi),$$ and $R_1>R_2$, $r_1<r_2$, ${\varepsilon}=1-\xi$, as before. First, we truncate all the contours in keeping only their interior parts $C^+(R_i,{\varepsilon}\rho_i,\xi)$, and then make the change of variables according to the rule , as we did before. The prefactor $(1-\xi)^2$ disappears, we compare the resulting integrand with that in , and check that their ratio has uniformly bounded modulus and converges pointwise to 1. This is done exactly as in step 3 of the proof of Proposition \[5.4\]. By virtue of Proposition \[6.3\], we get the desired convergence provided that the both contours in are truncated. Next, we check that the contribution from the remaining parts of the contours in is asymptotically negligible due to the prefactor $(1-\xi)^2$. Again, the proof goes as in the situation of Proposition \[5.4\]. We replace the integrand by its modulus, interchange summation and integration, and evaluate the double sum over $m$ and $n$ using Lemma \[5.3\]. This produces a factor growing like $(\log({\varepsilon}^{-1}))^2$, which we add to our small prefactor $(1-\xi)^2$. Due to this bound, the 4–fold integral reduces to the product of two double integrals, $$\oint\limits_{\{{\omega}_1\}}\oint\limits_{\{{\omega}_2\}} |F_{00;\xi}({\omega}_1,{\omega}_2)d{\omega}_1d{\omega}_2| \qquad \text{\rm and} \qquad \oint\limits_{\{{\omega}'_1\}}\oint\limits_{\{{\omega}'_2\}} |F'_{00;\xi}({\omega}'_1,{\omega}'_2)d{\omega}'_1d{\omega}'_2|,$$ where each of the 4 variables ranges over $C^\pm(R_i,{\varepsilon}\rho_i,\xi)$, and for at least one of them the corresponding superscript has to be “$-$”. These integrals were already examined in the proof of Proposition \[6.1\]. For each of the integrals, there are two possible cases: Either (a) both variables range over interior parts of the contours or (b) one of the variables ranges over the interior part while the other variable ranges over the exterior part. We know that in case (a) the integral grows as $(1-\xi)^{-1}$, while in case (b) the growth is suppressed by the small factor $(1-\xi)$, even if one adds the extra growing factor $(\log({\varepsilon}^{-1}))^2$. Since case (b) occurs for at least one of the double integrals, we see that the small prefactor $(1-\xi)^2$ suffices to make the whole contribution negligible. This completes the proof. [IIKS]{} S. Albeverio, Yu. G. Kondratiev, M. Röckner, [*Analysis and geometry on configuration spaces*]{}. J. Funct. Anal. [**154**]{} (1998), 444–500. A. Borodin, [*Determinantal point processes*]{}. In: The Oxford Handbook of Random Matrix Theory, to appear; arXiv:0911.1153. A. Borodin, A. Okounkov and G. Olshanski, [*Asymptotics of Plancherel measures for symmetric groups*]{}. J. Amer. Math. Soc. [**13**]{} (2000), 481–515; arXiv: math/9905032. A. Borodin and G. Olshanski, [*Distributions on partitions, point processes and the hypergeometric kernel*]{}. Comm. Math. Phys. [**211**]{} (2000), 335–358; arXiv: math.RT/9904010. A. Borodin and G. Olshanski, [*Random partitions and the Gamma kernel*]{}. Advances in Math. [**194**]{} (2005), 141–202; arXiv: math-ph/0305043. A. Borodin and G. Olshanski, [*Markov processes on partitions*]{}. Prob. Theory and Related Fields [**135**]{} (2006), 84–152; arXiv: math-ph/0409075. A. Borodin and G. Olshanski, [*Meixner polynomials and random partitions*]{}. Moscow Math. J. [**6**]{} (2006), 629–655; arXiv: math.PR/0609806. M. Brown, [*Discrimination of Poisson processes*]{}. Ann. Math. Stat. [**42**]{} (1971), 773–776. P. Deift [*Integrable operators*]{}. In: Differential operators and spectral theory: M. Sh. Birman’s 70th anniversary collection (V. Buslaev, M. Solomyak, D. Yafaev, eds.) American Mathematical Society Translations, Ser. 2, vol. [**189**]{}, AMS, Providence, R.I., 1999, pp. 69–84. A. Erdelyi (ed.), [*Higher transcendental functions. Bateman Manuscript Project*]{}, vol. I. McGraw-Hill, New York, 1953. A. R. Its, A. G. Izergin, V. E. Korepin, N. A. Slavnov, [*Differential equations for quantum correlation functions*]{}. Intern. J. Mod. Phys. [**B4**]{} (1990), 1003–1037. S. Kakutani, [*On equivalence of infinite product measures*]{}. Ann. Math. [**49**]{} (1948), 214–224. S. Kerov, G. Olshanski, and A. Vershik, [*Harmonic analysis on the infinite symmetric group*]{}. Invent. Math. [**158**]{} (2004), 551–642; arXiv: math.RT/0312270. H.-H. Kuo, [*Gaussian measures in Banach spaces*]{}. Lect. Notes Math. [**463**]{}. Springer, 1975. A. Lenard, [*Correlation functions and the uniqueness of the state in classical statistical mechanics*]{}. Comm. Math. Phys. [**30**]{} (1973), 35–44. I. G. Macdonald, [*Symmetric functions and Hall polynomials*]{}, 2nd edition. Oxford University Press, 1995. T. Miwa, M. Jimbo, and E. Date, [*Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras*]{}. Cambridge University Press, 2000. A. Okounkov, [*Infinite wedge and measures on partitions*]{}. Selecta Math. [**7**]{} (2001), 1–25; arXiv:math/9907127. A. Okounkov and N. Reshetikhin, [*Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram*]{}. J. Amer. Math. Soc. [**16**]{} (2003), 581–603; arXiv: math/0107056 G. Olshanski, [*Point processes related to the infinite symmetric group*]{}. In: The orbit method in geometry and physics: in honor of A. A. Kirillov (Ch. Duval et al., eds.), Progress in Mathematics [**213**]{}, Birkhäuser, 2003, pp. 349–393; arXiv: math.RT/9804086. G. Olshanski, [*Difference operators and determinantal point processes*]{}. Funct. Anal. Appl. [**42**]{} (2008), 317–329; arXiv:0810.3751. T. Shirai and H. J. Yoo, [*Glauber dynamics for fermion point processes*]{}. Nagoya Math. J. [**168**]{} (2002), 139–166. A. Soshnikov, [*Determinantal random point fields*]{}. Uspekhi Mat. Nauk [**55**]{} (2000), No. 5, 107–160 (Russian); English translation: Russian Math. Surveys [**55**]{} (2000), 923–975. A. M. Vershik and S. V. Kerov, [*Asymptotic theory of characters of the symmetric group*]{}. Funct. Anal. Appl. [**15**]{} (1981), 246–255. [^1]: At various stages of work, the present research was supported by the RFBR grants 07-01-91209 and 08-01-00110, by the project SFB 701 (Bielefeld University), and by the grant “Combinatorial Stochastic Processes" (Utrecht University). [^2]: A connection between quasi-invariance and existence of Markov dynamics, sometimes in hidden form, is present in various situations. See, e.g., [@AKR], [@SY]. [^3]: The pair $(z,z')$ should be viewed as an [*unordered*]{} pair of parameters, because the transposition $z\leftrightarrow z'$ does not affect the measure, see Theorem \[1.4\]. [^4]: The situation is formally similar to that of weak convergence and moment convergence of probability measures on ${\mathbb R}$: In general, the former does not imply the latter. [^5]: Another integral representation, given in [@BO4 Thm.3.3] and [@BO3 Thm. 6.3], seems to be less suitable for our purpose. [^6]: In [@BO3] the definition of the contour was slightly different: there we assumed that before and after going around 0, the path goes exactly on the positive real axis. In the context of the present section such a definition works equally well but it is not suitable for the integral representation appearing in Section \[6\]. This is why we have to slightly modify the definition of [@BO3]. [^7]: Incidentally I will also correct minor inaccuracies in [@BO3 §9].

--- abstract: 'We present a formulation of the operator product expansion that is infrared finite to all orders in the attendant massless non-Abelian gauge theory coupling constant, which we will oftentimes associate with the QCD theory, the theory that we actually have as our primary objective in view of the operation of the LHC at CERN. We make contact in this way with the recently introduced IR-improved DGLAP-CS theory and point-out phenomenological implications accordingly, with an eye toward the precision QCD theory for LHC physics.' author: - | B.F.L. Ward [^1]\ Baylor University\ `[email protected]` date: | BU-HEPP-12-01,\ Apr., 2012 title: 'IR-Improved Operator Product Expansions in non-Abelian Gauge Theory' --- **Introduction** {#intro} ================ With the start-up of the LHC the era of precision QCD, by which we mean predictions for QCD processes at the total precision tag of $1\%$ or better, is upon us and the need for exact, amplitude-based resummation of large higher order effects is becoming more and more acute. Methods to facilitate the realization of such resummation are then of particular interest. In this paper, we revisit the pioneering use of operator product expansion (OPE) methods, as presented by Wilson [@kgw] for short-distance limits of physical processes and as applied by Gross, Wilczek and Politzer in the QCD [@qcd] theory, especially as it is realized in the DGLAP-CS [@dglap; @cs] theory, from the standpoint of resummation of its large infrared effects with an eye toward the attendant application of the corresponding parton model representation to LHC precision physics. In this way, we make contact as well with the recently introduced IR-improved DGLAP-CS theory in Refs. [@irdglap1; @irdglap2; @herwiri]. Specifically, it is well-known [@djg-fw; @ghdp; @hdp1] that the usual formulation of the Wilson expansion in massless gauge theory is infrared divergent: the easiest way to realize this is to note that, already at one-loop, the respective leading twist operator matrix elements between fundamental particle states are in general infrared divergent and must be evaluated at off-shell (Euclidean) points in massless gauge theory – see for example Refs. [@djg-fw; @ghdp; @hdp1]. The result is that the coefficient functions of the Wilson operators in the OPE which encode the leading $Q^2$ dependence of the expansion are in general infrared divergent order-by-order in renormalized perturbation theory[^2]. Of course, all such infrared divergences cancel in physically observable (hadronic) matrix elements of the expansion so that, from the standpoint of such observables, the issue is one of choosing the best rearrangement of the large infrared effects that remain after all infrared divergences have canceled. Here, we will resum these large infrared effects. As a result, in what follows, we reformulate the OPE in such away that the respective expansion components are infrared finite. As a further result, we show how the new IR-improved DGLAP-CS theory in Ref. [@irdglap1; @irdglap2; @herwiri] arises naturally in this context. We argue that the IR-improved expansion should be closer to experiment for a given [*exact*]{} order in the loop expansion for the coefficient functions and respective operator matrix elements. The paper is organized as follows. In the next section, we recapitulate the formulation of the OPE following the arguments of Wilson as used in Refs. [@djg-fw; @ghdp; @hdp1] for the analysis of deep-inelastic lepton-nucleon scattering [@taylor], the proto-typical physical application of the method. In Section 3, we show how to improve it so that its hard coefficient functions are IR finite. We also make contact with the new IR-improved DGLAP-CS theory [@irdglap1; @irdglap2; @herwiri]. In Section 3, we also sum up with an eye toward phenomenological implications. Review of the OPE ================= For pedagogical reasons, we follow the historical development and use the deep inelastic electron-proton scattering problem discussed so effectively by Bjorken [@bj1] as our starting point: $e^-(\ell)+p(p_p)\rightarrow e^-(\ell')+X(p_X)$. Indeed, his discussion set the framework for the issues we address here. The kinematics and notation are summarized in the Fig. \[fig1\], so that we use $x\equiv x_{Bj}=Q^2/(2m_p\nu)$ for Bjorken’s scaling variable which has the interpretation in the attendant parton model as the struck parton’s momentum fraction when $\nu=qp_p/m_p$ with $q=\ell-\ell', \; Q^2=-q^2$. In the Fig. \[fig1\], the parton momenta are $p_i(p'_i)$ before(after) the hard interaction process. The limit of Bjorken is then of interest here, in which we take $Q^2 \rightarrow \infty$ with $x$ fixed. In this limit, where here we will for reasons of pedagogy focus on the photon exchange in Fig. \[fig1\] [^3], the standard methods can be used to represent the imaginary part of the attendant current-proton forward scattering amplitude as $$\begin{split} W^{EM}_{\alpha\beta}(p_p,q)&=\frac{1}{2\pi}\int d^4ye^{iqy}<p|[J^{EM}_\beta(y),J^{EM}_\alpha(0)]|p>\\ &=\qquad (-g_{\alpha\beta}+q_\alpha q_\beta/q^2)W_1(\nu,q^2)\\ &\qquad +\frac{1}{m_p^2}(p_p-qqp_p/q^2)_\alpha(p_p-qqp_p/q^2)_\beta W_2(\nu,q^2) \end{split} \label{eq1-ope}$$ Here, $J^{EM}_\alpha(y)$ is the hadronic electromagnetic current and $W_{1,2}$ are the usual deep inelastic the structure functions, which first were shown to exhibit Bjorken scaling by the SLAC-MIT experiments [@taylor] already at $Q^2\cong 1_+ \text{GeV}^2$, precocious scaling – we return to this point below. For our purposes here, henceforward we drop the superscript on $J^{EM}$ so that $J^{EM}\equiv J$ for ease of notation and we always understand the average over the spin of the proton even when we do not indicate so explicitly. In Bjorken’s limit, we have $$\begin{split} \lim_{Bj} m_pW_1(\nu,q^2)&=F_1(x)\\ \lim_{Bj} \nu W_2(\nu,q^2)&=F_2(x) \end{split} \label{eq2-ope}$$ where the scaling limits $F_{1,2}$ only depend on Bjorken’s variable $x$ and we denote $$\lim_{Bj} \equiv \lim_{Q^2\rightarrow\infty}|_{x-\text{fixed}}.$$ The QCD theory of Gross, Wilczek and Politzer [@qcd] provides a quantum field theoretic explanation of the observed Bjorken scaling behavior via Wilson’s OPE. Specifically, in Bjorken’s limit, the phase in integral over space-time in (\[eq1-ope\]) oscillates rapidly except in regions where it is bounded so that the value of the integral is dominated by the latter regions, which are well-known to correspond to the tip of the light-cone [@djg-sbt], the short-distance regime. Using Wilson’s expansion in this regime, we get the OPE [@djg-fw; @ghdp; @hdp1] [$$\begin{split} J_\beta(y)J_\alpha(0)&=\frac{1}{2}g_{\beta\alpha}\left(\frac{\partial}{\partial y}\right)^2\frac{1}{y^2-i\epsilon y_0}{\sum_{n=0}^{\infty}} \sum_{j} C^{(n)}_{j,1}(y^2 - i\epsilon y_0) O^j_{\mu_1 \cdots \mu_n} (0)y^{\mu_1} \cdots y^{\mu_n}\\ & + \frac{1}{y^2-i\epsilon y_0}{\sum_{n=0}^{\infty}} \sum_{j} C^{(n)}_{j,2}(y^2 - i\epsilon y_0) O^j_{\beta\alpha\mu_1 \cdots \mu_n} (0)y^{\mu_1} \cdots y^{\mu_n}+\cdots, \end{split} \label{eq3-ope}$$]{} where we have neglected gradient terms without loss of content for our purposes here and as usual $\epsilon \downarrow 0$. We also note that $\{ O^j_{\mu_1 \cdots \mu_n} (y)\}$ are traceless, symmetric spin $n$ operators of dimension $n+2$ or of twist = dimension -spin = 2 [@djg-sbt]. The $\cdots$ represent operators of higher twist that are suppressed by powers of $q^2$ to any finite order in perturbation theory. The coefficient c-number functions $\{ C^{(n)}_{j,k}\}$ are dimensionless and can be computed in renormalized perturbation theory. Continuing our recapitulation of the methods in Refs. [@djg-fw; @ghdp; @hdp1], if we define the spin averaged proton matrix elements of the operators $ O^j$ via $$<p| O^j_{\mu_1 \cdots \mu_n} (0)|p>|_{\text{spin averaged}}=i^n\frac{1}{m_p}{p_p}_{\mu_1} \cdots {p_p}_{\mu_n} M^n_j+\cdots, \label{eq4-ope}$$ where the second $\cdots$ denotes trace-terms, we get the following relationship [@djg-fw; @ghdp; @hdp1; @chm] between the moments of the structure functions and the Fourier transforms of the coefficient functions: $$\begin{split} \int_0^1dx x^n F_1(x,q^2)&=\sum_j \bar{C}^{(n+1)}_{j,1}(q^2)M^{n+1}_j, \\ \int_0^1dx x^n F_2(x,q^2)&=\sum_j \bar{C}^{(n)}_{j,2}(q^2)M^{n+2}_j, \end{split} \label{eq5-ope}$$ where [@djg-fw] $$\bar{C}^{(n)}_{j,k}(q^2)=\frac{1}{2}i(q^2)^{n+1}\left(-\frac{\partial}{\partial q^2}\right)^n \int d^4y e^{iqy}{\small \frac{C^{(n)}_{j,k}(y^2)}{y^2-i\epsilon y_0}} . \label{eq6-ope}$$ The $q^2$ dependence of the $\bar{C}^{(n)}$ is controlled by the Callan-Symanzik equation [@cs] which reads $$\left[\left(\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial g}\right)\delta_{ij}-\gamma^{(n)}_{ij}(g)\right]C^{(n)}_{j,k}=0 \label{eq7-ope}$$ where $\mu$ denotes the renormalization scale, $$\beta(g)=\mu\frac{\partial g}{\partial\mu}$$ for the attendant renormalized coupling $g$, and the anomalous dimension matrix $\gamma^{(n)}_{ij}(g)$ is given as $$\gamma^{(n)}_{ij}(g)=\left(Z_O^{-1} \mu\frac{\partial }{\partial\mu}Z_O\right)_{ij}|_{g_{0}, \text{regularization fixed}} \label{eq8-ope}$$ where the operators $O^{(n)}_j\equiv O^j_{\mu_1\cdots\mu_n}$ are renormalized via $$O^{(n)}_i\equiv O^{(n)}_{i,R}=\sum_j O^{(n)}_{j,bare}\left(Z_O^{-1}\right)_{ji} \label{eq9-ope}$$ so that they mix under renormalization in the well-known way [@djg-fw; @ghdp] and we use a standard notation of the renormalized, $R$, and unrenormalized , bare, operator representatives. $g_0$ is the bare coupling constant. It is well-known that the solution of (\[eq7-ope\]) leads to the conclusion that the asymptotic Bjorken limit is controlled by the operators with the smallest eigenvalue for their anomalous dimension matrix in an asymptotically free theory such as the QCD [@qcd] which we have in mind here. The implied behavior for the RHS of (\[eq5-ope\]) is in agreement with experiment [@taylor]. Here, we want to focus on the IR-improvement of the $C^{(n)}_{j,k},\; M^{n}_j$. IR-Improved OPE =============== The isolation of the infrared aspects of the $C^{(n)}_{j,k}$ is immediate if we use the fundamental particles in the respective Lagrangian quantum field theory, quarks and gluons in the case of QCD, to evaluate the essential anomalous dimension matrix elements $\gamma^{(n)}_{ij}(g)$, as this is equivalent to studying deep inelastic scattering from these fundamental particles and takes us immediately, at least conceptually, to the parton model perspective studied famously by many [@rpf-part; @bj-ps; @drlyn; @ellis-pol-mach; @fur-pet1; @fur-pet2]. Specifically, we then focus on the parton level version of hadronic tensor $W_{\alpha\beta}$ which for definiteness we associate with a fermion $F$ in the underlying asymptotically free theory(QCD): $$\begin{split} W^{F}_{\alpha\beta}(p_F,q)&=\frac{1}{2\pi}\int d^4ye^{iqy}<p_F|[J_\beta(y),J_\alpha(0)]|p_F>\\ &= (2\pi)^3\sum_{X}\delta(q+p_F-p_X)<p_F|J_\beta(0)|p_X><p_X|J_\alpha(0)|p_F>, \end{split} \label{eq10-ope}$$ where we use the fact that $q^0>0$ to drop the remaining term in the commutator and we always average over the spin of the fermion $F$, as we do for the proton$p$. We see clearly from (\[eq1-ope\]) that the RHS of (\[eq10-ope\]) and that of (\[eq1-ope\]) involve the same OPE. We first focus on the matrix element $${\cal M}_{X,\alpha}\equiv <p_X|J_\alpha(0)|p_F>. \label{eq11-ope}$$ Following Refs. [@irdglap1; @irdglap2], we isolate the dominant virtual IR divergences associated to the incoming line via the formula $${\cal M}_{X,\alpha} = e^{\alpha_s\; B_{QCD}} <p_X|J_\alpha(0)|p_F>_{IRI-virt}, \label{eq11a-ope}$$ where the virtual infrared function $B_{QCD}$ is given in Refs. [@irdglap1; @irdglap2]. The RHS of this last equation is valid to all orders in $\alpha_s\equiv g^2/(4\pi)$ so that one computes $<p_X|J_\alpha(0)|p_F>_{IRI-virt}$ from $<p_X|J_\alpha(0)|p_F>$ by comparing the coefficients of the powers of $\alpha_s$ on both sides of (\[eq11a-ope\]) iteratively. Introducing this result into (\[eq10-ope\]), we arrive at[$$\begin{split} W^{F}_{\alpha\beta}(p_F,q) &= (2\pi)^3\sum_{X}\delta(q+p_F-p_X)e^{2\alpha_s \Re B_{QCD}}\;\;{_{IRI-virt}\!<p_F|J_\beta(0)|p_X>}\\ & \qquad\qquad\;\;\;\; <p_X|J_\alpha(0)|p_F>_{IRI-virt}. \end{split} \label{eq12-ope}$$]{} We next isolate the leading soft, spin independent real emission infrared function associated to the incoming line as follows. We first separate $\{X\}$ into its multiple gluon subspaces via $$\{X\} = \{X: X = X'\otimes \{G_1\otimes\ldots\otimes G_n\}, \text{for some}\; n\ge 0, X'\; \text{is non-gluonic}\}. \label{eq13-ope}$$ Then we have [$$\begin{split} &e^{2\alpha_s \Re B_{QCD}}{_{IRI-virt}\!<p_F|J_\beta(0)|p_X>}<p_X|J_\alpha(0)|p_F>_{IRI-virt}\\ &=e^{2\alpha_s \Re B_{QCD}}\Big{[}\tilde S_{QCD}(k_1)\cdots \tilde S_{QCD}(k_n)\;\;{_{IRI-virt}<p_F|J_\beta(0)|p_{X'}>}\\ &<p_{X'}|J_\alpha(0)|p_F>_{IRI-virt}+\cdots+ {_{IRI-virt\&real}\!<p_F|J_\beta(0)|p_{X'},k_1,\cdots,k_n>}\\ &<p_{X'},k_1,\cdots,k_n|J_\alpha(0)|p_F>_{IRI-virt\&real}\Big{]}, \end{split} \label{eq14-ope}$$]{} where the real infrared function $\tilde S_{QCD}(k)$ is given in Refs. [@irdglap1; @irdglap2]. The IR-improved quantities $${_{IRI-virt\&real}\!<p_F|J_\beta(0)|p_{X}>}<p_{X}|J_\alpha(0)|p_F>_{IRI-virt\&real}$$ are defined iteratively from (\[eq11a-ope\]),(\[eq14-ope\]) to all orders in $\alpha_s$ and they no longer contain the infrared singularities from the initial line associated to $B_{QCD}$ and to $\tilde S_{QCD}$, although, because of the non-Abelian infrared algebra of the theory, they do contain other IR singularities which of course cancel in the structure functions by the KNL theorem for massless fundamental fermions. For massive fundamental fermions, these latter singularities also cancel provided we resum the theory as we are doing here accordingly – see Refs. [@bflw-nbn]. Introducing the representation in (\[eq14-ope\]) into (\[eq10-ope\]) we get $$\begin{split} W^{F}_{\beta\alpha}(p_F,q) & = (2\pi)^3\sum_{X}\delta(q+p_F-p_X)e^{2\alpha_s \Re B_{QCD}}\Big{[}\tilde S_{QCD}(k_1)\cdots \tilde S_{QCD}(k_n)\\ & \qquad\qquad{_{IRI-virt}<p_F|J_\beta(0)|p_{X'}>}<p_{X'}|J_\alpha(0)|p_F>_{IRI-virt}+\cdots\\ &\qquad\qquad\;\;+ {_{IRI-virt\&real}\!<p_F|J_\beta(0)|p_{X'},k_1,\cdots,k_n>}\\ &\qquad\qquad\;\;<p_{X'},k_1,\cdots,k_n|J_\alpha(0)|p_F>_{IRI-virt\&real} \Big{]}\\ & = \frac{1}{2\pi}\int d^4 y\sum_{X'}\sum_n\frac{1}{n!}\int\Pi_{j=1}^n \frac{d^3k_j}{k^0_j}e^{SUM_{IR}(QCD)}e^{iy(q+p_F-p_{X'}-\sum_j k_j)+D_{QCD}}\\ &\qquad\qquad\;\; {_{IRI-virt\&real}<p_F|J_\beta(0)|p_{X'},k_1,\cdots,k_n>}\\ &\qquad\qquad\;\;<p_{X'},k_1,\cdots,k_n|J_\alpha(0)|p_F>_{IRI-virt\&real}\\ & = \frac{1}{2\pi}\int d^4y e^{iqy}e^{SUM_{IR}(QCD)+D_{QCD}}\\ & \qquad\qquad\;\;{_{IRI-virt\&real}\!<p_F|}[J_\beta(y),J_\alpha(0)]|p_F>_{IRI-virt\&real}, \end{split} \label{eq15-ope}$$ where we have defined $${\rm SUM_{IR}(QCD)}=2\alpha_s \Re B_{QCD}+2\alpha_s\tilde B_{QCD}(\Kmax),$$ $$2\alpha_s\tilde B_{QCD}(\Kmax)=\int {d^3k\over k^0}\tilde S_\rQCD(k) \theta(\Kmax-k),$$ $$D_\rQCD=\int{d^3k\over k}\tilde S_\rQCD(k) \left[e^{-iy\cdot k}-\theta(\Kmax-k)\right],\label{eq15a-ope}$$ and we stress that (\[eq15-ope\]) does not depend on $K_{max}$. Using the standard partonic view, by which we have $$W_{\beta\alpha}= \sum_a \int_0^1\frac{dx}{x} {\cal F}_a(x)W^a_{\beta\alpha} \label{eq16-ope}$$ for appropriately defined parton distribution functions $\{{\cal F}_a\}$, we introduce the OPE in (\[eq3-ope\]) into (\[eq15-ope\]) and use (\[eq1-ope\]) to get the IR-improved results $$\begin{split} \int_0^1dx x^n F_1(x,q^2)&=\sum_j \tilde{\bar{C}}^{(n+1)}_{j,1}(q^2)\tilde{M}^{n+1}_j, \\ \int_0^1dx x^n F_2(x,q^2)&=\sum_j \tilde{\bar{C}}^{(n)}_{j,2}(q^2)\tilde{M}^{n+2}_j, \end{split} \label{eq17-ope}$$ where [@djg-fw] $$\tilde{\bar{C}}^{(n)}_{j,k}(q^2)=\frac{1}{2}i(q^2)^{n+1}\left(-\frac{\partial}{\partial q^2}\right)^n \int d^4y e^{iqy}e^{SUM_{IR}(QCD)+D_{QCD}}{\small \frac{\tilde{C}^{(n)}_{j,k}(y^2)}{y^2-i\epsilon y_0}} \label{eq18-ope}$$ and now $$\begin{split} <p| \tilde{O}^j_{\mu_1 \cdots \mu_n} (0)|p>|_{\text{spin averaged}}&\equiv\\ _{IRI-virt\&real}\!<p| O^j_{\mu_1 \cdots \mu_n} (0)|p>_{IRI-virt\&real}|_{\text{spin averaged}}&=i^n\frac{1}{m_p}{p_p}_{\mu_1} \cdots {p_p}_{\mu_n}\tilde{M}^n_j \\ & \qquad +\cdots, \end{split} \label{eq19-ope}$$ where the second $\cdots$ again denotes trace-terms and the $\{\tilde{C}^{(n)}_{j,k}\}$ are the respective (new) IR-improved OPE coefficient functions. The $q^2$ dependence of the $\tilde{\bar{C}}^{(n)}$ is also controlled by the Callan-Symanzik equation [@cs] which now reads $$\left[\left(\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial g}\right)\delta_{ij}-\tilde{\gamma}^{(n)}_{ij}(g)\right]\tilde{\bar{C}}^{(n)}_{j,k}=0 \label{eq20-ope}$$ where now the new matrix $\tilde{\gamma}^{(n)}_{ij}(g)$ is determined by the renormalization properties of the IR-improved matrix elements in (\[eq19-ope\]) as we will discuss presently. We need to stress that in writing (\[eq18-ope\]) we work to one-loop order in the various coefficients in this paper. A convenient starting point for obtaining the new matrix $\tilde{\gamma}^{(n)}_{ij}(g)$ is presented by the pioneering analysis of the authors in Ref. [@fur-pet1; @fur-pet2]. Working directly from the the representation in (\[eq16-ope\]), the authors in Ref. [@fur-pet1] make contact with the matrix $\gamma^{(n)}_{ij}(g)$ for the unimproved OPE as follows. Focusing for definiteness for the moment on the non-singlet operator [@djg-fw] $^N\!O^{F,b}(y)=\frac{1}{2}i^{N-1}S\bar{\psi}(y)\gamma_{\mu_1}\nabla_{\mu_2}\cdots\nabla_{\mu_N}\lambda^b\psi(y)-\text{trace terms}$, where $\nabla_\mu=\partial_\mu+ig\tau^aA^a_\mu$ is the covariant derivative, $\lambda^b$ is a flavor group generator and $S$ denotes symmetrization with respect to the indices $\mu_1\cdots\mu_n$, we have the matrix element between fundamental fermion states, where spin averaging is understood here as well, as $$<p|^N\!O^{F,b}(y)|p>=\; ^{F,b}\!O^{N}(\alpha_s,\epsilon)p_{\mu_1}\cdots p_{\mu_N}-\text{trace terms} \label{eq21-ope}$$ where we use $d=4-\epsilon$ dimensions for regularization and the notation $^{F,b}\!O^{N}(\alpha_s,\epsilon)\equiv M^N_{F,b}$ to make immediate contact with the arguments in Ref. [@fur-pet1]. The renormalized matrix element $^{F,b}\!O^{N}(\alpha_s,\epsilon)$ is related to the bare one as we have indicated in (\[eq9-ope\]): $$^{F,b}\!O^{N}(\alpha_s,\epsilon,p^2/\mu^2)=Z^{-1}_O(\alpha_s,\frac{1}{\epsilon})\; ^{F,b}\!O^{N}_{bare}((\alpha_s)(\mu^2/p^2)^\epsilon,\epsilon) \label{eq21a-ope}$$ so that collinear divergences are regularized by taking $p^2\ne 0$ in the approach in Ref. [@fur-pet1]. Using an arbitrary vector $\Delta$ with $\Delta^2=0$ we get $$^{F,b}\!O^{N}(\alpha_s,\epsilon)=<p|^N\!O^{F,b}_{\mu_1\cdots\mu_N}(y)|p>\Delta^{\mu_1}\cdots \Delta^{\mu_N}/(\Delta p)^N. \label{eq22-ope}$$ The pole part of $^{F,b}\!O^{N}$ which is the renormalization part $Z^{-1}_O$ can be determined in any gauge by gauge invariance. We set $\Delta=n$ where $x_{Bj}=np/np_p,\; x=kn/np$ and $nA^a=0$ so that we are in a light-like gauge. This allows us to write, following Ref. [@fur-pet1], $$^{F,b}\!O^{N}(\alpha_s,\epsilon)=\int_{-1}^{1} dx x^{N-1}\; {^{F,b}\!O(x,\alpha_s,\epsilon)} \label{eq23-ope}$$ where $${^{F,b}\!O(x,\alpha_s,\epsilon)}=Z_F[\delta(x-1)+ x\frac{\int d^d k}{(2\pi)^d}\delta(x-\frac{kn}{pn})[\frac{\not\!n}{4kn}T(p,k)\not\!p] \label{eq24-ope}$$ where we use the notation of Ref. [@fur-pet1] so that $T(p,k)$ is the respective fully connected four-point function and $[\not\!bB$ denotes $\sum_{\alpha\alpha'}b_{\alpha\alpha'}B^{\alpha\alpha'}_{\beta\beta'}$ with corresponding notation for $B\not\!b]$. $Z_F$ is the fermion field renormalization constant as usual. By first analytically continuing the LHS of (\[eq21a-ope\]) to $d=4+\epsilon$ dimensions with $\epsilon>0$ the authors in Ref. [@fur-pet1] note that the limit $p^2\rightarrow 0$ gives the RHS as just $Z^{-1}_O(\alpha_s,\epsilon)$ for an appropriate normalization of $\lambda^b$ . The RHS of (\[eq23-ope\]) may then related to the moments of the densities of partons in a quark, $\Gamma_S(x,\alpha_s,1/\epsilon)$ in the notation of Ref. [@fur-pet1], by writing a dispersion relation for $[(\not\!n/(4kn))T(p,k)\not\!p]$ and performing the attendant $k^2$ integral by closing the contour around the dispersive poles(see Sect. 4.2 of Ref. [@fur-pet1]), analytically continuing to $d=4+\epsilon$ dimensions with again $\epsilon>0$ and finally taking the limit $p^2\rightarrow 0$ to get $$Z^{-1}_O(\alpha_s,\frac{1}{\epsilon})=\int_{-1}^{1}dxx^{N-1}[\Gamma_{qq}(x,\alpha_s,\frac{1}{\epsilon})\theta(x)-\Gamma_{q\bar{q}}(-x,\alpha_s,\frac{1}{\epsilon})\theta(-x)] \label{eq25-ope}$$ where $\Gamma_{qq}(\Gamma_{q\bar{q}})$ is the respective parton density for a quark(anti-quark) in a quark. The coefficients of $\frac{1}{\epsilon}$ on both sides of this last equation then give the fundamental result, derived in Ref. [@fur-pet1], $$\begin{split} -\gamma^{(N)}(\alpha_s)&=2\int_{-1}^{1}dx x^{N-1}[P_{qq}(x,\alpha_s)\theta(x)-P_{q\bar{q}}(-x,\alpha_s)\theta(-x)]\\ &=2[ P_{qq}(N,\alpha_s)+(-1)^NP_{q\bar{q}}(N,\alpha_s)] \end{split} \label{eq26-ope}$$ where we define $$F(N)=\int_0^1 dx x^{N-1}F(x)$$ and the $P_{BA}$ are the usual DGLAP-CS [@dglap; @cs] splitting kernels defined in the convention of Ref. [@fur-pet1] and $\gamma^{(N)}(\alpha_s)$ is the respective anomalous dimension of the operator $^N\!O^{F,b}$. To apply this calculation to our new anomalous dimension matrix we IR-improve it at each step as we have shown above (and as we have shown for the IR-improved DGLAP-CS theory in Refs. [@irdglap1; @irdglap2]), so that we replace $$<p|^N\!O^{F,b}(y)|p>\rightarrow <p|^N\!\tilde{O}^{F,b}(y)|p>$$ as defined in (\[eq19-ope\]) with the corresponding substitution of $^{F,b}\!O^{N}(\alpha_s,\epsilon)$ by the analogous $^{F,b}\!\tilde{O}^{N}(\alpha_s,\epsilon)$. This leads to the relationship $$^{F,b}\!\tilde{O}^{N}(\alpha_s,\epsilon,p^2/\mu^2)=Z^{-1}_{\tilde{O}}(\alpha_s,\frac{1}{\epsilon})\; ^{F,b}\!\tilde{O}^{N}_{bare}((\alpha_s)(\mu^2/p^2)^\epsilon,\epsilon) \label{eq27-ope}$$ between the renormalized and bare IR-improved matrix elements. The analoga of (\[eq23-ope\]) and (\[eq24-ope\]) are then $$^{F,b}\!\tilde{O}^{N}(\alpha_s,\epsilon)=\int_{-1}^{1} dx x^{N-1}\; {^{F,b}\!\tilde{O}(x,\alpha_s,\epsilon)} \label{eq28-ope}$$ where $${^{F,b}\!\tilde{O}(x,\alpha_s,\epsilon)}=Z_F[\delta(x-1)+ x\frac{\int d^d k}{(2\pi)^d}\delta(x-\frac{kn}{pn})[\frac{\not\!n}{4kn}\tilde{T}(p,k)\not\!p] \label{eq29-ope}$$ and we continue to use the notation of Ref. [@fur-pet1] so that $\tilde{T}(p,k)$ is the respective IR-improved fully connected four-point function obtained from the unimproved one, $T(p,k)$, by using the master formula Eq.(1) in Refs. [@irdglap2] restricted to its QCD aspect, for example. This means that we get the analog of (\[eq25-ope\]) as $$Z^{-1}_{\tilde{O}}(\alpha_s,\frac{1}{\epsilon})=\int_{-1}^{1}dxx^{N-1}[\Gamma^{exp}_{qq}(x,\alpha_s,\frac{1}{\epsilon})\theta(x)-\Gamma^{exp}_{q\bar{q}}(-x,\alpha_s,\frac{1}{\epsilon})\theta(-x)] \label{eq30-ope}$$ where $\Gamma^{exp}_{qq},\; \Gamma^{exp}_{q\bar{q}}$ are the respective IR-improved parton densities. We get in this way the identification of the respective IR-improved anomalous dimension as $$-\tilde\gamma^{(N)}(\alpha_s)=2\frac{\alpha_s}{2\pi}[ P^{exp}_{qq}(N,\alpha_s)+(-1)^NP^{exp}_{q\bar{q}}(N,\alpha_s)] \label{eq31-ope}$$ where the $ P^{exp}_{qq},\; P^{exp}_{q\bar{q}}$ are the respective IR-improved kernels as introduced in Refs. [@irdglap1; @irdglap2], where we advise that the notation of Ref. [@fur-pet1] differs from that in Refs. [@irdglap1; @irdglap2] by whether or not one includes the factor $\alpha_s/(2\pi)$ on the RHS of (\[eq27-ope\]) in the definition of the kernels. This allows us to write at IR-improved one-loop level the identifications $$-\tilde\gamma^{(N)}(\alpha_s)_{ij}=2\frac{\alpha_s}{2\pi}P^{exp}_{ij}(N) \label{eq32-ope}$$ where the labels $i,j$ span the usual values for the one-loop anomalous dimension matrix for the evolution of the parton distributions as given in Refs. [@dglap; @cs; @djg-fw; @ghdp; @hdp1] for example. This establishes in a rigorous way the connection between the IR-improved DGLAP-CS theory in Ref. [@irdglap1; @irdglap2] and the OPE methods of Wilson as used by Refs. [@djg-fw; @ghdp; @hdp1] in the study of deep inelastic lepton-nucleon scattering. Evidently, this connection may be manifested in the analysis of other physical processes as well. We refer the reader to Refs. [@irdglap2; @herwiri] wherein the new precision-baseline MC Herwiri1.031 which realizes the IR-improved DGLAP-CS kernels has been introduced and compared to the Tevatron data [@d0pt; @galea] on single Z production. Its application to the various physical processes at LHC is in progress and will appear accordingly elsewhere [@elswh], where we need to stress that Herwiri1.031 can be applied to [*any process*]{} to which Herwig6.5 [@herwig] can be applied and that it interfaces to MC@NLO [@mcatnlo] the [*same way*]{} that does Herwig6.5. As we have shown in Refs. [@irdglap2; @herwiri], we have an improved agreement between the IR-improved MC’s shower and the Tevatron data with no need of an abnormally large intrinsic transverse momentum parameter, PTRMS$\sim 2$GeV in the notation of Herwig [@herwig], as it is required for similar agreement with Herwig6.5 [@mike]. We point-out that, consistent with the precociousness of Bjorken scaling, the IR-improved MC Herwiri.031 gives us a paradigm for reaching a precision QCD MC description of the LHC data, on an event-by-event basis with realistic hadronization from the Herwig6.5 environment, that does not involve an ad hoc hard scale parameter, where we define “hard” relative to the observed precociousness of Bjorken scaling. What we have shown in the discussion above is that this paradigm has a rigorous basis in quantum field theory. In closing, we thank Prof. Ignatios Antoniadis for the support and kind hospitality of the CERN TH Unit while part of this work was completed. [99]{} K. Wilson, Phys. Rev. [**179**]{} (1969) 1499; W. Zimmermann, in [*Lectures on Elementary Particles and Quantum Field Theory – 1970 Brandeis Summer Institute in Theoretical Physics, vol. 1*]{}, eds. S. Deser, H. Pendleton and M. Grisaru, (MIT Press, Cambridge, 1970) p. 395; R.A. Brandt and G. Preparata, Nucl. Phys. [**27**]{} (1971) 541. D. J. Gross and F. Wilczek, Phys. Rev. Lett. [**30**]{} (1973) 1343; H. David Politzer, [*ibid.*]{}[**30**]{} (1973) 1346; see also , for example, F. Wilczek, in [*Proc. 16th International Symposium on Lepton and Photon Interactions, Ithaca, 1993*]{}, eds. P. Drell and D.L. Rubin (AIP, NY, 1994) p. 593, and references therein. G. Altarelli and G. Parisi, [*Nucl. Phys.*]{} [**B126**]{} (1977) 298; Yu. L. Dokshitzer, [*Sov. Phys. JETP*]{} [**46**]{} (1977) 641; L. N. Lipatov, [*Yad. Fiz.*]{} [**20**]{} (1974) 181; V. Gribov and L. Lipatov, [*Sov. J. Nucl. Phys.*]{} [**15**]{} (1972) 675, 938; see also J.C. Collins and J. Qiu, [*Phys. Rev. D*]{}[**39**]{} (1989) 1398. C.G. Callan, Jr., [*Phys. Rev. D*]{}[**2**]{} (1970) 1541; K. Symanzik, [*Commun. Math. Phys.*]{} [**18**]{} (1970) 227, and in [*Springer Tracts in Modern Physics*]{}, [**57**]{}, ed. G. Hoehler (Springer, Berlin, 1971) p. 222; see also S. Weinberg, [*Phys. Rev. D*]{}[**8**]{} (1973) 3497. B.F.L. Ward, [*Adv. High Energy Phys.*]{} [**2008**]{} (2008) 682312. B.F.L. Ward, [*Ann. Phys.*]{} [**323**]{} (2008) 2147; B.F.L. Ward, S.K. Majhi and S.A. Yost, arXiv:1201.0515, in PoS([**RADCOR2011**]{}), in press. S. Joseph, S. Majhi, B.F.L. Ward and S.A. Yost, Phys. Lett. B[**685**]{} (2010) 283; Phys. Rev. D[**81**]{} (2010) 076008. D.J. Gross and F. Wilczek, Phys. Rev.D[**8**]{} (1973) 3633; ibid. [**8**]{} (1974) 980. H. Georgi and H.D. Politzer, Phys. Rev. D[**9**]{} (1974) 416, and references therein. H.D. Politzer, Phys. Rept. [**14**]{} (1974) 129. See for example R.E. Taylor, Phil. Trans. Roc. Soc. Lond. [**A359**]{} (2001) 225, and references therein. J. Bjorken, in [*Proc. 3rd International Symposium on the History of Particle Physics: The Rise of the Standard Model, Stanford, CA, 1992*]{}, eds. L. Hoddeson [*et al.*]{} (Cambridge Univ. Press, Cambridge, 1997) p. 589, and references therein. D.J. Gross and S.B. Treiman, Phys. Rev. D[**4**]{} (1971) 2105; [*ibid.*]{} [**4**]{} (1971) 1059, and references therein. N. Christ, B. Hasslacher and A.H. Mueller, Phys. Rev. D[**6**]{} (1972) 3543. R.P. Feynman, Phys. Rev. Lett. 23 (1969) 1415; [*Photon-Hadron Interactions*]{}, (Benjamin, New York, 1972). J.D. Bjorken and E.A. Paschos, Phys. Rev. [**185**]{} (1969) 1975; Phys.Rev. D[**1**]{} (1970) 3151, and references therein. S.D. Drell and T.-M. Yan, Phys. Rev. Lett. [**25**]{} (1970) 316; [*ibid.*]{}[**25**]{} (1970) 902; S.D. Drell, D.J. Levy and T.-M. Tan, Phys. Rev. D[**1**]{} (1970) 1617; [*ibid.*]{}[**1**]{} (1970) 1035; Phys.Rev. [**187**]{} (1969) 2159, and references therein. R.K. Ellis [*et al.*]{}, Phys. Lett. B [**78**]{} (1978) 281. C. Curci, W. Furmanski and R. Petronzio, Nucl.Phys. B[**175**]{} (1980) 27. W. Furmanski and R. Petronzio, Phys.Lett. B [**97**]{} (1980) 437. B.F.L. Ward, Phys.Rev. D[**78**]{} (2008) 056001; C. Di’Lieto, S. Gendron, I.G. Halliday, and C.T. Sachradja, Nucl. Phys.B[**183**]{}(1981) 223; R. Doria, J. Frenkel and J.C. Taylor, [*ibid.*]{} [**168**]{} (1980) 93; S. Catani, M. Ciafaloni and G. Marchesini, Nucl. Phys.B[**264**]{} (1986) 588; S. Catani, Z. Phys. [**C37**]{} (1988) 357, and references therein. V.M. Abasov et al., Phys. Rev. Lett. [**100**]{} (2008) 102002. C. Galea, in [*Proc. DIS 2008*]{}, London, 2008, http://dx.doi.org/10.3360/dis.2008.55. B.F.L. Ward, to appear. G. Corcella [*et al.*]{}, hep-ph/0210213; J. High Energy Phys. [**0101**]{} (2001) 010; G. Marchesini [*et al.*]{}, Comput. Phys. Commun.[**67**]{} (1992) 465. S. Frixione and B.Webber, J. High Energy Phys. [**0206**]{} (2002) 029; S. Frixione [*et al.*]{}, arXiv:1010.0568. M. Seymour, private communication. [^1]: Work supported in part by D.o.E. grant DE-FG02-09ER41600. [^2]: In Ref. [@kgw], Wilson pointed-out already that the coefficient functions in his expansion could be calculated order by order in perturbation theory and that the n-th order term could contain logarithms of $z^2m^2$ where $z$ is the space-time interval of the respective two operators and $m$ is the free field mass, so that these logs would be divergent at $m=0$. He also noted that an arbitrary subtraction constant $a$ could be introduced to convert the argument of these logarithms to $z^2a^2$. This is equivalent to what we have stated in the text with $a=\mu$ where $\mu$ is identified as a Euclidean point in an appropriate convention. [^3]: As it is well-known, adding in the effects of the Z exchange is straightforward and does not require any essentially new methods that are not already exhibited by what we do for the photon exchange case.

--- abstract: | An $n$-state deterministic finite automaton over a $k$-letter alphabet can be seen as a digraph with $n$ vertices which all have $k$ labeled out-arcs. @Grusho1973 proved that whp in a random $k$-out digraph there is a strongly connected component of linear size, i.e., a giant, and derived a central limit theorem. We show that whp the part outside the giant contains at most a few short cycles and mostly consists of tree-like structures, and present a new proof of @Grusho1973’s theorem. Among other things, we pinpoint the phase transition for strong connectivity. **Keywords:** random digraphs; deterministic finite automaton author: - | Xing Shi Cai, Luc Devroye\ School of Computer Science, McGill University of Montreal, Canada,\ `[email protected]`\ `[email protected]` bibliography: - 'citation.bib' title: 'The graph structure of a deterministic automaton chosen at random: full version' --- Introduction ============ The model and the history ------------------------- The deterministic finite automaton ([<span style="font-variant:small-caps;">dfa</span>]{}) is widely used in computational complexity theory. Formally, a [<span style="font-variant:small-caps;">dfa</span>]{} is a $5$-tuple $(Q, \Sigma, \delta, q_0, F)$, where $Q$ is a finite set called the set of states, $\Sigma$ is a finite set called the alphabet, $\delta:Q\times\Sigma \to Q$ is the transition function, $q_0 \in Q$ is the start state, and $F \subseteq Q$ is the set of accept states. If $q_0$ and $F$ are ignored, a [<span style="font-variant:small-caps;">dfa</span>]{} with $n$ states and a $k$-alphabet can be seen as a digraph with vertices $[n]\equiv\{1\ldots,n\}$ in which each vertex has $k$ out-arcs labeled by $1, \ldots, k$ (a *$k$-out digraph*). Note that such a digraph can have self-loops and multi-arcs. For a basic introduction to [<span style="font-variant:small-caps;">dfa</span>]{} and its applications, see [@Sipser2012]. Let ${{\cD_{n,k}}}$ denote a digraph chosen uniformly at random from all $k$-out digraphs of $n$ vertices. Equivalently $\cD_{n,k}$ is a random $k$-out digraph of $n$ vertices with the endpoints of its $kn$ arcs chosen independently and uniformly at random. When $k=1$, ${{\cD_{n,k}}}$ is equivalent to a uniform random mapping from $[n]$ to itself, which has been well studied by @Kolchin1986random, @Flajolet1990, and @Aldous1994. In $\cD_{n,1}$, the largest strongly connected component ([<span style="font-variant:small-caps;">scc</span>]{}) has expected size $\Theta(\sqrt{n})$, and so does the size of the longest cycle. However, as shown later, for $k \ge 2$, the largest [<span style="font-variant:small-caps;">scc</span>]{} has expected size $\Theta(n)$. From now on we assume that $k \ge 2$. Let ${\cS_{v}}$ (the *spectrum* of $v$) be the set of vertices in ${{\cD_{n,k}}}$ that are reachable from vertex $v$, including $v$ itself. In [-@Grusho1973] @Grusho1973 first proved that $({|{{\cS_{1}}}|}-{{\nu_k}}n)/\sigma_k \sqrt{n}$ converges in distribution to a standard normal, where ${{\nu_k}}$ and $\sigma_k$ are explicitly defined constants. Given a set of vertices $\cS \subseteq [n]$, call $\cS$ *closed* if there are no arcs that start from vertices in $\cS$ and end at vertices in $\cS^c \equiv [n] \setminus \cS$. Let ${{\cG_n}}$ be the set of vertices in the largest closed [<span style="font-variant:small-caps;">scc</span>]{} in ${{\cD_{n,k}}}$. (If the largest closed [<span style="font-variant:small-caps;">scc</span>]{} is not unique, let ${{\cG_n}}$ be the vertex set of the largest closed [<span style="font-variant:small-caps;">scc</span>]{} that contains the smallest vertex-label.) We call ${{\cG_n}}$ the giant. @Grusho1973 also proved that ${{|{{\cG_n}}|}}$ has the same limit distribution as ${|{{\cS_{1}}}|}$ by showing that with high probability (whp) ${{\cG_n}}$ is reachable from all vertices and that ${|{{\cS_{1}}}|}- {{|{{\cG_n}}|}}= o_p(\sqrt{n})$ (see [@Janson2011random] for the notation). His proof relies on a result by @Sevastyanov1967 which approximates the exploration of ${{\cS_{1}}}$ with a Gaussian process. In 2012 @Carayol12 proved a local limit theorem for ${|{{\cS_{1}}}|}$ by analyzing the limit behavior of the probability that ${|{{\cS_{1}}}|}= s$ for an $s$ close to ${{\nu_k}}n$. Their proof depends on a theorem by @Korshunov1978 which says that conditioned on every vertex having in-degree at least one, the probability that ${{\cS_{1}}}= [n]$ tends to some constant. @Carayol12 derived a simple and explicit formula of this constant from their theorem. (The same formula is also proved by @Lebensztayn2010 with a more analytic approach using Lagrange series.) Lately the simple random walk (SRW) on ${{\cD_{n,k}}}$ has gained some attention for its applications in machine learning. @Perarnau2014 studied the stationary distribution of the SRW by analyzing the distances in ${{\cD_{n,k}}}$. They proved that the diameter and the typical distance, rescaled by $\log n$, converge in probability to explicit constants. @Angluin2015 studied the rate of the convergence to the stationary distribution of the SRW. They also suggested an algorithm for learning a uniformly random [<span style="font-variant:small-caps;">dfa</span>]{} under Kearns’ statistical query model [@Kearns1998]. Our results and a sketch of proof --------------------------------- A digraph can be uniquely decomposed into [<span style="font-variant:small-caps;">scc</span>]{}s which form a directed acyclic graph (${\textsc{dag}}{}$) through a process called condensation that contracts every [<span style="font-variant:small-caps;">scc</span>]{} into a single vertex while keeping all the arcs between [<span style="font-variant:small-caps;">scc</span>]{}s [@Bang2009]. The condensation [<span style="font-variant:small-caps;">dag</span>]{} of ${{\cD_{n,k}}}$ is denoted by ${{\cD_{n,k}^{\mathrm A}}}$. Let ${{\cG_n^c}}\equiv [n] \setminus {{\cG_n}}$, i.e., ${{\cG_n^c}}$ is the set of vertices that are outside the giant. The structure of ${{\cD_{n,k}^{\mathrm A}}}$ depends on ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$, the digraph induced by ${{\cG_n^c}}$. Our analysis shows that in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$ the total number of cycles and the number of cycles of a fixed length both converge to Poisson distributions with constant means. So the number of cycles and the length of the longest cycle are both $O_p(1)$ (see [@Janson2011random]). Furthermore, these cycles are vertex-disjoint whp. Therefore, almost every vertex in ${{\cG_n^c}}$ is a [<span style="font-variant:small-caps;">scc</span>]{} itself and ${{\cD_{n,k}^{\mathrm A}}}$ is very much like ${{\cD_{n,k}}}$ with the giant contracted into a single vertex. The *$d$-core* of an undirected graph is the maximum induced subgraph in which all vertices have degree at least $d$. Similarly the *$d$-in-core* of a digraph can be defined as the maximum induced sub-digraph in which all vertices have in-degree at least $d$. Let ${\cO_n}$ denote the set of vertices in the one-in-core of ${{\cD_{n,k}}}$. Note that ${{\cG_n}}\subseteq {\cO_n}$ since a [<span style="font-variant:small-caps;">scc</span>]{} induces a sub-digraph with each vertex having in-degree at least one. Also note that cycles cannot exist outside ${\cO_n}$, for otherwise they contradict the maximality of ${\cO_n}$. Now assume that every vertex can reach ${{\cG_n}}$, which happens whp by @Grusho1973. Then ${{\cD_{n,k}}}$ can be divided into three layers: the center is ${{\cG_n}}$; then comes ${\cO_n}\setminus{{\cG_n}}$, which consists of cycles outside ${{\cG_n}}$ and paths from these cycles to ${{\cG_n}}$; the outermost is ${{\cO_n}^c}\equiv [n] \setminus {\cO_n}$, which is acyclic. at (0,0) []{}; at (3.4,2.7) [${{\cG_n}}$]{}; at (1.5,2.7) [${\cO_n}\! \setminus \! {{\cG_n}}$]{}; at (-0.1,2.7) [${{\cO_n}^c}$]{}; Since there cannot be many vertices in cycles outside the giant, the middle layer ${\cO_n}\setminus {{\cG_n}}$ must be very “thin”. Thus if we can prove $({|{\cO_n}|}- {{\nu_k}}n)/\sqrt{n}$ converges to a normal distribution, then we can also prove it for ${{|{{\cG_n}}|}}$. The event ${|{\cO_n}|}= s$ happens if and only if there is a set of vertices $\cS$ with $|\cS|=s$ such that: (a) ${{\cD_{n,k}}}[\cS]$, the sub-digraph induced by $\cS$, has minimum in-degree one (*surjective*) and there are no arcs going from $\cS$ to $\cS^c$ (*closed*), which we refer to as $\cS$ being a *$k$-surjection* (since ${{\cD_{n,k}}}[\cS]$ is equivalent to a surjective function from $[ks]$ to $[s]$); (b) ${{\cD_{n,k}}}[\cS^c]$ is acyclic. The probability of (a) can be computed by counting the number of surjective functions. And we are able to show that the probability of (b) converges to a constant. Note that for a fixed set [$\cS$]{} (a) and (b) are independent because they depend on the endpoints of two disjoint sets of arcs. Thus we can get the limit of $\p{{\cO_n}= \cS}$. Since the one-in-core of a digraph is unique, $\p{{|{\cO_n}|}= s} = \sum_{\cS \subseteq [n]:|\cS|=s} \p{{\cO_n}= \cS}$. Thus we can finish the proof by computing the characteristic function of $({|{\cO_n}|}- {{\nu_k}}n)/\sqrt{n}$. Note that although our formula for $\p{{|{\cO_n}|}=s}$ is inspired by and resembles @Carayol12’s formula for $\p{{|{{\cS_{1}}}|}= s}$, we actually prove the result from scratch without relying on previous work. Since we are able to derive explicit expressions of all the constants in our formula, the computation of the characteristic function becomes quite simple. Furthermore, to our knowledge this is the first self-contained proof. Thus in Section \[sec:giant\] we prove: \[thm:CLL\] Let $\cZ$ denote a standard normal random variable. Then as $n \to \infty$, $$\begin{aligned} \frac{{|{\cO_n}|}- {{\nu_k}}n}{\sigma_k \sqrt{n}} \inlaw \cZ, \qquad \frac{{{|{{\cG_n}}|}}- {{\nu_k}}n}{\sigma_k \sqrt{n}} \inlaw \cZ, \qquad \frac{\max_{v \in [n]} |{\cS_{v}}| - {{\nu_k}}n}{\sigma_k \sqrt{n}} \inlaw \cZ, \end{aligned}$$ where ${{\nu_k}}$ and $\sigma_k$ are constants defined by $${{\nu_k}}\equiv \frac {{{\tau_k}}} k, \qquad \qquad \qquad \sigma_k^2 \equiv \frac{{{{\tau_k}}}}{k e^{{{{\tau_k}}}}(1-ke^{{-{{\tau_k}}}})},$$ and ${{{\tau_k}}}$ is the unique positive solution of $1-{{{\tau_k}}}/k - e^{{-{{\tau_k}}}} = 0$. Equivalently, ${{\nu_k}}$ is the unique positive solution of $1-{{\nu_k}}=e^{-k {{\nu_k}}}$ and $$\sigma_k^2 = \frac{{{\nu_k}}(1-{{\nu_k}})}{1-k(1-{{\nu_k}})}.$$ Let $G(n,m)$ be a Erdős–Rényi random graph, i.e., a graph chosen uniformly at random from all graphs with $n$ vertices and $m$ edges [@Erdos60onthe]. It is well-known that for $k > 1$, $|\cC_{\max}^n|$—the size of the largest component in $G(n,m=nk/2)$—is $ ({{\nu_k}}+o(1)) n$ whp. Moreover, $ (|\cC_{\max}^n|-{{\nu_k}}n)/\sqrt{n}$ also converges in distribution to a normal random variable with variance $\sigma_k^2$ (see, e.g., @durrett2007random). Intuitively, this is because the in-degree of a vertex in ${{\cD_{n,k}}}$ has asymptotically a Poisson distribution of mean $k$. Thus a backward exploration process from vertex in ${{\cD_{n,k}}}$ is approximately a Galton-Watson process with survival probability ${{\nu_k}}$, as is the exploration process starting from a vertex in $G(n,m=nk/2)$. =\[draw,circle,fill=white,inner sep=0,minimum size=4pt\] =\[line width=3pt\] Section \[sec:dag\] studies the part of ${{\cD_{n,k}}}$ outside the giant, which determines the structure of ${{\cD_{n,k}^{\mathrm A}}}$ and supports the proof of Theorem \[thm:CLL\]. Our results are summarized in two theorems, where all our logarithms are natural: \[thm:cycle\] We have: (a) Let $L_n$ be the length of the longest cycle in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$. Then $L_n = O_p(1)$. (b) Let ${C_{n}}$ be the number of cycles in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$. Then $${C_{n}}\inlaw {\mathop{\mathrm{Poi}}}\left( \log \frac{1}{1-k {{e^{-{{\tau_k}}}}}} \right),$$ where ${\mathop{\mathrm{Poi}}}(x)$ denotes the Poisson distribution with mean $x$. (c) Let ${C_{n,\ell}}$ be the number of cycles of length $\ell$ in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$. Then for all fixed $\ell \ge 1$, $${C_{n,\ell}}\inlaw {\mathop{\mathrm{Poi}}}\left(\frac{(k {{e^{-{{\tau_k}}}}})^{\ell}}{\ell}\right).$$ \[thm:dag\] Let ${\cS^{\prime}_{v}} \equiv {\cS_{v}} \cap {{\cG_n^c}}$, i.e., ${\cS^{\prime}_{v}}$ is the spectrum of $v$ in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$. Let ${\mathop{\mathrm{dist}}}(v,u)$ be the distance from $v$ to $u$, i.e., the length of the shortest directed path from $v$ to $u$. Then (a) $\p{\cup_{v \in {{\cG_n^c}}}[{\mathop{\mathrm{arc}}}({{\cD_{n,k}}}[{\cS^{\prime}_{v}}]) - |{\cS^{\prime}_{v}}| \ge 1]} = o(1)$, where ${\mathop{\mathrm{arc}}}(\cdot)$ denotes the number of arcs. In other words, whp every spectrum in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$ is a tree or a tree plus an extra arc. (b) Let $S_n \equiv \max_{v \in {{\cG_n^c}}}|{\cS^{\prime}_{v}}|$. Let ${{\lambda_k}}\equiv (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1}$. Then $$\frac {S_n}{\log n} \inprob \frac 1{\log(1/{{\lambda_k}})}.$$ (c) Let $W_n \equiv \max_{v \in {{\cG_n^c}}} \min_{u \in {{\cG_n}}} {\mathop{\mathrm{dist}}}(v,u)$, i.e., the maximum distance to ${{\cG_n}}$. Then $$\frac {W_n}{\log_k \log n} \inprob 1.$$ (d) Let $M_n$ be the length of the longest path in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$. Then $$\frac {M_n}{\log n} \inprob \frac 1 {\log(e^{{{\tau_k}}}/k)}.$$ (e) Let $D_n \equiv \max_{v \in {{\cG_n^c}}} \max_{u \in {\cS^{\prime}_{v}}} {\mathop{\mathrm{dist}}}(v, u)$. Then $$\frac {D_n}{\log n} \inprob \frac 1 {\log(e^{{{\tau_k}}}/k)}.$$ The rest of the paper gives some other results regarding this model. Section \[sec:phase\] shows that ${{\cD_{n,k}}}$ exhibits a phase transition for strong connectivity. Section \[sec:simple\] extends some of our results to simple $k$-out digraphs. Section \[sec:typical:distance\] analyzes the typical distances in ${{\cD_{n,k}}}$ with a technique called path counting, which is very different from the method used by @Perarnau2014 in [@Perarnau2014]. Section \[sec:extension\] suggests some extensions of this model. Lemma \[lem:middle:layer\] shows that ${|{\cO_n}|}-{{|{{\cG_n}}|}}= O_p(1)$. The intuition is that a digraph with minimal in-degree and out-degree at least one is likely to have a large [<span style="font-variant:small-caps;">scc</span>]{}. This phenomenon is also observed in $D(n,p)$, which is a random digraph of $n$ vertices with each possible arc existing independently with probability $p$. @RSA:RSA20622 [thm. 1.3] showed that in $D(n,p)$ the $ (1,1)$-core—the maximal induced sub-digraph in which each vertex has in-degree and out-degree at least one—differs from the largest [<span style="font-variant:small-caps;">scc</span>]{} in size by at most $O((\log n)^8)$, whp. This intuition is also used for studying the asymptotic counts of strongly connected digraphs (see @RSA:RSA20416 [@RSA:RSA20433]). The size of the one-in-core {#sec:giant} =========================== The law of large numbers for the one-in-core -------------------------------------------- To prove Theorem \[thm:CLL\], we first need to narrow the range of ${|{\cO_n}|}$ to close to ${{\nu_k}}n$. \[thm:LLL\] For all fixed $\delta \in (0,1/2)$, $$\p{{|{\cO_n}|}\notin \cI_n} \le \frac {1+o(1)} n,$$ where $\cI_n \equiv {[{{\nu_k}}n-{n^{1/2+\delta}}, {{\nu_k}}n +{n^{1/2+\delta}}]}$. Thus ${|{\cO_n}|}/ n \inprob {{\nu_k}}$, which gives the theorem its name. Let $K_s$ be the number of $k$-surjections of size $s$ in ${{\cD_{n,k}}}$. Then it suffices to show that $\p{\sum_{s \notin \cI_n} K_s \ge 1} \le (1+o(1))/n$. As argued in the introduction, for a set of vertices $\cS$ to be the one-in-core, it must also be a $k$-surjection, i.e., every vertex in ${{\cD_{n,k}}}[\cS]$, the sub-digraph induced by $\cS$, must have minimum in-degree one ($\cS$ is *surjective*), and there are no arcs going from $\cS$ to $\cS^{c}$ ($\cS$ is *closed*). Thus $$\p{\cS \text{ is a \(k\)-surjection}} = \p{\cS \text{ is surjective}~|~\cS \text{ is closed}}\p{\cS \text{ is closed}}.$$ Computing the limit of the two factors shows that: We have $$\p{\sum_{s \notin {{\cI_n}}} K_s \ge 1} \le \frac {1+o(1)} n.$$ And for $s \in {{\cI_n}}$ $$\e{K_s} \sim \frac{1}{\sqrt{2 \pi(1-ke^{-{{\tau_k}}})n}} ~ g\left(\frac s n\right) ~ \left[ f\left(\frac s n \right) \right]^{n},$$ where $$g(x) \equiv \frac 1 {\sqrt{x(1-x)}}, \qquad \qquad f(x) \equiv \left[ \frac{x^{k-1} \gamma_k}{(1-x)^{(1-x)/x} } \right]^{x},$$ and $\gamma_k \equiv \left( \frac k {e{{{\tau_k}}}} \right)^k(e^{{{{\tau_k}}}} -1)$. \[lem:k:surj\] Theorem \[thm:LLL\] follows immediately. The proof of Lemma \[lem:k:surj\] is postponed to the appendix. (The two functions $f(x)$ and $g(x)$ are also studied by @Carayol12.) The central limit law of the one-in-core ---------------------------------------- In this section we prove the part of Theorem \[thm:CLL\] about ${|{\cO_n}|}$. The rest of the theorem appears as corollaries in Section \[sec:dag\]. Let ${{\partial \cO_n}}= {{|{\cO_n}|}- {{\nu_k}}n }$. Then ${{\partial \cO_n}}$ takes values in $ [n] - {{\nu_k}}n \equiv \{s : {{\nu_k}}n + s \in [n] \}. $ As Theorem \[thm:LLL\] shows, whp ${{\partial \cO_n}}\le {n^{1/2+\delta}}$ for all fixed $\delta \in (0,1/2) $. Thus it suffices to consider only the probability that ${{\partial \cO_n}}$ takes value in the set $$\cJ_n \equiv ([n] - {{\nu_k}}n) \cap \left[-{n^{1/2+\delta}},{n^{1/2+\delta}}\right],$$ for some fixed $\delta \in (0,1/2)$. Thus the characteristic function of ${{\partial \cO_n}}/\sqrt{n}$ is $$\begin{aligned} \phi_{n}(t) & = \sum_{s \in ([n]-{{\nu_k}}n) \setminus \cJ_n} e^{its/\sqrt{n}} \p{{{\partial \cO_n}}= s} + \sum_{s \in \cJ_n} e^{its/\sqrt{n}} \p{{{\partial \cO_n}}= s} \\ & = o(1) + \sum_{s \in \cJ_n} e^{its/\sqrt{n}} \p{{{\partial \cO_n}}= s}.\end{aligned}$$ Let $\cS$ be a set of vertices with $|\cS| = {{\nu_k}}n + s$ for some $s \in \cJ_n$. Recall that ${\cO_n}= \cS$ if and only if $\cS$ is a $k$-surjection and ${{\cD_{n,k}}}[\cS^c]$ is acyclic, two events that are independent. By Theorem \[thm:cyc:v\] in Section \[sec:sub:cycle\], $\p{\text{${{\cD_{n,k}}}[\cS^c]$ is acyclic}} \sim 1-ke^{{-{{\tau_k}}}}.$ Also recall that [$K_{x}$]{} counts the number of $k$-surjections of size $x$. It follows from Lemma \[lem:k:surj\] that $$\begin{aligned} \p{{{\partial \cO_n}}= s} & = \sum_{\cS \subseteq [n]:|\cS|={{\nu_k}}n + s} \p{{\cO_n}= \cS} \\ & = \sum_{\cS \subseteq [n]:|\cS|={{\nu_k}}n + s} \p{\text{\(\cS\) is a \(k\)-surjection}} \times \p{{{\cD_{n,k}}}[\cS^c] \text{ is acyclic}} \\ & \sim (1-k e^{{-{{\tau_k}}}}) \e{K_{{{\nu_k}}n + s}} \\ & = \sqrt{\frac{1-ke^{-{{\tau_k}}}}{2 \pi}} ~ \frac{1}{\sqrt{n}} ~ g\left({{\nu_k}}+ \frac s n\right) ~ \left[ f\left({{\nu_k}}+ \frac s n \right) \right]^{n},\end{aligned}$$ where $K_{x}$, $f(x)$ and $g(x)$ are defined as in the previous subsection. If $s \in \cJ_n$, then Lemma \[lem:function:h\] in the appendix shows that $$g\left( {{\nu_k}}+ \frac s n \right) = \left(1 + O\left( \frac{|s|}{{n}} \right) \right)\frac{1}{\sigma_k\sqrt{1-k e^{-{{\tau_k}}}}},$$ and $$f\left( {{\nu_k}}+ \frac s n \right) = \exp\left\{- \frac{s^2}{2 \sigma_k^2 n^2} \right\} + O\left( \frac{|s|^3}{{n^3}} \right).$$ Therefore, choosing $\delta$ small enough, e.g., $\delta = 1/9$, we have $$\begin{aligned} \sum_{s \in \cJ_n} e^{its/\sqrt{n}} \p{{{\partial \cO_n}}= s} & \sim \frac{1}{\sqrt{2 \pi \sigma_k^2}} \frac{1}{\sqrt{n}} \sum_{s \in \cJ_n} e^{its/\sqrt{n}} \exp\left\{-\frac{s^2}{2 \sigma_k n}\right\} \\ & = o(1) + \frac{1}{\sqrt{2 \pi \sigma_k^2}} ~ \int_{-n^{\delta}}^{n^{\delta}} e^{itx} \exp\left\{-\frac{x^2}{2 \sigma_k^2}\right\} ~ {\mathrm d}x \\ & = o(1) + \frac{1}{\sqrt{2 \pi \sigma_k^2}} ~ \int_{-\infty}^{\infty} e^{itx} \exp\left\{-\frac{x^2}{2 \sigma_k^2}\right\} ~ {\mathrm d}x \\ & = o(1) + \exp\left( \frac{\sigma_k^2 t^2}{2} \right).\end{aligned}$$ Thus the characteristic function of ${{\partial \cO_n}}/\sqrt{n}$ converges to $\exp(\sigma_k^2t^2/2)$, the characteristic function of $\sigma_k \cZ$. It follows from the central limit theorem that ${{\partial \cO_n}}/\sqrt{n}$ converges to $\sigma_k \cZ$ in distribution. Note that using the estimates of this section, we actually have a local limit theorem for [${|{\cO_n}|}$]{}. The structure of the directed acyclic graph {#sec:dag} =========================================== De-randomizing the giant ------------------------ Since a [<span style="font-variant:small-caps;">scc</span>]{} induces a sub-digraph in which each vertex has in-degree at least one, a closed [<span style="font-variant:small-caps;">scc</span>]{} is also a $k$-surjection. Lemma \[lem:k:surj\] implies that whp all $k$-surjections are of sizes in $\cI_n \equiv {[{{\nu_k}}n-{n^{1/2+\delta}}, {{\nu_k}}n +{n^{1/2+\delta}}]}$. When this happens, as ${{\nu_k}}> 1/2$ (Lemma \[lem:constant\]), there exists one and only one closed [<span style="font-variant:small-caps;">scc</span>]{} and it is ${{\cG_n}}$. And if ${{\cG_n}}$ is the only closed [<span style="font-variant:small-caps;">scc</span>]{}, then every vertex must be able to reach it. This can be summarized as: Whp ${{|{{\cG_n}}|}}\in \cI_n$ and ${{\cG_n}}$ is reachable from all vertices. \[lem:giant:size\] Since ${{e^{-{{\tau_k}}}}}\equiv 1-{{\tau_k}}/k \equiv 1 - {{\nu_k}}$, the above lemma implies that whp $|{{\left|\cG_n^c \right|}}- {{e^{-{{\tau_k}}}}}n| \le {n^{1/2+\delta}}$. Thus the structure of ${{\cD_{n,k}}}[{{\cG_n^c}}]$, the sub-digraph induced by ${{\cG_n^c}}\equiv[n]\setminus{{\cG_n}}$, should be close to that of a sub-digraph induced by a fixed set of vertices whose size is close to ${{e^{-{{\tau_k}}}}}n$. Formally, we have: Let $f_n$ be a sequence of integer-valued functions on a sequence of digraphs. Let $X$ be an integer-valued random variable. If there exists a sequence $\varepsilon_n \to 0$ such that $$\sup_{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in {{\cI_n}}}{{\left\|f_n({{\cD_{n,k}}}[{{\cV_n}^c}]), X\right\|_{\textsc{tv}}}} \le \varepsilon_n,$$ where ${{\cV_n}^c}\equiv [n] \setminus {\cV_n}$ and ${{\left\|\,\cdot\,,\,\cdot\right\|_{\textsc{tv}}}}$ denotes the total variation distance, then $$f_n({{\cD_{n,k}}}[{{\cG_n^c}}]) \inlaw X.$$ \[lem:giant:deterministic\] Define the event $E_n = [{{|{{\cG_n}}|}}\in {{\cI_n}}]$. Let $m$ be an integer, let ${\cV_n}\subseteq [n]$ be a fixed set of vertices with ${|{\cV_n}|}\in {{\cI_n}}$. Recall that since ${{\nu_k}}> 1/2$, ${|{\cV_n}|}> n/2 $ for large $n$. Thus the event $[{{\cG_n}}= {\cV_n}]$ depends only on the induced sub-digraph ${{\cD_{n,k}}}[{\cV_n}]$, which is independent of ${{\cD_{n,k}}}[{{\cV_n}^c}]$. Therefore the two events $[{{\cG_n}}= {\cV_n}]$ and $[f_n({{\cD_{n,k}}}[{{\cV_n}^c}]) = m]$ are independent. Using this observation and Lemma \[lem:giant:size\], we have $$\begin{aligned} & \p{f_n({{{\cD_{n,k}}}[{{\cG_n^c}}]}) = m} \\ & = \p{[f_n({{{\cD_{n,k}}}[{{\cG_n^c}}]}) = m] \cap E_n^c} + \p{[f_n({{{\cD_{n,k}}}[{{\cG_n^c}}]}) = m] \cap E_n} \\ & = o(1) + \sum_{{\cV_n}\subseteq [n]:|{\cV_n}|\in{{\cI_n}}} \p{f_n({{{\cD_{n,k}}}[{\cV_n}^c]}) = m ~|~ {{\cG_n}}= {\cV_n}} \p{{{\cG_n}}= {\cV_n}} \\ & \le o(1) + \sum_{{\cV_n}\subseteq [n]:|{\cV_n}|\in{{\cI_n}}} (\p{X = m} + \varepsilon_n) \p{{{\cG_n}}= {\cV_n}} \\ & \le o(1) + \p{X=m}. \end{aligned}$$ Similarly we have $\p{f_n({{{\cD_{n,k}}}[{{\cG_n^c}}]}) =m } \ge \p{X=m} + o(1)$. Since this applies to all integers $m$, $f_n({{{\cD_{n,k}}}[{{\cG_n^c}}]}) \inlaw X$. Let $\cE_n$ be a sequence of sets of digraphs. If there exists a sequence $\varepsilon_n \to 0$ such that $$\sup_{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in {{\cI_n}}}\p{{{{\cD_{n,k}}}[{\cV_n}^c]}\notin \cE_n} \le \varepsilon_n,$$ then whp ${{{\cD_{n,k}}}[{{\cG_n^c}}]}\in \cE_n$. \[cor:giant:determnistic\] This follows from the previous lemma by taking $X \equiv 1$ and $f_n$ to be the indicator function that a digraph is in $\cE_n$. The rest of this section proves Theorem \[thm:cycle\] and Theorem \[thm:dag\]. But instead of working on ${{\cG_n^c}}$ directly, we prove similar theorems on fixed sets of vertices, and then apply the above lemma or its corollary to get the final result. Cycles outside the giant {#sec:sub:cycle} ------------------------ In this subsection, we show the following: \[thm:cyc:v\] Let $\omega_n \to \infty$ be an arbitrary sequence. There exists a sequence $\varepsilon_n = o(1)$ such that for all fixed sets of vertices ${\cV_n}\subseteq [n]$ with ${|{\cV_n}|}\in {{\cI_n}}$, we have: (a) Let $L_{n}^{*}$ be the length of the longest cycle in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. Then $\p{L_n^{*} > \omega_n} \le \varepsilon_n$. (b) The probability that ${{{\cD_{n,k}}}[{\cV_n}^c]}$ contains vertex-intersecting cycles is at most $\varepsilon_n$. (c) Let ${C_{n,\ell}^*}$ be the number of cycles of length $\ell$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. Let $X_\ell = {\mathop{\mathrm{Poi}}}({(k {{e^{-{{\tau_k}}}}})^{\ell}}/{\ell})$. Then for all fixed $\ell$, ${{\left\|{C_{n,\ell}^*},X_{\ell}\right\|_{\textsc{tv}}}} \le \varepsilon_n.$ (d) Let ${C_n^*}$ be the number of cycles in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. Let $X = {\mathop{\mathrm{Poi}}}(\log\frac{1}{1-k {{e^{-{{\tau_k}}}}}})$. Then ${{\left\|{C_n^*},X\right\|_{\textsc{tv}}}} \le \varepsilon_n.$ As a result, $|\p{{C_n^*}= 0} - (1-ke^{-{{\tau_k}}})| \le 2 \varepsilon_n$. Theorem \[thm:cycle\] follows from the above theorem and Lemma \[lem:giant:deterministic\]. Our proof is inspired by @Cooper2004’s work on the directed configuration model [@Cooper2004]. Note that the Cooper-Frieze model is different from that studied by us. In their model, both in-degrees and out-degrees are predetermined, whereas we require all out-degrees to be $k$ but the in-degrees are random. The intuition behind Theorem \[thm:cyc:v\] is that when two cycles share vertices, they contain fewer vertices than arcs. So if we fix the “shape” of a pair of such cycles, the number of ways to label them times the probability that they both exist is $o(1)$. Thus whp cycles in ${{\cV_n}^c}$ are vertex-disjoint and the total number of cycles has a distribution close to a sum of independent indicator random variables. In the following proof, instead of finding the exact $\varepsilon_n$, we derive implicit $o(1)$ upper bounds for probabilities and total variation distances which only requires that ${|{\cV_n}|}\in {{\cI_n}}$. \[lem:cyc:long\] Let ${\overline{C_{n}^*}}\equiv \sum_{1 \le \ell \le \omega_n} {C_{n,\ell}^*}$. Then $\p{{C_n^*}\ne {\overline{C_{n}^*}}} = o(1)$. Define $(x)_{\ell} \equiv x(x-1)\cdots(x-\ell+1)$. Then the number of all possible cycles of length $\ell$ is $({|{\cV_n}^c|})_{\ell} k^{\ell}/\ell$. (Note that we are also considering the labels on arcs, which makes the counting easier.) And the probability that such a cycle exists is $n^{-\ell}$. Recalling that ${|{\cV_n}^c|}\in {[{{e^{-{{\tau_k}}}}}n-{n^{1/2+\delta}}, {{e^{-{{\tau_k}}}}}n +{n^{1/2+\delta}}]}$, we have $$\begin{aligned} \E{{C_{n,\ell}^*}} = \frac 1 \ell ({|{\cV_n}^c|})_\ell k^{\ell} \left( \frac{1}{n} \right)^{\ell} \le \left( k {{e^{-{{\tau_k}}}}}\left(1+ {{O\left( {n^{{-1}/2+\delta}}\right)}}\right) \right)^\ell. \label{eq:cyc:expe} \end{aligned}$$ Since $k {{e^{-{{\tau_k}}}}}\equiv k-{{\tau_k}}< 1$ (Lemma \[lem:constant\]), there exists a constant $c_1 < 1$ such that the above is less than $c_1^{\ell}$ for $n$ large enough. Since ${C_n^*}\ne {\overline{C_{n}^*}}$ if and only if $\sum_{\ell > \omega_n} {C_{n,\ell}^*}\ge 1$, $$\p{{C_n^*}\ne {\overline{C_{n}^*}}} = \p{\sum_{\ell > \omega_n} {C_{n,\ell}^*}\ge 1} \le \E{\sum_{\ell > \omega_n} {C_{n,\ell}^*}} \le {{O\left( c_1^{\omega_n} \right)}} = o(1). \tag*{\qedhere}$$ Since $L_n^{*} > \omega_n$ if and only if ${\overline{C_{n}^*}}\ne {C_n^*}$, part $(a)$ of Theorem \[thm:cyc:v\] follows. From now on let $\omega_n = \log \log n$. We show that: Let $X$ and $X_\ell$ be as in Theorem \[thm:cyc:v\]. Then ${{\left\|{\mathop{\mathrm{Poi}}}(\e{{\overline{C_{n}^*}}}), X\right\|_{\textsc{tv}}}} = o(1).$ And for all $\ell \le \omega_n$, ${{\left\|{\mathop{\mathrm{Poi}}}(\e{{C_{n,\ell}^*}}) ,X_\ell\right\|_{\textsc{tv}}}} = o(1). $ \[lem:cyc:poisson\] For all $\ell \le \omega_n$, by we have $$\begin{aligned} \e{{C_{n,\ell}^*}} = \frac 1 \ell \left({{e^{-{{\tau_k}}}}}n + {{O\left( {n^{1/2+\delta}}\right)}}\right)_\ell k^{\ell} \left( \frac{1}{n} \right)^{\ell} = \frac {(k {{e^{-{{\tau_k}}}}})^\ell} \ell (1 + O(\ell {n^{{-1}/2+\delta}})).\end{aligned}$$ Thus $$\begin{aligned} \e{{\overline{C_{n}^*}}} & = \sum_{1 \le \ell \le \omega_n} \E{{C_{n,\ell}^*}} = \log \left(\frac{1}{1-k {{e^{-{{\tau_k}}}}}}\right) + {{O\left( \omega_n {n^{{-1}/2+\delta}}\right)}}.\end{aligned}$$ Therefore $\e{{\overline{C_{n}^*}}} \to \e X$ and $\e{{C_{n,\ell}^*}} \to \e X_\ell$, which implies the lemma. By the two previous lemmas, it suffices to show that $${{\left\|{\overline{C_{n}^*}},{\mathop{\mathrm{Poi}}}(\e{{\overline{C_{n}^*}}})\right\|_{\textsc{tv}}}} = o(1), \quad {{\left\|{C_{n,\ell}^*},{\mathop{\mathrm{Poi}}}(\e{{C_{n,\ell}^*}})\right\|_{\textsc{tv}}}} = o(1)\quad \text{for all fixed \(\ell\)}.$$ We prove this by using a theorem of @Arratia1989 [@Arratia1989]. (A similar result is proved by @Barbour1992poisson). The method is known as the Chen-Stein method because it was first developed by @Chen1975 who applied @Stein1972’s theory [@Stein1972] on probability metrics to Poisson distributions. Let $\cC$ be the space of all possible cycles of length at most $\omega_n$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. For $\alpha \in \cC$, let $\cB_\alpha \subseteq \cC$ be the set of cycles that are vertex-intersecting with $\alpha$. Let $\indd{\alpha}$ be the indicator that a cycle $\alpha$ appears in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. Define $$\begin{aligned} b_1 \equiv \sum_{\alpha \in \cC} \sum_{\beta \in \cB_\alpha} \e \indd{\alpha} \e \indd{\beta}, \qquad b_2 \equiv \sum_{\alpha \in \cC} \sum_{\beta \in \cB_\alpha:\beta\ne\alpha} \E{\indd{\alpha}\indd{\beta}}, \qquad & b_3 \equiv \sum_{\alpha \in \cC} s_\alpha,\end{aligned}$$ where $$s_{\alpha} = \e \left| \E{\indd{\alpha}| \sigma\left( \indd{\beta}:\beta \in \cC \setminus \cB_{\alpha} \right)} - \e \indd{\alpha} \right|,$$ and $\sigma(\cdot)$ denotes the sigma algebra generated by $(\cdot)$. Theorem 1 of @Arratia1989 states that $${{\left\| {\overline{C_{n}^*}}, \, {\mathop{\mathrm{Poi}}}(\e{{\overline{C_{n}^*}}}) \right\|_{\textsc{tv}}}} \le 2(b_1 + b_2 + b_3).$$ If $\beta \in \cC \setminus \cB_{\alpha}$, then $\alpha$ and $\beta$ are vertex-disjoint. Thus $\indd{\alpha}$ and $\indd{\beta}$ are independent and $s_{\alpha} = 0$ for all $\alpha \in \cC$, i.e., $b_3 = 0$. It suffices to show that $b_1$ and $b_2$ are $o(1)$. Let $|\alpha|$ denote the length of a cycle $\alpha$. Fix $\ell_1 \le \omega_n$ and $\ell_2 \le \omega_n$. There are at most ${|{\cV_n}^c|}^{\ell_1} k^{\ell_1}$ cycles of length $\ell_1$. For $|\alpha| = \ell_1$, there are at most $\ell_1 {|{\cV_n}^c|}^{\ell_2-1} k^{\ell_2}$ cycles of length $\ell_2$ that share at least one vertex with $\alpha$. Since $({|{\cV_n}^c|})^{\ell} = (1+o(1))({{e^{-{{\tau_k}}}}}n)^{\ell}$ for $\ell \le \omega_n$, $$\begin{aligned} \sum_{\alpha \in \cC:|\alpha| = \ell_1} \sum_{\beta \in \cB_\alpha:|\beta| = \ell_2} \e \indd{\alpha} \e \indd{\beta} & \le (1+o(1)) \left[({{e^{-{{\tau_k}}}}}n)^{\ell_1} k^{\ell_1} \right] \left[\ell_1 ({{e^{-{{\tau_k}}}}}n)^{\ell_2-1} k^{\ell_2} \right] \left( \frac{1}{n} \right)^{\ell_1+\ell_2} \\ & = (1+o(1)) \frac 1 {{{e^{-{{\tau_k}}}}}n} \left[\ell_1 ({{e^{-{{\tau_k}}}}}k)^{\ell_1} \right] \left[({{e^{-{{\tau_k}}}}}k)^{\ell_2} \right].\end{aligned}$$ Therefore $$\begin{aligned} b_1 & = \sum_{1 \le \ell_1 \le \omega_n} \sum_{1 \le \ell_2 \le \omega_n} \sum_{\alpha \in \cC:|\alpha| = \ell_1} \sum_{\beta \in \cB_\alpha:|\beta| = \ell_2} \e \indd{\alpha} \e \indd{\beta} \\ & \le (1+o(1)) \frac 1 {{{e^{-{{\tau_k}}}}}n} \sum_{\ell_1 \ge 1} \sum_{\ell_2 \ge 1} \left[\ell_1 (k{{e^{-{{\tau_k}}}}})^{\ell_1} \right] \left[(k{{e^{-{{\tau_k}}}}})^{\ell_2} \right] \\ & \le (1+o(1)) \frac 1 {{{e^{-{{\tau_k}}}}}n} \left[\sum_{\ell_1 \ge 1}\ell_1 (k {{e^{-{{\tau_k}}}}})^{\ell_1} \right] \left[\sum_{\ell_2 \ge 1} (k {{e^{-{{\tau_k}}}}})^{\ell_2} \right]\end{aligned}$$ which is ${{O\left( 1/n \right)}}$ since both sums converge. To compute $b_2$, we upper bound the number of pairs of vertex-intersecting cycles that could possibly appear in ${{\cD_{n,k}}}[{{\cV_n}^c}]$ at the same time. Let $\alpha$ and $\beta$ be such a pair. Let $V(\alpha), A(\alpha), V(\beta), A(\beta)$ be the vertex set and (labeled) arc set of $\alpha$ and $\beta$ respectively. Let $\alpha \cup \beta$ be the digraph of vertex set $V = V(\alpha) \cup V(\beta)$ and arc set $A = A(\alpha) \cup B(\beta)$. Assume that $|V|=s$ and $|A|=s+t$. Note that as $\alpha$ and $\beta$ share at least one vertex, $t \ge 1$. Since $V \subset [n]$, we can relabel the $s$ vertices in $\alpha \cup \beta$ with $[s]$ such that the order of the vertex labels is maintained. The result is a digraph with vertex set $[s]$ and $s+t$ arcs labeled with $[k]$. There are at most $ (s^2)^{s+t} k^{s+t}$ such digraphs, since there are at most $s^2$ choices of endpoints and $k$ choices of labels for each of the $s+t$ arcs. Each digraph of this type corresponds to at most $\binom{{|{\cV_n}^c|}}{s} \le {|{\cV_n}^c|}^{s}$ pairs of cycles like $\alpha$ and $\beta$. Thus there are at most ${|{\cV_n}^c|}^{s} (s^2)^{s+t} k^{s+t}$ such pairs. Summing over $s$ and $t$, we have $$\begin{aligned} b_2 & \le \sum_{1 \le s \le 2 \omega_n} \sum_{1 \le t \le 2 \omega_n} {|{\cV_n}^c|}^{s} (s^{2})^{s+t} k^{s+t} \E{\indd{\alpha}\indd{\beta}} \\ & \le \sum_{1 \le s \le 2 \omega_n} \sum_{1 \le t \le 2 \omega_n} \left( e^{-{{\tau_k}}}n + {n^{1/2+\delta}}\right)^s (2 \omega_n)^{2 \times 4\omega_n} k^{s+t} \frac{1}{n^{s+t}} \\ & \le (2 \omega_n)^{8\omega_n} \sum_{1 \le s \le 2 \omega_n} \sum_{1 \le t \le 2 \omega_n} \frac{\left( n + e^{{{\tau_k}}} {n^{1/2+\delta}}\right)^s}{n^{s}} (k e^{-{{\tau_k}}})^{s} \frac{k^t}{n^t} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:b2} \\ & \le {{O\left( \frac{1}{n} \right)}} (2 \omega_n k)^{8\omega_n} \sum_{1 \le s \le 2 \omega_n} \sum_{1 \le t \le 2 \omega_n} (1+e^{{{\tau_k}}}n^{-1/2+\delta})^{2\omega_n} \qquad (ke^{-{{\tau_k}}} < 1/2) \\ & \le {{O\left( \frac{1}{n} \right)}} (2 \omega_n k)^{8\omega_n} (2 \omega_n)^{2} \left( 1 + {{O\left( n^{-1/2+\delta}\omega_n \right)}} \right) \to 0,\end{aligned}$$ where the last step we use that $\omega_n = \log \log n$. Thus part (d) of Theorem \[thm:cyc:v\] for ${C_n^*}$ is proved. We can prove part (c) for ${C_{n,\ell}^*}$ using the same method by limiting $\cC$ to contain only cycles of a fixed length $\ell$. Note that the above inequality shows that the probability that there exist vertex-intersecting cycles in ${{{\cD_{n,k}}}[{\cV_n}^c]}$ is $o(1)$, thus part (b) is also proved. The method used above can be easily adapted to prove similar results for undirected cycles, like the following lemma which is needed in the study of spectra in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$: Let $\psi_n \to \infty$ be an arbitrary sequence. There exists a sequence $\varepsilon_n = o(1)$ such that for all fixed sets of vertices ${\cV_n}$ with ${|{\cV_n}|}\in {{\cI_n}}$, we have: (a) The probability that ${{{\cD_{n,k}}}[{\cV_n}^c]}$ contains an undirected cycle of length greater than $\psi_n$ is at most $\varepsilon_n$. (b) The probability that ${{{\cD_{n,k}}}[{\cV_n}^c]}$ contains vertex-intersecting undirected cycles is at most $\varepsilon_n$. \[lem:cyc:undirected\] Let ${{U}_{\ell}}$ be the number of undirected cycles of length $\ell$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. Then $$\E{{{U}_{\ell}}} \le \frac 1 \ell ({|{\cV_n}^c|})^{\ell} (2k)^{\ell} \frac 1 {n^{\ell}} \le \left( 2k {{e^{-{{\tau_k}}}}}(1+ {{n^{{-1}/2+\delta}}}) \right)^\ell,$$ where the $2$ comes from the fact that each edge in an undirected cycle has two possible directions. Since $2k {{e^{-{{\tau_k}}}}}= 2(k - {{\tau_k}}) < 1$ (Lemma \[lem:constant\]), with exact the same argument of Lemma \[lem:cyc:long\], we can show that $\E{\sum_{\ell > \psi_n} {{U}_{\ell}}} = o(1)$ for all $\psi_n \to \infty$. Thus (a) is proved. Now choose $\psi_n = \log \log n$. Again we can show that whp there are no vertex-intersecting undirected cycles of length at most $\psi_n$ by repeating the computation of $b_2$ in the proof of Theorem \[thm:cyc:v\] with $ke^{-{{\tau_k}}}$ replaced by $2ke^{-{{\tau_k}}}$ in . Spectra outside the giant ------------------------- In this section, we prove Theorem \[thm:dag\] (spectra outside the giant). Instead of working on ${{\cG_n^c}}$ directly, we again prove similar results on a fixed set of vertices and then apply Lemma \[lem:giant:deterministic\] to finish the proof. ### The tree-like structure of some spectra We prove part (a) of Theorem \[thm:dag\]. Let ${\cV_n}\subseteq [n]$ with ${|{\cV_n}|}\in {{\cI_n}}\equiv {[{{\nu_k}}n-{n^{1/2+\delta}}, {{\nu_k}}n +{n^{1/2+\delta}}]}$ be a fixed set of vertices. For $v \in {{\cV_n}^c}\equiv [n] \setminus {\cV_n}$, let ${\cS^{*}_{v}}$ be the spectrum of $v$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$, the sub-digraph induced by ${{\cV_n}^c}$. The following lemma shows that whp every spectrum in ${{{\cD_{n,k}}}[{\cV_n}^c]}$ induces a sub-digraph that is a tree or a tree plus one extra arc: We have $$\sup_{{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in{{\cI_n}}}} \p{\cup_{v \in {{\cV_n}^c}}[{\mathop{\mathrm{arc}}}({{\cD_{n,k}}}[{\cS^{*}_{v}}]) - {|{\cS^{*}_{v}}|} \ge 1]} =o(1) ,$$ where ${\mathop{\mathrm{arc}}}(\cdot)$ denotes the number of arcs. \[lem:spec:backward\] For $v \in {{\cV_n}^c}$, if ${\mathop{\mathrm{arc}}}({{{\cD_{n,k}}}[\cS^{*}_{v}]}) \ge {|{\cS^{*}_{v}}|} + 1$, then ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$ must contain at least two undirected cycles. By Lemma \[lem:cyc:undirected\], whp all undirected cycles in ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$ are vertex-disjoint. Therefore, if ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$ contains two undirected cycles, then whp they are vertex-disjoint and connected by an undirected path. Let $X_{r,s,t}$ be the number of pairs of undirected cycles of length $r$ and $s$ respectively that are connected by an undirected path of length $t$. In such a structure the number of arcs is $r+s+t$ while the number of vertices is $r+s+t-1$. Since ${|{\cV_n}|}\in {{\cI_n}}$, we have ${|{\cV_n}^c|}= n - {|{\cV_n}|}\in {{\cI_n^c}}\equiv {[{{e^{-{{\tau_k}}}}}n-{n^{1/2+\delta}}, {{e^{-{{\tau_k}}}}}n +{n^{1/2+\delta}}]}$. Thus $$\begin{aligned} \e X_{r,s,t} \le ({|{\cV_n}^c|})^{r+s+t-1} (2k)^{r+s+t} \left( \frac 1 n \right)^{r+s+t} \le {{O\left( \frac 1 n \right)}} \left( 2 k {{e^{-{{\tau_k}}}}}+ \frac{2k}{n^{1/2-\delta}} \right)^{r+s+t}. \end{aligned}$$ Summing over all possible $r$, $s$ and $t$ shows that $$\begin{aligned} \sum_{1 \le r \le n} \sum_{1 \le s \le n} \sum_{1 \le t \le n} \e X_{r,s,t} & \le {{O\left( \frac 1 n \right)}} \sum_{1 \le r} \sum_{1 \le s} \sum_{1 \le t} \left(2 k {{e^{-{{\tau_k}}}}}+ \frac{2k}{n^{1/2-\delta}}\right)^{r+s+t} \\ & \le {{O\left( \frac 1 n \right)}} \left( \sum_{1 \le i} \left(2 k {{e^{-{{\tau_k}}}}}+\frac{2k}{n^{1/2-\delta}}\right)^{i} \right)^{3}, \end{aligned}$$ which is $o(1)$ since the sum in the brackets converges. ### The maximum size of spectra {#sec:dag:spec:size} This section proves part (b) of Theorem \[thm:dag\] (the sizes of spectra outside the giant). Let $\varepsilon > 0$ be a constant. Then $$\sup_{{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in{{\cI_n}}}} \p{\left| \frac{\max_{v \in {{\cV_n}^c}} {|{\cS^{*}_{v}}|}}{\log n} - \frac 1 {\log(1/\lambda_k)} \right| > \varepsilon} = o(1),$$ where ${{\lambda_k}}\equiv (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1}$. \[lem:spec:size\] The exploration of ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$ can be coupled with a colouring process. Initially, colour all vertices in ${\cV_n}$ green, all vertices in ${{\cV_n}^c}$ yellow, and all arcs white. Then: (i) Colour the vertex $v$ black, and colour the $k$ arcs that start from $v$ red. (Red arcs start from vertices in ${\cS^{*}_{v}}$ but their endpoints are not determined yet.) (ii) Pick an arbitrary red arc. Choose its endpoint uniformly at random from all the $n$ vertices. Colour this arc with the colour of its chosen endpoint vertex. (So a yellow arc goes to a vertex that is not already in ${\cS^{*}_{v}}$, a black arc goes to a vertex that is already in ${\cS^{*}_{v}}$.) If the chosen vertex is yellow, colour this vertex black and colour all its arcs red. (iii) If there are no red arcs left, terminate. Otherwise go to the previous step. In the end, ${\cS^{*}_{v}}$ consists of all black vertices, and arcs that start from vertices in ${\cS^{*}_{v}}$ have one of three colors: green arcs go to ${\cV_n}$; yellow arcs form a spanning tree of ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$ rooted at $v$; black arcs connect vertices in ${\cS^{*}_{v}}$ but they are not part of the yellow spanning tree, so they are in cycles in ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$. Figure \[fig:colour\] depicts the colouring process. at (0,0) [![The colouring process.[]{data-label="fig:colour"}](gfx/colouring.pdf "fig:")]{}; We use random variables $R_t$ and $Y_t$ to track the number of red arcs and yellow vertices after the $t$-th red arc is colored. Thus $R_0 = k$ and $Y_0 = {|{\cV_n}^c|}-1$. When a red arc is colored, if a yellow vertex is chosen as its endpoint, then the number of red arcs increases by $(k-1)$ and the number of yellow vertices decreases by one. Otherwise the number of red arcs decreases by one and the number of yellow vertices remains unchanged. Thus for $t \ge 1$, $$R_t = R_{t-1} + k \xi_{t} - 1 = k \sum_{i=1}^{t} \xi_i - (t-k), \quad \text{and} \quad Y_t = Y_{t-1} - \xi_{t} = {|{\cV_n}^c|}-1 - \sum_{i=1}^{t} \xi_i,$$ where $\xi_{t}$ are independent Bernoulli $Y_t/n$ (the probability that a yellow vertex is chosen). Let $T \equiv \min\{t:R_t \le 0\}$. Then ${|{\cS^{*}_{v}}|} = T/k$, since $T$ is the total number arcs that have been colored and ${|{\cS^{*}_{v}}|}$ is the total number of vertices that have been colored. Let $({\overline{\xi_{t}}})_{t \ge 1}$, be i.i.d. Bernoulli $({{e^{-{{\tau_k}}}}}+ {n^{{-1}/2+\delta}})$. Since $Y_t/n \le {|{\cV_n}^c|}/n \le {{e^{-{{\tau_k}}}}}+ {n^{{-1}/2+\delta}}$, we have ${\overline{\xi_{t}}} \succeq \xi_{t}$, where $\succeq$ denotes stochastically greater than (see [@Shaked2007]). Therefore there exists a coupling such that ${\overline{\xi_{t}}} \ge \xi_{t}$ for all $t$ almost surely. Let ${\overline{T}}_{t} \equiv \min\{t:k \sum_{i=1}^{t} {\overline{\xi_{i}}} - (t-k) \le 0\}$. Then ${\overline{T}}\ge T$ almost surely. (The random variable $T$ is called the total progeny of a Galton-Watson process with offspring distribution ${\overline{\xi_{1}}}$. For an introduction to Galton-Watson processes see [@Durrett2010probability]). It is well know that if $\e {\overline{\xi_{1}}} < 1$, which is true in this case, then $\e {\overline{T}}= k /(1-\e {\overline{\xi_{1}}}) = O(1)$. Thus $\e T = O(1)$. Let $\omega_n = \floor{(1+\varepsilon) \log n/ \log(1/\lambda_k)}+1$. Since ${\overline{T}}\ge T$, $$\begin{aligned} \p{T \ge k \omega_n} & \le \p{{\overline{T}}\ge k \omega_n} \le \p{\frac {\sum_{i=1}^{k {\omega_n}} {\overline{\xi_{i}}}} {k \omega_n} \ge \frac 1 {k_n}}\end{aligned}$$ where $k_n = k \omega_n/(\omega_n -1)$. @Hoeffding1963 showed that $$\p{\frac {{\mathop{\mathrm{Bin}}}(m,p)} m \ge p+x} \le \left\{ \left(\frac{p}{p+x} \right)^{p+x} \left(\frac{1-p}{1-p-x} \right)^{1-p-x} \right\}^{m}.$$ where ${\mathop{\mathrm{Bin}}}(m,p)$ denotes a binomial $(m,p)$ random variable. Recalling that $\e {\overline{\xi_{1}}} = {{e^{-{{\tau_k}}}}}+ {n^{{-1}/2+\delta}}\equiv 1-{{\tau_k}}/k + {n^{{-1}/2+\delta}}$ and ${{\lambda_k}}\equiv (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1}$, it follows from Hoeffding’s inequality that $\p{T \ge k \omega_n}$ is at most $$\begin{aligned} \left[ \left( \frac{\e {\overline{\xi_{1}}}}{1/k_n} \right) \left( \frac{1-\e {\overline{\xi_{1}}}}{1-1/k_n} \right)^{k_n-1} \right]^{\omega_n} & = \left[ (k-{{\tau_k}})\left(\frac {{\tau_k}}{k-1}\right)^{k-1} + O({n^{{-1}/2+\delta}}) \right]^{\omega_n+O(1)} \\ & = O(\lambda_k^{\omega_n}) \left(1 + O\left({n^{{-1}/2+\delta}}\right)\right)^{\omega_n} \\ & = {{O\left( n^{-(1+\varepsilon)} \right)}}. \label{eq:upper:bound:spectrum} {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ Since $k {|{\cS^{*}_{v}}|} = T$, by the union bound $$\p{\cup_{v \in {{\cV_n}^c}} {|{\cS^{*}_{v}}|} \ge \omega_n} \le n \p{{\overline{T}}\ge k \omega_n} = {{O\left( n^{-\varepsilon} \right)}}. \tag*{\qedhere}$$ Let $\psi_n \equiv \ceil{(1-\varepsilon)\log n/\log(1/{{\lambda_k}})}$. To show that whp there exists a $v \in {{\cV_n}^c}$ such that ${|{\cS^{*}_{v}}|} \ge \psi_n$, pick an arbitrary yellow vertex and run the colouring process. If at least $\psi_n$ vertices are colored black (success) in the process then terminate. Otherwise (failure) pick another yellow vertex and repeat the colouring process until one trial succeeds. If the colouring process is repeated for at most $t_n \equiv \floor{n/(\log n)^3}$ times, then at most $a_n \equiv t_n \psi_n = O(n/(\log n)^{2})$ vertices are colored black in the end. Therefore, the probability that the number of red arcs increases after colouring one red arc is at least $({|{\cV_n}^c|}- a_n)/n$. Let $({\underline{\xi_{i}}})_{i \ge 1}$ be i.i.d. Bernoulli $({|{\cV_n}^c|}- a_n-\psi_n)/n$. Let ${\underline{T}}= \min\{t:k \sum_{i=1}^{t} {\underline{\xi_{i}}} - (t-k) \le 0\}$. Then in each of the first $t_n$ iterations, the probability of a success is at least $\p{{\underline{T}}\ge k \psi_n} \ge \p{{\underline{T}}= k \psi_n}$. (For a detailed proof, see @Van2014randomV1’s discussion of the Erdős–Rényi model [@Van2014randomV1 chap. 4.2.2].) By the hitting-time theorem of Galton-Watson processes [@Van2008elementary], $$\p{{\underline{T}}= k \psi_n} = \frac {1} {\psi_n} \p{k \sum_{i = 1}^{k \psi_n} {\underline{\xi_{i}}} = k (\psi_n -1)}.$$ Since $\sum_{i=1}^{k \psi_n} {\underline{\xi_{i}}}$ is a binomial random variable, the above equals $$\frac 1 {\psi_n} \binom{k \psi_n}{\psi_n -1} \left( \frac{{|{\cV_n}^c|}- a_n - \psi_n}{n} \right)^{\psi_n -1} \left( 1-\frac{{|{\cV_n}^c|}- a_n - \psi_n}{n} \right)^{k {\psi_n} - \left( {\psi_n}-1 \right)} \equiv b_n.$$ By Stirling’s approximation [@Flajolet2009 pp. 407] $$\binom{k \psi_n}{\psi_n -1} = \Theta(1) \binom{k \psi_n}{\psi_n} = \frac 1 {\Theta\left( \sqrt{\psi_n} \right)} \left[ \frac k {(1-1/k)^{k-1}} \right]^{\psi_n}.$$ Recalling that $a_n \equiv {{O\left( n/(\log n)^2 \right)}}$ and $\psi_n \equiv \ceil{(1-\varepsilon)\log n/\log(1/{{\lambda_k}})}$, we have, in view of ${|{\cV_n}^c|}= e^{-{{\tau_k}}}n + {{O\left( n^{1/2+\delta} \right)}}$, $$\begin{aligned} \left( \frac{{|{\cV_n}^c|}- a_n - \psi_n}{n} \right)^{{\psi_n}-1} = \left({{e^{-{{\tau_k}}}}}- O\left(\frac 1 {(\log n)^2} \right)\right)^{{\psi_n} - 1} = \Theta\left( e^{-{{\tau_k}}\psi_n} \right),\end{aligned}$$ and $$\begin{aligned} \left( 1 - \frac{{|{\cV_n}^c|}- a_n - \psi_n}{n} \right)^{k {\psi_n} - \left( {\psi_n}-1 \right)} & = \left( 1- {{e^{-{{\tau_k}}}}}+ O\left( \frac 1 {(\log n)^2} \right) \right)^{k {\psi_n} - \left( {\psi_n}-1 \right)} \\ & = \Theta\left( \left(\frac{{{\tau_k}}}{k}\right)^{(k-1)\psi_n} \right).\end{aligned}$$ Recall that ${{e^{-{{\tau_k}}}}}\equiv 1 - {{\tau_k}}/k$. Therefore $${{\lambda_k}}\equiv (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1} = k {{e^{-{{\tau_k}}}}}\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1} = \frac{k}{\left( 1-1/k \right)^{k-1}} e^{-{{\tau_k}}} \left(\frac{{{\tau_k}}}{k}\right)^{k-1}.$$ Putting everything together, we have $$\begin{aligned} b_n = \Theta\left( \frac 1 {\psi_n} \frac{1}{\sqrt{\psi_n}} \left[ \frac{k}{\left( 1-1/k \right)^{k-1}} e^{-{{\tau_k}}} \left(\frac{{{\tau_k}}}{k}\right)^{k-1} \right]^{\psi_n} \right) = \Theta\left( \frac {\lambda_k^{\psi_n}}{\psi_n^{3/2}} \right) = \Theta\left( \frac {n^{-1+\varepsilon}}{\psi_n^{3/2}} \right).\end{aligned}$$ So the probability that all the first $t_n \equiv \floor{n/(\log n)^3}$ trials fail is at most $$(1-b_n)^{t_n} \le \exp\left\{-b_n t_n \right\} = \exp\left\{ \Theta\left(-\frac{n^{\varepsilon}}{(\log n)^{9/2}} \right) \right\} = o(1). \tag*{\qedhere}$$ By Lemma \[lem:giant:size\], whp ${{\cG_n}}$ is reachable from all vertices. When this happens, ${\cO_n}\setminus {{\cG_n}}$ consists of vertices either on cycles in ${{{\cD_{n,k}}}[{{\cG_n^c}}]}$ or on paths from these cycles to ${{\cG_n}}$. Since the number of such cycles and the length of the longest one of them are both $O_p(1)$, Lemma \[lem:spec:size\] implies that ${|{\cO_n}|}- {{|{{\cG_n}}|}}= O_{p}(\log n)$. Thus $$\frac{{{|{{\cG_n}}|}}- {{\nu_k}}n}{\sqrt{n}} = \frac{{|{\cO_n}|}- {{\nu_k}}n}{\sqrt{n}} - O_p\left( \frac {\log n}{\sqrt{n}} \right) \inlaw \cZ,$$ which is the second part of Theorem \[thm:CLL\]. In fact we can show that ${|{\cO_n}|}- {{|{{\cG_n}}|}}= O_p(1)$. This seems to be obvious since in ${{{\cD_{n,k}}}[{\cV_n}^c]}$ the expected size of a spectrum is $O(1)$ and the number of cycles is $O_{p}(1)$. However, it is not trivial because $\ind{v \text{ is on a cycle}}$ and ${|{\cS^{*}_{v}}|}$ are not independent. For a proof using Cayley’s formula, see Lemma \[lem:middle:layer\] in the next section (Section \[sec:mid\]). We can also use Lemma \[lem:spec:size\] to show that $$\frac{\max_{v \in [n]} |{\cS_{v}}|-{{|{{\cG_n}}|}}}{\log n} \inprob \frac 1 {\log(1/\lambda_k)} ,$$ which finishes the last part of Theorem \[thm:CLL\], i.e., $(\max_{v \in [n]} |{\cS_{v}}|-{{\nu_k}}n)/\sigma_k \sqrt{n} \inlaw \cZ$. Let $A_n$ be the event that every vertex can reach ${{\cG_n}}$. Assuming $A_n$ happens, ${{\cG_n}}\subseteq {\cS_{v}}$ for all $v \in [n]$. Thus for all $\varepsilon > 0$, $$\begin{aligned} & \p{ \left| \frac{\max_{v \in [n]} |{\cS_{v}}|-{{|{{\cG_n}}|}}}{\log n} - \frac 1 {\log(1/\lambda_k)} \right| > \varepsilon } \\ & \le \p{ \left[ \left| \frac{\max_{v \in [n]} |{\cS^{\prime}_{v}}|}{\log n} - \frac 1 {\log(1/\lambda_k)} \right| > \varepsilon \right] \cap A_n } + \p{A_n^c} = o(1) .\end{aligned}$$ Since ${|{{\cS_{1}}}|}\le \max_{v \in [n]} |{\cS_{v}}|$ and whp ${|{{\cS_{1}}}|}\ge {{|{{\cG_n}}|}}$, we also recover @Grusho1973’s central limit law of ${|{{\cS_{1}}}|}$. ### The size of the middle layer {#sec:mid} Lemma \[lem:middle:layer\] and Corollary \[cor:giant:determnistic\] imply that ${|{\cO_n}|}- {{|{{\cG_n}}|}}= O_{p}(1)$. Let $\omega_n \to \infty$ be an arbitrary sequence of nonnegative numbers. Then $$\sup_{{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in{{\cI_n}}}} \p{\sum_{v \in \cC({{\cV_n}^c})} {|{\cS^{*}_{v}}|} \ge \omega_n} = o(1) ,$$ where $\cC({{\cV_n}^c})$ denotes the set of vertices on cycles in ${{{\cD_{n,k}}}[{\cV_n}^c]}$, and ${\cS^{*}_{v}}$ is the spectrum of $v$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$, the sub-digraph induced by ${{\cV_n}^c}$. \[lem:middle:layer\] By Theorem \[thm:cyc:v\] and Lemma \[lem:spec:backward\], in ${{{\cD_{n,k}}}[{\cV_n}^c]}$ whp: (a) there are at most $\sqrt{\omega_n}$ vertices on cycles, i.e., $|\cC({{\cV_n}^c})| \le \sqrt{\omega_n}$; (b) every ${\cS^{*}_{v}}$ induces either a tree or a tree plus one extra arc; (c) $\max_{v \in {{\cG_n^c}}}{|{\cS^{*}_{v}}|} = O(\log n)$. Now assume all these events happen. If $\sum_{v \in \cC({{\cV_n}^c})} {|{\cS^{*}_{v}}|} \ge \omega_n$, then (a) implies there is at least one vertex $u \in \cC({{\cV_n}^c})$ with ${|{\cS^{*}_{u}}|} \ge {\sqrt{\omega_n}}$. By (b), ${\cS^{*}_{u}}$ induces a sub-digraph that consists of exactly one cycle and isolated trees with their roots on this cycle. If ${|{\cS^{*}_{u}}|} = \ell$, we call the induced sub-digraph an $\ell$-eye. Note that by (c) there are no $\ell$-eyes with $\ell > {(\log n)^2}$. at (0,0) [![The leftmost shaded part of this figure is an $\ell$-eye.[]{data-label="fig:eye"}](gfx/eye.pdf "fig:")]{}; Let $\cS \subseteq {{\cV_n}^c}$ with $|\cS|= \ell$ be a set of vertices. If $\cS$ induces an $\ell$-eye $\cD_e$, then there are $\ell$ arcs that start and end at specific vertices in $\cS$ decided by $\cD_e$, which happens with probability $(1/n)^{\ell}$. If $\cS={\cS^{*}_{u}}$ for some vertex $u \in \cS$, call $\cS$ a *partial spectrum*. For $\cS$ to be a partial spectrum, the other $(k-1)\ell$ arcs that start from $\cS$ must end at ${\cV_n}$, which happens with probability $ ({|{\cV_n}|}/n)^{(k-1)\ell}$. So the probability that $\cS$ induces a fixed $\cD_e$ and $\cS$ is a partial spectrum is $(1/n)^{\ell} ({|{\cV_n}|}/n)^{(k-1)\ell}$. By Cayley’s formula [@Biggs1976], there are $\ell^{\ell-1}$ ways that $\cS$ can form a rooted tree. In such a tree, there are at most $\ell^2$ ways to add an extra arc to make it an $\ell$-eye. In a vertex-labeled $\ell$-eye, there are at most $k^{\ell}$ ways to label the arcs. So the number of $\ell$-eyes can be induced by $\cS$ is less than $\ell^{\ell-1} \ell^{2} k^{\ell}$. And there are $\binom{{|{\cV_n}^c|}}{\ell}$ ways to choose $\cS$. Let $X_{\ell}$ be the number of $\ell$-eyes induced by partial spectra. Recall that ${{\nu_k}}\equiv {{\tau_k}}/k =1- {{e^{-{{\tau_k}}}}}$. Thus ${|{\cV_n}|}\in {{\cI_n}}\equiv {[{{\nu_k}}n-{n^{1/2+\delta}}, {{\nu_k}}n +{n^{1/2+\delta}}]}$ implies that ${|{\cV_n}^c|}\le {{e^{-{{\tau_k}}}}}n + {n^{1/2+\delta}}$. So for $\ell \le {(\log n)^{2}}$, by the above arguments, $$\begin{aligned} \e X_\ell & \le \binom{{|{\cV_n}^c|}}{\ell} \ell^{\ell-1} \ell^{2} k^{\ell} \left( \frac 1 n \right)^{\ell} \left( \frac{{|{\cV_n}|}}{n} \right)^{(k-1)\ell}\\ & \le \frac{({{e^{-{{\tau_k}}}}}n + {n^{1/2+\delta}})^{\ell}}{(\ell/e)^{\ell}} \ell^{\ell+1} k^{\ell} \left( \frac 1 n \right)^{\ell} \left( \frac{{{\tau_k}}}{k} + {n^{{-1}/2+\delta}}\right)^{(k-1)\ell} \\ & = \left[ e\left(e^{-{{\tau_k}}}+{n^{{-1}/2+\delta}}\right) k \left( \frac{{{\tau_k}}}{k} + {n^{{-1}/2+\delta}}\right)^{k-1} \right]^{\ell} \ell \\ & = \left(1 + {{O\left( \ell {n^{{-1}/2+\delta}}\right)}}\right) {\left(k e^{1-{{\tau_k}}} \left(\frac{{{\tau_k}}}k\right)^{k-1}\right)^{\ell}} \ell \\ & \equiv \left(1 + {{O\left( \ell {n^{{-1}/2+\delta}}\right)}}\right) \rho_k^\ell \ell . \end{aligned}$$ By Lemma \[lem:constant\], $\rho_{k} < 1$. Since ${\sqrt{\omega_n}} \to \infty$, $$\begin{aligned} \sum_{\sqrt{{\omega_n}} \le \ell \le (\log n)^2} \e X_\ell & \le \left[1+{{O\left( \frac{(\log n)^2}{{{n^{{1}/2-\delta}}}} \right)}}\right] \sum_{\sqrt{\omega_n} \le \ell}^{\infty} \ell (\rho_k)^{\ell} = o(1). \end{aligned}$$ Thus whp there are no $\ell$-eyes induced by partial spectra with $\ell \in [\sqrt{\omega_n},(\log n)^2]$. ### The distance to the giant This subsection proves part (c) of Theorem \[thm:dag\]. For all $\varepsilon > 0$, $$\sup_{{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in{{\cI_n}}}} \p{\left| \frac{\max_{v \in {{\cV_n}^c}} {W_{v}^{*}}}{\log_k \log n} - 1 \right| > \varepsilon} = o(1),$$ where ${W_{v}^{*}}\equiv \min_{u \in {\cV_n}}{\mathop{\mathrm{dist}}}(v,u)$, i.e., ${W_{v}^{*}}$ is the length of the shortest path from $v$ to ${\cV_n}$. \[lem:spec:width\] Let $v \in {{\cV_n}^c}$ be a vertex. If ${W_{v}^{*}}> 1$, then all neighbors of $v$ are in ${{\cV_n}^c}$, and most likely there are $k$ of them. So $\p{{W_{v}^{*}}> 1} \approx ({|{\cV_n}^c|}/n)^k \approx e^{-{{\tau_k}}k}$. If ${W_{v}^{*}}> 2$, then the neighbors of $v$’s neighbors are all in ${{\cV_n}^c}$, and most likely there are $k^2$ of them. So $\p{{W_{v}^{*}}> 2} \approx ({|{\cV_n}^c|}/n)^{k+k^2} \approx e^{-{{\tau_k}}(k+k^2)}$. Repeating this argument shows that $\p{{W_{v}^{*}}> x} \approx \exp\{-{{\tau_k}}(k + k^2 \dots k^x)\} = e^{-{{\tau_k}}\Theta(k^x)}$, which is $o(1/n)$ when $x \ge (1+\varepsilon) \log_k \log n$. To make the above intuition rigorous, the colouring process defined in the previous subsection needs to be slightly modified. Let $v$ be the vertex where the process has started. When choosing a red arc to colour, instead of choosing one arbitrarily from all red arcs, choose one arbitrarily from those that are closest to $v$. Thus at the end, the yellow arcs consist of not just a spanning tree but a breadth-first-search (bfs) spanning tree of ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$. If ${\cV_n}$ (the set of green vertices) is contracted into a single green vertex, then the green arcs together with yellow arcs form a [<span style="font-variant:small-caps;">dag</span>]{}. Let ${\cT_{v}}$ denote this [<span style="font-variant:small-caps;">dag</span>]{}. Then ${W_{v}^{*}}$ is the length of the shortest path from $v$ to the green vertex contracted from ${\cV_n}$. Figure \[fig:treev\] shows an example of ${\cT_{v}}$. at (0,0) [![An example of ${\cT_{v}}$.[]{data-label="fig:treev"}](gfx/treev.pdf "fig:")]{}; Let $\omega_n = \floor{(1+\varepsilon) \log_k \log n}$. Call the arcs whose endpoints are at distance $i$ to $v$ the $i$-th layer of ${\cT_{v}}$. The event ${W_{v}^{*}}> \omega_n$ implies that the first $\omega_n$ layers of arcs in ${\cT_{v}}$ are all yellow arcs and thus they form a tree of height $\omega_n$. By Lemma \[lem:spec:backward\], whp there are no $v \in {{\cV_n}^c}$ such that ${{{\cD_{n,k}}}[\cS^{*}_{v}]}$ contains more than one black arc. Thus whp in every ${\cT_{v}}$ all internal (non-leaf) vertices except at most one have out degree $k$. Let $A_n$ denote this event. Assuming $A_n$ happens, ${W_{v}^{*}}> \omega_n$ implies that there are at least $\Theta(k^{\omega_n}) = \Theta(\log n)^{1+\varepsilon}$ yellow arcs in the first $\omega_n$ layers of ${\cT_{v}}$. Thus in the colouring process, the first $\Theta(\log n)^{1+\varepsilon}$ arcs choose their endpoints in ${{\cV_n}^c}$. The probability that this happens is at most $({|{\cV_n}^c|}/n)^{\Theta(\log n)^{1+\varepsilon}}$. Since ${|{\cV_n}|}\in {{\cI_n}}$, ${|{\cV_n}^c|}= n - {|{\cV_n}|}\le {{e^{-{{\tau_k}}}}}n + {n^{1/2+\delta}}$. Then by the union bound, $$\begin{aligned} \p{\cup_{v \in {{\cV_n}^c}} [{W_{v}^{*}}> \omega_n]} & \le \sum_{v \in {{\cV_n}^c}} \p{[{W_{v}^{*}}> \omega_n] \cap A_n} + \p{A_n^c} \\ & \le n ({|{\cV_n}^c|}/n)^{\Theta(\log n)^{1+\varepsilon}} + o(1) \\ & \le n ({{e^{-{{\tau_k}}}}}+ {n^{{-1}/2+\delta}})^{\Theta(\log n)^{1+\varepsilon}} + o(1) = o(1) . \end{aligned}$$ Thus whp $\max_{v \in {{\cV_n}^c}} {W_{v}^{*}}\le \omega_n$. Let $\psi_n = \ceil{(1-\varepsilon) \log_k \log n}$. To show that whp there is a vertex $v$ with ${W_{v}^{*}}\ge \psi_n$, run the colouring process starting from an arbitrary yellow vertex $v$ until either an arc is colored black or green (failure), or the first $\psi_n-1$ layers of ${\cT_{v}}$ are colored yellow (success). So to succeed, the first $\psi_n-1$ layers of ${\cT_{v}}$ form a full $k$-ary tree, i.e., the first $k+k^2+\dots+k^{\psi_n-1} = \Theta(k^{\psi_n}) = \Theta(\log n)^{1-\varepsilon}$ arcs must be colored yellow. If the process fails, we pick another yellow vertex and try again until one trial succeeds. Since the colouring process stops before colouring the ${\psi_n}$ layer of ${\cT_{v}}$, each trial colors at most $\Theta(k^{\psi_n}) = \Theta(\log n)^{1-\varepsilon}$ vertices black. If the process is tried at most $\ceil{n/(\log n)^2}$ times, then at most $b_n \equiv \ceil{n/(\log n)^2} O(\log n)^{1-\varepsilon} = O(n/(\log n)^{1+\varepsilon})$ vertices are colored black. Therefore, each arc has probability at least $({|{\cV_n}^c|}- b_n)/n$ to be colored yellow during the first $\ceil{n/(\log n)^2}$ trials. Since ${|{\cV_n}|}\in {{\cI_n}}$, ${|{\cV_n}^c|}= n - {|{\cV_n}|}\ge {{e^{-{{\tau_k}}}}}n - {n^{1/2+\delta}}$. Thus the probability to succeed in one trial is at least $$\left( \frac{{|{\cV_n}^c|}- b_n}{n} \right)^{O(\log n)^{1-\varepsilon}} \ge \left[ {{e^{-{{\tau_k}}}}}- {{O\left( \frac 1 {(\log n)^{1+\varepsilon}} \right)}} \right]^{O(\log n)^{1-\varepsilon}} = e^{-{{O\left( \log n \right)}}^{1-\varepsilon}} .$$ Therefore, the probability that the first $\ceil{n/(\log n)^2}$ trials fail is at most $$\left( 1- e^{-O(\log n)^{1-\varepsilon}} \right)^{\ceil{n/(\log n)^2}} \le \exp\left\{- e^{-O(\log n)^{1-\varepsilon}} \frac n {(\log n)^2}\right\} = o(1).$$ Thus whp $\max_{v \in {{\cV_n}^c}} {W_{v}^{*}}\ge \psi_n$. ### The longest path outside the giant This subsection proves (d) and (e) of Theorem \[thm:dag\]. For all $\varepsilon > 0$, we have: $$\sup_{{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in{{\cI_n}}}} \p{ \left|\frac{m({{\cV_n}^c})}{\log n} - \frac{1}{\log(e^{{{\tau_k}}}/k) }\right| > \varepsilon} = o(1),$$ where $m({{\cV_n}^c})$ denotes the length of the longest path in ${{{\cD_{n,k}}}[{\cV_n}^c]}$; and $$\sup_{{{\cV_n}\subseteq [n]:{|{\cV_n}|}\in{{\cI_n}}}} \p{ \left|\frac{d({{\cV_n}^c})}{\log n} - \frac{1}{\log(e^{{{\tau_k}}}/k) }\right| > \varepsilon} = o(1).$$ where $d({{\cV_n}^c})$ denotes the maximal distance between two connected vertices in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. \[lem:path:len\] Since $m({{\cV_n}^c}) \ge d({{\cV_n}^c})$, it suffices to prove the upper bound for $m({{\cV_n}^c})$ and the lower bound for $d({{\cV_n}^c})$. Let $\omega_n=(1+\varepsilon) \log n / \log({e^{{{\tau_k}}}}/{k})$. Let $X_\ell$ be the number of labeled paths of length $\ell$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$. There are less than ${|{\cV_n}^c|}^{\ell+1}k^{\ell}$ possible such paths. Each of them exists with probability $(1/n)^{\ell}$. Recall that ${|{\cV_n}|}\in {{\cI_n}}$ implies ${|{\cV_n}^c|}\le {{e^{-{{\tau_k}}}}}n + {n^{1/2+\delta}}$. Thus $$\e X_\ell \le {|{\cV_n}^c|}^{\ell + 1} k^{\ell} \left( \frac 1 n \right)^{\ell} \le \left({{e^{-{{\tau_k}}}}}n + {n^{1/2+\delta}}\right) \left(k {{e^{-{{\tau_k}}}}}+ {k}{n^{{-1}/2+\delta}}\right)^{\ell}.$$ Since $k {{e^{-{{\tau_k}}}}}< 1$ (Lemma \[lem:constant\]), for $n$ large enough, $$\begin{aligned} \sum_{\omega_n < \ell < {|{\cV_n}^c|}} \e X_{\ell} \le n \sum_{\omega_n < \ell} (k {{e^{-{{\tau_k}}}}}+ k {n^{{-1}/2+\delta}})^{\ell} = {{O\left( n \left( k {{e^{-{{\tau_k}}}}}\right)^{\omega_n} \right)}} = {{O\left( n^{-\varepsilon} \right)}}. \end{aligned}$$ Thus $\p{m({{\cV_n}^c}) > \omega_n} = {{O\left( n^{-\varepsilon} \right)}}$. Let $ \psi_n \equiv \ceil*{(1-\varepsilon) {\log n}/{\log(1/k{{e^{-{{\tau_k}}}}})}}$. To show there are two vertices at distance within $[\psi_n, \infty)$, pick an arbitrary yellow vertex $v$ and run the colouring process until either a vertex at distance $\psi_n$ from $v$ has been colored (success), or $\ceil{(\log n)^2}$ vertices have been colored (failure), or the process terminates because all vertices that are reachable from $v$ in ${{{\cD_{n,k}}}[{\cV_n}^c]}$ has been discovered (failure). If the process fails, we pick another yellow vertex and try again until one trial succeeds. If at most $t_n \equiv \floor{n/(\log n)^4}$ trials are made, then at most $\ceil{(\log n)^2} t_n = {{O\left( n/(\log n)^2 \right)}}$ vertices are colored. So in the first $t_n$ trials, when an arc is colored, the probability that it is colored yellow is at least $\mu_n \equiv ({|{\cV_n}^c|}-{{O\left( n/(\log n)^2 \right)}})/n = {{e^{-{{\tau_k}}}}}- {{O\left( 1/(\log n)^2 \right)}}$. Let $(Z_m)_{m\ge 0}$ be a Galton-Watson process with offspring distribution ${\mathop{\mathrm{Bin}}}(k, \mu_n)$ and $Z_0 = 1$. In other words, $Z_{m+1} = \sum_{j =1}^{Z_m} X_{m,j}$, where $(X_{m,j})_{m \ge 0, j \ge 1}$ are i.i.d.  ${\mathop{\mathrm{Bin}}}(k, \mu_n)$. Then the probability that one trial succeeds is at least $\p{Z_{\psi_n} > 0}$ minus the probability that in a trial $\ceil{(\log n)^2}$ vertices have been colored, which is ${{O\left( n^{-1-\varepsilon} \right)}}$ by in Lemma \[lem:spec:size\]. Let $\varphi_m(y) = \e y^{Z_m}$, i.e., $\varphi_m(y)$ is the probability generating function of $Z_m$. Thus $\p{Z_m = 0} = \varphi_m(0)$. Since $k {{e^{-{{\tau_k}}}}}< 1/2$ (Lemma \[lem:constant\]), for $n$ large enough $k \mu_n < 1/2$. So we can apply Lemma \[lem:gen:f\] in the appendix to show that $$\varphi_m(0) \le 1 - (k \mu_n)^{m} + \left( 1- \frac{1}{2^{m}} \right) (k \mu_n)^{m+1} < 1- \frac 1 2 (k \mu_n)^{m}, \quad \text{for all $m \ge 0$}.$$ Recalling that $\psi_n \equiv \ceil*{(1-\varepsilon) {\log n}/{\log(1/k{{e^{-{{\tau_k}}}}})}}$, $$\p{Z_{\psi_n} > 0} = 1-\varphi_{\psi_n}(0) > \frac 1 2 \left( k {{e^{-{{\tau_k}}}}}- {{O\left( \frac{1}{(\log n)^2} \right)}} \right)^{\psi_n} = {{\Omega\!\left( n^{-1+\varepsilon} \right)}}.$$ So the probability that one trial succeeds is ${{\Omega\!\left( n^{-1+\varepsilon} \right)}} - {{O\left( n^{-1-\varepsilon} \right)}} = {{\Omega\!\left( n^{-1+\varepsilon} \right)}}$. (The ${{O\left( n^{-1-\varepsilon} \right)}}$ term is the probability that one trial colors too many vertices.) Thus the probability that the first $t_n \equiv \floor{n/(\log n)^4}$ trials fail is at most $$\left(1-{{\Omega\!\left( n^{-1+\varepsilon} \right)}}\right)^{t_n} \le \exp\left\{-\Omega\left( \frac {1} {n^{1-\varepsilon}} \floor*{\frac{n}{(\log n)^4}} \right)\right\} = \exp\left\{-\Omega\left( \frac {n^{\varepsilon}} {(\log n)^4} \right)\right\} = o(1).$$ Therefore whp $d({{\cV_n}^c}) \ge \psi_n$. Phase transition in strong connectivity ======================================= Now instead of assuming that $k$ is fixed, let $k \to \infty$ as $n \to \infty$. Let $K$ be a fixed integer. We can construct ${{\cD_{n,k}}}$ by first generating ${{\cD_{n,K}}}$ and then adding arcs with labels in $\{K+1,\ldots,k\}$ into it. By Lemma \[lem:giant:size\], for all $\varepsilon > 0$, there exists a $K$ depending only on $\varepsilon$ such that whp in ${{\cD_{n,K}}}$ the largest closed [<span style="font-variant:small-caps;">scc</span>]{} has size at least $ (1-\varepsilon) n$ and is reachable from all vertices. Since adding arcs can only increase the size of this [<span style="font-variant:small-caps;">scc</span>]{}, whp ${{\cD_{n,k}}}$ has a [<span style="font-variant:small-caps;">scc</span>]{} of size at least $ (1-\varepsilon)n $ that is reachable from all vertices. In fact, if $k$ increases fast enough, then whp ${{\cD_{n,k}}}$ is strongly connected. More precisely, ${{\cD_{n,k}}}$ exhibits a phase transition for strong connectivity similar to the analogous event in the Erdős–Rényi model [@Erdos1959random]. \[sec:phase\] \[thm:phase\] If $k-\log n \to -\infty$, then whp ${{\cD_{n,k}}}$ is not strongly connected. If $k - \log n \to \infty$, then whp ${{\cD_{n,k}}}$ is strongly connected. If there is a vertex with in-degree zero, then obviously the digraph is not strongly connected. Thus the following lemma proves the lower bound in Theorem \[thm:phase\]. \[lem:phase:lower\] If $k-\log n \to -\infty$, whp ${{\cD_{n,k}}}$ contains a vertex of in-degree zero. Let $\omega_n = \log n - k$. For vertex $i \in [n]$, let $X_i$ be the indicator that $i$ has in-degree zero. Let $N= \sum_{i=1}^n X_i$. We use second moment method to show that $N \ge 1$ whp. To have $X_1 = 1$, $ nk$ arcs need to avoid vertex $1$ as their endpoints. Thus $$\begin{aligned} \e X_1 & = \left(1-\frac{1}{n} \right)^{nk} \ge e^{ -nk\left({1}/{n}+{1}/{n^2} \right) } = e^{-k \left({1}+{1}/{n} \right)} = \left( \frac{e^{\omega_n}}{n} \right)^{1+1/n} . \end{aligned}$$ Since by assumption $\omega_n \to \infty$, $\e N = n \e X_1 = e^{\omega_n (1+1/n)}/n^{1/n} \to \infty$. To have $X_1 X_2 = 1$, $ nk$ arcs need to avoid vertices $1$ and $2$ as their endpoints. Thus $ \e X_1 X_2 = \left( 1-{2}/{n} \right)^{nk} . $ Therefore $$\begin{aligned} \frac{\E{X_1X_2}}{(\E{X_1})^2} & = \frac{(1-2/n)^{nk}}{(1-1/n)^{2nk}} = \left( \frac{n^2-2n}{n^2-2n+1} \right)^{nk} = \left( 1 - \frac{1}{(n-1)^2} \right)^{nk} \to 1 , \end{aligned}$$ since $nk/(n-1)^2 = o(1)$. Thus $$\begin{aligned} 1 \le \frac{\E{N^2}}{(\e{N})^2} = \frac{\e{N} + n(n-1)\E{X_1X_2}}{(\e N)^2} \le \frac{1}{\e N} + \frac{\E{X_1 X_2}}{(\e X_1)^2} \to 1. \end{aligned}$$ Therefore $\p{N = 0} \le \V{N}/(\e N)^2 = \E{N^2}/(\e N)^2 - 1 \to 0$. Given a set of vertices $\cS$, if there are no arcs that start from $\cS^c \equiv [n] \setminus \cS$ and end at $\cS$, then call $\cS$ a *non-leaf*. If ${{\cD_{n,k}}}$ is not strongly connected, then there must exist a non-leaf set of vertices $\cS$ with $|\cS| < n$. Thus the following lemma implies the upper bound in Theorem \[thm:phase\]. \[lem:phase:upper\] If $k-\log n \to +\infty$, whp there does not exist a non-leaf set of vertices $\cS$ with $|\cS| < n$. By the argument at the beginning of this subsection, whp ${{\cD_{n,k}}}$ contains a [<span style="font-variant:small-caps;">scc</span>]{} of size at least $n/2$ that is reachable form all vertices. So if $|\cS| \ge n/2$, then $\cS$ contains part of this [<span style="font-variant:small-caps;">scc</span>]{} and cannot be a non-leaf. Thus it suffices to prove the lemma for $\cS$ with $|\cS| < n/2$. Let $\omega_n = k - \log n$. For $s \in [ \floor{n/2}]$, let $X_s$ be the number of non-leaf sets of vertices of size $s$. Thus $$\begin{aligned} \e X_s & = \binom{n}{s}\left( 1-\frac{s}{n} \right)^{k(n-s)} \le \binom{n}{s} e^{-ks(1-s/n)} \label{eq:phase:low} {\addtocounter{equation}{1}\tag{\theequation}}. \end{aligned}$$ Therefore for $s < n / \log n$, $$\begin{aligned} \e X_s & \le \frac{n^s}{s!} e^{-ks(1-s/n)} \le \frac{1}{s!} \left( \frac{n}{e^{k(1-s/n)}} \right)^s \le \frac{1}{s!} \left( \frac{n}{(n e^{\omega_n} )^{1-1/\log n}} \right)^s \equiv \frac{\alpha_n^s}{s!} . \end{aligned}$$ By assumption $\omega_n \to \infty$. Thus $\alpha_n \equiv n^{1/\log n}/e^{\omega_n(1-1/\log n)} = e^{1-\omega_n(1-1/\log n)}=o(1)$. Therefore, $$\begin{aligned} \sum_{1 \le s < n/\log n} \e X_s \le \sum_{1 \le s} \frac{\alpha_n^s}{s!} = e^{\alpha_n} - 1 = o(1). \end{aligned}$$ On the other hand, it follows from that for $n/\log n \le s < n/2$, $$\begin{aligned} \e X_s \le \left( \frac{en}{s} \right)^s e^{-ks(1-s/n)} = \left( \frac{en}{s e^{k(1-s/n)}} \right)^s \le \left( \frac{en}{\frac{n}{\log n} e^{k/2}} \right)^s = \left( \frac{e \log n}{(ne^{\omega_n})^{1/2}} \right)^s \equiv \beta_n^s. \end{aligned}$$ Since $\beta_n = e \log n/(n e^{\omega_n})^{1/2} = o(1)$, $$\begin{aligned} \sum_{{n}/{\log n} \le s < n/2} \e X_s \le \sum_{1 \le s} \beta_n^s = {{O\left( \beta_n \right)}} = o(1). \end{aligned}$$ Thus $\p{\sum_{1 \le s < {n}/2} X_s \ge 1} \le \sum_{1 \le s < {n}/2} \e X_s = o(1). $ We omit the proofs of the above two lemmas as they use standard first and second moment methods. The simple digraph model, the number of self-loops and multiple arcs {#sec:simple} ==================================================================== A *simple* digraph is one in which there are no self-loops and there is no more than one arc from one vertex to another. Let ${{\cD_{n,k}^{*}}}$ denote a simple $k$-out digraph with $n$ vertices chosen uniformly at random from all such digraphs. ${{\cD_{n,k}^{*}}}$ can be viewed as ${{\cD_{n,k}}}$ restricted to the event that ${{\cD_{n,k}}}$ is simple. This section proves the following theorem: \[thm:simple\] The probability that ${{\cD_{n,k}}}$ is simple converges to $e^{-k-\binom{k}{2}}$ as $n \to \infty$. Theorem \[thm:simple\] can be proved directly as follows. Let $\indd{v}$ be the indicator that the $k$ arcs starting from vertex $v$ do not end at $v$ and do not end at the same vertex. Then $$\p{\indd{v} = 1} = \frac{(n-1)(n-2)\cdots (n-k)}{n^{k}} = 1 - \frac{k(k+1)}{2n} + {{O\left( \frac{1}{n^2} \right)}} .$$ Since ${{\cD_{n,k}}}$ is simple if and only if $\cap_{v = 1}^n [\indd{v}=1]$ happens, we have $$\begin{aligned} \p{\text{\({{\cD_{n,k}}}\) is simple}} & = \p{\cap_{v=1}^{n} \left[ \indd{v} = 1 \right]} = \prod_{v=1}^{n} \p{\indd{v} = 1} \\ & = \left( 1 - \frac{k(k+1)}{2n} + {{O\left( \frac{1}{n^2} \right)}} \right)^{n} \to e^{-k(k+1)/2} = e^{-k-\binom{k}{2}} .\end{aligned}$$ However, we can say more about self-loops and multiple arcs between vertices. Let $\cI \equiv [n]\times[k]$. For $ (v, i) \in \cI$, define the random variable $\indd{v,i}$ to be the indicator that the arc with label $i$ starting from vertex $v$ forms a self-loop. Let $ \cJ \equiv \{ (v,i,j) \in [n]\times[k]\times[k]:i <j \}. $ For $ (v,i,j) \in \cJ$, define the random variable $\indd{v,i,j}$ to be the indicator that the two arcs starting from vertex $v$ with labels $i$ and $j$ both end at the same vertex. Let $ S_n = \sum_{\alpha \in \cI} \indd{\alpha} \text{ and } M_n = \sum_{\alpha \in \cJ} \indd{\alpha} . $ Then $[S_n = 0] \cap [M_n = 0]$ if and only if ${{\cD_{n,k}}}$ is simple. \[lem:simple\] Let $S$ and $M$ be two independent Poisson random variables of means $k$ and $\binom{k}{2}$ respectively. Then $ (S_n, M_n) \inlaw (S,M)$ as $n \to \infty$. In fact, $${{\left\|(S_n, M_n), (S,M)\right\|_{\textsc{tv}}}} = {{O\left( \frac{1}{n} \right)}}.$$ Indeed the lemma implies that as $n \to \infty$, $$\p{{{\cD_{n,k}}}\text{ is simple}} = \p{S_n = M_n = 0} \to \p{S = 0} \p{M = 0} = e^{-k} e^{-\binom{k}{2}}.$$ @Bollobas1980311 proved a theorem similar to Lemma \[lem:simple\] for the configuration model (see also @bollobas2001random [sec. 2.4]). Many authors have extended this result under various conditions, see, e.g., @mckay1985asymptotics [@mckay1991asymptotic; @janson2009; @janson2014]. Our proof uses Stein’s method, which may also be applied to self-loops and multiple edges in the configuration model to get proofs shorter than previous ones. We use the Chen-Stein method [@Chen1975]. Since the probability that an arc forms a self-loop is $1/n$, $$\e S_n = \sum_{(v,i) \in \cI} \e \indd{v,i} = kn \frac{1}{n} = k.$$ Thus $\e S = k =\e S_n $. Since the probability that two arcs with the same start point have the same endpoint is also $1/n$, $$\e M_n = \sum_{v \in [n]} \sum_{1 \le i < j \le k} \e \indd{v,i,j} = n \binom{k}{2} \frac{1}{n} = \frac{k(k-1)}{2}.$$ Thus $\e M = k(k-1)/2 = \e M_n$. For $\alpha \in \cI \cup \cJ$, let $$\cB_{\alpha} = \{ \beta \in \cI \cup \cJ : \text{$\indd{\beta}$ and $\indd{\alpha}$ are dependent} \} .$$ (Note that $\indd{\alpha} \in \cB_{\alpha}$.) Define $$\begin{aligned} b_1 \equiv \sum_{\alpha \in \cI \cup \cJ} \sum_{\beta \in \cB_{\alpha}} \E{\indd{\alpha}} \E{\indd{\beta}} , \quad b_2 \equiv \sum_{\alpha \in \cI \cup \cJ} \sum_{\beta \in \cB_{\alpha}:\alpha \ne \beta} \E{\indd{\alpha}\indd{\beta}} , \quad b_3 \equiv \sum_{\alpha \in \cI \cup \cJ} s_\alpha, \end{aligned}$$ where $$s_{\alpha} = \e \left| \E{ \indd{\alpha} \,| \, \sigma\left(\indd{\beta}:{\beta} \in [\cI \cup \cJ] \setminus \cB_{\alpha} \right)} - \e \indd{\alpha} \right|.$$ By [@Chen1975 thm. 2], if $b_1 + b_2 + b_3 \to 0$, then $ (S_n, M_n) \inlaw \allowbreak (S,M)$. Since $\indd{\alpha}$ is independent of the random variables $\indd{\beta}$ with $\beta \in [\cI \cup \cJ] \setminus \cB_{\alpha}$, we have $s_\alpha = 0$ and thus $b_3 = 0$. For $ (v,i) \in \cI$, $\indd{v,i}$ depends on the random variables $\indd{v,r,s}$ with $ 1 \le r < s \le k$ and $i \in \{r,s\}$, of which there are $k-1$. Thus $|\cB_{v,i}| = 1 + (k-1) = k < 2k$. For $ (v,i,j) \in \cJ $, $\indd{v,i,j}$ depends on $\indd{v,i}$ and $\indd{v,j}$. It also depends on the random variables $\indd{v,r,s}$ with $ 1 \le r < s \le k$ and $ \{r,s\} \cap \{i,j\} \ne \emptyset$, of which there are $2(k-1)-1= 2k-3$. Thus $|\cB_{v,i,j}| = 2 + 2k-3 < 2k$. So for all $\alpha \in \cI \cup \cJ$, $|\cB_{\alpha}| < 2k$. Therefore $$\begin{aligned} b_1 & = \sum_{\alpha \in \cI} \sum_{\beta \in \cB_{\alpha}} \E{\indd{\alpha}} \E{\indd{\beta}} + \sum_{\alpha \in \cJ} \sum_{\beta \in \cB_{\alpha}} \E{\indd{\alpha}} \E{\indd{\beta}} \\ & < nk \times 2k \times \frac{1}{n} \times \frac{1}{n} + n \binom{k}{2} \times 2k \times \frac{1}{n} \times \frac{1}{n} = {{O\left( \frac{1}{n} \right)}} . \end{aligned}$$ Consider $ (v,i) \in \cI$. If $\beta \in \cB_{v,i} \cap \cI$, then $\beta = (v,i)$. If $\beta \in \cB_{v,i} \cap \cJ$, then $\beta = (v,r,s)$ for some $ (r,s) $ with $i \in \{r, s\}$. Then $\indd{v,i}\indd{v,r,s} = 1$ if and only if the two arcs starting from vertex $v$ labeled $r$ and $s$ respectively both end at $v$. Thus $\E{\indd{v,i}\indd{v,r,s}} = 1/n^2$. Therefore $$b_{2,\cI} \equiv \sum_{\alpha \in \cI} \sum_{\beta \in \cB_\alpha:\beta \ne \alpha} \E{\indd{\alpha}\indd{\beta}} = \sum_{\alpha \in \cI} \sum_{\beta \in \cB_\alpha \cap \cJ} \E{\indd{\alpha}\indd{\beta}} < nk \times 2k \times \frac{1}{n^2} = {{O\left( \frac{1}{n} \right)}} .$$ Consider $ (v,r,s) \in \cJ$. If $ (v,i) \in \cB_{v,r,s}$, then $ (v,r,s) \in \cB_{v,i}$. Thus by the above argument $\E{\indd{v,r,s}\indd{v,i}} = 1/n^2$. If $ (v,i,j) \in \cB_{v,r,s}$ and $ (i,j) \ne (r,s) $, then $|\{r,s\} \cup \{i,j\}| = 3$. So $\indd{v,r,s}\indd{v,i,j} = 1$ iff the three arcs starting from vertex $v$ with labels in $ \{r,s\} \cup \{i,j\} $ all end at the same vertex. Thus $\E{\indd{v,r,s}\indd{v,i,j}} = 1/n^2$. Therefore $$\begin{aligned} b_{2,\cJ} \equiv \sum_{\alpha \in \cJ} \sum_{\beta \in \cB_\alpha:\beta \ne \alpha} \E{\indd{\alpha}\indd{\beta}} < n \binom{k}{2} \times 2k \times \frac{1}{n^2} = {{O\left( \frac{1}{n} \right)}} . \end{aligned}$$ Thus $b_2 \equiv b_{2,\cI} + b_{2, \cJ} = O(1/n)$. \[cor:simple\] Let $\cE$ be a set of digraphs. If ${{\cD_{n,k}}}\in \cE$ whp, then ${{\cD_{n,k}^{*}}}\in \cE$ whp. We have $$\p{{{\cD_{n,k}^{*}}}\notin \cE} = \p{{{\cD_{n,k}}}\notin \cE \,|\, {{\cD_{n,k}}}\text{ is simple}} \le \frac{\p{{{\cD_{n,k}}}\notin \cE_n}}{\p{{{\cD_{n,k}}}\text{ is simple}}} \to 0. \tag*{\qedhere}$$ This corollary implies that all previous results in the form of “whp ${{\cD_{n,k}}}$ …” can be automatic translated into “whp ${{\cD_{n,k}^{*}}}$ …”. For example, the statement of Theorem \[thm:dag\] with ${{\cD_{n,k}}}$ replaced by ${{\cD_{n,k}^{*}}}$ is still true. \[cor:simple:unlabel\] Let ${{\cD_{n,k}^{**}}}$ be a digraph chosen uniformly at random from all simple and arc-unlabeled $k$-out digraphs with $n$ vertices. If whp ${{\cD_{n,k}}}$ has property [**P**]{} where [**P**]{} does not depend on arc-labels, then whp ${{\cD_{n,k}^{**}}}$ has property [**P**]{}. Note that: (a) for each digraph in the space of ${{\cD_{n,k}^{**}}}$, there $ (k!)^n $ ways to arc-label it to get $ (k!)^n $ different digraphs in the space of ${{\cD_{n,k}^{*}}}$; (b) no two different arc-unlabeled digraphs can be turned into the same digraph by arc-labeling. So there exists a $ (k!)^n $-to-one surjective mapping from the space of ${{\cD_{n,k}^{*}}}$ to the space of ${{\cD_{n,k}^{**}}}$. Thus ${{\cD_{n,k}^{**}}}$ can be viewed as ${{\cD_{n,k}^{*}}}$ with arc labels removed. Since [**P**]{} does not depend on arc-labels, it follows from Corollary \[cor:simple\] that whp ${{\cD_{n,k}^{**}}}$ has property [**P**]{}. The typical distance {#sec:typical:distance} ==================== The typical distance $H_n$ of ${{\cD_{n,k}}}$ is the distance between two vertices $v_1$ and $v_2$ chosen uniformly at random. If $v_1$ cannot reach $v_2$, then $H_n = \infty$. @Perarnau2014 proved that conditioned on $H_n < \infty$, $H_n/\log_k n \inprob 1$. This section[^1] gives an alternative proof using the path counting technique invented by @Van2014randomV2 [@Van2014randomV2 chap. 3.5]. For all $\varepsilon>0$, $$\p{ \left. \left| \frac{H_n}{\log_k n} - 1 \right| > \varepsilon ~ \right| ~ H_n < \infty } = o(1).$$ \[thm:typical:dist\] By Theorem \[thm:CLL\], $|{\cS_{v_1}}|/n \inprob {{\nu_k}}$, where ${\cS_{v_1}}$ is the spectrum of $v_1$. Thus $\p{H_n < \infty} = \p{v_2 \in {\cS_{v_1}}} \to {{\nu_k}}> 0$. Therefore $$\begin{aligned} \p{H_n < (1-\varepsilon)\log_k n~|~H_n < \infty} & = \frac {\p{H_n < (1-\varepsilon)\log_k n}} {\p{H_n < \infty}} \\ & \sim \frac {1} {{{\nu_k}}} {\p{H_n < (1-\varepsilon)\log_k n}} ,\end{aligned}$$ and $$\begin{aligned} \p{H_n > (1+\varepsilon)\log_k n~|~H_n < \infty} & = \frac {\p{(1+\varepsilon)\log_k n < H_n < \infty}} {\p{H_n < \infty}} \\ & \sim \frac {1} {{{\nu_k}}} {\p{(1+\varepsilon)\log_k n < H_n < \infty}} \equiv \frac {\p{B_n}} {{{\nu_k}}} .\end{aligned}$$ Thus it suffices to show that $\p{H_n < (1-\varepsilon)\log_k n}$ and $\p{B_n}$ are both $o(1)$. \[lem:typical:dist:lower\] For all $\varepsilon> 0$, $$\p{H_n < (1-\varepsilon)\log_k n} = o(1) .$$ Let $N_{\ell}$ denote the number of paths from $v_1$ to $v_2$ of length $\ell$. Consider such a path without labels on internal vertices and arcs. There are at most $n^{\ell-1}$ ways to label its internal vertices and there are at most $k^{\ell}$ ways to label its arcs. And the probability that such a labeled path appears is $ (1/n)^{\ell}$. Thus $$\e N_{\ell} \le n^{\ell-1} k^{\ell} \left(\frac{1}{n} \right)^{\ell} = \frac{k^{\ell}}{n} .$$ Let $\omega_n = (1-\varepsilon)\log_k n$. Then $$\sum_{\ell < \omega_n} \e N_{\ell} \le \sum_{\ell < \omega_n} \frac{k^{\ell}}{n} = \frac{{{O\left( k^{\omega_n} \right)}}}{n} = \frac{{{O\left( n^{1-\varepsilon} \right)}}}{n} = o(1).$$ Thus $\p{H_n < \omega_n} = \p{\sum_{\ell < \omega_n} N_{\ell} \ge 1} = o(1)$. The rest of this section is organized as follows: Subsection \[sec:typical:distance:tree\] shows that if $v_1$ can reach $v_2$ but only through a very long path, then it is very likely that $v_1$ can reach a lot of vertices and a lot of vertices can reach $v_2$. Subsection \[sec:typical:distance:counting\] computes a lower bound of the probability that there is a path of specific length from one large set of vertices to another large set of vertices. Finally, subsection \[sec:typical:dist:upper\] shows that these results together imply the upper bound in Theorem \[thm:typical:dist\], i.e., $\p{B_n} = o(1)$. Comparison to Galton-Watson processes {#sec:typical:distance:tree} ------------------------------------- Let $\cS_{m}^{+}(v)$ and $\cS_{m}^{-}(v)$ be the sets of vertices at distance exactly $m$ from or to vertex $v$ respectively. Let $\cS_{\le m}^{+}(v)$ and $\cS_{\le m}^{-}(v)$ be the sets of vertices at distance at most $m$ from or to vertex $v$ respectively. The following proposition shows that for fixed $m$, we can perfectly couple $(|{\cS_{t}^{+}(v_1)}|, |{\cS_{t}^{-}(v_2)}|)_{t=0}^{m} $ with two independent Galton-Watson processes. It is inspired by a similar result of the configuration model by @Van2014randomV2 [@Van2014randomV2 sec. 5.2], but the coupling method used here is new. \[prop:branching:approx\] Let $ (S_t)_{t \ge 0} $ be a Galton-Watson process with a binomial offspring distribution ${\mathop{\mathrm{Bin}}}(kn, 1/n)$. For all fixed $m \ge 1$, there exists a coupling $$\left[ \left(k^{t}, {Y}_{t} \right)_{t=0}^{m} , \left({{Y}^{+}}_t, {{Y}^{-}}_t \right)_{t=0}^{m} \right] ,$$ of $(k^t, {S}_{t})_{t=0}^{m}$ and $ (|{\cS_{t}^{+}(v_1)}|, |{\cS_{t}^{-}(v_2)}|)_{t=0}^{m} $, such that $$\p{ \left(k^{t}, {Y}_{t} \right)_{t=0}^{m} \ne \left({{Y}^{+}}_t, {{Y}^{-}}_t \right)_{t=0}^{m} } = o(1) .$$ We construct an incremental sequence of random digraphs, denoted by $ ({{\cD_{n,k}^{[t]}}})_{t \ge 0} $, through a signal spreading process. Let ${{\cD_{n,k}^{[0]}}}$ be a digraph of vertex set $[n]$ that has no arcs. Without loss of generality, let $v_1 = 1$ and $v_2 = 2$. At time $0$, put a $\oplus$ signal at $v_1$ and put a $\ominus$ signal at $v_2$. If a $\oplus$ signal reaches a vertex $v$ at time $t$, then at time $t+1/3$ the vertex $v$ grows $k$ out-arcs labeled $1,\ldots,k$ from itself and to $k$ endpoints chosen independently and uar from all the $n$ vertices. Then the $\oplus$ signal splits into $k$ $\oplus$ signals and each of them picks a different newly-grown out-arc and travels along the arc’s direction to reach its endpoint at time $t+1$. If a $\ominus$ signal reaches a vertex $v$ at time $t$, then at time $t+2/3$ the vertex $v$ grows a random number $X$ in-arcs from itself to $X$ random vertices as follows: Let $ (X_{i,j})_{i \in [n],j \in [k]}$ be i.i.d. Bernoulli $1/n$ random variables. If $X_{i,j} = 1$, then $v$ grows an in-arc from itself to vertex $i$ with label $j$. Thus in total $X \equiv \sum_{i\in[n],j\in[k]}X_{i,j}$ in-arcs are grown from $v$. Then the $\ominus$ signal splits into $X$ $\ominus$ signals and each of them picks a different newly-grown in-arc and travels against the arc’s direction to reach its starting vertex at time $t+1$. If $X=0$, then the $\ominus$ signal vanishes. Let ${{\cD_{n,k}^{[t]}}}$ be the digraph generated in the above process at time $t$. Let ${{\cY}^{+}}_{t}$ and ${{\cY}^{-}}_{t}$ be the sets of vertices that are visited by $\oplus$ and $\ominus$ signals at time $t$ respectively. Let ${{\cY}^{+}}_{\le t}$ and ${{\cY}^{-}}_{\le t}$ be the sets of vertices that have been visited by $\oplus$ and $\ominus$ signals before time $t+1$ respectively. At time $t$, if a signal visits a vertex in $[{{\cY}^{+}}_{\le t-1} \cup {{\cY}^{-}}_{\le t-1}]$ or if two signals visit the same vertex, then we say a *collision* happens. Let $T$ be the first time when a collision happens. Table 1 lists the types of events that make a collision happen. Three of them need special attention for reasons to be clear soon. First, if multiple $\ominus$ signals visit the same vertex $v$, then multiple arcs with the same label and $v$ as the starting point may grow. If this happens we pick an arbitrary arc among them and call the others *duplicate*. Second, a $\oplus$ signal may visit a vertex in ${{\cY}^{-}}_{\le T-1}$ through a newly-grown out-arc. Finally, a $\ominus$ signal may visit a vertex in ${{\cY}^{+}}_{\le T-1}$ through a newly-grown in-arc. We also call the newly-grown arcs being passed by in these two cases *biased*. =0.6mm [ c | c | c || c | c | c | c ]{} & &\ $\oplus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{{\tikz [baseline=-0.75ex,-latex,] \draw [fill=black!60,] (0,0) circle (2pt);}}{{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (1.0em,0.5ex) -- (0pt,0.5ex);}}{}}\oplus$ & $\ominus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{{\tikz [baseline=-0.75ex,-latex,] \draw [fill=black!60,] (0,0) circle (2pt);}}{{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (1.0em,0.5ex) -- (0pt,0.5ex);}}{}}\oplus$ & $\ominus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{{\tikz [baseline=-0.75ex,-latex,] \draw [fill=black!60,] (0,0) circle (2pt);}}{{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (1.0em,0.5ex) -- (0pt,0.5ex);}}{}}\ominus$ & $\oplus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{ { \tikz [semithick,baseline=-0.57ex,-latex] { \draw (0,0) -- (3.5ex,0ex); \draw (0,0) .. controls (0.4ex,1.2ex) .. (3ex,2ex); \draw (0,0) .. controls (0.4ex,-1.2ex) .. (3ex,-2ex); \draw [thin,fill=black!60,] (-2pt,0) circle (2pt); \draw [thin,fill=black!60,] (3.3ex,2ex) circle (2pt); \draw [thin,fill=black!60,] (3.3ex,-2ex) circle (2pt); \draw [thin,fill=black!60,] (3.8ex,0ex) circle (2pt); } } }$ & $\ominus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{ { \tikz [semithick,baseline=-0.57ex,-latex] { \draw (0,0) -- (3.5ex,0ex); \draw (0,0) .. controls (0.4ex,1.2ex) .. (3ex,2ex); \draw (0,0) .. controls (0.4ex,-1.2ex) .. (3ex,-2ex); \draw [thin,fill=black!60,] (-2pt,0) circle (2pt); \draw [thin,fill=black!60,] (3.3ex,2ex) circle (2pt); \draw [thin,fill=black!60,] (3.3ex,-2ex) circle (2pt); \draw [thin,fill=black!60,] (3.8ex,0ex) circle (2pt); } } }$ & $\oplus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{ { \tikz [semithick,baseline=-0.75ex,-latex] { \draw (3.5ex,0ex) -- (0,0); \draw (3ex,2ex) .. controls (0.6ex,1.2ex) .. (0,0); \draw (3ex,-2ex) .. controls (0.6ex,-1.2ex) .. (0,0); \draw [thin,fill=black!60,] (-2pt,0) circle (2pt); \draw [thin,fill=black!60,] (3ex,2ex) circle (2pt); \draw [thin,fill=black!60,] (3ex,-2ex) circle (2pt); \draw [thin,fill=black!60,] (3.5ex,0ex) circle (2pt); } } }$ & $\ominus {{{\tikz [baseline=-0.15ex,-latex, dashed,densely dashed] \draw [densely dashed] (0pt,0.5ex) -- (1.0em,0.5ex);}}{}}{ { \tikz [semithick,baseline=-0.75ex,-latex] { \draw (3.5ex,0ex) -- (0,0); \draw (3ex,2ex) .. controls (0.6ex,1.2ex) .. (0,0); \draw (3ex,-2ex) .. controls (0.6ex,-1.2ex) .. (0,0); \draw [thin,fill=black!60,] (-2pt,0) circle (2pt); \draw [thin,fill=black!60,] (3ex,2ex) circle (2pt); \draw [thin,fill=black!60,] (3ex,-2ex) circle (2pt); \draw [thin,fill=black!60,] (3.5ex,0ex) circle (2pt); } } }$\ We construct a random $k$-out graph ${{\widehat{\cD}_{n,k}}}$ as follows: First remove all duplicate and all biased arcs in ${{\cD_{n,k}^{[T]}}}$. Then for each pair $ (v,i) \in [n]\times[k]$, if vertex $v$ does not have an out-arc labeled $i$, then add such an out-arc with its endpoint chosen uar from $[n] \setminus {{\cY}^{-}}_{\le T-1} $. Denote the result digraph by ${{\widehat{\cD}_{n,k}}}$. The seemingly complicated ${{\widehat{\cD}_{n,k}}}$ is nothing but ${{\cD_{n,k}}}$ in disguise. In ${{\cD_{n,k}}}$, the endpoints of the arcs are chosen uar and simultaneously. In ${{\widehat{\cD}_{n,k}}}$, the endpoints of the arcs are still chosen uar but in several steps. First we mark the arcs whose end (start) vertices are at distance $t$ to $v_1$ (from $v_2$) for $t =1,\ldots, T$. To have ${{\widehat{\cD}_{n,k}}}\eql {{\cD_{n,k}}}$, obviously duplicate arcs must be removed. The biased arcs also cause trouble as their endpoints are chosen non-uniformly. For example, if at time $T$ a $\oplus$ signal visits a vertex in ${{\cY}^{-}}_{\le T-1}$, then an in-arc is added to a vertex whose in-arcs have already been decided by time $T-1$. Thus biased arcs must also be removed. Finally, we add arcs that are still missing in ${{\widehat{\cD}_{n,k}}}$ and choose their endpoints uar from $[n] \setminus {{\cY}^{-}}_{\le T-1}$, i.e., from these vertices whose in-arcs have not yet been marked. Thus we have ${{\widehat{\cD}_{n,k}}}\eql {{\cD_{n,k}}}$. Let ${{Y}^{+}}_t$ and ${{Y}^{-}}_t$ be the number of vertices in ${{\widehat{\cD}_{n,k}}}$ at distance $t$ from $v_1$ and to $v_2$ respectively. Then $$({{Y}^{+}}_t, {{Y}^{-}}_t)_{t=0}^{m} \eql (|{\cS_{t}^{+}(v_1)}|, |{\cS_{t}^{-}(v_2)}|)_{t=0}^{m}.$$ A $\oplus$ signal always splits into $k$ $\oplus$ signals after it arrives at a vertex. Thus at a non-negative integer time $t$ there are in total $k^t$ $\oplus$ signals. On the other hand, the number of $\ominus$ signals at time $t$, denoted by $Y_t$, is random. Each time a $\ominus$ signal splits, it splits into ${\mathop{\mathrm{Bin}}}(kn,1/n)$ signals. Because the splits are mutually independent, $ (Y_t)_{t \ge 0}$ has the same distribution as $ (S_t)_{t \ge } 0$, the Galton-Watson process with offspring distribution ${\mathop{\mathrm{Bin}}}(kn, 1/n)$. Assume that $T > m$. Then the part of ${{\widehat{\cD}_{n,k}}}$ within distance $m$ from $v_1$ or to $v_2$ is determined by ${{\cD_{n,k}^{[m]}}}$. Thus for $t \le m$, in ${{\widehat{\cD}_{n,k}}}$ a vertex is at distance $t$ from $v_1$ if and only if it has a $\oplus$ signal at time $t$ and a vertex is at distance $t$ to $v_2$ if and only if it has a $\ominus$ signal at time time $t$. This implies that $ \left(k^{t}, {Y}_{t} \right)_{t=0}^{m} = \left({{Y}^{+}}_t, {{Y}^{-}}_t \right)_{t=0}^{m} $. Thus to finish the proof, it suffices to show the following lemma: \[lem:typical:dist:signal\] For all fixed integers $m \ge 1$, whp $T > m$. The intuition is that since $m$ is fixed, for $t < m$, most likely $|{{\cY}^{+}}_{\le t} \cup {{\cY}^{-}}_{\le t}|$ is small. Thus it is unlikely that a collision happens at time $t+1$. See the end of this subsection for a detailed proof. \[cor:typical:dist:spectrum\] Let $\omega_n \to \infty$ be an arbitrary sequence. Let $M, \delta, \varepsilon$ be three arbitrary positive numbers. Let $\psi_n \equiv \floor{(1+\varepsilon)\log_k n}$. Let $$A_n(M,m) \equiv \left[M \le |{{\cS_{m}^{+}(v_1)}}| \right] \cap \left[M \le |{\cS_{m}^{-}(v_2)}| \right] \cap \left[|{{\cS_{\le m}^{-}(v_2)}}| \le \omega_n \right] .$$ Then there exists $m \ge 1$ such that $$\limsup_{n \to \infty} \p{A_{n}^c(M,m) \cap [\psi_n < H_n < \infty]} < \delta.$$ Let $\left(k^{t}, {Y}_{t} \right)_{t=0}^{m}$ be the coupling of $ (|{\cS_{t}^{+}(v_1)}|, |{\cS_{t}^{-}(v_2)}|)_{t=0}^{m} $ constructed in Proposition \[prop:branching:approx\]. Thus $ (Y_t)_{t \ge 0}$ is a Galton-Watson process with ${\mathop{\mathrm{Bin}}}(kn, 1/n)$ offspring distribution, i.e., $Y_0 = 1$ and $Y_t = \sum_{i = 1}^{Y_{t-1}} X_{t,i}$ for $t \ge 1$, where $X_{t,i}$’s are i.i.d. ${\mathop{\mathrm{Bin}}}(kn,1/n)$. Since $\e X_{1,1} = k > 1$, the survival probability of this process is a constant $\eta > 0$ (see [@Van2014randomV1 thm. 3.1]). For the same reason, $Y_{t}/k^{t} \to Y_{\infty}$ almost surely for some random variable $Y_{\infty}$ (see [@Van2014randomV1 thm. 3.9]). Since $\E{ X_{1,1}^2} < \infty$, by the Kesten-Stigum Theorem [@Van2014randomV1 thm. 3.10], $\p{Y_\infty > 0} = \eta$. Thus by the Bounded Convergence Theorem [@Durrett2010probability thm. 1.5.3], $$\lim_{m \to \infty} \p{Y_m > M} = \lim_{m \to \infty} \p{\frac{Y_m}{k^m} > \frac{M}{k^m}} = \p{Y_{\infty} > 0} = \eta.$$ For the same reason $\p{Y_{m} \ge 1} \to \eta$ as $m \to \infty$. Thus $$\lim_{m \to \infty} \p{1 \le Y_m < M} = \lim_{m \to \infty} \left( \p{Y_m \ge 1} - \p{Y_m \ge M} \right) = 0 .$$ Thus we can choose $m$ large enough such that $ \p{1 \le Y_m < M} < \delta/2 $ and that $k^m \ge M$. Recall that $B_n \equiv [\psi_n < H_n < \infty]$. When $n$ is large enough, $\psi_n > m$. Thus $B_n$ implies that $|{\cS_{m}^{+}(v_1)}| \ge 1$. Define the event $$C_n \equiv \left[ \left(k^{t}, {Y}_{t} \right)_{t=0}^{m} = \left( |{\cS_{t}^{+}(v_1)}|, |{\cS_{t}^{-}(v_2)}| \right)_{t=0}^{m} \right] .$$ By Proposition \[prop:branching:approx\], $\p{C_n^c} = o(1)$ as $n \to \infty$. Therefore $$\begin{aligned} \p{A_{n}(M,m)^c \cap B_n} & \le \p{C_n^c} + \p{A_{n}(M,m)^c \cap C_n \cap B_n} \\ & \le o(1) + \p{ [k^m < M] \cup [1 \le Y_m <M] \cup \left[ \omega_n < \sum_{t=0}^m Y_{t} \right] } \\ & \le o(1) + \p{k^m < M} + \p{1 \le Y_{m} < M} + \p{\omega_n < \sum_{t=0}^m Y_{t}} \\ & = o(1) + 0 + \delta/2 + o(1) , \end{aligned}$$ where the last equality is due to our choice of $m$ and that $ \E{\sum_{t=0}^m Y_{t}} = \sum_{t=0}^{m} k^t = O(1). $ Recall that ${{\cY}^{+}}_t$ and ${{\cY}^{-}}_t$ are the sets of vertices that are reached at time $t$ by a $\oplus$ signal or $\ominus$ signal respectively. Let $\cM_{m-1} = \cup_{t=0}^{m-1} [{{\cY}^{+}}_t \cup {{\cY}^{-}}_t]$. Define event $A_m \equiv \cap_{i \in [4]} E_{m,i}$ where $E_{m,i}$’s are defined as follows: - $E_{m,1}$ — The out-arcs that grow from vertices in ${{\cY}^{+}}_{m-1}$ all end at different vertices in $[n] \setminus \cM_{m-1}$. Thus at time $m$ all $\oplus$ signals visit different vertices and these vertices have never been visited by signals before. - $E_{m,2}$ — There are no in-arcs that grow from vertices in ${{\cY}^{-}}_{m-1}$ that have starting vertices in $\cM_{m-1} \cup {{\cY}^{+}}_{m}$. Thus at time $m$ all $\ominus$ signals visit vertices that have never been visited by signals before and that are not reached by $\oplus$ signals at time $m$. - $E_{m,3}$ — There are no two in-arcs that grow from vertices in ${{\cY}^{-}}_{m-1}$ that have the same starting vertex. Thus at time $m$ all $\ominus$ signals reach different vertices. - $E_{m,4}$ — $|{{\cY}^{-}}_m| \le (\log n)^{m}$. The event $A_t$ implies that no collision happens at time $t$. Thus ${\cap_{t=0}^{m} A_{t}}$ implies that no collision has happened by time $m$, and thus $T > m$. We show by induction that $\p{{\cap_{t=0}^{m} A_{t}}} = 1-o(1)$. Since $|{{\cY}^{-}}_{0}|=1$ and there are no arc-growing before time $0$, $\p{A_0} = 1$, which is the induction basis. Now assume that $\p{{\cap_{t=0}^{m-1} A_{t}}} = 1-o(1)$. Then $$\p{{\cap_{t=0}^{m} A_{t}}} = \p{ A_{m} \, | \, {\cap_{t=0}^{m-1} A_{t}}} \p{{\cap_{t=0}^{m-1} A_{t}}} = \p{A_m\,|\, {\cap_{t=0}^{m-1} A_{t}}}(1-o(1)).$$ Thus it suffices to show that $$\p{A_m^c\,|\,{\cap_{t=0}^{m-1} A_{t}}} = \p{[\cup_{i\in[4]} E_{m,i}^c]\,|\,{\cap_{t=0}^{m-1} A_{t}}} \le \sum_{i \in[4]} \p{E_{m,i}^c|{\cap_{t=0}^{m-1} A_{t}}} = o(1) .$$ The event ${\cap_{t=0}^{m-1} A_{t}}$ implies that $$|\cM_{m-1}| \le \sum_{t = 1}^{m-1} |{{\cY}^{+}}_{t}| + \sum_{t = 1}^{m-1} |{{\cY}^{-}}_{t}| \le \sum_{t = 1}^{m-1} k^{t} + \sum_{t = 1}^{m-1} (\log n)^{t} = {{O\left( \log n \right)}}^{m} .$$ For $E_{m,1}$ to happen, the $k^{m}$ arcs that grow out of ${{\cY}^{+}}_{m-1}$ must end at different vertices in $[n] \setminus \cM_{m-1}$. Thus $$\begin{aligned} \p{E_{m,1}|{\cap_{t=0}^{m-1} A_{t}}} = \prod_{0 \le i < k^m} \left[ \frac{n - |\cM_{m-1}| - i}{n} \right] \ge \left[ 1 - \frac{{{O\left( \log n \right)}}^m}{n} \right]^{k^m} = 1 - o(1) . \end{aligned}$$ For $E_{m,2}$ to happen, the vertices in ${{\cY}^{-}}_{m-1}$ cannot grow in-arcs that have starting vertex in in $\cM_{m-1} \cup {{\cY}^{+}}_{m}$. ${\cap_{t=0}^{m-1} A_{t}}$ implies that $|{{\cY}^{-}}_{m-1}| \le (\log n)^{m-1}$. Since deterministically $|{{\cY}^{+}}_{m}| = k^m$, $|\cM_{m-1} \cup {{\cY}^{+}}_{m}| ={{O\left( \log n \right)}}^{m}$. Thus the number of in-arcs that need to not grow at time $m-1/3$ to make sure that $E_{m,2}$ happens is at most $$k |{{\cY}^{-}}_{m-1}| |\cM_{m-1} \cup {{\cY}^{+}}_{m}| = {{O\left( \log n \right)}}^{2m} .$$ Since an in-arc does not grow with probability $1-1/n$, $$\begin{aligned} \p{E_{m,2}\,|\,{\cap_{t=0}^{m-1} A_{t}}} \ge \left( 1 - \frac{1}{n} \right)^{{{O\left( \log n \right)}}^{2m}} = 1 - o(1) . \end{aligned}$$ Let $X_v$ be the number of in-arcs that grow from ${{\cY}^{-}}_{m-1}$ and that have starting vertex $v$. Conditioned on ${{\cY}^{-}}_{m-1}$, $X_v \eql {\mathop{\mathrm{Bin}}}(k|{{\cY}^{-}}_{m-1}|, 1/n)$. Since ${\cap_{t=0}^{m-1} A_{t}}$ implies $|{{\cY}^{-}}_{m-1}| \le (\log n)^{m-1}$, $$\begin{aligned} \p{X_v \le 1\,|\,{\cap_{t=0}^{m-1} A_{t}}} & \ge \p{{\mathop{\mathrm{Bin}}}\left( k (\log n)^{m-1}, \frac{1}{n} \right) \le 1} \\ & = \left( 1-\frac{1}{n} \right)^{k(\log n)^{m-1}} + k(\log n)^{m-1} \frac{1}{n} \left( 1-\frac{1}{n} \right)^{k(\log n)^{m-1}-1} \\ & = 1 - {{O\left( \frac{(\log n)^{2(m-1)}}{n^2} \right)}} . \end{aligned}$$ Since for two different vertices $u$ and $v$, $X_u$ and $X_v$ depend on disjoint set of arcs, $ (X_{u})_{u \in [n]}$ are mutually independent. Thus $$\begin{aligned} \p{E_{m,3}|{\cap_{t=0}^{m-1} A_{t}}} & = \p{\cap_{v \in [n]} [X_v \le 1]\,|\,{\cap_{t=0}^{m-1} A_{t}}} \\ & \ge \left( 1 - {{O\left( \frac{(\log n)^{2(m-1)}}{n^2} \right)}} \right)^{n} = 1 - o(1) . \end{aligned}$$ Since $ (|{{\cY}^{-}}_{t}|)_{t \ge 1} $ is a Galton-Watson process with a ${\mathop{\mathrm{Bin}}}(kn, 1/n)$ offspring distribution, $\e |{{\cY}^{-}}_m| = k^m$. Thus $\p{|{{\cY}^{-}}_m| > (\log n)^m} = o(1)$. Therefore $$\p{E_{m,4}^c|{\cap_{t=0}^{m-1} A_{t}}} \equiv \p{|{{\cY}^{-}}_m| > (\log n)^{m}|{\cap_{t=0}^{m-1} A_{t}}} \le \frac { \p{|{{\cY}^{-}}_m| > (\log n)^{m}} } {\p{{\cap_{t=0}^{m-1} A_{t}}}} = o(1) ,$$ where the last equality is due to the induction assumption that $\p{{\cap_{t=0}^{m-1} A_{t}}} = 1-o(1)$. Path counting {#sec:typical:distance:counting} ------------- For three disjoint sets of vertices $\cA, \cB, \cC \subseteq [n]$, let ${N_{\ell}}$ denote the number of paths of length $\ell$ that start from $\cA$ and end at $\cB$, and that have all internal vertices in $\cC$. In the next subsection, we use the second moment method to lower bound $\p{{N_{\ell}}\ge 1}$, which requires estimates of $\E{{N_{\ell}}}$ and $\V{{N_{\ell}}}$. The following lemma does so by using the path counting technique [@Van2014randomV2 chap. 3.5]. \[prop:typical:dist:path\] Let $\omega$, $\ell$ and $M$ be three positive integers, possibly depending on $n$. Let $\cA, \cB, \cC \subseteq [n]$ be disjoint sets of vertices with $|\cA| = |\cB| = M \ge 1$ and $|\cC| \ge n-\omega$. There exist constants $C_1$ and $C_2$ such that $$\e {N_{\ell}}\ge \frac{k^{\ell}M^{2}}{n} \left( 1 - {\frac{(\omega+\ell)\ell}{n}} \right) , {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:n:expc}$$ and $$\V{{N_{\ell}}} \le \e {N_{\ell}}+ C_1 {\frac{k^{2\ell} M^3}{n^2}} + C_2 {\frac{k^{2\ell} M^4\ell^4}{n^3} } . {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:n:var}$$ Note that if $n \le (\omega + \ell) \ell$, then is trivially true. So we assume that $n > (\omega + \ell) \ell$. We simplify by contracting $\cA$ and $\cB$ into to two special vertices $v_a$ and $v_b$. The vertex $v_a$ has out-degree $kM$ and the vertex $v_b$ has probability $M/n$ to be chosen as the endpoint of each arc. Consider an unlabeled path of length $\ell \ge 1$ from $v_a$ to $v_b$. There are $kM$ ways to label the first arc. There are $k^{\ell-1}$ ways to label the other arcs. Recall that $ (x)_y \equiv (x-1)(x-2)\cdots(x-y+1) $. There are $ (|\cC|)_{\ell-1}$ ways to label the internal vertices of the path. The probability that a vertex-and-arc labeled path of length $\ell$ from $v_a$ to $v_b$ exists is $ (1/n)^{\ell-1} (M/n) $. Thus $$\begin{aligned} \e {N_{\ell}}& = (kM)k^{\ell-1} (|\cC|)_{\ell-1} \left(\frac{1}{n}\right)^{\ell-1} \left( \frac{M}{n} \right) \\ & \ge \frac{k^{\ell}M^{2}}{n} \left( 1 - \frac{\omega+\ell}{n} \right)^{\ell} \ge \frac{k^{\ell}M^{2}}{n} \left( 1 - {\frac{(\omega+\ell)\ell}{n}} \right) , \end{aligned}$$ where the last step is because $ (1-x)^y \ge 1 - xy $ when $x \ge 0, y \ge 1$. Let $\cL$ be the space of all possible arc-and-vertex labeled paths of length $\ell$ from $v_a$ to $v_b$ through $\cC$. In other words, if $\alpha \in \cC$, then $$\alpha = \left( {v_{0}^{[\alpha]}}\equiv v_a, \, {a_{0}^{[\alpha]}}, \, {v_{1}^{[\alpha]}}, \, {a_{1}^{[\alpha]}}, \, \ldots, \, {v_{\ell-1}^{[\alpha]}},\, {a_{\ell-1}^{[\alpha]}}, \, {v_{\ell}^{[\alpha]}} \equiv v_{b} \right),$$ where ${a_{0}^{[\alpha]}},\ldots, {a_{\ell-1}^{[\alpha]}}$ are arc labels and ${v_{1}^{[\alpha]}},\ldots, {v_{\ell-1}^{[\alpha]}}$ are different vertex labels in $\cC$. For $\alpha \in \cL$, let $\indd{\alpha}$ be the indicator that $\alpha$ appears. Given two paths $\alpha, \beta \in \cL$, call them *arc-disjoint* if there does not exist an $i$ such that ${v_{i}^{[\alpha]}} = {v_{i}^{[\beta]}}$ and ${a_{i}^{[\alpha]}} = {a_{i}^{[\beta]}}$. If two paths $\alpha$ and $\beta$ are arc-disjoint, then ${\indd{\alpha}}$ and ${\indd{\beta}}$ are independent, since they depend on the endpoints of two disjoint sets of arcs. Let $\alpha \sim \beta$ denote that $\alpha$ and $\beta$ are not arc-disjoint and that $\alpha$ and $\beta$ can both appear simultaneously. Then $$\begin{aligned} \V{{N_{\ell}}} & = \sum_{\alpha,\beta \in \cL}\left(\E{\indd{\alpha} \indd{\beta}} - \E{\indd{\alpha}} \E{\indd{\beta}} \right) \\ & \le \sum_{\alpha,\beta \in \cL}\ind{\alpha \sim \beta}\left[\E{\indd{\alpha} \indd{\beta}} - \E{\indd{\alpha}} \E{\indd{\beta}} \right] \\ & \le \e {N_{\ell}}+ \sum_{\alpha,\beta \in \cL} \ind{\alpha \sim \beta} \ind{\alpha \ne \beta} \E{\indd{\alpha} \indd{\beta}} \\ & \equiv \e {N_{\ell}}+ I . \end{aligned}$$ To bound $I$, we use a technique called path counting. Consider two paths $\alpha,\beta \in \cL$ with $\alpha \sim \beta$ and $\alpha \ne \beta$. First colour all vertices and arcs in $\alpha$ and $\beta$ white. Then colour all vertices and arcs shared by $\alpha$ and $\beta$ black. After this, $\alpha$ and $\beta$ both contain the same number, say $m$, of white paths separated by black paths (possibly a single black vertex). Since both $\alpha$ and $\beta$ start and end with black paths, each of them contains $m+1$ black paths. Define: 1. $\vec{x}_{m+1} = (x_1,\ldots, x_{m+1})$, where $x_i \ge 0$ denotes the length of the $i$-th black path in $\alpha$. 2. ${\vec{s}_m}= (s_1, \ldots, s_m)$, where $s_i > 0$ denotes the length of the $i$-th white path in $\alpha$. 3. ${\vec{t}_m}= (t_1, \ldots, t_m)$, where $t_i > 0$ denotes the length of the $i$-th white path in $\beta$. 4. $\vec{o}_{m+1} = (o_1, \ldots, o_{m+1})$ records the order in which black paths appear in $\beta$. Note that $o_1 \equiv 1$, $o_{m+1} \equiv m+1$, and $(o_2, \ldots, o_{m})$ is a permutation of $\{2,\ldots,m\}$. Define the shape of $\alpha$ and $\beta$ by ${\mathop{\mathrm{Sh}}}(\alpha,\beta) \equiv (\vec{x}_{m+1},{\vec{s}_m},{\vec{t}_m},\vec{o}_{m+1}).$ \[fig:path:counting\] (0,0) coordinate (astart) – node\[x label\] [$x_1$\ $o_1=1$]{} (2,0); (5,0) – node\[x label\] [$x_2$\ $o_2=2$]{} (8,0); (10,0) – node\[x label\] [$x_3$\ $o_3=4$]{} (14,0) coordinate (aend) ; (16,0) – node\[x label\] [$x_4$\ $o_4=3$]{} (18,0); (20,0) – node\[x label\] [$x_5$\ $o_5=5$]{} (23,0); (2,0) – node\[below\] [$s_1$]{} (5,0); (8,0) – node\[below\] [$s_2$]{} (10,0); (aend) – node\[below\] [$s_3$]{} ++(2,0); (18,0) – node\[below\] [$s_4$]{} (20,0); (0,0) coordinate (bstart) – (2,0); (5,0) – (8,0); (10,0) – (14,0) coordinate (bend) ; (16,0) – (18,0); (20,0) – (23,0); (2,0) to\[out=90,in=180\] ++(0.8,0.8) – node\[above\] [$t_1$]{} ($(5,0) + (-0.8,0.8)$) to\[out=0,in=90\] (5,0); (8,0) to\[out=90,in=180\] ++(1.9,1.9) – node\[above\] [$t_2$]{} ($(16,0)+(-1.9,1.9)$) to\[out=0, in=90\] (16,0); (18,0) to\[out=90,in=0\] ++(-0.8,0.8) – node\[near end, above\] [$t_3$]{} ($(10,0) + (0.8,0.8)$) to\[out=180,in=90\] (10,0); (14,0) to\[out=90,in=180\] ++(1.4,1.4) – node\[above\] [$t_4$]{} ($(20,0)+(-1.4,1.4)$) to\[out=0, in=90\] (20,0); at (-1.5,1.1) [$\beta$]{}; at (-0.5,0.1) [$v_a$]{}; at (23.5,0.1) [$v_b$]{}; at (-1.5,-1.1) [$\alpha$]{}; Let $r$ be the number of arcs shared by $\alpha$ and $\beta$, i.e., $r \equiv \sum_{i=1}^{m+1} x_i$. Since $\alpha \sim \beta$ and $\alpha \ne \beta$, $1 \le r < \ell$. Thus there are $\ell -r$ white arcs in $\alpha$. Since each white path contains at least one white arc, there are at most $\ell-r$ white paths in $\alpha$, i.e., $m \le \ell-r$. As $\alpha$ and $\beta$ must differ by at least one arc, $m \ge 1$. Let $\cS_{m, r}$ denote the set of shapes of two paths in $\cL$ that share $r$ arcs and each contains $m$ white paths. Then $I$ can be expressed as a sum over $r$, $m$ and $\cS_{m, r}$ by $$\begin{aligned} I = \sum_{1 \le r < \ell} \sum_{1 \le m \le \ell-r} \sum_{\sigma \in \cS_{m, r}} \sum_{\alpha,\beta \in \cL} \ind{{\mathop{\mathrm{Sh}}}(\alpha,\beta)=\sigma} \E{\indd{\alpha}\indd{\beta}} \equiv \sum_{1 \le m < \ell} \sum_{1 \le r < \ell-m} \sum_{\sigma \in \cS_{m, r}} J_{m,r,\sigma} . \end{aligned}$$ Now fix $m,r$ and a shape $\sigma = (\vec{x}_{m+1}, {\vec{s}_m}, {\vec{t}_m}, \vec{o}_{m+1}) \in \cS_{m, r}$. Consider arcs in two paths $\alpha, \beta \in \cL$ with $\cS(\alpha,\beta) = \sigma$. Call those starting from $v_a$ $a$-arcs, those ending at $v_b$ $b$-arcs, and other arcs middle-arcs. Let $z_a \equiv \ind{x_1 = 0}$ and $z_b \equiv \ind{x_{m+1} = 0}$. In other words, $z_a$ is the indicator that $\alpha$ and $\beta$ do not share an $a$-arc, and $z_b$ is the indicator that they do not share a $b$-arc. Then $\alpha$ and $\beta$ contain $1+z_a$ $a$-arcs and $1+z_b$ $b$-arcs. Since $\alpha$ and $\beta$ are both of length $\ell$ and they share $r$ arcs, they contain $2\ell - r$ arcs in total. Thus they contain $2\ell-r-(1+z_a)-(1+z_b)=2\ell -r -z_a-z_b-2$ middle-arcs. Recall that black paths are shared by $\alpha$ and $\beta$. Since the $i$-th black path is of length $x_i$, it contains $x_i+1$ black vertices. So the number of vertices shared by the two paths is $\sum_{i=1}^{m+1} (x_i+1) = r + m+ 1$. Therefore in total there are $2(\ell+1) - r - m - 1$ vertices in the two paths, and among them $2\ell -r -m -1$ are internal vertices. The above argument shows that, given two unlabeled path of the shape $\sigma$, there are at most $ n^{2\ell-r-m-1}$ ways to choose the internal vertices. There are at most$ (kM)^{1+z_a} $ ways to label $a$-arcs. There are $ k^{2 \ell - r- z_a - z_b -2 }$ ways to label middle-arcs. There are at most $ k^{1+z_b}$ ways to label $b$-arcs. Thus $$\begin{aligned} |\{(\alpha,\beta) \in \cL \times \cL: {\mathop{\mathrm{Sh}}}(\alpha,\beta) = \sigma\}| & \le n^{2\ell-r-m-1} (kM)^{1+z_a} k^{2 \ell - r- z_a - z_b -2 } k^{z_b + 1} \\ & = n^{2\ell-r-m-1} M^{1+z_a} k^{2 \ell - r} . \end{aligned}$$ And the probability that a pair of paths with shape $\sigma$ does appear is $$\left( \frac{1}{n} \right)^{1+z_a} \left( \frac{1}{n} \right)^{2\ell - r - z_a -z_b - 2} \left( \frac{M}{n} \right)^{1+z_b} = \frac{M^{1+z_b}}{n^{2\ell-r}} .$$ Together, $$\begin{aligned} J_{m,r,\sigma} & \equiv \sum_{\alpha,\beta \in \cL} \ind{{\mathop{\mathrm{Sh}}}(\alpha,\beta)=\sigma} \E{\indd{\alpha}\indd{\beta}} \le n^{2\ell-r-m-1} M^{1+z_a} k^{2 \ell - r} \frac{M^{1+z_b}}{n^{2\ell-r}} \\ & = \frac{k^{2\ell-r} M^{2+z_a+z_b}}{n^{m+1}} \equiv K_{m,r,z_a,z_b} {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:path:a:b} . \end{aligned}$$ Let $\cS_{m,r,z_a,z_b}$ be the set of shapes with parameters $m,r,z_a,z_b$. Then we have $\cS_{m,r} = \cup_{z_a,z_b \in \{0,1\}} \cS_{m,r,z_a,z_b}$, where the sets in the union are disjoint. Thus $$\begin{aligned} I & = \sum_{1 \le m < \ell} \sum_{z_a,z_b \in \{0,1\}} \sum_{1 \le r < \ell-m} \sum_{\sigma \in \cS_{m, r,z_a,z_b}} J_{m,r,\sigma} \\ & \le \sum_{1 \le m < \ell} \sum_{z_a,z_b \in \{0,1\}} \sum_{1 \le r < \ell-m} |\cS_{m, r,z_a,z_b}| K_{m,r,z_a,z_b} \\ & = \sum_{z_a,z_b \in \{0,1\}} \sum_{1 \le r < \ell-m} |\cS_{1, r,z_a,z_b}| K_{1,r,z_a,z_b} + \sum_{2 \le m < \ell} \sum_{z_a,z_b \in \{0,1\}} \sum_{1 \le r < \ell-m} |\cS_{m, r,z_a,z_b}| K_{m,r,z_a,z_b} \\ & \equiv I^{[1]} + I^{[\ge 2]} . \end{aligned}$$ By counting the choices of $\vec{x}_{m+1}, {\vec{s}_m}, {\vec{t}_m}, \vec{o}_{m+1}$, we can upper bound $|\cS_{m,r,z_a,z_b}|$: If $m \ge z_a + z_b$, then $$|\cS_{m,r,z_a,z_b}| = (r+1)^{m-z_a-z_b} \binom{\ell-r-1}{m-1} \binom{\ell-r-1}{m-1} (m-1)!. \label{eq:path:shape}$$ If $m < z_a + z_b$, then $|\cS_{m,r,z_a,z_b}| = 0$. \[lem:typical:dist:smrab\] First consider $m \ge 2$, which implies that $m \ge z_a + z_b$. When $z_a = 1$, $x_1 = 0$. When $z_b = 1$, $x_{m+1} = 0$. Thus the number of ways to choose $\vec{x}_{m+1}$ equals the number of ways to choose $m+1-z_a-z_b \ge 1$ ordered non-negative integers such that they sum to $r$, which is well known to be $ (r+1)^{m-z_a-z_b}$, which explains the first factor in . Similarly the second term and the third term are the numbers of ways to choose ${\vec{s}_m}$ and ${\vec{t}_m}$ respectively. The last term is the number of ways to choose $\vec{o}_{m+1}$ since $o_2,\ldots,o_m$ is a permutation of $\{2,\ldots,m\}$. Now assume $m = 1$. If $z_a + z_b \le m = 1$, the above argument still works. If $z_a + z_b > 1$, then $z_a = z_b = 1$. In other words, the two paths do not share arcs at the beginning and at the end, and they must meet at least one internal vertex. So in this shape, there must be at least two white sub-paths in each of the two paths, i.e., $m \ge 2$, which is a contradiction. Therefore, $S_{1,r,1,1} = \emptyset$. $I^{[1]} \le 6{{ k^{2\ell}M^3}/{n^2}}$. \[lem:typical:dist:Ilmone\] By and the above lemma, $$\begin{aligned} \sum_{1 \le r < \ell-1} |\cS_{1,r,0,0}| \times K_{1,r,0,0} & = \sum_{1 \le r < \ell-1} (r+1) \left[\binom{\ell-r-1}{0} \right]^{2} 0! \frac{k^{2\ell-r} M^{2}}{n^{2}} \\ & \le \frac{k^{2\ell}M^2}{n^2} \sum_{1 \le r} \frac{r+1}{k^{r}} \le \frac{k^{2\ell}M^2}{n^2} \left[ \sum_{1 \le r} \frac{1}{2^{r}} + \sum_{1 \le r} \frac{r}{2^{r}} \right] \\ & = \frac{k^{2\ell}M^2}{n^2} \left( 1 + \frac{1}{2} + \sum_{2 \le r} \frac{r}{2^{r}} \right) \le 4 {\frac{k^{2\ell}M^2}{n^2}} , \end{aligned}$$ where the last step is because $\sum_{2\le r} r/2^r \le \int_{1}^{\infty} x/2^x {\mathrm d}x \le 2$. Similarly, $$\begin{aligned} \sum_{1 \le r < \ell-1} |\cS_{1,r,0,1}| \times K_{1,r,0,1} & = \sum_{1 \le r < \ell-1} |\cS_{1,r,1,0}| \times K_{1,r, 1,0} \\ & = \sum_{1 \le r < \ell-1} (r+1)^{0} \left[\binom{\ell-r-1}{0} \right]^{2} 0! \frac{k^{2\ell-r} M^{3}}{n^{2}} \\ & \le \frac{ k^{2\ell}M^3}{n^2} \sum_{1 \le r} \frac{1}{k^{r}} \\ & \le \frac{ k^{2\ell}M^3}{n^2} \sum_{1 \le r} \frac{1}{2^{r}} = {\frac{ k^{2\ell}M^3}{n^2}} . \end{aligned}$$ Also by Lemma \[lem:typical:dist:smrab\], $\cS_{1, r, 1,1} = \emptyset$. Thus $$\begin{aligned} I^{[1]} & \equiv \sum_{z_a,z_b \in \{0,1\}} \sum_{1 \le r < \ell-1} |\cS_{1,r,z_a,z_b}| \times K_{1,r, z_a,z_b} \\ & \le 4 {\frac{ k^{2\ell}M^2}{n^2}} + 2 {\frac{ k^{2\ell}M^3}{n^2}} + 0 \le 6 {\frac{ k^{2\ell}M^3}{n^2}} . \tag*{\qedhere} \end{aligned}$$ $I^{[\ge 2]} = 4{{\ell^4 k^{2\ell}M^{4}}/{n^{3}}}$. \[lem:typical:dist:Ilmtwo\] By Lemma \[lem:typical:dist:smrab\], for $r \in [1,\ell)$, $$\begin{aligned} & \sum_{z_a,z_b \in \{0,1\}} |\cS_{m,r,z_a,z_b}| \times K_{m,r,z_a,z_b} \\ & = \sum_{z_a,z_b \in \{0,1\}} (r+1)^{m-z_a-z_b} \left[\binom{\ell-r-1}{m-1} \right]^2 (m-1)! \frac{k^{2\ell-r} M^{2+z_a+z_b}}{n^{m+1}} \\ & \le \ell^{m} \frac{\ell^{2(m-1)}}{(m-1)!} \frac{k^{2\ell-r} }{n^{m+1}} \sum_{z_a,z_b \in \{0,1\}} M^{2+z_a+z_b} \\ & \le \frac{\ell^{3m-2}k^{2\ell-r}}{(m-1)!n^{m+1}} 4{M^4} . \end{aligned}$$ Thus $$\begin{aligned} \sum_{1 \le r < \ell-m} \sum_{z_a,z_b \in \{0,1\}} |\cS_{m,r,z_a,z_b}| \times K_{m,r,z_a,z_b} & \le \sum_{1 \le r < \ell-m} \frac{\ell^{3m-2}k^{2\ell-r}}{(m-1)!n^{m+1}} 4{M^4} \\ & \le \frac{\ell^{3m-2}k^{2\ell}}{(m-1)!n^{m+1}} 4{M^4} \sum_{1 \le r } \frac{1}{k^r} \\ & \le \frac{\ell^{3m-2}k^{2\ell}}{(m-1)!n^{m+1}} 4{M^4} . \end{aligned}$$ Therefore, $$\begin{aligned} I^{[\ge 2]} & \equiv \sum_{2 \le m <\ell} \sum_{1 \le r < \ell-m} \sum_{z_a,z_b \in \{0,1\}} |\cS_{m,r,z_a,z_b}| \times K_{m,r,z_a,z_b} \\ & \le \sum_{2 \le m} \frac{\ell^{3m-2}k^{2\ell}}{(m-1)!n^{m+1}} 4M^4 \\ & \le \frac{\ell k^{2\ell}4M^{4}}{n^{2}} \sum_{2 \le m} \frac{\ell^{3(m-1)}}{n^{m-1}(m-1)!} \\ & \le \frac{\ell k^{2\ell}4M^{4}}{n^{2}} \left( \exp\left\{\frac{\ell^{3}}{n}\right\}-1 \right) \le 4 { \frac{\ell^4 k^{2\ell}M^{4}}{n^{3}} } . \tag*{\qedhere} \end{aligned}$$ By Lemma \[lem:typical:dist:Ilmone\] and Lemma \[lem:typical:dist:Ilmtwo\], $$\begin{aligned} I = I^{[1]} + I^{[\ge 2]} \le 6 { \frac{ k^{2\ell}M^3}{n^2} } + 4 { \frac{\ell^4 k^{2\ell}M^{4}}{n^{3}} } . \end{aligned}$$ Thus $\V{N_{\ell}} \le \E{N_{\ell}} + I = \E{N_{\ell}} + 6 { {k^{2\ell}M^3}/{n^2} } + 4 { {\ell^4 k^{2\ell}M^{4}}/{n^{3}} } $. Finishing the proof of Theorem \[thm:typical:dist\] {#sec:typical:dist:upper} --------------------------------------------------- We can assume $\varepsilon < 1/2$. Recall that $\psi_n \equiv \floor{(1+\varepsilon)\log_k n}$ and that $B_n = [\psi_n < H_n < \infty] $. As argued at the beginning of this section, to finish the proof of Theorem \[thm:typical:dist\], it suffices to show that $\p{B_n} = o(1)$. Let $\omega_n \equiv \psi_n$. Let $M, m$ be two positive integers which are picked later. Recall that $\cS_{i}^{+}(v)$ and $\cS_{i}^{-}(v)$ are the sets of vertices at distance exactly $i$ from or to vertex $v$ respectively, and that $\cS_{\le i}^{+}(v)$ and $\cS_{\le i}^{-}(v)$ are the sets of vertices at distance at most $i$ from or to $v$ respectively. The following argument shows that by properly choosing $M$ and $m$, the probability that there exists a path of length exactly $\psi_n - 2m$ from ${{\cS_{m}^{+}(v_1)}}$ to ${{\cS_{\le m}^{-}(v_2)}}$ is at least $1-\delta$ for $n$ large enough, where $\delta > 0$ is arbitrary and fixed. at (0,0) [![${\cS_{\le m-1}^{+}(v_1)}, {{\cS_{m}^{+}(v_1)}}$, and ${{\cS_{\le m}^{-}(v_2)}}$.[]{data-label="fig:distance"}](gfx/typical-distance.pdf "fig:")]{}; Let the event $A_n(M,m)$ be defined as in Corollary \[cor:typical:dist:spectrum\], i.e., $$A_n(M,m) \equiv \left[M \le |{{\cS_{m}^{+}(v_1)}}| \right] \cap \left[M \le |{\cS_{m}^{-}(v_2)}| \right] \cap \left[|{{\cS_{\le m}^{-}(v_2)}}| \le \omega_n \right] .$$ Since each vertex has out-degree exactly $k \ge 2$, deterministically, $$|{\cS_{\le m-1}^{+}(v_1)}| \le 1 + k + \cdots + k^{m-1} < k^m, \qquad |{\cS_{m}^{+}(v_1)}| \le k^m.$$ Since $\psi_n > 2m$ for $n$ large enough, $B_n$ implies ${\cS_{\le m}^{+}(v_1)}$ and ${\cS_{\le m}^{-}(v_2)}$ are disjoint. Thus the event $A_n(M,m) \cap B_n$ implies that $ ( {\cS_{\le m-1}^{+}(v_1)}, {{\cS_{m}^{+}(v_1)}}, {\cS_{m}^{-}(v_2)}, {\cS_{\le m-1}^{-}(v_2)}) \in \cA $, where $\cA$ is a set of quadruples of disjoint sets of vertices defined by $$\begin{aligned} \cA \equiv \{ (\cS_1, \cS_2, \cS_3, \cS_4):\, & v_1 \in \cS_1; v_2 \in \cS_4; \\ & |\cS_1| < k^{m}; M \le |\cS_2| \le k^m; M \le |\cS_3|; |\cS_3 \cup \cS_4| \le \omega_n \} . \end{aligned}$$ For ${\vec{\cS}}= (\cS_1, \cS_2, \cS_3, \cS_4)\in \cA$, define the event $$A'_{n}({\vec{\cS}}) \equiv \left[ {\cS_{\le m-1}^{+}(v_1)} = \cS_1 \right] \cap \left[ {{\cS_{m}^{+}(v_1)}}= \cS_2 \right] \cap \left[ {\cS_{m}^{-}(v_2)} = \cS_3 \right] \cap \left[ {\cS_{\le m-1}^{-}(v_2)} = \cS_4 \right] .$$ Thus $[B_n \cap A_{m}(M,m)] \subseteq \cup_{{\vec{\cS}}\in \cA} [B_n \cap A'_{n}({\vec{\cS}})]$ and the events in the union are disjoint. Now fix a ${\vec{\cS}}\in \cA$. Let $\cA_{{\vec{\cS}}}$ and $\cB_{{\vec{\cS}}}$ be arbitrary subsets of $\cS_2$ and $\cS_3$ respectively with $|\cA_{{\vec{\cS}}}| = M$ and $|\cB_{{\vec{\cS}}}| = M$. Let $N_{{\vec{\cS}}}$ be the number of paths of length $\psi_n - 2m$ that start from $\cA_{{\vec{\cS}}}$ and end at $\cB_{{\vec{\cS}}}$, and that contain internal vertices only in $\cC_{{\vec{\cS}}} \equiv [n] \setminus \cup_{i\in[4]} \cS_i$. Thus there are $ \left| \cC_{{\vec{\cS}}} \right| = n-|\cup_{i \in [4]} \cS_i| \ge n - (\omega_n + 2k^{m}) $ vertices that can be internal vertices of these paths. By of Proposition \[prop:typical:dist:path\], $$\begin{aligned} \e N_{{\vec{\cS}}} & \ge \frac{k^{\psi_n - 2m} M^2}{n} \left( 1 - \frac {(\omega_n + 2k^{m} + \psi_n - 2m)(\psi_n - 2m)} {n} \right) \\ & \ge \frac{k^{(1+\varepsilon)\log_k(n) - 1 - 2m} M^2}{n} \left( 1 - \frac {2 \psi_n^2} {n} \right) \ge \frac{n^{\varepsilon}M^2}{k^{2m+1}} \frac{1}{2} , \end{aligned}$$ for $n$ large enough. By of Proposition \[prop:typical:dist:path\], $$\begin{aligned} \V{N_{{\vec{\cS}}}} & \le \e N_{{\vec{\cS}}} + C_1 { \frac {k^{2(\psi_n-2m)}M^3} {n^2} } + C_2 { \frac {k^{2(\psi_n-2m)}M^4(\psi_n - 2m)^4} {n^3} } \\ & \le \e N_{{\vec{\cS}}} + C_1 {\frac{n^{2(1+\varepsilon)}M^3}{n^2 k^{4m}}} + C_2 {\frac{n^{2(1+\varepsilon)}M^4\psi_n^4}{n^3 k^{4m}}} \\ & \le \e N_{{\vec{\cS}}} + C_1 { \frac {n^{2\varepsilon} M^3} {k^{4m}} } + C_3 {\frac{M^4}{k^{4m}}} {\frac{ (\log n)^{4}}{n^{1-2\varepsilon}}} , \end{aligned}$$ where $C_3$ is a constant that does not depend on $M$ or $m$. Thus $$\begin{aligned} \p{N_{{\vec{\cS}}} = 0} & \le \frac {\V{N_{{\vec{\cS}}}}} {\left( \e N_{{\vec{\cS}}} \right)^2} \le \frac{2 k^{2m+1}}{n^{\varepsilon}M^2} + \frac {C_1 {n^{2\varepsilon} M^3k^{-4m}}} {\left( n^{\varepsilon} M^22^{-1}k^{-2m-1} \right)^2} + \frac {C_3 {{M^4 (\log n)^{4}n^{2\varepsilon-1}} {k^{-4m}}}} {\left( n^{\varepsilon} M^22^{-1}k^{-2m-1} \right)^2} \\ & \le \frac{2 k^{2m+1}}{n^{\varepsilon}M^2} + \frac {4 k^2 C_1} {M} + \frac {4 k^2 C_3 (\log n)^{4}} {n} . \end{aligned}$$ Later $m$ is chosen solely depending on $M$. Thus we can pick $M$ large enough such that for $n$ large enough, $\p{N_{{\vec{\cS}}} = 0} \le \delta / 2$ for all ${\vec{\cS}}\in \cA$. If $H_n > \psi_n$, then there cannot exist paths of length $\psi_n - 2m$ from ${{\cS_{m}^{+}(v_1)}}$ to ${\cS_{m}^{-}(v_2)}$. Thus $B_n \cap A_n'({\vec{\cS}})$ implies that $[N_{{\vec{\cS}}} = 0] \cap A_n'({\vec{\cS}})$. A crucial observation is that $$\p{N_{{\vec{\cS}}}=0\left|A'_{n}({\vec{\cS}})\right.} \le \p{N_{{\vec{\cS}}}=0}.$$ This is because $A'_{n}({\vec{\cS}})$ implies that arcs starting from vertices in $\cC_{{\vec{\cS}}}$ cannot choose vertices in ${\cS_{\le m-1}^{-}(v_2)} = \cS_{4}$ as their endpoints. Whereas when we compute $\p{N_{{\vec{\cS}}}=0}$ without any condition, arcs starting from vertices in $\cC_{{\vec{\cS}}}$ are allowed to choose all vertices as their endpoints. Thus some of these arcs are possibly “wasted” by choosing their endpoints in $\cS_4$. This increases the probability that $N_{{\vec{\cS}}} = 0$. Thus $$\begin{aligned} \p{B_n \cap A'_n({\vec{\cS}})} & \le \p{[N_{{\vec{\cS}}} = 0] \cap A'_n({\vec{\cS}})} = \p{N_{{\vec{\cS}}} = 0\left|A'_n({\vec{\cS}})\right.}\p{A'_n({\vec{\cS}})} \\ & \le \p{N_{{\vec{\cS}}} = 0} \p{A'_n({\vec{\cS}})} \le \frac{\delta}{2} \p{A'_n({\vec{\cS}})} . \end{aligned}$$ Therefore $$\begin{aligned} \p{B_n \cap A_n(M,m)} & \le \sum_{{\vec{\cS}}\in \cA} \p{B_n \cap A'_n({\vec{\cS}})} \le \frac{\delta}{2} \sum_{{\vec{\cS}}\in \cA} \p{A'_n({\vec{\cS}})} \\ & \le \frac{\delta}{2} \p{({\cS_{\le m-1}^{+}(v_1)}, {\cS_{m}^{+}(v_1)}, {\cS_{m}^{-}(v_2)}, {\cS_{\le m-1}^{-}(v_2)}) \in \cA} \le \frac \delta 2 . \end{aligned}$$ By Corollary \[cor:typical:dist:spectrum\], we can choose $m$ depending on $M$ such that for $n$ large enough, $\p{B_n \cap A_n^c(M,m)} < \delta/2$. Thus $$\limsup_{n \to \infty} \p{B_n} = \limsup_{n \to \infty} \left( \p{B_n \cap A_n(M,m)} + \p{B_n \cap A_n^c(M,m)} \right) \le \delta. \tag*{\qedhere}$$ Extensions {#sec:extension} ========== The typical distance $H_n$ of ${{\cD_{n,k}}}$ is the distance between two vertices $v_1$ and $v_2$ chosen uniformly at random. If $v_1$ cannot reach $v_2$, then $H_n = \infty$. @Perarnau2014 proved that conditioned on $H_n < \infty$, $H_n/\log_k n \inprob 1$. We have found an alternative proof for this result using the path counting technique invented by @Van2014randomV2 [@Van2014randomV2 chap. 3.5]. The proof can be found in a longer version of this paper at \[<http://arxiv.org/abs/1504.06238>\]. @Perarnau2014 also proved that the diameter of the giant component divided by $\log n$ converges in probability to $1/\log(k) + 1/\log(1/\lambda_k)$. Recall that the longest path outside the giant divided by $\log n$ converges in probability to $1/\log(1/\lambda_k)$. This seems to be a strong indication that it might be possible to derive a new proof for the diameter of the giant. Recall that ${{\cD_{n,k}^{*}}}$ is a simple $k$-out digraph with $n$ vertices chosen uniformly at random from all such digraphs. Section \[sec:simple\] proved that if whp ${{\cD_{n,k}}}$ has property [**P**]{}, then whp ${{\cD_{n,k}^{*}}}$ has property [**P**]{}. But results like Theorem \[thm:CLL\], the central limit law of the one-in-core, cannot be transferred to ${{\cD_{n,k}^{*}}}$ automatically. We believe that it might be possible to achieve get the same result for ${{\cD_{n,k}^{*}}}$ following the line of @Janson2008’s treatment of the configuration model [@Janson2008]. A natural generalization of ${{\cD_{n,k}}}$ is to have a deterministic out-degree sequence, as in the directed configuration model, instead of requiring each vertex to have out-degree exactly $k$. With some constraints on the out-degree sequence, most of our results should hold for this generalized model. Furthermore, we could let each vertex choose its out-degree independently at random from an out-degree distribution. Again by adding some restrictions on the out-degree distribution, most of our results should still hold. The problem of generating a uniform random surjective function with fixed domain size is an open problem. Theorem \[thm:CLL\] implies a simple algorithm for choosing a $[km] \to [m]$ surjective function uniformly at random. Let $n =\ceil{m/{{\nu_k}}}$. Then we generate a ${{\cD_{n,k}}}$. If ${|{\cO_n}|}= m$, i.e., if the one-in-core in ${{\cD_{n,k}}}$ contains $m$ vertices, then ${{\cD_{n,k}}}[{\cO_n}]$ is equivalent to a uniform random sample of a $[km] \to [m]$ surjective function. Otherwise we try again until ${|{\cO_n}|}= m$. Theorem \[thm:CLL\] shows that $\p{{|{\cO_n}|}= m} = \Theta(1/\sqrt{m})$. Thus the expected number of ${{\cD_{n,k}}}$ needed to be generated is $\Theta(\sqrt{m})$. Since generating a ${{\cD_{n,k}}}$ takes $\Theta(m)$ time, the expected running time of the whole algorithm is $\Theta(m^{3/2})$. But we believe that $\Theta(m)$ should be achievable. Acknowledgment {#acknowledgment .unnumbered} ============== The authors thank Laura Eslava, Hamed Hatami, Guillem Perarnau, Bruce Reed, Henning Sulzbach and Yelena Yuditsky for valuable comments on this work, and Denis Thérien for pointing out the importance of the model. Appendix {#appendix .unnumbered} ======== 1. Inequalities for constants {#inequalities-for-constants .unnumbered} ----------------------------- Assume that $k \ge 2$. (a) There exists exactly one ${{\tau_k}}> 0$ such that $1-{{{\tau_k}}}/k - e^{-{{{\tau_k}}}} = 0$; (b) $0 < k - {{{\tau_k}}} < 1/2$; (c) $1/2 < 1-\frac{1}{2k} < {{\nu_k}}\equiv {{{\tau_k}}}/k < 1$; (d) ${{\lambda_k}}\equiv (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1} < \lambda_k' \equiv (k-{{{\tau_k}}}) e^{1-k+{{{\tau_k}}}} < 1$; (e) $\gamma_k \equiv \left( \frac k {e{{{\tau_k}}}} \right)^k(e^{{{{\tau_k}}}} -1) < 1$; (f) $\rho_{k} \equiv k e^{1-{{\tau_k}}} \left( \frac {{\tau_k}}k \right)^{k-1} < 1$; (g) ${{\lambda_k}}= \Theta(ke^{-k})$ as $k \to \infty$. \[lem:constant\] Let $\eta(x) = 1 - x/k - e^{-x}$. Since $\eta''(x) = -e^{-x} < 0$, $\eta(x)$ is strictly concave. Since $\eta(k-1/2) > 0$, and $\eta(k) < 0$, $\eta(x) = 0$ must have exactly one positive solution and this solution must be in $(k-1/2,k)$. Thus (a) and (b) are proved. (c) follows since ${{{\tau_k}}}/k > 1 -1/k \ge 1/2$. For (d) note that ${{\lambda_k}}< {{\lambda_k}}'$ as $1-x < e^{-x}$ for all $x \ne 0$. For $\lambda_k' < 1$ note that $$\begin{aligned} \log {{\lambda_k}}' = \log(k-{{{\tau_k}}}) + 1 - (k-{{{\tau_k}}}) = \log\left[1- (1- (k-{{{\tau_k}}})) \right] + 1 - (k-{{{\tau_k}}}) < 0, \end{aligned}$$ since $\log(1-x) < -x$ for all $x \in (0,1)$. For (e), first use ${{{\tau_k}}}/k \equiv 1-e^{-{{{\tau_k}}}}$ to get $$\begin{aligned} \gamma_k = \frac 1 {e^{k}( 1 - e^{-{{{\tau_k}}}} )^{k}} e^{{{{\tau_k}}}} (1-e^{-{{{\tau_k}}}}) = e^{{{{\tau_k}}}-k}( 1-e^{-{{{\tau_k}}}} )^{1-k}. \end{aligned}$$ Then use $k e^{-{{\tau_k}}} \equiv k - {{\tau_k}}$ to get $$\begin{aligned} \log \gamma_k & = {{{\tau_k}}} - k + (1-k) \log ( 1 - e^{-{{{\tau_k}}}}) \\ & = ({{{\tau_k}}} - k) + \log(1-e^{-{{{\tau_k}}}}) -k \log(1-e^{-{{{\tau_k}}}}) \\ & < ({{{\tau_k}}} - k) - e^{-{{{\tau_k}}}} +k (e^{-{{{\tau_k}}}} +e^{-2{{{\tau_k}}}}) \\ & = ({{{\tau_k}}} - k) + (k-{{{\tau_k}}}) + e^{-{{{\tau_k}}}}(k-{{{\tau_k}}} - 1) < 0, \end{aligned}$$ since $-x > \log(1-x) > -x -x^2$ for all $x \in (0, 1/2)$ and $e^{-{{\tau_k}}}=1-{{\nu_k}}\in (0,1/2)$. For (f), use ${{\tau_k}}< k$ from (a) to get $${{\tau_k}}\equiv k(1-e^{-{{\tau_k}}}) < k(1- e^{-k}) . \label{eq:k:tau:lower}$$ Therefore, $$\frac{{{\tau_k}}}{k} \equiv 1-e^{-{{\tau_k}}} < 1- \exp\left\{-k\left(1- e^{-k}\right)\right\} .$$ Again by (a), ${{\tau_k}}> k-1/2$. Thus $${{\tau_k}}\equiv k(1-e^{-{{\tau_k}}}) > k(1- e^{-k+\frac 1 2}) . \label{eq:k:tau:upper}$$ Therefore, $$ke^{-{{\tau_k}}} < k\exp\left\{-k\left(1- e^{-k+\frac 1 2}\right)\right\} .$$ The above bounds imply that $$\begin{aligned} \rho_{k} \equiv k e^{1-{{\tau_k}}} \left( \frac {{\tau_k}}k \right)^{k-1} < k \exp \left\{1 - k\left( 1-e^{-k + \frac 1 2} \right) \right\} \left( 1- \exp\left\{ -k \left( 1-e^{-k} \right) \right\} \right)^{k-1} . \end{aligned}$$ Using this bound, numeric computations show that $\rho_2 < 0.945651$. When $k \ge 3$, the above upper bound is less than $$k \exp \left\{1 - k\left( 1-e^{-\frac 5 2} \right) \right\},$$ which takes its maximal value at $k = 3$ for $k \in [3, \infty)$. This maximal value is about $0.52$. Thus $\rho_k < 1$ for all $k \ge 2$. By and , $k - {{\tau_k}}= ke^{-k + O(1)}$ and ${{\tau_k}}/k = 1- e^{-k+O(1)}$ as $k \to \infty$. Therefore $$\begin{aligned} {{\lambda_k}}& \equiv (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k-1} \right)^{k-1} \\ & = (k-{{{\tau_k}}})\left( \frac{{{{\tau_k}}}}{k} \right)^{k-1} \left( \frac{k}{k-1} \right)^{k-1} \\ & = ke^{-k+O(1)} \left(1- e^{-k+O(1)} \right)^{k-1} e(1+o(1)) = ke^{-k+O(1)} . \end{aligned}$$ Thus (g) is proved. 2. The sizes of $k$-surjections {#the-sizes-of-k-surjections .unnumbered} ------------------------------- In this section we prove Lemma \[lem:k:surj\]. Recall that $K_s$ is the number of $k$-surjections of size $s$ in ${{\cD_{n,k}}}$. We first deal the case that $s$ is small: \[lem:surj:one\] $\p{K_1 \ge 1} \le 1/{n^{k-1}} \le 1/n$. A single vertex is a $k$-surjection if and only if all its $k$ arcs are self-loops. Thus $$\p{K_1 \ge 1} \le \sum_{v \in [n]} \p{v \text{ has only self-loops}} = n \left( \frac 1 n \right)^k \le \frac 1 {n^{k-1}} \le \frac 1 n. \tag*{\qedhere}$$ \[lem:surj:small\] $ \p{\sum_{2 \le s \le a n} K_s \ge 1} = {{o\left( {1}/{n} \right)}}, $ for all fixed $a \in \left(0, e^{-1/(k-1)}\right)$. We can choose $\varepsilon \in (0,1)$ such that $2(k-1)(1-\varepsilon) > 1$ since $k \ge 2$. Let $J = \{2,\ldots, \lfloor a n \rfloor\}$. Then $$\begin{aligned} \p{\sum_{s \in J} K_s \ge 1} & \le \sum_{s \in J} \sum_{\cS \subseteq [n]:|\cS|=s} \p{\cS \text{ is closed}} \\ & = \sum_{s \in J} \binom{n}{s} \left( \frac s n \right)^{ks} \\ & \le \sum_{s \in J} \left( \frac {en}{s} \right)^s \left( \frac s n \right)^{ks} \qquad (\text{Stirling's approximation}) \\ & = \sum_{2 \le s \le n^{\varepsilon}} \left[ e \left( \frac s n \right)^{k-1} \right]^{s} + \sum_{n^{\varepsilon} < s < an } \left[ e \left( \frac s n \right)^{k-1} \right]^{s} \\ & \le \left[ e \left( \frac{n^{\varepsilon}}{n} \right)^{k-1} \right]^{2} \sum_{2 \le s+2} \left[ e \left( \frac{n^{\varepsilon}}{n} \right)^{k-1} \right]^{s} + \sum_{n^{\varepsilon} < s} \left( e \times a^{k-1} \right)^{s} \\ & = {{O\left( n^{-2(k-1)(1-\varepsilon)} \right)}} + {{O\left( (e a^{k-1})^{n^{\varepsilon}} \right)}}, \end{aligned}$$ where both terms are $o(1/n)$ due to our choice of $\varepsilon$ and $a$. When $s$ is large, we need to take into account the probability that $\cS$ is surjective. Let $\stirling{x}{y}$ denote Stirling’s number of the second kind, i.e., the number of ways to put $x$ balls into $y$ unordered bins such that there are no empty bins [@Flajolet2009 pp. 64]. Then $$\p{\cS \text{ is surjective}~|~\cS \text{ is closed}} = \frac{\stirling{ks}{s} s!}{s^{ks}},$$ where the numerator is the number of ways to choose endpoints for the $ks$ arcs in $\cS$ so that minimum in-degree is one, and the denominator is the total number of ways to choose endpoints for $ks$ arcs in $\cS$. Thus $$\begin{aligned} \p{\cS \text{ is a \(k\)-surjection}} & = \p{\cS \text{ is surjective}~|~\cS \text{ is closed}}\p{\cS \text{ is closed}} \\ & = \frac{\stirling{ks}{s} s!}{s^{ks}} \left(\frac{s}{n} \right)^{ks} = \frac{\stirling{ks}{s} s!}{n^{ks}}.\end{aligned}$$ @Good1961 established an asymptotic estimation of Stirling’s numbers of the second kind $$\stirling{k s}{s} \sim \frac{(k s)!}{s!} \frac{(e^{{{{\tau_k}}}}-1)^{s}}{{{{\tau_k}}}^{k s} \sqrt{2 \pi k s (1-k e^{-k})}}. $$ Applying this and Stirling’s approximation for factorials, we have $$\begin{aligned} \p{\cS \text{ is a \(k\)-surjection}} & \sim \frac{(k s)!}{s!} \frac{(e^{{{{\tau_k}}}}-1)^{s}}{{{{\tau_k}}}^{k s} \sqrt{2 \pi k s (1-k e^{-k})}} \frac{s!}{n^{ks}} \\ & \sim \frac{1}{\sqrt{1-ke^{-{{{\tau_k}}}}}} \left[ \left( \frac{s}{n} \right)^{k} \gamma_k \right]^{s}, {\addtocounter{equation}{1}\tag{\theequation}}\label{prob:comm}\end{aligned}$$ where $\gamma_k \equiv \left({k}/{e{{{\tau_k}}}} \right)^{k}(e^{{{{\tau_k}}}}-1) < 1$ (see Lemma \[lem:constant\]). \[lem:surj:big\] There exists a constant $b \in ({{\nu_k}}, 1)$ such that $\p{\sum_{bn \le s \le n} K_s \ge 1} = o\left( 1/n \right)$. Let $b > {{\nu_k}}$ be a constant decided later. If $|\cS| = s \in [bn, n]$, then by $$\begin{aligned} \p{\cS \text{ is a \(k\)-surjection}} = {{O\left( \left[\left(\frac{s}{n}\right)^{k}\gamma_k \right]^{s} \right)}} \le {{O\left( \gamma_k^s \right)}} \le {{O\left( \gamma_k^{bn} \right)}}. \end{aligned}$$ Since $b > {{\nu_k}}> 1/2$ (Lemma \[lem:constant\]), $$\binom{n}{s} \le \binom{n}{bn} = O\left(\frac 1 {\sqrt n} \left[ \frac{1}{b^{b}(1-b)^{1-b}} \right]^{n}\right).$$ Therefore $$\begin{aligned} \p{K_s \ge 1} \le \binom{n}{s} \p{\cS \text{ is a \(k\)-surjection}} \le {{O\left( \left[ \frac{\gamma_k^{b}}{b^{b}(1-b)^{1-b}} \right]^{n} \right)}}. \end{aligned}$$ Since the quantity in the square brackets goes to $\gamma_k < 1$ as $b \to 1$, we can pick a $b$ close enough to one such that $\p{\sum_{bn \le s \le n} K_s \ge 1} = o\left( 1/n \right)$. Let $a \in (0,{{\nu_k}})$ and $b \in ({{\nu_k}},1)$ be two constants such that the upper bounds in Lemma \[lem:surj:small\] and \[lem:surj:big\] hold. If $|\cS| = x n$ with $x \in (a, b)$ and $xn$ integer-valued, then by and Stirling’s approximation $$\begin{aligned} \e {K_{xn}} & = \binom{n}{xn} \p{\cS \text{ is a \(k\)-surjection}} \\ & \sim \frac{1}{\sqrt{2 \pi x (1- x) n}} \left[ \frac{1}{\left( x \right)^{x} (1-x)^{1-x}} \right]^{n} \frac{1}{\sqrt{1-ke^{-{{{\tau_k}}}}}} \left(x^k \gamma_k \right)^{x n} \\ & = \frac{1}{\sqrt{2 \pi (1-k e^{-{{{\tau_k}}}})n}} g\left( x \right) \left[f\left( x \right) \right]^{n} {\addtocounter{equation}{1}\tag{\theequation}}\label{expec:comm}\end{aligned}$$ where $$g(x) \equiv \frac 1 {\sqrt{x(1-x)}}, \qquad \qquad f(x) \equiv \left[ \frac{x^{k-1} \gamma_k}{(1-x)^{(1-x)/x} } \right]^{x}.$$ \[lem:surj:middle\] For all fixed $a \in (0, {{\nu_k}})$, $b \in ({{\nu_k}}, 1)$ and $\delta \in (0,1/2)$, $\p{\sum_{s \in J} K_s \ge 1} = o(1/n)$, where $J = [an, {{\nu_k}}n - n^{\frac 1 2 + \delta}] \cup [{{\nu_k}}n + n^{\frac 1 2 + \delta}, bn]$. Let $h(x) \equiv \log f(x)$. Lemma \[lem:function:h\] shows that as $x \to {{\nu_k}}$, $$h(x) = -\frac{(x-{{\nu_k}})^2}{2 \sigma_k^2}+{{O\left( |x- {{\nu_k}}|^3 \right)}},$$ and that $h(x)$ is strictly increasing on $(a,{{\nu_k}})$ and strictly decreasing on $({{\nu_k}},b)$. It follows from $|s/n - {{\nu_k}}| > {n^{{-1}/2+\delta}}$ that $h\left( s/n \right) \le -{n^{2\delta-1}}/{2 \sigma_k^2} + {{O\left( n^{3\delta-3/2} \right)}}. $ As for $g(x)$, it is bounded on $(a,b)$. Thus by and Markov’s inequality $$\begin{aligned} \log ( n^2 \p{K_{s} \ge 1}) & \le \log ( n^2 \e{K_{s}}) \\ & = \log \left(n^2 {{O\left( n^{-1/2} \right)}} f\left( \frac s n \right)^{n} \right) \\ & = {{O\left( \log n \right)}} + n h\left( \frac s n \right) \\ & \le {{O\left( \log n \right)}} -\frac{n^{2\delta}}{2 \sigma_k^2} + O\left( n^{3\delta-1/2} \right), \end{aligned}$$ which goes to $-\infty$. In other words, $ \p{K_s \ge 1} = o\left( 1/{n^2} \right). $ So $\p{\sum_{s \in J} K_s \ge 1} = o\left( 1/n \right)$. Lemma \[lem:k:surj\] follows immediately from Lemma \[lem:surj:one\], \[lem:surj:small\], \[lem:surj:big\], and \[lem:surj:middle\]. 3. Special functions {#special-functions .unnumbered} -------------------- Let $f(x)$, $g(x)$ and $h(x)$ be defined as in the previous subsection. Let ${{\nu_k}}$, ${{\tau_k}}$ and $\sigma_k$ be as in Lemma \[lem:constant\]. Then (a) As $x \to {{\nu_k}}$, $g\left( x \right) = g({{\nu_k}}) + {{O\left( |x-{{\nu_k}}| \right)}} = \left(1 + {{O\left( |x-{{\nu_k}}| \right)}} \right) /{(\sigma_k\sqrt{1-k e^{-{{\tau_k}}}})}$. (b) $h(x)$ and $f(x)$ are strictly increasing on $(1-\frac{1}{k},{{\nu_k}})$ and strictly decreasing on $\left({{\nu_k}}, 1 \right)$. (c) As $x \to {{\nu_k}}$, $$h(x) = h({{\nu_k}}) + O(|x-{{\nu_k}}|^3) = - \frac{(x-{{\nu_k}})^2}{2 \sigma_k^2} + O(|x-{{\nu_k}}|^3),$$ which implies that $$f(x) = e^{h(x)} = \exp\left\{- \frac{(x-{{\nu_k}})^2}{2 \sigma_k^2}\right\} + O(|x-{{\nu_k}}|^3).$$ \[lem:function:h\] For (a), recall that $ \sigma_k^2 \equiv {{{{\tau_k}}}}/(k e^{{{{\tau_k}}}}(1-ke^{{-{{\tau_k}}}})).$ Thus $\sigma_k^2 (1-k e^{-{{\tau_k}}}) = {{\nu_k}}(1-{{\nu_k}}).$ Then $ g({{\nu_k}})= {1}/{\sqrt{{{\nu_k}}(1-{{\nu_k}})}} = {1}/{\sigma_k \sqrt{1-ke^{-{{\tau_k}}}}}. $ Since $g'(x)$ is bounded around ${{\nu_k}}$, by Taylor’s theorem, $$g(x) = g({{\nu_k}}) + O(|x - {{\nu_k}}|) = (1+{{O\left( |x-{{\nu_k}}| \right)}})\frac{1}{\sigma_k \sqrt{1-ke^{-{{\tau_k}}}}}, \qquad \text{as \(x \to {{\nu_k}}\)}.$$ Let $r(x) = \log \left( f(x)^{1/x} \right) = h(x)/x$. Using ${{\tau_k}}/k \equiv 1-e^{-{{\tau_k}}} \equiv {{\nu_k}}$ shows that $$\gamma_k = \left( \frac{1}{e {{\nu_k}}} \right)^{k} e^{{{\tau_k}}} {{\nu_k}}= {{\nu_k}}^{-k+1} e^{-k+{{\tau_k}}} = {{\nu_k}}^{-k+1} (e^{-{{\tau_k}}})^{(k-{{\tau_k}})/{{\tau_k}}} = {{\nu_k}}^{-k+1} (1-{{\nu_k}})^{(1-{{\nu_k}})/{{\nu_k}}}.$$ Then $ r({{\nu_k}}) = \log\left( {{{\nu_k}}^{k-1}}{\left( 1-{{\nu_k}}\right)^{({{\nu_k}}-1)/{{\nu_k}}}} \gamma_k \right) = \log(1) = 0, $ $$r'(x) = \frac k x + \frac 1 {x^2} \log(1-x),\qquad \text{and} \qquad r''(x) = - \frac k {x^2} - \frac {2 \log(1-x)} {x^3} - \frac 1 {x^2(1-x)}.$$ Therefore $r'({{\nu_k}}) = 0$ and $r''({{\nu_k}})=-1/({{\nu_k}}\sigma_k^2)$. Since $h(x) = x r(x)$, $$h'(x) = r(x) + x r'(x), \qquad h''(x) = 2 r'(x) + x r''(x) = \frac k x - \frac 1 {x(1-x)}.$$ Thus $h({{\nu_k}}) = 0$, $h'({{\nu_k}}) = 0$ and $h''({{\nu_k}}) = -1/\sigma_k^2$. Also recalling that $1-\frac{1}{k} < 1 - \frac{1}{2k} < {{\nu_k}}< 1$ (Lemma \[lem:constant\]), $h(x)$ is strictly concave on $ (1-\frac{1}{k}, 1)$, reaching maximum at ${{\nu_k}}$. Thus (b) is proved. The two asymptotic equations in (c) follow from Taylor’s theorem. 4. Probability generating functions of Galton-Watson processes {#probability-generating-functions-of-galton-watson-processes .unnumbered} -------------------------------------------------------------- Let $\mu \in (0, \frac{1}{2k})$ be a constant where $k \ge 2$. Let $(Z_{m})_{m \ge 0}$ be a Galton-Watson process with $Z_0 \equiv 1$ and offspring distribution ${\mathop{\mathrm{Bin}}}(k, \mu)$. Let $\varphi_m(y) \equiv \e y^{Z_m}$. Then $$\varphi_{m}(0) \le 1 - (k \mu)^{m} + \left( 1- \frac{1}{2^{m}} \right) (k \mu)^{m+1}.$$ \[lem:gen:f\] We use induction. Let $c_m = 1 - 1/2^{m}$. For $m=1$, $$\varphi_1(y) = \e y^{Z_1} = (1 - \mu(1-y))^{k}.$$ Since $\mu > 0$ and $k \ge 2$, by Taylor’s theorem, $$\varphi_1(0) = (1-\mu)^k \le 1 - k \mu + \frac{(k \mu)^2}{2} = 1 - k \mu + c_1 (k \mu)^2.$$ It is well known that for $m > 1$, $\varphi_{m}(y) = \varphi_1(\varphi_{m-1}(y))$ (see [@Durrett2010probability]). Assuming the lemma holds for $m$, then $$\begin{aligned} \varphi_{m+1}(0) & = \varphi_1(\varphi_{m}(0)) = \left( 1- \mu \left( 1-\varphi_m(0) \right) \right)^{k} \\ & \le \left( 1- \mu\left( (k\mu)^{m} - c_m(k \mu)^{m+1} \right) \right)^k \\ & \le 1 - k \mu \left( (k\mu)^{m} - c_m(k \mu)^{m+1} \right) + \frac{k^2}{2} \mu^2 \left( (k\mu)^{m} - c_m(k \mu)^{m+1} \right)^2 \\ & = 1 - (k\mu)^{m+1} + c_m (k \mu)^{m+2} + \frac{(k \mu)^m}{2}(1- c_m k \mu)^2 (k \mu)^{m+2} \\ & \le 1 - (k\mu)^{m+1} + c_{m+1}(k \mu)^{m+2} , \end{aligned}$$ since $k \mu < 1/2$ and $c_{m+1} = c_m + 1/2^{m+1}$. [^1]: In a shorter version of this paper, this section is omitted.

--- abstract: 'Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. In our construction, the space fails to be even locally compact.' author: - 'Roman Glebov[^1]' - 'Daniel Král’[^2]' - 'Jan Volec[^3]' title: 'Compactness and finite forcibility of graphons[^4]' --- Introduction {#sec:intro} ============ Recently, a theory of limits of combinatorial structures emerged and attracted substantial attention. The most studied case is that of limits of dense graphs initiated by Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi [@bib-borgs08+; @bib-borgs+; @bib-borgs06+; @bib-lovasz06+; @bib-lovasz10+], which we also address in this paper. A sequence of graphs is convergent if the density of every graph as a subgraph in the graphs contained in the sequence converges. A convergent sequence of graphs can be associated with an analytic object (graphon) which is a symmetric measurable function from the unit square $[0,1]^2$ to $[0,1]$. Graph limits and graphons are also closely related to flag algebras introduced by Razborov [@bib-razborov07], which were successfully applied to numerous problems in extremal combinatorics [@bib-flag1; @bib-flag2; @bib-flagrecent; @bib-flag3; @bib-flag3half; @bib-flag4; @bib-flag5; @bib-flag6; @bib-flag7; @bib-flag8; @bib-flag9; @bib-flag10; @bib-razborov07; @bib-flag11; @bib-flag12]. The development of the graph limit theory is also reflected in a recent monograph by Lovász [@bib-lovasz-book]. In this paper, we are concerned with finitely forcible graphons, i.e., those that are uniquely determined (up to a natural equivalence) by finitely many subgraph densities. Such graphons are related to uniqueness of extremal configurations in extremal graph theory as well as to other problems. For example, the classical result of Chung, Graham and Wilson [@bib-chung89+] asserting that a large graph is pseudorandom if and only if the homomorphic densities of $K_2$ and $C_4$ are the same as in the Erdős-Rényi random graph $G_{n,1/2}$ can be cast in the language of graphons as follows: The graphon identically equal to $1/2$ is uniquely determined by homomorphic densities of $K_2$ and $C_4$, i.e., it is finitely forcible. Another example that can be cast in the language of finite forcibility is the asymptotic version of the theorem of Turán [@bib-turan]: There exists a unique graphon with edge density $\frac{r-1}{r}$ and zero density of $K_{r+1}$, i.e., it is finitely forcible. The result of Chung, Graham, and Wilson [@bib-chung89+] was generalized by Lovász and Sós [@bib-lovasz08+] who proved that any graphon that is a stepfunction is finitely forcible. This result was further extended and a systematic study of finitely graphons was initiated by Lovász and Szegedy [@bib-lovasz11+] who found first examples of finitely forcible graphons that are not stepfunctions. In particular, they observed that every example of a finitely forcible graphon that they found had a somewhat finite structure. Formalizing this intuition, they associated typical vertices of a graphon $W$ with the topological space $T(W)\subseteq L_1[0,1]$ and observed that their examples of finitely forcible graphons $W$ have compact and at most $1$-dimensional $T(W)$. This led them to make the following conjectures [@bib-lovasz11+ Conjectures 9 and 10]. \[conj:1\] If $W$ is a finitely forcible graphon, then $T(W)$ is a compact space. They noted that they could not even prove that $T(W)$ had to be locally compact. We give a construction of a finitely forcible graphon $W$ such that $T(W)$ fails to be locally compact, in particular, $T(W)$ is not compact. \[thm:main\] There exists a finitely forcible graphon $W_R$ such that the topological space $T(W_R)$ is not locally compact. They also made the following conjecture on the dimension of $T(W)$ of a finitely forcible graphon $W$. \[conj:2\] If $W$ is a finitely forcible graphon, then $T(W)$ is finite dimensional. Lovász and Szegedy noted in their paper that they did not want to specify the notion of dimension they had in mind. Since every non-compact subset of $L_1[0,1]$ has infinite Minkowski dimension, Theorem \[thm:main\] also provides a partial answer to Conjecture \[conj:2\]. Since we believe that other notions of dimension than the Minkowski dimension (e.g., the Lebesgue dimension) are more appropriate measures of dimension for $T(W)$, we do not claim to disprove this conjecture in this paper. We discuss Conjecture \[conj:2\] in more details in Section \[sec:concl\], where we also mention other applications of techniques used in this paper. Notation ======== In this section, we introduce notation related to concepts used in this paper. A [*graph*]{} is a pair $(V,E)$ where $E\subseteq {V\choose 2}$. The elements of $V$ are called [*vertices*]{} and the elements of $E$ are called [*edges*]{}. The [*order*]{} of a graph $G$ is the number of its vertices and it is denoted by $|G|$. The [*density*]{} $d(H,G)$ of $H$ in $G$ is the probability that $|H|$ randomly chosen distinct vertices of $G$ induce a subgraph isomorphic to $H$. If $|H|>|G|$, we set $d(H,G)=0$. A sequence of graphs $(G_i)_{i\in\NN}$ is [*convergent*]{} if the sequence $(d(H,G_i))_{i\in\NN}$ converges for every graph $H$. We now present basic notions from the theory of dense graph limits as developed in [@bib-borgs08+; @bib-borgs+; @bib-borgs06+; @bib-lovasz06+]. A [*graphon*]{} $W$ is a symmetric measurable function from $[0,1]^2$ to $[0,1]$. Here, symmetric stands for the property that $W(x,y)=W(y,x)$ for every $x,y\in [0,1]$. A [*$W$-random graph*]{} of order $k$ is obtained by sampling $k$ random points $x_1,\ldots,x_k\in [0,1]$ uniformly and independently and joining the $i$-th and the $j$-th vertex by an edge with probability $W(x_i,x_j)$. Since the points of $[0,1]$ play the role of vertices, we refer to them as to vertices of $W$. To simplify our notation further, if $A\subseteq [0,1]$ is measurable, we use $|A|$ for its measure. The [*density $d(H,W)$*]{} of a graph $H$ in a graphon $W$ is equal to the probability that a $W$-random graph of order $|H|$ is isomorphic to $H$. Clearly, the following holds: $$d(H,W)=\frac{|H|!}{|\Aut(H)|}\int\limits_{[0,1]^{|H|}} \prod_{(i,j)\in E(H)} W(x_i,x_j) \prod_{(i,j)\not\in E(H)} (1-W(x_i,x_j)) \,\dif \lambda_{|H|},$$ where $\Aut(H)$ is the automorphism group of $H$. One of the key results in the theory of dense graph limits asserts that for every convergent sequence $(G_i)_{i\in\NN}$ of graphs with increasing orders, there exists a graphon $W$ (called the [*limit*]{} of the sequence) such that for every graph $H$, $$d(H,W)=\lim_{i\to\infty} d(H,G_i)\;\mbox{.}$$ Conversely, if $W$ is a graphon, then the sequence of $W$-random graphs with increasing orders converges with probability one and its limit is $W$. Every graphon can be assigned a topological space corresponding to its typical vertices [@bib-lovasz10++]. For a graphon $W$, define for $x\in [0,1]$ a function $f^W_x(y)=W(x,y)$. For an open set $A\subseteq L_1[0,1]$, we write $A^W$ for ${\left}\{x\in [0,1],\, f_x^W\in A{\right}\}$. Let $T(W)$ be the set formed by the functions $f\in L_1[0,1]$ such that ${\left}|U^W{\right}|>0$ for every neighborhood $U$ of $f$. The set $T(W)$ inherits topology from $L_1[0,1]$. The vertices $x\in [0,1]$ with $f^W_x\in T(W)$ are called [*typical vertices*]{} of a graphon $W$. Notice that almost every vertex is typical [@bib-lovasz10++]. Two graphons $W_1$ and $W_2$ are [*weakly isomorphic*]{} if $d(H,W_1)=d(H,W_2)$ for every graph $H$. If $\varphi:[0,1]\to[0,1]$ is a measure preserving map, then the graphon $W^\varphi(x,y):=W(\varphi(x),\varphi(y))$ is always weakly isomorphic to $W$. The opposite is true in the following sense [@bib-borgs10+]: if two graphons $W_1$ and $W_2$ are weakly isomorphic, then there exist measure measure preserving maps $\varphi_1:[0,1]\to [0,1]$ and $\varphi_2:[0,1]\to [0,1]$ such that $W_1^{\varphi_1}=W_2^{\varphi_2}$ almost everywhere. A graphon $W$ is [*finitely forcible*]{} if there exist graphs $H_1,\ldots,H_k$ such that every graphon $W'$ satisfying $d(H_i,W)=d(H_i,W')$ for $i\in\{1,\ldots,k\}$ is weakly isomorphic to $W$. For example, the result of Diaconis, Homes, and Janson [@DiHoJa8] asserts that the half graphon $W_{\Delta}(x,y)$ defined as $W_{\Delta}(x,y)=1$ if $x+y\ge 1$, and $W_{\Delta}=0$, otherwise, is finitely forcible. Also see [@bib-lovasz11+] for further results. When dealing with a finitely forcible graphon, we usually give a set of equality constraints that uniquely determines $W$ instead of specifying the finitely many subgraphs that uniquely determine $W$. A [*constraint*]{} is an equality between two density expressions where a [*density expression*]{} is recursively defined as follows: a real number or a graph $H$ are density expressions, and if $D_1$ and $D_2$ are two density expression, then the sum $D_1+D_2$ and the product $D_1\cdot D_2$ are also density expressions. The value of the density expression is the value obtained by substituting for every subgraph $H$ its density in the graphon. Observe that if $W$ is a unique (up to weak isomorphism) graphon that satisfies a finite set $\CC$ of constraints, then it is finitely forcible. In particular, $W$ is the unique (up to weak isomorphism) graphon with densities of subgraphs appearing in $\CC$ equal to their densities in $W$. This holds since any graphon with these densities satisfies all constraints in $\CC$ and thus it must be weakly isomorphic to $W$. We extend the notion of density expressions to rooted density expressions following the ideas from the concept of flag algebras from [@bib-razborov07]. A subgraph is [*rooted*]{} if it has $m$ distinguished vertices labeled with numbers $1,\ldots,m$. These vertices are referred to as roots while the other vertices are non-roots. Two rooted graphs are [*compatible*]{} if the subgraphs induced by their roots are isomorphic through an isomorphism mapping the roots with the same label to each other. Similarly, two rooted graphs are [*isomorphic*]{} if there exists an isomorphism that maps the $i$-th root of one of them to the $i$-th root of the other. A [*rooted density expression*]{} is a density expression such that all graphs that appear in it are mutually compatible rooted graphs. We will also speak about compatible rooted density expressions to emphasize that the rooted graphs in all of them are mutually compatible. The value of a rooted density expression is defined in the next paragraph. Fix a rooted graph $H$. Let $H_0$ be the graph induced by the roots of $H$, and let $m=|H_0|$. For a graphon $W$ with $d(H_0,W)>0$, we let the auxiliary function $c:[0,1]^m\to [0,1]$ denote the probability that an $m$-tuple $(x_1,\ldots, x_m)\in[0,1]^m$ induces a copy of $H_0$ in $W$ respecting the labeling of vertices of $H_0$: $$c(x_1,\ldots, x_m)= {\left}(\prod_{(i,j)\in E(H_0)} W(x_i,x_j){\right}) \cdot{\left}(\prod_{(i,j)\not\in E(H_0)} (1-W(x_i,x_j)) {\right}).$$ We next define a probability measure $\mu$ on $[0,1]^m$. If $A\subseteq [0,1]^m$ is a Borel set, then: $$\mu(A)= \frac{\int\limits_A c(x_1,\ldots, x_m) \dif\lambda_m}{\int\limits_{[0,1]^m} c(x_1,\ldots, x_m) \dif\lambda_m}.$$ When $x_1,\ldots,x_m\in [0,1]$ are fixed, then the density of a graph $H$ with root vertices $x_1,\ldots, x_m$ is the probability that a random sample of non-roots yields a copy of $H$ conditioned on the roots inducing $H_0$. Noticing that an automorphism of a rooted graph has all roots as fixed points, we obtain that this is equal to $$\frac{(|H|-m)!}{|\Aut(H)|}\int\limits_{[0,1]^{|H|-m}} \prod\limits_{(i,j)\in E(H)\setminus E(H_0)} W(x_i,x_j)\prod\limits_{(i,j)\not\in E(H)\cup \binom{H_0}{2}} (1-W(x_i,x_j)) \dif \lambda_{|H|-m}.$$ For different choices of $x_1,\ldots, x_m$, we obtain different values. The value of a rooted density expression is a random variable determined by the choice of the roots according to the probability distribution $\mu$. We now consider a constraint such that both left and right hand sides $D$ and $D'$ are compatible rooted density expression. Such a constraint should be interpreted to mean that it holds $D-D'=0$ with probability one. It can be shown (see, e.g., [@bib-razborov07]) that the expected value of a rooted density expression $D$ with roots inducing $H_0$ is equal to ${\left\llbracket D\right\rrbracket_{}}/d(H_0,W)$, where ${\left\llbracket D\right\rrbracket_{}}$ is an ordinary density expression independent of $W$. Observe that if $D$ and $D'$ are compatible rooted density expressions, then a graphon satisfies $D=D'$ if and only if it satisfies the (ordinary) constraint ${\left\llbracket (D-D')\times (D-D')\right\rrbracket_{}}=0$ . Since this allows us to express constraints involving rooted density expressions as ordinary constraints, we will not distinguish between the two types of constraints in what follows. Partitioned graphons ==================== In this section, we introduce partitioned graphons. Some of the methods presented in this section are analogous to those used by Lovász and Sós in [@bib-lovasz08+] and by Norine [@bib-norine-comm] (see the construction in Section \[sec:concl\]). In particular, they used similar types of arguments to specialize their constraints to parts of graphons they were forcing as we do in this section. However, since it is hard to refer to any particular lemma in their paper instead of presenting a full argument, we decided to give all details. A [*degree*]{} of a vertex $x\in [0,1]$ of a graphon $W$ is equal to $$\int_{[0,1]}W(x,y)\dif y\;\mbox{.}$$ Note that the degree is well-defined for almost every vertex of $W$. A graphon $W$ is [*partitioned*]{} if there exist $k\in \mathbb{N}$ and positive reals $a_1,\ldots,a_k$ with $\sum_i a_i=1$ and distinct reals $d_1,\ldots,d_k \in [0,1]$ such that the the set of vertices of $W$ with degree $d_i$ has measure $a_i$. We will often speak just about partitioned graphons when having in mind fixed values of $k$, $a_1,\ldots,a_k$, and $d_1,\ldots,d_k$. Having a fixed partition can be finitely forced as given in the next lemma. \[lm-partition\] Let $a_1,\ldots,a_k$ be positive real numbers summing to one and let $d_1$,$\ldots$,$d_k$ be distinct reals between zero and one. There exists a finite set of constraints $\CC$ such that any graphon $W$ satisfying $\CC$ also satisfies the following: $$\mbox{The set of vertices of $W$ with degree $d_i$ has measure $a_i$.}$$ In other words, such $W$ must be a partitioned graphon with parts of sizes $a_1,\ldots,a_k$ and degrees $d_1,\ldots,d_k$. The graphon is forced by the following set of constraints: $$\prod\limits_{i=1}^k (e_1-d_i)=0\;\mbox{, and}$$ $${\left\llbracket \prod\limits_{i=1,\, i\neq j}^k (e_1-d_i)\right\rrbracket_{}}=a_j\prod\limits_{i=1,\, i\neq j}^k(d_j-d_i) \mbox{~~for every $j$, $1\leq j\leq k$,}$$ where $e_1$ is an edge with one root and one non-root. The first constraint says that the degree of almost every vertex is equal to one of the numbers $d_1,\ldots,d_k$. For $j\leq k$, the left hand side of the second constraint before applying the ${\left\llbracket \cdot\right\rrbracket_{}}$-operator is non-zero only if the degree of the root is $d_j$. Hence, the left hand side is equal to $$\prod_{i=1,i\not=j}^k(d_j-d_i)$$ in that case. Therefore, the measure of vertices of degree $d_j$ is forced to be $a_j$. Assume that $W$ is a partitioned graphon. We write $A_i$ for the set of vertices of degree $d_i$ for $i$, $1\leq i\leq k$ and identify $A_i$ with the interval $[0,a_i)$ (note that the measure of $A_i$ is $a_i$). This will be convenient when defining partitioned graphons. For example, we can use the following when defining a graphon $W$: $W(x,y)=1$ if $x\in A_1$, $y\in A_2$ and $x\ge y$. A graph $H$ is [*decorated*]{} if its vertices are labeled with parts $A_1,\ldots,A_k$. The density of a decorated graph $H$ is the probability that randomly chosen $|H|$ vertices induce a subgraph isomorphic to $H$ with its vertices contained in the parts corresponding to the labels. For example, if $H$ is an edge with vertices decorated with parts $A_1$ and $A_2$, then the density of $H$ is the density of edges between $A_1$ and $A_2$, i.e., $$d(H,W)=\int\limits_{A_1}\int\limits_{A_2}W(x,y)\,\dif x\,\dif y\;\mbox{.}$$ Similarly as in the case of non-decorated graphs, we can define rooted decorated subgraphs. A constraint that uses (rooted or non-rooted) decorated subgraphs is referred to as decorated. The next lemma shows that decorated constraints are not more powerful than non-decorated ones, and therefore they can be used to show that a graphon is finitely forcible. We will always apply this lemma after forcing a graphon $W$ to be partitioned using Lemma \[lm-partition\]. Before proving it, we introduce convention for drawing density expressions: edges of graphs are always drawn solid, non-edges dashed, and if two vertices are not joined, then the picture represents the sum over both possibilities. If a graph contains some roots, the roots are depicted by square vertices, and the non-root vertices by circles. If there are more roots from the same part, then the squares are rotated to distinguish the roots. Decorations of vertices are always drawn inside vertices. \[lm-partitioned\] Let $k\in \mathbb{N}$, and let $a_1,\ldots,a_k$ be positive real numbers summing to one and let $d_1,\ldots,d_k$ be distinct reals between zero and one. If $W$ is a partitioned graphon with $k$ parts formed by vertices of degree $d_i$ and measure $a_i$ each, then any decorated (rooted or non-rooted) constraint can be expressed as a non-decorated constraint, i.e., $W$ satisfies the decorated constraint if and only if it satisfies the non-decorated constraint. By the argument analogous to the non-decorated case, it is enough to show that the density of a non-rooted decorated subgraph can be expressed as a combination of densities of non-decorated subgraphs. Let $H$ be a non-rooted decorated subgraph with vertices $v_1,\ldots,v_n$ such that $v_i$ is labeled with a part $A_{\ell_i}$. Let $H_i$ be the sum of all rooted non-decorated graphs on $n+1$ vertices with $n$ roots such that the roots induce $H$ with the $j$-th vertex being $v_j$ for $j=1,\ldots,n$ and the only non-root is always adjacent to $v_i$ (an example is given in Figure \[fig-lm-partitioned\]). We claim that the density of $H$ is equal to the following: $$\label{(1)} \frac{|H|!}{|\Aut(H)|}{\left\llbracket \prod_{i=1}^n \prod_{j=1,\, j\neq\ell_i}^k \frac{H_i-d_j}{d_{\ell_i}-d_j}\right\rrbracket_{}} \;\mbox{.}$$ Indeed, if the $n$ roots are chosen on a copy of $H$ such that the $i$-th root is not from $A_{\ell_i}$, then the second product of the above expression is zero. Otherwise, the second product is one, possibly except for a set of measure zero. Hence, the value of (\[(1)\]) is exactly the probability that randomly chosen $n$ vertices induce a labeled copy of $H$ such that the $i$-th vertex belong to $A_{\ell_i}$. Since decorated constraints are not more powerful than non-decorated ones, we will not distinguish between decorated and non-decorated constraints in what follows. We finish this section with two lemmas that are straightforward corollaries of Lemma \[lm-partitioned\]. The first one says that we can finitely force a finitely forcible graphon on a part of a partitioned graphon. \[lm-gadget\] Let $W_0$ be a finitely forcible graphon. Then for every choice of $k\in \mathbb{N}$, positive reals $a_1,\ldots,a_k$ summing to one, distinct reals $d_1,\ldots,d_k$ between zero and one, and $\ell\leq k$, there exists a finite set of constraints $\CC$ such that the graphon induced by the $\ell$-th part of every graphon $W$ that is a partitioned graphon with $k$ parts $A_1,\ldots, A_k$ of measures $a_1,\ldots,a_k$ and degrees $d_1,\ldots,d_k$, respectively, and that satisfies $\CC$ is weakly isomorphic to $W_0$. More precisely, there exist measure preserving maps $\varphi$ and $\varphi'$ from $A_\ell$ to itself such that $W_0(\varphi(x)/|A_\ell|,\varphi(y)/|A_\ell|)=W(\varphi'(x),\varphi'(y))$ for almost every $x,y\in A_\ell$. Assume that $W_0$ is forced by constraints of the form $$d(H_i,W)=d_i$$ for $i\in [m]$. The set $\CC$ is then formed by constraints of the form $$d(H'_i,W)=a_{\ell}^{|H_i|}d_i\; \mbox{,}$$ where $H'_i$ is the graph $H_i$ with all vertices decorated with $A_{\ell}$. The second lemma asserts finite forcibility of pseudorandom bipartite graphs between different parts of a partitioned graphon. \[lm-pseudobipartite\] For every choice of $k\in \mathbb{N}$, positive reals $a_1,\ldots,a_k$ summing to one, distinct reals $d_1,\ldots,d_k$ between zero and one, $\ell,\ell'\leq k$, $\ell\neq \ell'$, and $p\in [0,1]$, there exists a finite set of constraints $\CC$ such that every graphon $W$ that is a partitioned graphon with $k$ parts $A_1,\ldots, A_k$ of measures $a_1,\ldots,a_k$ and degrees $d_1,\ldots,d_k$, respectively, and that satisfies $\CC$ also satisfies that $W(x,y)=p$ for almost every $x\in A_\ell$ and $y\in A_{\ell'}$. Let $H$ be a rooted edge with the root decorated with $A_{\ell}$ and the non-root decorated with $A_{\ell'}$, let $H_1$ be a triangle with two roots such that the roots are decorated with $A_{\ell}$ and the non-root with $A_{\ell'}$, and let $H_2$ be a cherry (a path on three vertices) with two roots on its non-edge such that the roots are decorated with $A_{\ell}$ and the non-root with $A_{\ell'}$. The set $\CC$ is formed by three constraints: $H=p$, $H_1=p^2$, and $H_2=p^2$ (also see Figure \[fig:CS\]). These constraints imply that $$\int\limits_{A_{\ell'}}W(x,y)\,\dif y=a_{\ell'}p\qquad\mbox{and}\qquad \int\limits_{A_{\ell'}}W(x,y)\cdot W(x',y)\,\dif y=a_{\ell'}p^2$$ for almost every $x,x'\in A_\ell$. Following the reasoning given in [@bib-lovasz11+ proof of Lemma 3.3], the second equation implies that $$\int\limits_{A_{\ell'}}W^2(x,y)\,\dif y=a_{\ell'}p^2$$ for almost every $x\in A_\ell$. Cauchy-Schwarz inequality yields that $W(x,y)=p$ for almost every $x\in A_\ell$ and $y\in A_{\ell'}$. Rademacher Graphon ================== In this section, we introduce a graphon $W_{R}$ which we refer to as [*Rademacher graphon*]{}. The name comes from the fact that the adjacencies between its parts $A$ and $C$ resembles Rademacher system of functions (such adjacencies also appear in [@bib-lovasz-book Example 13.30]). We establish its finite forcibility in the next section. The graphon $W_{R}$ has eight parts. Instead of using $A_1, \dots, A_8$ for its parts, we use $A$, $A'$, $B$, $B'$, $B''$, $C$, $C'$ and $D$. All the parts except for $C$ have the same size $a=1/9$; the size of $C$ is $2a=2/9$. For $x \in [0, 1)$, let us denote by $[x]$ the smallest integer $k$ such that $x + 2^{-k} < 1$. The graphon $W_{R}$ is then defined as follows (also see Figure \[fig-chess\]). Let $x$ and $y$ be two of its vertices. The value $W_{R}(x, y)$ is equal to $1$ in the following cases: - $x, y \in A$ and $[x/a] \neq [y/a]$, - $x, y \in A'$ and $[x/a] \neq [y/a]$, - $x \in A$, $y \in A'$ and $[x/a] = [y/a]$, - $x \in A$, $ y \in B$ and $x+y \le a$, - $x \in A$, $ y \in B''$ and $x+y \ge a$, - $x \in A'$, $ y \in B'$ and $x+y\le a$, - $x \in A'$, $ y \in B''$ and $y\le x$, - $x, y \in B$ and $x + y \ge a$, - $x, y \in B'$ and $x + y \ge a$, - $x \in A$, $ y \in C$ and $\left\lfloor \frac{y}{2a} \cdot 2^{[x/a]}\right\rfloor$ is even, - $x \in A'$, $y\in C'$ and $(1-2^{-[x/a]}-x/a)2^{[x/a]} + y/a \le 1$, - $x, y \in C'$ and $x + y \ge a$. If $x\in A'$, $y\in C$ and $\left\lfloor \frac{y}{2a} \cdot 2^{[x/a]}\right\rfloor$ is even, then $W_{R}(x,y)=(1-2^{-[x/a]}-x/a)2^{[x/a]}$. If $x,y\in C$, then $W_{R}(x,y)=3/4$ if $x+y\ge 2a$. If $y\in D$, then $$W_{R}(x,y)=\left\{ \begin{array}{cl} 0.2 & \mbox{if $x\in A'$ or $x\in B'$,} \\ 0.4 & \mbox{if $x\in B''$, and } \\ 0.8 & \mbox{if $x\in C'$.} \end{array} \right.$$ Finally, $W_{R}(x,y)=0$ if neither $(x,y)$ nor the symmetric pair fall in any of the described cases. Part $A$ $A'$ $B$ $B'$ $B''$ $C$ $C'$ $D$ -------- ------- --------- ------- -------- -------- -------- -------- -------- Degree $3a$ $3.2a$ $a$ $1.2a$ $1.4a$ $1.5a$ $1.8a$ $1.6a$ $3/9$ $16/45$ $1/9$ $2/15$ $7/45$ $1/12$ $1/5$ $8/45$ : The degrees of vertices in the nine parts of Rademacher graphon $W_{R}$.[]{data-label="tbl-chess"} The degrees of vertices in the eight parts of Rademacher graphon $W_{R}$ are routine to compute and they are given in Table \[tbl-chess\]. We finish this section with establishing that Rademacher graphon, assuming its finite forcibility, yields Theorem \[thm:main\]. \[prop-non-compact\] The topological space $T(W_{R})$ is not locally compact. We understand the interval $[0,1]$ to be partitioned by the intervals $A$, $A'$, $B$, $B'$, $B''$, $C$, $C'$ and $D$. Let $g:[0,1]\to [0,1]$ be the function defined as follows: $$g(x)=\left\{ \begin{array}{cl} 1 & \mbox{if $x\in A'\cup B''\cup C'$,} \\ 0.2 & \mbox{if $x\in D$, and} \\ 0 & \mbox{otherwise.} \end{array}\right.$$ Further, let $g_{i,\delta}:[0,1]\to [0,1]$ for $i\in\NN$ and $\delta\in [0,1]$ be defined as follows: $$g_{i,\delta}(x)=\left\{ \begin{array}{cl} 1 & \mbox{if $x\in A$ and $[x/a]=i$,} \\ 1 & \mbox{if $x\in A'$ and $[x/a]\not=i$,} \\ 1 & \mbox{if $x\in B'$ and $x\le (1+\delta)2^{-i}$,} \\ 1 & \mbox{if $x\in B''$ and $x\le 1-(1+\delta)2^{-i}$,} \\ \delta & \mbox{if $x\in C$ and $\lfloor 2^i\cdot x/2a\rfloor$ is even,} \\ 1 & \mbox{if $x\in C'$ and $x/a\le1-\delta$,} \\ 0.2 & \mbox{if $x\in D$, and} \\ 0 & \mbox{otherwise.} \end{array}\right.$$ Observe that $W_{R}{\left}(x,2/9-(1+\delta)2^{-i}{\right})=g_{i,\delta}(x)$ for every $i\in\NN$, $\delta\in (0,1)$, and $x\in [0,1]$. The following two estimates on the distances between $g$ and $g_{i,\delta}$ are straightforward to obtain: $$\begin{array}{rcll} \| g_{i,\delta}-g\|_1 & =& \frac{(4+2\delta)\cdot 2^{-i}+2\delta}{9}& \\ \| g_{i,\delta}-g_{i',\delta'}\|_1 & > & \frac{\delta+\delta'}{18} &\mbox{for $i\neq i'$.} \end{array}$$ Hence, since $g_{i,\delta}\in T(W_{R})$ for every $i\in\NN$ and $\delta\in (0,1)$, we obtain that $g\in T(W_{R})$. However, for every $\varepsilon>0$, all the functions $g_{i,\varepsilon}$ with $i>\log_2\varepsilon^{-1}$ are at $L_1$-distance at most $\varepsilon$ from $g$ and the $L_1$-distance between any pair of them is at least $\varepsilon/9$. We conclude that no neighborhood of $g$ in $T(W)$ is compact. Forcing ======= In this section, we prove that Rademacher graphon $W_{R}$ is finitely forcible. We first describe the set of constraints. We give names to the different kinds of these constraints to refer to them later. The whole set of constraints is denoted by $\CC_{R}$ in what follows. - The [*partition constraints*]{} forcing the existence of eight parts of sizes as in $W_{R}$ and with vertex degrees as in $W_{R}$ (the existence of such constraints follows from Lemma \[lm-partition\]), - the [*zero constraints*]{} setting the edge density inside $B''$ and $D$ to zero as well as setting the edge density between the following pairs of parts to zero: $A$ and $C'$, $A$ and $D$, $A'$ and $B$, $B$ and $B'$, $B$ and $B''$, $B$ and $C$, $B$ and $C'$, $B$ and $D$, $B'$ and $B''$, $B'$ and $C$, $B'$ and $C'$, $B''$ and $C$, $B''$ and $C'$, and $C$ and $D$, - the [*triangular constraints*]{} forcing the half graphons on $B$, $B'$, $C$, and $C'$ with densities $1$, $1$, $1$ and $3/4$ (see Lemma \[lm-gadget\] and [@bib-lovasz11+ Corollaries 3.15 and 5.2] for their existence), respectively, - the [*pseudorandom constraints*]{} forcing the pseudorandom bipartite graph between $D$ and the parts $A'$, $B'$, $B''$, and $C'$ with densities $0.2$, $0.2$, $0.4$, and $0.8$, respectively (see Lemma \[lm-pseudobipartite\] for their existence), - the [*monotonicity constraints*]{} depicted in Figure \[fig-mono\], - the [*split constraints*]{} depicted in Figure \[fig-split\], - the [*infinitary constraints*]{} depicted in Figure \[fig-infin\], and - the [*orthogonality constraints*]{} depicted in Figure \[fig-ortho\]. The existence of the corresponding monotonicity, split, infinitary, and orthogonality constraints as ordinary constraints follows from Lemma \[lm-partitioned\]. Also note that the first five monotonicity constraints imply that the graphon has values zero and one almost everywhere between the corresponding parts (also see [@bib-lovasz11+ Lemma 3.3] for further details). \[thm-main\] If $W$ is a graphon satisfying all constraints in $\CC_{R}$, then there exist measure preserving maps $\varphi,\psi:[0,1]\to [0,1]$ such that $W^{\varphi}$ and $W^{\psi}_{R}$ are equal almost everywhere. Since $W$ satisfies the partition constraints contained in $\CC_{R}$, Lemma \[lm-partition\] yields that the interval $[0,1]$ can be partitioned into eight parts all but one having measure $1/9$ and the remaining one with measure $2/9$ such that almost all vertices in the parts have degrees as those in the corresponding parts of $W_{R}$. In particular, there exists a measure preserving map $\varphi:[0,1]\to [0,1]$ such that the subintervals of $[0,1]$ corresponding to the parts of $W_R$ are mapped to the corresponding parts of $W$. From now on, we use $A$, ${A'}$, $B$, ${B'}$, ${B''}$, $C$, ${C'}$, and $D$ for the subintervals of $[0,1]$ corresponding to the parts. We next construct a measure preserving map $\psi$ consisting of measure preserving maps on the intervals $A$, ${A'}$, $B$, ${B'}$, ${B''}$, $C$ and ${C'}$. We choose these maps such that there exist decreasing functions $f_A:A\to [0,1]$ and $f_{A'}:{A'}\to [0,1]$, and increasing functions $f_B:B\to [0,1]$, $f_{B'}:{B'}\to [0,1]$, $f_{B''}:{B''}\to [0,1]$, $f_C:C\to [0,1]$ and $f_{C'}:{C'}\to [0,1]$ such that the following holds almost everywhere (the existence of such maps and functions follows from Monotone Reordering Theorem): $$\begin{array}{ccclcccl} \forall x\in A & f_A(\psi(x)) & = & \int\limits_{B} W^{\varphi}(x,y) \dif y & \forall x\in {A'} & f_{A'}(\psi(x)) & = & \int\limits_{{B'}} W^{\varphi}(x,y) \dif y \\ \forall x\in B & f_B(\psi(x)) & = & \int\limits_{B} W^{\varphi}(x,y) \dif y & \forall x\in {B'} & f_{B'}(\psi(x)) & = & \int\limits_{{B'}} W^{\varphi}(x,y) \dif y \\ &&&& \forall x\in {B''} & f_{B''}(\psi(x)) & = & \int\limits_{{A}} W^{\varphi}(x,y) \dif y \\ \forall x\in C & f_C(\psi(x)) & = & \int\limits_{C} W^{\varphi}(x,y) \dif y & \forall x\in {C'} & f_{C'}(\psi(x)) & = & \int\limits_{{C'}} W^{\varphi}(x,y) \dif y \end{array}$$ In the rest of the proof, we establish that $W^\varphi$ and $W^\psi_{R}$ are equal almost everywhere. The pseudorandom and zero constraints in $\CC_{R}$ imply that $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $D\times [0,1]$ and $[0,1]\times D$. The zero and triangular constraints and the choice of $\psi$ on $B$, ${B'}$, $C$, and ${C'}$ yield the same conclusion for $(B\cup {B'}\cup {B''}\cup C\cup {C'})^2$, $A\times B'$, $A'\times B$, $B\times A'$, and $B'\times A$. Let us now introduce some additional notation. If $x$ is a vertex and $Y$ is one of the parts, let $N_Y(x)$ denote the set of $y\in Y$ such that $W^\varphi(x,y)>0$. If $x$ and $y$ belong to the same part, then we write $x\preceq y$ iff $\psi(x)\le\psi(y)$. Observe that the monotonicity constraint (a) from Figure \[fig-mono\] and the choice of $\psi$ implies the existence of a set $Z$ of measure zero such that $N_B(x')\setminus N_B(x)$ has measure zero for $x,x'\in A\setminus Z$ if and only if $x\preceq x'$. Since the degree of every vertex in $B$ is $1/9$, this yields that the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $A\times B$. The same reasoning applies to $A'$ and $B'$. Thus, we conclude that the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $(A\cup {A'})\times (B\cup {B'})$ and $(B\cup {B'})\times (A\cup {A'})$. We now apply the same reasoning using the monotonicity constraint (b) and the split constraints (b) to deduce the existence of a zero measure set $Z$ such that $N_{B''}(x)\setminus N_{B''}(x')$ has measure zero if and only if $x\preceq x'$ for $x,x'\in A\setminus Z$. The monotonicity constraint also imply that $W^\varphi$ has only values zero and one almost everywhere on $A\times B''$. Since the measure of $N_{B}(x)\cup N_{B''}(x)$ is $1/9$ for almost all $x\in A$ by the split constraint (b), the choice of $\psi$ on $B''$ implies that the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $A\times B''$. The degree regularity in $B''$, the split constraint (d), and the monotonicity constraint (d), which yields that $W^\varphi$ has values zero and one almost everywhere on $A'\times B''$, yield the agreement almost everywhere on $A'\times B''$. Symmetrically, they agree almost everywhere on $B''\times (A\cup A')$. We now focus on the graphon $W^\varphi$ on $A^2$. The monotonicity constraints (f) and (h) from Figure \[fig-mono\] imply that there exists a set $Z$ of measure zero such that every point $x\in A\setminus Z$ can be associated with a unique open interval $J_x\subseteq A$ such that $W^\varphi(x,x')=0$ for almost every $x'\in\psi^{-1}(J_x)$, $W^\varphi(x,x')=1$ for almost every $x'\in A\setminus\psi^{-1}(J_x)$, and the intervals $J_x$ and $J_{x''}$ for all points $x,x''\in A\setminus Z$ are either the same or disjoint. The interval $J_x$ can be empty for some choice of $x$. Recall that $|J_x|$ is the measure of the interval $J_x$, and let $\JJ$ be the set of all intervals $J_x$, $x\in A\setminus Z$, with $|J_x|>0$. Since the intervals in $\JJ$ are disjoint, the set $\JJ$ is equipped with a natural linear order. Let us now focus on the infinitary constraint (b) from Figure \[fig-infin\]. Fix three vertices (two from $A$ and one from $B$) as in the figure and let $x$ be the left vertex from $A$. Observe that if $x\in A$ is fixed, then the set of choices of the other two vertices has non-zero measure unless $\psi(x)=\sup J_x$. The left hand side of the constraint is equal to the measure of $J_x$, i.e., $\sup J_x-\inf J_x$. The right hand side is equal to $1/9-\sup J_x$. We conclude that $\inf J_x=1/9-2|J_x|$. This implies that the set $\JJ$ is well-ordered and countable. Let us write $J_k$ for the $k$-th interval contained in $\JJ$. Furthermore, for $k\geq 1$, define $$\beta_k= \frac{2(1-9\inf J_{k+1})}{1-9\inf J_k} = \frac{2|J_{k+1}|}{|J_k|}\;\mbox{,}$$ and let $\beta_0$ be equal to $1-9\inf J_1$. Note that by the observations made in the last paragraph and since $\inf J_{k+1}\geq \sup J_k$, we obtain $\beta_k\leq 1$ for every $k\geq 0$. In case that $\JJ$ is finite, we define $\beta_k=0$ for $k>|\JJ|$. We can now express the density of non-edges with both end-vertices in $A$ as $$\sum_{J\in \JJ} |J|^2=\sum_{k=1}^\infty {\left}(\frac{1}{9\cdot 2^{k}}\prod\limits_{k'=0}^{k-1}\beta_{k'}{\right})^2\;\mbox{.}$$ Since the sum is forced to be $1/243$ by the infinitary constraint (a), we get that $\beta_k=1$ for every $k$. This implies that for every $k$, $J_k={\left}(\frac{1-2^{-k+1}}{9},\frac{1-2^{-k}}{9}{\right})$. In particular, the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $A^2$. The same reasoning as for $A^2$ yields that the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $A'^2$. Let $\JJ'$ be the corresponding set of intervals for $A'$ and let $J'_1,J'_2,\ldots$ be their ordering. The split constraints (e) and (f) from Figure \[fig-split\] imply that for almost every $x\in A$ with ${\left}|N_{A'}(x){\right}|>0$, there exists $J'\in \JJ'$ such that $N_{A'}(x)\mathop{\Delta}\psi^{-1}(J')$ has measure zero and $W^\varphi(x,y)=1$ for almost every $y\in \psi^{-1}(J')$. Let $x\in\psi^{-1}(J_k)$. The split constraint (a) from Figure \[fig-split\] yields that ${\left}|N_{A'}(x){\right}|=\frac{1}{2^k\cdot 9}$. Consequently, $N_{A'}(x)\mathop{\Delta}\psi^{-1}(J'_k)$ has measure zero for almost every $x\in\psi^{-1}(J_k)$ and $W(x,x')=1$ for almost every $x\in\psi^{-1}(J_k)$ and $x'\in\psi^{-1}(J'_k)$. We conclude that the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $A\times A'$ and $A'\times A$. The orthogonality constraints (a) and (b) from Figure \[fig-ortho\] yield that there exist measurable subsets $I_k\subseteq C$ with $|I_k|=1/9$ for every $k\geq 1$ such that it holds for almost every $x\in\psi^{-1}(J_k)$ that $N_C(x)$ differs from $I_k$ on a set of measure zero and $W^\varphi(x,y)=1$ for almost every $y\in I_k$. The construction of $\psi$ and the split constraint (h) from Figure \[fig-split\] imply that $|N_A(x)|=1/9-\psi(x)/2$ for almost every $x\in C$. Since $\psi^{-1}(J_1)\setminus N_A(x)$ has measure zero for almost every $x\in I_1$, we get that $|J_1|\le |N_A(x)|$ for almost every $x\in I_1$. This implies that $I_1$ and $\psi^{-1}([0,1/9])$ differ on a set of measure zero (also see Figure \[fig-AvsC\]). Since $\psi^{-1}(J_2)\setminus N_A(x)$ has measure zero for almost every $x\in I_2$ and $J_1\cap J_2$ has measure zero, we get that $|J_1|+|J_2|\le |N_A(x)|$ for almost every $x\in I_1\cap I_2$ and that $|J_2|\le |N_A(x)|$ for almost every $x\in I_2\setminus I_1$. This implies that $I_2$ and $\psi^{-1}([0,1/18]\cup [1/9,1/6])$ differ on a set of measure zero. Iterating the argument, we obtain that $I_k$ differs from the preimage with respect to $\psi$ of the set $$\bigcup_{i=1}^{2^{k-1}}\left[\frac{2i-2}{9\cdot 2^{k-1}},\frac{2i-1}{9\cdot 2^{k-1}}\right]$$ on a set of measure zero for every $k\in\NN$. This yields that the graphons $W^\varphi$ and $W^\psi_{R}$ agree almost everywhere on $A\times C$. The orthogonality constraint (c) from Figure \[fig-ortho\] implies that $(C\setminus N_C(x))\cap N_C(x')$ has measure zero for every $k$, almost every $x\in A\setminus \psi^{-1}(J_k)$, and almost every $x'\in \psi^{-1}(J'_k)$. In particular, almost every $x'\in \psi^{-1}(J'_k)$ satisfies that $N_C(x')\setminus I_k$ has measure zero, i.e., $W^\varphi(x',y)=0$ for almost every $x'\in \psi^{-1}(J'_k)$ and $y\not\in I_k$. We now interpret the orthogonality constraint (d) from Figure \[fig-ortho\]. Fix an integer $k\geq 1$ and a typical vertex $x'\in \psi^{-1}(J'_k)$. The left term in the product on the left hand side of the constraint is equal to the square of $\int\limits_C W^{\varphi}(x',y)\dif y = \int\limits_{I_k} W^{\varphi}(x',y)\dif y\;\mbox{.}$ The right term in the product is equal to the square of $|J'_k|=2^{-[\psi(x')/a]}/9$. The term on the right hand side is equal to the probability that randomly chosen $x''$ and $y$ satisfy $x''\in A'$, $y\in B'$, $x''\in \psi^{-1}(J'_k)$, and $\psi(x')\le \psi(y)<\psi(x'')$. This is equal to $$\frac{\left(1-2^{-[\psi(x')/a]}-\psi(x')/a\right)^2}{2\cdot 9^2}\;\mbox{.}$$ We deduce that almost every $x'\in\psi^{-1}(J'_k)$ satisfies $$\int\limits_{I_k} W^{\varphi}(x',y)\dif y=\frac{1-2^{-[\psi(x')/a]}-\psi(x')/a}{9\cdot 2^{-[\psi(x')/a]}}\;\mbox{.} \label{eq-cs1}$$ We apply the same reasoning to the orthogonality constraint (e) from Figure \[fig-ortho\] and deduce that almost every pair of vertices $x',x''\in\psi^{-1}(J'_k)$ satisfies $$\begin{aligned} &\frac{9^2}{4}\cdot\left(\int\limits_{I_k}W^{\varphi}(x',y)W^{\varphi}(x'',y)\dif y\right)^2\cdot \left(2^{-[\psi(x')/a]}\right)^4 =\\ &\qquad\frac{\left(1-2^{-[\psi(x')/a]}-\psi(x')/a\right)^2}{2}\cdot \frac{\left(1-2^{-[\psi(x'')/a]}-\psi(x'')/a\right)^2}{2}\;\mbox{.}\end{aligned}$$ This implies (similarly as in the proof of Lemma \[lm-pseudobipartite\]) that almost every $x'\in\psi^{-1}(J'_k)$ satisfies: $$\left(\int\limits_{I_k} W^{\varphi}(x',y)^2\dif y\right)^{1/2}=\frac{1-2^{-[\psi(x')/a]}-\psi(x')/a}{3\cdot 2^{-[\psi(x')/a]}}\;\mbox{.} \label{eq-cs2}$$ Using Cauchy-Schwartz Inequality, we deduce from (\[eq-cs1\]) and (\[eq-cs2\]) (recall that $|I_k|=1/9$) that the following holds for almost every $x'\in\psi^{-1}(J'_k)$ and $y\in I_k$, $$W^{\varphi}(x',y)=\frac{1-2^{-[\psi(x')/a]}-\psi(x')/a}{2^{-[\psi(x')/a]}}\;\mbox{.}$$ In other words, $W^{\varphi}(x',y)$ is constant almost everywhere on $I_k$ for almost every $x'\in\psi^{-1}(J'_k)$ and its value linearly decreases from one to zero almost everywhere inside $\psi^{-1}(J'_k)$. Hence, the graphons $W^{\varphi}$ and $W^\psi_{R}$ agree almost everywhere on $A'\times C$ and $C\times A'$ (recall that $W^\varphi(x',y)=0$ for almost every pair $x'\in\psi^{-1}(J'_k)$ and $y\not\in I_k$). The monotonicity constraint (e) from Figure \[fig-mono\] yields that at least one of the sets $N_{C'}(x)\setminus N_{C'}(x')$ or $N_{C'}(x')\setminus N_{C'}(x)$ has measure zero for every $k$ and almost every pair $x,x'\in A'$ and the graphon $W^{\varphi}$ has values zero and one almost everywhere on $A'\times C'$. This and the regularity on $A'$ imply that the graphons $W^{\varphi}$ and $W^\psi_{R}$ agree almost everywhere on $A'\times C'$. Since the graphon $W^{\varphi}$ is zero almost everywhere on $A\times C'$ by one of the zero constraints, we have shown that the graphons $W^{\varphi}$ and $W^\psi_{R}$ agree almost everywhere on $(A\cup A')\times (C\cup C')$ and $(C\cup C')\times (A\cup A')$. Since these were the last subsets of their domains that remained to be analyzed, we proved that the graphon $W^{\varphi}$ is equal to $W^\psi_{R}$ almost everywhere. Theorem \[thm-main\] immediately yields the following. \[cor-main\] The graphon $W_{R}$ is finitely forcible. Conclusion {#sec:concl} ========== It is quite clear that the construction of Rademacher graphon can be modified to yield other graphons $W$ with non-compact $T(W)$. Some of these modifications can yield such graphons with a smaller number of parts at the expense of making the argument that the graphon is finitely forcible less transparent. In [@bib-lovasz11+], Lovász and Szegedy considered finite forcibility inside two classes of functions. Conjecture \[conj:1\], which we addressed in this paper, relates to the class they refer to as $\WW_0$. This class consists of symmetric measurable functions from $[0,1]^2$ to $[0,1]$. A larger class referred to as $\WW$ in [@bib-lovasz11+] is the class containing all symmetric measurable functions from $[0,1]^2$ to $\RR$. It is not hard to see that Rademacher graphon $W_R$ is also finitely forcible inside this larger class. Also note that stronger constraints involving multigraphs were used in [@bib-lovasz11+] but we have used only constraints involving simple graphs in this paper. In [@bib-lovasz-book], an analogue of the space $T(W)$ with respect to the following metric is also considered. If $f,g\in L_1[0,1]$, then $$d_W(f,g):=\int\limits_{[0,1]}\left|\;\int\limits_{[0,1]} W(x,y)(f(y)-g(y))\dif y\right| \dif x\;\mbox{.}$$ However, the appropriate closure of $T(W)$ always form a compact space [@bib-lovasz-book Corollary 13.28]. As mentioned in Section \[sec:intro\], Rademacher graphon $W_R$ also provides a partial answer to [@bib-lovasz11+ Conjecture 10] in the sense that the Minkowski dimension of $T(W_R)$ is infinite. However, the dimension is finite when several other notions of dimension are considered. For instance, its Lebesgue dimension is only one. In [@bib-our-next-paper], the first two authors and Klimošová disprove Conjecture \[conj:2\] in a more convincing way: they construct a finitely forcible graphon $W$ such that a subspace of $T(W)$ is homeomorphic to $[0,1]^\infty$. The construction is also based on partitioned graphons used in this paper. We finish by presenting a construction of a finitely forcible graphon $W_d$ with a part of $T(W_d)$ positive measure isomorphic to $[0,1]^d$; the construction is analogous to one found earlier by Norine [@bib-norine-comm]. Fix a positive integer $d$. We construct a graphon $W_d$ with $2d+2$ parts $A$, $B_1,\ldots,B_{2d}$, and $C$, each of size $(2d+2)^{-1}$. If $x,y\in B_i$, then $W_d(x,y)=1$ if $x+y\ge (2d+2)^{-1}$, i.e., $W_d$ is the half graphon on each $B_i^2$. If $x\in B_i$ and $y\in C$, then $W_d(x,y)=W_d(y,x)=i/4d$. Fix now a measure preserving map $\varphi$ from $[0,1]$ to $[0,1]^d$. If $x\in A$ and $y\in B_i$, $i\le d$, then $W_d(x,y)=W_d(y,x)=1$ if $\varphi((2d+2)x)_i\ge (2d+2)y$. Finally, if $x\in A$ and $y\in B_i$, $i\ge d+1$, then $W_d(x,y)=W_d(y,x)=1$ if $1-\varphi((2d+2)x)_i\ge (2d+2)y$. The graphon $W_d$ is equal to zero for other pairs of vertices. Clearly, $W_d$ is a partitioned graphon with $2d+2$ parts with vertices inside each part having the same degree and vertices in different parts having different degrees. Using the techniques presented in this paper and generalizing arguments from [@bib-kral12+], one can show that $W_d$ is finitely forcible. Since the subspace of $T(W_d)$ formed by typical vertices from $A$ is homeomorphic to $[0,1]^d$, the Lebesgue dimension of $T(W_d)$ is at least $d$. This shows that finitely forcible graphons can have arbitrary large finite dimension. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Jan Hladký, Tereza Klimošová, Serguei Norine, and Vojtěch Tma for their comments on the topics discussed in the paper. [99]{} R. Baber: Turán densities of hypercubes, preprint available on <http://arxiv.org/abs/1201.3587>. R. Baber and J. Talbot: Hypergraphs do jump, [*Combinatorics, Probability and Computing*]{} [**20**]{} (2011), 161–171. R. Baber and J. Talbot: A solution to the $2/3$ conjecture, preprint available on <http://arxiv.org/abs/1306.6202>. J. Balogh, P. Hu, B. Lidický, and H. Liu: Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube, to appear in [*European Journal on Combinatorics*]{}. C. Borgs, J.T. Chayes, and L. Lov[á]{}sz: Moments of two-variable functions and the uniqueness of graph limits, [*Geometric and Functional Analysis*]{} [**19**]{} (2010), 1597–1619. C. Borgs, J.T. Chayes, L. Lov[á]{}sz, V.T. S[ó]{}s, and K. Vesztergombi: Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing, [*Advances in Mathematics*]{} [**219**]{} (2008), 1801–1851. C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós, and K. Vesztergombi: Convergent sequences of dense graphs II. Multiway cuts and statistical physics, [*Annals of Mathematics*]{} [**176**]{} (2012), 151–219. C. Borgs, J. Chayes, L. Lov[á]{}sz, V.T. S[ó]{}s, B. Szegedy, and K. Vesztergombi: Graph limits and parameter testing, in: [*Proceedings of the 38rd Annual ACM Symposium on the Theory of Computing (STOC), ACM, New York, 2006*]{}, 261–270. F.R.K. Chung, R.L. Graham, and R.M. Wilson: Quasi-random graphs, [*Combinatorica*]{} [**9**]{} (1989), 345–362. P. Diaconis, S. Holmes, and S. Janson: Threshold graph limits and random threshold graphs, [*Internet Mathematics*]{} [**5**]{} (2008), 267–318. R. Glebov, T. Klimošová, and D. Král’: Infinite dimensional finitely forcible graphon, in preparation. R. Glebov, D. Král’, J. Volec: A problem of Erdos and Sos on 3-graphs, preprint available on <http://arxiv.org/abs/1303.7372>. A. Grzesik: On the maximum number of five-cycles in a triangle-free graph, [*Journal of Combinatorial Theory, Series B*]{}, [**102**]{} (2012), 1061–1066. H. Hatami, J. Hladký, D. Král’, S. Norine, and A. Razborov: Non-three-colorable common graphs exist, [*Combinatorics, Probability and Computing*]{} [**21**]{} (2012), 734–742. H. Hatami, J. Hladký, D. Král’, S. Norine, and A. Razborov: On the number of pentagons in triangle-free graphs, [*Journal of Combinatorial Theory, Series A*]{}, [**120**]{} (2013), 722–732. D. Král’, C.-H. Liu, J.-S. Sereni, P. Whalen, and Z. Yilma: A new bound for the 2/3 conjecture, [*Combinatorics, Probability and Computing*]{} [**22**]{} (2013), 384–393. D. Král’, L. Mach, and J.-S. Sereni: A new lower bound based on Gromov’s method of selecting heavily covered points, [*Discrete & Computational Geometry*]{} [**48**]{} (2012), 487–498. D. Kr[á]{}l’, O. Pikhurko: Quasirandom permutations are characterized by 4-point densities, [*Geometric and Functional Analysis*]{} [**23**]{} (2013), 570–579. L. Lovász: [**Large networks and graph limits**]{}, AMS, Providence, RI, 2012. L. Lovász and V.T. Sós: Generalized quasirandom graphs, [*Journal of Combinatorial Theory, Series B*]{}, [**98**]{} (2008), 146–163. L. Lov[á]{}sz and B. Szegedy: Finitely forcible graphons, [*Journal of Combinatorial Theory, Series B*]{}, [**101**]{} (2011), 269–301. L. Lov[á]{}sz and B. Szegedy: Limits of dense graph sequences, [*Journal of Combinatorial Theory, Series B*]{}, [**96**]{} (2006), 933–957. L. Lov[á]{}sz and B. Szegedy: Regularity partitions and the topology of graphons, in: I. Bárány, J. Solymosi (eds.): An Irregular Mind. Szemerédi is 70, Springer J. Bolyai Mathematical Society, 2010, 415–446. L. Lov[á]{}sz and B. Szegedy: Testing properties of graphs and functions, [*Israel Journal of Mathematics*]{} [**178**]{} (2010), 113–156. S. Norine: private communication. O. Pikhurko and A. Razborov: Asymptotic structure of graphs with the minimum number of triangles, preprint available on <http://arxiv.org/abs/1204.2846>. O. Pikhurko and E.R. Vaughan: Minimum number of k-cliques in graphs with bounded independence number, preprint available on <http://arxiv.org/abs/1203.4393>. A. Razborov: Flag algebras, [*Journal of Symbolic Logic*]{} [**72**]{} (2007), 1239–1282. A. Razborov: On the minimal density of triangles in graphs, [*Combinatorics, Probability and Computing*]{} [**17**]{} (2008), 603–618. A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, [*SIAM Journal on Discrete Mathematics*]{} [**24**]{} (2010), 946–963. P. Turán: On an extremal problem in graph theory, [*Matematikai és Fizikai Lapok*]{} [**48**]{} (1941), 436–452. \[in Hungarian\] (Also see: On the theory of graphs, [*Colloquium Mathematicum*]{} [**3**]{} (1954), 19–30.) [^1]: Department of Mathematics, ETH, 8092 Zurich, Switzerland. E-mail: [[email protected]]{}. Previous affiliation: Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK. [^2]: Mathematics Institute, DIMAP and Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. E-mail: [[email protected]]{}. [^3]: Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK. E-mail: [[email protected]]{}. [^4]: The work leading to this invention has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385.

--- abstract: | $Revision: 1.38 $, compiled\ Rigidity analysis using the “pebble game” can usefully be applied to protein crystal structures to obtain information on protein folding, assembly and the structure-function relationship. However, previous work using this technique has not made clear how sensitive rigidity analysis is to small structural variations. We present a comparative study in which rigidity analysis is applied to multiple structures, derived from different organisms and different conditions of crystallisation, for each of several different proteins. We find that rigidity analysis is best used as a comparative tool to highlight the effects of structural variation. Our use of multiple protein structures brings out a previously unnoticed peculiarity in the rigidity of trypsin. address: ' $^1$Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, United Kingdom $^2$Department of Systems Biology, University of Warwick, Coventry, CV4 7AL, United Kingdom\' author: - 'Stephen A. Wells$^{1}$, J. Emilio Jimenez-Roldan$^{1,2}$, Rudolf A. Römer$^{1}$' date: '$Revision: 1.38 $, compiled ' title: 'Sensitivity of protein rigidity analysis to small structural variations: a large-scale comparative analysis' --- Introduction {#sec-intro} ============ The “pebble game” [@JacT95] is an integer algorithm for rigidity analysis. By matching degrees of freedom against constraints, it can rapidly divide a network into rigid regions and floppy “hinges” with excess degrees of freedom. The algorithm is applicable to protein crystal structures if these are treated as molecular frameworks in which bond lengths and angles are constant but dihedral angles may vary; this application, and the program “FIRST” which implements the algorithm, have been described in the literature [@JacRKT01]. The rigid units in a protein structure may be as small as individual methyl groups or large enough to include entire protein domains containing multiple secondary-structure units (alpha helices and beta sheets). Rigidity analysis thus generates a natural multi-scale coarse graining of a protein structure [@GohT06]. The division of a structure into rigid units is referred to as a Rigid Cluster Decomposition (RCD). This coarse graining has been used as the basis of simulation methods aiming to explore the large-amplitude flexible motion of proteins, first in the ROCK algorithm [@ThoLRJ01] and more recently in the “FRODA” geometric simulation algorithm [@WelMHT05]. FIRST/FRODA has been used to examine the inherent mobility of a protein crystal structure for comparison to NMR ensembles [@WelMHT05], to examine possible mechanisms for the assembly of a protein complex [@JolWHT06], to fit the structure of the bacterial chaperonin GroEL to low-resolution cryo-EM data [@JolWFT08], and to examine the relation between protein flexibility and function in enzymes such as IDO [@MacNBC07] and myosin [@SunRAJ08]. The coarse-graining provided by the rigidity analysis dramatically reduces the computational cost of simulating flexible motion while retaining all-atom steric detail, making geometric simulation a promising complement to other methods such as molecular dynamics (MD). Rigidity analysis itself (without motion simulation) has been used to study the formation of a virus capsid by assembly of protein subunits [@HesJT04] and, in particular, to examine the process of protein folding, considered as a transition from a floppy to a rigid state. The results of the rigidity analysis depend upon the set of constraints that are included in the network. While covalent bonds and hydrophobic tethers are always included in the analysis, the hydrogen bonds in the protein structure are assigned an energy using a potential based on the geometry of the bond [@DahGM97], and only hydrogen bonds with energies below a user-defined “cutoff” are included in the constraint network. It is therefore possible to perform a “rigidity dilution” on a protein crystal structure, in which the “cutoff” is gradually reduced so that fewer and fewer hydrogen bonds are included in the analysis and the structure gradually loses rigidity. A study by Rader et al. [@RadHKT02] analysed a set of 26 different protein structures using rigidity dilution, drawing an analogy between the loss of rigidity during this dilution and the thermal denaturation of a protein. A similar study by Hespenheide et al. [@HesRTK02] made predictions for the “folding core” of several proteins based on the regions that retained rigidity longest during the dilution, obtaining a promising correlation with experimental data. This variation of rigidity as different constraints are included, however, is in a sense the Achilles heel of the method if it is to be used as a coarse-graining approach and a basis for simulation. Firstly, it is unclear if there is a ‘correct’ value of the hydrogen-bond energy cutoff, which can be used to obtain a physically realistic simulation of the flexible motion of a protein; the studies referenced above have used a wide variety of cutoff values on an [*ad hoc*]{} basis. Secondly, it is unclear how the RCD is affected by small variations in structure — as, for example, if the same protein is crystallised under slightly different conditions, or if examples of a given protein from multiple different organisms are compared. Thus, we do not know if the RCD obtained for a given crystal structure and energy cutoff is in some sense ‘typical’ of the protein, or if it is likely to vary dramatically compared to that of an apparently very similar structure and cutoff. If the RCD of a protein is robust to small structural variations, this justifies the use of rigidity-based coarse graining for simulations of conformational change between structures [@JolWFT08] and structural variation due to flexible motion [@WelMHT05; @MacNBC07; @SunRAJ08]. However, if small changes in the structure cause dramatic changes to RCDs, then the RCD of one structure may not be representative. In this case, the rigidity analysis can best be used in a comparative mode; that is, to draw attention to parts of the structure where rigidity has changed, or to determine whether or not a given constraint (such as an interaction between two particular residues) is consistent with a given flexible motion. This was the approach taken by [@JolWHT06] when studying a major conformational change during assembly of a large protein complex. Previous studies on large numbers of proteins [@RadHKT02; @HesRTK02] have considered single examples of multiple unrelated proteins, and hence do not provide the comparative information we need. A study by Mamonova et al. [@MamHST05] examined the variation in RCD of a protein structure in the course of a 10ns MD simulation, finding that the RCD varied quite dramatically for different snapshots of the MD trajectory. An additional motivation for our study is to determine the typical pattern of rigidity loss during dilution. A recent study on rigidity percolation in glassy networks [@SarWHT07] found that the transition between rigid and floppy states could display either first-order or second-order behaviour. In the first case, the gradual introduction of constraints into the network led to a sharp transition from an entirely floppy state to one in which the entire system became rigid. In the second, rigidity initially developed in a percolating rigid cluster involving only a small proportion of the network, which then gradually increased in size as more constraints were introduced. By performing rigidity dilution on a very large number of protein structures and observing the pattern of rigidity loss, we should be able to determine whether the loss is typically sudden or gradual, and whether different types of protein display different behaviours. Our initial hypothesis was that proteins such as cytochrome C, whose function depends on retaining its structure to protect a contained heme group from exposure to solvent, would retain a large amount of rigidity over a wide range of cutoff values and then lose rigidity suddenly, while one whose function depends on conformational flexibility, e.g. an enzyme such as trypsin, would display a gradual variation in rigidity. As we shall see, the results we obtained were very different from our expectations.[^1] Materials and Methods {#sec-method} ===================== Protein selection {#sec-select} ----------------- In this study we take a deliberately comparative approach by first choosing a set of globular proteins, and then, for each protein in the set, obtaining multiple crystal structures. We have particularly sought for (i) examples of the same protein from different organisms, [*e.g.*]{} cytochrome C proteins from multiple different eukaryotic mitochondria, and (ii) protein structures obtained under different conditions of crystallisation, e.g. in complex with different ligands or substrates. The default treatment of hydrogen bonds and hydrophobic tethers in FIRST is based on the assumption that the protein exists in a polar solvent, e.g. cytoplasm, rather than being within a hydrophobic or amphiphilic environment as for membrane-bound proteins. Proteins in a membrane environment can still be handled but this requires hand-editing of the constraint network. In this study we have therefore confined ourselves to examining a set of non-membrane proteins. Our starting point was the set of proteins considered by Hespenheide et al. [@HesRTK02]. We then searched for proteins with similar structures, i.e. those listed under the same domain under SCOP classification and/or as homologous under CATH classification. We note that the RCSB Protein Data Bank provides these cross-references as derived information for each protein crystal structure [@BerWFG00]. Rigidity analysis is best carried out on crystal structures with high resolution, so that we can have confidence in the accuracy of the atomic positions when constructing the hydrogen-bond geometries. We therefore eliminated NMR solution structures and concentrated on X-ray crystal structures with resolutions of better than $2.5$ Ångstroms. We also limited ourselves to wild-type structures rather than including engineered mutants. To widen our data set we included several proteins not used in the earlier study. We included hemoglobin by analogy with myoglobin. The presence of a BPTI structure in the initial dataset let us to include a set of trypsins. In this paper we discuss results on 12 cytochrome C structures, 11 hemoglobin structures, 3 myoglobins, 6 alpha-lactalbumins and 18 trypsin structures. The full set of all data generated from our study (from over seventy crystal structures) is available as electronic supplementary information. Rigidity analysis method {#sec-software} ------------------------ We used the FIRST rigidity analysis software [@JacRKT01] to perform rigidity dilution (as described for example in [@RadHKT02]) on a wide selection of crystal structures obtained from the Protein Data Bank online database [@BerWFG00]. Our choice of protein crystal structures is described in section \[sec-select\]. Each structure was processed as follows. From the crystal structure as recorded in the PDB, we extracted a single protein main chain, eliminating all crystal water molecules, but retaining important hetero groups such as the porphyrin/heme units of cytochrome C and hemoglobin. The “PyMOL” visualisation software [@Del____] proved very useful for this purpose. From multimeric crystal structures such as tetrameric hemoglobins, we extracted and used only a single protein main chain, reserving consideration of protein complexes to a future study which is now in preparation. We processed the resulting protein structure to add the hydrogens that are generally absent from X-ray crystal structures, using the “REDUCE” software [@WorLRR99] which also performs necessary flipping of side chains. After the addition of hydrogens we renumbered the atoms using “PyMOL” [@Del____] again to produce files usable as input to FIRST. The program FIRST was then run over the resulting processed PDB file with appropriate command line options The energy of each potential hydrogen bond in the processed structure is calculated by FIRST using the Mayo potential [@DahGM97], performing an initial rigidity analysis including all bonds with energies of $0$ kcal/mol or lower (option [-E 0]{}). We also perform a dilution in which bonds are then removed in order of strength, gradually reducing the rigidity of the structure (option [-dil 1]{}). The initial selection of proteins from the PDB was done by hand, but all subsequent stages were automated using bash scripts in a Linux environment with the jobs farmed out over a distributed cluster of workstations. This automation and the rapidity of the rigidity analysis software gave very high throughput for processing structures. ![Example of an RCD plot which is a result of the rigidity analysis. This plot is for the bacterial protein barnase, from the 1A2P.pdb structure. The labels have been lightly edited from the original FIRST output for clarity. The columns $E$ and $\langle r \rangle$ give the values of energy cutoff and network mean coordination, respectively, at which changes in the backbone rigidity occur. []{data-label="stripy1"}](\figdir/barnaseEdit.eps){width="99.00000%"} Running the rigidity dilution for a given protein produces an RCD plot (also called ‘stripy plot’) as illustrated in figure \[stripy1\]. In this plot the horizontal axis represents the linear protein backbone. Flexible areas are shown as a thin black line while areas lying within a large rigid cluster are shown as thicker coloured blocks. The colour is used to show which residues belong to which rigid clusters; obviously the three-dimensional protein fold makes it possible for residues that are widely separated along the backbone to be spatially adjacent and form a single rigid cluster. The vertical axis on the RCD plot represents the dilution of constraints by progressively lowering the cutoff energy for inclusion of hydrogen bonds in the constraint network. Each time the rigid cluster analysis of the mainchain alpha-carbons changes as a result of the dilution, a new line is drawn on the RCD plot labelled with the energy cutoff and with the network mean coordination for the protein at that stage. We should stress that the RCD is always performed over the entire protein structure (mainchain and sidechain atoms) and a dilution is performed for every hydrogen bond removed from the set of constraints, typically several hundred bonds for a small globular protein. The RCD plot is then a summary concentrating on the rigid-cluster membership of the alpha-carbon atoms defining the protein backbone. Tracking rigidity {#sec-tracking} ----------------- We can compare the results of the rigidity dilution on similar crystal structures in several ways. The most detailed form of comparison is to view the RCD plots for each structure side by side. This highlights the effect of structural variations; for example, a change in structure may lead what was previously a single rigid unit such as a helix to break up into two smaller rigid units connected by a flexible region. This is visible in the RCD plot as the division of a single coloured block into two. In this paper we will use this technique of detailed comparison to examine the rigidity of several mitochondrial cytochrome C structures. This form of comparison, however, becomes unwieldy when comparing large numbers of structures. It is also not suitable for comparing results between different proteins whose structures are not closely related, as of course their rigidity-analysis RCD plots will not resemble each other at all. For larger-scale comparison we therefore adopt an approach inspired by recent studies on the rigidity properties of glassy networks. In this approach we extract a quantity measuring the degree of rigidity of the structure, and plot it as a function of either the hydrogen-bond energy cutoff, $E$, or the mean coordination of the network, $\langle r \rangle$. The latter quantity is the average number of covalently-bonded neighbours that each atom possesses in the network. In previous studies of glassy networks [@SarWHT07] the measure used was the fraction of atoms lying in the largest spanning rigid cluster in a model of a glassy network with periodic boundary conditions. This exact measure is inappropriate here as each protein structure is a finite network; we cannot define a spanning rigid cluster. Instead we extract the number of alpha-carbon atoms belonging to each of the largest rigid clusters, so that our measure of overall rigidity is the proportion of atoms that are found in large rigid units. The rigidity analysis sorts rigid clusters by size so that rigid cluster $1$ (RC1) is the largest. The first measure that comes to mind is to track the number of number of alpha carbons lying within the single largest rigid cluster, $n$(RC1). However, this measure is clearly vulnerable to large variations. Consider the case of a large rigid cluster which divides, due to the removal of a constraint. At one extreme, a small portion of the cluster may become flexible leaving the cluster size only slightly reduced. At the other extreme, the cluster may divide into two clusters of roughly equal size linked by a flexible hinge; in this case the size of the largest rigid cluster is roughly halved. It would be hard, however, to argue that one case is a greater loss of protein rigidity than the other. We have therefore chosen a less sensitive measure; the number of alpha-carbon atoms found within the first [*five*]{} rigid clusters, denoted as $n$(RC1-5). Examination of the data files confirms that our results would not change significantly if we were to choose the first four or siz rigid clusters instead; the use of the first five largest rigid clusters appears, for these small globular proteins, to capture all clusters containing any significant number of backbone alpha-carbon atoms. This measure is not directly reported by the FIRST software during a dilution. We obtained our data by first running the dilution over all structures as described in section \[sec-method\]; then, for each structure, we extracted the values of the energy cutoff at which changes in the backbone rigidity were observed. We then again run rigidity analyses at each of these cutoff values and extract our rigidity measure. We can plot rigidity as a function either of the hydrogen-bond energy cutoff $E$, or as a function of the network mean coordination $\langle r \rangle$ (cp. figure \[fig-example-en\]). In the latter case we see results reminiscent of glassy networks [@SarWHT07], with the structure being largely rigid when the network mean coordination exceeds $2.4$ and becoming more flexible at lower mean coordinations. In the case of proteins, however, finite-size effects and open boundary conditions mean that flexibility can persist at mean coordinations above $2.4$. In this study we will concentrate on the behaviour of rigidity as a function of energy cutoff, the variable which is directly under the user’s control when using the FIRST software. ![Rigidity $n$(RC1-5) versus hydrogen-bond energy cutoff $E$ and network mean coordination $\langle r \rangle$ for human ubiquitin, 1UBI. Here and in all following such graphs, lines are guides to the eyes only.[]{data-label="fig-example-en"}](\figdir/UBI_EN.eps "fig:"){width="49.00000%"} ![Rigidity $n$(RC1-5) versus hydrogen-bond energy cutoff $E$ and network mean coordination $\langle r \rangle$ for human ubiquitin, 1UBI. Here and in all following such graphs, lines are guides to the eyes only.[]{data-label="fig-example-en"}](\figdir/UBI_RN.eps "fig:"){width="49.00000%"} \[fig-example-rn\] Results and discussion {#sec-results} ====================== We begin with a detailed comparison of cytochrome C structures in section \[sec-cytC\]. Examination of four horse mitochondrial cytochrome C structures (1HRC, 1WEJ, 1CRC and 1U75) obtained under different conditions of crystallisation, and then of four tuna mitochondrial cytochrome C structures (1I55, 1I54, 5CYT and 1LFM) with different heme-group metal ions, gives us a sense of how much variation in rigidity can result from small variations in crystal structure. We then compare a wide set of cytochrome C structures from eukaryotic mitochondria (1I55, 1CYC, 1YCC, 2YCC and 1CCR) and a bacterium (1A7V) to determine if there is any correlation between the phylogenetic similarity of proteins and their rigidity behaviour. In section \[sec-multi\] we will examine a set of hemoglobin structures including four human structure in different oxidation states (1A3N deoxy, 2DN1 oxy, 2DN2 deoxy and 2DN3 carbonmonoxy) and a selection of structure from other eukaryotes including goose (1A4F), bovine (1G09), worm (1KR7), clam (1MOH), alga (1DLY), protozoa (1DLW) and rice (1D8U). We will also consider a set of myoglobins from sea turtle (1LHS), sperm whale (1HJT) and horse (1DWR) and a set of alpha-lactalbumin structure derived from several mammals: human (1HML, 1A4V), baboon (1ALC), goat (1HFY), bovine (1F6R) and guinea-pig (1HFX). Finally, in section \[sec-trypsin\], we will consider a wide-ranging set of trypsin structures from humans (1H4W, 1TRN), rat (1BRA, 1BRB, 1BRC, 3TGI), pig (1AVW, 1AVX, 1LDT), salmon (1A0J, 1BZX) and bovine (in complex with various inhibitors — 1AQ7, 1AUJ, 1AZ8 — and small molecules — 1K1I, 1K1J, 1K1M, 1K1N, 1K1O, 1K1P). The behaviour of trypsin structures during rigidity dilution appears distinctively different from the other proteins we have considered. Cytochrome C {#sec-cytC} ------------ ![ A graph of hydrogen bond energy cutoff $E$ versus rigidity for horse cytochromes. The different symbols and lines denote different types. \[fig-cyto-en-horse1\]](\figdir/CYTC_EN_horse1.eps){width="99.00000%"} In figure \[fig-horse-stripy\] we show rigidity breakdowns for four forms of horse cytochrome C, namely, the uncomplexed 1HRC structure, the 1WEJ structure which is crystallised in complex with an antibody, the 1U75 structure from a complex with cytochrome C peroxidase, and the low-ionic-strength 1CRC form. The overall rigidity of the four structures is shown in figure \[fig-cyto-en-horse1\]. Significant differences are visible in the rigidity dilution behaviour of the four structures. For example, the 1HRC and 1WEJ structures retain a significant amount of rigidity (about 25% of alpha carbons lying within large rigid clusters) at cutoff values as low as around -3 kcal/mol, where the 1U75 and 1CRC structures have lost almost all rigidity. Let us now investigate the structural variations which give rise to this variation in rigidity behaviour. We can quantify the differences between the structures by aligning the alpha-carbon atoms defining the protein backbone and obtaining the root-mean-square deviation (RMSD) between alpha-carbon positions. We carry this out using the PyMOL “align” command [@Del____]. The results are given in table \[tab-horse-1\]. The variations are remarkably small, the largest being 0.572 Å between 1U75 and 1WEJ. It thus appears that quite large variations in rigidity-dilution behaviour can arise from relatively modest structural variations. [ c c c c ]{} From$\backslash$To:&1HRC&1CRC&1WEJ1CRC&0.32&—&—1WEJ&0.318&0.321&—1U75&0.472&0.53&0.572 ![ A graph of hydrogen bond energy cutoff $E$ versus rigidity for four tuna cytochrome C structures. Note the considerable differences in behaviour between, for example, 5CYT and 1I55, especially in the $-1$ to $-2$ kcal/mol energy range. \[fig-cyto-en-tuna1\]](\figdir/CYTC_EN_tuna1.eps){width="99.00000%"} In figure \[fig-tuna-stripy\] we show RCD plots for the rigidity dilutions of four mitochondrial cytochrome C structures obtained from tuna, chosen as they differ in their heme-group metal content. 5CYT is a normal ferrocytochrome C with an Fe atom in the heme group. 1I55 and 1I54 were crystallised from mixed-metal cytochromes, with 2Zn:1Fe and 2Fe:1Zn respectively. Evidently a given heme group can contain only one metal atom — from the pdb files we selected chains containing Fe-heme groups, to maximise comparability with 5CYT. 1LFM, meanwhile, has Co replacing Fe in the heme group. It is clear from comparison of the plots in figure \[fig-tuna-stripy\] and the corresponding graph of rigidity versus cutoff energy (Figure \[fig-cyto-en-tuna1\]) that there are noticeable differences between the rigidity behaviour of these four structures. It is particularly instructive to compare the behaviour of 5CYT against that of 1I55 in figure \[fig-cyto-en-tuna1\]; in the energy range of -1 to -2 kcal/mol, 1I55 remains largely rigid while 5CYT has lost a considerable amount of rigidity. The alpha-carbon RMSD differences between the structures are given in table \[tab-tuna-1\] and are even smaller than those among the horse cytochrome structures, none being higher than 0.3 Å. [ c c c c ]{} From$\backslash$To:&5CYT&1I55&1I541I55&0.27&—&—1I54&0.2668&0.041&—1LFM&0.286&0.116&0.087 In figure \[fig-cyto-en-other\] we show the rigidity versus cutoff energy behaviour of a selection of cytochrome C structures from several different branches of the tree of life. The structures 1I55 (tuna) and 1CYC (bonito) are animal; 1YCC and 2YCC (yeast) are fungal; 1CCR (rice), a plant; and 1A7V, a bacterial cytochrome C structure. It appears that the range of behaviours spans the gamut of possibilities, with some structures rapidly losing rigidity at cutoffs around $-0.5$ kcal/mol and other retaining a large amount of rigidity at cutoffs below $-2$ kcal/mol. In terms of retention of rigidity the 1A7V bacterial structure and the 1I55 tuna structure appear similar, whereas by structure the bacterial protein is quite dissimilar to any of the eukaryotic mitochondrial cytochrome C structures. ![ A graph of hydrogen bond energy cutoff $E$ versus rigidity for various cytochromes: 1CCR, rice; 1CYC, bonito; 1A7V, bacterial; 1I55, tuna; 1YCC and 2YCC, yeast. Note the wide variety in rigidity behaviour. \[fig-cyto-en-other\]](\figdir/CYTC_EN_OTHER.eps){width="99.00000%"} Our initial hypothesis of a correlation between rigidity behaviour and protein function thus appears to have been disconfirmed by the cytochrome C data. Instead, protein rigidity appears to depend on quite small structural variations; it varies widely within a family of closely related, isostructural and isofunctional proteins. Hemoglobin, myoglobin and lactalbumin {#sec-multi} ------------------------------------- ![ A graph of hydrogen bond energy cutoff $E$ versus rigidity for a variety of hemoglobins, showing considerable variation in the pattern of rigidity loss during dilution. 1A3N, 2DN1-alpha, 2DN2 and 2DNS are all human hemoglobin alpha subunits; 2DN2-beta is a beta subunit. \[fig-hemo-en\]](\figdir/HEMO_EN.eps){width="70.00000%"} ![ A graph of hydrogen bond energy cutoff $E$ versus rigidity for a variety of myoglobins and alpha-lactalbumins, showing considerable variation in the pattern of rigidity loss during dilution. \[fig-multi-en\]](\figdir/MULTI_EN.eps){width="99.00000%"} [ c c c c ]{} From$\backslash$To:&2DN1&2DN2&2DN31A3N&0.445&0.219&0.4462DN1&—&0.527&0.2152DN2&—&—&0.479 In figure \[fig-hemo-en\] we show rigidity versus cutoff energy $E$ for 11 hemoglobin structures displaying a wide variation in rigidity behaviour, while in figure \[fig-multi-en\] we show results for a set of three myoglobin structures and six alpha-lactalbumins from a variety of mammals. In each case we see wide variation in the pattern of rigidity loss during the dilution, with some members of each family largely losing rigidity by the time the cutoff energy $E$ drops to around -2 kcal/mol and with others retaining considerable rigidity down to much lower cutoff values. For the hemoglobins, we note that our dataset includes four human hemoglobin structures. Those identified in Figure \[fig-hemo-en\] as 1A3N, 2DN1-alpha, 2DN2 and 2DN3 are alpha subunits of human deoxy, oxy, deoxy and carbonmonoxy-hemoglobin respectively. Interestingly, their rigidities differ for cutoff values in the range 0 to -2 kcal/mol but are all similar for lower cutoff values, suggesting that their structural differences (see table \[tab-hemo-1\] for alpha-carbon RMSD values) mainly affect the weaker hydrogen bonds. For comparison, we also show data for 2DN1-beta, a beta subunit from the 2DN1 structure, whose rigidity clearly differs from any of the human alpha subunits, resembling that of the unrelated rice hemoglobin structure 1D8U. Trypsin {#sec-trypsin} ------- ![A graph of hydrogen bond energy cutoff $E$ versus rigidity for a wide variety of trypsin structures. Notice, in contrast to previous figures, that all trypsins lose rigidity for cutoffs in the region of -1 to -2 kcal/mol. \[fig-trypsin\]](\figdir/TRYPSIN_EN.eps){width="99.00000%"} The presence of a trypsin inhibitor, BPTI, in the Hespenheide dataset [@HesRTK02] suggested to us that we should include trypsin itself in our study. Many crystal structures of trypsins have been reported from different species and under multiple conditions of crystallisation (e.g. in complex with BPTI or various small molecules); we assembled a dataset of 18 structures from five different species- human, bovine, rat, pig and salmon. The trypsins display a unique feature in their rigidity-dilution behaviour which sets them aside from any of our other proteins, as shown in Figure \[fig-trypsin\]. They display a steep loss of rigidity at relatively low values of the hydrogen-bond energy cutoff, in the range of -1 to -2 kcal/mol. Comparison to previous figures shows clearly how much this behaviour differs from the other protein groups we have examined. Trypsin is the largest of the proteins in our set, typical structures having about 220 residues. This suggests that the structure might divide into a larger number of medium-sized rigid clusters during the dilution, causing us to ‘lose’ some atoms when we count only the five largest rigid clusters. Inspection of the RCD plots, however, shows that trypsin does not appear very different to other proteins in our set and does not divide into a larger number of sizeable rigid clusters. It thus appears that there is genuinely something different about the behaviour of the trypsin fold during rigidity dilution. The distinctiveness of the trypsins is only evident because of our strategy of comparing multiple related structures from different organisms and conditions. A single trypsin structure might not have appeared unusual when compared against single examples of other proteins. Our comparative approach, however, shows clearly that our other protein families display much more variation in the rigidity dilution behaviours and generally include some members which retain rigidity down to much lower energy cutoffs, e.g. $-3$ to $-5$ kcal/mol. This result thus confirms the value of the present comparative approach. Conclusion and outlook {#sec-concl} ====================== Our motivation in this study was twofold. Firstly, we wished to clarify a methodological issue in the use of rigidity analysis on protein structures, by determining the robustness of RCDs against small structural variations. Secondly, we wished to obtain, from our large sample of protein structures, an insight into the “typical” pattern of rigidity loss during hydrogen-bond dilution. On the first point, we find that there is considerable variation in the RCDs of structurally similar proteins during dilution. Figure \[fig-cyto-en-other\], for example, shows that among a group of cytochrome C structures drawn from various eukaryotic mitochondria, energy cutoffs in the range from 0 to $-2$ kcal/mol (such as have typically been used for FIRST/FRODA simulations of flexible motion [@WelMHT05; @JolWHT06; @JolWFT08; @MacNBC07]) can produce anything from near-total rigidity to near-total flexibility. We conclude that the results of rigidity analysis on individual crystal structures should not be over-interpreted as being “the” RCD for a protein. This implies that rigidity analysis is best used for comparison of structures and for hypothesis testing. [*Comparison*]{} of the RCDs of two similar structures can bring out the significance of structural variation, by drawing attention to those parts of the protein where the constraint network has changed. In combination with simulation of flexible motion, a [*hypothesis testing*]{} approach can reveal whether the presence or absence of a particular interaction is significant in allowing or forbidding a flexible motion to occur. An early use of this approach was the study by Jolley et al. [@JolWHT06], which compared two hypothetical pathways for formation of a protein complex by examining the changes in rigidity and flexibility which each pathway would entail. On the second point, we found that for most of the protein structures in our survey, rigidity loss occurs gradually and continuously as a function of energy cutoff during dilution. The only case which stood out was that of the enzyme trypsin. We surveyed a large number of trypsin structures from multiple organisms and found that all of the structures displayed near-total loss of rigidity before the cutoff dropped to $-2$ kcal/mol. No other protein group in our survey displayed such behaviour. We cannot at present account for this distinctive phenomenon, which may imply a difference in the structure-function relationship in the trypsins compared to other proteins. Our results in this paper suggest two avenues for further enquiry. Firstly, the rigidity of protein monomers extracted from complexes should be compared with their rigidity within the complex, which will be affected by interactions between monomers. Secondly, the robustness of flexible motion simulations based on rigidity analysis must now be investigated. A recent study of the flexible motion of myosin [@SunRAJ08] found that the flexible motion of the myosin structure appeared qualitatively similar over a wide range of cutoff values covering both highly flexible and highly rigid structures. This suggests that rigidity analysis retains its value as a natural coarse-graining for simulations even if the rigidity behaviour during dilution is as variable as we have found. We thankfully acknowledge discussions with R. Freedman and T. Pinheiro. SAW and RAR gratefully acknowledge the Leverhulme Trust (grant F/00 215/AH) for financial support. [10]{} D. J. Jacobs and M. F. Thorpe. Generic rigidity percolation: The pebble game. , 75:4051–4054, 1995. D. J. Jacobs, A. J. Rader, L. A. Kuhn, and M. F. Thorpe. Protein flexibility predictions using graph theory. , 44:150–165, 2001. H. Gohlke and M. F. Thorpe. A natural coarse graining for simulating large biomolecular motion. , 91:2115–2120, 2006. M. F. Thorpe, M. Lei, A. J. Rader, D. J. Jacobs, and L. A. Kuhn. Protein flexibility and dynamics using constraint theory. , 19:60–69, 2001. S. A. Wells, S. Menor, B. M. Hespenheide, and M. F. Thorpe. Constrained geometric simulation of diffusive motion in proteins. , 2:S127–S136, 2005. C. C. Jolley, S. A. Wells, B. M. Hespenheide, M. F. Thorpe, and P. Fromme. Docking of photosystem i subunit c using a constrained geometric simulation. , 128:8803–8812, 2006. C. C. Jolley, S. A. Wells, P. Fromme, and M. F. Thorpe. Fitting low-resolution cryo-em maps of proteins using constrained geometric simulations. , 94:1613–1621, 2008. A. Macchiarulo, R. Nuti, D. Bellochi, E. Camaioni, and R. Pellicciari. Molecular docking and spatial coarse graining simulations as tools to investigate substrate recognition, enhancer binding and conformational transitions in indoleamine-2,3-dioxygenase (ido). , 1774:1058–1068, 2007. M. Sun, M. B. Rose, S. K. Ananthanarayanan, D. J. Jacobs, and C. M. Yengo. Characterisation of the pre-force-generation state in the actomyosin cross-bridge cycle. , 105:8631–8636, 2008. B. M. Hespenheide, D. J. Jacobs, and M.F. Thorpe. Structural rigidity and the capsid assembly of cowpea chlorotic mottle virus. , 16:S5055–S5064, 2004. B. I. Dahiyat, D. B. Gordon, and S. L. Mayo. Automated design of the surface positions of protein helices. , 6:1333–1337, 1997. A. J. Rader, B. M. Hespenheide, L. A. Kuhn, and M. F. Thorpe. Protein unfolding: Rigidity lost. , 99:3540–3545, 2002. B. M. Hespenheide, A. J. Rader, M. F. Thorpe, and L. A. Kuhn. Identifying protein folding cores: Observing the evolution of rigid and flexible regions during unfolding. , 21:195–207, 2002. T. Mamonova, B. Hespenheide, R. Straub, M.F. Thorpe, and M. Kurnikova. Protein flexibility using constraints from molecular dynamics simulations. , 2:S137–S147, 2005. A. Sartbaeva, S. A. Wells, A. Huerta, and M. F. Thorpe. Local structural variability and the intermediate phase window in network glasses. , 75:224204, 2007. H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov, and P. E. Bourne. The protein data bank. , 28:235–242, 2000. http://www.rcsb.org. W. DeLano. The pymol molecular graphics system. www.pymol.org. J. M. Word, S. C. Lovell, J. S. Richardson, and D. C. Richardson. Asparagine and glutamine: Using hydrogen atoms contacts in the choice of side-chain amide orientation. , 285:1735–1747, 1999. http://kinemage.biochem.duke.edu/software/reduce.php. [^1]: “Give me fruitful error any time, bursting with its own corrections.” — Vilfredo Pareto.

--- abstract: 'The correlation functions of the spin-${\frac{1}{2}}$ $XXZ$ spin chain in the ground state are expressed in the form of the multiple integrals. For ${-1< \Delta <1}$, they were obtained by Jimbo and Miwa in 1996. Especially the next nearest-neighbour correlation functions are given as certain three-dimensional integrals. We shall show these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions. We have also found that the remaining one-dimensinal integrals can be evaluated analytically, when ${\nu = \cos^{-1}(\Delta)/\pi}$ is a rational number.' author: - Go Kato - Masahiro Shiroishi - Minoru Takahashi - Kazumitsu Sakai date: 'April 22, 2003' title: 'Next Nearest-Neighbor Correlation Functions of the Spin-1/2 XXZ Chain at Critical Region' --- The spin-1/2 ${XXZ}$ chain is one of the fundamental models in the study of the low-dimensional magnetism. The Hamiltonian is given by $$\begin{aligned} H=\sum_{j=-\infty}^{\infty} \left\{ S^x_j S^x_{j+1} + S^y_j S^y_{j+1} + \Delta S^z_j S^z_{j+1} \right\}, \label{XXZ}\end{aligned}$$ where $S= \sigma /2$ and $\sigma$ are Pauli matrices. The model can be solved by Bethe ansatz method and diverse physical properties have been investigated with varying the anisotropy parameter $\Delta$ [@Bethe31; @Hulthen38; @TakahashiBook; @KorepinBook]. Especially, in the region ${-1 < \Delta \le 1}$, the ground state is critically disordered and the excitation spectrum is gapless. Then the long-distance asymptotics of two-point the correlation functions such as $\langle S^{\alpha}_j S^{\alpha}_{j+k} \rangle_{k \gg 1} \ \ {\alpha=x,y,z}$ are shown to decay as a power law via field theoretical approach (see, for example, [@Lukyanov03] and the references therein). However, if possible, it is more desirable\ (1) to calculate $\langle S^{\alpha}_j S^{\alpha}_{j+k} \rangle$ for finite ${k}$ first and\ (2) to derive its asymptotic behavior exactly.\ Unfortunately such a program has not been achieved except for the ${\Delta=0}$ case [@Lieb61; @McCoy68]. In 1996, Jimbo and Miwa [@Jimbo96] obtained the multiple integral representation for the arbitrary correlation functions of the ${XXZ}$ chain for ${-1 < \Delta < 1}$. For example, the Emptiness Formation Probability (EFP) [@Korepin94] defined by $$\begin{aligned} P(n) = \Big\langle \prod_{j=1}^n \left(S_j^z + \frac{1}{2} \right) \Big\rangle, \label{EFP}\end{aligned}$$ has the integral representation $$\begin{aligned} P(n)&= \left(- \nu \right)^{-\frac{n(n-1)}{2}} \int_{-\infty}^{\infty} \frac{{\rm d} x_1}{2 \pi} \cdots \int_{-\infty}^{\infty} \frac{{\rm d} x_n}{2 \pi} \prod_{a>b} \frac{\sinh (x_a-x_b)} {\sinh \left( \left(x_a-x_b - {\rm i} \pi \right) \nu \right)}\nonumber \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \prod_{k=1}^{n} \frac{\sinh^{n-k} \left( \left(x_k + {\rm i} \pi/2 \right) \nu \right) \sinh^{k-1} \left( \left(x_k - {\rm i} \pi/2 \right) \nu \right)}{\cosh^n x_k}, \label{EFPintegral}\end{aligned}$$ where the parameter ${\nu}$ is related to the anisotropy $\Delta$ as $$\begin{aligned} \nu = \frac{1}{\pi} \cos^{-1}(\Delta). \label{nu}\end{aligned}$$ Recently there have been an increasing number of researches concerning the properties of the EFP [@Korepin94; @Essler95n1; @Essler95n2; @Kitanine00; @Razumov01; @Kitanine02n1; @Shiroishi01; @Boos01; @Boos02; @BKNS02; @Boos03; @Abanov02; @Korepin03; @Kitanine02n2]. Particularly, in the isotropic limit ${\Delta \to 1 (\nu \to 0)}$, the general method to evaluate the multiple integral was recently developed by Boos and Korepin [@Boos01; @Boos02; @BKNS02]. It is of some importance to note that ${P(2)}$ and ${P(3)}$ are related to the nearest and next nearest-neighbor correlation functions : $$\begin{aligned} \langle S_{j}^z S_{j+1}^z \rangle &= P(2)-1/4, \label{P2} \\ \langle S_{j}^z S_{j+2}^z \rangle &= 2 (P(3)-P(2)+1/8). \label{P3} \end{aligned}$$ One of our purposes in this letter is to evaluate ${\langle S_{j}^z S_{j+2}^z \rangle}$ through the integral representations of ${P(3)}$ and ${P(2)}$. Similarly the nearest and the next nearest-neighbor transverse correlation functions have the integral representations, $$\begin{aligned} \langle S_{j}^x S_{j+1}^x \rangle = -\frac{1}{2 \nu} \int_{-\infty}^{\infty} \frac{{\rm d} x_1}{2 \pi} \int_{-\infty}^{\infty} \frac{{\rm d} x_2}{2 \pi} \frac{\sinh (x_2-x_1)}{\sinh \left( \left(x_2-x_1 \right) \nu \right)} \frac{\sinh \left( \left( x_1 + {\rm i} \pi/2 \right) \nu \right) \sinh \left( \left(x_2-{\rm i} \pi/2 \right) \nu \right)}{\cosh^2 x_1 \cosh^2 x_2}, \label{transverse1}\end{aligned}$$ and $$\begin{aligned} \langle S_{j}^x S_{j+2}^x \rangle &= - \frac{1}{\nu^3} \prod_{k=1}^{3} \int_{-\infty}^{\infty} \frac{{\rm d} x_k}{2 \pi} \frac{\sinh^{3-k} \left( \left(x_k + {\rm i} \pi/2 \right) \nu \right) \sinh^{k-1} \left( \left(x_k - {\rm i} \pi/2 \right) \nu \right)}{\cosh^3 x_k} \nonumber \\ & \hspace{2cm} \times \frac{\sinh (x_2-x_1)} {\sinh \left(\left(x_2-x_1 \right) \nu \right)} \frac{\sinh (x_3-x_1)} {\sinh \left(\left(x_3-x_1 \right) \nu \right)} \frac{\sinh (x_3-x_2)} {\sinh \left( \left(x_3-x_2 - {\rm i} \pi \right) \nu \right)}, \label{transverse2}\end{aligned}$$ respectively. It was already shown by Jimbo and Miwa that the two-dimensional integral for ${P(2)}$ reduces to the one-dimensional one for arbitrary ${\nu}$ as $$\begin{aligned} & P(2)= \frac{1}{2} + \frac{1}{2 \pi^2 \sin \pi \nu} \frac{\partial}{\partial \nu} \left\{ \sin \pi \nu \int_{-\infty}^{\infty} \frac{\sinh(1-\nu) w}{\sinh w \cosh \nu w} {\rm d} w \right\}, \label{P2new}\end{aligned}$$ which leads to the one-dimensional integral representation of ${\langle S_{j}^z S_{j+1}^z \rangle}$ via the relation (\[P2\]). The result coincides with a different derivation of ${\langle S_{j}^z S_{j+1}^z \rangle}$ from the ground-state energy per site ${e_0}$ : $$\begin{aligned} \langle S_{j}^z S_{j+1}^z \rangle &= \frac{\partial e_0}{\partial \Delta} = - \frac{1}{\pi \sin \pi \nu} \frac{\partial e_0}{\partial \nu} \nonumber \\ &= \frac{1}{4} + \frac{\cot \pi \nu}{2 \pi} \int_{-\infty}^{\infty} \frac{{\rm d} w }{\sinh w} \frac{\sinh(1-\nu) w}{ \cosh \nu w} - \frac{1}{2 \pi^2} \int_{-\infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{ w \cosh w}{(\cosh \nu w)^2}, \label{nextZZ}\end{aligned}$$ where $$\begin{aligned} e_0 =\frac{\Delta}{4} - \frac{\sin \pi \nu}{2 \pi} \int_{-\infty}^{\infty} \frac{\sinh(1- \nu) w}{\sinh w \cosh \nu w} {\rm d} w. \end{aligned}$$ Similarly we can get the one-dimensional integral representation for ${\langle S_{j}^x S_{j+1}^x \rangle }$ as $$\begin{aligned} \langle S_{j}^x S_{j+1}^x \rangle &= \frac{1}{2} \left( e_0 - \Delta \langle S_{j}^z S_{j+1}^z \rangle \right) \nonumber \\ &= - \frac{1}{4 \pi \sin \pi \nu} \int_{-\infty}^{\infty} \frac{{\rm d} w }{\sinh w} \frac{\sinh(1-\nu) w}{ \cosh \nu w} + \frac{\cos \pi \nu}{4 \pi^2} \int_{-\infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{ w \cosh w}{(\cosh \nu w)^2}. \label{nextXX}\end{aligned}$$ Thus, for the nearest-neighbor correlation functions, we have known the two-dimensional integrals by Jimbo-Miwa formula can be reduced to one-dimensional ones. The main result of this letter is that [*three-dimensional integrals for ${P(3)}$ and ${\langle S_{j}^x S_{j+2}^x \rangle}$ can also be reduced to one-dimensional integrals*]{}. In other words, we have succeeded in performing the integrals for ${P(3)}$ and ${\langle S_{j}^x S_{j+2}^x \rangle}$ twice. Our results are $$\begin{aligned} P(3)&= \frac{1}{2} + \int_{- \infty- {\rm i} \delta }^{\infty- {\rm i} \delta} \frac{{\rm d} x}{\sinh x} \Bigg[ \frac{1- \cos 2 \pi \nu}{16 \pi^2} \frac{\partial}{\partial \nu} \left\{ \frac{\cosh 3 \nu x}{(\sinh \nu x)^3} \right\} + \frac{3 \tan \pi \nu}{8 \pi} \frac{\cosh 3\nu x} {(\sinh \nu x)^3} \nonumber \\ & \hspace{3cm} - \frac{4- \cos 2 \pi \nu}{4 \pi^2} \frac{\partial \left(\coth \nu x \right)}{\partial \nu} - \frac{\left(4- \cos 2 \pi \nu \right) \coth \nu x} {2 \pi \sin 2 \pi \nu} \Bigg], \label{P3new}\end{aligned}$$ and $$\begin{aligned} \langle S_{j}^x S_{j+2}^x \rangle &= \int_{-\infty- {\rm i} \delta}^{\infty - {\rm i} \delta} \frac{{\rm d} x}{\sinh x} \Bigg[ - \frac{\left(\sin \pi \nu \right)^2}{8 \pi^2} \frac{\partial}{\partial \nu} \left\{ \frac{\cosh 3 \nu x}{(\sinh \nu x)^3} \right\} - \frac{3 \cos 2 \pi \nu(1- \cos 2 \pi\nu)}{8 \pi \sin 2 \pi \nu} \frac{\cosh 3 \nu x}{(\sinh \nu x)^3} \nonumber \\ & \hspace{1.5cm} + \frac{1}{4 \pi^2} \frac{\partial \left(\coth \nu x \right)}{\partial \nu} + \frac{\left\{ 1+ 3\cos 2 \pi \nu -3 \left(\cos 2 \pi \nu \right)^2 \right\} \coth \nu x} {2 \pi \sin 2 \pi \nu} \Bigg]. \label{transnew}\end{aligned}$$ Here ${\delta \ (0<|\delta| <\pi)}$ should take some non-zero real value, which is introduced to avoid the singularity at the origin. We, however, note that the singular term may be subtracted from the integrand in principle, as its residue vanishes. Or alternatively, by applying the Fourier transform, we can express the one-dimensional representations (\[P3new\]) and (\[transnew\]) as $$\begin{aligned} P(3)&= \frac{1}{2} + \int_{- \infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{\sinh (1-\nu)w}{\cosh \nu w} \left[ \frac{1+ 2 \cos 2 \pi \nu}{2 \pi \sin 2 \pi \nu} + \frac{3 \tan \pi \nu}{4 \pi^3} w^2 \right] \nonumber \\ & \ \ \ - \int_{- \infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{\cosh w}{(\cosh \nu w)^2} \left[ \frac{3}{4 \pi^2} w + \frac{(\sin \pi \nu)^2}{4 \pi^4} w^3 \right], \label{P3w}\end{aligned}$$ and $$\begin{aligned} \langle S_{j}^x S_{j+2}^x \rangle &= -\int_{- \infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{\sinh (1-\nu)w}{\cosh \nu w} \left[ \frac{1}{2 \pi \sin 2 \pi \nu} + \frac{3 \cos 2 \pi \nu \tan \pi \nu}{4 \pi^3} w^2 \right] \nonumber \\ & \ \ \ + \int_{- \infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{\cosh w}{(\cosh \nu w)^2} \left[ \frac{\cos 2 \pi \nu}{4 \pi^2} w + \frac{(\sin \pi \nu)^2}{4 \pi^4} w^3 \right]. \label{transw}\end{aligned}$$ Note that in the representations (\[P3w\]) and (\[transw\]), there are no singularities at the origin. This is in the similar situation as (\[nextZZ\]) and (\[nextXX\]). Combining (\[P2new\]) and (\[P3w\]), we can also write down the one-dimensional integral representation for ${\langle S_{j}^z S_{j+2}^z \rangle}$ through the relation (\[P3\]), $$\begin{aligned} \langle S_{j}^z S_{j+2}^z \rangle &= \frac{1}{4}+ \int_{- \infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{\sinh (1-\nu)w}{\cosh \nu w} \left[ \frac{\cot 2 \pi \nu}{\pi} + \frac{3 \tan \pi \nu}{2 \pi^3} w^2 \right] \nonumber \\ & \ \ \ - \int_{- \infty}^{\infty} \frac{{\rm d} w}{\sinh w} \frac{\cosh w}{(\cosh \nu w)^2} \left[ \frac{1}{2 \pi^2} w + \frac{(\sin \pi \nu)^2}{2 \pi^4} w^3 \right]. \label{longiw}\end{aligned}$$ Now let us discuss some properties of the obtained one-dimensional representations (\[transw\]) and (\[longiw\]). At a first glance, some integrands in (\[transw\]) and (\[longiw\]) are divergent at ${\nu=1/2}$ and also in the limit ${\nu \to 0}$ and ${\nu \to 1}$, due to the the factor ${1/\sin{2 \pi \nu}}$. However, by investigating the integrands carefully, we have found these singular terms cancel each other, therefore yielding definite finite values for the correlation functions : - ${\nu=1/2}$ : $$\begin{aligned} \langle S_{j}^x S_{j+2}^x \rangle = \dfrac{1}{\pi^2}, \ \ \ \ \langle S_{j}^z S_{j+2}^z \rangle = 0, \label{Delta0}\end{aligned}$$ - ${\nu \to 0}$ : $$\begin{aligned} \langle S_{j}^x S_{j+2}^x \rangle, \ \ \langle S_{j}^z S_{j+2}^z \rangle \to \frac{1}{12}- \frac{4}{3} \ln 2 + \frac{3}{4} \zeta(3), \label{Delta1}\end{aligned}$$ - ${\nu \to 1}$ : $$\begin{aligned} \langle S_{j}^x S_{j+2}^x \rangle \to \dfrac{1}{8}, \ \ \langle S_{j}^z S_{j+2}^z \rangle \to 0. \label{Deltam1}\end{aligned}$$ It is especially intriguing to observe that the ${\nu \to 0}$ limit (\[Delta1\]) reproduces the known result by one of the authors in [@Takahashi77]. More generally, we have found when ${\nu}$ takes a rational value, the one-dimensional integrals can be evaluated analytically. Some of our explicit results are summarized in Table 1. One can see that the correlation functions are in general a polynomial of ${1/\pi}$. Particularly, when ${\nu=1/3}$, we expect all the correlation functions are given solely as a single rational number (c.f. [@Razumov01; @Kitanine02n1]). In Fig. \[fig:1\] and Fig. \[fig:2\], plotted are the numerical values of the nearest and the next nearest-neighbor correlation functions calculated from the one-dimensional integral representations. For comparison, the analytical values in Table 1 are represented by the filled circles. We have obtained (\[P3new\]) and (\[transnew\]) by generalizing the method developed by Boos and Korepin [@Boos01; @Boos02], which allows us to calculate the multiple integrals for the correlation functions of the ${XXX}$ model (${\Delta=1}$). Below we outline the derivation of (\[P3new\]) and (\[transnew\]) quite briefly. The details of the calculations will be published in a separate paper [@Kato03]. First we introduce the following convenient notation for ${P(3)}$ : $$\begin{aligned} P(3) = \prod_{k=1}^{3} \int_{-\infty - {\rm i}/2}^{\infty-{\rm i}/2} \frac{{\rm d} \lambda_{k}}{2 \pi {\rm i}} U_3(\lambda_1,\lambda_2,\lambda_3) T_3(\lambda_1,\lambda_2,\lambda_3), \label{notation} \end{aligned}$$ where $$\begin{aligned} U_3(\lambda_1,\lambda_2,\lambda_3) = \frac{\pi^3 \prod_{1 \le k< j \le3} \sinh \pi \left(\lambda_j-\lambda_k \right)} {\nu^3 \prod_{j=1}^{3} \sinh^3 \pi \lambda_j}, \end{aligned}$$ and $$\begin{aligned} T_3(\lambda_1,\lambda_2,\lambda_3) = \frac{(q z_1-1)^2 (q z_2-1)(z_2-1) (z_3-1)^2} {8 (z_2-q z_1)(z_3-q z_1)(z_3-q z_2)}, \end{aligned}$$ with ${q \equiv {\rm e}^{2 \pi {\rm i} \nu}, z_i \equiv {\rm e}^{2 \pi \nu {\lambda_i}}, \ (i=1,2,3)}$. Then after similar but more complicated calculation as Boos and Korepin [@Boos02], we can transform the integrand ${ T_3(\lambda_1,\lambda_2,\lambda_3)}$ into a certain [*canonical form*]{} without changing the value of the integral as $$\begin{aligned} T_c = P_0 + \frac{P_1}{z_2-z_1}, \end{aligned}$$ where $$\begin{aligned} P_0 =& \frac{(1+q)^2}{8q}\frac{z_1}{z_3}, \\ P_1 =& \frac{3(1+q)}{2 q} - \frac{(1+10q+q^2)z_1}{8 q} - \frac{3(1+q)^2}{8q^2 z_1} \nonumber \\ &+ z_3 \left\{ - \frac{1+10q+q^2}{8q} + \frac{3(1+q) z_1}{8}+ \frac{3(1+q)}{8q z_1} \right\} \nonumber \\ &+ \frac{1}{z_3} \left\{ - \frac{3(1+q)^2}{8q^2} + \frac{3(1+q) z_1}{8q}+ \frac{(1+q)(1+q+q^2)}{8q^3 z_1} \right\}. \end{aligned}$$ We find the first part of ${T_c}$, namely ${P_0}$, can be integrated easily $$\begin{aligned} \prod_{k=1}^{3} \int_{-\infty - {\rm i}/2}^{\infty-{\rm i}/2} \frac{{\rm d} \lambda_{k}}{2 \pi {\rm i}} U_3(\lambda_1,\lambda_2,\lambda_3) \frac{(1+q)^2 {\rm e}^{2 \pi \nu (\lambda_1-\lambda_3)}}{8q} = \frac{1}{2}. \label{P0integral}\end{aligned}$$ The second part ${P_1/(z_2-z_1)}$ can be integrated twice, namely with respect to ${\lambda_3}$ and ${ a \equiv (\lambda_2 + \lambda_1)/2 }$. Finally the remaining one-dimensional integration with respect to ${ d \equiv (\lambda_2-\lambda_1)/2}$ together with (\[P0integral\]) gives the expression (\[P3new\]). Similarly for ${\langle S_{j}^x S_{j+2}^x \rangle}$, ${T_3(\lambda_1,\lambda_2,\lambda_3)}$ in (\[notation\]) is replaced by $$\begin{aligned} T_3(\lambda_1,\lambda_2,\lambda_3) = \frac{(q z_1-1)^2 (q z_2-1)(z_2-1) (z_3-1)^2} {8q (z_2-z_1)(z_3-z_1)(z_3-q z_2)}. \end{aligned}$$ As a corresponding canonical form, we have found $$\begin{aligned} T_c = \frac{P_1}{z_2-z_1}, \end{aligned}$$ with $$\begin{aligned} P_1 =& - \frac{(1+q)(3+q^2)}{8q^2} + \frac{(3- 2q+3 q^2) z_1}{8q} + \frac{(1+q)^2}{8 q^3 z_1} \nonumber \\ &+ z_3 \left\{ \frac{3-q}{4q} - \frac{(1+q) z_1}{8q}+ \frac{(1+q)(-2+q)}{8q^2 z_1} \right\} \nonumber \\ &+ \frac{1}{z_3} \left\{\frac{1+q}{2q} - \frac{(1+q) z_1}{8}- \frac{1+q}{8q^2 z_1} \right\}. \end{aligned}$$ Again after integrating with respect to ${\lambda_3}$ and ${ a \equiv (\lambda_2 + \lambda_1)/2 }$, we arrive at the one-dimensional integral representation (\[transnew\]). In conclusion, we have shown the multiple integrals for the correlation functions of the ${XXZ}$ chain at the critical region ${-1<\Delta<1}$, can be reduced to the one-dimensional ones in the case of the next nearest-neighbor correlation functions. This property will be generalized to other higher-neighbor correlations as well as the correlations in the massive region (${\Delta>1}$) [@Jimbo92; @JimboBook]. In this respect, we like to refer to the recent work by Boos, Korepin and Smirnov [@BKS03], where they have shown the reducibility of the multiple integrals in the case of the ${XXX}$ chain. In our future work, we are particularly interested in calculating the third-neighbor correlation functions ${\langle S_{j}^x S_{j+3}^x \rangle}$ and ${\langle S_{j}^z S_{j+3}^z \rangle}$ for general ${\Delta}$. For ${\Delta=1}$, they were recently calculated in [@Sakai03] from the multiple integrals [@Korepin94; @Nakayashiki94] by Boos-Korepin method . We are grateful to H.E. Boos, M. Jimbo, V.E. Korepin, Y. Nishiyama, J. Suzuki and M. Wadati for valuable discussions. This work is in part supported by Grant-in-Aid for the Scientific Research (B) No. 14340099 from the Ministry of Education, Culture, Sports, Science and Technology, Japan. GK and KS are supported by the JSPS research fellowships for young scientists. MS is supported by Grant-in-Aid for young scientists No. 14740228. [|c||c|c|c|c|]{} ${\displaystyle \nu}$ & $\langle S_j^xS_{j+1}^x \rangle$ & $\langle S_j^z S_{j+1}^z \rangle$ & $\langle S_j^xS_{j+2}^x \rangle$ & $\langle S_j^zS_{j+2}^z \rangle$\ & $\frac{1}{12}-\frac{\ln 2}{3}$&$\frac{1}{12}-\frac{\ln 2}{3}$& $\frac{1}{12}-\frac{4\ln2}{3}+ \frac{3\zeta(3)}{4}$& $\frac{1}{12}-\frac{4\ln2}{3}+ \frac{3\zeta(3)}{4}$\ $\frac{1}{2}$& $-\frac{1}{2 \pi}$&$-\frac{1}{\pi^2}$&$\frac{1}{\pi^2}$ &\ $\frac{1}{3}$& $-\frac{5}{32}$&$-\frac{1}{8}$&$\frac{41}{512}$ &$\frac{7}{256}$\ $\frac{1}{4}$ &$-\frac{\sqrt{2}}{2 \pi}+ \frac{\sqrt{2}}{2 \pi^2}$ & $-\frac{1}{4}+\frac{1}{\pi}-\frac{2}{\pi^2}$ &$\frac{3}{16} - \frac{1}{\pi} + \frac{2}{\pi^2}$& $-\frac{5}{8}+\frac{4}{\pi}-\frac{6}{\pi^2}$\ $\frac{1}{5}$&$-\frac{3}{64} - \frac{3 \sqrt{5}}{64}$ &$-\frac{19}{8}$ + &$\frac{7737}{1024} - \frac{3429 \sqrt{5}}{1024}$ &$-\frac{3529}{512} + \frac{1589 \sqrt{5}}{512}$\ $\frac{1}{6}$ &$\frac{\sqrt{3}}{48} - \frac{1}{\pi} + \frac{3 \sqrt{3}}{4 \pi^2}$ &$-\frac{7}{18} + \frac{\sqrt{3}}{\pi}-\frac{3}{\pi^2}$ &$\frac{247}{576} - \frac{17 \sqrt{3}}{12 \pi} +\frac{33}{ 8 \pi^2}$ &$-\frac{283}{288}+\frac{11 \sqrt{3}}{3 \pi} - \frac{39}{4 \pi^2}$\ $\frac{2}{3}$&$\frac{47}{128}-\frac{\sqrt{3}}{4}-\frac{9}{32 \pi}$ & $\frac{23}{32} - \frac{\sqrt{3}}{4} - \frac{9}{8 \pi}$ &$ - \frac{2719}{8192} +\frac{47 \sqrt{3}}{256} + \frac{441}{1024 \pi}$ & $\frac{8671}{4096} - \frac{49 \sqrt{3}}{64} - \frac{1305}{512 \pi}$\ $\frac{3}{4}$& ---------------------------------------------------------------- $- \frac{4 \sqrt{6}}{27} + \frac{64 \sqrt{2}}{243} - \frac{\sqrt{2}}{6\pi}$ $\qquad-\frac{8 \sqrt{6}}{81\pi} - \frac{\sqrt{2}}{18 \pi^2} $ ---------------------------------------------------------------- : Some analytical values of the correlation functions when ${\nu}$ takes rational values []{data-label="table1"} & ----------------------------------------------------- $\frac{781}{972} -\frac{8 \sqrt{3}}{27} -\frac{1}{3\pi}$ $\quad-\frac{32 \sqrt{3}}{81\pi}-\frac{2}{9 \pi^2}$ ----------------------------------------------------- : Some analytical values of the correlation functions when ${\nu}$ takes rational values []{data-label="table1"} & ----------------------------------------------------------- $-\frac{22111}{34992} + \frac{8 \sqrt{3}}{27} + \frac{1}{3\pi}$ $\qquad+ \frac{160 \sqrt{3}}{729\pi} + \frac{2}{9 \pi^2}$ ----------------------------------------------------------- : Some analytical values of the correlation functions when ${\nu}$ takes rational values []{data-label="table1"} & --------------------------------------------------------- $\frac{36169}{17496} -\frac{160 \sqrt{3}}{243} -\frac{4}{3\pi}$ $\qquad-\frac{608 \sqrt{3}}{729\pi}- \frac{2}{3 \pi^2}$ --------------------------------------------------------- : Some analytical values of the correlation functions when ${\nu}$ takes rational values []{data-label="table1"} \ &$-\frac{1}{8}$&&$ \frac{1}{8}$ &\ ![\[fig:1\] Nearest-neighbor correlation functions for the XXZ chain](XXZnearest.eps){width="10cm"} ![\[fig:2\] Next nearest-neighbor correlation functions for the XXZ chain](XXZnextnearest.eps){width="10cm"} H.A. Bethe, Z. Phys. [**71**]{} (1931) 205. L. Hulthén, Ark. Mat. Astron. Fys. A [**26**]{} (1938) 1. M. Takahashi, *Thermodynamics of One-Dimensional Solvable Models*, (Cambridge University Press,Cambridge, 1999). V.E Korepin, N.M. Bogoliubov and A.G. Izergin, *Quantum Inverse Scattering Method and Correlation Functions*, (Cambridge University Press, Cambridge, 1993). S. Lukyanov and V. Terras, Nucl. Phys. B [**654**]{} (2003) 323. E.H. Lieb, T. Schultz and D. Mattis, A. Phys. (N.Y.) [**16**]{} (1961) 417. B.M. McCoy, Phys. Rev. [**173**]{} (1968) 531. M. Jimbo and T. Miwa, J. Phys. A [**29**]{} (1996) 2923. V.E. Korepin, A.G. Izergin, F.H.L. Essler and D.B. Uglov Phys. Lett. A [**190**]{} (1994) 182. F.H.L. Essler, H. Frahm, A.G. Izergin and V.E. Korepin, Commun. Math. Phys. [**174**]{} (1995) 191. F.H.L. Essler, H. Frahm, A.R. Its and V.E. Korepin, Nucl Phys [**B 446**]{}, (1995) 448. N. Kitanine, J. M. Maillet and V. Terras, Nucl. Phys. [**B 567**]{} (2000) 554. A.V. Razumov and Yu.G. Stroganov, J. Phys. A: Math. Gen. [**34**]{} (2001) 3185, *ibid.* 5335. N. Kitanine, J.M. Maillet, N.A. Slavnov, and V. Terras, J. Phys. A. [**35**]{} (2002) L385. M. Shiroishi, M. Takahashi and Y. Nishiyama, J. Phys. Soc. Jpn. [**70**]{} (2001) 3535. H.E. Boos and V.E. Korepin, J. Phys. A [**34**]{} (2001) 5311. H.E. Boos and V.E. Korepin, *Integrable models and Beyond*, edited by M. Kashiwara and T. Miwa (Birkhäuser, Boston, 2002); hep-th/0105144. H.E. Boos, V.E. Korepin, Y. Nishiyama and M. Shiroishi, J. Phys. A [**35**]{} (2002) 4443. H.E. Boos, V.E. Korepin and F.A. Smirnov, Nucl. Phys. B. [**658**]{} (2003) 417. A.G. Abanov and V.E. Korepin, Nucl. Phys. [**B 647**]{} (2002) 565. V.E. Korepin, S. Lukyanov, Y. Nishiyama and M. Shiroishi, hep-th/0210140, to appear in Phys. Lett. A (2003). N. Kitanine, J.M. Maillet, N.A. Slavnov, and V. Terras, J. Phys. A. [**35**]{} (2002) L753. M. Takahashi, J. Phys. C [**10**]{} (1977) 1289. G. Kato, K. Sakai, M. Shiroishi and M. Takahashi, in preparation. M. Jimbo, K. Miki, T. Miwa and A. Nakayashiki, Phys. Lett. A [**168**]{}, (1992) 256. M. Jimbo and T. Miwa, *Algebraic Analysis of Solvable Lattice Models*, (American Mathematical Society, Providence, RI, 1995). H. Boos, V.E. Korepin and F. Smirnov, *New Formulae for the solutions of quantum Knizhnik-Zamolodchikov equations on level-4*, hep-th/0304077. K. Sakai, M. Shiroishi, Y. Nishiyama and M. Takahashi, *Third Neighbor Correlators of one-dimensional Spin-1/2 Heisenberg Antiferromagnet*, cond-mat/0302564. A. Nakayashiki, Int. J. Mod. Phys. A [**9**]{}, (1994) 5673.

--- abstract: 'In earlier papers the author studied some classes of equations with Carlitz derivatives for ${\mathbb F_q}$-linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider equations containing self-compositions $u\circ u\circ \cdots \circ u$ of the unknown function. As an algebraic background, imbeddings of the composition ring of ${\mathbb F_q}$-linear holomorphic functions into skew fields are considered.' author: - | ANATOLY N. KOCHUBEI[^1]\ Institute of Mathematics, National Academy\ of Sciences of Ukraine, Tereshchenkivska 3, Kiev, 01601 Ukraine title: Strongly Nonlinear Differential Equations with Carlitz Derivatives over a Function Field --- INTRODUCTION ============ Let $K$ be the set of formal Laurent series $t=\sum\limits_{j=N}^\infty \xi_jx^j$ with coefficients $\xi_j$ from the Galois field ${\mathbb F_q}$, $\xi_N\ne 0$ if $t\ne 0$, $q=p^\upsilon $, $\upsilon \in \mathbf Z_+$, where $p$ is a prime number. It is well known that $K$ is a locally compact field of characteristic $p$, with natural operations over power series, and the topology given by the absolute value $|t|=q^{-N}$, $|0|=0$. The element $x$ is a prime element of $K$. Any non-discrete locally compact field of characteristic $p$ is isomorphic to such $K$. Below we denote by ${\overline{K}_c}$ the completion of an algebraic closure $\overline{K}$ of $K$. The absolute value $|\cdot |$ can be extended in a unique way onto ${\overline{K}_c}$. An important class of functions playing a significant part in the analysis over ${\overline{K}_c}$ is the class of ${\mathbb F_q}$-linear functions. A function $f$ defined on a ${\mathbb F_q}$-subspace $K_0$ of $K$ (or ${\overline{K}_c}$), with values in ${\overline{K}_c}$, is called ${\mathbb F_q}$-[*linear*]{} if $f(t_1+t_2)=f(t_1)+f(t_2)$ and $f(\alpha t)=\alpha f(t)$ for any $t,t_1,t_2\in K_0$, $\alpha \in {\mathbb F_q}$. A typical example is a ${\mathbb F_q}$-linear polynomial $\sum c_kt^{q^k}$ or, more generally, a power series $\sum\limits_{k=0}^\infty c_kt^{q^k}$, where $c_k\in {\overline{K}_c}$ and $|c_k|\le C^{q^k}$, convergent on a neighbourhood of the origin. In the theory of differential equations over $K$ initiated in [@K2; @K3] (which deals also with some non-analytic ${\mathbb F_q}$-linear functions) the role of a derivative is played by the operator $$d=\sqrt[q]{}\circ \Delta ,\quad (\Delta u)(t)=u(xt)-xu(t),$$ introduced by Carlitz [@C1] and used subsequently in various problems of analysis in positive characteristic [@C2; @G1; @K1; @Th; @W]. The differential equations considered so far were analogs of linear ordinary differential equations, though the operator $d$ is only ${\mathbb F_q}$-linear and the meaning of a polynomial coefficient in the function field case is not a usual multiplication by a polynomial, but the action of a polynomial in the ${\mathbb F_q}$-linear operator $\tau$, $\tau u=u^q$. Note that ${\mathbb F_q}$-linear polynomials form a ring with respect to the composition $u\circ v$ (the usual multiplication violates the ${\mathbb F_q}$-linearity), so that natural classes of equations with stronger nonlinearities must contain expressions like $u\circ u$ or, more generally, $u\circ u\cdots \circ u$. An investigation of such “strongly nonlinear” Carlitz differential equations is the main aim of this paper. However we have to begin with algebraic preliminaries of some independent interest (so that not all the results are used in the subsequent sections) regarding the ring ${\mathcal R_K}$ of locally convergent ${\mathbb F_q}$-linear holomorphic functions. The ring is non-commutative, and the algebraic structures related to strongly nonlinear Carlitz differential equations are much more complicated than their classical counterparts. So far their understanding is only at its initial stage. Here we show that ${\mathcal R_K}$ is imbedded into a skew field of ${\mathbb F_q}$-linear “meromorphic” series containing terms like $t^{q^{-k}}$. Note that a deep investigation of bi-infinite series of this kind convergent on the whole of ${\overline{K}_c}$ has been carried out by Poonen [@Po]. We also prove an appropriate version of the implicit function theorem. After the above preparations we consider general strongly nonlinear first order ${\mathbb F_q}$-linear differential equations (resolved with respect to the derivative of the unknown function) and prove an analog of the classical Cauchy theorem on the existence and uniqueness of a local holomorphic solution of the Cauchy problem. In our case the classical majorant approach (see e.g. [@Hi]) does not work, and the convergence is proved by direct estimates. We also consider a class of Riccati-type equations possessing ${\mathbb F_q}$-linear solutions which are meromorphic in the above sense. Skew fields of ${\mathbb F_q}$-linear power series ================================================== Let $\mathcal R_K$ be the set of all formal power series $a=\sum\limits_{k=0}^\infty a_kt^{q^k}$ where $a_k\in K$, $|a_k|\le A^{q^k}$, and $A$ is a positive constant depending on $a$. In fact each series $a=a(t)$ from ${\mathcal R_K}$ converges on a neighbourhood of the origin in $K$ (and ${\overline{K}_c}$). ${\mathcal R_K}$ is a ring with respect to the termwise addition and the composition $$a\circ b=\sum\limits_{l=0}^\infty \left( \sum\limits_{n=0}^l a_nb_{l-n}^{q^n}\right) t^{q^l},\quad b=\sum\limits_{k=0}^\infty b_kt^{q^k},$$ as the operation of multiplication. Indeed, if $|b_k|\le B^{q^k}$, then, by the ultra-metric property of the absolute value, $$\left| \sum\limits_{n=0}^la_nb_{l-n}^{q^n}\right| \le \max\limits_{0\le n\le l}A^{q^n}\left( B^{q^{l-n}}\right)^{q^n}\le C^{q^l}$$ where $C=B\max (A,1)$. The unit element in ${\mathcal R_K}$ is $a(t)=t$. It is easy to check that ${\mathcal R_K}$ has no zero divisors. If $a\in {\mathcal R_K}$, $a=\sum\limits_{k=0}^\infty a_kt^{q^k}$, is such that $|a_0|\le 1$ and $|a_k|\le A^{q^k}$, $|A|\ge 1$, for all $k$, then we may write $$|a_k|\le A_1^{q^k-1},\quad k=0,1,2,\ldots ,$$ if we take $A_1\ge A^{q^k/(q^k-1)}$ for all $k\ge 1$. If also $b=\sum\limits_{k=0}^\infty b_kt^{q^k}$, $|b_k|\le B_1^{q^k-1}$, $B_1\ge 1$, then for $a\circ b=\sum\limits_{l=0}^\infty c_lt^{q^l}$ we have $$|c_l|\le \max_{i+j=l}A_1^{q^i-1}\left( B_1^{q^j-1}\right)^{q^i}\le C_1^{q^l-1}$$ where $C_1=\max (A_1,B_1)$. In particular, in this case the coefficients of the series for $a^n$ (the composition power) satisfy an estimate of this kind, with a constant independent of $n$. The ring ${\mathcal R_K}$ is a left Ore ring, thus it possesses a classical ring of fractions. [*Proof*]{}. By Ore’s theorem (see [@Her]) it suffices to show that for any elements $a,b\in {\mathcal R_K}$ there exist such elements $a',b'\in {\mathcal R_K}$ that $b'\ne 0$ and $$a'\circ b=b'\circ a.$$ We may assume that $a\ne 0$, $$a=\sum\limits_{k=m}^\infty a_kt^{q^k},\quad b=\sum\limits_{k=l}^\infty b_kt^{q^k},$$ $m,l\ge 0$, $a_m\ne 0$, $b_l\ne 0$. Without restricting generality we may assume that $l=m$ (if we prove (1) for this case and if, for example, $l<m$, we set $b_1=t^{q^{m-l}}\circ b$, find $a'',b'$ in such a way that $a''\circ b_1=b'\circ a$, and then set $a'=a''\circ t^{q^{m-l}}$), and that $a_l=b_l=\alpha$, so that $$a=\alpha t^{q^l}+\sum\limits_{k=l+1}^\infty a_kt^{q^k},\quad b=\alpha t^{q^l}+\sum\limits_{k=l+1}^\infty b_kt^{q^k},$$ $\alpha \ne 0$. We seek $a',b'$ in the form $$a'=\sum\limits_{j=l}^\infty a_j't^{q^j},\quad b'=\sum\limits_{j=l}^\infty b_j't^{q^j}.$$ The coefficients $a_j',b_j'$ can be defined inductively. Set $a_l'=b_l'=1$. If $a_j',b_j'$ have been determined for $l\le j\le k-1$, then $a_k',b_k'$ are determined from the equality of the $(k+l)$-th terms of the composition products: $$a_k'\alpha^{q^k}+\sum\limits_{\genfrac{}{}{0pt}{}{i+j=k+l}{j\ne l}}a_i'b_j^{q^i}=b_k'\alpha^{q^k}+\sum\limits_{\genfrac{}{}{0pt}{}{i+j=k+l}{j\ne l}}b_i'a_j^{q^i}$$ (the above sums do not contain non-trivial terms with $a_i',b_i'$, $i\ge k$, since $a_j=b_j=0$ for $j<l$). In particular, we may set $b_k'=0$, $$a_k'=\alpha^{-q^k}\left\{ \sum\limits_{\genfrac{}{}{0pt}{}{i+j=k+l}{i<k,j\ne l}}\left( a_i'b_j^{q^i}-b_i'a_j^{q^i}\right) \right\} .$$ If this choice is made for each $k\ge l+1$, then we have $b_i'=0$ for every $i\ge l+1$, so that $$a_k'=\alpha^{-q^k}\sum\limits_{\genfrac{}{}{0pt}{}{i+j=k+l}{i<k,j\ne l}}a_i'b_j^{q^i}.$$ Denote $C_1=|\alpha |^{-1}$. We have $|b_j|\le C_2^{q^j}$ for all $j$. Denote, further, $C_3=\max (1,C_1,C_2)$, $C_4=C_3^{q^{l+2}}$. Let us prove that $$\left| a_k'\right| \le C_4^{q^k}.$$ Suppose that $\left| a_i'\right| \le C_4^{q^i}$ for all $i$, $l\le i\le k-1$ (this is obvious for $i=1$, since $a_l'=1$). By (2), $$\begin{gathered} \left| a_k'\right| \le C_1^{q^k}\max \limits_{\genfrac{}{}{0pt}{}{i+j=k+l}{i<k,j\ne l}}C_4^{q^i}C_2^{q^{i+j}}\le C_1^{q^k}C_4^{q^{k-1}}C_2^{q^{k+l}}\\ \le C_3^{q^k+q^{k+l+1}+q^{k+l}}=C_3^{(1+q^l+q^{l+1})q^k}\le \left( C_3^{q^{l+2}}\right)^{q^k}=C_4^{q^k},\end{gathered}$$ as desired. Thus $a'\in {\mathcal R_K}$. $\qquad \blacksquare$ Every non-zero element of ${\mathcal R_K}$ is invertible in the ring of fractions ${\mathcal A_K}$, which is actually a skew field consisting of formal fractions $c^{-1}d$, $c,d\in {\mathcal R_K}$. Each element $a=c^{-1}d\in {\mathcal A_K}$ can be represented in the form $a=t^{q^{-m}}a'$ where $t^{q^{-m}}$ is the inverse of $t^{q^m}$, $a'\in {\mathcal R_K}$. [*Proof*]{}. It is sufficient to prove that any non-zero element $c\in {\mathcal R_K}$ can be written as $c=c'\circ t^{q^m}$ where $c$ is invertible in ${\mathcal R_K}$. Let $c=\sum\limits_{k=m}^\infty c_kt^{q^k}$, $c_m\ne 0$, $|c_k|\le C^{q^k}$. Then $$c=c_m\left( t+\sum\limits_{l=1}^\infty c_m^{-1}c_{m+l}t^{q^l}\right) \circ t^{q^m}$$ where $\left| c_m^{-1}c_{m+l}\right| \le C_1^{q^l-1}$ for all $l\ge 1$, if $C_1$ is sufficiently large. Denote $$w=\sum\limits_{l=1}^\infty c_m^{-1}c_{m+l}t^{q^l},\quad c'=c_m(t+w).$$ The series $(t+w)^{-1}=\sum\limits_{n=0}^\infty (-1)^nw^n$ converges in the standard non-Archimedean topology of formal power series (see [@Pierce], Sect. 19.7) because the formal power series for $w^n$ begins from the term with $t^{q^m}$; recall that $w^n$ is the composition power, and $t$ is the unit element. Moreover, $w^n=\sum\limits_{j=n}^\infty a_j^{(n)}t^{q^j}$ where $\left| a_j^{(n)}\right| \le C_1^{q^j-1}$ for all j, with the same constant independent of $n$. Using the ultra-metric inequality we find that the coefficients of the formal power series $(t+w)^{-1}=\sum\limits_{j=0}^\infty a_jt^{q^j}$ (each of them is, up to a sign, a finite sum of the coefficients $a_j^{(n)}$) satisfy the same estimate. Therefore $(c')^{-1}\in {\mathcal R_K}$. $\qquad \blacksquare$ The skew field of fractions ${\mathcal A_K}$ can be imbedded into wider skew fields where operations are more explicit. Let ${K_{\text{perf}}}$ be the perfection of the field $K$. Denote by ${\mathcal A_{K_{\text{perf}}}}^\infty$ the composition ring of ${\mathbb F_q}$-linear formal Laurent series $a=\sum\limits_{k=m}^\infty a_kt^{q^k}$, $m\in \mathbb Z$, $a_k\in {K_{\text{perf}}}$, $a_m\ne 0$ (if $a\ne 0$). Since $\tau$ is an automorphism of ${K_{\text{perf}}}$, ${\mathcal A_{K_{\text{perf}}}}^\infty$ is a special case of the well-known ring of twisted Laurent series [@Pierce]. Therefore ${\mathcal A_{K_{\text{perf}}}}^\infty$ is a skew field. Let ${\mathcal A_{K_{\text{perf}}}}$ be a subring of ${\mathcal A_{K_{\text{perf}}}}^\infty$ consisting of formal series with $|a_k|\le A^{q^k}$ for all $k\ge 0$. Just as in the proof of Proposition 2, we show that ${\mathcal A_{K_{\text{perf}}}}$ is actually a skew field. Its elements can be written in the form $t^{q^{-m}}\circ c$ where $c$ is an invertible element of the ring ${\mathcal R_{K_{\text{perf}}}}\in {\mathcal A_{K_{\text{perf}}}}$ of formal power series $\sum\limits_{k=0}^\infty a_kt^{q^k}$. In contrast to the case of the skew field ${\mathcal A_K}$, in ${\mathcal A_{K_{\text{perf}}}}$ the multiplication of $t^{q^{-m}}$ by $c$ is indeed the composition of (locally defined) functions, so that ${\mathcal A_{K_{\text{perf}}}}$ consists of fractional power series understood in the classical sense. Of course, ${\mathcal A_{K_{\text{perf}}}}$ can be extended further, by considering $\overline{K}$ or ${\overline{K}_c}$ instead of ${K_{\text{perf}}}$. The above reasoning carries over to these cases (we can also consider the ring $\mathcal R_{\overline{K}_c}$ of locally convergent ${\mathbb F_q}$-linear power series as the initial ring). In each of them the presence of a fractional composition factor $t^{q^{-m}}$ is a ${\mathbb F_q}$-linear counterpart of a pole of the order $m$. Recurrent relations =================== In our investigations of strongly nonlinear equations and implicit functions we encounter recurrent relations of the same form $$c_{i+1}=\mu_i\sum\limits_{\genfrac{}{}{0pt}{}{j+l=i}{l\ne 0}} \sum\limits_{k=1}^\infty B_{jkl}\left( \sum\limits_{n_1+\cdots +n_k=l}c_{n_1}c_{n_2}^{q^{n_1}}\cdots c_{n_k}^{q^{n_1+\cdots +n_{k-1}}}\right)^{q^{j+\lambda}}+a_i,\quad i=1,2,\ldots ,$$ (here and below $n_1,\ldots ,n_k\ge 1$ in the internal sum), with coefficients from ${\overline{K}_c}$, such that $|\mu_i|\le M$, $M>0$, $|B_{jkl}|\le B^{kq^j}$, $B\ge 1$, $|a_i|\le M$ for all $i,j,k,l$; the number $\lambda$ is either equal to 1, or $\lambda =0$, and in that case $|B_{01l}|\le 1$. For an arbitrary element $c_1\in {\overline{K}_c}$, the sequence determined by the relation (3) satisfies the estimate $|c_n|\le C^{q^n}$, $n=1,2,\ldots$, with some constant $C\ge 1$. [*Proof*]{}. Set $c_n=\sigma d_n$, $|\sigma |<1$, $n=1,2,\ldots$, and substitute this into (3). We have $$\begin{gathered} d_{i+1}=\mu_i\Biggl\{ \sum\limits_{\genfrac{}{}{0pt}{}{j+l=i}{l\ne 0}} \sum\limits_{k=1}^\infty B_{jkl}\sum\limits_{n_1+\cdots +n_k=l} \sigma^{\left( 1+q^{n_1}+\cdots +q^{n_1+\cdots +n_{k-1}}\right)^{q^{j+\lambda }}-1}\\ \times \left( d_{n_1}d_{n_2}^{q^{n_1}}\cdots d_{n_k}^{q^{n_1+\cdots +n_{k-1}}}\right)^{q^{j+\lambda}}\Biggr\} +\sigma^{-1}a_i.\end{gathered}$$ Here $$\left| \sigma^{\left(1+q^{n_1}+\cdots +q^{n_1+\cdots +n_{k-1}}\right)^{q^{j+\lambda }}-1}\right| \le |\sigma|^{kq^{j+\lambda }-1},$$ and (under our assumptions) choosing such $\sigma$ that $|\sigma |$ is small enough we reduce (3) to the relation $$d_{i+1}=\mu_i\sum\limits_{\genfrac{}{}{0pt}{}{j+l=i}{l\ne 0}} \sum\limits_{k=1}^\infty b_{jkl}\sum\limits_{n_1+\cdots +n_k=l}\left( d_{n_1}d_{n_2}^{q^{n_1}}\cdots d_{n_k}^{q^{n_1+\cdots +n_{k-1}}}\right)^{q^{j+\lambda}}+\sigma^{-1}a_i,\quad i=1,2,\ldots ,$$ where $|b_{jkl}|\le 1$. It follows from (4) that $$|d_{i+1}|\le M\max\limits_{\genfrac{}{}{0pt}{}{j+l=i}{l\ne 0}} \sup\limits_{k\ge 1}\max\limits_{n_1+\cdots +n_k=l}\max \Biggl\{ \left( |d_{n_1}|\cdot |d_{n_2}^{q^{n_1}}|\cdots |d_{n_k}|^{q^{n_1+\cdots +n_{k-1}}}\right)^{q^{j+\lambda}},M^{-1}\left| \sigma^{-1}a_i\right| \Biggr\} .$$ Let $B=\max \left\{ 1,M,|d_1|,M^{-1}\sup\limits_i|\sigma^{-1}a_i|\right\}$. Let us show that $$|d_n|\le B^{q^{n-1}+q^{n-2}+\cdots +1},\quad n=1,2,\ldots .$$ This is obvious for $n=1$. Suppose that we have proved (5) for $n\le i$. Then $$\begin{gathered} |d_{i+1}|\le M\max\limits_{j+l=i} \sup\limits_{k\ge 1}\max\limits_{n_1+\cdots +n_k=l}\left( B^{q^{n_1-1}+q^{n_1-2}+\cdots +1}\cdot B^{q^{n_1+n_2-1}+q^{n_1+n_2-2}+\cdots +q^{n_1}}\cdots \right. \\ \left. \times \cdots B^{q^{n_1+\cdots +n_{k-1}+n_k-1}+q^{n_1+\cdots +n_{k-1}+n_k-2}+\cdots +1}+q^{n_1+\cdots +n_{k-1}}\right)^{q^{j+1}} \le M\max\limits_{j+l=i}B^{q^{j+l}+\cdots +q^{j+1}}\\ \le B\cdot B^{q^i+q^{i-1}+\cdots +q}=B^{q^i+q^{i-1}+\cdots +1},\end{gathered}$$ and we have proved (5). Therefore $$|c_n|\le |\sigma |B^{\frac{q^n-1}{q-1}}\le C^{q^n}$$ for some $C$, as desired. $\qquad \blacksquare$ Implicit functions of algebraic type ==================================== In this section we look for ${\mathbb F_q}$-linear locally holomorphic solutions of equations of the form $$P_0(t)+P_1(t)\circ z+P_2(t)\circ (z\circ z)+\cdots +P_N(t)\circ \underbrace{(z\circ z\circ \cdots \circ z)}_N=0$$ where $P_0,P_1,\ldots P_N\in {\mathcal R_{{\overline{K}_c}}}$. Suppose that the coefficient $P_k(t)=\sum\limits_{j\ge 0}a_{jk}t^{q^j}$ is such that $a_{00}=0$, $a_{01}\ne 0$; these assumptions are similar to the ones guaranteeing the existence and uniqueness of a solution in the classical complex analysis. Then (see Sect. 2) $P_1$ is invertible in ${\mathcal R_{{\overline{K}_c}}}$, and we can rewrite (6) in the form $$z+Q_2(t)\circ (z\circ z)+\cdots +Q_N(t)\circ \underbrace{(z\circ z\circ \cdots \circ z)}_N=Q_0(t)$$ where $Q_0,Q_2,\ldots ,Q_N\in {\mathcal R_{{\overline{K}_c}}}$, that is $$Q_k(t)=\sum\limits_{j=0}^\infty b_{jk}t^{q^j},\quad |b_{jk}|\le B_k^{q^j},$$ for some constants $B_k>0$, and $b_{00}=0$. The equation (6) has a unique solution $z\in {\mathcal R_{{\overline{K}_c}}}$ satisfying the “initial condition” $$\frac{z(t)}t\longrightarrow 0,\quad t\to 0.$$ . Let us look for a solution of the transformed equation (7), of the form $$z(t)=\sum\limits_{i=1}^\infty c_it^{q^i},\quad c_i\in {\overline{K}_c};$$ our initial condition is automatically satisfied for a function (8). Substituting (8) into (7) we come to the system of equalities $$c_i=-\sum\limits_{k=2}^N\sum\limits_{\genfrac{}{}{0pt}{}{j+l=i}{j\ge 0,l\ge 1}} b_{jk}\left( \sum\limits_{\genfrac{}{}{0pt}{}{n_1+\cdots +n_k=l}{n_j\ge 1}} c_{n_1}c_{n_2}^{q^{n_1}}\cdots c_{n_k}^{q^{n_1+\cdots +n_{k-1}}}\right)^{q^j}+b_{i0},\quad i\ge 1.$$ In each of them the right-hand side depends only on $c_1,\ldots ,c_{i-1}$, so that the relations (9) determine the coefficients of a solution (8) uniquely. By Proposition 3, $z\in {\mathcal R_{{\overline{K}_c}}}$. $\qquad \blacksquare$ More generally, let $$P_1(t)=\sum\limits_{j\ge \nu}a_{j1}t^{q^j},\quad \nu \ge 0,\quad a_{\nu 1}\ne 0.$$ Then the equation (6) has a unique solution in ${\mathcal R_{{\overline{K}_c}}}$, of the form $$z(t)=\sum\limits_{i=\nu +1}^\infty c_it^{q^i},\quad c_i\in {\overline{K}_c}.$$ The proof is similar. Equations with Carlitz derivatives ================================== Let us consider the equation $$dz(t)=\sum\limits_{j=0}^\infty \sum\limits_{k=1}^\infty a_{jk}\tau^j\underbrace{(z\circ z\circ \cdots \circ z)}_k(t)+\sum\limits_{j=0}^\infty a_{j0}t^{q^j}$$ where $a_{jk}\in {\overline{K}_c}$, $|a_{jk}|\le A^{kq^j}$ ($k\ge 1$), $|a_{j0}|\le A^{q^j}$, $A\ge 1$. We look for a solution in the class of ${\mathbb F_q}$-linear locally holomorphic functions of the form $$z(t)=\sum\limits_{k=1}^\infty c_kt^{q^k},\quad c_k\in {\overline{K}_c},$$ thus assuming the initial condition $t^{-1}z(t)\to 0$, as $t\to 0$. A solution (11) of the equation (10) exists with a non-zero radius of convergence, and is unique. . We may assume that $$|a_{j0}|\le 1,\quad a_{j0}\to 0,\quad \text{as $j\to \infty$}.$$ Indeed, if that is not satisfied, we can perform a time change $t=\gamma t_1$ obtaining an equation of the same form, but with the coefficients $a_{j0}\gamma^{q^j}$ instead of $a_{j0}$, and it remains to choose $\gamma$ with $|\gamma |$ small enough. Note that, in contrast with the case of the usual derivatives, the operator $d$ commutes with the above time change. Assuming (12) we substitute (11) into (10) using the fact that $d\left( c_kt^{q^k}\right) =c_k^{1/q}[k]^{1/q}t^{q^{k-1}}$, $k\ge 1$, where $[k]=x^{q^k}-x$. Comparing the coefficients we come to the recursion $$c_{i+1}=[i+1]^{-1}\sum\limits_{\genfrac{}{}{0pt}{}{j+l=i}{j\ge 0,l\ge 1}}\sum\limits_{k=1}^\infty a_{jk}^q\left( \sum\limits_{n_1+\cdots +n_k=l} c_{n_1}c_{n_2}^{q^{n_1}}\cdots c_{n_k}^{q^{n_1+\cdots +n_{k-1}}}\right)^{q^{j+1}}+a_{i0},\quad i\ge 1,$$ where $c_1=[1]^{-1}a_{00}^q$. This already shows the uniqueness of a solution. The fact that $|c_i|\le C^{q^i}$ for some $C$ follows from Proposition 3. $\qquad \blacksquare$ Using Proposition 4 we can easily reduce to the form (10) some classes of equations given in the form not resolved with respect to $dz$. As in the classical case of equations over $\mathbb C$ (see [@Hi]), some of equations (10) can have also non-holomorphic solutions, in particular those which are meromorphic in the sense of Sect. 2. As an example, we consider Riccati-type equations $$dy(t)=\lambda (y\circ y)(t)+(P(\tau )y)(t)+R(t)$$ where $\lambda \in {\overline{K}_c}$, $0<|\lambda |\le q^{-1/q^2}$, $$(P(\tau )y)(t)=\sum\limits_{k=1}^\infty p_ky^{q^k}(t),\quad R(t)=\sum\limits_{k=0}^\infty r_kt^{q^k},$$ $p_k,r_k\in {\overline{K}_c}$, $|p_k|\le q^{-1/q^2}$, $|r_k|\le q^{-1/q^2}$ for all $k$. Under the above assumptions, the equation (13) possesses solutions of the form $$y(t)=ct^{1/q}+\sum\limits_{n=0}^\infty a_nt^{q^n},\quad c,a_n\in {\overline{K}_c},\ c\ne 0,$$ where the series converges on the open unit disk $|t|<1$. . For the function (14) we have $$dy(t)=c^{1/q}[-1]^{1/q}t^{q^{-2}}+\sum\limits_{n=1}^\infty a_n^{1/q}[n]^{1/q}t^{q^{n-1}},\quad [-1]=x^{1/q}-x,$$ $$\begin{gathered} (y\circ y)(t)=c\left( ct^{1/q}+\sum\limits_{n=0}^\infty a_nt^{q^n}\right)^{1/q}+\sum\limits_{n=0}^\infty a_n\left( ct^{1/q}+\sum\limits_{m=0}^\infty a_mt^{q^m}\right)^{q^n}\\ =c^{1+\frac{1}q}t^{q^{-2}}+\left( ca_0^{1/q}+ca_0\right) t^{q^{-1}}+\sum\limits_{n=0}^\infty \left( ca_{n+1}^{1/q}+c^{q^{n+1}}a_{n+1}\right) t^{q^n} +\sum\limits_{l=0}^\infty t^{q^l}\sum\limits_{\genfrac{}{}{0pt}{}{m+n=l}{m,n\ge 0}}a_na_m^{q^n}.\end{gathered}$$ Finally, $$(P(\tau )y)(t)=\sum\limits_{k=0}^\infty p_{k+1}c^{q^{k+1}}t^{q^k} +\sum\limits_{l=0}^\infty t^{q^l}\sum\limits_{\genfrac{}{}{0pt}{}{i+j=l}{i\ge 1,j\ge 0}}p_ia_j^{q^i}.$$ Comparing the coefficients we find that $$c=\lambda^{-1}[-1]^{1/q},\quad a_0^{1/q}+a_0=0,$$ $$a_{l+1}^{1/q}([l+1]^{1/q}-\lambda c)-\lambda c^{q^{l+1}}a_{l+1}=\lambda \sum\limits_{\genfrac{}{}{0pt}{}{m+n=l}{m,n\ge 0}}a_na_m^{q^n}+\sum\limits_{\genfrac{}{}{0pt}{}{i+j=l}{i\ge 1,j\ge 0}}p_ia_j^{q^i}+r_l,\quad l\ge 0.$$ By (15), we have $|c|\ge 1$, and either $a_0=0$, or $|a_0|=1$. Next, (16) is a recurrence relation (with an algebraic equation to be solved at each step) giving values of $a_l$ for all $l\ge 1$. Let us prove that $|a_j|\le 1$ for all $j$. Suppose we have proved that for $j\le l$. It follows from (16) that $$\left| a_{l+1}[l+1]-\lambda^qc^qa_{l+1}-\lambda^qc^{q^{l+2}}a_{l+1}^q\right| \le q^{-1/q}.$$ Suppose that $|a_{l+1}|>1$. We have $\lambda^qc^q=[-1]$, so that $|\lambda^qc^q|=q^{-1/q}$, and since $|[l+1]|=q^{-1}$ and $|c|\ge 1$, we find that $$\left| a_{l+1}[l+1]\right| <\left| \lambda^qc^qa_{l+1}\right| <\left| \lambda^qc^{q^{l+2}}a_{l+1}^q\right| .$$ Therefore the left-hand side of (17) equals $\left| \lambda^qc^q\right| \cdot \left| c^{q^{l+1}}\right| \cdot \left| a_{l+1}^q\right| >q^{-1/q}$, and we have come to a contradiction. $\qquad \blacksquare$ [999]{} L. Carlitz, On certain functions connected with polynomials in a Galois field, [*Duke Math. J.*]{} [**1**]{} (1935), 137–168. L. Carlitz, Some special functions over $GF(q,x)$, [*Duke Math. J.*]{} [**27**]{} (1960), 139–158. D. Goss, Fourier series, measures, and divided power series in the theory of function fields, [*K-Theory*]{} [**1**]{} (1989), 533–555. I. N. Herstein, [*Noncommutative Rings*]{}, The Carus Math. Monograph No. 15, Math. Assoc. of America, J. Wiley and Sons, 1968. E. Hille, [*Lectures on Ordinary Differential Equations*]{}, Addison-Wesley, Reading, 1969. A. N. Kochubei, ${\mathbb F_q}$-linear calculus over function fields, [*J. Number Theory*]{} [**76**]{} (1999), 281–300. A. N. Kochubei, Differential equations for ${\mathbb F_q}$-linear functions, [*J. Number Theory*]{} [**83**]{} (2000), 137–154. A. N. Kochubei, Differential equations for ${\mathbb F_q}$-linear functions, II: Regular singularity, [*Finite Fields Appl.*]{} [**9**]{} (2003), 250–266. R. S. Pierce, [*Associative Algebras*]{}, Springer, New York, 1982. B. Poonen, Fractional power series and pairings on Drinfeld modules, [*J. Amer. Math. Soc.*]{} [**9**]{} (1996), 783–812. D. Thakur, Hypergeometric functions for function fields II, [*J. Ramanujan Math. Soc.*]{} [**15**]{} (2000), 43–52. C. G. Wagner, Linear operators in local fields of prime characteristic, [*J. Reine Angew. Math.*]{} [**251**]{} (1971), 153–160. [^1]: This research was supported in part by CRDF under Grants UM1-2421-KV-02 and UM1-2567-OD-03.

--- abstract: 'In this paper we study homeomorphisms of the circle with several critical points and bounded type rotation number. We prove complex a priori bounds for these maps. As an application, we get that bi-cubic circle maps with same bounded type rotation number are $C^{1+\alpha}$ rigid.' address: - 'Instituto de Matemática, Universidade Federal do Rio de Janeiro' - 'Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo' - 'Department of Mathematics, University of Toronto' author: - Gabriela Estevez - Daniel Smania - Michael Yampolsky title: Complex a priori bounds for multicritical circle maps with bounded type rotation number --- Introduction ============ Complex [*a priori*]{} bounds have emerged as a key analytic tool in one-dimensional dynamics. They provide the analytic foundation for the results in one-dimensional Renormalization theory, rigidity, density of hyperbolicity, and local connectivity of Julia sets and the Mandelbrot set. Speaking informally, they are the bounds on the size of the domains of the analytic continuations of the first return maps corresponding to renormalizations of one-dimensional dynamical systems. In this paper we prove complex [*a priori*]{} bounds for multicritical circle maps with rotation numbers of bounded type. This generalizes the results of [@Yam2019], where they were obtained under the assumption that the rotation number is a quadratic irrational, which is a particular case of bounded type. Similarly to [@Yam2019], we apply the bounds to the case of bi-cubic circle maps, and prove that such maps with irrational rotation numbers of bounded type are $C^{1+\alpha}$-rigid: that is, a topological conjugacy which maps critical points to critical points must be $C^{1+\alpha}$-regular. Preliminaries ============= We will refer to the affine manifold ${{\Bbb T}}={{\Bbb R}}/{{\Bbb Z}}$ as the circle, and will identify it as needed with the unit circle $S^1$ via the exponential map $x\mapsto e^{2\pi i x}$. For a homeomorphism $f:{{\Bbb T}}\to{{\Bbb T}}$ we will denote by $\rho(f)\in(0,1)$ its rotation number. For $\alpha>0$, we let $$G(\alpha)\equiv \left\{\frac{1}{\alpha}\right\}$$ be the Gauss map. Starting with $\alpha\in(0,1)$ we consider the orbit $$\alpha_0\equiv \alpha, \, \dots \,, \alpha_{n}=G(\alpha_{n-1}) \, ,\dots$$ It is finite if and only if $\alpha$ is rational, in which case we will end it at the last non-zero term. The numbers $a_n=[1/\alpha_n]$ for $n\geq 0$ are the coefficients of the continued fraction expansion of $\alpha$ with positive terms (which is defined uniquely if and only if $\alpha\notin{{\Bbb Q}}$). We will denote such continued fraction as $$\alpha=[a_0,a_1,\ldots].$$ We say that $\alpha\in[0,1]\setminus{{\Bbb Q}}$ is of a [*type bounded by* ]{}$K\in{{\Bbb N}}$ if $\sup a_i\leq K$. We will refer to the union of such numbers for all $K\in{{\Bbb N}}$ as irrationals of [*bounded type*]{}, note that this class coincides with Diophantine numbers of order $2$. We will let $R_\alpha(x)\equiv x+\alpha \,{\operatorname{mod}}{{\Bbb Z}}$ denote the rigid rotation by angle $\alpha$. Given two positive numbers $a$, $b$ we say that they are $C$-commensurable for $C>1$, and we denote it by $a \asymp_C b$, if $$\frac{1}{C}\leq \frac{a}{b}\leq C.$$ We will say that $a$ and $b$ are universally commensurable, or simply commensurable, if the constant $C$ is universal. In that case we denote it by $a \asymp b$. Two sets in the plane are $C$-commensurable if their diameters are $C$-commensurable. We will use ${\operatorname{diam}}(A)$ to denote the Euclidean diameter of a bounded set $A\subset {{\Bbb C}}$. We let ${{\Bbb D}}_r(z)$ be the open disk of radius $r$ centered at $z\in{{\Bbb C}}$; ${{\Bbb D}}$ will stand for the unit disk. $$\displaystyle U_r(A)=\cup_{z\in A}{{\Bbb D}}_r(z)$$ will stand for the $r$-neighborhood of a set $A$. We denote $${\operatorname{dist}}(A,B)=\inf\{r>0\;|\; U_r(A)\cap B\neq \emptyset\}$$ the Euclidean distance between $A$ and $B$. ![\[fig:poincare\]A Poincaré neighborhood $D_\theta(J)$.](poincare.eps){width="77.00000%"} Given $J=(a,b)$, a subinterval in the real line, let ${\ensuremath{\mathbb{C}}}_{J} =({\ensuremath{\mathbb{C}}}\setminus {\ensuremath{\mathbb{R}}}) \cup J$. Following Sullivan [@Su], we let the [*P*oincaré neighborhood]{} of $J$ of radius $r>0$ to be the set of points in ${\ensuremath{\mathbb{C}}}_{J}$ such that their hyperbolic distance in ${\ensuremath{\mathbb{C}}}_J$ to $J$ is less or equal to $r$. A Poincaré neighborhood is an ${{\Bbb R}}$-symmetric union of two Euclidean disks with a common chord $J$. If we denote the external angle between one of the boundary circles of such a neighborhood with ${{\Bbb R}}$ by $\theta\in (0,\pi)$, then $$r=r(\theta)=\log \cot (\theta/4).$$ It is more convenient for us to identify a Poincaré neighborhood by the external angle $\theta$ rather than the hyperbolic radius $r$, so we will use the notation $D_\theta(J)$ (see Figure \[fig:poincare\]). Clearly, if $\theta_1<\theta_2$ then $D_{\theta_2}(J) \subset D_{\theta_1}(J)$ and so $r(\theta_2) <r(\theta_1)$. We let $D_{\pi/2}(J)\equiv D(J)$; this is the Euclidean disk with diameter $J$. We note that $$\label{eq:diamD} \displaystyle {\operatorname{diam}}(D_{\theta}(J))= \left(\frac{1+\cos \theta}{\sin \theta} \right)|J|.$$ Multicritical circle maps ------------------------- We say that $f$ is a $C^{3}$ multicritical circle map if it is a $C^3$ orientation preserving circle homeomorphism with a finite number of [*non-flat*]{} critical points. That means that for each critical point $c$ there exist $d \in 2{\ensuremath{\mathbb{N}}}+1$ ($d$ is called the criticality of $c$), a neighbourhood $W$ of $c$ and an orientation preserving diffeomorphism $\phi$ satisfying $\phi(c)=0$ such that for all $x \in W$, $$f(x)=f(c)+\big(\phi(x)\big)^{d}.$$ In the space of analytic maps, we say that an analytic multicritical circle map is just an analytic orientation preserving circle homeomorphism with a finite number of critical points. We assume that the rotation number of $f$ is irrational. By a result of Yoccoz [@Yoc1984], this implies that $f$ is topologically conjugate with the rigid rotation by the angle $\rho(f)$. This clearly implies the existence of a unique ergodic $f$-invariant measure, which is the pullback of the Lebesgue measure by the conjugacy. We denote this measure by $\mu_f$. We define the *signature* of a multicritical circle map $f$ to be the $(2N+2)$-tuple $$(\rho(f) \,;N;\,d_0,d_1,\ldots,d_{N-1};\,\delta_0,\delta_1,\ldots,\delta_{N-1}),$$ where $N$ is the number of critical points, $\rho(f)$ is the rotation number of $f$, $d_i$ is the criticality of the critical point $c_i$, and $\delta_i=\mu_f[c_i,c_{i+1})$ (with the convention that $c_{N}=c_0$). Dynamical partitions -------------------- Let $f$ be a multicritical circle map with irrational rotation number $\rho(f)$, and continued fraction expansion given by $\rho(f) \;=\; [a_{0} , a_{1} , \cdots ]$. Let us consider the continued fraction convergents $p_n/q_n$ obtained by truncating the expansion at level $n-1$, that is, $p_n/q_n\;=\;[a_0,a_1, \cdots ,a_{n-1}]$. The sequence of denominators $\{q_n\}_{n\in{\ensuremath{\mathbb{N}}}}$ satisfies the recursive formula $$q_{0}=1, \hspace{0.4cm} q_{1}=a_{0}, \hspace{0.4cm} q_{n+1}=a_{n}\,q_{n}+q_{n-1} \hspace{0.3cm} \text{for all $n \geq 1$} .$$ Moreover, for $x \in S^1$ and $n \in {\ensuremath{\mathbb{N}}}$, the iterates $\{f^{q_n}(x)\}$ are closest returns of $x$ in the following sense: denote $I_n(x)=[x,f^{q_n}(x)]$ the arc of the circle connecting these two points and not containing $f^{q_{n+1}}(x)$. Then $[x,f^{q_n}(x)]$ does not contain any iterates smaller than $q_n$ in the orbit of $x$. For $a,b\in{{\Bbb T}}$ we will denote $[a,b]$ the arc of the circle obtained as $\psi^{-1}(A)$ where $\psi$ is a conjugacy between $f$ and $R_{\rho(f)}$ and $A$ is the shorter of the two arcs connecting $\psi(a)$ with $\psi(b)$. The collection of intervals $$\mathcal{P}_n(x) \ = \ \left\{ f^{i}(I_n(x)):\;0\leq i\leq q_{n+1}-1 \right\} \;\bigcup\; \left\{ f^{j}(I_{n+1}(x)):\;0\leq j\leq q_{n}-1 \right\}$$ is a partition of the circle by closed intervals intersecting only at their endpoints. It is called the [*$n$-th dynamical partition*]{} associated to the point $x$ (see [@dMvS Section 1.1, Lemma 1.3, page 26] or [@EdFG Appendix]). For each $n \in {\ensuremath{\mathbb{N}}}$, we will refer to the intervals $I_n(x)$ and $I_{n+1}(x)$ as [*fundamental intervals*]{} of the dynamical partition $\mathcal{P}_n(x)$. The dynamical partitions $\mathcal{P}_n(x)$ form a sequence of (non-strict) refinements: the intervals $I_n^j(x)$ for $0\leq j\leq q_{n+1}-1$ are subdivided by exactly $a_{n+1}$ intervals belonging to $\mathcal{P}_{n+1}(x)$ while the intervals $I_{n+1}^i(x)$ for each $0\leq i\leq q_n-1$ remain invariant, see Figure \[fig:partition\] below. \[\]\[\]\[1\][$x$]{} \[\]\[\]\[1\][$f^{q_{n+1}}(x)$]{} \[\]\[\]\[1\][$f^{q_n}(x)$]{} \[\]\[\]\[1\][$\mathcal{P}_n(x)$]{} \[\]\[\]\[1\][$\mathcal{P}_{n+1}(x)$]{} \[\]\[\]\[1\][$I_n$]{} \[\]\[\]\[1\][$I_{n+1}$]{} \[\]\[\]\[1\][$I_{n+2}$]{} \[\]\[\]\[1\][$I_{n+1}^{q_n}$]{} \[\]\[\]\[1\][$I_{n+1}^{q_n+q_{n+1}}$]{} \[\]\[\]\[1\][$\cdots$]{} ![\[fig:partition\] Two consecutive dynamical partitions.](partition.eps "fig:"){width="4.1in"} Following the convention introduced by Sullivan [@Su], we say that for a map $f$ a quantity is “beau” (which translates as “bounded and eventually universally (bounded)”), if it is bounded and the bound becomes universal (that is, independent of the map). Let $c \in {\operatorname{Crit}}(f)$, from now on we will consider $n$ bigger enough such that the adjacent intervals $I_{n+1}(c)$ and $I_n(c)$, do not contain any other critical point of $f$. The following fundamental geometric control was obtained by Herman [@H] and Światek [@G] in the 1980’s. A detailed proof of Theorem \[teobeau\] can be found in [@EdFG]. \[teobeau\] Let $f$ be a multicritical circle map with irrational rotation number and $N$ critical points, and let $c$ be any of its critical points. Then there exists $n_0 \in \mathbb{N}$ such that for all $n\geq n_0$ the iterate $f^{q_{n+1}}|_{I_n(c)}$ is decomposed as $$f^{q_{n+1}}|_{I_n(c)}= \psi_{m+1} \circ p_m \circ \psi_m\circ p_{m-1} \circ \dots \circ \psi_1 \circ p_0 \circ \psi_0,$$ where $m \leq N +1$, $p_j(x)=x^{d_j}$ for $d_j$ an odd integer, and each $\psi_j$ is an interval diffeomorphism with beau distortion. An immediate corollary of Theorem \[teobeau\] is the following result: \[corollarybeau\] Given $N \in {\ensuremath{\mathbb{N}}}$ and $d>1$ let $\mathcal{F}_{N,d}$ be the family of multicritical circle maps with at most $N$ critical points whose maximum criticality is bounded by $d$. There exists a beau constant $C=C(N,d)>1$ with the following property: for any given $f \in \mathcal{F}_{N,d}$ and $c \in {\operatorname{Crit}}(f)$ there exists $n_0 \in \mathbb{N}$ such that for all $n\geq n_0$ each pair of adjacent intervals $I,J \in \mathcal{P}_{n}(c)$ is $C$-commensurable. Renormalization of multicritical circle maps -------------------------------------------- In this section, we recall the notion of *multicritical commuting pair*, which is a generalization of *critical commuting pair* introduced in [@ORSS]. This notion will let us to define the renormalization of a multicritical circle map. \[multcommpairs\] A $C^3$ (or $C^{\infty}$) multicritical commuting pair is a pair $\zeta=(\eta,\xi)$ consisting of two $C^3$ orientation preserving interval homeomorphisms $\xi:I_\xi \rightarrow \xi(I_\xi)$ and $\eta : I_\eta \rightarrow \eta(I_{\eta})$ satisfying: 1. $I_{\xi}=[\eta(0), 0]$ and $I_{\eta}=[0, \xi(0)]$ are compact intervals in the real line; 2. the origin has odd integer criticality for $\eta$ and for $\xi$; 3. $\xi$ and $\eta$ satisfy the commuting property: $(\eta \circ \xi)(0)=(\xi \circ \eta)(0) \neq 0$; 4. $\xi$ and $\eta$, contain others critical points (with odd integers criticalities) in theirs domains, $I_\xi$ and $I_\eta$; 5. both $\xi$ and $\eta$, have homeomorphic extensions to some interval neighborhoods $V_{\xi}$ and $V_{\eta}$, of $I_\xi$ and $I_\eta$, with same smoothness $C^3$ (or $C^{\infty}$) preserving the commuting property. Let $f$ be a $C^r$ multicritical circle map with irrational rotation number $\rho(f)$ and critical points $c_{0}, \dots, c_{N-1}$. For each critical point $c_j$, we can define a multicritical commuting pair in the following way: let $\widehat{f}$ be the lift of $f$ (under the universal covering $t \mapsto c_j\cdot\exp(2\pi i t)$) such that $0< \widehat{f}(0)<1$ (and note that $D\widehat{f}(0)=0$). For $n\geq 1$, let $\widehat{I}_{n}(c_j)$ be the closed interval in ${\ensuremath{\mathbb{R}}}$, containing the origin as one of its extreme points, which is projected onto $I_{n}(c_j)$. We define $\xi: \widehat{I}_{n+1}(c_j) \rightarrow {\ensuremath{\mathbb{R}}}$ and $\eta: \widehat{I}_{n}(c_j) \rightarrow {\ensuremath{\mathbb{R}}}$ by $\xi= T^{-p_{n}}\circ \widehat{f}^{q_{n}}$ and $\eta= T^{-p_{n+1}}\circ \widehat{f}^{q_{n+1}}$, where $T$ is the unit translation $T(x)=x+1$. Then the pair $(\eta|_{\widehat{I}_{n}(c_j)}, \xi|_{\widehat{I}_{n+1}(c_j)})$ is a multicritical commuting pair, that we denote by $(f^{q_{n+1}}|_{I_n(c_j)}, f^{q_n}|_{I_{n+1}(c_j)})$.\ We restrict our attention to *normalized* multicritical commuting pairs: for any given pair $\zeta=(\eta,\xi)$ we denote by $\widetilde{\zeta}$ the pair $(\widetilde{\eta}|_{\widetilde{I_{\eta}}}, \widetilde{\xi}|_{\widetilde{I_{\xi}}})$, where tilde means linear rescaling by the factor $1/|I_{\xi}|$. Note that $|\widetilde{I_{\xi}}|=1$ and $\widetilde{I_{\eta}}$ has length equal to the ratio between the lengths of $I_{\eta}$ and $I_{\xi}$. Equivalently $\widetilde{\eta}(0)=-1$ and $\widetilde{\xi}(0)=|I_{\eta}|/|I_{\xi}|=\xi(0)/\big|\eta(0)\big|$. We define the *height* of the pair $\zeta=(\eta, \xi)$ as the natural number $a$ such that$$\eta^{a+1}(\xi(0)) <0\leq \eta^{a}(\xi(0)),$$when such number exists, and we denote it by $\chi(\zeta)$. If such $a$ does not exist, that is, when $\eta$ has a fixed point, we define $\chi(\zeta)=\infty$. \[defren\] Let $\zeta=(\eta, \xi)$ be a multicritical commuting pair with $(\xi \circ \eta)(0) \in I_{\eta}$ and $\chi(\zeta)=a< \infty$. We define the *pre-renormalization* of $\zeta$ as the pair$$p\mathcal{R}(\zeta) =(\eta|_{[0, \eta^{a}(\xi(0)) ]} \ , \ \eta^{a}\circ \xi|_{I_{\xi}} ).$$ Moreover, we define the *renormalization* of $\zeta$ as the normalization of $p \mathcal{R}(\zeta)$:$$\mathcal{R}(\zeta)= \left(\widetilde{\eta}|_{[0,\widetilde{\eta^{a}(\xi(0))} ]} \ , \ \widetilde{\eta^{a}\circ \xi}|_{\widetilde{I}_{\xi}} \right).$$ If $\zeta$ is a multicritical commuting pair with $\chi(\mathcal{R}^{j}\zeta)< \infty$ for $0 \leq j \leq n-1$, we say that $\zeta$ is *$n$-times renormalizable*, otherwise, if $\chi(\mathcal{R}^{j}\zeta)< \infty$ for all $j \in {\ensuremath{\mathbb{N}}}$, we say that $\zeta$ is *infinitely renormalizable*. In the last case, we define the *rotation number* of the multicritical commuting pair $\zeta$, and denote it by $\rho(\zeta)$, as the irrational number whose continued fraction expansion is given by$$[\chi(\zeta), \chi(\mathcal{R}\zeta), \cdots, \chi(\mathcal{R}^{n}\zeta), \cdots ].$$ Its normalization will be denoted by $\mathcal{R}_i^{n}f$, that is:$$\mathcal{R}_i^{n}f=\left(\widetilde{f}^{q_{n+1}}|_{\widetilde{I_n}(c_i)}, \widetilde{f}^{q_{n}}|_{\widetilde{I_{n+1}}(c_i)}\right).$$ Observe that $\rho(\mathcal{R}(\zeta)))=G(\rho(\zeta)))$, where $G$ is the Gauss map. Complex [*a priori*]{} bounds {#sec:complexbounds} ============================= Holomorphic commuting pairs --------------------------- We recall the definition of a holomorphic commuting pair given in [@Yam2019], which generalizes the orginal definition of de Faria [@dF2]. \[def:hcp\] Given an analytic multicritical commuting pair $\zeta=(\eta|_{I_{\eta}}, \xi_{I_{\xi}})$, we say that it extends to a *holomorphic commuting pair* $\mathcal{H}$, if there exist three simply-connected and ${\ensuremath{\mathbb{R}}}-$symmetric domains $D,U,V\subseteq {\ensuremath{\mathbb{C}}}$, whose intersections with the real line are denoted by $I_U=U \cap {\ensuremath{\mathbb{R}}}$, $I_V=V \cap {\ensuremath{\mathbb{R}}}$ and $I_D=D \cap {\ensuremath{\mathbb{R}}}$ and a simply connected ${\ensuremath{\mathbb{R}}}-$symmetric Jordan domain $\Delta$ that satisfy the following, 1. the endpoints of $I_U$ and $I_V$ are critical points of $\eta$ and $\xi$, respectively; 2. $\overline{D}, \overline{U}, \overline{V}$ are contained in $\Delta$; $\overline{U}\cap \overline{V} = \{0\} \subseteq D$; the sets $U \setminus D, V\setminus D, D \setminus U$ and $D \setminus V$ are non-empty, connected and simply-connected; $I_{\eta} \subset I_U \cup \{0\}$, $I_{\xi}\subset I_V \cup \{0\}$; 3. $U \cap \mathbb{H}$, $V \cap \mathbb{H}$ and $D \cap \mathbb{H}$ are Jordan domains; 4. the maps $\eta$ and $\xi$ have analytic extensions to $U$ and $V$, respectively, so that $\eta$ is a branched covering map of $U$ onto $(\Delta \setminus {\ensuremath{\mathbb{R}}}) \cup \eta(I_U)$, and $\xi$ is a branched covering map of $V$ onto $(\Delta \setminus {\ensuremath{\mathbb{R}}})\cup \xi(I_V)$, with all the critical points of both maps contained in the real line; 5. the maps $\eta:U \to \Delta$ and $\xi: V \to \Delta$ can be extended to analytic maps $\widehat{\eta}:U \cup D \to \Delta$ and $\widehat{\xi}:V \cup D \to \Delta$, so that the map $\nu=\widehat{\eta}\circ \widehat{\xi}= \widehat{\xi}\circ \widehat{\eta}$ is defined in $D$ and is a branched covering of $D$ onto $(\Delta\setminus {\ensuremath{\mathbb{R}}})\cup \nu(I_D)$ with only real branched points. \[\]\[\]\[1\][$\Delta$]{} \[\]\[\]\[1\][$V$]{} \[\]\[\]\[1\][$U$]{} \[\]\[\][$D$]{} \[\]\[\]\[1\][$\xi$]{} \[\]\[\]\[1\][$\eta$]{} \[\]\[\]\[1\][$\nu$]{} \[\]\[\]\[1\][$I_{\xi}$]{} \[\]\[\][$I_{\eta}$]{} \[\]\[\][$I_{D}$]{} \[\]\[\][$I_{V}$]{} \[\]\[\][$I_{U}$]{} ![\[fig:hcp\] A holomorphic commuting pair.](hcp.eps "fig:"){width="4in"} We shall identify a holomorphic pair ${{\mathcal H}}$ with a triple of maps ${{\mathcal H}}=(\eta,\xi, \nu)$, where $\eta\colon U\to\Delta$, $\xi\colon V\to\Delta$ and $\nu\colon D\to\Delta$. We shall also call $\zeta$ the [*commuting pair underlying ${{\mathcal H}}$*]{}, and write $\zeta\equiv \zeta_{{\mathcal H}}$. When no confusion is possible, we will use the same letters $\eta$ and $\xi$ to denote both the maps of the commuting pair $\zeta_{{\mathcal H}}$ and their analytic extensions to the corresponding domains $U$ and $V$. The sets $\Omega_{{\mathcal H}}=D\cup U\cup V$ and $\Delta\equiv\Delta_{{\mathcal H}}$ will be called *the domain* and *the range* of a holomorphic pair ${{\mathcal H}}$. We will sometimes write $\Omega$ instead of $\Omega_{{\mathcal H}}$, when this does not cause any confusion. We can associate to a holomorphic pair ${{\mathcal H}}$ a piecewise defined map $S_{{\mathcal H}}\colon\Omega\to\Delta$: $$S_{{\mathcal H}}(z)=\begin{cases} \eta(z),&\text{ if } z\in U,\\ \xi(z),&\text{ if } z\in V,\\ \nu(z),&\text{ if } z\in\Omega\setminus(U\cup V). \end{cases}$$ De Faria [@dF2] calls $S_{{\mathcal H}}$ the [*shadow*]{} of the holomorphic pair ${{\mathcal H}}$. We can naturally view a holomorphic pair ${{\mathcal H}}$ as three triples $$(U,\xi(0),\eta),\;(V,\eta(0),\xi),\;(D,0,\nu).$$ We say that a sequence of holomorphic pairs converges in the sense of Carath[é]{}odory convergence, if the corresponding triples do. We denote the space of triples equipped with this notion of convergence by ${\Bbb H}$. We let the [*m*odulus of a holomorphic commuting pair]{} ${{\mathcal H}}$, which we denote by ${\operatorname{mod}}({{\mathcal H}})$ to be the modulus of the largest annulus $A\subset \Delta$, which separates ${{\Bbb C}}\setminus\Delta$ from $\overline\Omega$. \[H\_mu\_def\] For $\mu\in(0,1)$ let ${\Bbb H}(\mu)\subset{\Bbb H}$ denote the space of holomorphic commuting pairs ${{{\mathcal H}}}:\Omega_{{{{\mathcal H}}}}\to \Delta_{{{\mathcal H}}}$, with the following properties: 1. ${\operatorname{mod}}({{\mathcal H}})\ge\mu$; 2. [$|I_\eta|=1$, $|I_\xi|\ge\mu$]{} and $|\eta^{-1}(0)|\ge\mu$; 3. ${\operatorname{dist}}(\eta(0),\partial V_{{\mathcal H}})/{\operatorname{diam}}V_{{\mathcal H}}\ge\mu$ and ${\operatorname{dist}}(\xi(0),\partial U_{{\mathcal H}})/{\operatorname{diam}}U_{{\mathcal H}}\ge\mu$; 4. [the domains $\Delta_{{\mathcal H}}$, $U_{{\mathcal H}}\cap{{\Bbb H}}$, $V_{{\mathcal H}}\cap{{\Bbb H}}$ and $D_{{\mathcal H}}\cap{{\Bbb H}}$ are $(1/\mu)$-quasidisks.]{} 5. ${\operatorname{diam}}(\Delta_{{{\mathcal H}}})\le 1/\mu$; Let the [*degree*]{} of a holomorphic pair ${{\mathcal H}}$ denote the maximal topological degree of the covering maps constituting the pair. Denote by ${\Bbb H}^K(\mu)$ the subset of ${\Bbb H}(\mu)$ consisting of pairs whose degree is bounded by $K$. The following is an easy generalization of Lemma 2.17 of [@Yam3]: \[bounds compactness\] For each $K\geq 3$ and $\mu\in(0,1)$ the space ${\Bbb H}^K(\mu)$ is sequentially compact. We say that a real commuting pair $\zeta=(\eta,\xi)$ with an irrational rotation number has [*complex [a priori]{} bounds*]{}, if there exists $\mu>0$ such that all renormalizations of $\zeta=(\eta,\xi)$ extend to holomorphic commuting pairs in ${\Bbb H}(\mu)$. The existense of complex [*a priori*]{} bounds is a key analytic issue of renormalization theory. \[A\_r\_com\_pair\_def\] For $S\subset{{\Bbb C}}$ and $r>0$, we let $N_r(S)$ stand for the $r$-neighborhood of $S$ in ${{\Bbb C}}$. For each $r>0$ we introduce a class ${{\mathcal A}}_r$ consisting of pairs $(\eta,\xi)$ such that the following holds: - $\eta$, $\xi$ are real-symmetric analytic maps defined in the domains $$U_r([0,1])\text{ and }U_{r|\eta(0)|}([\eta(0),0])$$ respectively, and continuous up to the boundary of the corresponding domains; - the pair $$\zeta\equiv (\eta|_{[0,1]},\xi|_{[\eta(0),0]})$$ is a multicritical commuting pair. \[rem:domains\] For simplicity, if $\zeta$ is as above, we will write $\zeta\in{{\mathcal A}}_r$. But it is important to note that viewing our multicritical commuting pair $\zeta$ as an element of ${{\mathcal A}}_r$ imposes restrictions on where we are allowed to iterate it. Specifically, we view such $\zeta$ as undefined at any point $z\notin U_r([0,\xi(0)])\cup U_{r|\eta(0)|}([0,\eta(0)])$ (even if $\zeta$ can be analytically continued to $z$). Similarly, when we talk about iterates of $\zeta\in{{\mathcal A}}_r$ we iterate the restrictions $\eta|_{U_r([0,\xi(0)])}$ and $\xi|_{U_{r|\eta(0)|}([0,\eta(0)]}$. In particular, we say that the first and second elements of $p{{\mathcal R}}\zeta=(\eta^a\circ\xi,\eta)$ are defined in the maximal domains, where the corresponding iterates are defined in the above sense. Complex bounds for pairs of bounded type ---------------------------------------- We will denote ${{\mathcal A}}_r^L$ the subset of ${{\mathcal A}}_r$ consisting of pairs whose degree is bounded by $L$. The following statement generalizes Theorem 3.2 of [@Yam2019] to all pairs of bounded type. \[thm:bounds\] Let $L\geq 3$, $B\in{{\Bbb N}}$. There exists a constant $\mu>0$ such that the following holds. For every positive real number $r>0$ and every pre-compact family $S\subset\mathcal A_r^L$ of multicritical commuting pairs, there exists $N=N(r, S)\in{{\Bbb N}}$ such that if $\zeta\in S$ is a commuting pair whose rotation number $\rho(\zeta)$ is of type bounded by $B$ then $p{{\mathcal R}}^n\zeta$ restricts to a holomorphic commuting pair ${{\mathcal H}}_n:\Omega_n\to\Delta_n$ with $\Delta_n\subset U_r(I_\eta)\cup U_r(I_\xi)$, for all $n\geq N$. Furthermore, the range $\Delta_n$ is a Euclidean disk, and the appropriate affine rescaling of ${{\mathcal H}}_n$ is in ${\Bbb H}(\mu)$. Below we will give a proof of this theorem which generalizes the proof in [@Yam1999] and strengthens the proof in [@Yam2019]. For simplicity of notation, we will assume that the pair $\zeta$ is a pre-renormalization of a multicritical circle map $f$, that is $$\zeta=(f^{q_n},f^{q_{n+1}}).$$ This will allow us to write explicit formulas for long compositions of terms $\eta$ and $\xi$, which will greatly streamline the exposition. Theorem \[thm:bounds\] follows from Theorem \[maintheorem\] which we state below. Let us further assume that $f$ has $N$ critical points, namely $c_0, c_1, \dots, c_{N-1}$, and that the critical point $c_0=0$ (which is the one at which we renormalize) has criticality equal to $d\in 2{{\Bbb N}}+1$. Let $U$ be a ${{\Bbb T}}$-symmetric annulus contained in the domain of analicity of $f$. We consider the dynamical partitions associated to the critical point $0$, and denote the fundamental domains of the $n-th$ partition by $I_n$ and $I_{n+1}$. Moreover, the $n-th$ renormalization of $f$ at $0$ is denoted by $\mathcal{R}^nf$. Our main goal in this section is to prove the following result \[maintheorem\] There exist universal positive constants $r_0,b,c$ such that the following holds. Let $f$ be as above and $r\geq r_0$. There exists $m_0=m_0(f,r) \in {\ensuremath{\mathbb{N}}}$ such that for all $n\geq m_0$, if we denote by $\mathcal{R}^nf=(\eta,\xi)$ then there exists subdomains $V_{\eta} \supset I_{\eta}$, $V_{\xi} \supset I_{\xi}$ containing the origin, such that the maps $\eta, \xi$ are branched covering $V_{\eta} \to \{z \in {\ensuremath{\mathbb{C}}}: |z|<r \} \cap {\ensuremath{\mathbb{C}}}_{\eta(I_{\eta})}$ and $V_{\xi} \to \{z \in {\ensuremath{\mathbb{C}}}: |z|<r \} \cap {\ensuremath{\mathbb{C}}}_{\eta(I_{\xi})}$. Moreover, for each $z \in V_{\eta}\cap V_{\xi}$ we have $$|\mathcal{R}^{n}f(z)|\geq c|z|^{d} +b,$$ where the left-hand side stands for $\eta$ on $V_{\eta}$ and $\xi$ on $V_{\xi}$. Finally, the constant $m_0$ can be chosen to be depending only on $r$ in any pre-compact family of maps in the compact-open topology on the annulus $U$. Theorem \[maintheorem\] is telling us that inverse branches of deep renormalizations, around the critical point $0$, behave as roots of degree $d$. Hence, these inverse branches map a large disk (containing $0$) well within itself, and therefore the modulus of the annuli between the disk and its image is bigger than a certain positive constant. Following [@Yam1999], Theorem \[maintheorem\] will clearly follows from Lemma \[mainlemma\] below. \[mainlemma\] There exist constants $B_1,B_2$ and $M>0$ such that for any $n\geq M$ the inverse branch $f^{-{q_{n+1}+1}}$ is well defined and univalent over $$\Omega_{n,M}=(D_{n-M}\setminus {\ensuremath{\mathbb{R}}}) \cup f^{q_{n+1}}(I_n),$$ and for any $z \in \Omega_{n,M}$ we have the following $$\label{ineqmainlemma} \frac{{\operatorname{dist}}(z_{-(q_{n+1}-1)}, f(I_n))}{|f(I_n)|} \leq B_1 \, \frac{{\operatorname{dist}}(z,I_n)}{|I_n|} + B_2.$$ Proof of Lemma \[mainlemma\] ---------------------------- In this subsection we give a proof of Lemma \[mainlemma\] based in the proof of [@Yam2019 Lemma 4.2]. Let us introduce some notation. Let $M>n_0$, where $n_0$ is given by Corollary \[corollarybeau\] and let $n \geq M$. We define $H_n$ to be the interval $$H_n= [f^{q_{n+1}}(0), f^{q_{n}-q_{n+1}}(0)],$$ and $D_n$ the Euclidean disk whose intersection with the real line is the interval $H_n$. Note that, by Corollary \[corollarybeau\], ${\operatorname{diam}}(D_n)=|H_n|\asymp |I_n|$. Also, consider the inverse orbit: $$\label{J-orbit} J_0 = f^{q_{n+1}}(I_n), \ J_{-1}= f^{q_{n+1}-1}(I_n), \cdots , J_{-(q_{n+1}-1)} = f(I_n).$$ For any point $z \in D_M$, we say that $$\label{z-orbit} z_0 = z, z_{-1} , \cdots , z_{-(q_{n+1} -1)}$$ is a corresponding inverse orbit if each $z_{-(k+1)}$ is obtained by applying to $z_{-k}$ a univalent inverse branch of $f|_W$, where $W$ is a sub-interval of $J_{-(k+1)}$. We will need four lemmas, the first one is the following result from [@Yam2019]. \[lemmadifeo\] For each $n \geq1$ there exist $K_n \geq1$ and $\theta_n >0$ with $K_n \to 1$ and $\theta_n\to 0$ as $n \to \infty$ such that the following holds. Let $\theta \geq \theta_n$, and let $0 \leq i < j \leq q_{n+1}$ be such that the restriction $f^{j-i} : f^{i}(I_n) \to f^{j}(I_n)$ is a diffeomorphism on the interior. Then the inverse branch $f^{-(j-i)}|_{f^{j}(I_n)}$ is well-defined over $D_{\theta}(f^{j}(I_n))$ and maps it univalently into the Poincaré neighborhood $D_{\theta/K_n}(f^{i}(I_n))$. Let $J=[a,b]$, for a point $z \in \overline{{\ensuremath{\mathbb{C}}}}_{J}$ we define the angle between $z$ and $J$, which is denoted by $\widehat{(z,J)}$, as the least of the angles between the intervals $[a,z],[b, z]$ and the corresponding rays $(a, -\infty), [b, +\infty)$ of the real line, measured in the range $0\leq \theta \leq \pi$. Next result is the same as [@Yam2019 Lemma 4.7]. We provide a more detailed proof for reader’s convenience. \[lemmaineqonelevel\] Fix $n \geq M $, $\varepsilon_1>0$ and $B>0$. Let $0<i<k<q_{n+1}$ and consider two intervals of the inverse orbit given in \[J-orbit\], namely $J=J_{-i}$ and $J'=J_{-k}$. Let $z,z'$ be the corresponding points of the orbit \[z-orbit\]. Assume that $\widehat{(z,J)}\geq \varepsilon_1$ and ${\operatorname{dist}}(z,J) \leq B\, |J|$. Then $$\frac{{\operatorname{dist}}(z',J')}{|J'|} \leq C\, \frac{{\operatorname{dist}}(z,J)}{|J|},$$ for some constant $C=C(\varepsilon_1,B)>0$. Since the orbit $\{J_{-i}\}_{0<i\leq q_{n+1}-1}$ forms part of the dynamical partition $\mathcal{P}_n(f)$, then there are at most one critical point for the iterate $f^{k-i}|_{J'}$. That critical point belongs to the interior or to the boundary of some interval $\Delta$ of the next dynamical partition $\mathcal{P}_{n+1}(f)$ in $J'$. Note that $|\Delta|\asymp |J'|$, see [@EdF Lemma $4.2$] and [@EdF Proposition $4.1$]. Therefore, there exists at most two intervals $I'_1,I'_2\subset J'$ comparable with $J'$ and such that $f^{k-i}: I'_j \to I_j$, for $j=1,2$, is a diffeomorphism. Let $D_{\theta_j}(I_j)$ be the smallest closed hyperbolic neighborhood enclosing $z$, for $j=1,2$. Observe that $\theta_j=\theta_j(\varepsilon_1,B)$, $ {\operatorname{diam}}(D_{\theta_j}(I_j)) \asymp {\operatorname{diam}}(D_{\theta_j}(J))$ and that there exists a constant $\tilde{C}_j=\tilde{C}_j(\varepsilon_1,B)>0$ such that ${\operatorname{diam}}(D_{\theta_j}(I_j)) \leq \tilde{C}_j \, {\operatorname{dist}}(z,I_j)$, see [@Yam1999 Lemma 2.1]. By Lemma \[lemmadifeo\] there exists $K_n>1$ such that $f^{-(k-i)}(D_{\theta_j}(I_j)) \subseteq D_{\theta_j/K_n}(I'_j)$. Then for some $j\in \{1,2\}$ [ $$\dfrac{{\operatorname{dist}}(z', J')}{|J'|} \leq \dfrac{C_1\, {\operatorname{diam}}(D_{\theta_j/K_n}(I'_j))}{|I'_j|} \leq \dfrac{C_2\, {\operatorname{diam}}( f^{-(k-i)}(D_{\theta_j}(I_j)))}{|I'_j|}$$ ]{} where the constants $C_1,C_2$ are beau. The lemma follows since [ $$\dfrac{C_2\, {\operatorname{diam}}( f^{-(k-i)}(D_{\theta_j}(I_j)))}{|I'_j|} \leq \dfrac{C_3\, {\operatorname{diam}}(D_{\theta_j}(I_j))}{|I_j|} \leq \dfrac{C_4 \, {\operatorname{dist}}(z, J)}{|I_j|} \leq \dfrac{C_5\, {\operatorname{dist}}(z,J)}{|J|},$$ ]{} where $C_3$ is beau and $C_4, C_5$ depend on $\varepsilon_1$ and $B$. Next result is borrowed from [@dFdM2 Page 345, Lemma 2.2]. \[lemmaJ-orbitcontained\] Let $n > M$ and consider the inverse orbit defined by \[J-orbit\]. Given $m$ with $n>m \geq M$, let $P_0, \dots P_{-k}$ be the moments in the backward orbit \[J-orbit\] of $I_n$ before the first return to $I_{m+1}$ such that $P_{-i}\subseteq I_m$. Then $k=a_{m+1}$, $P_0 \subseteq I_{m+2}$ and $$P_{-i}\subseteq f^{q_m+(a_{m+1}-i)q_{m+1}}(I_{m+1}).$$ \[\]\[\]\[1\][$I_{m+1}$]{} \[\]\[\]\[1\][$I_{m}$]{} \[\]\[\]\[1\][$I_{m+2}$]{} \[\]\[\][$I_{m+1}^{q_{m+2}-q_{m+1}}$]{} \[\]\[\]\[1\][$I_{m+1}^{q_m}$]{} \[\]\[\]\[1\][$P_0$]{} \[\]\[\]\[1\][$P_{-a_{m+1}}$]{} \[\]\[\]\[1\][$c$]{} \[\]\[\][$\Large{\dots}$]{} ![\[fig:lemmaJ-orbitcontained\] The moments in the backward orbit \[J-orbit\] of $I_n$ before the first return to $I_{m+1}$.](figlemmaJorbitcontained.eps "fig:"){width="4.5in"} Next result and its proof is a version of [@Yam2019 Lemma 4.8], for bounded type rotation numbers. We remark that this result and its proof is the main difference in the proof of complex [*a priori*]{} bounds for multicritical circle maps with bounded type rotation number and multicritical circle maps with irrational quadratic rotation number. Since we are interested in following the orbit of $I_n$ by the inverse branch of the map $f^{q_{m+1}}$ in $D_m$, we decompose the inverse branch by a finite composition of diffeomorphisms and inverse images of $f$. \[lemmaineqseverallevels\] There exists $\varepsilon_2>0$ such that for each $n>m$ the following holds. Let $J=J_{-k}$, $J'=J_{-k-q_{m+1}}$ be two consecutive returns of the backward orbit \[J-orbit\] of $I_n$, before the first return to $I_{m+1}$, and let $\zeta,\zeta'$ be the corresponding points of the orbit \[z-orbit\]. Suppose that $\zeta \in D_m$, then either $\zeta' \in D_m$ or the following holds. There is $k<s\leq k+q_{m+1}$ such that $|J_{-s}|>C_0|J|$ and for the point $z_{-s}$ in the orbit \[z-orbit\] we have $\widehat{(z_{-s},J_{-s})}>\varepsilon_2$, Moreover, $\widehat{(\zeta',J')}>\varepsilon_2$ and hence ${\operatorname{dist}}(\zeta', J')<C\,|I_m|$, where $C$ is beau. Firstly, replacing $f$ by its renormalization if need be, we can ensure that every element of the dynamical partition of level $\geq M$ contains at most one critical value of $f$. Now, if the iterate $f^{q_{m+1}}$ does not have a critical point in the interior of the interval $I_m$, then we are in the situation of [@Yam1999 Lemma 4.2] and the same proof applies. Therefore, let us assume that there are critical points of $f^{q_{m+1}}$ in the interior of $I_m$. For simplicity, let us assume that there is only one such point, otherwise the argument below will need to be repeated at most $N-1$ times (recall that $N$ is the number of critical points of $f$). Let us denote by $\beta$ the critical value of $f^{q_{m+1}}$ in the interior of $f^{q_{m+1}}(I_m)$; there exists $\ell<q_{m+1}$ such that $\beta =f^{\ell}(c_1)$, for $c_1\neq 0$ being a critical point of $f$ with criticality $d_1\in2{{\Bbb N}}+1$. Below we will distinguish two scenarios. In the first one, the distance between $\beta$ and $J$ is commensurable with the size of $I_m$. Then we are in the case described in [@Yam2019 Lemma 4.8], and we proceed accordingly. In the other case, the pull-back $J_{-k-(\ell-1)}\mapsto J_{-k-\ell}$ will factor as $\phi\circ s$ where $w\mapsto s(w)$ is a root of degree $d_1$, and $\phi$ is univalent, and $T\equiv \phi(J_{-k-(\ell-1)})$ is near to $0$. If the point $z_{-k-\ell}$ in the orbit \[z-orbit\] “jumps” at some definite angle from the real line, then the relative distance ${\operatorname{dist}}(z_{-k-\ell},J_{-k-\ell})/|J_{-k-\ell}|$ will not increase – it will actually [*decrease*]{} up to a constant by a root of degree $d_1$[^1]. Hence the argument can be completed using Lemma \[lemmaineqonelevel\]. We proceed with a formal discussion below. Let ${\Delta}$ be the interval of $\mathcal{P}_{m+1}$ containing $\beta$. Note that by our assumption on $\beta$, is not possible to have ${\Delta}=I_{m+1}^{q_m}$. Therefore there are two cases for ${\Delta}$: either ${\Delta}\subseteq I_m \setminus (I_{m+2} \cup I_{m+1}^{q_m})$ or ${\Delta}=I_{m+2}$. 1. Let ${\Delta}\subseteq I_m\setminus (I_{m+2} \cup I_{m+1}^{q_m})$. In this case, there exists $r>0$ which is beau commensurable with $|I_m|$ such that $$\Delta_1\equiv U_r(\Delta)\subset D_m\text{ and }\Delta_1\cap U_r(f^{q_m}(0))=\emptyset.$$ Let us observe that by our assumptions $J \subseteq \Delta_1$. In the case when $\zeta \notin \Delta_1$, we are in the same situation as in [@Yam2019 Lemma 4.8], and the argument there applies [*verbatim*]{}. Therefore, let us assume that $\zeta\in\Delta_1$. Note that $J'\subset [f^{-q_{m+1}(0)},f^{q_m-q_{m+1}}(0)]$ and the endpoint $f^{q_m-q_{m+1}}(0)$ is a critical value of the iterate $f^{q_{m+1}}$. This easily implies (cf. [@Yam2019 Figure 3]) that there exists a beau ${\varepsilon}>0$ such that if $\widehat{(z_{-j},J_{-j})}>{\varepsilon}$ for all $j$ between $k$ and $k+q_{m+1}$, then $\zeta'\in D_m$. As we are pulling back by the iterate $f^{q_{m+1}}$, our interval $H_m$ will go through a critical value of $f$ three times: twice through $f(0)$, for the pullbacks by $f^{q_m-1}$ and $f^{q_{m+1}-1}$, and once through $f(c_1)$ via pullback by $f^{\ell-1}$. Suppose $\widehat{(\zeta,J)}<{\varepsilon}$ (otherwise, we would be done by Lemma \[lemmaineqonelevel\]). By considerations of Koebe Distortion Theorem, there exist ${\varepsilon}_2>0$ beau and also beau constants of commensurability such that one of the following possibilities holds: - $\zeta'\in D_m$; - ${\operatorname{dist}}(z_{-k-(q_{m}-1)},J_{-k-(q_m-1)}) \asymp |f^{-(q_m-1)}(I_m)|$,\ $\widehat{(z_{-k-(q_m-1)},J_{-k-(q_m-1)})}<{\varepsilon}$ and $\widehat{(z_{-k-q_m},J_{-k-q_m})}>{\varepsilon}_2$; - ${\operatorname{dist}}(z_{-k-(q_{m+1}-q_m-1)},J_{-k-(q_{m+1}-q_m-1)}) \asymp |f^{-(q_{m+1}-q_m-1)}(I_m)|$,\ $\widehat{(z_{-k-(q_{m+1}-q_m-1)},J_{-k-(q_{m+1}-q_m-1)})}<{\varepsilon}$ and\ $\widehat{(z_{-k-(q_{m+1}-q_m)},J_{-k-(q_{m+1}-q_m)})}>{\varepsilon}_2$; - ${\operatorname{dist}}(z_{-k-(\ell-1)},J_{-k-(\ell-1)}) \asymp |f^{-(\ell-1)}(I_m)|$,\ $\widehat{(z_{-k-(\ell-1)},J_{-k-(\ell-1)})}<{\varepsilon}$ and $\widehat{(z_{-k-\ell},J_{-k-\ell})}>{\varepsilon}_2$. Since we have assumed that $J$ is far from the critical values $f^{q_m}(0)$, $f^{q_{m+1}-q_m}(0)$, in case b) we have $|J_{-k-q_m}| \asymp |J'|$, and the statement follows from Lemma \[lemmaineqonelevel\]. Case c) is handled in the same way. It remains to discuss case d). Now, since we have assumed that $\zeta$ is close to $\beta$, we will have $$\frac{{\operatorname{dist}}(z_{-k-\ell},J_{-k-\ell})}{|J_{-k-\ell}|} \, < \, C_1 \, \sqrt[d_1]{\frac{{\operatorname{dist}}(z_{-k-(\ell-1)},J_{-k-(\ell-1)})}{|J_{-k-(\ell-1)}|}}+C_2,$$ for $C_1,C_2$ beau constants. The claim follows from Lemma \[lemmaineqonelevel\], see Figure \[fig:cased\] \[\]\[\]\[1\][$0$]{} \[\]\[\]\[1\][$D_{m}$]{} \[\]\[\][$J_{-k-(\ell-1)}$]{} \[\]\[\]\[1\][$f^{q_{m+1}-\ell}$]{} \[\]\[\]\[1\][$f^{\ell-1}$]{} \[\]\[\]\[1\][$c_1$]{} \[\]\[\]\[1\][$\tilde{D}_m$]{} \[\]\[\]\[1\][$f(c_1)$]{} \[\]\[\]\[1\][$\phi \circ s$]{} \[\]\[\]\[1\][$J'$]{} \[\]\[\]\[1\][$J_{-k-\ell}$]{} \[\]\[\]\[1\][$\hat{D}_m$]{} \[\]\[\]\[1\][$\Delta_1$]{} \[\]\[\]\[1\][$D'_m$]{} \[\]\[\]\[1\][$J$]{} ![\[fig:cased\] Proof of Lemma \[lemmaineqseverallevels\], case d).](cased.eps "fig:"){width="4.6in"} 2. If ${\Delta}=I_{m+2}$ then for $J$ as in the statement we can have that $J$ is between $0$ and $\beta$, or $\beta \in J$, or $J$ is between $\beta$ and $f^{q_{m+2}}(0)$. In any case, we can repeat the strategy used in item 1) and obtain the result. Next result will let us to obtain the step of induction that we will use in the proof of Main Lemma. We refer the reader to [@Yam2019 Lemma 4.9]. \[lemmaineqinductionstep\] There exists $\varepsilon_3>0$ such that for each $n>m$ we have the following. Let $J$ be the last return of the backward orbit \[J-orbit\] to $I_n$ before the first return to $I_{m+1}$. Let $J'$ and $J''$ be the first two returns of \[J-orbit\] to $I_{m+1}$ and $\zeta,\zeta'$ be the corresponding moments in the backward orbit \[z-orbit\], in other words, $\zeta=f^{q_m}(\zeta')$ and $\zeta'=f^{q_{m+2}}(\zeta'')$. Suppose that $\zeta \in D_m$, then either $\zeta'' \in D_{m+1}$, or $\widehat{(\zeta'', I_{m+1})}>\varepsilon_3$ (and ${\operatorname{dist}}(\zeta'', J'')< C|I_{m+1}|$ where $C$ is beau). By our assumption in the proof of Lemma \[lemmaineqseverallevels\], if $\Delta$ is the interval belonging to $\mathcal{P}_{n+1}$ containing the point $\beta=f^{\ell}(c_1)$, then $\Delta\neq I_{m+1}^{q_m}$. We have two cases; $\Delta=I_{m+2}$ or $\Delta\neq I_{m+2}$. In Figure \[fig:lemmainductionstep\] we show the case $\Delta=I_{m+2}$ and $q_m<\ell$, the other cases are similar. Let $D^{''}_m=f^{-(q_m-q_{m+2})}(D_m)$. Using an analogous argument used in the proof of Lemma \[lemmaineqseverallevels\], that is, decomposing the iterate $f^{q_{m+2}-q_m}$ as compositions of diffeomorphisms and one iterate of $f$ we get the following: there exist $\varepsilon_3$ beau and $I \subseteq H_{m+1}$ with $|I|\asymp |I_{m+1}|$, such that $D^{''}_m \subseteq D_{m+1} \cup D_{\varepsilon_3}(I)$, see Figure \[fig:lemmainductionstep\] below. \[\]\[\]\[1\][$D_{m}$]{} \[\]\[\]\[1\][$D_{m+1}$]{} \[\]\[\][$f^{q_m}$]{} \[\]\[\]\[1\][$f^{\ell-q_m}$]{} \[\]\[\]\[1\][$f^{q_m+q_{m+1}-\ell}$]{} \[\]\[\]\[1\][$f^{q_{m+2}-q_{m+1}}$]{} \[\]\[\]\[1\][$f^{q_{m+1}}(0)$]{} \[\]\[\]\[1\][$\ f^{q_m-q_{m+1}}(0)$]{} \[\]\[\]\[1\][$f^{q_{m+1}-q_{m+2}}(0)$]{} \[\]\[\]\[1\][$f^{-q_m}(\beta)$]{} \[\]\[\]\[1\][$\beta$]{} \[\]\[\]\[1\][$D'_m$]{} \[\]\[\]\[1\][$0$]{} \[\]\[\]\[1\][$f^{q_{m+1}-q_m}(0)$]{} \[\]\[\]\[1\][$c_{1}$]{} \[\]\[\]\[1\][$f^{-\ell+q_m}(0)$]{} \[\]\[\]\[1\][$0$]{} \[\]\[\]\[1\][$f^{-q_{m+1}}(0)$]{} \[\]\[\]\[1\][$D^{''}_m$]{} \[\]\[\]\[1\][$\tilde{D}_m$]{} \[\]\[\]\[1\][$\widehat{D}_m$]{} \[\]\[\]\[1\][$I_{m+2}$]{} \[\]\[\]\[1\][$I_{m+1}^{q_m}$]{} ![\[fig:lemmainductionstep\] Proof of Lemma \[lemmaineqinductionstep\].](figlemmainductionstep.eps "fig:"){width="4.8in"} Then either $\zeta'' \in D_{m+1}$ or $\widehat{(\zeta'', I_{m+1})}>\varepsilon_3$. In the last case, by equation we have $${\operatorname{dist}}(\zeta'', J') \leq {\operatorname{diam}}(D_{\varepsilon_3}(I)) = C\,|I|\asymp |I_{m+1}|,$$ for $C=C(\varepsilon_3)>0$ and therefore beau. Now, with Lemma \[lemmaineqseverallevels\] and Lemma \[lemmaineqinductionstep\] at hand we proceed to prove our Main Lemma. Let $z \in D_M$. Let $m$ be the largest number such that $z \in D_m$. We always will have two cases, and in each case we will get the inequality for each $n \geq m$. Let $P_0, \dots, P_{-k}$ be the consecutive returns of the backward orbit \[J-orbit\] of $I_n$ to $I_m$ before the first return to $I_{m+1}$ and denote by $z=\zeta_0, \dots \zeta_{-k}=\zeta'$ the corresponding points of the orbit \[z-orbit\]. By Lemma \[lemmaineqseverallevels\] there exist beau constants $C>0$ and $\varepsilon_2>0$ such that $\widehat{(\zeta', J')}>\varepsilon_2$ and ${\operatorname{dist}}(\zeta',P_{-k})<C\, |I_m|$, or $\zeta' \in D_m$. In the first case, by Lemma \[lemmaineqonelevel\] we obtain inequality . In the second case, we consider the point $\zeta''$ corresponding to the second return of the orbit \[z-orbit\] to $I_{m+1}$. By Lemma \[lemmaineqinductionstep\], there exist beau constants $\varepsilon_3>0$ and $C>0$ such that either $\widehat{(\zeta'',I_{m+1})}>\varepsilon_3$ and ${\operatorname{dist}}(\zeta'',I_{m+1})<C\, |I_{m+1}|$, or $\zeta'' \in D_{m+1}$. In the first case, we obtain the inequality by Lemma \[lemmaineqonelevel\]. In the second case, we repeat the previous argument this time for $m+1$ instead $m$. Applications to bi-cubic maps ============================= Let us now specialize to the case when a multicritical circle map $f$ has exactly two critical points, both of which are of criticality $3$. We will call such maps [*bi-cubic*]{}, and place one of these points at $0$ to fix the ideas; we will denote the other critical point by $c$. The renormalizations of such map will then be defined with respect to the critical point at $0$. The following is a generalization of [@Yam2019 Theorem 2.8], the proof applies [*verbatim*]{}, and will be omitted: \[th:renconv\] For each $K\in{{\Bbb N}}$ there exists $\lambda\in(0,1)$ such that the following holds. Suppose $f$ and $g$ are two bi-cubic critical circle maps with the same signature, and assume that $\rho(f)=\rho(g)$ is of a type bounded by $K$. Then $${\operatorname{dist}}({{\mathcal R}}^jf,{{\mathcal R}}^jg)=o(\lambda^j)$$ in the uniform norm on a neighborhood of their intervals of definition. The following result is Main Theorem in [@EG] \[rigidityEG\] There exists a full Lebesgue measure set $\mathcal{A}\subset(0,1)$ of irrational numbers (which includes the set of bounded type numbers) with the following property. Let $f$ and $g$ be $C^3$ multicritical circle maps with the same signature and such that its common rotation number belongs to the set $\mathcal{A}$. If the renormalizations of $f$ and $g$ around corresponding critical points converge together exponentially fast in the $C^1$ topology, then $f$ and $g$ are conjugate to each other by a $C^{1+\alpha}$ diffeomorphism. Therefore, Theorem \[th:renconv\] and Theorem \[rigidityEG\] imply the following result, \[th:rigidity\] Let $K\in{{\Bbb N}}$. There exists $\alpha>0$ such that the following holds. Suppose $f$ and $g$ are two bi-cubic circle maps whose signatures are the same, and furthermore, $\rho(f)=\rho(g)$ is of a type bounded by $K$. Then $f$ and $g$ are $C^{1+\alpha}$ conjugate on ${{\Bbb T}}$. [99]{} E. de Faria, Asymptotic rigidity of scaling ratios for critical circle mappings. *Ergodic Theory Dynam. Systems*, [**19**]{} (1999), 995–1035 E. de Faria, and W. de Melo, Rigidity of critical circle mappings I. *J. Eur. Math. Soc. (JEMS)*, [**1**]{} (1999), 339–392 E. de Faria and W. de Melo, Rigidity of critical circle mappings II. *J. Amer. Math. Soc.*, [**[13]{}**]{} (2000), 343–370. W. de Melo and S. van Strien, One dimensional dynamics. *Springer-Verlag*, 1993. G. Estevez and E. de Faria, Real bounds and quasisymmetric rigidity of multicritical circle maps. *Trans. Amer. Math. Soc.* [**[370]{}**]{}, (8), 2018, 5583–5616. G. Estevez and E. de Faria and P. Guarino, Beau bounds for multicritical circle maps. *Indag. Math.* [**[29]{}**]{}, 2018, 842–859. G. Estevez and P. Guarino, Renormalization for multicritical circle maps. Work in progress. G. Światek, Rational rotation numbers for maps of the circle. *Comm. Math.Phys* [119]{} (1988), 109–128. M. Herman, Conjugaison quasi-simétrique des homéomorphismes du cercle à des rotations (manuscript), 1988. A. Khinchin, Continued fractions. Dover Publications Inc. (reprint of the 1964 translation), 1997. D. Sullivan, Bounds, quadratic differentials, and renormalization conjectures. *AMS Centennial Publications*, [**[2]{}**]{}, Mathematics into the Twenty-first Century, 1988. S. Ostlund, D. Rand, J. Sethna, and E. Siggia, Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. *Physica D: Nonlinear Phenomena.* 8(3) (1983), 303–342. M. Yampolsky, Complex bounds for renormalization of critical circle maps. *Ergod.Th. and Dynam. Sys.* (1999), [**[19]{}**]{}, 227–257. M. Yampolsky, The attractor of renormalization and rigidity of towers of critical circle maps. *Comm. Math. Phys.* [**[218]{}**]{} (2001), 537–568. M. Yampolsky, Hyperbolicity of renormalization of critical circle maps. *Publ. Math. IHES.* [**[96]{}**]{} (2002), 1–41. M. Yampolsky, Renormalization horseshoe for critical circle maps. *Commun. Math. Phys.* [**[240]{}**]{} (2003), 75–96. M. Yampolsky, Renormalization of bi-cubic circle maps. *C. R. Math. Rep. Acad. Sci. Canada.* (2019), Vol. 41 (4), 57–83. J.-C. Yoccoz, Il n’y a pas de contre-exemple de Denjoy analytique. *C.R. Acad. Sc. Paris*, [**[298]{}**]{}, 1984, 141–144. [^1]: It is helpful to think here what happens in the limiting situation, that is, when $\beta$ is one of the endpoints of ${{\mathcal P}}_m$. Then $f$ renormalizes to a commuting pair with a critical point of criticality $d\times d_1$ at $0$ and the estimate in Theorem \[maintheorem\] is [*improved*]{}.

--- abstract: 'High resolution x-ray magnetic scattering has been used to determine the variation with temperature of the magnetic modulation vector, ${\mathbf{\tau}}$, in ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$ to study the interplay between the weakly ferromagnetic (WFM) phase and proposed lock-in transitions in these materials. At temperatures below the WFM transitions, the modulation wave vectors are within the resolution limit of the commensurate values $11/20$ and $6/11$ for ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$, respectively.' author: - 'C. Detlefs' - 'C. Song' - 'S. Brown' - 'P. Thompson' - 'A. Kreyssig' - 'S. L. Bud’ko' - 'P. C. Canfield' date: Version title: 'Lock-in transitions in ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$' --- [^1] Investigations of the physical properties of the superconducting rare-earth nickel boride-carbides, $R{\mathrm{Ni_2B_2C}}$ ($R$ = ${\mathrm{Gd}}$–${\mathrm{Lu}}$, ${\mathrm{Y}}$), continue to provide insight into the interplay between superconductivity and magnetism (For recent overviews, see refs. .) Recently, ${\mathrm{ErNi_2B_2C}}$ attracted special attention when the possible coexistence of superconductivity and weak ferromagnetism (WFM) was indicated by several measurements[@Canfield96; @Kawano99; @Gammel00]. A similar WFM state also exists in ${\mathrm{TbNi_2B_2C}}$[@Cho96; @Dervenagas96; @Song01b]. The similarity of the crystallographic and magnetic properties of these compounds suggests a common origin of the WFM[@Walker03]. In order to study the interplay between WFM and the dominant antiferromagnetic (AFM) order we have performed a comparative study of these two compounds using the technique of x-ray resonant magnetic scattering. In ${\mathrm{ErNi_2B_2C}}$, superconductivity is observed below $T_C = 10.5 {\,\mathrm{K}}$. Neutron diffraction experiments [@Sinha95; @Zarestky95] show that below $T_N=6.0{\,\mathrm{K}}$ it orders in a transverse spin density wave with modulation wave vector ${\mathbf{\tau}}_a \approx (0.55, 0, 0)$ and magnetic moments aligned parallel to the $(0,1,0)$-axis of the crystal. As the temperature is lowered, higher harmonic satellites develop, indicating that the spin density wave squares up; the modulation wave vector was reported to be approximately independent of temperature[@Sinha95; @Zarestky95]. A phase transition into a state with a weak ferromagnetic (WFM) component of about $0.33{\,\mathrm{\mu_B}}/{\mathrm{Er}}$ (at $T=2{\,\mathrm{K}}$) is observed at $T_{\mathrm{WFM}}\approx2.3{\,\mathrm{K}}$[@Canfield96; @Kawano99; @Gammel00]. Whereas this transition is clearly resolved in zero-field specific heat measurements[@Canfield96], the magnetic moment of the ground state has to be extrapolated from magnetization measurements at finite fields above $H_{C1}$[@Canfield96]. The structure and origin of this WFM state remain unclear. Recently, neutron scattering[@Choi01; @Kawano02] showed that the WFM state is intimately linked to the appearance of *even* order harmonic components of the spin wave and a lock-in of the modulation wave vector, ${\mathbf{\tau}}$, onto a commensurate position. However, the values of ${\mathbf{\tau}}$ these two groups cite do not agree: Whereas @Choi01[@Choi01] find ${\mathbf{\tau}}=0.548$, @Kawano02[@Kawano02] report ${\mathbf{\tau}}=11/20 = 0.55$. Finally, an increase of the scattered intensity at nuclear Bragg peaks confirmed the presence of a WFM component[@Kawano99; @Choi01]. Because of the striking similarities in their magnetic properties, it is useful to compare ${\mathrm{ErNi_2B_2C}}$ to ${\mathrm{TbNi_2B_2C}}$. Whereas the latter is not superconducting, it displays the same local moment anisotropy in the paramagnetic phase (easy axes $(1,0,0)$ and $(0,1,0)$) and forms an AFM structure closely related to that of ${\mathrm{ErNi_2B_2C}}$. In neutron scattering experiments[@Dervenagas96] the magnetic modulation wave vector is found to decrease from ${\mathbf{\tau}} =(0.551, 0, 0)$ at $T_N=14.9{\,\mathrm{K}}$ to $(0.545, 0, 0)$ below $T_{\mathrm{WFM}}\approx7{\,\mathrm{K}}$. Unlike in the ${\mathrm{Er}}$ compound, the ${\mathrm{Tb}}$ magnetic moments are aligned *parallel* to the modulation wave vector, forming a *longitudinal* spin wave. Furthermore, a phase transition to a WFM state similar to that of ${\mathrm{ErNi_2B_2C}}$, albeit in the absence of superconductivity, occurs around $T_{\mathrm{WFM}}\approx 7{\,\mathrm{K}}$[@Cho96; @Sanchez98], with magnetic moment $\approx 0.55{\,\mathrm{\mu_{B}}}/{\mathrm{Tb}}$ at $T=2{\,\mathrm{K}}$. Again, the neutron experiments[@Dervenagas96] indicate that the appearance of a WFM moment may be related to a lock-in of the AFM spin wave to a commensurate propagation vector, in this case $\tau = 6/11 = 0.545$. A recent Ginzburg-Landau type analysis[@Walker03] shows that a lock-in to ${\mathbf{\tau}}=M/N {\mathbf{G}}$ (${\mathbf{G}}$ = reciprocal lattice vector) with $M$ even and $N$ odd directly induces weak ferromagnetism as secondary order. This result is confirmed by a mean field analysis[@Jensen02] of ${\mathrm{ErNi_2B_2C}}$. In both compounds the magnetic ordering transitions are accompanied by structural distortions which lower the crystal symmetry from tetragonal (space group $I4/mmm$) to orthorhombic ($Immm$)[@Detlefs97b; @Song99; @Kreyssig01]. This distortion breaks the symmetry between the $(1, 0, 0)$ and $(0, 1, 0)$ crystallographic directions and thus leads to an unique easy axis of the magnetic moment ($\mu \parallel (0, 1, 0)$ in ${\mathrm{ErNi_2B_2C}}$, and $\mu \parallel (1, 0, 0)$ in ${\mathrm{TbNi_2B_2C}}$) and the modulation wave vector ($\tau \parallel (1, 0, 0)$ in both compounds[@Detlefs99; @Song01]). Since these magneto-elastic effects were not reported in the earlier studies[@Dervenagas96; @Kawano02; @Choi01], it appears doubtful that they had the resolution necessary to clearly resolve a lock-in transition into a commensurate state. We therefore performed high-resolution resonant magnetic scattering experiments using synchrotron x-rays[@Gibbs88; @Hannon88] on both ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$. ![\[fig.rawdata\] Scans along the $(H,0,0)$ direction of reciprocal space through the $(-1+\tau,0,7)$ (top), $(-\tau,0,8)$ (middle), and $(1_{A,B},0,7)$ (bottom) reflections of ${\mathrm{TbNi_2B_2C}}$ at selected temperatures. The horizontal scale is given by the experimental orientation matrix defined at $T=1.7{\,\mathrm{K}}$. Note that the shift in position of the magnetic peaks is comparable to the shift of the structural peak due to the magneto-elastic effects. The width of the magnetic peaks is approximately $1.6\times 10^{-3} \frac{2\pi}{a}$.](fig_rawdata){width="0.95\columnwidth"} ![\[fig.er.rawdata\] Longitudinal scans through the $(2\pm\tau,0,0)$ (top, bottom) and $(2,0,0)$ (middle) reflections of ${\mathrm{ErNi_2B_2C}}$ at selected temperatures. The width of the peaks is approximately $2.0\times 10^{-3}\frac{2\pi}{a}=3.6\times 10^{-3}{\,\mathrm{\AA^{-1}}}$.](fig_er_rawdata){width="0.95\columnwidth"} Single crystals of ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$ were grown at the Ames Laboratory using a high-temperature flux growth technique[@Canfield94; @Cho95b]. Platelets extracted from the flux were examined by x-ray diffraction and were found to be high quality single crystals with the $(0, 0, 1)$-axis perpendicular to their flat surface. The ${\mathrm{ErNi_2B_2C}}$ sample was cut perpendicular to the $(1, 0, 0)$ direction and the resulting face was mechanically polished to obtain a flat, oriented surface for x-ray diffraction. The sample dimensions after polishing were approximately $2 \times 1.5 \times 0.5 {\,\mathrm{mm^3}}$. The ${\mathrm{TbNi_2B_2C}}$ sample, the same one used in our earlier studies[@Song99; @Song01], had a $(0, 0, 1)$ polished face. The synchrotron experiments were carried out at XMAS CRG and at the Tro[ï]{}ka undulator beamline (ID10C) of the European Synchrotron Radiation Facility (ESRF). The samples were mounted on the cold finger of a Displex closed cycle refrigerator equipped with an additional Joule-Thompson stage[^2]. The base temperature of this configuration was approximately $1.7{\,\mathrm{K}}$. Fig. \[fig.rawdata\] shows scans through selected magnetic and charge reflections of ${\mathrm{TbNi_2B_2C}}$. The variation of the $a$-axis lattice parameter is significant and has to be taken into account when calculating the modulation wave vector[@Detlefs99] (the horizontal scale is identical in all three panels). The corresponding raw data for ${\mathrm{ErNi_2B_2C}}$ are shown in Fig. \[fig.er.rawdata\]. The data presented in Fig. \[fig.rawdata\] were fitted to a Lorentzian-squared line shape. $\tau$ in units of the reciprocal lattice was calculated from $$\begin{aligned} \frac{2\pi}{a} & = & - Q_H(-1 + \tau, 0, 7) - Q_H(-\tau, 0, 8) \\ \tau & = & \frac{ Q_H(-\tau, 0, 8) }{ Q_H(-1 + \tau, 0, 7) + Q_H(-\tau, 0, 8) },\end{aligned}$$ where ${\mathbf{Q}}=(Q_H,Q_K,Q_L)$ is the scattering vector. The resulting data are presented in Fig. \[fig.tau\]. For the case of ${\mathrm{ErNi_2B_2C}}$ a similar calculation was applied to the measured positions of the $(2 \pm \tau, 0, 0)$ magnetic Bragg reflections[@Detlefs99]: $$\begin{aligned} \frac{2\pi}{a} & = & \frac{1}{2}\left[Q(2+\tau,0,0) + Q(2-\tau,0,0)\right] \\ \tau & = & \frac{ \left[Q(2+\tau,0,0) - Q(2-\tau,0,0)\right] }{ \left[Q(2+\tau,0,0) + Q(2-\tau,0,0)\right] }.\end{aligned}$$ The resulting data are presented in Fig. \[fig.er.tau\]. For both samples, the directly measured lattice parameter is in good agreement with the one calculated from the position of the magnetic reflections and with our earlier experiments [@Detlefs97; @Detlefs99; @Song99; @Song01]. We estimate the relative systematic errors to be below $1\times 10^{-3}$, as indicated in Figs. \[fig.tau\] and \[fig.er.tau\]. Clearly, these are systematic errors – the noise level is at least a factor of 10 smaller. The importance of using this procedure is underlined by the discrepancy between our value and that given by @Choi01 [@Choi01]. The latter, $\tau_{\mathrm{Choi}}=0.548$, can be reproduced from our raw data by simply dividing the low temperature $Q$-value of the $(2+\tau,0,0)$ reflection, $Q=4.5823{\,\mathrm{\AA^{-1}}}$ by the reciprocal lattice parameter at $T=6.1{\,\mathrm{K}}$, $\frac{2\pi}{a}=\frac{1}{2}\cdot Q(2,0,0)=\frac{1}{2}\cdot 3.5988{\,\mathrm{\AA^{-1}}}=1.7994{\,\mathrm{\AA^{-1}}}$. Using the high temperature value of $Q(2,0,0)$ corresponds to taking the average of the $Q$-values of the $(2,0,0)$ and $(0,2,0)$ peaks. We obtain $Q/\frac{2\pi}{a} = 2.547 \approx 2 + \tau_{\mathrm{Choi}}$. In both compounds the variation of $\tau$ vs.[ ]{}temperature flattens out dramatically as $T_{\mathrm{WFM}}$ is approached; within the error cited above their modulation wave vectors agree with the commensurate values indicated by the dashed lines in Figs. \[fig.tau\] and \[fig.er.tau\]. Their absolute values, $\tau_{{\mathrm{Er}}} = 11/20$, and $\tau_{{\mathrm{Tb}}} = 6/11$, probably differ because of small difference in the lattice parameters and electronic structures, resulting in different spacings between the nested parts of the Fermi surfaces. However, there is no sign of a discontinuous change of the modulation at the proposed WFM transition. On the contrary, it appears as if the modulation wave vector was changing continuously, and as if it was not constant, i.e.[ ]{}not locked to a commensurate value, below $T_{\mathrm{WFM}}$. This behavior is confirmed by the neutron data[@Choi01; @Kreyssig03]. Our results are consistent with a second order lock-in transition to the commensurate values given above[@Walker03]. In this scenario, the magnetic structure is composed of commensurate blocks separated by domain walls often called “discommensurations”[@Walker03]. Across these domain walls the phase of the spin density wave shifts by a defined value, in a way similar to the “spin slip” structures observed in ${\mathrm{Ho}}$[@Gibbs85]. The modulation wave vector $\tau$ of the structure is obtained from the Fourier transform over several blocks, including the domain walls. The discrepancy between the actual $\tau$ and the commensurate value is then proportional to the average phase shift per unit length along the modulation wave vector, i.e.[ ]{}the density of domain walls. When the temperature of the system is varied the domain walls shift until in the limit $T \rightarrow 0$ the distance between them becomes infinite and the wave vector truly commensurate. For ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$, this theory has to be modified to take into account domain wall pinning which is expected to be significant because of the large magneto-elastic strain induced by the orthorhombic distortion associated with the AFM phase transition[@Detlefs97b; @Song99; @Kreyssig01]. The net ferromagnetic moment per formula unit resulting from the above commensurate structures would be $2/N$ of the saturation moment, i.e., $1/10{\,\mathrm{\mu_{\mathrm{Er}}}}$ and $2/11{\,\mathrm{\mu_{\mathrm{Tb}}}}$, respectively. These numbers have to be compared to the measured fractions (at $T=2{\,\mathrm{K}}$), $0.33{\,\mathrm{\mu_{B}}}/7.8{\,\mathrm{\mu_{B}}}\approx1/24$ for ${\mathrm{ErNi_2B_2C}}$[@Canfield96], and $0.55{\,\mathrm{\mu_{B}}}/9.5{\,\mathrm{\mu_B}}\approx 1/17$ for ${\mathrm{TbNi_2B_2C}}$[@Cho96]. That the observed WFM moment is significantly smaller than the predicted value might indicate that the saturated, completely ordered state was not reached at the measurement temperature, or that only a finite volume fraction of the sample undergoes the WFM transition. Furthermore, in ${\mathrm{ErNi_2B_2C}}$ the observed magnetization is lowered by the diamagnetism of the superconducting state. Note also that $\tau_{{\mathrm{Er}}} = 11/20$ does not have the form $N/M\vec{G}$ with $N$ even and $M$ odd, so that the lock-in does not necessarily induce weak ferromagnetism[@Walker03]. Finally, we note that the specific heat of ${\mathrm{ErNi_2B_2C}}$ exhibits a broad anomaly near the WFM transition (Fig. \[fig.er.tau\](c)), whereas the specific heat of ${\mathrm{TbNi_2B_2C}}$ (Fig. \[fig.tau\](c)) shows no indication of the WFM transition. This might be related to the different form of the commensurate value, odd/even vs.[ ]{}even/odd, for the ${\mathrm{Er}}$ and ${\mathrm{Tb}}$ compound, respectively [@Walker03]. In summary, we have performed high resolution magnetic x-ray diffraction measurements to study the AFM host structures of ${\mathrm{ErNi_2B_2C}}$ and ${\mathrm{TbNi_2B_2C}}$ close to the weak ferromagnetic transition at $T_{\mathrm{WFM}}$. We determined the modulation wave vector $\tau$ with high precision, taking into consideration the magneto-elastic distortion associated with the antiferromagnetic order. In both materials above $T_{\mathrm{WFM}}$ $\tau$ varies strongly with temperature, whereas below $T_{\mathrm{WFM}}$ the variations are much smaller, but still finite and observable. Across the whole temperature range the variations are continuous, so that first order transitions can be excluded. Our observations are consistent with a second order lock-in transition in the presence of discommensurations[@Walker03]. The authors are grateful to A. I. Goldman, H. Furukawa-Kawano, P. L. Gammel, and M. B. Walker for stimulating discussions. We also wish to thank the ESRF and the beamline staff of ID10C for assistance with the experiments. Ames Laboratory is operated for the U. S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was supported by the Director for Energy Research, Office of Basic Sciences. [27]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , eds., ** (, , ). , ****, (). , , , ****, (). , , , , ****, (). , , , , , , , ****, (). , , , ****, (). , , , , , , ****, (). , , , , , , , , , , ****, (). , ****, (). , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , ****, (). , ****, (). , , , , , , , , ****, (). , , , , , , ****, (). , , , , , , , in [@Natobook], p. . , , , , ****, (). , , , , , , ****, (). , , , , , , ****, (). , , , , ****, (). , , , , , ****, (). , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , ****, (). [^1]: Current address: Electron Spin Science Center, Department of Physics, Pohang University of Science & Technology, Pohang, 790-784, South Korea. [^2]: This cryostat is a prototype of a system now commercially available from A.S. Scientific Products Ltd, Abingdon, UK.

--- abstract: | The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $\alpha\in(0,1]$. We turn to the case of very heavy tailed reproduction distribution $\alpha=0$ assuming Zubkov’s regularity condition with parameter $\beta\in(0,\infty)$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.\ **Keywords:** Reduced branching process, critical branching process, heavy tail, regular variation.\ **MSC:** Primary 60J80, secondary 60F05. author: - 'Andreas N. Lagerås[^1] [^2]' - Serik Sagitov title: Reduced branching processes with very heavy tails --- Introduction ============ A single type branching process describes a population of particles with independent and identical reproduction laws. In the Markov branching process with continuous time each particle lives an exponential time with mean one and at death splits into a random number of daughter particles. If we assume that the branching system starts at time zero from a single particle with $\nu$ daughters, then the whole process is defined by the distribution of the random variable $\nu$. In the critical case when the average number of daughters is exactly one ${\mathbb{E}}[\nu]=1$, the limit behavior of the branching process is studied under the following Zolotarev-Slack regularity condition (cf.  [@Zo] and [@Sl]). The generating function $f(s)={\mathbb{E}}[s^\nu]$ is assumed to satisfy $$\label{alpha} f(s)=s+(1-s)^{1+\alpha}L \left({1\over1-s}\right),\ 0\le\alpha\le1,$$ where $L$ is slowly varying at infinity. This condition is valid with $\alpha=1$ if the variance of $\nu$ is finite, while the case $0<\alpha<1$ is usually referred to as the infinite variance case and is well studied in the literature. We turn to the less studied case $\alpha=0$, focussing on a special class of slowly varying functions (which was initially introduced by Zubkov [@Zu]) $$\label{beta} L(x)\sim(\ln x)^{-\beta}L_1(\ln x),\ \beta>0,\ x\to\infty,$$ where $L_1$ is another slowly varying function. Given and $\alpha=0$, Zubkov’s regularity condition is equivalent to the next requirement on the tail distribution function (see Lemma \[l1\]) $$\label{beta1} {\mathbb{P}}(\nu>k)\sim\beta k^{-1}(\ln k)^{-1-\beta}L_1(\ln k),\ k\to\infty.$$ Notice that in this case ${\mathbb{E}}[\nu(\ln_+\nu)^{\beta-{\varepsilon}}]<\infty$ and ${\mathbb{E}}[\nu(\ln_+\nu)^{\beta+{\varepsilon}}]=\infty$ for all ${\varepsilon}>0$. This is a consequence of [@BGT Thm 8.1.8]. The main characteristic of the branching process is the number of particles $Z(t)$ alive at time $t$. The key issues of the asymptotics of the non-extinction probability $Q(t)={\mathbb{P}}(Z(t)>0)$ as $t\to\infty$ and the limit behavior of $Z(t)$ conditioned on non-extinction in the case $\alpha=0$ were recently addressed by Nagaev and Wachtel [@NW]. They consider the discrete time version of the Markov branching process and obtain general limit results without the extra assumption . By repeating Zubkov’s arguments [@Zu p. 607], our results could be carried over to discrete time with no change. In Section \[s4\] we give a direct proof (which is more straightforward than the counterpart of the Nagaev-Wachtel proof) of the following result with the extra condition. \[th\] If holds with $\alpha=0$ and $L$ satisfies , then $$\label{Q} Q(t)=\exp\left(-t^{1/(1+\beta)} L_q(t)\right),$$ where $L_q(t)$ is such a slowly varying function as $t\to\infty$ that $$L_q^{1+\beta}(t)\sim(1+\beta)L_1(t^{1/(1+\beta)} L_q(t)).$$ Furthermore, there exists a regularly varying function (see ) $$\label{c} c(t)=t^{\beta(1+\beta)^{-2}}L_c(t)$$ such that for all $x\ge0$ $$\label{lt} P(Z(t)\le e^{xc(t)}|Z(t)>0)\to1-e^{-x^{\beta+1}},\ t\to\infty.$$ There is a striking feature in the asymptotics of $Z(t)$ which was pointed out to us by V. Wachtel. Notice that the regular variation index $\beta(1+\beta)^{-2}$ of the scaling function $c(t)$ increases as $\beta$ goes from infinity down to $\beta=1$. As the reproduction tail becomes heavier this is what we expect to happen, namely to have larger asymptotic value for the population size $Z(t)$ at survival. What is puzzling about however, is that as $\beta$ falls below the threshold value 1, the corresponding scaling function $c(t)$ attributes smaller size for the surviving population despite the fact that the reproduction tail becomes even heavier. Some light on this phenomenon is shed by the following seminal results by Zubkov [@Zu]. If $\tau(t)$ is the time to the most common ancestor for all particles alive at time $t$, then under condition with $0<\alpha\le1$ $$\label{zu0} {\mathbb{P}}(\tau(t)\le tx|Z(t)>0)\to x,\ t\to\infty,$$ while under the conditions of Theorem \[th\] $$\label{zu} {\mathbb{P}}(\tau(t)\le tx|Z(t)>0)\to x^{\beta/(1+\beta)},\ t\to\infty.$$ The latter means that the ratio $\tau(t)/t$ is asymptotically distributed over $[0,1]$ with the density function $$\label{fi} \phi_\beta(x)=\beta(1+\beta)^{-1}x^{-1/(1+\beta)}.$$ That is, if $\beta$ is changed towards smaller values, then the time to the most recent common ancestor will become shorter and one can expect an eventual drop in the size of the surviving population as $\beta$ goes below a certain threshold value. Why the threshold value should be $\beta=1$ is an interesting *open problem*. Section \[s3\] presents the so-called reduced branching process describing the genealogy of the particles alive at time $t$. In this section we recall the known limit processes for the reduced branching processes in the cases $0<\alpha\le1$ obtained in [@FSS] and [@Ya]. Then we state the main result of this paper, Theorem \[the\], giving a new limit structure as $t\to\infty$ for the reduced branching process in the case $\alpha=0$ and . Our Theorem \[the\] is an extension of in the same manner as the results by [@FSS] and [@Ya] are extensions of . It is worth mentioning that in Theorem \[the\] we do not loose much generality by assuming in the case $\alpha=0$ since by Zubkov’s proof it actually follows that non-degenerate limit distributions of $\tau(t)/t$ must be either of the form or for critical branching processes. In Section \[s2\] we establish some preliminary results, and in Section \[s4\] we prove Theorems \[th\] and \[the\]. Limit theorem for the reduced branching process {#s3} =============================================== Let $Z(u,t)$ stand for the number of particles at time $u$ which stay alive or have descendants at a later time $t\ge u$. For a given time horizon $t$ the process $\{Z(u,t), 0\le u\le t\}$ is called the reduced branching process. The term *reduced* reflects the fact that we count only those branches in the full genealogical tree that reach to the time of observation $t$. Clearly, $$\label{sam} {\mathbb{P}}(Z(u,t)=1)={\mathbb{P}}(\tau(t)\le t-u),$$ where $\tau(t)$ is the time to the most recent common ancestor. If holds with $0<\alpha\le1$, then the limit theorem for the time to the most recent ancestor is extended to the following limit theorem for the reduced branching process by [@FSS] and [@Ya] $$\label{lpa} (Z(tx,t)|Z(t)>0)\overset{d}{\to} Z_{\alpha}(-\log(1-x)),$$ where the convergence of the processes over the time interval $0\le x<1$ holds in the Skorohod sense. Here the limit is a time transformed Markov branching process $Z_{\alpha}(t)$, whose generating functions can be written down explicitly. The particles of this process live exponential times with mean one and at the moment of death produce offspring with the generating function $$f_{\alpha}(s) = s + \frac{1}{\alpha}(1-s)((1-s)^{\alpha}-1).$$ This process is supercritical with the offspring number $\nu_\alpha$ having mean $f'_{\alpha}(1)=1+1/\alpha$ and a distribution over $k=2,3,\dots$ with probabilities $$\label{na} {\mathbb{P}}(\nu_\alpha=k) = \left\{\begin{aligned} &\mathbf{1}_{\{k=2\}}, &&\text{if }\alpha = 1, \\ &\frac{(1+\alpha)\Gamma(k-1-\alpha)}{\Gamma(1-\alpha)\Gamma(k+1)}, &&\text{if }0<\alpha <1, \end{aligned}\right.$$ where $\Gamma$ is the Gamma function. It is easily checked that the following generating function $$F_{\alpha}(s,t) = 1-(1-e^{-t}+e^{-t}(1-s)^{-\alpha})^{-1/\alpha}$$ solves the forward Kolmogorov equation $${\partial F_{\alpha}(s,t)\over \partial t}=(f_{\alpha}(s)-s){\partial F_{\alpha}(s,t)\over\partial s}$$ thus providing the generating function of $Z_\alpha(t)$. Note that $Z_1(t)$ is the well-known Yule process of binary splitting with $$F_1(s,t) = \frac{s e^{-t}}{1-(1-e^{-t})s},$$ implying that the distribution of $Z_1(t)$ is shifted geometric for all $t$. To state our main result we introduce a Markov branching process $Z_0(t)$ with infinite mean for the offspring number. Here the reproduction law is given by $$\label{n0} {\mathbb{P}}(\nu_0=k) =\frac{1}{k(k-1)},\ k=2,3,\ldots$$ corresponding to the generating function $$f_0(s) = s + (1-s)\log(1-s).$$ Thus the part of the formula given for $0<\alpha<1$ is also valid for $\alpha=0$. The generating function of $Z_0(t)$ is $$\label{limit_dist} F_{0}(s,t) = 1-(1-s)^{e^{-t}}$$ which can be used to compute $$\label{1} {\mathbb{P}}(Z_0(t)=1)=e^{-t}.$$ \[the\] Under the conditions of Theorem \[th\] the weak convergence of the processes over the time interval $0\le x<1$ $$(Z(tx,t)|Z(t)>0)\overset{d}{\to} Z_{0}\left(-{\beta\over1+\beta}\log(1-x)\right)$$ holds in the Skorohod sense as $t\to\infty$. From this result it easy to recover using and . The limit process $Z_{0}\left(-\beta(1+\beta)^{-1}\log(1-x)\right)$ gives the following algorithm defining the genealogical tree: - start with a single particle at time 0 which lives a random time $1-\tau_0$, where $\tau_0$ has density function , - at the time $1-\tau_0$ split the initial particle into a random number of daughter particles according to the distribution , - given $\tau_0$, let each daughter particle, independently of other particles, mimic the life of its mother, namely let it live a time $(1-\tau_1)\tau_0$, where $\tau_1$ has density and then split it using , - given $\tau_0$ and $\tau_1$, let a granddaughter particle live a time $(1-\tau_2)\tau_1\tau_0$, where $\tau_2$ has density and then split it using , and so on. This should be compared with a similar algorithm describing the limit process in for $0<\alpha\le1$, where the density is replaced by the uniform density over $[0,1]$, and the offspring number distribution is replaced by . Figure \[figuren\] clearly indicates an interesting transformation of the limit law for the genealogical tree as the parameter $\alpha$ decreases from 1 to 0 and then at $\alpha=0$ the new parameter $\beta$ goes from $\infty$ down to 0. The $\alpha$-model has common branch length distribution and the value of parameter $\alpha$ determines the reproduction law, which smoothly changes from the deterministic splitting at $\alpha=1$ via to the distribution with infinite mean at $\alpha=0$. It is easy to verify the stochastic domination property: if $0\leq \alpha_1 \leq \alpha_2 \leq 1$, then ${\mathbb{P}}(\nu_{\alpha_2}>k)\leq {\mathbb{P}}(\nu_{\alpha_1}>k)$ for all $k$. This property is nicely illustrated by simulations on the left part of Figure \[figuren\]: the lower is the value of the parameter $\alpha$ the faster is the growth of the genealogical tree. At $\alpha=0$ when the new parameter $\beta$ takes over the control, the dynamics of tree behavior drastically changes. As $\beta$ goes from larger to smaller values it is the branch length (and not the reproduction) that undergoes transformation. Since the reproduction law is common for all $\beta\in(0,\infty)$, we observe an opposite development of the tree growth: the closer $\beta$ is to zero (and therefore the heavier is the tail of the original reproduction law), the closer the splitting times are located to the observation time. Here we have another example of the phenomenon mentioned earlier concerning the critical value $\beta=1$. We observe a growth pattern (this time in terms of genealogical trees) which reaches its top and then is followed by a monotone decline. **Remarks** This development in the tree growth depending on decreasing values of $\beta$ indicates that at the region $\beta=0$ the linear time scaling for the reduced process should be replaced by a non-linear one in agreement with [@Zu Thm 4(b)]. It was pointed out in [@Sa] that the limit reproduction law for the critical reduced branching process with $\alpha\in(0,1]$ is related to the merging law of the $\Lambda$-coalescent with $\Lambda(dx) = (1-\alpha)x^{-\alpha}dx$. Namely, if $Y_n$ is the size of the next merger given there currently are $n$ branches, then $${\mathbb{P}}(Y_n = k ) = {\mathbb{P}}(\nu_{\alpha} = k| \nu_{\alpha} \leq n).$$ Theorem \[the\] shows that there is a similar link between the reduced processes with $\alpha = 0$ and the $\Lambda$-coalescent with uniform $\Lambda$, i.e. the Bolthausen-Sznitman coalescent (see [@Pi]). ![Six blocks with ten realizations each of the limit process for different parameter values. Three blocks on the left correspond to the values: top $\alpha = 1$, middle $\alpha=0.3$, bottom $\alpha=0.1$. Three blocks on the right correspond to the values: top $\beta = 5$, middle $\beta=1$, bottom $\beta=0.2$. For $\beta=0.2$, some trajectories are too close to 1 to be visible. Time to the most recent common ancestor is the horizontal distance from the right end of the time interval $[0,1)$ to the point where the trajectory leaves the state $1 = 10^0$.[]{data-label="figuren"}](a1.eps "fig:"){width="6.5cm"}![Six blocks with ten realizations each of the limit process for different parameter values. Three blocks on the left correspond to the values: top $\alpha = 1$, middle $\alpha=0.3$, bottom $\alpha=0.1$. Three blocks on the right correspond to the values: top $\beta = 5$, middle $\beta=1$, bottom $\beta=0.2$. For $\beta=0.2$, some trajectories are too close to 1 to be visible. Time to the most recent common ancestor is the horizontal distance from the right end of the time interval $[0,1)$ to the point where the trajectory leaves the state $1 = 10^0$.[]{data-label="figuren"}](b5.eps "fig:"){width="6.5cm"} ![Six blocks with ten realizations each of the limit process for different parameter values. Three blocks on the left correspond to the values: top $\alpha = 1$, middle $\alpha=0.3$, bottom $\alpha=0.1$. Three blocks on the right correspond to the values: top $\beta = 5$, middle $\beta=1$, bottom $\beta=0.2$. For $\beta=0.2$, some trajectories are too close to 1 to be visible. Time to the most recent common ancestor is the horizontal distance from the right end of the time interval $[0,1)$ to the point where the trajectory leaves the state $1 = 10^0$.[]{data-label="figuren"}](a03.eps "fig:"){width="6.5cm"}![Six blocks with ten realizations each of the limit process for different parameter values. Three blocks on the left correspond to the values: top $\alpha = 1$, middle $\alpha=0.3$, bottom $\alpha=0.1$. Three blocks on the right correspond to the values: top $\beta = 5$, middle $\beta=1$, bottom $\beta=0.2$. For $\beta=0.2$, some trajectories are too close to 1 to be visible. Time to the most recent common ancestor is the horizontal distance from the right end of the time interval $[0,1)$ to the point where the trajectory leaves the state $1 = 10^0$.[]{data-label="figuren"}](b1.eps "fig:"){width="6.5cm"} ![Six blocks with ten realizations each of the limit process for different parameter values. Three blocks on the left correspond to the values: top $\alpha = 1$, middle $\alpha=0.3$, bottom $\alpha=0.1$. Three blocks on the right correspond to the values: top $\beta = 5$, middle $\beta=1$, bottom $\beta=0.2$. For $\beta=0.2$, some trajectories are too close to 1 to be visible. Time to the most recent common ancestor is the horizontal distance from the right end of the time interval $[0,1)$ to the point where the trajectory leaves the state $1 = 10^0$.[]{data-label="figuren"}](a01.eps "fig:"){width="6.5cm"}![Six blocks with ten realizations each of the limit process for different parameter values. Three blocks on the left correspond to the values: top $\alpha = 1$, middle $\alpha=0.3$, bottom $\alpha=0.1$. Three blocks on the right correspond to the values: top $\beta = 5$, middle $\beta=1$, bottom $\beta=0.2$. For $\beta=0.2$, some trajectories are too close to 1 to be visible. Time to the most recent common ancestor is the horizontal distance from the right end of the time interval $[0,1)$ to the point where the trajectory leaves the state $1 = 10^0$.[]{data-label="figuren"}](b02.eps "fig:"){width="6.5cm"} Preliminary results {#s2} =================== If $f(s)>s$ for $0\le s<1$, the function $$\pi(s)=\int_0^s\frac{dv}{f(v)-v}$$ is obviously monotone and consequently $\rho(x) = \pi(1-e^{-x})$ as well. Let hold with $\alpha=0$ and put $g(x)=L(e^x)$, then condition is equivalent to $$\label{g} g(x)\sim x^{-\beta}L_1(x),\ x\to\infty.$$ Note that $$\begin{aligned} \pi(s)&=\int_{1-s}^1\frac{dt}{f(1-t)-1+t}\\ &=\int_{1-s}^1\frac{1}{L(1/t)}\frac{dt}{t}=\int_0^{-\ln(1-s)}\frac{dw}{L(e^{w})}=\int_0^{-\ln(1-s)}\frac{dw}{g(w)},\end{aligned}$$ therefore $$\label{rg} \rho(x)=\int_0^xdw/g(w),$$ and it follows from and [@BGT Thm 1.5.8] that $$\label{r} \rho(x)\sim {x^{\beta+1}\over(\beta+1)L_1(x)},\ x\to\infty.$$ \[l1\] Put $q(t) = -\ln Q(t)$, then $$\begin{aligned} \rho(q(t)) &=&t, \label{rq}\\ q'(t) &=& g(q(t)).\label{q'} \end{aligned}$$ The generating function $F(s,t) ={\mathbb{E}}[s^{Z(t)}]$ of the Markov branching process satisfies the backward Kolmogorov equation $$\frac{\partial F(s,t)}{\partial t}=f(F(s,t))-F(s,t)$$ with the boundary condition $F(s,0)=s$. It follows that $$\label{raz} \pi(F(s,t)) = \pi(s) + t.$$ Putting $s=0$ yields the asserted equality $$\label{dva} \pi(1-Q(t)) = t,$$ since $\pi(0)=0$ and $Q(t)=1-F(0,t)$. Furthermore, and give an important representation $$\label{F_och_Q} 1-F(s,t) = Q(\pi(s)+t).$$ After differentiating both sides of we find $$\label{to} \rho'(q(x))=\frac{1}{q'(x)}$$ which together with implies . \[l2\] If holds with $\alpha=0$, then implies . On the other hand, if ${\mathbb{E}}[\nu]=1$, then implies with $\alpha=0$ and . First notice that $$1-f(s)=(1-s)\sum_{k=0}^\infty {\mathbb{P}}(\nu>k)s^k$$ and similarly, since ${\mathbb{E}}[\nu]=\sum_{k= 0}^{\infty}{\mathbb{P}}(\nu>k)=1$, $$\label{ff} \frac{f(s)-s}{1-s}=(1-s)\sum_{k=0}^\infty \left(\sum_{i>k}{\mathbb{P}}(\nu>i)\right)s^k.$$ Therefore, condition with $\alpha=0$ is equivalent to $$\sum_{k=0}^\infty \left(\sum_{i>k}{\mathbb{P}}(\nu>i)\right)s^k=(1-s)^{-1}L \left({1\over1-s}\right).$$ According to [@BGT Cor. 1.7.3] the latter is equivalent to $$\sum_{i>k}{\mathbb{P}}(\nu>i)\sim L (k),\ k\to\infty$$ or in the integral form $$\int_{x}^\infty{\mathbb{P}}(\nu>y)dy\sim L (x),\ x\to\infty.$$ Now given we apply [@BGT Thm 1.7.2b] to see that $$\int_{z}^\infty{\mathbb{P}}(\nu>e^y)e^ydy\sim z^{-\beta}L_1(z),\ z\to\infty$$ entails $${\mathbb{P}}(\nu>e^z)e^z\sim \beta z^{-1-\beta}L_1(z),\ z\to\infty$$ which is . To prove the assertion in the opposite direction one should apply [@BGT Thm 1.5.11]. \[l3\] Under conditions of Theorem \[th\] as $x\to\infty$ $$\begin{aligned} g'(x)&\sim& -\frac{\beta g(x)}{x},\label{g'}\\ q''(x)&\sim& \frac{\beta q'(x)^2}{q(x)},\label{le2}\\ \rho''(q(x))&\sim& \frac{\beta}{q'(x)q(x)}. \label{le3}\end{aligned}$$ If we define $c(t)$ by $$\label{cc} {1\over c(t)}=q\left({1\over g(q(t))}\right),$$ then it will satisfy . In the critical case we have $$\begin{aligned} 1-f'(s) &=& (1-s)\sum_{k=0}^\infty \left(\sum_{i=k+2}^{\infty}i{\mathbb{P}}(\nu=i)\right)s^k \\ &=& (1-s)\sum_{k=0}^\infty {\mathbb{E}}\left[\nu \mathbf{1}_{\{\nu\geq k+2\}}\right]s^k, \end{aligned}$$ and on the other hand, due to $$\begin{aligned} f(s)-s &=& (1-s)^2\sum_{k=0}^\infty \left(\sum_{i>k}{\mathbb{P}}(\nu>i)\right)s^k \\ &=& (1-s)^2\sum_{k=0}^\infty {\mathbb{E}}\left[(\nu-k-1) \mathbf{1}_{\{\nu\geq k+2\}}\right]s^k, \end{aligned}$$ These two relations together with $L((1-s)^{-1})=(f(s)-s)/(1-s)$ yield $$\begin{aligned} \frac{1}{(1-s)^2}L'\left(\frac{1}{1-s}\right)&={f'(s)-1\over1-s}+{f(s)-s\over(1-s)^2}\\ &= -\sum_{k=0}^\infty \left({\mathbb{E}}\left[\nu \mathbf{1}_{\{\nu\geq k+2\}}\right]-{\mathbb{E}}\left[(\nu-k-1) \mathbf{1}_{\{\nu\geq k+2\}}\right]\right)s^k\\ &= -\sum_{k=0}^\infty (k+1){\mathbb{P}}(\nu >k+1)s^k, \end{aligned}$$ thus due to as $s\to1$ $$\begin{aligned} L'\left(\frac{1}{1-s}\right)&\sim&-\beta(1-s)^2\sum_{k=0}^\infty (\ln k)^{-1-\beta}L_1(\ln k)s^k. \end{aligned}$$ Since due to [@BGT Prop. 1.5.8] $$\begin{aligned} \sum_{k=0}^n (\ln k)^{-1-\beta}L_1(\ln k)\sim n(\ln n)^{-1-\beta}L_1(\ln n),\end{aligned}$$ we derive from the previous relation applying [@BGT Cor. 1.7.3] $$\begin{aligned} {1\over1-s}L'\left({1\over1-s}\right)&\sim&-\beta|\ln (1-s)|^{-1-\beta}L_1(-\ln (1-s)). \end{aligned}$$ Now follows from the relations $g'(x)=e^xL'(e^x)$ and . To derive it is enough to observe that $q''(x)=g'(q(x))q'(x)$ and use . From we get $\rho''(q(x))=-q''(x)/q'(x)^3$. This and give . The last assertion of the lemma is a simple consequence of the basic properties of regular varying functions. Proof of Theorems \[th\] and \[the\] {#s4} ==================================== The asymptotics of $Q(t)$ stated in Theorem \[th\] follows from and in view of [@BGT Thm 1.5.12]. We prove and Theorem \[the\] after deriving an expression for the generating function of the reduced process $$\tilde{F}(s;u,t) = {\mathbb{E}}[s^{Z(u,t)}]$$ in terms of $F(s,t)$. The survival probability at time $t$ for a branching process starting from a single particle at time $u$ is equal to $Q(t-u)$. Since the total number of particles alive at time $u$ is described by $F(s,u)$ we can write $$\tilde{F}(s;u,t) = F(1-Q(t-u)+Q(t-u)s,u).$$ Combining this with the obvious relation $$\tilde{F}(s;u,t)=1-Q(t)+{\mathbb{E}}[s^{Z(u,t)}|Z(t)>0]Q(t)$$ we deduce $$\begin{aligned} {\mathbb{E}}[s^{Z(u,t)}|Z(t)>0] &= \frac{\tilde{F}(s;u,t)-1+Q(t)}{Q(t)} \notag\\ &= 1- \frac{1-F(1-Q(t-u)+Q(t-u)s,u)}{Q(t)}.\end{aligned}$$ In view of it follows that $$\begin{aligned} {\mathbb{E}}[s^{Z(u,t)}|Z(t)>0] &= 1- \frac{Q(\pi(1-Q(t-u)(1-s))+u)}{Q(t)}\notag\\ &=1-\exp\left\{q(t)-q(t+\Delta(s,t-u))\right\},\label{Fq}\end{aligned}$$ where $$\Delta(s,t)=\rho(q(t)-\ln(1-s))-t.$$ Using with $u=t$ we get $$\label{Fred_Rtt} -\ln(1-{\mathbb{E}}[s^{Z(t)}|Z(t)>0]) = q(\rho(-\ln(1-s))+t)-q(t).$$ To prove we study with $s=1-e^{-x/c(t)}$, where $x$ is a fixed positive number and $c(t)$ is defined by . According to Lemma \[l3\] and $$\rho(x/c(t))\sim x^{1+\beta}/q'(t),\ t\to\infty,$$ and therefore, by a Taylor expansion around $t$, $$\begin{aligned} -\ln(1-&{\mathbb{E}}[(1-e^{-x/c(t)})^{Z(t)}|Z(t)>0]) \\ &= q(\rho(x/c(t))+t)-q(t) \\ &= \rho(x/c(t))q'(t)+\rho(x/c(t))^2O(q''(t)) \\ &\to x^{1+\beta}.\end{aligned}$$ Thus $${\mathbb{E}}[(1-e^{-x/c(t)})^{Z(t)}|Z(t)>0] \to 1-e^{-x^{1+\beta}}$$ and by the arguments of Darling [@Darling:1970] and Seneta [@Seneta:1973], or Nagaev and Wachtel [@NW], this implies that $${\mathbb{P}}(\ln Z(tx,t) \leq xc(t)|Z(t)>0)\to 1-e^{-x^{1+\beta}},$$ which finishes our proof of Theorem \[th\]. The proposed limit process $R(x)=Z_0(-\beta(1+\beta)^{-1}\log(1-x))$ in Theorem \[the\] is a time inhomogeneous Markov branching process, and from we find that $${\mathbb{E}}[s^{R(y)}|R(x)=1] = 1 - (1-s)^{\left(\frac{1-y}{1-x}\right)^{\beta/(1+\beta)}},\, 0\leq x \leq y < 1.$$ This yields $$\label{FSS-koppling} {\mathbb{P}}(R(y)=1|R(x)=1)=\left(\frac{1-y}{1-x}\right)^{\beta/(1+\beta)}.$$ By similar arguments that led to we have $${\mathbb{E}}[s^{Z(v,t)}|Z(u,t)=1, Z_t>0] = 1-\exp\{q(t-u)-q(\Delta(s,t-v)+t-u)\},$$ for $0\leq u \leq v < t$. In order to prove convergence of finite dimensional distributions it suffices to show $$q(\Delta(s,xt)+yt)-q(yt)\to -\left[\log(1-s)^{(x/y)^{\beta/(1+\beta)}}\right],\, 0< x\leq y \leq 1.$$ We do this in two steps. First we find the asymptotics of $\Delta(s,t)$ by Taylor expansion of the function $\rho$ around $q$ $$\begin{aligned} \Delta(s,t) &=&\rho(q(t)-\ln(1-s))- \rho(q(t))\\ &=& -\ln(1-s)\rho'(q(t))+O(\rho''(q(t))) \\ &\sim& -\ln(1-s)/q'(t).\end{aligned}$$ And second we do another Taylor expansion, this time of the function $q$ around $yt$ $$\begin{aligned} q(yt+\Delta(s,xt))-q(yt)&= -\ln(1-s)\frac{q'(yt)}{q'(xt)}+(q'(t))^{-2}O(q''(t)) \\ &\to -\ln (1-s)\left(\frac{x}{y}\right)^{\beta/(1+\beta)},\end{aligned}$$ according to Lemma \[l3\]. This finishes the proof of convergence of finite dimensional distributions in Theorem \[the\]. To show the convergence in the Skorohod sense, we can now copy the proof in [@FSS] almost verbatim. The only difference is that we have with $0<\beta/(1+\beta)<1$, whereas they in our notation have the expression $(1-y)/(1-x)$ for the corresponding probability. The difference in the exponent has no consequence for the proof. **Acknowledgement.** We thank V. Wachtel for stimulating discussions of his recent paper [@NW]. [99]{} <span style="font-variant:small-caps;">Bingham, N.H., Goldie, C.M. and Teugels, J.L.</span> (1987). *Regular variation*. Cambridge University Press. <span style="font-variant:small-caps;">Darling, D.A.</span> (1970). The Galton-Watson process with infinite mean. *J. Appl. Probab.* **7**, 455–456. <span style="font-variant:small-caps;">Fleischmann, K. and Siegmund-Schultze, R.</span> (1977). The structure of reduced critical Galton-Watson processes. *Math. Nachr.* **79**, 233–241. <span style="font-variant:small-caps;">Nagaev, S.V. and Wachtel, V.</span> (2007). The critical Galton-Watson process without further power moments. *J. Appl. Probab.* **44**, 753–769. <span style="font-variant:small-caps;">Pitman, J.</span> (1999). Coalescents with multiple collisions. *Ann. Probab.* **27**, 1870–1902. <span style="font-variant:small-caps;">Sagitov, S.</span> (1999). The general coalescent with asynchronous mergers of ancestral lines. *J. Appl. Probab.* **36**, 1116–1125. <span style="font-variant:small-caps;">Seneta, E.</span> (1973). The simple branching process with infinite mean. I. *J. Appl. Probab.* **10**, 206–212. <span style="font-variant:small-caps;">Slack, R.S.</span> (1968). A branching process with mean one and possibly infinite variance. *Z. Wahrscheinlichkeitstheorie und Verw. Gebiete* **9**, 139–145. <span style="font-variant:small-caps;">Yakymiv, A.L.</span> (1980). Reduced branching processes. *Theory Probab. Appl.* **25**(3), 584–588. <span style="font-variant:small-caps;">Zolotarev, V.M.</span> (1957). More exact statements of several theorems in the theory of branching processes. *Theory Probab. Appl.* **2**(2), 245–253. <span style="font-variant:small-caps;">Zubkov, A.M.</span> (1975). Limit distributions of the distance to the nearest common ancestor. *Theory Probab. Appl.* **20**, 602–612. [^1]: Centre for Theoretical Biology, Göteborg University. [^2]: Address: Mathematical Sciences, Chalmers University of Technology, 412 96 Göteborg, Sweden.

--- abstract: 'We consider within-host virus models with $n\geq 2$ strains and allow mutation between the strains. If there is no mutation, a Lyapunov function establishes global stability of the steady state corresponding to the fittest strain. For small perturbations this steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Moreover, using a perturbation result due to Smith and Waltman [@smith-waltman], we show that this steady state also preserves global stability.' author: - | Patrick De Leenheer[^1] and Sergei S. Pilyugin[^2]\ Department of Mathematics\ University of Florida, Gainesville, FL 32611-8105, USA\ [*To our mentor and good friend Hal Smith, on the occasion of his $60$th birthday.*]{} title: | Multi-strain virus dynamics with mutations:\ A global analysis --- Introduction ============ The study of the dynamics of within-host virus disease models has been a very fruitful area of research over the past few decades. Of particular importance has been the work on mathematical models of HIV infection by Perelson and coauthors [@perelson-etal; @perelson-nelson] and Nowak and coauthors [@nowak-may]. It has spurred more recent research by among others Hal Smith with one of us [@hiv], [@wang-li] and [@zhilan]. For single-strain virus models, the understanding of the global behavior has been largely based on the fact that they are competitive [@hiv] and the use of particular mathematical tools developed by Muldowney; see Li and Muldowney [@li-muldowney] for an application of these techniques to the classical SEIR model in epidemiology. Of course it is well known that for globally stable systems there is a Lyapunov function, but finding such a function is often difficult, as illustrated by the following quote from Smith and Waltman’s classical work on chemostats [@smith-waltman-chem] on p. 37: 0.1in *Considerable ingenuity, intuition, and perhaps luck are required to find a Liapunov function.* 0.1in One of the purposes of this paper is to find such Lyapunov functions for various within host virus models following the ingenuity from [@korobeinikov] and [@iggidr]. Another purpose of the paper is to investigate what happens if we include mutation effects in the model by allowing different virus strains to mutate into each other. This is very relevant in the context of HIV where mutations have profound impact on treatment, see for instance [@zhilan] where a two-strain model is considered. Mathematically we will treat the model with mutations as a perturbation of the original model. It turns out that the structural properties of the mutation matrix that describes the possible mutations (in particular, whether this matrix is irreducible or not), dictate which single strain steady states of the unperturbed model persist in the perturbed model, and which don’t. An obvious problem is to determine if the globally stable single strain steady state of the unperturbed model persists. We will show that this is always the case, regardless of the mutation matrix. Moreover, taking advantage of the perturbation result developed by Smith and Waltman in [@smith-waltman], we will show that this steady state remains globally stable for small values of the mutation parameter. In order to apply this perturbation result we will first need to establish a particular persistence property, uniform in the perturbation parameter, and to achieve this we invoke the theory developed by Hutson [@hutson1; @hutson2], see also [@hofbauer], which uses the notion of an average Lyapunov function. It will be shown that a rather simple -in fact, linear- average Lyapunov function exists. The paper is organized as follows. In Section \[single\] we present a Lyapunov function to establish global stability of the disease equilibrium of a single-strain virus model. This is extended in Section \[multi\] to a global stability result for a multi-strain model which does not include mutations. In biological terms, we demonstrate that in the absence of mutations the fittest strain of the virus drives all other viral strains to extinction. In Section \[mutation\] we investigate what happens if mutations are taken into account for two different models. Finally, in the Appendix we extend all our results to a slightly modified model which includes an often neglected loss term in the virus equation. Single-strain {#single} ============= In this paper, we consider the basic model of the form $$\begin{aligned} \label{hiv1} {\dot T}&=&f(T)-kVT\nonumber \\ {\dot T^*}&=&kVT-\beta T^*\nonumber \\ {\dot V}&=&N\beta T^*-\gamma V, \end{aligned}$$ where $T$, $T^*$, $V$ denote the concentrations of uninfected (healthy) and infected host cells, and free virions, respectively. Equations (\[hiv1\]) describe a general viral infection where the viral replication is limited by the availability of target cells $T$. In this model, we assume that all infected cells $T^*$ are virus-producing cells, that is, we do not include any intermediate stage(s) corresponding to latently infected cells. In addition, we do not explicitly consider the impact of the immune response. Implicitly, the immune response can be accounted for by the removal term $-\beta T^*$. The rate of viral production is assumed proportional to the removal of infected cells. In case of lytic viruses, $N$ represents the average burst size of a single infected cell; whereas in case of budding viruses, $N$ can be thought of as the average number of virions produced over a lifetime of an infected cell. For different infections, the actual class of the target cells in (\[hiv1\]) may vary from the $CD4+$ $T$ lymphocytes (in case of HIV), to the epithelial cells (in case of Influenza), to the red blood cells (in case of Malaria). The $T$, $T^*$, $V$ notation is adopted from the classical HIV model [@perelson-nelson]. All parameters are assumed to be positive. The parameters $\beta$ and $\gamma$ are the removal rates of the infected cells and virus particles respectively. Following [@perelson-nelson; @nowak-may], we neglect the term in the $V$-equation that represents the loss of a virus particle upon infection. But all subsequent results hold when this loss term is included, in which case the $V$-equation reads: $${\dot V}=N\beta T^*-\gamma V-kVT.$$ These results will be presented in the Appendix. The growth rate of the uninfected cell population is given by the smooth function $f(T):{\mathbb{R}}_+ \rightarrow {\mathbb{R}}$, which is assumed to satisfy the following: $$\label{T0} \exists \; T_0>0 \;\; :\;\; f(T)(T-T_0)<0,\;\; T\neq T_0.$$ Since continuity of $f$ implies that $f(T_0)=0$, it is easy to see that $$E_0=(T_0,0,0),$$ is an equilibrium of $(\ref{hiv1})$. Effectively, $T_0$ is the carrying capacity for the healthy cell population. A second, positive equilibrium may exist if the following quantities are positive: $$\label{eq} {\bar T}=\frac{\gamma}{kN},\;\; {\bar T^*}=\frac{f({\bar T})}{\beta},\;\; {\bar V}=\frac{f({\bar T})}{k{\bar T}}.$$ Note that this is the case if and only if $f(\frac{\gamma}{kN})>0$, or equivalently by $(\ref{T0})$ that ${\bar T}=\frac{\gamma}{kN}<T_0$. In terms of the basic reproduction number $${\cal R}^0:=\frac{kN}{\gamma}T_0=\frac{T_0}{\bar T_0},$$ existence of a positive equilibrium is therefore equivalent to ${\cal R}^0>1.$ We assume henceforth that ${\cal R}^0>1$ and denote the disease steady state by $E=({\bar T},{\bar T^*},{\bar V})$. Let us introduce the following sector condition: $${\bf (C)}\;\; (f(T)-f({\bar T}))\left(1-\frac{{\bar T}}{T}\right)\leq 0.$$ Note that this condition is satisfied when $f(T)$ is a decreasing function, independently of the value of ${\bar T}$. For instance, [@nowak-may] considers $f(T)=c_1-c_2T$, where $c_i$ are positive constants. Another example [@perelson-nelson] is $f(T)=s+rT(1-\frac{T}{K})$ provided that $f(0)=s\geq f({\bar T})$. \[1strain\] Let [**(C)**]{} hold. Then the equilibrium $E$ is globally asymptotically stable for $(\ref{hiv1})$ with respect to initial conditions satisfying $T^*(0)+V(0)>0$. Consider the following function on ${\operatorname{int}}({\mathbb{R}}^3_+)$: $$W=\int_{{\bar T}}^T \left(1-\frac{{\bar T}}{\tau}\right) d \tau +\int_{{\bar T^*}}^{T^*} \left(1-\frac{{\bar T^*}}{\tau}\right)d \tau+ \frac{\beta}{N\beta}\int_{{\bar V}}^V \left(1-\frac{{\bar V}}{\tau}\right) d \tau.$$ Then $$\begin{aligned} {\dot W}&=&(f(T)-kVT)\left(1-\frac{{\bar T}}{T} \right)+(kVT-\beta T^*)\left(1-\frac{{\bar T^*}}{T^*} \right)+ \frac{1}{N}(N\beta T^*-\gamma V)\left(1-\frac{{\bar V}}{V} \right)\\ &=&f(T)\left(1-\frac{{\bar T}}{T} \right)+kV{\bar T}-kVT\frac{{\bar T^*}}{T^*}+\beta {\bar T^*}-\beta T^*\frac{{\bar V}}{V} -\frac{\gamma}{N}V+\frac{\gamma}{N}{\bar V}\\ \end{aligned}$$ Since from $(\ref{eq})$ we have that $\beta{\bar T^*}=k{\bar V}{\bar T}=\frac{\gamma}{N}{\bar V}$, it follows that $$\begin{aligned} {\dot W}&=&f(T)\left(1-\frac{{\bar T}}{T} \right)+\beta {\bar T^*}\frac{V}{{\bar V}} -\beta {\bar T^*}\frac{{\bar T^*}VT}{T^*{\bar V}{\bar T}}+\beta {\bar T^*}-\beta {\bar T^*}\frac{{\bar V}T^*}{V{\bar T^*}} -\beta T^*\frac{V}{{\bar V}}+\beta T^*\\ &=&(f(T)-f({\bar T}))\left(1-\frac{{\bar T}}{T} \right)+\beta T^*\left(1-\frac{{\bar T}}{T} \right) +\beta {\bar T^*}\frac{V}{{\bar V}} -\beta {\bar T^*}\frac{{\bar T^*}VT}{T^*{\bar V}{\bar T}}+\beta {\bar T^*}-\beta {\bar T^*}\frac{{\bar V}T^*}{V{\bar T^*}} -\beta T^*\frac{V}{{\bar V}}+\beta T^*\\ &=&(f(T)-f({\bar T}))\left(1-\frac{{\bar T}}{T} \right)- \beta {\bar T^*}\left[\frac{{\bar T}}{T}+\frac{{\bar T^*}VT}{T^*{\bar V}{\bar T}}+\frac{{\bar V}T^*}{V{\bar T^*}}-3\right] \end{aligned}$$ The first term is non-positive by [**(C)**]{}. The second term is non-positive as well since the geometric mean of $3$ non-negative numbers is not larger than the arithmetic mean of those numbers. Hence, ${\dot W}\leq 0$ in ${\operatorname{int}}({\mathbb{R}}^3_+)$, and the local stability of $E$ follows. Notice that ${\dot W}$ equals zero iff both the first term and the second term are zero, and using ${\bf (C)}$, this happens at points where: $$\frac{{\bar T}}{T}=1\textrm{ and } \frac{{\bar T^*}V}{T^*{\bar V}}=1.$$ Then LaSalle’s Invariance Principle [@lasalle] implies that all bounded solutions in ${\operatorname{int}}({\mathbb{R}}^3_+)$ converge to the largest invariant set in $$M=\{(T,T^*,V)\in {\operatorname{int}}({\mathbb{R}}^3_+)\;|\; \frac{{\bar T}}{T}=1,\;\; \frac{{\bar T^*}V}{T^*{\bar V}}=1\}.$$ Firstly, boundedness of all solutions follows from Lemma $\ref{bounded}$ which is proved later in a more general setting. Secondly, it is clear that the largest invariant set in $M$ is the singleton $\{E\}$. Finally, note that forward solutions starting on the boundary of ${\mathbb{R}}^3_+$ with either $T_1(0)$ or $V_1(0)$ positive, enter ${\operatorname{int}}({\mathbb{R}}^3_+)$ instantaneously. This concludes the proof. Competitive exclusion in a multi-strain model {#multi} ============================================= Let us now consider a multi-strain model: $$\begin{aligned} {\dot T}&=&f(T)-\sum_{i=1}^nk_iV_iT\label{multi1}\\ {\dot T^*_i}&=&k_iV_iT-\beta_iT_i^*,\;\; i=1,\dots,n\label{multi2}\\ {\dot V_i}&=&N_i\beta_iT_i^*-\gamma_iV_i,\;\; i=1,\dots,n\label{multi3} \end{aligned}$$ where all parameters are positive. Similar calculations as in the single-strain model show there is a unique disease-free equilibrium $E_0=(T_0,0,0)$. For each $i$, there is a corresponding single-strain equilibrium $E_i$ with positive $T$, $T^*_i$ and $V_i$ components and zero components otherwise if and only if $$1<{\cal R}^0_i.$$ Here, ${\cal R}^0_i$ is the basic reproduction number for strain $i$ which is defined by $${\cal R}^0_i=\frac{T_0}{{\bar T}^i}.$$ The positive components of $E_i$ are then given by $$\label{poscomp} {\bar T}^i=\frac{\gamma_i}{k_iN_i},\;\; {\bar T}^*_i=\frac{f({\bar T}^i)}{\beta_i},\;\; {\bar V}_i=\frac{f({\bar T}^i)}{k_i{\bar T}^i}.$$ We assume that all $E_i$ exist and assume without loss of generality (by possibly reordering components) that $$\label{T's} {\bar T}^1<{\bar T}^2\leq \dots\leq {\bar T}^{n-1}\leq {\bar T}^n <T_0,$$ or equivalently, that $$\label{R's} 1<{\cal R}^0_n\leq {\cal R}^0_{n-1}\leq \dots\leq {\cal R}^0_2<{\cal R}^0_1.$$ and will prove the following competitive exclusion principle. It asserts that the strain with the lowest target cell concentration at steady state (or equivalently, with highest basic reproduction number) will ultimately dominate, provided that such strain is present initially. \[multistrain\] Assume that all $E_i$ exist for $(\ref{multi1})-(\ref{multi3})$, that ${\bf (C)}$ holds with ${\bar T}^1$ instead of ${\bar T}$, and that $(\ref{T's})$ holds. Then $E_1$ is globally asymptotically stable for $(\ref{multi1})-(\ref{multi3})$ with respect to initial conditions satisfying $T_1^*(0)+V_1(0)>0$. Consider the following function on $U:=\{(T,T^*_1,\dots,T^*_n,V_1,\dots,V_n)\in {\mathbb{R}}^{2n+1}\;|\; T,T_1^*,V_1>0\}$: $$W=\int_{{\bar T}^1}^T \left(1-\frac{{\bar T}^1}{\tau}\right) d \tau +\int_{{\bar T^*}_1}^{T^*_1} \left(1-\frac{{\bar T^*}_1}{\tau}\right)d \tau+ \frac{1}{N_1}\int_{{\bar V}_1}^{V_1} \left(1-\frac{{\bar V}_1}{\tau}\right) d \tau+\sum_{i=2}^n \left(T^*_i+\frac{1}{N_i}V_i\right).$$ Then $$\begin{aligned} {\dot W}&=&(f(T)-\sum_{i=1}^nk_iV_iT)\left(1-\frac{{\bar T}^1}{T} \right)+(k_1V_1T-\beta_1 T_1^*)\left(1-\frac{{\bar T_1^*}}{T_1^*} \right)+ \frac{1}{N_1}(N_1\beta_1 T_1^*-\gamma_1 V_1)\left(1-\frac{{\bar V_1}}{V_1} \right)\\ &&+\sum_{i=2}^n \left(k_iV_iT-\frac{\gamma_i}{N_i}V_i\right)\\ &=&(f(T)-k_1V_1T)\left(1-\frac{{\bar T}^1}{T} \right)+(k_1V_1T-\beta_1 T_1^*)\left(1-\frac{{\bar T_1^*}}{T_1^*} \right)+ \frac{1}{N_1}(N_1\beta_1 T_1^*-\gamma_1 V_1)\left(1-\frac{{\bar V_1}}{V_1} \right)\\ &&-\sum_{i=2}^n \left(-k_iV_i {\bar T}^1+\frac{\gamma_i}{N_i}V_i\right)\\ \end{aligned}$$ Notice that the first three terms can be simplified in a way similar as in the proof of Theorem $\ref{1strain}$, and using the expression for ${\bar T}^i$ in $(\ref{poscomp})$, we find that $$\begin{aligned} {\dot W}&=&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3\right]\\ &&-\sum_{i=2}^n k_iV_i({\bar T}^i-{\bar T}^1) \end{aligned}$$ Each of the first two terms is non-positive as was shown in the proof of Theorem $\ref{1strain}$. The third part is also non-positive by $(\ref{T's})$. Thus ${\dot W}\leq 0$, establishing already stability of $E_1$. An application of LaSalle’s Invariance Principle shows that all bounded solutions in $U$ (and as before, boundedness follows from Lemma $\ref{bounded}$ which is proved later) converge to the largest invariant set in $$\left\{(T,T^*_1,\dots,T^*_n,V_1,\dots,V_n)\in U\;|\;\frac{{\bar T^1}}{T}=1,\;\; \frac{{\bar T^*_1}V_1}{T^*_1{\bar V_1}}=1 ,\;\;V_i=0,\;\; i>2 \right\},$$ which is easily shown to be the singleton $\{E_1\}$. Finally, solutions on the boundary of $U$ with $T_1^*(0)+V_1(0)>0$ enter $U$ instantaneously, which concludes the proof. Perturbations by mutations {#mutation} ========================== In this section we expand model $(\ref{multi1})-(\ref{multi3})$ to account for mutations between the $n$ strains. In fact, we will study two different ways in which mutations occur. Our first extended model can be written compactly as follows: $$\begin{aligned} {\dot T}&=&f(T)-k'VT, \quad T \in {\mathbb{R}}_+\label{multic1}\\ {\dot T^*}&=&P(\mu)KVT-B T^*, \quad T^* \in {\mathbb{R}}^n_+\label{multic2}\\ {\dot V}&=&{\hat N}B T^*-\Gamma V, \quad V \in {\mathbb{R}}^n_+\label{multic3}, \end{aligned}$$ while the second is written as $$\begin{aligned} {\dot T}&=&f(T)-k'VT, \quad T \in {\mathbb{R}}_+\label{multic4}\\ {\dot T^*}&=&KVT-B T^*, \quad T^* \in {\mathbb{R}}^n_+\label{multic5}\\ {\dot V}&=&P(\mu){\hat N}B T^*-\Gamma V, \quad V \in {\mathbb{R}}^n_+\label{multic6}, \end{aligned}$$ In both models $K={\operatorname{diag}}(k)$, $B={\operatorname{diag}}(\beta)$, ${\hat N}={\operatorname{diag}}(N)$ and $\Gamma={\operatorname{diag}}(\gamma)$, and the matrix $P(\mu)$ with $\mu \in [0,1]$ is defined as follows: $$P(\mu)=I+\mu Q,$$ where $Q$ is a matrix with $q_{ij}>0$ if strain $j$ can mutate to $i$ (for $i\neq j$) so that different magnitudes of $q_{ij}$ reflect the possible differences in the specific mutation rates. The diagonal entries of $Q$ are such that each column of $Q$ sums to zero. Notice that $P$ is a stochastic matrix (all its entries are in $[0,1]$ and all its columns sum to one) provided that $\mu \leq -1/q_{ii}$ for all $i$ (which is assumed henceforth), and that $P(0)=I$. \[bounded\] Both system $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ are dissipative, i.e. there exists a forward invariant compact set $K \subset {\mathbb{R}}^{2n+1}_+$ such that every solution eventually enters $K$. From $(\ref{multic1})$ and $(\ref{multic4})$ follows that ${\dot T}\leq f(T)$, hence $$\label{estimate} \limsup_{t\rightarrow \infty}T(t)\leq T_0,$$ provided solutions to both systems are defined for all $t\geq 0$. To see that this is indeed the case, we argue by contradiction and let $(T(t),T^*(t),V(t))$ be a solution with bounded maximal interval of existence ${\cal I}_+:=[0,t_{\max})$. Then necessarily $T(t)\leq \max (T(0),T_0):=T_{\max}$ for all $t\in {\cal I}_+$. This implies that on ${\cal I}_+$, the following differential inequality holds for the solution of system $(\ref{multic1})-(\ref{multic3})$: $$\begin{aligned} {\dot T^*}&\leq&P(\mu)KVT_{\max}-B T^*\\ {\dot V}&\leq&{\hat N}B T^*-\Gamma V, \end{aligned}$$ or for system $(\ref{multic4})-(\ref{multic6})$ $$\begin{aligned} {\dot T^*}&\leq&KVT_{\max}-B T^*\\ {\dot V}&\leq&P(\mu){\hat N}B T^*-\Gamma V, \end{aligned}$$ respectively. Notice that the right hand sides in the above inequalities are cooperative and linear vector fields. By a comparison principle for such inequalities we obtain that $T(t)\leq {\tilde T}(t)$ and $V(t)\leq {\tilde V}(t)$ (interpreted componentwise) for all $t$ in the intersection of the domains where the solutions are defined. Here, $({\tilde T}(t),\; {\tilde V}(t))$ is the solution to the linear system whose vector field appears in the right hand side of the above inequalities, hence these solutions are defined for all $t\geq 0$. But then $T(t)$ and $V(t)$ can be extended continuously to the closed interval $[0,T_{\max}]$, contradicting maximality of ${\cal I}_+$. Inequality $(\ref{estimate})$ implies that for an arbitrary small $\epsilon>0$, there holds that $T(t)\leq T_0+\epsilon$ for all sufficiently large $t$. Now consider the behavior of the quantity $T+1'T^*$ along solutions of both system $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$: $$\frac{d}{dt} \left( T+1'T^*\right)=f(T)-1'BT^*\leq f(T)-b1'T^*,$$ where $b:=\min_i (\beta_i)$. By continuity of $f$ on the compact interval $[0,T_0+\epsilon]$, there exists (sufficiently large) $a>0$ such that $$f(T)+b T \leq a ,\;\; \textrm{ for all } T\in [0,T_0+\epsilon].$$ Therefore, for all sufficiently large $t$, there holds that $$\frac{d}{dt} \left( T+1'T^*\right)\leq a-bT-b1'T^*\leq a-b (T+1'T^*),$$ and hence $$\limsup_{t\rightarrow \infty} T(t)+1'T^*(t)\leq \frac{a}{b}.$$ Finally, from $(\ref{multic3})$ and $(\ref{multic6})$ follows that $$\limsup_{t\rightarrow \infty}V(t)\leq \frac{a}{b}\Gamma^{-1}{\hat N}B,$$ and $$\limsup_{t\rightarrow \infty}V(t)\leq \frac{a}{b}\Gamma^{-1}P(\mu){\hat N}B,$$ respectively, where the $\limsup$ of a vector function is to be understood componentwise. Dissipativity now follows by observing that all the above bounds are independent of the initial condition. \[invertible\] For $\mu =0$, let all single strain equilibria $E_1,E_2\dots,E_n$ exist for either $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$, and assume that $$\label{c1} {\bar T^1}<{\bar T^2}<\dots<{\bar T^n}<{\bar T^{n+1}}:=T_0,$$ and $$\label{c2} f'({\bar T}^j)\leq 0,\textrm{ for all }j=1,\dots, n+1.$$ Then the Jacobian matrices of $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$, evaluated at any of the $E_i$’s, $i=1,\dots, n+1$ (where $E_{n+1}:=E_0$) have the following properties: $J(E_i)$ has $i-1$ eigenvalues (counting multiplicities) in the open right half plane and $2(n+1)-i$ eigenvalues in the open left half plane. In particular, $J(E_1)$ is Hurwitz. Note that when $\mu=0$, the Jacobian matrix associated to both model $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ is the same and given by: $$J=\begin{pmatrix} f'(T)-k'V&0&-k'T\\ KV&-B&KT\\ 0&{\hat N}B&-\Gamma \end{pmatrix}.$$ To evaluate the Jacobian at any of the $E_i$’s it is more convenient to reorder the components of the state vector by means of the following permutations: 1. For $i=1,\dots,n$ we use $(T,T^*,V)\rightarrow (T,T^*_i,V_i,T^*_1,V_1,\dots,T^*_{i-1},V_{i-1},T^*_{i+1},V_{i+1},\dots, T^*_n,V_n)$. 2. For $i=n+1$ we use $(T,T^*,V)\rightarrow (T,T^*_1,V_1,T^*_2,V_2,\dots,T^*_n,V_n)$. Then the Jacobian matrices have the following structure: 1. For $i=1,\dots,n$, $$J(E_i)= \begin{pmatrix} A_1^i&*&\dots&*&*&\dots&*\\ 0&B_1^i&\dots&0&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\dots&0\\ 0&0&\dots&B^i_{i-1}&0&\dots&0\\ 0&0&\dots&0&B^i_{i+1}&\dots&0\\\ \vdots&\vdots&\dots&\vdots&\vdots&\ddots&0\\ 0&0&\dots&0&0&\dots&B_n^i \end{pmatrix},$$ where $$A_1^i=\begin{pmatrix} f'({\bar T}^i)-k_i{\bar V}_i& 0& -k_i{\bar T}^i\\ k_i{\bar V}_i& -\beta_i & k_i {\bar T}^i\\\ 0&N_i\beta_i&-\gamma_i \end{pmatrix}\textrm{ and } B_l^i=\begin{pmatrix} -\beta_l&k_l{\bar T}^i\\ N_l\beta_l& -\gamma_l \end{pmatrix},\; l\neq i,$$ and therefore the eigenvalues of $J(E_i)$ coincide with those of $A_1^i$ and $B_l^i$, $l\neq i$. Since $f'({\bar T}^i)\leq 0$ it follows from lemma $3.4$ in [@hiv] that the eigenvalues of $A_1^i$ are in the open left half plane. The matrices $B_l^i$ are quasi-positive, irreducible matrices, hence by the Perron-Frobenius Theorem they have a simple real eigenvalue $\lambda_l^i$ with corresponding (componentwise) positive eigenvector. Notice that $${\operatorname{tr}}(B_l^i)<0, \textrm{ and } {\operatorname{det}}(B_l^i)=\beta_l \gamma_l\left(1-\frac{{\bar T}^i}{{\bar T}^l} \right),$$ and thus by $(\ref{c1})$ that $$\lambda_l^i\begin{cases} <0,\textrm{ for all } l>i,\\ >0,\textrm{ for all } l<i. \end{cases}$$ There are $i-1$ unstable $B$-blocks on the diagonal of $J(E_i)$, each of which contributes one positive eigenvalue to $J(E_i)$. 2. For $i=n+1$, $$J(E_{n+1})= \begin{pmatrix} A_1^{n+1}&*&\dots&*\\ 0&B_1^{n+1}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&B_n^{n+1} \end{pmatrix},$$ where $$A_1^{n+1}=\begin{pmatrix} f'({\bar T}^{n+1}) \end{pmatrix}\textrm{ and } B_l^{n+1}=\begin{pmatrix} -\beta_l&k_l{\bar T}^{n+1}\\ N_l\beta_l& -\gamma_l \end{pmatrix},\; l=1,...,n.$$ Notice that by a similar argument as in the previous case, all $n$ $B$-blocks on the diagonal of $J(E_{n+1})$ are unstable with one positive and one negative eigenvalue. When $\mu \neq 0$, the question arises as to what happens to the equilibria $E_1,\dots,E_{n+1}$. The previous Lemma allows us to apply the Implicit Function Theorem which for small positive $\mu$ establishes the existence of (unique) equilibria $E_j(\mu)$ near each $E_j$. Indeed, denoting the vector field of either $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$ by $F(X,\mu)$, we have that for all $j=1,\dots, n+1$, there holds that $F(E_j,0)=0$, and under the conditions of the previous Lemma we also have that $\partial F / \partial X (E_j,0)$ is invertible. It is clear that $E_{n+1}(\mu)=E_{n+1}(0)$ for all $\mu \geq 0$, i.e. the disease-free equilibrium is not affected by mutations. The main issue is of course whether or not the remaining equilibria $E_j(\mu),\ j=1,...,n$ are non-negative. We study this problem next and derive results in terms of the properties of the mutation matrix $Q$. For the steady-state analysis, we will need the following Lemma which is a relevant modification of Theorem A.12 (ii) in [@smith-waltman-chem]. \[improve\] Let $M$ be an irreducible square matrix with non-negative off-diagonal entries and let $s(M)$ be the stability modulus of $M$. Suppose that there exist $x,r \geq 0$ such that $Mx+r=0$. Then the following hold: 1. If $s(M)>0$, then $x=r=0$; 2. If $s(M)=0$, then $r=0$ and $x$ is a multiple of the positive eigenvector of $M$. Due to Perron-Frobenius Theorem, $s(M)$ is the principal eigenvalue of $M$. It is also the principal eigenvalue of $M'$. Since $M'$ is also irreducible and non-negative off-diagonal, there exists $v>0$ such that $M' v =s(M) v$. Equivalently, $v'M=s(M)v'$. Hence $$0 = v'(M x+r)=s(M)v'x+v'r.$$ If $s(M)>0$, then both non-negative products $v'x$ and $v'r$ must be zero which implies $x=r=0$. If $s(M)=0$, then $v'r=0$ which implies $r=0$. Hence $Mx=0=s(M)x$ so that $x$ is a multiple of the positive eigenvector of $M$. For convenience, we introduce the following notation. We define $A(\mu):=\Gamma^{-1} {\hat N}P(\mu)K$ and assume (by renumbering the strains if necessary) that the strains are numbered in such a way that the matrix $A(\mu)$ has the lower block-triangular form $$\label{Amu} A(\mu)=\begin{pmatrix} A_1(\mu)& 0 & \dots &0 \\ \mu B_{2,1} & A_2(\mu) & \dots & 0\\ \vdots & \vdots &\ddots & \vdots \\ \mu B_{k,1} & \mu B_{k,2}& \dots &\mu A_k(\mu) \end{pmatrix},$$ where each diagonal block $$A_i(\mu)={\operatorname{diag}}\biggl(\frac{1}{\bar T_{i1}},\ldots, \frac{1}{\bar T_{is_i}} \biggr)+\mu B_i$$ is such that $B_i,\ i=1,...,k$ are irreducible with non-negative off-diagonal entries. The off-diagonal blocks $B_{i,j},\ i>j$ are non-negative. We note that the diagonal entries of $A(0)$ are a permutation of $$0< \frac{1}{\bar T_{n}} < \frac{1}{\bar T_{n-1}} < \dots < \frac{1}{\bar T_{1}}.$$ We say that the strain group $j$ is reachable from strain group $i$ if there exists a sequence of indices $i=l_1<l_2<...<l_m=j$ such that all matrices $B_{l_{s+1},l_{s}}$ are nonzero. Our first result is as follows: \[eigenvectors\] Let the assumptions of Lemma $\ref{invertible}$ hold, then the following hold: 1. For all sufficiently small $\mu>0$, matrix $A(\mu)$ admits $n$ distinct positive eigenvalues given by $$\frac{1}{\hat T_{n}(\mu)} < \frac{1}{\hat T_{n-1}(\mu)} < \dots < \frac{1}{\hat T_{1}(\mu)},$$ such that $\hat T_i(0)=\bar T_i$ for $i=1,...,n$; 2. Matrix $A(\mu),\ \mu>0$ admits a positive eigenvector $(v_1,v_2,...,v_k)$ if and only if $\frac{1}{\hat T_{1}(\mu)}$ is a principal eigenvalue of $A_1(\mu)$, and all strain groups $j \geq 2$ are reachable from strain group $1$; 3. Matrix $A(\mu),\ \mu>0$ admits a non-negative eigenvector $(v_1,v_2,...,v_k)$ for each eigenvalue $\frac{1}{\hat T_{r}(\mu)}$ such that $\frac{1}{\hat T_{r}(\mu)}$ is a principal eigenvalue of some diagonal block $A_i(\mu)$, and $s(A_j(\mu))< \frac{1}{\hat T_{r}(\mu)}$ for all $j=i+1,...,k$ such that strain group $j$ is reachable from strain group $i$. The component $v_j$ is positive (zero) if group $j$ is reachable (not reachable) from strain group $i$. 4. All other eigenvectors of $A(\mu),\ \mu>0$ are not sign definite. The first assertion follows readily because $A(0)$ has $n$ real distinct eigenvalues and $A(\mu)$ is continuous (actually, linear) in $\mu$. The continuity of eigenvalues with respect to $\mu$ implies that $\hat T_i(0)=\bar T_i$ for $i=1,...,n$. To prove the second assertion, we begin with sufficiency of the condition. Let $\mu>0$ be small and suppose that $\frac{1}{\hat T_{1}(\mu)}$ is a principal eigenvalue of $A_1(\mu)$, and all strain groups $j \geq 2$ are reachable from strain group $1$. Since $A_1(\mu)$ is irreducible with non-negative off-diagonal entries, Perron-Frobenius Theorem implies that the eigenvector $v_1$ associated with $\frac{1}{\hat T_{1}(\mu)}$ is positive. Since $\frac{1}{\hat T_{1}(\mu)}$ is also the principal eigenvalue of $A(\mu)$, it follows that $$s(A_j(\mu)-\frac{1}{\hat T_{1}(\mu)}I)<0, \quad j\geq 2,$$ hence $(A_j(\mu)-\frac{1}{\hat T_{1}(\mu)}I)^{-1}<0$ (see e.g. Theorem A.12 (i) in [@smith-waltman-chem]). The remaining components $v_2,...,v_k$ of the eigenvector satisfy the triangular system $$\begin{array}{ccl} 0 &= & \mu B_{2,1} v_1 + (A_2(\mu)-\frac{1}{\hat T_{1}(\mu)}I) v_2,\\ 0 &= & \mu B_{3,1} v_1 + \mu B_{3,2} v_2 + (A_3(\mu)-\frac{1}{\hat T_{1}(\mu)}I) v_3,\\ \vdots & \vdots & \vdots\\ 0 & = & \mu B_{k,1} v_1 + \dots + \mu B_{k,k-1} v_{k-1} + (A_k(\mu)-\frac{1}{\hat T_{1}(\mu)}I) v_k, \end{array}$$ Solving this system recursively, we obtain $$v_j = -(A_j(\mu)-\frac{1}{\hat T_{1}(\mu)}I)^{-1}(\mu B_{j,1} v_1 + \dots + \mu B_{j,j-1} v_{j-1}), \quad j=2,...,k.$$ Since the strain group $2$ is reachable from strain group $1$, the vector $\mu B_{2,1} v_1 \geq 0$ is nonzero. Positivity of the matrix $-(A_2(\mu)-\frac{1}{\hat T_{1}(\mu)}I)^{-1}$ then implies that $v_2>0$. By induction on $j$, it follows that $v_j>0$ for all $j=2,...,k$, and hence $v=(v_1,v_2,...,v_k)$ is a positive eigenvector. To prove the converse (the necessary condition), let $v=(v_1,v_2,...,v_k)$ be a positive eigenvector of $A(\mu)$ and let $\frac{1}{\hat T_{q}(\mu)}$ be the corresponding eigenvalue. Since $(A_1(\mu)-\frac{1}{\hat T_{q}(\mu)}I)v_1=0$ and $v_1>0$, $\frac{1}{\hat T_{q}(\mu)}$ must be the principal eigenvalue of $A_1(\mu)$ (Perron-Frobenius Thm). It remains to prove that $s(A_j(\mu)) < \frac{1}{\hat T_{q}(\mu)}$ for all $j\geq 2$. Consider $j=2$, and for the sake of contradiction suppose that $s(A_j(\mu)-\frac{1}{\hat T_{q}(\mu)}I) \geq 0$. Since the eigenvalues are real and distinct for small $\mu>0$, this actually implies $s(A_j(\mu)-\frac{1}{\hat T_{q}(\mu)}I) > 0$. Then we have that $$(A_j(\mu)-\frac{1}{\hat T_{q}(\mu)}I) v_2 +\mu B_{2,1} v_1=0$$ holds with non-negative vectors $v_2$ and $\mu B_{2,1} v_1$ which are both nonzero. By Lemma \[improve\], we have $v_2=0$, a contradiction. Hence $s(A_2(\mu)-\frac{1}{\hat T_{q}(\mu)}I) < 0$. Proceeding by induction on $j$, we find that $s(A_j(\mu)-\frac{1}{\hat T_{q}(\mu)}I) < 0$ for all $j \geq 2$. Therefore, $\frac{1}{\hat T_{q}(\mu)}$ must be the principal eigenvalue of $A(\mu)$, that is, $\frac{1}{\hat T_{q}(\mu)}=\frac{1}{\hat T_{1}(\mu)}$. This proves the second assertion. To prove the third assertion, we again start with sufficient condition. Suppose that $\frac{1}{\hat T_{r}(\mu)}$ is a principal eigenvalue of some diagonal block $A_i(\mu)$, and $s(A_j(\mu))< \frac{1}{\hat T_{r}(\mu)}$ for all $j=i+1,...,k$ such that strain group $j$ is reachable from strain group $i$. It follows immediately that all matrices $A_l(\mu)-\frac{1}{\hat T_{r}(\mu)}I,\ l<i$ are nonsingular, and thus $v_l=0,\ l<i$. The component $v_i$ is the eigenvector of $A_i(\mu)-\frac{1}{\hat T_{r}(\mu)}I$ and it is positive by Perron-Frobenius Theorem. Let $j=i+1$, then one of the following holds. If $i+1$ is not reachable from $i$, that is, $B_{i+1,i}=0$ so that $$(A_{i+1}(\mu)-\frac{1}{\hat T_{r}(\mu)}I)v_{i+1}=0$$ implies $v_{i+1}=0$ because $A_{i+1}(\mu)-\frac{1}{\hat T_{r}(\mu)}I$ is nonsingular. If $i+1$ is reachable from $i$ and $s(A_{i+1}(\mu)-\frac{1}{\hat T_{r}(\mu)}I) < 0$, then $$v_{i+1}=-(A_{i+1}(\mu)-\frac{1}{\hat T_{r}(\mu)}I)^{-1} \mu B_{i+1,i} v_i > 0.$$ By induction on $j$, it follows that $v_j=0$ for all $j>i$ that are not reachable from $i$ and $v_j> 0$ for all $j>i$ that are reachable from $i$. Hence $v=(0,...,0,v_i,v_{i+1},...,v_k)$ is a non-negative eigenvector. Now we prove the necessary condition of the third assertion. Let $v=(v_1,v_2,...,v_k)$ be a non-negative eigenvector of $A(\mu)$ associated with eigenvalue $\frac{1}{\hat T_{r}(\mu)}$. Let $v_i \geq 0$ be the first nonzero component of $v$, that is, $v=(0,...,0,v_i,...,v_k)$. Then $v_i$ satisfies $(A_i(\mu)-\frac{1}{\hat T_{r}(\mu)}I)v_i=0$ hence $\frac{1}{\hat T_{r}(\mu)}$ must be an eigenvalue of $A_i(\mu)-\frac{1}{\hat T_{r}(\mu)}I$. Moreover, by Perron-Frobenius Theorem, it must be the principal eigenvalue and $v_i>0$. Now consider $j=i+1$ and the equation $$(A_{i+1}(\mu)-\frac{1}{\hat T_{r}(\mu)}I)v_{i+1} + \mu B_{i+1,i} v_i=0.$$ The vectors $v_{i+1}$ and $\mu B_{i+1,i} v_i$ are non-negative. If $s(A_{i+1}(\mu)-\frac{1}{\hat T_{r}(\mu)}I)>0$ then by Lemma \[improve\], $\mu B_{i+1,i} v_i=0$. Since $\mu>0$ and $v_i>0$, this implies $B_{i+1,i}=0$. Equivalently, $j=i+1$ is not reachable from $i$. An induction argument concludes the proof of the third assertion. The final assertion of this Theorem is a simple one. Let $\frac{1}{\hat T_{r}(\mu)}$ be an eigenvalue of $A_i(\mu)$ but not the principal eigenvalue and let $v=(v_1,v_2,...,v_n)$ be the corresponding eigenvector. Since all eigenvalues of $A(\mu)$ are distinct, the matrices $A_{l}(\mu)-\frac{1}{\hat T_{r}(\mu)}I, l<i$ are nonsingular so that $v_l=0,\ l<i$. Then $v_i$ must be an eigenvector of $A_i(\mu)$ and it cannot be sign definite due to Perron-Frobenius theorem. It follows that $v$ is not sign definite. Our second result concerns the existence and the number of non-negative equilibria for the systems $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ with small $\mu>0$. \[reduce\] Let the assumptions of Lemma $\ref{invertible}$ hold and suppose that the strains are renumbered so that $A(\mu)$ has the form (\[Amu\]). Let $E_j(\mu)=(\hat T_j(\mu),\hat T^*_j(\mu),\hat V_j(\mu))$ denote the nontrivial equilibria of both $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ for small $\mu>0$. Then 1. $E_j(\mu)$ is positive if and only if $\frac{1}{\hat T_j(\mu)}$ is an eigenvalue of $A(\mu)$ with a positive eigenvector $V_j$. 2. $E_j(\mu)$ is non-negative if and only if $\frac{1}{\hat T_j(\mu)}$ is an eigenvalue of $A(\mu)$ with a non-negative eigenvector $V_j$. 3. $E_j(\mu) \notin {\mathbb{R}}^{2n+1}_+$ if and only if $\frac{1}{\hat T_j(\mu)}$ is an eigenvalue of $A(\mu)$ with eigenvector $V_j$ which is not sign-definite. We will prove the Proposition only for system $(\ref{multic1})-(\ref{multic3})$ (the proof for $(\ref{multic4})-(\ref{multic6})$ is similar). Observe that the equilibrium relation following from $(\ref{multic3})$, can be expressed as $\hat T^*_j(\mu)=({\hat N}B)^{-1}\Gamma {\hat V_j}(\mu)$. Hence, the signs of the corresponding components of $\hat T^*_j(\mu)$ and ${\hat V_j}(\mu)$ are the same. Substituting the above expression into $(\ref{multic2})$ and $(\ref{multic3})$, we find that ${\hat V_j}(\mu)$ must satisfy $$[\Gamma^{-1} {\hat N}P(\mu)K-\frac{1}{{\hat T_j}(\mu)}I]{\hat V_j}(\mu)= [A(\mu)-\frac{1}{{\hat T_j}(\mu)}I]{\hat V_j}(\mu)=0.$$ Thus for each nontrivial equilibrium $E_j(\mu)$, the quantity $\frac{1}{{\hat T_j}(\mu)}$ must be an eigenvalue of $A(\mu)$ and ${\hat V_j}(\mu)$ must be a multiple of the corresponding eigenvector $V_j$. If $V_j$ is not sign definite, it follows that $E_j(\mu) \notin {\mathbb{R}}^{2n+1}_+$. For all $V_j \geq 0$, the components of $E_j(\mu)$ are uniquely determined via $$\hat V_j(\mu)=\frac{f(\hat T_j(\mu))}{\hat T_j(\mu) k' V_j} V_j, \quad \hat T^*_j(\mu)=({\hat N}B)^{-1}\Gamma {\hat V_j}(\mu).$$ Hence $E_j(\mu)$ is positive (non-negative) if and only if $V_j$ is positive (non-negative). An immediate corollary to Propositions \[eigenvectors\] and \[reduce\] is that if the mutation matrix $Q$ is irreducible, then $A(\mu)$ is irreducible and systems $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ with small $\mu>0$ admit a unique positive equilibrium $E_1(\mu)$ and no other nontrivial non-negative equilibria. If the mutation matrix $Q$ is reducible, then positive equilibrium exists if and only the fittest strain (with lowest value $\bar T_1$) belongs to strain group 1 and all other strain groups are reachable from group 1, meaning that the fittest strain can eventually mutate into any other strain. In addition, nontrivial non-negative equilibria which are not positive are feasible for $\mu>0$ only if $Q$ is reducible. Specifically, if the strains can be numbered according to (\[Amu\]), then at most $k$ nontrivial non-negative equilibria exist. One extreme case is when the fittest strain belongs to group $k$, in which no positive and only one non-negative equilibrium exist. The opposite extreme case is $k=n$ where $A(\mu)$ is lower-triangular, the diagonal entries of $A(\mu)$ are arranged in decreasing order, and for any pair $i<j$, strain $j$ is reachable from strain $i$. In this case, there is a single positive equilibrium and $n-1$ non-negative equilibria. On uniform strong repellers --------------------------- Inspired by Thieme [@thieme93], we make the following definition. \[repellor\] Consider a system $$\dot x=F(x) \label{sys}$$ on a compact forward invariant set $K \subset {\mathbb{R}}^m$ with a continuous flow $\phi(t,x)$. Let $K_0 \subset K$ be a closed forward invariant subset of $K$. Let $d(x,A)$ denote the distance from a point $x$ to the set $A$. We say that $K_0$ is a uniform strong repeller in $K$ if there exists a $\delta>0$ such that for all solutions $\phi(t,x) \in K \backslash K_0$, $\liminf_{t\to\infty} d(\phi(t,x),K_0) \geq \delta$. \[patrickdoeshofbauer\] Let $\Pi: K \to {\mathbb{R}}^+$ be a continuously differentiable function such that $\Pi(x)=0$ if and only if $x\in K_0$. Suppose there exists a lower semi-continuous function $\psi: K \to {\mathbb{R}}$ such that $$\frac{\dot \Pi}{\Pi}=\psi,\quad \forall x\in K\backslash K_0. \label{logdot}$$ Suppose that the following condition holds $$(H) \quad \forall x \in K_0, \ \exists T>0: \ \langle \psi(\phi(T,x)) \rangle >0.$$ Then $K_0$ is a uniform strong repellor in $K$. [*Step 1*]{}. Note that by lower semi-continuity of $\psi$ and continuity of $\phi$, for every $p\in K_0$ we can find an open set $U_p$ containing $p$, and a lower semi-continuous map $T_p:U_p \rightarrow (0,+\infty)$ so that for every $q \in U_p$, (H) holds with $x=q$ and $T=T_p(q)$. Choose for every $p\in K_0$ a non-empty open set $V_p$ with ${\bar V_p}\subset U_p$. Then by lower semi-continuity of each map $T_p$ and compactness of $\bar V_p$, $$\inf_{q\in {\bar V_p}}T_p(q) >0$$ is achieved in ${\bar V_p}$. Since $\cup_{p\in K_0}V_p$ is an open cover of $K_0$, we may choose a finite open subcover $\cup_{i=1,\dots ,n}V_{p_i}$. Let $\tau_i=\inf_{q\in {\bar V_{p_i}}}T_{p_i}(q)>0$ and set $$\tau=\min_{i=1,\dots,n}\tau_i>0.$$ Note that for every $p\in K_0$, there is a $T\geq \tau$ so that (H) holds with $x=p$. That is, $\tau$ is a uniform (in $K_0$) lower bound for $T$’s for which (H) holds. [*Step 2*]{}. Let $h>0$ be given. Define $$\label{layer} U_h=\{x\in K\;|\;\exists \,T> \tau:\langle \psi(\phi(T,x))\rangle > h\}$$ We claim that $U_h$ is open. Fix $z\in U_h$. Then there is some $T> \tau$ so that $$\epsilon:=\langle \psi(\phi(T,z))\rangle -h>0.$$ Then by continuity of $\phi$ and lower semi-continuity of $\psi$ (and therefore uniform lower semi-continuity of $\psi$ on compact sets), it follows that there exists an open set $W_z$ containing $z$ such that for all $z'\in W_z$ holds that $$\label{bounds} \psi(\phi(t,z')>\psi(\phi(t,z))-\epsilon,\;\; \forall t\in [0,T].$$ Now since $$\langle \psi(\phi(T,z))\rangle=\epsilon +h,$$ it follows from $(\ref{bounds})$ that for all $z'\in W_z$: $$\langle \psi(\phi(T,z'))\rangle>h,$$ and thus that $W_z\subset U_h$, establishing our claim. [*Step 3*]{}. Define $T_h:U_h\rightarrow [\tau,+\infty)$ as $$T_h(z)=\inf\{T>\tau\;|\; \langle \psi(\phi(T,z)) \rangle>h\}.$$ We claim that $T_h$ is upper semi-continuous. Fix $z\in U_h$ and let $\epsilon'>0$ be given. Then there is some $T>\tau$ so that $$\langle \psi(\phi(T,z))\rangle>h,$$ so that $$\label{first} T<T_h(z)+\epsilon'$$ By the argument in Step 2, there is some open set $W_z$ containing $z$, such that for all $z'\in W_z$ holds that: $$\langle \psi(\phi(T,z'))\rangle>h,$$ and thus that for all $z'\in W_z$: $$\label{second} T_h(z')\leq T$$ Our claim follows by combining $(\ref{first})$ and $(\ref{second})$. [*Step 4*]{}. The nested family $\{U_h\}_{h>0}$ is decreasing (under set inclusion), and forms an open cover of $K_0$. Hence, there is some ${\bar h}$ so that $U_{{\bar h}}$ covers $K_0$. Since ${\tilde K}:=K \setminus U_{{\bar h}}$ is compact, and $\Pi$ is continuous, $\Pi$ attains its minimal value $m>0$ on ${\tilde K}$. Choose $p\in (0,m)$ and define: $$I_p=\{z\in K\;|\;\Pi(z)\in (0,p]\}.$$ Then $I_p \subset U_{{\bar h}}$. [*Step 5*]{}. We claim that every forward solution starting in $I_p$, eventually leaves $I_p$, that is: $$\forall z\in I_p, \exists t_z>0:\;\; \phi(t_z,z)\notin I_p.$$ By contradiction, if $\phi(t,z)\in I_p$ for all $t\geq 0$, then $\phi(t,z)\in U_{{\bar h}}$ for all $t\geq 0$, and thus: $$\exists T_t\geq \tau:\;\; \frac{1}{T_t}\int_t^{t+T_t}\psi(\phi(s,z))ds > {\bar h}.$$ Then integrating equation (\[logdot\]) from $t$ to $t+T$ yields that: $$\ln\left(\frac{\Pi(\phi(t+T_t,z))}{\Pi(\phi(t,z))} \right) >{\bar h}T_t,$$ and thus that $$\label{div-seq} \Pi(\phi(t+T_t,z))>e^{{\bar h}T_t}\Pi(\phi(t,z)).$$ Set $t_0=0$ and $t_k=t_{k-1}+T_{t_{k-1}}$ for $k=1,2,\dots$. Since each $T_{t_k}\geq \tau >0$ it follows that $t_k\rightarrow \infty$. Then by $(\ref{div-seq})$ and since $t_k\geq \tau$ for all $k$, we have that: $$\Pi(\phi(t_k,z))>e^{{\bar h}T_{t_{k-1}}}\Pi(\phi(t_{k-1},z))>e^{k \tau}\Pi(z),$$ so that $\Pi(\phi(t_k,z))\rightarrow \infty$ as $k\rightarrow \infty$. This contradicts boundedness of $\Pi$ on the compact set $K$. [*Step 6*]{}. Let $${\tilde I}_p=I_p \cup K_0.$$ We will show that there is some $q\in (0,p)$ so that forward solutions starting outside ${\tilde I}_p$, never reach $I_q$, that is: $$\exists q\in (0,p):\;\; z\notin {\tilde I}_p\Rightarrow \phi(t,z)\notin I_q,\;\; \forall t\geq 0.$$ Consider a forward solution $\phi(t,z)$ with $z\notin {\tilde I}_p$. If $\phi(t,z)\notin {\tilde I}_p$ for all $t\geq 0$, then we are done since ${\tilde I}_q \subset {\tilde I}_p$, so let us assume that for some $t_z>0$, holds that $\phi(t_z,z)\in {\tilde I}_p$. Denote the first time this happens by $t_0$: $$t_0=\min \{t>0\;|\; \phi(t,z)\in {\tilde I}_p\}.$$ Set $z^*=\phi(t_0,z)$ and note that $\Pi(z^*)=p$. Denote $\inf_{z\in K_0}\psi(z)$ by $m'$. If $m'\geq 0$, then (\[logdot\]) implies that $\Pi(\phi(t,z^*))\geq \Pi(z^*)=p$ for all $t\geq 0$, so that we’re done. If on the the other hand $m'<0$, we first define $${\bar T}=\max_{z\in {\tilde I}_p} T_h(z) (\geq \tau>0).$$ Notice that this maximum is indeed achieved on the compact set ${\tilde I}_p$, since $T_h$ is upper semi-continuous. Now we define $$q=pe^{m'{\bar T}},$$ and notice that $q$ is independent of the chosen solution $z(t)$. We will show that for this choice of $q$, our claim is established. We have that: $$\forall t\in (0,{\bar T}):\;\; \frac{1}{t}\int_0^t\psi(\phi(s,z^*))ds\geq m',$$ and thus by (\[logdot\]) that $$\label{one-hand} \forall t\in (0,{\bar T}):\;\; \Pi(\phi(t,z^*))\geq \Pi(z^*)e^{m't}>q,$$ which implies that during the time interval $(0,{\bar T})$, the solution $\phi(t,z^*)$ has not reached $I_q$. On the other hand, during that same time interval $(0,{\bar T})$, the solution $\phi(t,z^*)$ must have left ${\tilde I}_p$. If this were not the case, then by the argument in Step 5, there would be some $T^*\in [\tau , {\bar T})$ so that $$\Pi(\phi(T^*,z^*))\geq \Pi(z^*)e^{{\bar h}T}>p,$$ and thus that $\phi(T^*,z^*)\notin {\tilde I}_p$, a contradiction to our assumption. This process can be repeated iteratively and leads to the conclusion that the forward solution $\phi(t,z)$ which did not start in ${\tilde I}_p$, will never reach $I_q$. So far we have shown that for any solution $\phi(t,x) \notin K_0$, inequality $\Pi(\phi(t,x)) \geq q>0$ for all sufficiently large $t$. The sets $K_0=\Pi^{-1}(\{0\})$ and $\Pi^{-1}([q,+\infty)) \cap K$ are compact and disjoint. Therefore, there exists $\delta>0$ such that $d(\phi(t,x),K_0) \geq \delta$ for all $x\notin K_0$ and all sufficiently large $t$. Global stability for small $\mu>0$ ---------------------------------- The following Lemmas will be used to prove global stability of the positive equilibrium for small $\mu>0$. \[liminf\] Let $a: {\mathbb{R}}^m \to {\mathbb{R}}^n$ be continuous and let $b \in {\rm int}({\mathbb{R}}^n_+)$. Let $f: {\mathbb{R}}^m \times ({\mathbb{R}}^n_{+} \backslash \{0\}) \to {\mathbb{R}}$ be defined as $$f(x,y) =\frac{a'(x)y}{b'y}.$$ Then $$\liminf_{x\to x_0, y\to 0+} f(x,y)=\min_{i\in\{1,...,n\}} \frac{a_i(x_0)}{b_i}, \label{LIM}$$ furthermore, if we define $f(x,0)=\min_{i\in\{1,...,n\}} \frac{a_i(x)}{b_i},$ then $f(x,y)$ becomes a lower semi-continuous function on ${\mathbb{R}}^m \times {\mathbb{R}}^n_{+}$ whose restriction on ${\mathbb{R}}^m \times \{0\}$ is continuous. Extending the function $f(x,y)$ by defining $f(x_0,0)=\liminf_{x\to x_0, y\to 0+} f(x,y)$ clearly produces a lower semi-continuous function. Furthermore, since $a(x)$ is continuous, the function $\min_{i\in\{1,...,n\}} \frac{a_i(x)}{b_i}$ is continuous as well. So it remains to show that (\[LIM\]) holds. Without loss of generality, we may assume that $\min_{i\in\{1,...,n\}} \frac{a_i(x_0)}{b_i} =\frac{a_1(x_0)}{b_1}$. Setting $x=x_0$ and $y_2=y_3=...=y_n=0$ and letting $y_1\to 0^+$, we find that $f(x_0,y_1,0,...,0) \to \frac{a_1(x_0)}{b_1}.$ Hence, $\liminf_{x\to x_0, y\to 0+} f(x,y)\leq \frac{a_1(x_0)}{b_1}.$ We also observe that as long as $y\in {\mathbb{R}}^n_+ \backslash \{0\}$, the value $$\frac{a'(x)y}{b'y} = \sum_{i=1}^n \frac{a_i(x)}{b_i} \frac{b_i y_i}{b_1 y_1+ \cdots + b_n y_n}$$ is a convex linear combination of the values $\frac{a_i(x)}{b_i},\ i=1,...,n$. By continuity of $a(x)$, for any $\varepsilon >0$ there exists $\delta>0$ such that $\forall i\in\{1,...,n\}$ and $\forall x \in B_{\delta}(x_0)$, we have $a_i(x) > a_i(x_0)- \varepsilon b_i$. Hence, for all $x \in B_{\delta}(x_0)$ and for all $y\in {\mathbb{R}}^n_+\backslash \{0\}$, $f(x,y) \geq \frac{a_1(x_0)}{b_1} -\varepsilon$. We have established that $$\frac{a_1(x_0)}{b_1} \geq \liminf_{x\to x_0, y\to 0+} f(x,y)\geq \frac{a_1(x_0)}{b_1}-\varepsilon.$$ Since $\varepsilon>0$ is arbitrary, (\[LIM\]) follows. \[lowerbound\] Suppose that (\[T’s\]) holds. Then there exist $\eta,\mu_0>0$ such that $$\liminf_{t\to\infty} 1' V(t) \geq \eta >0$$ for any $\mu\in[0,\mu_0]$ and for any solution of $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ with $1' V(t)>0$. We will prove the claim for system $(\ref{multic1})-(\ref{multic3})$ (the proof for $(\ref{multic4})-(\ref{multic6})$ is similar). The proof consists of two parts. We first show that there exist $\eta_0,\mu_0>0$ such that $\liminf_{t\to\infty} 1'(T^*(t)+V(t)) \geq \eta_0 >0$ for all solutions with $T^*(t),V(t) \not=0$. We choose $n$ positive numbers $\tilde N_i$ so that $\frac{\gamma_i}{k_i T_0} < \tilde N_i<N_i$ for all $1 \leq i \leq n$. This is possible since we assume $\bar T_i =\frac{\gamma_i}{k_i N_i} < T_0$. Let $v=(\tilde N,1)$. It follows that $$v'\begin{pmatrix} -B & K T_0\cr \hat N B & -\Gamma\cr\end{pmatrix}= \biggl(b_1(N_1-\tilde N_1),...,b_n(N_n-\tilde N_n), k_1 T_0 \tilde N_1 - \gamma_1,..., k_n T_0 \tilde N_n - \gamma_n\biggr)$$ is a positive vector. By continuity, there exists a $\mu_0>0$ such that $$v'M(T,\mu),\quad {\rm where}\ M(T,\mu):=\begin{pmatrix} -B & P(\mu) K T\cr \hat N B & -\Gamma\cr\end{pmatrix}$$ is a positive vector for all $\mu \in [0,\mu_0]$. Consider a system $$\begin{aligned} {\dot T}&=&f(T)-k'VT,\ T \in {\mathbb{R}}_+\label{pers1}\\ {\dot T^*}&=&P(\mu)KVT-B T^*,\ T^* \in {\mathbb{R}}^n_+,\label{pers2}\\ {\dot V}&=&{\hat N}B T^*-\Gamma V,\ V\in {\mathbb{R}}^n_+,\label{pers3}\\ {\dot \mu}&=& 0,\ \mu \in [0,\mu_0].\label{pers4} \end{aligned}$$ Let $K'$ be the forward invariant compact set for $(\ref{multic1})-(\ref{multic3})$ established in Lemma \[bounded\] and define $K=K'\times[0,\mu_0]$. It is clear that $K$ is compact and forward invariant under $(\ref{pers1})-(\ref{pers4})$ The set $$K_0:=\left([0,T_0]\times \{0\} \times \{0\} \times [0,\mu_0]\right) \cap K$$ is clearly a compact forward invariant subset of $K$. Let $\Pi(T^*,V):=v'(T^*,V)$. The function $\Pi$ is clearly smooth, zero on $K_0$, and positive on $K\backslash K_0$. Furthermore, $$\frac{\dot \Pi}{\Pi}= \psi := \frac{v'M(T,\mu)(T^*,V)}{v'(T^*,V)}$$ is lower semi-continuous on $K$ by Lemma \[liminf\] once we define the value of $\psi$ on $K_0$ as $$\psi(T,\mu)=\min_{i=1,...,n} \frac{v'M(T,\mu)_i}{v_i} .$$ We note that the function $\psi(T,\mu)$ is continuous in $(T,\mu)$. Since all solutions of $(\ref{pers1})-(\ref{pers4})$ in $K_0$ have the property that $\lim_{t\to\infty}T(t)=T_0$, it implies that $\psi(T(t),\mu)>0$ for all sufficiently large $t$. Hence by Theorem \[patrickdoeshofbauer\], the set $K_0$ is a uniform strong repeller in $K$. If we use the $L^1$-norm of $(T^*,V)$ as the distance function to $K_0$, we find that there exists an $\eta_0>0$ such that $$\liminf_{t\to\infty} 1'(T^*+V) \geq \eta_0$$ for all solutions of $(\ref{pers1})-(\ref{pers4})$ in $K\backslash K_0$. To complete the proof, we need to show that there exists $\eta>0$ such that $\liminf_{t\to\infty} 1'V(t) \geq \eta >0$ for all solutions with $1'V(t)>0$. Observe that $1'V(t)>0$ implies that $1'T^*(t)>0$. Hence by the result of part one, we have that $\liminf_{t\to\infty} 1'(T^*(t)+V(t)) \geq \eta_0 >0$, or equivalently, $1'T^*(t) > \eta_0/2 - 1'V(t)$ for all sufficiently large $t$. We substitute this inequality into (\[multic3\]) and find that $$1'\dot V \geq A_0 \biggl(\frac{\eta_0}{2} - 1'V(t)\biggr)-A_1 1'V(t), \quad A_0:=\min_{i} (N_i \beta_i)>0, \ A_1:=\max_{i} (\gamma_i) >0$$ holds for large $t$. It follows immediately that $$\liminf_{t\to\infty} 1'V(t) \geq \eta=\frac{\eta_0 A_0}{2(A_0+A_1)}>0.$$ \[minilemma\] Let $$\sigma(x,y,z):= x+ y + \frac{z}{xy}-3 z^\frac{1}{3}.$$ Then for any $z_0,M>0$, there exists $\delta>0$ such that $\sigma(x,y,z)>M$ for all $0<x<\delta$, all $y>0$, and all $z>z_0$. Observe that the minimum of the function $\sigma(x,\cdot,z)$ on the set $y\in (0,+\infty)$ is achieved at $y=\sqrt{z/x}$. Hence for all $y>0$, it holds that $$f(x,y,z) \geq f(x,\sqrt{\frac{z}{x}},z)=x+ 2 \sqrt{\frac{z}{x}} -3 z^\frac{1}{3}.$$ Let $z_0>0$ and define $$\delta:=\frac{4 z_0}{\left( M+ 3 z_0^\frac{1}{3} \right)^2}.$$ Then for all $0<x<\delta$, all $y>0$, and all $z>z_0$, it holds that $$f(x,y,z) \geq 2 \sqrt{\frac{z}{x}} -3 z^\frac{1}{3} =z^\frac{1}{2} \left( \frac{2}{x^{\frac{1}{2}}}-3 z^{-\frac{1}{6}} \right) > z_0^\frac{1}{2} \left( \frac{2}{\delta^\frac{1}{2}}-3 z_0^{-\frac{1}{6}} \right)=M.$$ \[compactset\] Let $K$ be the absorbing compact set established in Lemma \[bounded\], and let $$U=\{ (T,T^*,V) \in {\mathbb{R}}^{2n+1}_+ | T,T^*_1,V_1>0 \}.$$ Suppose that ${\bf (C)}$ holds with ${\bar T_1}$ instead of ${\bar T}$. Then there exist $\mu_1>0$ and a compact set $K_{\delta} \subset U$ such that for any $\mu\in[0,\mu_1]$ and for any solution of $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$ in $U$, there exists a $t_0>0$ such that $ (T(t),T^*(t),V(t)) \in K_{\delta}$ for all $t >t_0$. Both for system $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$, the proof will be based on the same Lyapunov function $$W=\int_{{\bar T}^1}^T \left(1-\frac{{\bar T}^1}{\tau}\right) d \tau +\int_{{\bar T^*}_1}^{T^*_1} \left(1-\frac{{\bar T^*}_1}{\tau}\right)d \tau+ \frac{1}{N_1}\int_{{\bar V}_1}^{V_1} \left(1-\frac{{\bar V}_1}{\tau}\right) d \tau+\sum_{i>1}T^*_i+\frac{1}{N_i}V_i$$ that we used to show competitive exclusion with $\mu=0$. [**Case 1**]{}: System $(\ref{multic1})-(\ref{multic3})$. Computing $\dot W$ for system $(\ref{multic1})-(\ref{multic3})$, we obtain after some simplifications $$\begin{aligned} {\dot W}&=&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3\right]\\ &&-\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) +\frac{T^*_1-\bar T^*_1}{T^*_1} \mu \sum_{j=1}^n q_{1j} k_j V_j T + \mu \sum_{i=2}^n \sum_{j=1}^n q_{ij} k_j V_j T. \end{aligned}$$ Recombining the terms, we further obtain $$\begin{aligned} {\dot W}&=&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3\right]\\ &&-\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) -\frac{\bar T^*_1}{T^*_1} \mu \sum_{j=1}^n q_{1j} k_j V_j T + \mu \sum_{i=1}^n \sum_{j=1}^n q_{ij} k_j V_j T. \end{aligned}$$ We note that $$\sum_{i=1}^n \sum_{j=1}^n q_{ij} k_j V_j T = \sum_{j=1}^n \biggl( \sum_{i=1}^n q_{ij}\biggr) k_j V_j T=0$$ since all column sums of $Q$ are zero. Hence, $$\begin{aligned} {\dot W}&=&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3\right]\\ &&-\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) - \frac{\bar T^*_1}{T^*_1} \mu q_{11} k_1 V_1 T - \frac{\bar T^*_1}{T^*_1}\mu \sum_{j=2}^n q_{1j} k_j V_j T. \end{aligned}$$ We rewrite $\dot W$ as $$\begin{aligned} {\dot W}&=&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+(1+q_{11}\mu)\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && + 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) - \frac{\bar T^*_1}{T^*_1}\mu \sum_{j=2}^n q_{1j} k_j V_j T. \end{aligned}$$ Note that the last term of $\dot W$ is non-positive, hence[^3] $$\begin{aligned} {\dot W}&\leq &(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+(1+q_{11}\mu)\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && + 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1). \end{aligned}$$ By Lemma \[lowerbound\], there exist $\eta,\mu_a>0$ such that $1'V(t) > \eta$ for all $\mu \in [0,\mu_a]$ and all sufficiently large $t$. Let $\alpha=\min_{i \geq 2} k_i({\bar T}^i-{\bar T}^1)>0$, then $$\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) \geq \alpha \sum_{i=2}^n V_i \geq \alpha (\eta- V_1).$$ Thus, by shifting time forward if necessary, we have the inequality $$\begin{aligned} {\dot W}&\leq &(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+(1+q_{11}\mu)\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && + 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\alpha \eta + \alpha V_1. \end{aligned}$$ Let $\mu_b>0$ be such that for all $\mu \in [0,\mu_b]$ , $$1+q_{11} \mu \in \left[\frac{1}{2},1\right], \quad 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\alpha \eta \leq -\frac{\alpha \eta}{2}.$$ Let $\mu_1=\min(\mu_a,\mu_b)$ and choose sufficiently large $L>0$ so that $$3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\alpha \eta + \alpha V_1<L$$ for all solutions of $(\ref{multic1})-(\ref{multic3})$ in $K$ and all $\mu \in [0,\mu_1]$. For any $\mu \in [0,\mu_1]$, we have that $$\begin{aligned} {\dot W}&\leq &(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+(1+q_{11}\mu)\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && -\frac{\alpha \eta}{2} + \alpha V_1, \end{aligned}$$ where the first two terms are non-positive and $1+q_{11}\mu \in \left[\frac{1}{2},1\right]$. Inspecting the first term in $\dot W$, we find that there exists $\delta_0>0$ such that $$(f(T)-f({\bar T}^1)\left(1-\frac{{\bar T}^1}{T} \right)<-(L+1)$$ for all $T<\delta_0$ and all $\mu \in [0,\mu_1]$. Now we inspect the the second term in $\dot W$. Using Lemma \[minilemma\] with $$x=\frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}, \quad y= \frac{{\bar T}^1}{T}, \quad z=1+q_{11}\mu, \quad z_0=\frac{1}{2},$$ we conclude that there exists $\delta_1>0$ such that $$-\beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+(1+q_{11}\mu)\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]<-(L+1)$$ for all $\frac{T^*_1}{V_1}<\delta_1$ and all $\mu \in [0,\mu_1]$. Finally, there exists $\delta_2>0$ such that $-\frac{\alpha\eta}{2}+ \alpha V_1 <-\frac{\alpha \eta}{4}$ for all $V_1<\delta_2$ and all $\mu \in [0,\mu_1]$. Let $$\hat K_{\delta}=\{ (T,T^*,V) \in K \cap U | T \geq \delta_0, V_1 \geq \delta_2, T^*_1 \geq \delta_1 V_1\}.$$ Consider $(T,T^*,V) \in (K \cap U) \backslash \hat K_{\delta}$ and let $\mu \in [0,\mu_1]$, then at least one of the following holds: - $T <\delta_0$, in which case $ \dot W \leq -(L+1) + L \leq -1$; - $T^*_1/V_1 < \delta_1$, in which case $ \dot W \leq -(L+1) + L \leq -1$; - $ V_1 < \delta_2$, in which case $\dot W \leq -\frac{\alpha \eta}{4}$; Hence, for all $(T,T^*,V) \in (K \cap U) \backslash \hat K_{\delta}$ and all $\mu \in [0,\mu_1]$, we have $$\dot W \leq -\min(1,\frac{\alpha \eta}{4})<0.$$ We postpone the rest of the proof until we have showed that a similar inequality holds for system $(\ref{multic4})-(\ref{multic6})$. [**Case 2**]{}: System $(\ref{multic4})-(\ref{multic6})$. Computing $\dot W$ for system $(\ref{multic4})-(\ref{multic6})$, we obtain after some simplifications $$\begin{aligned} {\dot W}&=&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3\right]\\ &&-\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) + \mu\left(\frac{V_1-{\bar V_1}}{V_1} \right)\sum_{j=1}^n q_{1j}\frac{N_j}{N_1}\beta_j T^*_j+ \mu\sum_{i=2}^n\sum_{j=1}^n q_{ij}\frac{N_j}{N_i}\beta_j T^*_j. \end{aligned}$$ Note that the $\mu$ dependent terms can be rearranged as follows: $$\mu\left(\sum_{i=1}^nq_{ii} \beta_i T^*_i - \frac{{\bar V_1}}{V_1}\sum_{j=2}^nq_{1j}\frac{N_j}{N_1}\beta_j T^*_j\right) +\mu\left(\sum_{j=2}^nq_{1j}\frac{N_j}{N_1}\beta_j T^*_j +\sum_{i=2}^n \sum_{j\neq i}^n q_{ij}\frac{N_j}{N_i}\beta_j T^*_j\right) -\mu\frac{{\bar V_1}}{V_1}q_{11}\beta_1 T^*_1.$$ In the above the first term is non-positive, and the second term can be re-written as follows: $$\mu\sum_{i=1}^n\alpha_iT^*_i,$$ for suitable $\alpha_i\geq 0$, and the third term will be absorbed in the square bracket $[ \;\;]$ term in ${\dot W}$. We find that $$\begin{aligned} {\dot W}&\leq&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ (1+q_{11}\mu)\frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && + 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})+\mu \sum_{i=1}^n \alpha_i T^*_i -\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1). \end{aligned}$$ By Lemma \[lowerbound\], there exist $\eta,\mu_a>0$ such that $1'V(t) > \eta$ for all $\mu \in [0,\mu_a]$ and all sufficiently large $t$. Let $\alpha=\min_{i \geq 2} k_i({\bar T}^i-{\bar T}^1)>0$, then $$\sum_{i=2}^nk_iV_i({\bar T}^i-{\bar T}^1) \geq \alpha \sum_{i=2}^n V_i \geq \alpha (\eta- V_1).$$ Thus, by shifting time forward if necessary, we have the inequality $$\begin{aligned} {\dot W}&\leq&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ (1+q_{11}\mu)\frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && + 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})+\mu \sum_{i=1}^n \alpha_i T^*_i -\alpha \eta + \alpha V_1. \end{aligned}$$ Since solutions are in the compact set $K$ for sufficiently large times, there is some $\mu'_a>0$ such that $$\mu\sum_{i=1}^n \alpha_i T^*_i \leq \frac{\alpha \eta}{2},\;\; \forall \; \mu \in [0,\mu_a'],$$ and therefore $$\begin{aligned} {\dot W}&\leq&(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ (1+q_{11}\mu)\frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && + 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\frac{\alpha \eta}{2} + \alpha V_1. \end{aligned}$$ Let $\mu_b>0$ be such that for all $\mu \in [0,\mu_b]$, $$1+q_{11} \mu \in \left[\frac{1}{2},1\right], \quad 3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\frac{\alpha \eta}{2} \leq -\frac{\alpha \eta}{4}.$$ Let $\mu_1=\min(\mu_a,\mu_a',\mu_b)$ and choose sufficiently large $L>0$ so that $$3\beta_1 {\bar T^*}_1(1-(1+q_{11}\mu)^{1/3})-\alpha \eta + \alpha V_1<L$$ for all solutions of $(\ref{multic4})-(\ref{multic6})$ in $K$ and all $\mu \in [0,\mu_1]$. For any $\mu \in [0,\mu_1]$, we have that $$\begin{aligned} {\dot W}&\leq &(f(T)-f({\bar T}^1))\left(1-\frac{{\bar T}^1}{T} \right)- \beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ (1+q_{11}\mu)\frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]\\ && -\frac{\alpha \eta}{4} + \alpha V_1, \end{aligned}$$ where the first two terms are non-positive and $1+q_{11}\mu \in \left[\frac{1}{2},1\right]$. Inspecting the first term in $\dot W$, we find that there exists $\delta_0>0$ such that $$(f(T)-f({\bar T}^1)\left(1-\frac{{\bar T}^1}{T} \right)<-(L+1)$$ for all $T<\delta_0$ and all $\mu \in [0,\mu_1]$. Inspecting the second term in $\dot W$, we use Lemma \[minilemma\] with $$x=(1+q_{11}\mu) \frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}, \quad y= \frac{{\bar T}^1}{T}, \quad z=1+q_{11}\mu, \quad z_0=\frac{1}{2},$$ and conclude that there exists $\delta_1>0$ such that $$-\beta_1 {\bar T^*}_1\left[\frac{{\bar T}^1}{T}+\frac{{\bar T^*}_1V_1T}{T^*_1{\bar V}_1{\bar T}^1}+ (1+q_{11}\mu)\frac{{\bar V}_1T^*_1}{V_1{\bar T^*}_1}-3(1+q_{11}\mu)^{1/3}\right]<-(L+1)$$ for all $\frac{T^*_1}{V_1}<\delta_1$ and all $\mu \in [0,\mu_1]$. Finally, there exists $\delta_2>0$ such that $-\frac{\alpha\eta}{4}+ \alpha V_1 <-\frac{\alpha \eta}{8}$ for all $V_1<\delta_2$ and all $\mu \in [0,\mu_1]$. Let $$\hat K_{\delta}=\{ (T,T^*,V) \in K \cap U | T \geq \delta_0, V_1 \geq \delta_2, T^*_1 \geq \delta_1 V_1\}.$$ Consider $(T,T^*,V) \in (K \cap U) \backslash \hat K_{\delta}$ and let $\mu \in [0,\mu_1]$, then at least one of the following holds: - $T <\delta_0$, in which case $ \dot W \leq -(L+1) + L \leq -1$; - $T^*_1/V_1 < \delta_1$, in which case $ \dot W \leq -(L+1) + L \leq -1$; - $ V_1 < \delta_2$, in which case $\dot W \leq -\frac{\alpha \eta}{8}$; Hence, for all $(T,T^*,V) \in (K \cap U) \backslash \hat K_{\delta}$ and all $\mu \in [0,\mu_1]$, we have $$\dot W \leq -\min(1,\frac{\alpha \eta}{8})<0.$$ The remainder of the proof is the same for both of the above two cases and presented next. The non-negative function $W(T,T^*,V,\mu)$ is continuous and bounded from above on the set $\hat K_{\delta} \times [0,\mu_1]$ because $T,T_1^*,V_1$ are bounded away from zero. Hence it attains a finite positive maximum $$w:=\max_{\hat K_{\delta} \times [0,\mu_1]} W(T,T^*,V,\mu)>0.$$ Define a new set $$K_{\delta}=\{ (T,T^*,V) \in K \cap U | W(T,T^*,V,\mu) \leq w, \forall \mu \in [0,\mu_1]\}.$$ By construction, we have that $\hat K_{\delta} \subset K_{\delta} \subset K \cap U$. The continuity of $W$ implies that $K_{\delta}$ is closed, and therefore compact in $U$. It remains to show that all solutions of $(\ref{multic1})-(\ref{multic3})$ in $U$ enter and remain in $K_{\delta}$ for all sufficiently large times. Since $K \cap U$ is an absorbing set for all $\mu \geq 0$ (Lemma \[bounded\]), without loss of generality we need to prove this for all solutions in $K \cap U$. Let $\Phi(t)=(T(t),T^*(t),V(t)) \in K \cap U$ be a solution of $(\ref{multic1})-(\ref{multic3})$ for some fixed $\mu \in [0,\mu_1]$. Observe that in the set $(K \cap U) \backslash \hat K_{\delta}$, the inequality $\dot W \leq -\min(1,\frac{\alpha \eta}{8})<0$ holds. Since $W \geq 0$, there exists $t_0 \geq 0$ such that $\Phi(t_0) \in \hat K_{\delta} \subset K_{\delta}$. We will show that $\Phi(t) \in K_{\delta}$ for all $t \geq t_0$. For the sake of contradiction, let us suppose that there exists $t_1>t_0$ such that $\Phi(t_1) \notin K_{\delta}$. Then there exists $t_2 \in [t_0,t_1)$ such that $\Phi(t_2) \in K_{\delta}$ and $\Phi(t) \notin K_{\delta}$ for all $t\in(t_2,t_1]$. On the one hand, we have that $$W(\Phi(t_2),\mu) \leq w < W(\Phi(t_1),\mu)$$ by definition of $K_{\delta}$. On the other hand, for all $t\in(t_2,t_1]$, we have $\Phi(t) \notin K_{\delta}$ and consequently $\Phi(t) \notin \hat K_{\delta}$ so that $\frac{d}{dt} W(\Phi(t),\mu)=\dot W <0$. This contradiction shows that $\Phi(t) \in K_{\delta}$ for all $t \geq t_0$ and concludes the proof of the Theorem. \[global\] Let the assumptions of Lemma \[invertible\] hold, let $U$ be the set from Theorem $\ref{compactset}$, and define $$U'=\{ (T,T^*,V) \in {\mathbb{R}}^{2n+1}_+ | \; \; T^*_1+V_1>0 \}\supset U.$$ Then there exist $\mu_0>0$ and a continuous map $E: [0,\mu_0] \rightarrow U$ such that 1. $E(0)=E_1$ (where $E_1$ is the same as in Lemma \[invertible\]), and $E(\mu)$ is an equilibrium of $(\ref{multic1})-(\ref{multic3})$ or of $(\ref{multic4})-(\ref{multic6})$ for all $\mu \in [0,\mu_0]$; 2. For each $\mu \in [0,\mu_0]$, $E(\mu)$ is a globally asymptotically stable equilibrium of $(\ref{multic1})-(\ref{multic3})$ or of $(\ref{multic4})-(\ref{multic6})$ in $U'$. To prove the first assertion, we begin by noting that for $\mu=0$, $E_1$ is a stable hyperbolic equilibrium of $(\ref{multic1})-(\ref{multic3})$ or of $(\ref{multic4})-(\ref{multic6})$ by Lemma $\ref{invertible}$. Since the vector field of $(\ref{multic1})-(\ref{multic3})$ and $(\ref{multic4})-(\ref{multic6})$ is linear in $\mu$, by the Implicit Function Theorem there exist $h>0$ and a continuous map $E:(-h,h) \rightarrow {\mathbb{R}}^{2n+1}$ such that $E(\mu)$ is an equilibrium of $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$ for all $\mu \in (-h,h)$. The fact that $E(\mu) \in U$ for all $\mu \in [0,h)$ follows from Proposition \[reduce\] and the fact that $\bar T_1 < \bar T_i,\ i \geq 2$. Note that for $\mu>0$, $E(\mu)$ may be positive (if $Q$ is irreducible) or non-negative (if $Q$ is reducible). Nevertheless, in both cases, $\mu>0$ implies $E(\mu) \in U$. The proof of the second assertion is based on the result of Smith and Waltman (Corollary 2.3 in [@smith-waltman]). We have already established the fact that $E(0)$ is a stable hyperbolic equilibrium of $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$. By Theorem \[multistrain\], $E(0)$ is globally asymptotically stable in $U'$ for $\mu=0$. In addition, by Theorem \[compactset\] there exist $\mu_0>0$ and a compact set $K_{\delta} \subset U$ such that for each $\mu \in[0,\mu_0]$, and each solution $(T(t),T^*(t),V(t))$ of $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$ in $U$, there exists $t_0>0$ such that $(T(t),T^*(t),V(t)) \in K_{\delta}$ for all $t>t_0$. Hence, the condition (H1) of Corollary 2.3 in [@smith-waltman] holds. The Proposition 2.3 itself then implies the global stability of $E(\mu)$ in $U$ for all sufficiently small $\mu\geq 0$. Finally, solutions of $(\ref{multic1})-(\ref{multic3})$ or $(\ref{multic4})-(\ref{multic6})$ starting in $U'$ enter $U$ instantaneously, hence global stability of $E(\mu)$ in $U'$ follows as well. Appendix: Inclusion of loss of virus in the model {#appendix-inclusion-of-loss-of-virus-in-the-model .unnumbered} ================================================= Single-strain {#single-strain .unnumbered} ------------- When taking the loss of the virus particle upon infection into account, model $(\ref{hiv1})$ becomes $$\begin{aligned} \label{hiv1a} {\dot T}&=&f(T)-kVT\nonumber \\ {\dot T^*}&=&kVT-\beta T^*\nonumber \\ {\dot V}&=&N\beta T^*-\gamma V-kVT, \end{aligned}$$ We still assume that the growth rate of the healthy cell population is given by $(\ref{T0})$, hence $E_0=(T_0,0,0)$ is still an equilibrium of $(\ref{hiv1a})$. A second, positive equilibrium may exist if the following quantities are positive: $$\label{eqa} {\bar T}=\frac{\gamma}{k(N-1)},\;\; {\bar T^*}=\frac{f({\bar T})}{\beta},\;\; {\bar V}=\frac{f({\bar T})}{k{\bar T}}.$$ Note that this is the case iff $N>1$ and $f\left(\frac{\gamma}{k(N-1)}\right)>0$, or equivalently by $(\ref{T0})$ that ${\bar T}=\frac{\gamma}{k(N-1)}<T_0$. In terms of the basic reproduction number $${\cal R}^0:=\frac{k(N-1)}{\gamma}T_0 =\frac{T_0}{\bar T},$$ existence of a positive equilibrium is therefore equivalent to ${\cal R}^0>1$. Assuming that ${\cal R}^0>1$, we will still denote this disease steady state by $E=({\bar T},{\bar T^*},{\bar V})$. We introduce the following condition. $${\bf (C')}\;\;f'(c)+\frac{k}{\gamma}f({\bar T})\leq 0, \textrm{ for all } c\in [0,T_0].$$ Note that this condition is satisfied when $f(T)$ is a decreasing function with sufficiently large negative derivative. \[1straina\] Let ${\bf (C')}$ hold. Then the equilibrium $E$ is globally asymptotically stable for $(\ref{hiv1a})$ with respect to initial conditions satisfying $T^*(0)+V(0)>0$. Consider the following function on ${\operatorname{int}}({\mathbb{R}}^3_+)$: $$W=(N-1)\int_{{\bar T}}^T \left(1-\frac{{\bar T}}{\tau}\right) d \tau +N\int_{{\bar T^*}}^{T^*} \left(1-\frac{{\bar T^*}}{\tau}\right)d \tau+ \int_{{\bar V}}^V \left(1-\frac{{\bar V}}{\tau}\right) d \tau.$$ Then $$\begin{aligned} {\dot W}&=&(N-1)(f(T)-kVT)\left(1-\frac{{\bar T}}{T} \right)+N(kVT-\beta T^*)\left(1-\frac{{\bar T^*}}{T^*} \right)+ (N\beta T^*-\gamma V-kVT)\left(1-\frac{{\bar V}}{V} \right)\\ &=&(N-1)f(T)\left(1-\frac{{\bar T}}{T}\right)-NkVT\frac{{\bar T^*}}{T^*}+N\beta{\bar T^*}-N\beta T^* \frac{{\bar V}}{V}+\gamma {\bar V} +k{\bar V}T\\ &=&(N-1)(f(T)-f({\bar T}))\left(1-\frac{{\bar T}}{T}\right)+(N-1)f({\bar T})\left(1-\frac{{\bar T}}{T}\right) +N\beta{\bar T^*}\left[2-\frac{VT{\bar T^*}}{{\bar V}{\bar T}T^*}-\frac{T^*{\bar V}}{{\bar T^*}V}\right] -\beta{\bar T^*}+\beta {\bar T^*}\frac{T}{{\bar T}}\\ &=&(N-1)(f(T)-f({\bar T}))\left(1-\frac{{\bar T}}{T}\right)+(N-1)\beta {\bar T^*}\left(1-\frac{{\bar T}}{T}\right) +N\beta{\bar T^*}\left[2-\frac{VT{\bar T^*}}{{\bar V}{\bar T}T^*}-\frac{T^*{\bar V}}{{\bar T^*}V}\right] -\beta{\bar T^*}+\beta {\bar T^*}\frac{T}{{\bar T}}\\ &=&(N-1)(f(T)-f({\bar T}))\left(1-\frac{{\bar T}}{T}\right)+\beta{\bar T^*}\left(-2+\frac{{\bar T}}{T}+\frac{T}{{\bar T}}\right) +N\beta{\bar T^*}\left[3-\frac{VT{\bar T^*}}{{\bar V}{\bar T}T^*}-\frac{T^*{\bar V}}{{\bar T^*}V}-\frac{{\bar T}}{T}\right]\\ &=&\left[(N-1)(f(T)-f({\bar T})){\bar T}+\beta {\bar T^*}(T-{\bar T}) \right]\frac{(T-{\bar T})}{T{\bar T}} +N\beta{\bar T^*}\left[3-\frac{VT{\bar T^*}}{{\bar V}{\bar T}T^*}-\frac{T^*{\bar V}}{{\bar T^*}V}-\frac{{\bar T}}{T}\right]\\ \end{aligned}$$ where we used $(\ref{eqa})$ repeatedly; in particular in the second, third and fourth equation. By the mean value theorem there is some $c\in (T,{\bar T})$ or $({\bar T},T)$ such that $$f(T)-f({\bar T})=f'(c)(T-{\bar T}),$$ hence using $(\ref{eqa})$ once more $${\dot W}=(N-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T}) \right]\frac{(T-{\bar T})^2}{T} +N\beta{\bar T^*}\left[3-\frac{VT{\bar T^*}}{{\bar V}{\bar T}T^*}-\frac{T^*{\bar V}}{{\bar T^*}V}-\frac{{\bar T}}{T}\right].$$ The first term is non-positive by ${\bf (C')}$ and because we can assume that $T\leq T_0$ by dissipativity (see Lemma $\ref{boundedKVT}$ later). The second term is non-positive as well since the geometric mean of $3$ non-negative numbers is not larger than the arithmetic mean of those numbers. We conclude that ${\dot W}\leq 0$ in ${\operatorname{int}}({\mathbb{R}}^3_+)$, hence local stability of $E$ follows. Notice that ${\dot W}$ equals zero if and only if both the first term and the second term are zero, This happens at points where: $$\frac{{\bar T}}{T}=1\textrm{ and } \frac{{\bar T^*}V}{T^*{\bar V}}=1.$$ Then LaSalle’s Invariance Principle implies that all bounded solutions (and as before, solutions are easily shown to be bounded, see also Lemma \[boundedKVT\] later) in ${\operatorname{int}}({\mathbb{R}}^3_+)$ converge to the largest invariant set in $$M=\{(T,T^*,V)\in {\operatorname{int}}({\mathbb{R}}^3_+)\;|\; \frac{{\bar T}}{T}=1,\;\; \frac{{\bar T^*}V}{T^*{\bar V}}=1\}.$$ It is clear that the largest invariant set in $M$ is the singleton $\{E\}$. Finally, note that forward solutions starting on the boundary of ${\mathbb{R}}^3_+$ with either $T_1(0)$ or $V_1(0)$ positive, enter ${\operatorname{int}}({\mathbb{R}}^3_+)$ instantaneously. This concludes the proof. Competitive exclusion {#competitive-exclusion .unnumbered} --------------------- Now we modify the multi-strain model $(\ref{multi1})-(\ref{multi3})$ to $$\begin{aligned} {\dot T}&=&f(T)-kVT, \quad T \in {\mathbb{R}}_+\label{hiv2a} \\ {\dot T^*}&=&KVT-B T^*,\quad T^* \in {\mathbb{R}}^n_+\label{hiv2b} \\ {\dot V}&=&\hat N B T^*-\Gamma V-KVT, \quad V \in {\mathbb{R}}^n_+\label{hiv2c}, \end{aligned}$$ where $k=(k_1,...,k_n)$, $K={\operatorname{diag}}(k_1,...,k_n)$, $B={\operatorname{diag}}(\beta_1,...,\beta_n)$, $\hat N={\operatorname{diag}}(N_1,...,N_n)$, and $\Gamma={\operatorname{diag}}(\gamma_1,...,\gamma_n)$. Suppose that each strain is capable to persist at steady state by itself, that is, $N_i>1$ and $\bar T_i =\frac{\gamma_i}{k_i(N_i-1)} < T_0$ and denote the corresponding equilibria also by $E_1,\dots, E_n$. Assume that $$\label{nogeens} 0< \bar T_1 \leq \bar T_2 \leq \ldots \leq \bar T_n < T_0.$$ In addition, suppose that ${\bf (C')}$ holds with $\bar T=\bar T_1$. Then we have the following. \[multistrain2\] The single strain equilibrium $E_1$ is globally asymptotically stable for $(\ref{hiv2a})-(\ref{hiv2c})$ with respect to initial conditions satisfying $T_1^*(0)+V_1(0)>0$. Consider the function $W$ defined on $U:=\{(T,T^*,V)\in {\mathbb{R}}^{2n+1}\;|\; T,T_1^*,V_1>0\}$ as $$\begin{aligned} W &=& (N_1-1)\int_{{\bar T_1}}^T \left(1-\frac{{\bar T_1}}{\tau}\right) d \tau +N_1\int_{{\bar T_1^*}}^{T_1^*} \left(1-\frac{{\bar T_1^*}}{\tau}\right)d \tau+ \int_{{\bar V_1}}^{V_1} \left(1-\frac{{\bar V_1}}{\tau}\right) d \tau\\ && + \sum_{i=2}^n \frac{N_1-1}{N_i-1}(N_i T^*_i +V_i). \end{aligned}$$ Computing $\dot W$, we find that $$\begin{aligned} \dot W &=& (N_1-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T_1}) \right]\frac{(T-{\bar T_1})^2}{T} +N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & +\sum_{i=2}^n \left(-k_i V_i (T-\bar T_1) + \frac{(N_1-1)}{N_i-1} (N_i k_i V_i T-N_i \beta_i T^*_i + N_i \beta_i T^*_i - \gamma_i - k_i V_i T) \right). \end{aligned}$$ After simplifications, we have $$\begin{aligned} \dot W &=& (N_1-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T_1}) \right]\frac{(T-{\bar T_1})^2}{T} +N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -(N_1-1) \sum_{i=2}^n k_i V_i (\bar T_i -\bar T_1). \end{aligned}$$ The first term is non-positive since ${\bf (C')}$ with ${\bar T}={\bar T_1}$ holds and because $T\leq T_0$ by disspiativity (see Lemma $\ref{boundedKVT}$ later). The second term is non-positive is well, and so is the third by $(\ref{nogeens})$. Thus ${\dot W}\leq 0$ which already implies that $E_1$ is stable. An application of LaSalle’s Invariance Principle shows that all bounded solutions in $U$ (boundedness follows from Lemma $\ref{boundedKVT}$ which is proved later) converge to the largest invariant set in $$\left\{(T,T^*_1,\dots,T^*_n,V_1,\dots,V_n)\in U\;|\;\frac{{\bar T^1}}{T}=1,\;\; \frac{{\bar T^*_1}V_1}{T^*_1{\bar V_1}}=1 ,\;\;V_i=0,\;\; i>2 \right\},$$ which is easily shown to be the singleton $\{E_1\}$. Finally, solutions on the boundary of $U$ with $T_1^*(0)+V_1(0)>0$ enter $U$ instantaneously, which concludes the proof. Adding mutations {#adding-mutations .unnumbered} ---------------- We modify the model $(\ref{hiv2a})-(\ref{hiv2c})$ to account for mutations. Again, we consider two alternative models $$\begin{aligned} \label{hiv3a} {\dot T}&=&f(T)-kVT,\quad T \in {\mathbb{R}}_+\nonumber \\ {\dot T^*}&=&P(\mu)KVT- B T^*, \quad T^* \in {\mathbb{R}}^n_+\nonumber \\ {\dot V}&=&\hat N B T^*-\Gamma V-KVT, \quad V \in {\mathbb{R}}^n_+, \end{aligned}$$ and $$\begin{aligned} \label{hiv3b} {\dot T}&=&f(T)-kVT,\quad T \in {\mathbb{R}}_+\nonumber \\ {\dot T^*}&=&KVT- B T^*, \quad T^* \in {\mathbb{R}}^n_+\nonumber \\ {\dot V}&=&P(\mu)\hat N B T^*-\Gamma V-KVT,\quad V \in {\mathbb{R}}^n_+, \end{aligned}$$ where $k,K,B,\hat N, \Gamma$ are the same as before, and $P(\mu)=I +\mu Q$ and $Q$ is a stochastic matrix with non-negative off-diagonal entries. \[boundedKVT\] Both systems $(\ref{hiv3a})$ and $(\ref{hiv3b})$ are dissipative, i.e. there is some compact set $K$ such that every solution eventually enters $K$ and remains in $K$ forever after. The proof is similar to the proof of Lemma \[bounded\] and will be omitted. \[invertibleKVT\] For $\mu =0$, let all single strain equilibria $E_1,E_2\dots,E_n$ exist for either $(\ref{hiv3a})$ or $(\ref{hiv3b})$, and assume that $$\label{c1KVT} {\bar T^1}<{\bar T^2}<\dots<{\bar T^n}<{\bar T^{n+1}}:=T_0,$$ and $$\label{c2KVT} f'({\bar T}^j)\leq 0,\textrm{ for all }j=1,\dots, n+1.$$ Then the Jacobian matrices of $(\ref{hiv3a})$ or $(\ref{hiv3b})$, evaluated at any of the $E_i$’s, $i=1,\dots, n+1$ (where $E_{n+1}:=E_0$) have the following properties: $J(E_i)$ has $i-1$ eigenvalues (counting multiplicities) in the open right half plane and $2(n+1)-i$ eigenvalues in the open left half plane. In particular, $J(E_1)$ is Hurwitz. The proof is similar to that of Lemma $\ref{invertible}$. The only difference is that the entries of the Jacobian matrices change. In particular, the $(3,1)$ and $(3,3)$ entry of $A_1^i$ now become $-k_i{\bar V_i}$ and $-\gamma_i-k{\bar T^i}$ respectively, but by $(\ref{c1KVT})$ and Lemma $3.4$ in [@hiv], $A_1^i$ is still Hurwitz. To study equilibria of systems $(\ref{hiv3a})$ and $(\ref{hiv3b})$, we introduce the matrix $$A(\mu) =\Gamma^{-1}(\hat N P(\mu) - I)K, \label{AKVT}$$ which has non-negative off-diagonal entries for $\mu>0$ and $$A(0)={\operatorname{diag}}\left(\frac{k_1(N_1-1)}{\gamma_1},\ldots, \frac{k_n(N_n-1)}{\gamma_n} \right)= {\operatorname{diag}}\left(\frac{1}{\bar T_1},\ldots, \frac{1}{\bar T_n} \right).$$ Clearly, Proposition \[eigenvectors\] holds with $A(\mu)$ given by (\[AKVT\]). Hence, we have the following. \[reduceKVT\] Let the assumptions of Lemma $\ref{invertibleKVT}$ hold and suppose that the strains are renumbered so that $A(\mu)$ has the form (\[Amu\]). Let $E_j(\mu)=(\hat T_j(\mu),\hat T^*_j(\mu),\hat V_j(\mu))$ denote the nontrivial equilibria of both $(\ref{hiv3a})$ and $(\ref{hiv3b})$ for small $\mu>0$. Then 1. $E_j(\mu)$ is positive if and only if $\frac{1}{\hat T_j(\mu)}$ is an eigenvalue of $A(\mu)$ with a positive eigenvector $V_j$. 2. $E_j(\mu)$ is non-negative if and only if $\frac{1}{\hat T_j(\mu)}$ is an eigenvalue of $A(\mu)$ with a non-negative eigenvector $V_j$. 3. $E_j(\mu) \notin {\mathbb{R}}^{2n+1}_+$ if and only if $\frac{1}{\hat T_j(\mu)}$ is an eigenvalue of $A(\mu)$ with eigenvector $V_j$ which is not sign-definite. We will prove the Proposition only for system $(\ref{hiv3a})$ (the proof for $(\ref{hiv3b})$ is similar). Observe that at equilibrium, $\hat T^*_j(\mu)=({\hat N}B)^{-1}(\Gamma+K \hat T_j(\mu)) {\hat V_j}(\mu)$. Hence, the the signs of the corresponding components of $\hat T^*_j(\mu)$ and ${\hat V_j}(\mu)$ are the same. Substituting the above expression into $(\ref{hiv3a})$, we find that ${\hat V_j}(\mu)$ must satisfy $$[\Gamma^{-1} ({\hat N}P(\mu)-I)K-\frac{1}{{\hat T_j}(\mu)}I]{\hat V_j}(\mu)= [A(\mu)-\frac{1}{{\hat T_j}(\mu)}I]{\hat V_j}(\mu)=0.$$ Thus for each nontrivial equilibrium $E_j(\mu)$, the quantity $\frac{1}{{\hat T_j}(\mu)}$ must be an eigenvalue of $A(\mu)$ and ${\hat V_j}(\mu)$ must be a multiple of the corresponding eigenvector $V_j$. If $V_j$ is not sign definite, it follows that $E_j(\mu) \notin {\mathbb{R}}^{2n+1}_+$. For all $V_j \geq 0$, the components of $E_j(\mu)$ are uniquely determined via $$\hat V_j(\mu)=\frac{f(\hat T_j(\mu))}{\hat T_j(\mu) k' V_j} V_j, \quad \hat T^*_j(\mu)=({\hat N}B)^{-1}(\Gamma+ K \hat T_j(\mu)) {\hat V_j}(\mu).$$ Hence $E_j(\mu)$ is positive (non-negative) if and only if $V_j$ is positive (non-negative). Lower bounds {#lower-bounds .unnumbered} ------------ \[lowerboundKVT\] Suppose that $(\ref{c1KVT})$ holds. Then there exist $\eta,\mu_0>0$ such that $$\liminf_{t\to\infty} 1' V(t) \geq \eta >0$$for any $\mu\in[0,\mu_0]$ and for any solution of $(\ref{hiv3a})$ and $(\ref{hiv3b})$ with $1' V(t)>0$. We will prove the claim for system $(\ref{hiv3a})$(the proof for $(\ref{hiv3b})$ is similar). The proof consists of two parts. We first show that there exist $\eta_0,\mu_0>0$ such that $\liminf_{t\to\infty} 1'(T^*(t)+V(t)) \geq \eta_0 >0$ for all solutions with $T^*(t),V(t) \not=0$. We choose $n$ positive numbers $\tilde N_i$ so that $\frac{\gamma_i +k_i T_0}{k_i T_0} < \tilde N_i<N_i$ for all $1 \leq i \leq n$. This is possible since we assume $\bar T_i =\frac{\gamma_i}{k_i (N_i-1)} < T_0$ which is equivalent to $N_i > \frac{\gamma_i +k_i T_0}{k_i T_0}$. Let $v=(\tilde N,1)$. It follows that $$v'\begin{pmatrix} -B & K T_0\cr \hat N B & -\Gamma-KT_0\cr\end{pmatrix}= \biggl(b_1(N_1-\tilde N_1),...,b_n(N_n-\tilde N_n), k_1 T_0 \tilde N_1 - (\gamma_1+k_1 T_0),..., k_n T_0 \tilde N_n - (\gamma_n+k_n T_0)\biggr)$$ is a positive vector. By continuity, there exists a $\mu_0>0$ such that $$v'M(T_0,\mu),\quad {\rm where}\ M(T,\mu):=\begin{pmatrix} -B & P(\mu) K T\cr \hat N B & -\Gamma-K T\cr\end{pmatrix}$$ is a positive vector for all $\mu \in [0,\mu_0]$. Consider a system $$\begin{aligned} {\dot T}&=&f(T)-k'VT,\ T \in {\mathbb{R}}_+\label{persKVT1}\\ {\dot T^*}&=&P(\mu)KVT-B T^*,\ T^* \in {\mathbb{R}}^n_+,\label{persKVT2}\\ {\dot V}&=&{\hat N}B T^*-\Gamma V -KVT,\ V\in {\mathbb{R}}^n_+,\label{persKVT3}\\ {\dot \mu}&=& 0,\ \mu \in [0,\mu_0].\label{persKVT4} \end{aligned}$$ Let $K'$ be the forward invariant compact set for $(\ref{hiv3a})$ established in Lemma \[boundedKVT\] and define $K=K'\times[0,\mu_0]$. It is clear that $K$ is compact and forward invariant under $(\ref{persKVT1})-(\ref{persKVT4})$ The set $K_0=([0,T_0]\times 0 \times 0 \times [0,\mu_0]) \cap K$ is clearly a compact forward invariant subset of $K$. Let $\Pi(T^*,V):=v'(T^*,V)$. The function $\Pi$ is clearly smooth, zero on $K_0$, and positive on $K\backslash K_0$. Furthermore, $$\frac{\dot \Pi}{\Pi}= \psi := \frac{v'M(T,\mu)(T^*,V)}{v'(T^*,V)}$$ is lower semi-continuous on $K$ by Lemma \[liminf\] once we define the value of $\psi$ on $K_0$ as $$\psi(T,\mu)=\min_{i=1,...,n} \frac{v'M(T,\mu)_i}{v_i} .$$ We note that the function $\psi(T,\mu)$ is continuous in $(T,\mu)$. Since all solutions of $(\ref{persKVT1})-(\ref{persKVT4})$ in $K_0$ have the property that $\lim_{t\to\infty}T(t)=T_0$, it implies that $\psi(T(t),\mu)>0$ for all sufficiently large $t$. Hence by Theorem \[patrickdoeshofbauer\], the set $K_0$ is a uniform strong repellor in $K$. If we use the $L^1$-norm of $(T^*,V)$ as the distance function to $K_0$, we find that there exists an $\eta_0>0$ such that $$\liminf_{t\to\infty} 1'(T^*+V) \geq \eta_0$$ for all solutions of $(\ref{persKVT1})-(\ref{persKVT4})$ in $K\backslash K_0$. To complete the proof, we need to show that there exists $\eta>0$ such that $\liminf_{t\to\infty} 1'V(t) \geq \eta >0$ for all solutions with $1'V(t)>0$. Observe that $1'V(t)>0$ implies that $1'T^*(t)>0$. Hence by the result of part one, we have that $\liminf_{t\to\infty} 1'(T^*(t)+V(t)) \geq \eta_0 >0$, or equivalently, $1'T^*(t) > \eta_0/2 - 1'V(t)$ for all sufficiently large $t$. From (\[persKVT3\]), we have that $$1'\dot V \geq \sum_{i=1}^n N_i \beta_i T^*_i - \sum_{i=1}^n (\gamma_i + k_i T) V_i \geq \sum_{i=1}^n N_i \beta_i T^*_i - \sum_{i=1}^n (\gamma_i + k_i T_0) V_i.$$Hence, $$1'\dot V \geq A_0 \biggl(\frac{\eta_0}{2} - 1'V(t)\biggr)-A_1 1'V(t), \quad A_0:=\min_{i} (N_i \beta_i)>0, \ A_1:=\max_{i} (\gamma_i+k_i T_0) >0$$ holds for large $t$. It follows immediately that $$\liminf_{t\to\infty} 1'V(t) \geq \eta=\frac{\eta_0 A_0}{2(A_0+A_1)}>0.$$ Existence of absorbing compact set for small $\mu>0$. {#existence-of-absorbing-compact-set-for-small-mu0. .unnumbered} ----------------------------------------------------- \[compactset-KVT\] Let $K$ be the absorbing compact set established in Lemma \[boundedKVT\], and let $$U=\{ (T,T^*,V) \in {\mathbb{R}}^{2n+1}_+ | T,T^*_1,V_1>0 \}.$$ Suppose that there exists $\epsilon>0$ such that $${\bf (C_{\epsilon})}\;\;f'(c)+\frac{k_1}{\gamma_1}f({\bar T_1})\leq -\epsilon <0, \textrm{ for all } c\in [0,T_0].$$ Then there exist $\mu_1>0$ and a compact set $K_{\delta} \subset U$ such that for any $\mu\in[0,\mu_1]$ and for any solution of system $(\ref{hiv3a})$ in $U$, there exists a $t_0>0$ such that $ (T(t),T^*(t),V(t)) \in K_{\delta}$ for all $t >t_0$.\ An identical statement holds for system $(\ref{hiv3b})$. \(a) We first prove the statement for system $(\ref{hiv3a})$. Consider the function $$\begin{aligned} W &=& (N_1-1)\int_{{\bar T_1}}^T \left(1-\frac{{\bar T_1}}{\tau}\right) d \tau +N_1\int_{{\bar T_1^*}}^{T_1^*} \left(1-\frac{{\bar T_1^*}}{\tau}\right)d \tau+ \int_{{\bar V_1}}^{V_1} \left(1-\frac{{\bar V_1}}{\tau}\right) d \tau\\ && + \sum_{i=2}^n \frac{N_1-1}{N_i-1}(N_i T^*_i +V_i). \end{aligned}$$ Computing $\dot W$ for the system $(\ref{hiv3a})$, we obtain $$\begin{aligned} \dot W &=& (N_1-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T_1}) \right]\frac{(T-{\bar T_1})^2}{T} +N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -(N_1-1) \sum_{i=2}^n k_i V_i (\bar T_i -\bar T_1) + \mu N_1 \frac{T^*_1-\bar T^*_1}{T^*_1} \sum_{j=1}^n q_{1j} k_j V_j T +\mu (N_1-1) \sum_{i=2}^n \frac{N_i}{N_i-1} \sum_{j=1}^n q_{ij} k_j V_j T, \end{aligned}$$ Recombining the terms, we find that $$\begin{aligned} \dot W &=& (N_1-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T_1}) \right]\frac{(T-{\bar T_1})^2}{T} +N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -(N_1-1) \sum_{i=2}^n k_i V_i (\bar T_i -\bar T_1) +\mu (N_1-1) \sum_{i=1}^n \frac{N_i}{N_i-1} \sum_{j=1}^n q_{ij} k_j V_j T \\ && -\mu N_1 q_{11} \frac{\bar T^*_1 V_1 T}{T^*_1} - \mu N_1 \frac{\bar T^*_1}{T^*_1} \sum_{j=1}^n q_{1j} k_j V_j T, \end{aligned}$$ where the last term is clearly non-positive. Let $$\begin{aligned} \alpha &= &(N_1-1) \min_{i\geq 2} k_i (\bar T_i -\bar T_1) >0,\\ L & =& \sup_{K} (N_1-1) \sum_{i=1}^n \frac{N_i}{N_i-1} \sum_{j=1}^n q_{ij} k_j V_j T \geq 0. \end{aligned}$$ By Lemma \[lowerboundKVT\], there exist $\eta,\mu_a>0$ such that $1'V(t) > \eta$ for all $\mu \in [0,\mu_a]$ and all sufficiently large $t$. Hence, by shifting time forward if necessary, we have the inequality $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -\alpha (\eta -V_1) +\mu L -\mu N_1 q_{11} \frac{\bar T^*_1 V_1 T}{T^*_1}, \end{aligned}$$ which holds in $K$ for all $\mu \in [0,\mu_a]$. We combine the second and the last terms to obtain $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*}\left[3- (1+q_{11}\mu)\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]\\ & & -\alpha (\eta -V_1) +\mu L. \end{aligned}$$ Further, we rewrite the above inequality as $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*} \left[3(1+q_{11}\mu)^{1/3}- (1+q_{11}\mu)\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]\\ & & -\alpha (\eta -V_1) +\mu L + 3 N_1\beta_1{\bar T_1^*}\left[1-(1+q_{11}\mu)^{1/3}\right]. \end{aligned}$$ Let $\mu_b>0$ be such that for all $\mu \in [0,\mu_b]$, $$(1+q_{11}\mu) \in [\frac{1}{2},1], \quad -\alpha \eta +\mu L + 3 N_1\beta_1{\bar T_1^*}\left[1-(1+q_{11}\mu)^{1/3}\right] \leq -\frac{\alpha \eta}{2} .$$ Now we let $\mu_1=\min[\mu_a,\mu_b]$, so that for all $\mu \in [0,\mu_1]$ and all points in $K$, $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*} \left[3(1+q_{11}\mu)^{1/3}- (1+q_{11}\mu)\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]\\ & & -\frac{\alpha \eta}{2} +\alpha V_1. \end{aligned}$$ Let $L_1=\alpha \sup_{K} V_1$. Inspecting the first term in $\dot W$, we find that there exists $\delta_0>0$ such that $$-\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}<-L_1$$ for all $T<\delta_0$. Similarly, inspecting the second term in $\dot W$ and using Lemma \[minilemma\], we find that there exists $\delta_1>0$ such that $$N_1\beta_1{\bar T_1^*} \left[3(1+q_{11}\mu)^{1/3}- (1+q_{11}\mu)\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]<-L_1$$ for all $\frac{T^*_1}{V_1}<\delta_1$ and all $\mu \in [0,\mu_1]$. Finally, there exists $\delta_2>0$ such that $-\frac{\alpha\eta}{2}+ \alpha V_1 <-\frac{\alpha \eta}{4}$ for all $V_1<\delta_2$. Let $$\hat K_{\delta}=\{ (T,T^*,V) \in K \cap U | T \geq \delta_0, V_1 \geq \delta_2, T^*_1 \geq \delta_1 V_1\}.$$ Consider $(T,T^*,V) \in (K \cap U) \backslash \hat K_{\delta}$ and let $\mu \in [0,\mu_1]$, then at least one of the following holds: - $T <\delta_0$, in which case $ \dot W \leq -L_1 -\frac{\alpha \eta}{2} + L_1 \leq -\frac{\alpha \eta}{2}$; - $T^*_1/V_1 < \delta_1$, in which case $ \dot W \leq -L_1 -\frac{\alpha \eta}{2} + L_1 \leq -\frac{\alpha \eta}{2}$; - $ V_1 < \delta_2$, in which case $\dot W \leq -\frac{\alpha \eta}{4}$; Hence, for all $(T,T^*,V) \in (K \cap U) \backslash \hat K_{\delta}$ and all $\mu \in [0,\mu_1]$, we have $ \dot W \leq -\frac{\alpha \eta}{4}<0.$ From this point forward, the proof is identical to the proof of Theorem \[compactset\], so it will be omitted. \(b) Now we consider system $(\ref{hiv3b})$. Let $W$ be the same as in part (a). Computing $\dot W$ for the system $(\ref{hiv3b})$, we obtain $$\begin{aligned} \dot W &=& (N_1-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T_1}) \right]\frac{(T-{\bar T_1})^2}{T} +N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -(N_1-1) \sum_{i=2}^n k_i V_i (\bar T_i -\bar T_1) + \mu \frac{V_1-\bar V_1}{V_1} \sum_{j=1}^n q_{1j} N_j \beta_j T^*_j +\mu \sum_{i=2}^n \frac{N_1-1}{N_i-1} \sum_{j=1}^n q_{ij} N_j \beta_j T^*_j, \end{aligned}$$ Recombining the terms, we find that $$\begin{aligned} \dot W &=& (N_1-1)\left[f'(c)+\frac{k}{\gamma}f({\bar T_1}) \right]\frac{(T-{\bar T_1})^2}{T} +N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -(N_1-1) \sum_{i=2}^n k_i V_i (\bar T_i -\bar T_1) +\mu \sum_{i=1}^n \frac{N_1-1}{N_i-1} \sum_{j=1}^n q_{ij} N_j \beta_j T^*_j \\ && -\mu q_{11} \frac{\bar V_1 N_1 \beta_1 T^*_1}{V_1} - \mu \frac{\bar V_1}{V_1} \sum_{j=2}^n q_{1j} N_j \beta_j T^*_j, \end{aligned}$$ where the last term is clearly non-positive. Let $$\begin{aligned} \alpha &= &(N_1-1) \min_{i\geq 2} k_i (\bar T_i -\bar T_1) >0,\\ L & =& \sup_{K} \sum_{i=1}^n \frac{N_1-1}{N_i-1} \sum_{j=1}^n q_{ij} N_j \beta_j T^*_j \geq 0. \end{aligned}$$ By Lemma \[lowerboundKVT\], there exist $\eta,\mu_a>0$ such that $1'V(t) > \eta$ for all $\mu \in [0,\mu_a]$ and all sufficiently large $t$. Hence, by shifting time forward if necessary, we have the inequality $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*}\left[3-\frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}-\frac{{\bar T_1}}{T}\right]\\ & & -\alpha (\eta -V_1) +\mu L -\mu q_{11} \frac{\bar V_1 N_1 \beta_1 T^*_1}{V_1}, \end{aligned}$$ which holds in $K$ for all $\mu \in [0,\mu_a]$. We combine the second and the last terms to obtain $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*}\left[3- \frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-(1+q_{11}\mu)\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]\\ & & -\alpha (\eta -V_1) +\mu L. \end{aligned}$$ Further, we rewrite the above inequality as $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*} \left[3(1+q_{11}\mu)^{1/3}- \frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-(1+q_{11}\mu)\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]\\ & & -\alpha (\eta -V_1) +\mu L + 3 N_1\beta_1{\bar T_1^*}\left[1-(1+q_{11}\mu)^{1/3}\right]. \end{aligned}$$ Let $\mu_b>0$ be such that for all $\mu \in [0,\mu_b]$, $$(1+q_{11}\mu) \in [\frac{1}{2},1], \quad -\alpha \eta +\mu L + 3 N_1\beta_1{\bar T_1^*}\left[1-(1+q_{11}\mu)^{1/3}\right] \leq -\frac{\alpha \eta}{2} .$$ Now we let $\mu_1=\min[\mu_a,\mu_b]$, so that for all $\mu \in [0,\mu_1]$ and all points in $K$, $$\begin{aligned} \dot W &\leq & -\epsilon (N_1-1)\frac{(T-{\bar T_1})^2}{T}+N_1\beta_1{\bar T_1^*} \left[3(1+q_{11}\mu)^{1/3}- \frac{V_1 T{\bar T_1^*}} {{\bar V_1}{\bar T_1}T_1^*}-(1+q_{11}\mu)\frac{T_1^*{\bar V_1}}{{\bar T_1^*}V_1}- \frac{{\bar T_1}}{T}\right]\\ & & -\frac{\alpha \eta}{2} +\alpha V_1. \end{aligned}$$ From this point forward, the proof is identical to the proof of part (a), so it will be omitted. \[global2\] Let the assumptions of Lemma \[invertibleKVT\] hold, let $U$ be the set from Theorem $\ref{compactset-KVT}$, and define $$U'=\{ (T,T^*,V) \in {\mathbb{R}}^{2n+1}_+ | \; \; T^*_1+V_1>0 \}\supset U.$$ Then there exist $\mu_0>0$ and a continuous map $E: [0,\mu_0] \rightarrow U$ such that 1. $E(0)=E_1$ (where $E_1$ is the same as in Lemma \[invertibleKVT\]), and $E(\mu)$ is an equilibrium of $(\ref{hiv3a})$ or of $(\ref{hiv3b})$ for all $\mu \in [0,\mu_0]$; 2. For each $\mu \in [0,\mu_0]$, $E(\mu)$ is a globally asymptotically stable equilibrium of $(\ref{hiv3a})$ or of $(\ref{hiv3b})$ in $U'$. The proof is similar to that of Theorem $\ref{global}$. [199]{} P. De Leenheer, and H.L. Smith, Virus dynamics: a global analysis, SIAM J. Appl. Math. 64 (2003), 1313-1327. J. Hofbauer, and K. Sigmund, Evolutionary Games and Replicator Dynamics, Cambridge University Press, 1998. V. Hutson, A theorem on average Liapunov functions, Mh. Math. 98 (1984), 267-275. V. Hutson, and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosc. 111 (1992), 1-71. A. Iggidr, J.-C. Kamgang, G. Sallet, and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle, SIAM J. Appl. Math. 67 (2006), 260-278. A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, IMA Math. Med. Biol. 21 (2004), 75-83. J.P. LaSalle, Stability theory for ordinary differential equations, J. Diff. Eqns. 4 (1968), 57-65. M.Y. Li, and J.S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosc. 125 (1995), 155-164. M.A. Nowak, and R.M. May, Virus dynamics, Oxford University Press, New York, 2000. A.S. Perelson, D.E. Kirschner, and R. De Boer, Dynamics of HIV infection of CD$4^+$ T cells, Math. Biosc. 114 (1993), 81-125. A.S. Perelson, and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999), 3-44. L. Rong, Z. Feng, and A.S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, preprint. H.L. Smith, and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1994. H.L. Smith, and P. Waltman, Perturbation of a globally stable steady state, Proc. Amer. Math. Soc. 127 (1999), 447-453. H.R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal. 24 (1993), 407-435. L. Wang, and M.Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD$4^+$ T cells, Math. Biosc. 200 (2006), 44-57. [^1]: email: [[email protected]]{}. Supported in part by NSF grant DMS-0614651. [^2]: email: [[email protected]]{}. Supported in part by NSF grant DMS-0517954. [^3]: Incidentally, if $q_{11}=0$, we obtain global stability of the boundary equilibrium $E_1$ for all $\mu>0$.

--- abstract: | The high density of the recently discovered close-in extrasolar planet HD149026b suggests the presence of a huge core in the planet, which challenges planet formation theory. We first derive constraints on the amount of heavy elements and hydrogen/helium present in the planet: We find that preferred values of the core mass are between 50 and $80{{\rm M}_\oplus}$, although a minimum value of the core mass is $\sim 35{{\rm M}_\oplus}$ in the extreme case of formation of the planet at $> 0.5\,$ AU, followed by late inward migration after $>$ 1Ga and negligible reheating due to tidal dissipation. We then investigate the possibility of subcritical core accretion as envisioned for Uranus and Neptune. We show that a massive core surrounded by an envelope in hydrostatic equilibrium with the gaseous disk may indeed grows beyond $30{{\rm M}_\oplus}$ provided the core accretion rate remains larger than $\sim 2\times 10^{-5}{{\rm M}_\oplus}\,\rm yr^{-1}$. However, we find the subcritical accretion scenario is very unlikely in the case of HD149026b for at least two reasons: (i) Subcritical planets are such that the ratio of their core mass to their total mass is above $\sim 0.7$, in contradiction with constraints for all but the most extreme interior models of HD149026b; (ii) High accretion rates and large isolation mass required for the formation of a subcritical $>35{{\rm M}_\oplus}$ core are possible only at specific orbital distances in a disk with a surface density of dust equal to at least 10 times that of the minimum mass solar nebula. This value climbs to 30 when considering a $50{{\rm M}_\oplus}$ core. These facts point toward two main routes for the formation of this planet: (i) Gas accretion that is limited by a slow viscous inflow of gas in an evaporating disk; (ii) A significant modification of the composition of the planet after gas accretion has stopped. These two routes are not mutually exclusive. Illustrating the second route, we show that for a wide range of impact parameters, giant impacts lead to a loss of the gas component of the planet and thus may lead to planets that are highly enriched in heavy elements. Alternatively, the planet may be supplied with heavy elements by planetesimals by secular perturbations. Both in the giant impact and the secular perturbation scenarios, we expect an outer giant planet to be present. Observational studies by imaging, astrometry and long term interferometry of this system are needed to better narrow down the ensemble of possibilities. author: - 'M. Ikoma' - 'T. Guillot' - 'H. Genda, T. Tanigawa, & S. Ida' title: On the Origin of HD149026b --- INTRODUCTION \[sec:introduction\] ================================= The combined detection of extrasolar planets by radial velocimetry and transit photometry provides unique information by the measurement of both their mass and radius. Further spectroscopic studies of the system in and off transit can then even inform us on chemical species in the planet’s atmosphere, its possible evaporation, global atmospheric temperature, possible presence of winds, etc. Nine transiting extrasolar planets have been discovered so far (see www.obspm.fr/planets). Eight of them have radii of 1.0–1.3 Jupiter’s radius (${\rm R}{_{\rm J}}$), but one stands out: HD149026b has a radius of only $0.73\pm 0.03 \rm R_{J}$ for a mass of $0.36 \pm 0.03$ Jupiter’s mass ($= 110 \pm 10 {{\rm M}_\oplus}$) and an orbital distance of 0.042AU [@Sato05; @Charbonneau06]. The parent star is believed to be 1.3M$_\odot$, and is metal-rich with $\rm [Fe/H]=0.36\pm 0.05$. Contrary to the other transiting extrasolar planets, believed to be mainly formed with hydrogen and helium [@Burrows00; @Bodenheimer01; @GuillotShowman02; @Laughlin05; @Guillot05; @Baraffe06], HD149026b is [*clearly*]{} made of a significant fraction of heavy elements. Indeed, based on evolution models, @Sato05 and then @Fortney06 have shown the planet has a huge rock/ice core of $\sim 70{{\rm M}_\oplus}$. As noted by @Sato05, such a big core points toward formation of the planet in the core accretion scenario [e.g., @Mizuno80; @BP86], rather than in the disk instability scenario [e.g., @Boss97]. However, HD149026b may even challenge the core accretion scenario, because the core mass is far larger than that of Jupiter or Saturn, and the “canonical” critical core mass as derived by @Mizuno80 is around $10{{\rm M}_\oplus}$. The purpose of this paper is to constrain the structure of HD149026b, to examine how the planet may have formed based on these constraints, and, hopefully, to stimulate further studies of the formation of close-in giant planets. In section \[sec:present\_structure\], we attempt to constrain the structure of HD149026b based on various evolution models. In section \[sec:accretion\], we then examine whether these constraints are compatible with the properties of growing protoplanetary cores embedded in a gaseous circumstellar disk. On these bases, various possibilities for the formation of this planet are explored in section \[sec:possibilities\]. Section \[sec:summary\] summarizes the results and proposes two possible formation scenarios that account for the properties of HD149026b. PRESENT STRUCTURE: HOW MUCH HEAVY ELEMENTS? \[sec:present\_structure\] ====================================================================== As already evident from the measurements and from previous models of the evolution of the planet [@Sato05; @Fortney06], HD149026b is surprisingly small, and should possess either a big core or a large amount of heavy elements in its interior. In this work, we attempt to constrain the core mass by using a variety of evolutionary models beyond those explored by Sato et al. and Fortney et al. In doing so, we will show that HD149026b indeed contains a significant amount of heavy elements, in apparent contradiction with current models of the formation of giant planets. Standard Models \[sec:standard\_model\] --------------------------------------- The evolution of a giant planet is essentially governed by a Kelvin-Helmholtz cooling and contraction similar to a stellar pre-main sequence evolution but slightly modified by degeneracy effects, and by the intense stellar irradiation that slows the cooling through the growth of a deep radiative region that can extend to kbar levels [@Guillot96; @Guillot05]. The level of the irradiation received by the planet is calculated from the observations by @Sato05: We adopt values of the stellar radius ($R_*=1.45\,\rm R_\odot$) and temperature ($T_*=6150\,$K) and the orbital distance of $0.046\,$ AU to derive zero-albedo equilibrium temperature $T_{\rm eq}^*=1740\,$K. We use this value as a minimum for the atmospheric temperature at the 1 bar level (the corresponding temperature found by @Fortney06 is between 2000 and 2200K, and using the model of @IBG05, we find a value of 1980K). We use the 1 bar level as outer boundary condition for the evolution models [see @Guillot05]. Standard models are calculated on the basis of structure consisting of a central rock/ice core and of an envelope of solar composition, and further assuming that the planet has either formed [*in situ*]{} or moved rapidly to its present location, so that throughout its evolution it has received a constant stellar heat flux to slow its cooling. For the core, we use the equations of state for “rocks” and “ices” obtained by @HM89. For the envelope, we use the H-He EOS from @SCVH95. The structure of the core with @HM89’s EOS is independent of its temperature, but this simplification is reasonable because it has much smaller effects than changes in the (unknown) core composition. Although it has little effect on the models, we assume that the core luminosity is due both to radioactive decay (with $l_{\rm chondritic}\approx 10^{20}\rm\,erg\,s^{-1}\,{{\rm M}_\oplus}^{-1}$) and cooling (assuming that in the core the temperature is uniform and the specific heat is $c_v\approx 10^7\rm\,erg\,g^{-1}\,K^{-1}$). Internal opacities are calculated from Rosseland opacity tables provided by @Allard01. The calculations are made for various core masses and core compositions and compared to the observational constraints in Fig. \[fig:evol\_std\]. As discussed by @Sato05 and by @Fortney06, the small planetary radius implies that it contains a large fraction of its mass in heavy elements. Difference in composition of the core—whether these heavy elements are mostly “ices” (a mixture of water, ammonia and methane in their \[unknown\] high pressure, relatively high temperature form) or “rocks” (a mixture of refractory materials including mostly silicates)—affects the radius of the planet by 10 to 20%. For comparison, the interiors of Uranus and Neptune are consistent with being mostly made of “ices” [e.g., @PPM00]. Figure \[fig:evol\_std\] thus confirms the need for a substantial amount of heavy elements in HD149026b when adopting a standard evolution scenario. Quantitatively, this implies core masses between 41 and 83M$_\oplus$ in good agreement with @Sato05 and [@Fortney06]. In particular, the latter yields values between 60 and 93M$_\oplus$. The difference is due to the fact that in order to constrain more strictly the minimum mass of heavy elements present in the planet, we used a low value of the atmospheric temperature. The Cold Storage Hypothesis \[sec:cold\_storage\_hypothesis\] ------------------------------------------------------------- We now investigate the hypothesis that the planet may have cooled far from its parent star before suddenly being sent into the presently observed 0.046 AU orbit. The reason for this sudden event is to be determined, but could be due to dynamical planet-planet interactions (see section \[sec:late\_Z\_supply\]). In terms of evolution, the planet is allowed to cool more rapidly during the time-period when it is far from the parent star. The sudden inward migration indeed yield an expansion of the planet but it may be limited to the outer layers and thus be relatively small [@Burrows00]. We want to investigate whether the present mass and radius are compatible with a relatively small core. We thus calculate evolution models similar to those in the previous section, but using $T_{\rm eq}=100\,$K and an atmospheric model based on simplified radiative transfer calculations of an isolated atmosphere [see @Saumon96; @Guillot05]. Based on these calculations, we obtain generally a much faster contraction of the planet; a less than 30${{\rm M}_\oplus}$ core can account for the observed radius. However, we have to include reheating of the outer shell of the envelope. This shell is approximatively defined by the region whose temperature is lower than the new equilibrium temperature, 1740K. Based on our calculations, this corresponds to a pressure level of $\sim$10kbar. A simple estimation of the reheating timescale of the outer shell suggests that we have to include this effect. Using equations of radiative diffusion and energy conservation, we estimate that the reheating timescale of a layer of pressure $P$, temperature $T$, opacity $\kappa$ and heat capacity $c_p$ is to first order independent of its initial temperature and equal to: $$\tau_{\rm rad}\approx {3c_p\over 4ac}{\kappa P^2\over g^2 T^3},$$ where $a$ is the radiation density constant, $c$ the speed of light, and $g$ the planet’s gravity (assumed uniform). This expression is consistent with direct simulations by @IBG05. This timescale is about $10^8\,$years for $\kappa\approx 1\,\rm cm^2\,g^{-1}$, $P\approx 10$kbar, $g\approx 1550\rm\,cm\,s^{-2}$, $T\approx 2000\,{\rm K}$ and $c_p\approx 8\times 10^7\rm\,erg\,g^{-1}\,K^{-1}$, which implies that the reheating is relatively fast and should involve most of the shell that we defined. However, the expansion due to the reheating is relatively small. The $P^2$ dependence ensures that most of this outer shell will be affected, and it will expand by a radius estimated to be: $$\Delta R\approx H_p\Delta\ln P,$$ where $H_p\approx{\cal R}T/\mu g$ is the new pressure scale height and $\Delta\ln P$ measures the depth of the expanding shells in pressure units. Based on our simulations, a rough estimate is that the planet should expand by $\sim 4\times 10^8\,$cm ($= 0.06 R_{\rm J}$) when considering levels between 10kbar and 1 bar. Note that this estimate does not include factors such as global reheating due to tidal dissipation caused by the necessary circularization of the planet’s spin and eccentricity. If such dissipation occurs sufficiently deep in the planet, this will lead to an expansion of the planet’s deeper layers that could completely erase the planet’s cold evolution phase [see @Guillot05 and references therein]. Thus, this very possibility that a planet can contract more efficiently far away from the star and then be brought in with little consequence on its radius is very much in question. We choose however to consider it because it gives a sense of the robustness of the conclusion that HD149026b contains a large fraction of heavy elements. [cccc]{} & $M_Z \rm\ (M_\oplus)$ & $M_{\rm H\,He}\rm\ (M_\oplus)$ & $M_Z/M_{\rm p}$\ \   ices & $50-83$ & $27-54$ & 0.52–0.72\   rocks & $41-69$ & $38-64$ & 0.43–0.60\ \   ices & $37-77$ & $34-66$ & 0.39–0.67\   rocks & $33-61$ & $46-76$ & 0.33–0.53\ \ & $33-83$ & $27-76$ & 0.33–0.72\ Adding $\Delta R$ ($\sim 4 \times 10^8$ cm) to the radius obtained by the evolution calculation of the isolated planet (not including the global reheating), we derive masses of heavy elements that reproduce the observed radius and show these in Table \[tab:constraints\]. Even in the case of a long storage ($\ga 10^8$ years) of the planet in a cold environment (corresponding to more than 10 AU), a large amount of heavy elements (strictly more than 33M$_\oplus$) is required to reproduce the observed radius of HD149026b. Heavy Elements: In the Core or in the Envelope? \[sec:core\_or\_envelope\] -------------------------------------------------------------------------- We have thus far assumed all heavy elements to lie within a central core. In order to see how the results are dependent on this assumption, we compare our previous model with a 60M$_\oplus$ ice core to a similar model with a 30M$_\oplus$ ice core and an envelope which is enriched by 30M$_\oplus$ of ices. For the ices in the envelope, we use the EOS described by @SG04. Figure \[fig:evol\_opa\] shows the results of the calculations in the two cases: (i) with unchanged opacities (i.e., as calculated for a solar-composition mixture) and (ii) with 30 times larger opacities to mimic the effect of the enrichment of heavy elements in the envelope. Basically, when using an unchanged opacity table (the dotted line), there is very little difference between a planet that has all its heavy elements in the core and a planet that has them mixed throughout its envelope. However, large differences arise when the opacities are affected proportionally to the amount of heavy elements that are mixed (the dashed line). In that case, the cooling and contraction timescale, which is dominated by radiative transport in the outer radiative zone, becomes long, and prevents the rapid contraction of the planet. These results confirm again that our estimates of the amount of heavy elements in HD149026b are lower limits and that the planet indeed must contain a significant amount of heavy elements. If most of the heavy elements are located below the radiative zone, which is not unlikely, then we expect our constraints to be relatively accurate. On the other hand, if the material is mixed in the outer radiative zone, we expect that its interior contains more heavy elements by up to $\sim 20\,\rm M_\oplus$ than calculated here. Impact of the Various Parameters \[sec:various\_parameters\] ------------------------------------------------------------ We have used a rather simplified approach for the calculations of the interior structure. On the other hand, reality is without doubt more complex. However, this should not affect significantly the global constraints that are derived in Table \[tab:constraints\]. First, the two extreme compositions used for the heavy elements (“ices” and “rocks”) ensure that most variations due to improved EOS, for example, are likely to fall in between the range of values that are considered. Our external boundary condition is extremely simplified. Although it agrees with more detailed atmospheric models, one has to account for the fact that the atmospheric temperatures may be higher, in particular if the atmosphere is itself enriched in heavy elements [see @Fortney06]. A higher atmospheric temperature leads to a slightly larger radius, everything else being the same. However, given the decrease of the opacity with increasing temperature in that regime, the radius increase is very limited. Other factors point toward a slightly larger amount of heavy elements than calculated here: (i) as discussed, the likely increase of the opacities in an enriched envelope; (ii) the presence of any other energy source such as the one that is required to reproduce the radius of HD209458b; (iii) thermally-dependent EOS for the core. Again, given that the core mass is already quite large, we expect these factors not to change the masses of heavy elements that are inferred above. A SUBCRITICAL CORE ACCRETION SCENARIO? \[sec:accretion\] ======================================================== The structure of HD149026b as derived in the previous section is certainly puzzling: It contains as much hydrogen and helium as Saturn, but at least twice as much heavy elements. In other words, the ratio of heavy elements to hydrogen and helium is intermediate between those of Saturn and Uranus/Neptune. In the core accretion scenario, when core mass exceeds a critical value (called the *critical core mass*), the Kelvin-Helmholtz contraction of the envelope takes place, resulting in substantial disk-gas accretion onto the planet [e.g., @Mizuno80; @BP86]. A widespread idea is that Jupiter and Saturn experienced the substantial gas accretion beyond the critical core mass, while the cores of Uranus and Neptune remained subcritical because of progressive accretion of planetesimals until the gaseous circumstellar disk disappeared [@P96]. In this section, we investigate whether HD149026b may have formed in a way similar to what envisioned for Uranus and Neptune. Critical Core Mass and Gas Accretion Rate \[sec:critical\_mass\] ---------------------------------------------------------------- Figure \[fig:critical\_core\_mass\] shows the critical core mass ($M_{\rm c,crit}$) as a function of core accretion rate for several different choices of distance from the parent star and local density of disk gas. In the numerical simulations we have used the H-He EOS from @SCVH95, the grain opacity from @PMC85, and the gas opacity from @AF94. The integration method is basically the same as that of @Ikoma00 who used simpler forms of EOS and gas and grain opacities. The input physics used here is somewhat different from that used in section \[sec:present\_structure\]. First we include grain opacity that was not included in the simulations in section 2, because an accreting envelope contains low temperature parts in which grain opacity is dominant, unlike the fully-formed planet ($T \ge 1700$ K). Although the fully-formed planet also contains such low temperature parts in the cold storage model, the planet is almost fully convective. Second the gas opacity used in this simulation is different from that used in the previous section. This difference has little influence on the results obtained in this section, because most of the part of the envelope where gas opacity is dominant relative to grain opacity is convective. Finally, although constant core density is assumed here, the critical core mass is known to be insensitive to core density [@MNH78]. Figure \[fig:critical\_core\_mass\] shows a large subcritical core can be formed if core accretion rate is high. High core accretion rates stabilize the envelope, yielding progressively larger critical core masses [@Ikoma00]. This, however, remains true up to a point for which the envelope becomes fully convective; the structure of a fully convective envelope is almost adiabatic and thus independent of energy flux supplied by incoming planetesimals (i.e., core accretion rate). At that point, the critical core mass reaches its maximum value. In this case, this [*maximal*]{} critical core mass depends on the local disk conditions (i.e., the local density and temperature of disk gas and the distance from the parent star) [@Wuchterl93; @Ikoma01]. In the minimum-mass solar nebula (MSN) proposed by @Hayashi81, the entropy of the disk gas decreases as the distance to the parent star decreases. That is why the critical core mass is smaller at locations closer to the parent star: In a denser gas disk, the critical core mass is smaller for the same reason [@Ikoma01]. As a result, formation of a large subcritical core of 50–80 ${{\rm M}_\oplus}$ requires not only high core accretion rate, but also a location not too close to its parent star and not too massive gaseous disk. Although a large core of 50–80 ${{\rm M}_\oplus}$ can, in principle, be formed by subcritical growth as shown above, the ratio of the critical core mass to the corresponding total (core + envelope) mass suggests that the core mass of HD149026b was likely to be supercritical. Figure \[fig:critical\_ratio\] shows the ratio of core mass to planetary total mass as a function of core mass (normalized by the critical core mass) for three different sets of core accretion rate and distance from the parent star. This ratio is found to be rather insensitive to values of the parameters; it is always $\sim 0.7$ at the critical core mass. The evolution model of HD149026b obtained in section \[sec:present\_structure\] shows that the current ratio of core mass to planetary mass is smaller in most cases (see Table \[tab:constraints\]). Although there are a few cases where the current ratio is as large as $\sim 0.7$, the probability of the formation of such planets seems to be low. From Fig. \[fig:critical\_ratio\] we also find the duration during which the ratio of core mass to planetary mass is around $\sim 0.7$ is quite short in the total core formation time. It follows from this consideration that HD149026b is likely to have experienced disk-gas accretion due to the Kelvin-Helmholtz contraction of the envelope. The rate of the gas accretion determined by the Kelvin-Helmholtz contraction of the envelope beyond the critical core mass—the gas accretion is hereafter called the KH gas accretion—depends strongly on the critical core mass. Figure \[fig:KH\] shows the typical timescale ($\tau_{\rm g}$) for the KH gas accretion defined by $$\tau_{\rm g} = C \frac{G M_{\rm c,crit} M_{\rm e,crit}} {R_{\rm conv} L}, \label{eq: KH time}$$ where $G$ is the gravitational constant, $M_{\rm c,crit}$ is the critical core mass, $M_{\rm e,crit}$ is the corresponding envelope mass, $R_{\rm conv}$ is the outer radius of the inner convective region, and $L$ is the luminosity [@Ikoma00]. The numerical factor, $C$, is chosen to be $3/2$ based on the numerical simulations by @IG06. In cases where core accretion is suddenly stopped (corresponding to core isolation; see below), $C$ should be $1/3$ [@IG06]. On this timescale the envelope mass increases by a factor of $e$. The timescale for the KH gas accretion is approximately expressed by $$\tau_{\rm g} \sim 5 \times 10^{10} \left(\frac{M_{\rm c,crit}}{{{\rm M}_\oplus}} \right)^{-3.5} {\rm years} \label{eq:KHtime_formula}$$ in this range[^1]. This relation implies that if the core mass is small, the timescale of the KH gas accretion is much longer than the disk lifetime ($\sim 10^6$-$10^7$ years). For discussion in section \[sec:subcritical\], we have also drawn lines representing the time for formation of a critical core with constant core accretion rate, $$\tau_{\rm c,acc,crit} = \frac{M_{\rm c}}{\dot{M}_{\rm c}} \mid_{M_{\rm c}=M_{\rm c,crit}}, \label{eq:core_growth_time}$$ for three different sets of the parameters. Subcritical Core Accretion \[sec:subcritical\] ---------------------------------------------- We examine the subcritical core accretion to know how large core can be formed before the onset of the KH gas accretion. Core mass is limited by the isolation mass that is determined by the total mass of solid components in the core’s feeding zone, unless planetesimals are supplied from outside the feeding zone by migration of planetesimals or the core itself. Merging between the isolated cores is not likely until disk gas is severely depleted [@Iwasaki02; @Kominami02], because the eccentricity damping due to disk-planet interaction [@Artymowicz93; @Ward93] is too strong to allow orbital crossing of the isolated cores. The core isolation mass is thus given by [@IL04a] $$M_{\rm c,iso} \simeq 0.16 f_{\rm d}^{3/2} \eta_{\rm ice}^{3/2} \left(\frac{a}{1{\rm AU}}\right)^{3/4} \left(\frac{M_*}{{\rm M}_{\odot}}\right)^{-1/2} {{\rm M}_\oplus}, \label{eq:M_c_iso}$$ where $a$ is semimajor axis of the core. The scaling factor $f_{\rm g}$ and $f_{\rm d}$ express enhancement of disk surface density of gas ($\Sigma$) and dust ($\sigma$) components relative to those of MSN, defined by $$\begin{array}{lll} \Sigma & = & \displaystyle{2400 f_{\rm g} \left(\frac{a}{1{\rm AU}}\right)^{-3/2} {\rm g/cm}^2} \\ \sigma & = & \displaystyle{10 f_{\rm d} \eta_{\rm ice} \left(\frac{a}{1{\rm AU}}\right)^{-3/2} {\rm g/cm}^2}. \end{array}$$ In the solar metallicity cases, the values of these factors may have a range between 0.1 and 10 [@IL04a]. For metal-rich disks, the range of $f_{\rm d}$ may shift to larger values by a factor of $10^{\rm [Fe/H]}$, while that of $f_{\rm g}$ does not change [@IL04b]. $\eta_{\rm ice}$ expresses an enhancement of disk surface density due to ice condensation, which is 3–4 beyond an ice line at $\sim 3$ AU. Equation (\[eq:M\_c\_iso\]) shows large cores form at large $a$, in particular beyond the ice line, in disks with large $f_{\rm d}$. As seen from Fig. \[fig:critical\_core\_mass\], smaller core accretion rate yields smaller critical core mass, which implies core isolation triggers the KH gas accretion. For a core of $M_{\rm c}$ not to be isolated, that is, $M_{\rm c} < M_{\rm c, iso}$, $$f_{\rm d} > 30 \left(\frac{M_{\rm c}}{30 {{\rm M}_\oplus}}\right)^{2/3} \eta_{\rm ice}^{-1} \left(\frac{a}{1{\rm AU}}\right)^{-1/2} \left(\frac{M_*}{{\rm M}_{\odot}}\right)^{1/3}. \label{eq:f_iso}$$ Next, we consider the condition for a core to avoid the KH gas accretion until its isolation. Figure 3 shows that $M_{\rm c,crit}$ is expressed approximately by $$M_{\rm c,crit} \simeq 10 \left(\frac{\dot{M}_{\rm c}} {10^{-7}{{\rm M}_\oplus}/{\rm yr}} \right)^{0.2} {{\rm M}_\oplus}, \label{eq:m_crit}$$ except for fully convective cases. Note that consideration of fully convective cases does not change our conclusion (see below). The condition for a core to avoid the KH gas accretion is $M_{\rm c} < M_{\rm c, crit}$. Using $\tau_{\rm c,acc} \equiv M_{\rm c}/\dot{M}_{\rm c}$ and Eq. (\[eq:m\_crit\]), this condition is rewritten by (see Fig. \[fig:KH\]) $$\tau_{\rm c,acc} < \tau_{\rm c,acc,crit} \simeq 1 \times 10^{6} \left(\frac{M_{\rm c}}{30 {{\rm M}_\oplus}}\right)^{-4}{\rm years}. \label{eq:m_crit_time}$$ For cores of 30, 45 and $70 {\rm M}_{\oplus}$, $\tau_{\rm c,acc}$ must be shorter than $1 \times 10^{6}$ years, $2 \times 10^5$ years, and $3 \times 10^4$ years, respectively. The core growth timescale due to planetesimal accretion is given by [@IL04a] $$\tau_{\rm c,acc} \simeq 9.3 \times 10^5 f_{\rm d}^{-1} f_{\rm g}^{-2/5} \eta_{\rm ice}^{-1} \left(\frac{a}{1{\rm AU}}\right)^{27/10} \left(\frac{M_{\rm c}}{30 {{\rm M}_\oplus}}\right)^{1/3} \left(\frac{M_*}{{\rm M}_{\odot}}\right)^{-1/6} \left(\frac{m}{10^{21} \rm g}\right)^{2/15} \mbox{years}, \label{eq:tau_c_acc}$$ where $m$ is the mass of a planetesimal. The dependence on $f_{\rm g}$ appears because damping of velocity dispersion of planetesimals due to aerodynamical drag is taken into account. Substituting Eq. (\[eq:tau\_c\_acc\]) into Eq. (\[eq:m\_crit\_time\]), we have $$f_{\rm d} > 0.95 \left(\frac{M_{\rm c}}{30 {{\rm M}_\oplus}}\right)^{65/21} \left(\frac{f_{\rm d}}{f_{\rm g}}\right)^{2/7} \eta_{\rm ice}^{-5/7} \left(\frac{a}{1{\rm AU}}\right)^{27/14} \left(\frac{M_*}{{\rm M}_{\odot}}\right)^{-5/42} \left(\frac{m}{10^{21} \rm g}\right)^{2/21}. \label{eq:f_tau}$$ This inequality indicates that large $f_{\rm d}$ is needed to keep core accretion rate high enough to avoid the KH gas accretion. For subcritical core accretion, both Eqs. (\[eq:f\_iso\]) and (\[eq:f\_tau\]) must be satisfied. For $M_* = 1.3 {\rm M}_\odot$ and \[Fe/H\]=0.36 ($f_{\rm d}/f_{\rm g} = 2.3$), which correspond to HD149026, the conditions are plotted in Fig. \[fig:condition\]. Subcritical core accretion could be possible far from the parent star in relatively massive disks for $30{\rm M}_{\oplus}$ cores. However, for $45$ and $70{\rm M}_{\oplus}$ cores, extremely massive or extremely metal-rich disks with $f_{\rm d} > 30-50$ are required. Even for those large values of $f{_{\rm d}}$, since the critical core mass reaches its maximal value in cases of high core accretion rate, subcritical formation of 45 and 70 ${{\rm M}_\oplus}$ is impossible in some cases (see Fig. \[fig:critical\_core\_mass\]). Although formation of a $30{\rm M}_{\oplus}$ core may not be very difficult in relatively metal-rich disks, $30{\rm M}_{\oplus}$ is the lowest estimated value for $M_Z$ of HD149026b in the cold storage model; $50$–$80{\rm M}_{\oplus}$ is most likely. Therefore, heavy elements must be supplied after substantial accretion of the envelope. This issue is discussed in section \[sec:Z\_supply\]. SEVERAL POSSIBILITIES FOR THE FORMATION OF HD149026b \[sec:possibilities\] ========================================================================== The investigations in sections \[sec:present\_structure\] and \[sec:accretion\] have revealed that we have to explain at least three properties of HD149026b: its small orbital radius, its high metalicity, and its moderate amount of hydrogen and helium. A possible scenario that explains the proximity of the planet to its parent star is the [*in situ*]{} formation or migration of the planet. The [*in situ*]{} formation is however unlikely, because the core isolation mass is so small at $\sim$ 0.04 AU (see eq. \[\[eq:M\_c\_iso\]\]) that the planet can get only a tiny amount of hydrogen/helium within the disk lifetime because of too long timescale for the KH gas accretion (see eq. \[\[eq:KHtime\_formula\]\]). Section \[sec:subcritical\] has shown cores of $\ga 30 {{\rm M}_\oplus}$ can form only far from the parent star in reasonably massive and/or metal-rich disks. Except for the cold storage model (section \[sec:cold\_storage\_hypothesis\]), residual heavy elements of 20–$50{\rm M}_{\oplus}$ must be supplied to the planet. In section \[sec:Z\_supply\], we examine two possible scenarios; supply of heavy elements (planetesimals or another planet) during gas accretion (§ \[sec:concurrent\_Z\_supply\]) and after disk gas depletion (§ \[sec:late\_Z\_supply\]). To account for the moderate amount of hydrogen and helium of HD149026b, we examine limited supply of disk gas (§ \[sec:limited\_gas\_supply\]) and loss of the envelope gas (§ \[sec:envelope\_loss\]). Supply of Heavy Elements \[sec:Z\_supply\] ------------------------------------------ ### During gas accretion \[sec:concurrent\_Z\_supply\] First we briefly discuss the possibility that the large amount of heavy elements in HD149026b were supplied *during* substantial accretion of the envelope that happened far from the parent star. An increase in the planet’s mass due to gas accretion expands its feeding zone. If the planet can efficiently capture the planetesimals in the feeding zone, supply of heavy elements during gas accretion is possible [@P96; @Alibert04]. However, the cross section of gravitational scattering is much larger than that of collision in outer regions. Most planetesimals are thus scattered before accretion onto the planet. The gravitational scattering combined with eccentricity damping by disk-gas drag ends up opening a gap in the planetesimal disk: This phenomenon is called “shepherding” [e.g., @TI97][^2]. Although fast migration or growth of the planet can avoid the gap opening, unrealistically fast migration or growth may be required for a Saturn-mass planet to avoid the gap opening [@TI99; @Ida00]. Thus, supply of a large amount of planetesimals may be unlikely to happen during gas accretion. However, much more detailed studies are needed for this possibility. If the shepherding completely prevented the planetesimal accretion throughout the gas accretion phase, the well-known enrichment of heavy elements in the envelopes of Jupiter and Saturn would be unable to be accounted for. A possibility is accretion of small fragments. Small fragments are produced by disruptive collisions of planetesimals in the vicinity of a large core [@IWI03]. Since atmospheric-gas drag is strong for the fragments, they are accreted onto the core [@II03]. Because these studies are limited to the subcritical core case, we need to extend the calculation to the case of supercritical cores, in which the planet mass rapidly increases and non-uniformity of disk gas is pronounced. ### After disk gas depletion \[sec:late\_Z\_supply\] We next explore the possibility of enrichment of heavy elements in HD149026b (i.e., bombardment of another planet or planetesimals) *after* substantial accretion of the envelope. If the late bombardment happened, it is likely to have occurred after the planet migrated to its current location: The close-in planet collided with another planet(s) or planetesimals that had been gravitationally scattered by an outer giant planet(s). At $\sim$ 0.05 AU the planet’s Hill radius is only several times as large as its physical radius, while the former is much larger (by 100–1000) than the latter in outer regions ($\ga 1$ AU). That means the ratio of collision to scattering cross sections is much larger at $\sim$ 0.05 AU relative to in outer regions. The scattering cross section is further reduced by high speed encounter. The impact velocities ($v_{\rm imp}$) of the scattered bodies in nearly parabolic orbits are as large as the local Keplerian velocity at $\sim$ 0.05 AU, which is a few times larger than surface escape velocity of the inner Saturn-mass planet ($v_{\rm esc}$). The collision cross section is, in general, larger than the scattering cross section for $v_{\rm imp} > v_{\rm esc}$. Note that the shepherding does not work for highly eccentric orbits. To know how efficiently such scattered bodies collide with the close-in giant planet, we perform the following numerical simulation. We consider an inner planet of $0.5M_{\rm J}$ in a circular orbit at 0.05 AU and a planetesimal (test particle) or another giant planet of $0.5M_{\rm J}$ in a nearly parabolic orbit of $e \simeq 1$ and $i = 0.01$ with initial semimajor axis of $a = 1$ AU. For each initial pericenter distance ($q$), 100 cases with random angular distributions are numerically integrated by 4th order Hermite integrator for 100 Keplerian periods at 1 AU. Then, collision probability with the inner planet ($P_{\rm col}$), that with the parent star ($P_{\rm col}^*$), and ejection probability ($P_{\rm ejc}$) are counted. The residual fraction, $1 - (P_{\rm col} + P_{\rm col}^* + P_{\rm ejc})$, corresponds to the cases in which the planetesimal/planet is still orbiting after 100 Keplerian periods. We also did longer calculations and found that $P_{\rm col}/P_{\rm ejc}$ is similar, although individual absolute values of $P_{\rm col}$ and $P_{\rm ejc}$ increases. The results are insensitive to $a$ as long as $a \gg 0.05$ AU, because velocity and specific angular momentum of the incoming planetesimal/planet are given by $v \simeq \sqrt{2GM_*/q}$ and $L \simeq \sqrt{2 GM_* q}$ that are independent of initial semimajor axis $a$ of incoming bodies. Figure \[fig:scat\] shows the probabilities of the three outcomes of encounters between the inner planet and a planetesimal (panel a) or another giant planet (panel b). As shown in Fig. \[fig:scat\], when $0.01 {\rm AU} < q < 0.06 {\rm AU}$, the incoming bodies closely approach the inner planet. Although some fraction of them are ejected, the comparable fraction results in collision with the inner planet. In particular, $P_{\rm col} > P_{\rm ejc}$ in the planet-planet case (panel b), in which the ratio of geometrical cross section to Hill radius is larger than in the planet-planetesimal case (panel a). If $q$ is smaller than or close to the parent star’s physical radius (which is 0.01 AU in the calculations here), most of the incoming bodies hit the parent star rather than collide with the inner planet. If $q> 0.06$ AU, they do not closely encounter with the inner planet, so that both collision and ejection probabilities almost vanish ($P_{\rm col}, P_{\rm ejc} \ll 1$). The results shown in Fig. \[fig:scat\] demonstrate that efficient supply of heavy elements to the inner giant planet requires a very limited range (close to unity) of eccentricities of the incoming bodies; for example, $0.94 < e < 0.99$ for $a = 1$ AU, since $q = a(1-e)$. In the case of chaotic scattering by outer giant planets, the probability to acquire $q$ in such a narrow range would be very small [e.g., @Ford2001]. On the other hand, if the scattering comes from secular perturbation, $e$ is increased secularly while $a$ is kept constant. Accordingly, $q$ decreases secularly, and the incoming bodies eventually collide with the inner planet. For example, the following scenario is possible. More than three giant planets in addition to the inner one are formed in nearly circular orbits. They start orbit crossing on timescales longer than their formation timescales [@RF96; @WM96; @LI97; @MW02]. In many cases, one giant planet is ejected and the residual giant planets remain in stable eccentric orbits after the orbit crossing [@MW02]. In about 1/3 cases, $q$ of a giant planet decreases secularly to $\la 0.05$ AU due to the Kozai effect [@Kozai62] during the orbit crossing [@nagasawa]. This scenario requires formation of at least four giant planets. Origin of the Small Amount of Hydrogen and Helium \[sec:origin\_of\_small\_envelope\] ------------------------------------------------------------------------------------- ### Limited disk gas supply \[sec:limited\_gas\_supply\] Gas accretion onto a core can be truncated by gap opening or global depletion of disk gas. A gap may be formed, if both the viscous and thermal conditions are satisfied [e.g., @LP85; @LP93; @CMM06]. The former is that the torque exerted by the planet overwhelms viscous torque, and given by [@IL04a], $$M_{\rm p} > M_{\rm p,vis} = 30 \left(\frac{\alpha}{10^{-3}}\right) \left(\frac{a}{1{\rm AU}}\right)^{1/2} \left(\frac{M_*}{{\rm M}_{\odot}}\right) {{\rm M}_\oplus}, \label{eq:vis_condition}$$ where $M_{\rm p}$ is the planet mass and $\alpha$ is the parameter for the $\alpha$-prescription for disk viscosity [@alpha]. The latter condition is that the Hill radius becomes larger than disk scale height, and given by[^3] [@IL04a] $$M_{\rm p} > M_{\rm p,th} = 400 \left(\frac{a}{1{\rm AU}}\right)^{3/4} \left(\frac{M_*}{{\rm M}_{\odot}}\right) {{\rm M}_\oplus}, \label{eq:therm_condition}$$ assuming an optically thin disk in which $T = 280(a/1{\rm AU})^{-1/2}$ K. Hence, the actual truncation condition may be the thermal condition and it is very unlikely that gas accretion is truncated by the gap opening at $M_{\rm p} \sim 110 {{\rm M}_\oplus}$ (which corresponds to HD149026b) far from the parent star, as long as a sufficient amount of disk gas remains. Furthermore, since the gap is replenished by viscous diffusion, gas accretion may not completely be truncated by the gap opening [@D'Angelo03]. If a sufficient amount of disk gas remains, one way to truncate gas accretion at relatively small planetary mass is that the planet migrates to the vicinity of its parent star before the planet accretes a large amount of gas. Both $M_{\rm p,vis}$ and $M_{\rm p,th}$ are small in the inner regions (Eqs. \[\[eq:vis\_condition\]\] and \[\[eq:therm\_condition\]\]). However, this is unlikely, even if type II migration occurs when $M_{\rm p} \sim 30 {{\rm M}_\oplus}$. The timescale for the KH gas accretion $\la 3 \times 10^4$ years for a core of $\ga 30 {{\rm M}_\oplus}$ (Fig. \[fig:KH\]), while the timescale for the migration is much longer ($\sim 10^6$–$10^7$ years). Furthermore, it is not clear if the gap opening can stop gas accretion completely. If disk gas is globally depleted when $M_{\rm c}$ reaches $M_{\rm c,crit}$, gas accretion onto the core can be limited by viscous diffusion of disk gas, not by the Kelvin-Helmholtz contraction of the envelope [e.g., @GuillotHueso06]. That could be possible to account for the small amount of H/He. However, whether such a small amount of disk gas can bring the planet to the vicinity of the parent star should be examined. If it does not work, different migration mechanism such as gravitational scattering during orbital crossing of giant planets is required. ### Loss of envelope gas \[sec:envelope\_loss\] Another way to explain the small amount of the H/He gas of HD149026b is loss of the envelope gas. There are three possibilities for loss of the envelope gas; photoevaporation driven by incident UV flux from the parent star, the Roche lobe overflow, and impact erosion by a collision with another gas giant planet. Photoevaporation process for gas-rich planets is normally faster for smaller planetary masses, because less massive planets are more expanded and their envelope gas is more loosely bound [@Baraffe06]. This means the envelope quickly disappears once the evaporation occurs, so that it should be relatively rare to observe a planet at a stage when a relatively small amount (30–60 ${{\rm M}_\oplus}$) of envelope gas remains. However, this possibility is not excluded at present, because $Z$-rich planets such as HD149026b can be more compact and their envelope gas is not necessarily more loosely bound [e.g., @BACB06]. Envelope gas can also be lost by the Roche lobe overflow. When the Roche lobe is filled with the envelope gas, the gas overflows through the inner Lagrangian point to the inner disk. The Roche lobe overflow takes place if a planet is very close to its parent star because of inflation by stellar tidal heating as well as reduction of the Roche lobe [e.g., @Trilling98], and subsequent tidal inflation instability of the envelope [e.g., @Gu2003]. The spilled gas from the Roche lobe first goes to the inner disk and is eventually going to fall on the parent star by the gravitational interaction with the planet. At the same time, the planet gains angular momentum as the counteraction and migrates outward. Because of the increase in the orbital radius, the evaporation process slows down. Thus a state of a 30–60 ${{\rm M}_\oplus}$ envelope could be possible. This possibility is, however, in question because it is unclear if sufficient angular momentum is lost. @Sato05 suggested that collision between two giant planets could account for the internal structure of HD149026b. If cores with individual mass of $\sim 30 {{\rm M}_\oplus}$ are merged while significant amounts of their envelopes evaporate, a planet similar to HD149026b is formed. Here we examine if significant loss of the envelope gas occurs through SPH simulations of a collision between two giant planets. High impact velocity is expected. Suppose that one giant planet orbits at 0.03–0.05 AU and another giant planet in a highly eccentric orbit is sent from an outer region (for the mechanism to send planets, see section \[sec:late\_Z\_supply\]). Since the impactor has high orbital eccentricity ($e \sim 1$), collision velocity is similar to local Keplerian velocity at 0.03–0.05 AU around a $1.3M_{\odot}$ star, 150–190 km/s. This velocity is 2–3 times as high as the two-body surface escape velocity. Since specific angular momentum of the impactor is given by $L = \sqrt{G M_* a(1-e^2)} \simeq \sqrt{2 G M_* q}$, the impactor and the target have similar $L$. Hence, the semimajor axis of the merged planet would be similar to that of the target planet unless envelope evaporation changes specific angular momentum significantly. Previous SPH simulations of a collision between Mars-size or Earth-size rocky planets [e.g., @Canup04; @AA04] show almost no loss of material occurs by a low-velocity impact ($v_{\rm imp}$) of $1.0$–$1.5$ times as high as the two-body surface escape velocity ($v_{\rm esc}$) defined by $v_{\rm esc}^2 = 2G(M_{\rm t}+M_{\rm i})/(R_{\rm t}+R_{\rm i})$, where $M$ and $R$ are the planetary mass and radius and subscriptions t and i represent the target and the impactor. Collisions with higher impact velocity lose larger amounts of material; for example, 40% of the total mass is lost when $v_{\rm imp} \sim 2.5 v_{\rm esc}$ in their simulations. For gas giant planets, we will have qualitatively similar results, but can expect more significant loss of envelope gas because the envelope is less tightly bound compared to rocky planets. In order to simulate the collision between two gas giant planets, we use a SPH method with the modified spline kernel function [@ML85] and artificial viscosity [@MG83]. Our SPH simulation code was checked to reproduce the results of @AA04 for Mars-size planets using the Tillotson EOS. In this paper, we assume that the gas giant planets are composed only of ideal gas with the adiabatic exponent of 2; this value reproduces the present structure of Jupiter. We consider two hydrostatically equilibrated planets (each being composed of about 30,000 SPH particles) with Jupiter’s mass and radius for initial conditions. We perform the simulations of head-on collisions with various impact velocities (1.0, 1.5, 2.0, 2.25, 2.5, and 3.0$v_{\rm esc}$). In our calculations, $v_{\rm esc}$ is 60 km/s. We calculate the gravitational force for each SPH particle using the special-purpose computer for gravitational N-body systems, GRAPE-6. Figure \[fig8\] shows the mass fraction lost by collisions with various impact velocities. Higher impact velocity results in higher loss fraction. Almost no escape occurs for a low-velocity impact ($v_{\rm imp} \sim v_{\rm esc}$). When the impact velocity is higher than $2.5 v_{\rm esc}$, almost all SPH particles ($\sim$ 90%) are lost. Although we are unable to know exactly the loss fraction of the core with our current simulation, we can estimated it approximately by tracking the motion of the SPH particles which are initially located in inner part of the planet. Figure \[fig8\] also shows the loss fractions of the inner parts that are initially located inside 1/2 and 1/4 of the planetary radius. The inner part of the planet tends to remain in the merged planet. For example, in the case of $2.0 v_{\rm esc}$, about 50% of materials is totally lost, but almost no escape occurs for the material inside of 1/4 initial radius. A head-on collision of two gas giant planets with $v_{\rm imp} \sim 2$–$2.5 v_{\rm esc}$ possibly results in the considerable depletion of envelope and merging of core. The strongest point of this model for envelope loss is that we need no additional process to supply heavy elements to the planet. The collision increases $M_Z$ up to $\sim 60 {{\rm M}_\oplus}$ as well as decreases $M_{\rm H\,He}$; as we showed in section \[sec:subcritical\], it is possible to form cores of $\sim 30{\rm M}_{\oplus}$ by ordinary planetesimal accretion in relatively massive disks, in particular, metal-rich disks. However, the numerical simulations performed here is still preliminary. Since this mechanism may be promising, we should perform detailed simulations in the future. Other Possibilities \[sec:discussion\] -------------------------------------- Another possibility of formation of a metal-rich giant planet is the formation in a disk with originally high dust/gas ratio. In general, dust grains migrate inward in a gas disk to change the local dust/gas ratio [e.g., @Stepinski97]. If the dust/gas ratio is enhanced in planet-forming regions at $\la 10$ AU, the metal-enriched envelope could be formed without any planetesimal/planet supply. The high dust/gas ratio also makes possible planetesimal formation from dust through self-gravitational instability against Kelvin-Helmholtz instability [@Youdin02] and avoids type-I migration of terrestrial planets [@Kominami05]. However, it is not clear if the dust/gas ratio can become large enough to account for $M_Z$ of HD149026b. In principle, it is possible for a $70{\rm M}_{\oplus}$ core to form at $\sim 0.05$ AU through accretion of many Earth-size to Uranian-size cores that migrate close to the parent star. In the vicinity of the parent star, gas accretion should be truncated at a relatively small mass (see eq. \[\[eq:therm\_condition\]\]). We may need to carry out N-body simulations of the migrating protoplanets in order to examine whether they can pass through resonant trapping (or shepherding) to merge or not. However, we should keep in mind that incorporation of type-I migration without any conditions causes severe inconsistency with observed data of extrasolar planets and our Solar System. If we rely on type-I migration model, we need to clarify the condition for the occurrence of type-I migration at the same time. SUMMARY AND CONCLUSIONS \[sec:summary\] ======================================= The high density of the close-in extrasolar giant planet HD149026b recently discovered by @Sato05 challenges theories of planet formation. In this paper, we have attempted to derive robust constraints on the planet’s composition and infer possible routes to explain its formation. We have first simulated the evolution of HD149026b more extensively than previous workers [@Sato05; @Fortney06] and confirmed that the planet contains a substantial amount of heavy elements. Preferred values of the total mass of heavy elements are 50–80 ${{\rm M}_\oplus}$ (section \[sec:standard\_model\]), which is consistent with the previous calculations. We showed that the results are unchanged for heavy elements located in the central core, or distributed inside the envelope, provided they remain deeper than the external radiative zone. In the event of a significant enrichment of the outer layers, slightly higher values of heavy elements content are possible (section \[sec:core\_or\_envelope\]). In order to derive minimum values of the mass of heavy elements, we have explored the possibility that the planet was stored in a relatively cold environment for some time before migrating near to the planet. This strict minimum is $\sim 35 {{\rm M}_\oplus}$, but is regarded as unlikely because it requires a late migration and no reheating of the planet by tidal circulization (section \[sec:cold\_storage\_hypothesis\]). We have then investigated the possibility of subcritical core accretion as envisioned for Uranus and Neptune to account for the small envelope mass as well as the large core mass. In principle a large core of 50–80 ${{\rm M}_\oplus}$ can be formed by subcritical core accretion. However we have found it very unlikely for at least two reasons: (i) A subcritical core accretion results in a ratio of the core mass to the total mass above $\sim 0.7$ (section \[sec:critical\_mass\]), whereas our evolution calculations showed such a high ratio to be possible in a very limited range of parameters (see Table \[tab:constraints\]); (ii) The subcritical formation of a 50–80 ${{\rm M}_\oplus}$ core requires an extremely massive or metal-rich disk with dust surface density 30-50 times the values obtained for the minimum mass solar nebula (section \[sec:subcritical\]). A reasonably massive and/or metal-rich disk can form cores of at most $\sim 30 {{\rm M}_\oplus}$ far from the parent star. Those facts require us to consider (i) the migration of the planet, (ii) the supply of heavy elements to the planet during or after the gas accretion phase, and (iii) a limited supply of disk gas or loss of the envelope gas to account for the properties of HD149026b. In section \[sec:Z\_supply\], we have discussed how the heavy elements can be delivered to the planet during or after the gas accretion phase according to these scenarios. An efficient delivery during the gas accretion phase needs to be re-investigated in much more details, because the shepherding tends to prevent the planet from accreting planetesimals (section \[sec:concurrent\_Z\_supply\]). On the other hand, scattering of planetesimals/planets by one or several outer giant planets was shown to lead to an efficient accretion by a close-in giant planet, and is a promising explanation for the formation of metal-rich planets (section \[sec:late\_Z\_supply\]). The relatively small amount of hydrogen and helium present in HD149026b is also to be explained by the relatively slow formation of the planet in a low-density environment (section \[sec:limited\_gas\_supply\]). Another possibility is related to erosion of the envelope during giant impacts. A reasonable impact velocity of 2–2.5 times the two-body surface escape velocity was found to result in a substantial loss of the envelope gas, while solid cores are probably merged (section \[sec:envelope\_loss\]). In summary, we can propose at least two scenarios for the origin of HD149026b. - A giant planet with a core of $\sim 30 {{\rm M}_\oplus}$ forms far from its parent star in a relatively massive and/or metal-rich disk. Then it moves to the vicinity of the parent star through type-II migration [@L96]. In the outer regions additional more than three giant planets form, which may be likely in massive/metal-rich disks [@IL04a; @IL04b]. They starts orbit crossing and the innermost one is temporally detached from the outer two. The perturbation from the outer two secularly pumps up the eccentricity of the innermost one [@nagasawa]. The associated secular decrease of its pericenter distance results in collision with the close-in planet. The impactor also has a core of $\sim 30 {{\rm M}_\oplus}$. Their solid cores are merged to form a core of $\sim 60 {{\rm M}_\oplus}$, while envelope gas is severely eroded to $M_{\rm H He} \sim 50 {{\rm M}_\oplus}$. The orbital eccentricity of the merged body is damped by tidal interaction with the parent star. - A $\sim 30 {{\rm M}_\oplus}$ core forms in an evaporating disk that was originally massive and/or metal rich. The core thus becomes supercritical, but the accretion of the envelope is limited by viscous diffusion in an evolved, extended gaseous disk [see @GuillotHueso06]. The planet subsequently accretes heavy elements through the delivery of planetesimals and/or planets, either because of migration due to interactions with the gas disk or to secular perturbations with a massive outer planet. Above two scenarios are different from each other in that the former leaves at least one outer giant planet while the latter does not necessarily need any outer giant planets. Long-term radial velocity observations of HD149026b are thus needed to determine the presence of other planets in the system. We thank the anonymous referee for fruitful comments. This research was supported by Ministry of Education, Culture, Sports, Science and Technology of Japan, Grant-in-Aid for Scientific Research on Priority Areas, “Development of Extra-solar Planetary Science”. Agnor, C. & Asphaug, E. 2004, , 613, L157 Allard, F., Hauschildt, P. H., Alexander, D. R., Tamanai, A., & Schweitzer, A. 2001, , 556, 357 Alexander, D. R. & Ferguson, J. W. 1994, , 437, 879 Alibert, Y., Mordasini, C., & Benz, W. 2004, , 417, L25 Artymowicz, P. 1993, , 419, 166 Baraffe, I., et al. 2005, , 436, L47 Baraffe, I., Alibert, Y., Chabrier, G., & Benz, W. 2006, , 450, 1221 Bodenheimer, P., & Pollack, J. B. 1986, , 67, 391 Bodenheimer, P., Lin, D. N. C., & Mardling, R. A. 2001, , 548, 466 Boss, A. P. 1997, Science, 276, 1836 Burrows, A., Guillot, T., Hubbard, W. B., Marley, M. S., Saumon, D., Lunine, J. I., & Sudarsky, D. 2000, , 534, L97 Canup, R. M. 2004, , 168, 433 Charbonneau, D., et al. 2006, , 636, 445 Crida, A., Morbidelli, A., & Masset, F. 2006, , 181, 587 D’Angelo, G., Kley, W., & Henning, T. 2003, , 586, 540 Ford, E. B., Havlickova, M., & Rasio, F. A. 2001, , 150, 303 Fortney, J. J., Saumon, D., Marley, M. S., Lodders, K., & Freedman, R. S. 2006, , in press (astro-ph/0507422) Guillot, T., Burrows, A., Hubbard, W.B., Lunine, J.I. & Saumon,D. 1996, , 459, L35 Guillot, T. 2005, AREPS, 33, 493 Guillot, T., & Hueso R. 2006, MNRAS, 367, L47 Guillot, T., & Showman, A. P. 2002, , 385, 156 Gu, P-G, Lin, D. N. C., & Bodenheimer, P. 2003, , 588, 509 Hayashi, C. 1981, Prog. Theor. Phys. Suppl., 70, 35 Hubbard, W. B., & Marley, M. S. 1989, , 78, 102 Ida, S., Bryden, G., Lin, D. N. C., & Tanaka, H. 2000, , 534, 428 Ida, S., & Lin, D. N. C. 2004a, , 604, 388 Ida, S., & Lin, D. N. C. 2004b, , 616, 567 Ikoma, M., Emori, H., & Nakazawa, K. 2001, , 553, 999 Ikoma, M., & Genda, H. 2006, , in press Ikoma, M., Nakazawa, K., & Emori, H. 2000, , 537, 1013 Inaba, S., & Ikoma, M. 2003, , 410, 711 Inaba, S., Wetherill, G. W., & Ikoma, M. 2003, , 166, 46 Iro, N., Bézard, B., & Guillot, T. 2005, , 436, 719 Iwasaki, K., Emori, H., Nakazawa, K., & Tanaka, H. 2002, , 54, 471 Kary, D. M., Lissauer, J. J., & Greenzweig, Y. 1993, 106, 228 Kominami, J., & Ida, S. 2002, , 157, 43 Kominami, J., Tanaka, H., & Ida, S. 2006, , submitted Kozai, Y. 1962, , 67, 591 Laughlin, G., Wolf, A., Vanmunster, T., Bodenheimer, P., Fischer, D., Marcy, G., Butler, P., & Vogt, S. 2005, , 621, 1072 Lin, D. N. C., Bodenheimer, P., & Richardson, D. 1996, , 380, 606 Lin, D. N. C., & Ida, S. 1997, , 477, 781 Lin, D. N. C., & Papaloizou, J. C. B. 1985, Protostars and Planets II, ed. D. C. Black & M. S.Matthew (Tucson: Univ. of Arizona Press), 981 Lin, D. N. C., & Papaloizou, J. C. B. 1993, in Protostars and Planets III, ed. E. H. Levy and J. I. Lunine (Tucson:Univ. of Arizona Press), 749 Monaghan, J. J., & Gingold, R. A. 1983, J. Comp. Phys., 52, 375 Monaghan, J. J., & Lattanzio, J. C. 1985, , 149, 135 Marzari, F., & Weidenschilling, S. 2002, Icarus, 156, 570 Mizuno, H. 1980, Prog. Theor. Phys. Suppl., 64, 54 Mizuno, H., Nakazawa, K., & Hayashi, C. 1978, Prog. Theor. Phys., 60, 699 Nagasawa, M., Bessho, T., Tanaka, H., & Ida, S., in preparation Podolak, M., Podolak, J. I., & Marley, M. S. 2000, , 48, 143 Pollack, J. B., McKay, C. P., & Christofferson, B. M. 1985, , 64, 471 Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J., Podolak, M., & Greenzweig, Y. 1996, , 124, 62 Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954 Sato, B., et al. 2005, , 633, 465 Saumon, D., Chabrier, G., & Van Horn, H. M. 1995, , 99, 713 Saumon, D., & Guillot, T. 2004, , 609, 1170 Saumon, D., Hubbard, W. B., Burrows, A., Guillot, T., Lunine, J. I., & Chabrier, G. 1996, , 460, 993 Shakura, N. I., & Sunyaev, R. A. 1973, , 24, 337 Stepinski, T. F., & Valageas, P. 1997, , 319, 1007 Tanaka, H., & Ida, S. 1997, , 125, 302 Tanaka, H., & Ida, S. 1999, , 139, 350 Trilling, D. E., Benz, W., Guillot, T., Lunine, J. I., Hubbard, W. B., & Burrows, A. 1998, , 500, 428 Ward, W. 1993, , 106, 274 Weidenschilling, S. J., & Davis, D. R. 1985, 62, 16 Weidenschilling, S. J., & Marzari, F. 1996, , 384, 619 Wuchterl, G. 1993, , 106, 323 Youdin, A. N., & Shu, F. H. 2002, , 580, 494 [^1]: The form of equation (\[eq:KHtime\_formula\]) is different from that of equation (19) of @Ikoma00. The difference comes mainly from the difference in grain opacity [see @IG06 for the detailed discussion]. [^2]: The shepherding by isolated mean-motion resonances is called “resonant trapping” [e.g., @WD85; @KLG93]. [^3]: Here we assume the Hill radius ($r{_{\rm H,c}}$) is equal to 1.5 times disk scale height ($h$), roughly taking into account late gas accretion with a reduced rate after $r{_{\rm H, c}} = h$, following @IL04a.

--- abstract: 'We consider manipulation of the transmission coefficient for a quantum particle moving in one dimension where the shape of the potential is taken as the control. We show that the control landscape—the transmission as a functional of the potential—has no traps, i.e., any maxima correspond to full transmission.' author: - | Alexander N. Pechen$^{1,2}$[^1] and David J. Tannor$^1$\ \ $^1$Department of Chemical Physics, Weizmann Institute of Science\ Rehovot 76100, Israel\ $^2$Steklov Mathematical Institute of Russian Academy of Sciences\ Gubkina str. 8, Moscow 119991, Russia title: Control of quantum transmission is trap free --- Keywords: Quantum control, transmission coefficient, control landscape [*This article is part of a Special Issue dedicated to Professor Paul Brumer in recognition of his contributions to chemistry.*]{} Introduction ============ Control of atomic and molecular scale systems obeying quantum equations of motion is an important branch of modern science. Applications range from selective excitation of atomic or molecular states to laser control of chemical reactions and high harmonic generation [@Tannor1985; @Rice2000; @Brumer2003; @Fradkov2003; @Tannor2007; @Letokhov2007; @Brif2012]. One of the major questions in quantum control theory is whether for a given objective the control landscape has traps, that is, local maxima with values less than the global maximum [@Rabitz2004; @Pechen2011]. Much effort has been directed towards the study of control landscapes of $n$-level systems. Despite this effort, the proof of trap free behavior has been obtained so far only for two-level systems [@Pechen2012]. The case of systems with an infinite dimensional Hilbert space has not been treated at all. Here we consider control of transmission of a quantum particle moving through a potential barrier where the shape of the potential is used as a control parameter. This is relevant, for example, to control of tunneling [@Brumer1999; @Moiseev2002PRL; @Moiseev2002IEEE]. We show that the landscape of the transmission coefficient of a quantum particle as a functional of the potential is trap free, i.e., any maxima correspond to full transmission. Formulation =========== Consider a particle of a fixed energy $E$ scattering on a barrier of potential $V(x)$ which is assumed to have compact support ($V(x)=0$ when $|x|>a$ for some $a$). The particle wavefunction satisfies the time-independent Schrödinger equation $$\label{eq1} H_V\Psi(x)=E\Psi(x)$$ where $$\begin{aligned} H_V=-\frac{d^2}{dx^2}+V(x) .\end{aligned}$$ We take the mass $m=1/2$ and $\hbar=1$. The second-order differential equation (\[eq1\]) has two independent solutions. We are free to choose linear combinations of the solutions that behave as [@Note] \^0\_1(x)&=& { [l]{} e\^[i k\_E x]{}+A\_Ee\^[-i k\_Ex]{},x&lt;-a\ B\_Ee\^[i k\_E x]{},x&gt;a .\[eq2a\]\ \^0\_2(x)&=& { [l]{} D\_Ee\^[-i k\_Ex]{},x&lt;-a\ e\^[-i k\_E x]{}+C\_Ee\^[i k\_E x]{},x&gt;a . .\[eq2b\] Here $k_E=\sqrt{E}$. The solution $\Psi^0_1$ describes the particle incident on the barrier from the left. The particle is partially reflected and partially transmitted trough the barrier. Thus the wavefunction on the left, far away from the barrier, is a sum of the incoming and reflected waves, $\Psi^0_1(x)=e^{i k_Ex}+A_E e^{-i k_Ex}$ ($x\to -\infty$), whereas on the right of the barrier the wavefunction is an outgoing wave, $\Psi^0_1(x)=B_Ee^{i k_Ex}$ ($x\to +\infty$). The coefficients $A_E$ and $B_E$ determine the probabilities of reflection and transmission, respectively. The transmission coefficient is defined as the amplitude of the transmitted wave, $T_E[V]=|B_E|^2$, and describes the probability of transmission through the barrier. Similarly, the solution $\Psi^0_2$ describes the particle incident on the barrier from the right, which is partially reflected back to the right and partially transmitted to the left [@Ivanov2012]. Kinematic control landscape --------------------------- The general solution of Eq. (\[eq1\]) as $x\to -\infty$ is a sum of incoming and reflected waves $\Psi(x)=A'e^{i k_Ex}+Ae^{-i k_Ex}$ and as $x\to +\infty$ is $\Psi(x)=B e^{i k_Ex}+ B'e^{-i k_Ex}$. The coefficients $B$ and $B'$ are linearly related to the coefficients $A'$ and $A$ by a $2\times 2$ matrix $M$ which is called the [*monodromy operator*]{}: $$\begin{aligned} \left(\begin{array}{c} B \\ B' \end{array}\right) =M \left(\begin{array}{c} A' \\ A \end{array}\right)\, .\end{aligned}$$ The monodromy operator is an element of the [*special $(1,1)$ unitary group*]{} $SU(1,1)$ also called the [*real symplectic group of second order*]{} $Sp(1,\mathbb R)$ [@Arnold]. Any element of this group can be represented as $$\begin{aligned} M= \left(\begin{array}{cc} \sqrt{1+|z|^2}e^{i\phi} & z \\ \bar z & \sqrt{1+|z|^2}e^{-i\phi} \end{array}\right)\end{aligned}$$ where $z\in\mathbb C$ and $\phi\in[0,2\pi)$. Consider a wave incident on the potential from left infinity. Then $A'=1$, $B'=0$ and the equality $$\begin{aligned} \left(\begin{array}{c} B \\ 0 \end{array}\right) = \left(\begin{array}{cc} M_{11} & M_{12} \\ M_{21} & M_{22} \end{array}\right) \left(\begin{array}{c} 1 \\ A \end{array}\right)\end{aligned}$$ implies $A=-M_{21}/M_{22}$ and $B=1/M_{22}$. This gives for the transmission coefficient (as a function of $M$) the [*kinematic*]{} expression $$\label{eq2} T(M)=|B|^2=\frac{1}{|M_{22}|^2}=\frac{1}{1+|z|^2}$$ \[theorem1\] The only extrema of $T(M)$ over $M\in SU(1,1)$ are global maxima. These occur at $z=0$, where $$\begin{aligned} M= \left(\begin{array}{cc} e^{i\phi} & 0 \\ 0 & e^{-i\phi} \end{array}\right),\qquad \phi\in[0,2\pi).\end{aligned}$$ [**Proof.**]{} The theorem follows from eq. (\[eq2\]) and the domain of $z$. Theorem \[theorem1\] shows that the control landscape of the transmission coefficient has no kinematic traps and that its only kinematic extrema are global maxima corresponding to full transmission. Dynamic control landscape ------------------------- What is of ultimate interest is to know if the [*dynamic*]{} landscape of the transmission coefficient has traps, i.e. whether the transmission coefficient as a functional of the potential $V(x)$, has any local maxima or only a global maximum for full transmission. In this section we prove that there are no traps, i.e. all extrema of the transmission coefficient $T_E[V]$ as a functional of the potential are only global maxima. We will use the known result that for sufficiently smooth functions $f(E)$ and $S(E)$ $$\int\limits_\mathbb R \frac{e^{ix S(E_f)}f(E_f)}{E_f-E_i-i0}dE_f = i\pi[1+{\rm sgn}\,S'(E_i)] f(E_i) e^{ix S(E_i)}+O\left(x^{-\infty}\right)\label{lemma1}$$ provided $S'(E_0)\ne 0$ [@Fedoruk1987; @Avrimidi]. The only extrema of the objective $J[V]=T_E[V]$ are global maxima. [**Proof.**]{} Let $\Psi^0_{\alpha,E_i}(x)$ ($\alpha=1,2$) be two eigenfunctions of $H_V$ with energy $E$. Consider a small variation of the potential $V(x)\to V(x)+\delta v(x)$. The modification of the eigenfunction with energy $E$ due to the variation of the potential can be computed using perturbation theory for continuous spectrum as follows (we omit a sum over the discrete spectrum since the transmission coefficient depends only on the behavior of the wave function at infinity, where wavefunctions corresponding to the discrete spectrum vanish) $$\Psi_{1,E_i}=\Psi^0_{1,E_i}+\underbrace{\int\frac{\langle\Psi^0_{1,E_f} , \delta v\Psi^0_{1,E_i}\rangle}{E_i-E_f+i0}\Psi^0_{1,E_f}dE_f}\limits_{\delta\Psi_1(x)} +\underbrace{\int\frac { \langle\Psi^0_ { 2 , E_f } , \delta v\Psi^0_{1,E_i}\rangle}{E_i-E_f+i0}\Psi^0_{2,E_f}dE_f}\limits_{ \delta\Psi_2(x)}+o(\|\delta v\|)\label{eq5}$$ Here $\langle\Psi^0_{\alpha,E_f},\delta v\Psi^0_{1,E_i}\rangle=\int_{\mathbb R}\overline{\Psi}^0_{\alpha,E_f}(x)\delta v(x)\Psi^0_{1,E_i}(x)dx$ for $\alpha=1,2$. The transmission coefficient at energy $E_i$ for the modified potential $V+\delta v$ up to linear order in $\delta v$ can be computed as $$\begin{aligned} T_{E_i}[V+\delta v]&=&\lim\limits_{x\to +\infty}|\Psi_{1,E_i}(x)|^2\\ &=&\lim\limits_{x\to+\infty}\Bigl\{|\Psi^0_{1,E_i} (x)|^2 +2\Re\Bigl[\overline{\Psi}^0_{1,E_i} (x)\Bigl(\delta\Psi_1(x)+\delta\Psi_2(x)\Bigr)\Bigr]\Bigr\} +o(\|\delta v\|)\\ &=&T_{E_i}[V]+\delta J(V)+o(\|\delta v\|)\end{aligned}$$ By eqs. (\[eq2a\]), (\[lemma1\]), and (\[eq5\]) $$\begin{aligned} \lim\limits_{x\to+\infty}2\Re[\overline{\Psi}^0_{1,E_i} (x)\delta\Psi_1(x) ]&=&-2\Re\lim\limits_{x\to +\infty}\int\frac{\langle\Psi^0_{1,E_f},\delta v\Psi^0_{1,E_i}\rangle}{E_f-E_i-i0}B^*_{E_i} B_{E_f}e^{i (k_{E_f}-k_{E_i})x}\\ &=&-4\pi\Im[\langle\Psi^0_{1,E_i},\delta v\Psi^0_{1,E_i}\rangle|B_{E_i}|^2]=0\end{aligned}$$ Here the second equality follows from (\[lemma1\]) with $S(E_f)=k_{E_f}-k_{E_i}$ which gives ${\rm sgn}\, S'(E_f)|_{E_f=E_i}=1$, and the last equality follows from the fact that diagonal matrix elements of $\delta v$ are real. Similarly, $$\begin{aligned} \lefteqn{\lim\limits_{x\to+\infty}2\Re[\overline{\Psi}^0_{1,E_i} (x)\delta\Psi_2(x) ]} \\ &=&-2\Re\lim\limits_{x\to+\infty} \int\frac{\langle\Psi^0_{2,E_f},\delta v\Psi^0_{1,E_i}\rangle}{E_f-E_i-i0}B^*_{E_i}e^{-i k_{E_i}x}[e^{-i k_{E_f}x}+C_{E_f}e^{i k_{E_f}x}]dE_f\\ &=&-2\Re\lim\limits_{x\to+\infty} \int\frac{\langle\Psi^0_{2,E_f},\delta v\Psi^0_{1,E_i}\rangle}{E_f-E_i-i0}B^*_{E_i}[e^{-i (k_{E_f}+k_{E_i})x}+C_{E_f}e^{i (k_{E_f}-k_{E_i})x}]dE_f\end{aligned}$$ The term with $e^{-i (k_{E_f}+k_{E_i})x}$ in the square brackets gives zero contribution in the limit since for this term $S(E_f)=-k_{E_f}-k_{E_i}$ gives ${\rm sgn}\, S'(E_f)=-1$ and the integral is $O(x^{-\infty})$ according to (\[lemma1\]). The contribution of the term with $e^{i (k_{E_f}-k_{E_i})x}$ in the square brackets can be computed using equality (\[lemma1\]) as follows: $$\begin{aligned} -2\lim\limits_{x\to+\infty}\Re\int\frac{\langle\Psi^0_ {2,E_f} , \delta v\Psi^0_{1,E_i}\rangle}{E_f-E_i-i0}B^*_{E_i}C_{E_f}e^{i (k_{E_f}-k_{E_i})x}dE_f &=&-4\pi\Re\left[i\langle\Psi^0_{2,E_i},\delta v\Psi^0_{1,E_i}\rangle B^*_{E_i}C_{E_i}\right]\vphantom{\frac{A^*_{E_i}}{B^*_{E_i}}}\\ &=&-4\pi |B_{E_i}|^2\Im\left[\langle\Psi^0_{2,E_i},\delta v\Psi^0_{1,E_i}\rangle\frac{A^*_{E_i}}{B^*_{E_i}}\right]\\ &=&-4\pi T_{E_i}[V]\Im\left[\langle\Psi^0_{2,E_i},\delta v\Psi^0_{1,E_i}\rangle\frac{A^*_{E_i}}{B^*_{E_i}}\right]\\&=&\delta J(V)\end{aligned}$$ Here we have used the fact for any $E$ $$C(E)=-\frac{B(E) A^*(E)}{B^*(E)}$$ (see Eqs. (7.84) and (7.86) in [@Tannor2007]). A critical potential $V(x)$ should satisfy $\delta J(V)=0$ for any $\delta v$. Since for any $E$ $B(E)\ne 0$ and $T_E[V]\ne 0$, this is possible only if $A(E_i)=0$. That corresponds to $T_{E_i}[V]=1$, i.e., any critical potential leads to a global maximum of the transmission coefficient. This concludes the proof of the theorem. Comparison with the landscape for coherent control by lasers ============================================================ Quantum control landscapes for $n$-level systems controlled by lasers or electro-magnetic fields have been extensively studied in recent years. In this section, we put our findings about the landscape for control of transmission in the context of what is known about the landscape for coherent control of $n$-level systems by lasers. We assume that the $n$-level system interacts only with the laser and is isolated from other environments, i.e., is a closed quantum system. The evolution equation for a system controlled by a laser field ${\varepsilon}(t)$ is the Schrödinger equation $$i\frac{dU^{\varepsilon}_t}{dt}=(H_0+{\varepsilon}(t)V)U^{\varepsilon}_t,\qquad U^{\varepsilon}_0=\mathbb I$$ Here $H_0$ and $V$ are the free and interaction Hamiltonians, respectively. The solution of this equation is a unitary matrix, $U^{\varepsilon}_T\in U(n)$. The overall phase of the unitary evolution operator is physically meaningless, so that $U$ and $e^{i\phi}U$ describe the same physics. Thus the space of kinematic controls for laser control is the special unitary group $SU(n)$, instead of the special $(1,1)$ unitary group $SU(1,1)$ for control of the transmission coefficient. The objective that corresponds to the transmission coefficient is the probability of transition from some initial state $\psi_i$ to some final state $\psi_f$, $J(U_T^{\varepsilon})=|\langle\psi_f|U^{\varepsilon}_T|\psi_i\rangle|^2$. The kinematic landscape for the transition probability $J(U)$ (considered as a function of $U\in SU(n)$) is trap-free [@Rabitz2004]. However, this result does not imply the absence of traps for the corresponding dynamic landscape $J(U^{\varepsilon}_T)$ (considered as a functional of ${\varepsilon}$) due to the existence of non-regular controls (i.e. controls where the rank of the Jacobian $\delta U_T/\delta{\varepsilon}(t)$ is not maximal). To see this, consider the chain rule $$\frac{\delta J({\varepsilon})}{\delta{\varepsilon}(t)}=\frac{\delta J(U)}{\delta U}\biggl|_{U=U^{\varepsilon}_T} \frac{\delta U^{\varepsilon}_T}{\delta{\varepsilon}(t)}.$$ The absence of traps for $J(U)$ implies the absence of traps for $J({\varepsilon})$ only if the Jacobian $\delta U^{\varepsilon}_T/\delta{\varepsilon}(t)$ has full rank, i.e. has no zero eigenvalues. The analogous full rank criterion for control of the transmission coefficient is that rank of the Jacobian $\delta M_V/\delta V(x)$ is maximal, where $M_V$ is the monodromy operator for potential $V(x)$. The only rigorous proof of the absence of dynamical traps for coherent laser control is for $n=2$ [@Pechen2012]. Interestingly, the dimension of the corresponding kinematic control space $SU(2)$ is the same as the dimension of the kinematic control space $SU(1,1)$ for control of transmission. While the resulting conclusion of trap-free dynamics is the same for these two cases, the proofs are fundamentally different. We summarize the comparison of landscape-related notions for laser control and for control of transmission in Table 1. \[table1\] Coherent control by laser Control by potential --------------------- ----------------------------------------------------------------------------- ------------------------------------------------------------------------------ Dynamic control Laser pulse ${\varepsilon}(t)$ Potential $V(x)$ Kinematic control $U\in SU(n)$ $M\in SU(1,1)$ Objective $J(U)=|\langle\psi_f|U|\psi_i\rangle|^2$ $J(M)=\vphantom{\int\limits_A^B} \frac{\textstyle 1}{\textstyle |M_{22}|^2}$ Kinematic landscape No traps, No traps, ${\rm max}\, J=1$, ${\rm min}\, J=0$ ${\rm max}\, J=1$, ${\rm inf}\, J=0$ Full rank criterion Jacobian $\delta U^{\varepsilon}_T/\delta{\varepsilon}(t)$ has maximal rank Jacobian $\delta M_V/\delta V(x)$ has maximal rank Dynamic landscape Generally unknown. Trap-free for $n=2$. Trap-free. : Comparison of landscape-related notions for coherent control by lasers and control by potential. Acknowledgments {#acknowledgments .unnumbered} =============== A.N. Pechen acknowledges support of the Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme. This research is made possible in part by the historic generosity of the Harold Perlman family. [99]{} Tannor, D. J.; Rice, S. A. [*J. Chem. Phys.*]{} **1985**, [*83*]{}, 5013. Rice, S. A.; Zhao, M. [*Optical Control of Molecular Dynamics*]{}; Wiley: New York, **2000**. Brumer, P. W.; Shapiro, M. [*Principles of the Quantum Control of Molecular Processes*]{}; Wiley-Interscience: **2003**. (in Russian), Eds. Fradkov A. L. and Yakubovskii O. A. (Institute for Computer Studies, Moscow-Izhevsk, 2003). Tannor, D. J. [*Introduction to Quantum Mechanics: A Time Dependent Perspective*]{}; University Science Press: Sausalito, **2007**. Letokhov, V. S. [*Laser Control of Atoms and Molecules*]{}; Oxford University Press: USA, **2007**. Brif, C.; Chakrabarti, R.; Rabitz H., in [*Advances in Chemical Physics*]{}, edited by S. A. Rice and A. R. Dinner (Wiley, New York, 2012), vol. 148, p. 1. Rabitz, H. A.; Hsieh, M. M.; Rosenthal C. M. [*Science*]{} **2004**, [*303*]{}, 1998. Pechen, A. N.; Tannor, D. J. [*Phys. Rev. Lett.*]{} **2011**, [*106*]{}, 120402. Pechen, A. N.; Il’in, N. B. [*Phys. Rev. A*]{} **2012**, [*86*]{}, 052117. Frishman, E.; Shapiro, M.; Brumer, P. [*J. Chem. Phys.*]{} **1999**, [*110*]{}, 9-11. Averbukh, V; Osovski, S; Moiseyev, N. [*Phys. Rev. Lett.*]{} **2002**, [*89*]{}, 253201. Kenis, A. M.; Cederbaum, L. S.; Moiseyev, H. [*IEEE J. Quant. Electronics*]{} **2002**, [*38*]{}, 1638. For a somewhat more detailed discussion than usual see [@Tannor2007]. Ivanov, M. G. [*How to Understand Quantum Mechanics*]{}; R&C Dynamics: Moscow, **2012** (in Russian). Arnold, V. I. [*Geometrical Methods in the Theory of Ordinary Differential Equations*]{}; Springer: New York, **1988**. Fedoruk, M. V. [*Asymptotics, Integrals, and Series*]{}, p. 174; Nauka: Moscow, **1987**. Avrimidi, I. [*Lecture Notes on Asymptotic Expansions*]{}, p. 30. [^1]: Corresponding author. E-mail: <[email protected]>; Webpage: [mathnet.ru/eng/person17991](http://mathnet.ru/eng/person17991)

--- abstract: 'This paper reports turbulent boundary layer measurements made over open-cell reticulated foams with varying pore size [[[and thickness, but constant porosity ($\epsilon \approx 0.97$)]{}]{}]{}. The foams were flush-mounted into a cutout on a flat plate. A Laser Doppler Velocimeter (LDV) was used to measure mean streamwise velocity and turbulence intensity immediately upstream of the porous section, and at [[[multiple]{}]{}]{} measurement stations along the porous substrate. The friction Reynolds number upstream of the porous section was $Re_\tau \approx 1690$. For all [[[but the thickest]{}]{}]{} foam tested, the internal boundary layer was fully developed by $<10 \delta$ downstream from the porous transition, where $\delta$ is the boundary layer thickness. Fully developed mean velocity profiles showed the presence of a substantial slip velocity at the porous interface ($>30\%$ of the free stream velocity) and a mean velocity deficit relative to the canonical smooth-wall profile further from the wall. While the magnitude of the mean velocity deficit increased with average pore size, the slip velocity remained approximately constant. Fits to the mean velocity profile suggest that the logarithmic region is shifted relative to a smooth wall, and that this shift increases with pore size until it becomes comparable to substrate thickness $h$. For all foams, the turbulence intensity was found to be elevated further into the boundary layer to $y/ \delta \approx 0.2$. An outer peak in intensity was also evident for the largest pore sizes. Velocity spectra indicate that this outer peak is associated with large-scale structures resembling Kelvin-Helmholtz vortices that have streamwise length scale $2\delta-4\delta$. Skewness profiles suggest that these large-scale structures may have an amplitude-modulating effect on the interfacial turbulence.' author: - 'Christoph Efstathiou$^{1}$' - 'Mitul Luhar$^{1}$[^1]' bibliography: - 'Efstathiou2017.bib' title: 'Mean turbulence statistics in boundary layers over high-porosity foams' --- Introduction {#sec:intro} ============ Turbulent flows of scientific and engineering interest are often bounded by walls that are not smooth, solid, or uniform. Manufacturing techniques, operational requirements and natural evolution often lead to non-uniform, rough, and porous boundaries. Examples include flows over heat exchangers, forest canopies, bird feathers and river beds [e.g., @finnigan2000turbulence; @jimenez2004turbulent; @ghisalberti2009obstructed; @favier2009passive; @manes2011turbulent; @jaworski2013aerodynamic; @chandesris2013direct; @kim2016experimental]. Porous boundaries also enable active flow control through suction and blowing for use in drag reduction and delaying transition from laminar to turbulent flow [e.g., @parikh2011passive]. Despite the potential applications, relatively little is known about the relationship between turbulent flows and porous substrates. For example, it is unclear how established features of smooth-wall flows, such as the self-sustaining near-wall cycle, the logarithmic region in the mean profile, the larger-scale structures found further from the wall, and the interaction between the inner and outer regions of the flow [e.g. the amplitude modulation phenomenon; @marusic2010predictive] are modified over porous surfaces. As a result, there are few models that can predict how a porous substrate of known geometry will influence the mean flow field and turbulent statistics. A key limitation is that few experimental and numerical datasets exist for turbulent flows over porous media. Further, most previous experimental datasets (motivated primarily by flows over packed sediment beds or canopies) include limited near-wall measurements, while previous numerical simulations have been restricted to relatively low Reynolds numbers. [[[In addition, almost all previous studies on turbulent flow over porous substrates have employed relatively thick media, such that flow penetration into the substrate depends only on pore size or permeability [see e.g., @manes2011turbulent]. However, in many natural and engineered systems, the porous substrate can be of finite thickness and bounded by a solid boundary. Examples include feathers or fur in natural locomotion [@itoh2006turbulent; @jaworski2013aerodynamic], and heat exchangers employing metal foams [@mahjoob2008synthesis]. In such systems, substrate thickness can also have an important influence on flow development and the eventual equilibrium state. At the limit where porous medium thickness becomes comparable to pore size, the substrate can essentially be considered a rough wall. The transition from this rough-wall limit to more typical porous medium behaviour is not fully understood. The experimental study described in this paper seeks to address some of the limitations described above.]{}]{}]{} Previous studies {#sec:literature} ---------------- Previous numerical efforts investigating the effect of porous boundaries include the early direct numerical simulations (DNS) performed by @jimenez2001turbulent, which employed an effective admittance coefficient linking the wall-pressure and wall-normal velocity to model the porous wall. More recent DNS efforts have either employed the volume-averaged Navier-Stokes equations, where the porous substrate is modeled as a resistive medium [@breugem2006influence], or explicitly modeled the porous medium as an array of cubes [@chandesris2013direct]. In particular, the results of @breugem2006influence showed important deviations from turbulent channel flow over smooth walls. The mean velocity was significantly reduced across much of the channel for the cases with high porosity ($\epsilon = 0.8$ and $0.95$) and this was accompanied by a skin friction coefficient increase of up to $30\%$. The presence of the porous substrate also led to substantial changes in flow structure. Large spanwise rollers with streamwise length scale comparable to the channel height were observed at the porous interface. Given the presence of an inflection point in the mean velocity profile near the porous interface, @breugem2006influence attributed the emergence of these large-scale structures to a Kelvin-Helmholtz type of instability mechanism [see also @jimenez2001turbulent; @chandesris2013direct]. In addition, the near-wall cycle comprising streaks and streamwise vortices, a staple of smooth wall turbulent boundary layers, was substantially weakened over the porous substrate. This weakening was linked to a reduction in the so-called wall-blocking effect and enhanced turbulent transport across the interface. @breugem2006influence also found that the root mean square (rms) of the wall-normal velocity fluctuations did not exhibit outer layer similarity and suggested two potential causes for this (i) more vigorous sweeps and ejections due to the absence of an impermeable wall and (ii) insufficient scale separation between the channel half-height and the penetration distance into the porous substrate. @chandesris2013direct noted broadly similar trends in their DNS, which also considered thermal transport. [[[@rosti2015direct performed DNS studies of turbulent channel flow at $Re_\tau = 180$ using a VANS formulation that allowed for the porosity and permeability to be decoupled. These simulations showed that even relatively low wall permeabilities led to substantial modification of the turbulent flow in the open channel. Further, despite a substantial variation in the porosities tested ($\epsilon = 0.3-0.9$), the flow was found to be much more sensitive to permeability. @motlagh2016pod performed a VANS Large Eddy Simulation (LES) of channel flow over porous substrates with $\epsilon=0, 0.8$ and $0.95$ at bulk Reynolds number 5500. Based on proper orthogonal decomposition (POD) of the flow field, they showed that large-scale features prevalent in smooth-walled flows were still present but beginning to break down for the lower porosity case. Consistent with the results of @breugem2006influence, for the higher porosity case, these structures were replaced by spanwise-elongated counter-rotating vortices.]{}]{}]{} [[[Recently, @kuwata2016lattice and @kuwata2017direct have performed Lattice-Boltzman simulations over rough walls, staggered arrays of cubes, as well as anisotropic porous media. In particular, @kuwata2017direct considered a model system in which the streamwise, spanwise, and wall-normal permeabilities could be altered individually. These simulations showed that the streamwise permeability is instrumental in preventing the high- and low-speed streaks associated with the near-wall cycle. The simulations also indicated that, unlike the streamwise and spanwise permeabilities, the wall-normal permeability does not significantly enhance turbulence intensities.]{}]{}]{} Similar to turbulent flows over smooth and rough walls, previous studies suggest that a logarithmic region in the mean profile $U(y)$ can also be expected in turbulent flows over porous media. This logarithmic region is often parametrized as: $$\label{eq:log-law} U^+ = \frac{1}{\kappa} \ln{\left(\frac{y + y_d}{k_0} \right)} = \frac{1}{\kappa} \ln{(y^+ + y_d^+)} + B -\Delta U^+,$$ where $\kappa$ and $B$ are the von Karman and additive constants, $y$ is the wall-normal distance from the porous interface, $y_d$ is the shift of the logarithmic layer (or zero-plane displacement height), $k_0$ is the equivalent roughness height, and $\Delta U^+$ is the roughness function [@jimenez2004turbulent]. A superscript $+$ indicates quantities normalized by the friction velocity $u_\tau$ and the kinematic viscosity $\nu$. Fits to the mean velocity profiles obtained in DNS by @breugem2006influence suggest that the von Karman constant decreases from $\kappa \approx 0.4$ for the smooth wall case to $\kappa = 0.23$ for the most porous substrate. However, further tests were recommended at higher Reynolds number to confirm this effect. Experimental efforts in this realm have considered flows over beds of packed spheres, perforated sheets, foams, as well as seal fur [e.g., @ruff1972turbulent; @zagni1976channel; @kong1982turbulent; @itoh2006turbulent; @suga2010effects; @manes2011turbulent; @kim2016experimental]. Interestingly, the seal fur experiments show a reduction in skin friction though the exact mechanism behind this remains to be understood [@itoh2006turbulent]. @kong1982turbulent investigated the effect of small scale roughness over smooth, rough, and porous surfaces on turbulent boundary layers over bluff bodies. The porous boundaries consisted of perforated sheets and mesh screens. These experiments showed that the turbulent Reynolds stresses increased near the interface, as did the skin friction. However, it is difficult to separate roughness effects from permeability effects for these experiments. The mean velocity profile was shifted by $\Delta U^+ \approx 3-4$, which was similar to the shift obtained over an impermeable rough wall of similar geometry. @suga2010effects studied laminar and turbulent channel flow over foamed ceramics with porosity $\epsilon \approx 0.8$ and varying pore sizes via Particle Image Velocimetry (PIV). These experiments were carried out at relatively low Reynolds numbers (bulk Reynolds number $Re_b \le 10,200$). The measurements indicated that the transition to turbulence occurs at lower Reynolds number over the porous media. Further, turbulence intensities were generally enhanced over the porous medium, and the displacement and roughness heights in the modified logarithmic law were found to increase with increasing pore size and permeability. [[[@detert2010synoptic also used PIV to study the flow over packed spheres and gravel from the river Rhine in an open channel flow with friction Reynolds numbers $Re_\delta = 1.88-14.7\times 10^3$. At the relatively low porosities tested ($\epsilon = 0.26-0.33$), the flow exhibited many of the large-scale structures observed in turbulent boundary layers over smooth walls, including hairpin vortex packages. The observed flow patterns were also found to be relatively insensitive to Reynolds number.]{}]{}]{} [@manes2011turbulent] made open channel turbulent flow measurements over very porous ($\epsilon > 0.96$) polyurethane foam mattresses with 10-60 pores per inch (ppi) at Reynolds number $Re_\tau>2000$ using an LDV. In contrast to [@breugem2006influence], the results obtained by @manes2011turbulent supported the outer layer similarity hypothesis. This suggests that the lack of outer-layer similarity in the DNS carried out by @breugem2006influence resulted from insufficient scale separation between the inner and outer layers of the flow, which is analogous to the breakdown of outer-layer similarity in shallow boundary layers over rough walls [@jimenez2004turbulent]. Further, @manes2011turbulent reported a reduction in the streamwise fluctuation intensity near the wall and an increase in the intensity of wall-normal fluctuations, which they attributed to reduced wall blocking. For the logarithmic region in the mean profile, a fitting procedure similar to the one employed by @breugem2006influence and @suga2010effects yielded $\kappa \approx 0.3$ over the porous media and an equivalent roughness height $k_0$ that generally increased with increasing pore size. However, the mean profiles did not exhibit a clear logarithmic region over the 10 ppi foam. Since the flow is expected to penetrate further into the foam as pore size increases, @manes2011turbulent suggested that a lack of scale separation between the penetration distance and the water depth may have influenced the results. Finally, note that there are substantial similarities in turbulent flows over porous media and vegetation or urban canopies [e.g., @finnigan2000turbulence; @poggi2004effect; @white2007shear]. For example, large-scale structures resembling Kelvin-Helmholtz vortices play an essential role in dictating mass and momentum transport at the interface [see e.g., @finnigan2000turbulence] and a shifted log law of the form shown in equation (\[eq:log-law\]) provides a reasonable fit for the velocity profile above the canopy. In a comparative study of obstructed shear flows, @ghisalberti2009obstructed suggested that an inflection point exists in the mean profile if the distance to which the flow penetrates into the porous medium or canopy is much smaller than the height of the medium. This penetration distance is expected to scale as $\sqrt{k}$ for porous substrates [e.g., @battiato2012self], where $k$ is the permeability, and $(C_D a)^{-1}$ for canopies where, $C_D$ is a drag coefficient and $a$ is the frontal area per unit volume of the canopy. For densely packed or tall substrates where an inflection point is observed, the interfacial dynamics are dominated by structures resembling Kelvin-Helmholtz vortices. In such cases, the slip velocity at the interface depends primarily on the friction velocity, with little dependence on substrate geometry. Further, the interfacial turbulence also tends to be more isotropic, such that intensity of the wall-normal velocity fluctuations is comparable to the intensity of the streamwise fluctuations. Contribution and outline {#sec:outline} ------------------------ The present study builds on the experiments pursued by @manes2011turbulent to provide further insight into the near-wall flow physics over high-porosity surfaces. An LDV was used to measure streamwise velocity profiles over commercially-available foams with systematically varying pore sizes [[[and thicknesses]{}]{}]{} at moderately high Reynolds number. For reference, the friction Reynolds number over the smooth wall upstream of the porous section was $Re_\tau = u_\tau \delta/\nu \approx 1690$, where $\delta$ is the $99\%$ boundary layer thickness. The velocity profiles include measurements very close to the porous interface (corresponding to 2-3 viscous units over the smooth wall). [[[Unfortunately, wall-normal velocities were not measured due to instrumentation limitations.]{}]{}]{} Morphologically, the foams tested in this study are similar to those considered by [@manes2011turbulent]. However, one important distinction is the thickness of the foam layer, $h$. In an effort to isolate the effects of pore size, $s$, and permeability, $k$, @manes2011turbulent considered very thick porous media with $h \gg \sqrt{k}$ and $h \gg s$. The scale separation is more limited in the present study, with [[[$h/s \approx 4.3-44$ and $h/\sqrt{k} \approx 36-160$]{}]{}]{}. As a result, porous layer thickness does influence the flow for foams with the largest pore sizes and permeabilities. In other words, the present study provides insight into the transition between conditions in which the shear flow penetrates across the entire porous domain and conditions in which the shear layer only reaches a small distance into the porous medium. In a sense, this is analogous to the transition between sparse and dense canopy behaviour observed in vegetated shear flows [@luhar2008interaction; @ghisalberti2009obstructed]. In recent years, the amplitude modulation phenomenon observed in smooth- and rough-walled turbulent flows has led to a promising class of predictive models [e.g., @mathis2009large; @marusic2010predictive; @mathis2013estimating; @pathikonda2017inner]. Specifically, it has been observed that the so-called very-large-scale motions (VLSMs) prevalent in the logarithmic region of the flow at high Reynolds number [@smits2011high] have a modulating influence on the intensity of the near-wall turbulence. Further, it has been shown that there is an intrinsic link between this phenomenon and the skewness of the streamwise velocity fluctuations [@mathis2011relationship; @duvvuri2015triadic]. Therefore, by considering the skewness of the streamwise velocity, we also evaluate whether such interactions between the inner and outer region persist in turbulent flows over porous media that may be dominated by a different class of large-scale structure, i.e. Kelvin-Helmholtz vortices. The remainder of this paper is structured as follows: §\[sec:expts\] describes the experiments, providing details on the flow facility, porous substrates, and diagnostic techniques; §\[sec:development\] shows results on flow development over the porous foams; §\[sec:poresize\] illustrates the effect of pore size on mean turbulence statistics; [[[§\[sec:thickness\] explores the effect of substrate thickness; §\[sec:spectra\] presents velocity spectra over all the different foams tested; ]{}]{}]{}§\[sec:discussion\] tests whether a shifted logarithmic region exists over the porous surfaces, evaluates the relative effect of pore size and substrate thickness, and also considers how porous substrates may affect the amplitude modulation phenomenon. Conclusions are presented in §\[sec:conclusions\]. Experimental methods {#sec:expts} ==================== Flow facility and flat plate apparatus {#sec:facility} -------------------------------------- ![Schematics showing flat plate apparatus in the wall normal-spanwise plane ($y-z$, top) and the streamwise-wall normal plane ($x-y$, bottom). Measurement positions marked 1-4 were located at $x/h = 11, 21, 42, 53$, where $h=12.7$ mm is the baseline substrate thickness, and $x=0$ is defined as the smooth-porous wall transition. The smooth wall reference measurements were made at a location $x/h=-21.5$. Note that the wall-normal coordinate is $y$.[]{data-label="fig:diag"}](Figures/Fig1-Exp.pdf){width="\textwidth"} All experiments were conducted in the USC water channel, a free-surface, recirculating facility with glass along both sidewalls and at the bottom to allow for unrestricted optical access. The water channel has a test section of length 762 cm, width 89 cm, and height 61 cm, and is capable of generating free-stream velocities up to 70 cm/s with background turbulence levels $<1\%$ at a water depth of 48 cm. The temperature for all experiments was $23\pm 0.5^\circ C$ for which the kinematic viscosity is $\nu = 0.93 \times 10^{-2}$ cm$^2$/s. Figure \[fig:diag\] provides a schematic of the experimental setup in the wall normal-spanwise (top) and streamwise-wall normal (bottom) planes. A 240 cm long flat plate was suspended from precision rails at a height $H=30$ cm above the test section bottom. To avoid free-surface effects, measurements were made below the flat plate. The nominal free-stream velocity was set at $U_e = 58$ cm/s for all the experiments. The confinement between the flat plate and bottom of the channel naturally led to a slightly favorable pressure gradient and slight free stream velocity increase along the plate $(\leq 3\%)$, however the non-dimensional acceleration parameter, $\Lambda = \frac{\nu}{U_e^2} \frac{dU_e}{dx}$ was on the order of $10^{-7}$ suggesting any pressure gradient effects are likely to be mild [@PatelPreston1965; @de2000reynolds; @schultz2007rough]. A cutout of length 89 cm and width 60 cm was located 130 cm downstream of the leading edge. Smooth and porous surfaces were substituted into the cutout, flush with the smooth plate around it. The porous test specimens, described in further detail below, were bonded to a solid Garolite^TM^ sheet to provide a rigid structure and prevent bleed through. [[[The setup was designed to accommodate porous substrates of thicknesses $h=6.35, 12.7$ and $25.4$ mm.]{}]{}]{} Care was taken to minimize gaps and ensure a smooth transition from solid to porous substrate. Velocity profiles were measured over the smooth section upstream of the cutout and at four additional locations over the porous walls. The flow was tripped by a wire of 0.5 mm diameter located 15 cm downstream of the leading edge. Velocity measurement {#sec:diagnostics} -------------------- Measurements of [[[streamwise velocity, $u$,]{}]{}]{} were made at the channel centerline using a Laser Doppler Velocimeter (LDV, MSE Inc.) with a 50 cm standoff distance and a measurement volume of $300 \mu$m by $150 \mu$m by $1000 \mu$ m ($x$ by $y$ by $z$). The LDV was mounted on a precision traverse capable of 16 $\mu$m resolution. Polyamide seeding particles (PSP) with an average size of 5 $\mu$m were used to seed the flow. While the large standoff distance enabled measurements at the channel centerline, it also limited data rates to 50 Hz in the free-stream and less than 1 Hz at the stations closest to the wall. A minimum of 800 data points in the regions with lowest velocity and 4000-25000 data points in regions of higher velocity were collected to ensure fully converged statistics. In all cases, a preliminary coarse velocity profile was measured to determine the boundary layer thickness and the approximate location of the wall. This preliminary profile was used to generate a finer logarithmically-spaced vertical grid for the actual measurements. The nominal smooth-wall location ($y^\prime = 0$) was identified as the position where the data rate dropped to zero. For all the profiles, the vertical grid resolution was reduced to 30 $\mu$m in the near-wall region. Given the 150$\mu$m measurement volume and the $30\mu$m vertical resolution, the nominal estimate for wall-normal location suffers from an uncertainty of $\le 75\mu$m. LDV measurements suffer from two significant distortions: velocity gradients across the measurement volume and velocity biasing [@durst1976principles; @degraaff2001high]. The former is significant in regions where large velocity gradients are present across the measurement volume, as is the case near solid walls. The latter occurs because, in turbulent flows, more high velocity particles move through the measurement volume in a given period compared to low-velocity particles (assuming uniform seeding density). To correct for this bias, an inverse velocity weighting factor was used to correct mean statistics [@mclaughlin1973biasing]. The mean streamwise velocity was estimated using the following relationship: $$\label{eq:weight-U} U = \frac{\sum\limits_{i=1}^{N}b_i u_i}{\sum\limits_{i=1}^{N} b_i},$$ where $u_i$ is an individual velocity sample, $N$ is the total number of samples, and $b_i = 1/|u_i|$ is the weighting factor. Similarly, the weighted streamwise turbulence intensity was estimated as: $$\label{eq:weight-u} \overline{u^2}=\frac{\sum\limits_{i=1}^{N} b_i \left[u_i-U\right]^2}{\sum\limits_{i=1}^{N}b_i}.$$ Alternative methods of correcting for velocity biasing are discussed and evaluated in [@herrin1993investigation]. These methods include using the inter-arrival time ($b_i = t_i-t_{i-1}$) as the weighting factor, or the sample-and-hold technique ($b_i=t_{i+1}-t_i$). Figure \[fig:weights\] shows how these different correction techniques affect a representative mean velocity ($U$) and streamwise turbulent intensity ($\overline{u^2}$) profile. Inverse velocity weighting leads to the largest correction relative to the raw data in the near-wall region where the sampling is more intermittent (n.b. this is also consistent with previous studies). However, the overall trends remain very similar in all cases and the correction is minimal in the outer region of the flow (see e.g. $y/\delta \ge 0.1$ in figure \[fig:weights\]). ![Mean velocity $U$ and streamwise turbulence intensity $\overline{u^2}$ profiles corrected by inverse velocity magnitude and inter-arrival time weighting. The wall-normal coordinate is normalized using the local 99% boundary layer thickness, $\delta$.[]{data-label="fig:weights"}](Figures/Fig2-Weights.pdf) While the corrections accounting for velocity biasing do introduce uncertainty, this uncertainty is likely to be correlated across all cases. Therefore, for comparison across cases, the uncertainty in mean velocity is taken to be the larger of the instrument uncertainty ($0.1$%, MSE Inc.) and the standard error, given by $\overline{\sigma_u}= \sigma_u/\sqrt{N}$, where $\sigma_u$ is the standard deviation of the measurements. In all cases, the standard error was the larger contributer to uncertainty in the near-wall region due to the limited number of samples acquired. As shown in figure \[fig:uncertainty\], the standard error was typically $\overline{\sigma_u} \le 1$% across the entire boundary layer. ![Profile showing standard error of mean velocity (in $\%$) over upstream smooth wall section. The discontinuity in $\overline{\sigma_u}$ between $y/\delta = 0.01$ and $0.02$ corresponds to the location where the measurement duration for each point was reduced from 45 minutes to 10 minutes.[]{data-label="fig:uncertainty"}](Figures/Fig3-StandardError.pdf) The friction velocity, $u_\tau$, over the smooth-wall section upstream of the porous cutout was estimated by fitting the following relationship to the near-wall mean velocity measurements: $U(y) = (u_\tau^2/\nu)(y^\prime + y_0)$, in which $y_0$ represents an offset from the nominal wall location where the data rate falls to zero, $y^\prime = 0$. The smooth-wall velocity profiles reported below correct for this offset. In other words, the true wall-normal distance is assumed to be $y = y^\prime + y_0$, such that the near-wall velocity profile is consistent with the theoretical relation $U^+ = y^+$ (see figure \[fig:dev-u\]). No such correction was made for the porous substrate. For reference, the friction velocity upstream of the porous section was estimated to be $u_\tau = 2.3 \pm 0.05$ cm/s. The uncertainty is estimated by fitting the relationship in the viscous sublayer to different ranges of 5-10 points near the wall. The offset was $y_0 = 40\mu$m, which translates into approximately one viscous length scale. Note that the above estimate for the friction velocity is also consistent with estimates obtained from fits to the logarithmic region in the mean velocity profile using $\kappa =0.39$ [@marusic2013logarithmic]. The estimates for $u_\tau$ derived above also agreed with the third method utilizing the velocity gradient. See figure \[fig:log-law\] and related discussion in §\[sec:log-shift\] for further detail. Reynolds number ranges and spatio-temporal resolution {#sec:Re} ----------------------------------------------------- The 99% boundary layer thickness, estimated via interpolation, was $\delta = 6.83 \pm 0.07$ cm for the smooth-wall profile, and so the friction Reynolds number was $\delta^+ = u_\tau \delta/\nu = 1690 \pm 70$ upstream of the porous section. Over the porous sections, the boundary layer thickness increased; the maximum measured value was $\delta = 11.2 \pm 0.1$ cm. As a result, the Reynolds number based on the nominal free-stream velocity ranged from $Re = U_e \delta/\nu = 42600-69900$. The estimated friction velocity $u_\tau = 2.3 \pm 0.05$ cm/s translates into a viscous length-scale $\nu/u_\tau \approx 40 \mu$m and viscous time-scale $\nu/u_\tau^2 \approx 1.7$ ms. Thus, the 16 $\mu$m precision of the traverse provides adequate vertical resolution for profiling purposes. However, the LDV measurement volume ($150 \mu$m in $y$) extends across 3 viscous units in the wall-normal direction, which means that the near-wall measurements reported below suffer from distortion due to velocity gradients. In the near-wall region, the LDV sampling frequency was approximately 0.5 Hz, which corresponds to an average sampling time of 1200 viscous units. In the outer region of the flow ($y/\delta \gtrsim 0.05$), the sampling frequencies were as high as 50 Hz, which translates into an average sampling time of 12 viscous units. In other words, time-resolved velocity measurements are only expected in the outer region of the flow. Porous Substrates {#sec:foams} ----------------- Boundary layer measurements were made adjacent to four different types of open-cell reticulated polyurethane foams. Per the manufacturer, the porosity of all the foams was $\epsilon \approx 0.97$. This was confirmed to within $0.5$% via measurements that involved submerging the foams in water to measure solid volume displacements. The nominal pore sizes corresponded to 10, 20, 60, and 100 pores per inch (ppi, see figure \[fig:foam\]). Pore size distributions for each foam were estimated from photographs of thin foam sheets via image analysis routines in Matlab (Mathworks Inc.). The measured average pore sizes ranged from $s = 2.1 \pm 0.3$ mm for the 10 ppi foam to $s = 0.29 \pm 0.02$ mm for the 100 ppi foam (see Table \[tab:foam\]). These measurements are generally within $\pm 20\%$ of the nominal pore sizes. Pore size measurements for the 10 and 20 ppi foam were also made using precision calipers. These caliper-based measurements were consistent with the imaging-based estimates to within uncertainty ($s = 2.2 \pm 0.1$mm for 10 ppi and $s = 1.7 \pm 0.1$ mm for 20 ppi, where uncertainties correspond to standard error). Note that all of the pore sizes discussed above and listed in Table. \[tab:foam\] correspond to the *exposed* streamwise-spanwise plane of the foam. Caliper-based measurements suggest that the pore structure may be anisotropic. For the 10 ppi foam, average pore sizes were approximately 14% larger than the listed values in the spanwise-wall normal plane of the foam ($2.5 \pm 0.1$ mm) and 21% larger in the streamwise-wall normal plane ($2.7 \pm 0.1$ mm). Similarly, for the 20 ppi foam, average pore sizes were approximately 10% larger in the spanwise-wall normal plane and 23% larger in the streamwise-wall normal plane. Another important length scale arises from the permeability, $k$, of the porous medium. Specifically, $\sqrt{k}$ determines the distance to which the shear penetrates into the porous medium [@battiato2012self]. [[[Permeabilities were estimated from pressure drop experiments using Darcy’s law. These estimates ranged from $k = 6.6 \pm 0.6 \times 10^{-9}$ m$^2$ for the 100 ppi foam to $k = 46 \pm1 \times 10^{-9}$ for the 10 ppi foam.]{}]{}]{} Based on the friction velocity measured upstream of the plate, the inner-normalized pore sizes range from $s^+ \approx 7$ for the 100 ppi foam to $s^+ \approx 52$ for the 10 ppi foam. [[[Similarly, the Reynolds number based on permeability varies between $Re_k = \sqrt{k}u_\tau / \nu \approx 2.0 $ for the 100 ppi foam to $Re_k \approx 5.3 $ for the 10 ppi foam. The baseline thickness tested for all foams was $h=12.7$ mm. For the 20 ppi foam, two additional thicknesses, $h = 6.35$ mm and $h = 25.4$ mm, were also considered. This means that the ratio of foam thickness to average pore size ranged between $h/s = 4.3$ for the thin 20 ppi foam to $h/s = 44$ for the 100 ppi foam. Finally, keep in mind that despite having the same nominal pore sizes, the 10 and 60 ppi foams tested here are not identical to those tested by @manes2011turbulent. For example, the 10 ppi foam tested by @manes2011turbulent had a pore size ($s=3.9$ mm) approximately twice that of the 10ppi foam used here, and a permeability ($k = 160 \times 10^-9$ m$^2$) almost four times higher.]{}]{}]{} Table \[tab:foam\] lists physical properties for all the foams tested in the experiments, along with related dimensionless parameters. ![Photographs showing thin sheets of the 10, 20, 60, and 100 ppi foams (from left to right). Each image represents a 2 cm $\times$ 2 cm cross section.[]{data-label="fig:foam"}](Figures/Fig4-Foams.jpg){width="\textwidth"} ------------------- --------------------- ------------------- ----------------- -------- ------- -------------- ------- -- -- -- Foam $k$($10^{-9}$m$^2$) $\epsilon$ $s$ (mm) $Re_k$ $s^+$ $h/\sqrt{k}$ $h/s$ \[0.1cm\] 10 ppi $46\pm1$ $0.976 \pm 0.003$ $2.1 \pm 0.3$ $5.3$ 52 59 6 (160) (3.9) \[0.1cm\] 20 ppi 73 8.5 20 ppi thin $30\pm2$ $0.972 \pm 0.003$ $1.5 \pm 0.2$ $4.3$ 37 37 4.3 20 ppi thick 147 17 \[0.1cm\] 60 ppi $7.9\pm0.6$ $0.965 \pm 0.005$ $0.40 \pm 0.03$ $2.2$ 10 143 32 (6) (0.5) \[0.1cm\] 100 ppi $6.6\pm 0.6$ $0.967 \pm 0.005$ $0.29 \pm 0.02$ $2.0$ 7 156 44 ------------------- --------------------- ------------------- ----------------- -------- ------- -------------- ------- -- -- -- : [[[Permeability ($k$), porosity ($\epsilon$), average pore sizes ($s$), and related dimensionless parameters for tested foams. Permeability and pore size values from [@manes2011turbulent] are noted in parenthesis for the 10 and 60 ppi foams. Note that $s^+ = s u_\tau /\nu$ and $Re_k = \sqrt{k} u_\tau/\nu$ are defined using the friction velocity upstream of the porous section. Porosity ($\epsilon$) was estimated from solid volume displacement in water and permeability was estimated from pressure drop experiments.]{}]{}]{}[]{data-label="tab:foam"} Results {#sec:results} ======= Boundary layer development {#sec:development} -------------------------- First, we consider boundary layer development over the porous foams. The results presented below correspond to the 20 ppi foam [[[of thickness $h=12.7$mm]{}]{}]{}. Similar trends were observed for all the porous substrates. As illustrated schematically in figure \[fig:diag\], the transition from the smooth wall to the porous substrate leads to the development of an internal layer, which starts at the transition point and grows until it spans the entire boundary layer thickness. When this internal layer reaches the edge of the boundary layer, a new equilibrium boundary layer profile is established. This new profile reflects the effects of the porous substrate. Internal layers have been studied extensively in the context of smooth to rough wall transitions in boundary layers [e.g., @antonia1971response; @jacobi2011new]. In particular, previous literature suggests that turbulent boundary layers adjust relatively quickly for transitions from smooth to rough walls; the adjustment occurs over a streamwise distance of $O(10\delta)$. The profiles of mean velocity ($U/U_e$) and streamwise intensity ($\overline{u^2}/U_e^2$) shown in figure \[fig:dev-u\] suggest that the adjustment from smooth to porous wall velocity profiles occurs over a similarly short streamwise distance. ![Streamwise mean velocity (a) and turbulence intensity (b) measured at streamwise locations $x/h={-21.5,11,21,42,53}$ relative to the transition from smooth wall to porous substrate. These data correspond to the 20 ppi foam with $h=12.7$mm. The dashed lines correspond to a linear profile of the form $U^+ = y^+$ in the near-wall region and a logarithmic profile of the form $U^+ = (1/\kappa)\ln(y^+)+B$ in the overlap region, with $\kappa = 0.39$ and $B = 4.3$. The friction velocity was estimated from fitting a linear slope to the near-wall measurements.[]{data-label="fig:dev-u"}](Figures/Fig5-Development.pdf) Upstream of the porous section, the measured profiles are consistent with previous smooth wall literature. In the near-wall region ($y/\delta < 5\times 10^{-3}$), the measurements agree reasonably well with the fitted linear velocity profile $U^+ = y^+$. The measured mean velocities are higher at the first two measurement locations, which can be attributed to the bias introduced by velocity gradients across the LDV measurement volume. In the overlap region ($0.02 \le y/\delta \le 0.2$), the mean velocity profile is consistent with the logarithmic law: $U^+ = (1/\kappa)\ln (y^+) + B$. The streamwise intensity profile shows the presence of a distinct inner peak at $y/\delta \approx 0.006$, or $y^+ \approx 10$, which is also consistent with previous studies. The presence of the porous substrate substantially modifies the mean velocity and streamwise intensity profiles. Figure \[fig:dev-u\] shows clear evidence of substantial slip at the porous interface ($U(y\approx 0) > 0.3U_e$). Farther from the interface, there is a velocity deficit relative to the upstream, smooth wall profile. This velocity deficit increases with distance along the porous substrate, and appears to saturate for the final two profiles measured at $x/h \ge 42$. The mean velocity profiles collapse together for $y/\delta \ge 0.5$, suggesting that the outer part of the wake region remains unchanged over the porous substrate. Consistent with the mean velocity profiles, the streamwise intensity profiles also show a substantial departure from the smooth case. Although there is some scatter in the measurements closest to the interface, the inner peak is replaced by an elevated plateau at $\overline{u^2}/U_e^2 \approx 0.01$, which extends from the porous interface to $y/\delta \approx 0.01$. Further, an outer peak appears near $y/\delta \approx 0.1$. The origin of this outer peak is discussed further in §\[sec:poresize\] below. In general, the streamwise intensity profiles also converge for $x/h \ge 42$. ![Normalized boundary layer thickness $\delta/h$ for the 20 ppi foam measured at streamwise locations $x/h={-21.5,11,21,42,53}$ relative to the substrate transition point.[]{data-label="fig:dev-delta"}](Figures/Fig6-Delta.pdf) The streamwise evolution of the boundary layer thickness is presented in figure \[fig:dev-delta\]. Boundary layer growth over the first two measurement locations ($x/h \le 21$) past the smooth to porous transition is relatively slow and appears unchanged from the smooth wall boundary layer, suggesting that the internal layer does not yet span the boundary layer thickness at these locations. For the last two measurement locations, $x/h \geq 42$, the boundary layer thickness grows much faster, suggesting that the flow adjustment is complete and that the effects of the porous substrate extend across the entire boundary layer. These observations are consistent with the mean velocity and streamwise intensity profiles shown in figure \[fig:dev-u\]. As an example, the profiles at measurement location $x/h = 11$ show that the internal layer only extends to $y/\delta \approx 0.2$. For $y/\delta \ge 0.2$, the mean velocity and streamwise intensity collapse onto the smooth-wall profiles. Development data for the remaining foams with are not presented here for brevity. [[[For all the foams of thickness $h \le 12.7$ mm]{}]{}]{}, the velocity measurements and boundary layer thickness data suggest that flow adjustment is complete by the measurement station at $x/h = 42$. Since the incoming boundary layer thickness is $\delta \approx 5.5 h$, the streamwise adjustment happens over $x \approx 8\delta$, which is consistent with previous literature on the transition from smooth to rough walls. [@antonia1971response] [[[However, this analogy to flow adjustment over rough-wall flows breaks down for the the thick 20 ppi foam with $h = 25.4$ mm. For the thick foam, flow adjustment was not complete even at the last measurement location (see results presented in §\[sec:thickness\]). This observation is still in broad agreement with the results presented above since the last measurement location only yields a dimensionless development length of $x/h \approx 26$ for the thick foam, while figure \[fig:dev-delta\] suggests that $x/h \ge 30$ is required for adjustment.]{}]{}]{} [[[The specific setup considered here may also be seen as flow over a backward facing step, with the region beyond the step filled with a highly porous ($\epsilon > 0.96$) and permeable ($Re_k>1$) material. Step Reynolds numbers in the present experiment range from $Re_h=U_e h/\nu = 4-16\times 10^3$. DNS by @le1997direct and LDV measurements by @jovic1994backward at $Re_h=5100$ for a canonical (i.e., unfilled) backward facing step indicate that the mean velocity profile behind the step does not return to a log-law for $x/h=20$ beyond the step. This is comparable to the development length observed here over the flush-mounted highly porous substrates ($x/h>30$). Thus, it appears that both $\delta$ and $h$ play a role in dictating flow adjustment for the system considered here.]{}]{}]{} Unfortunately, since the present study was limited to four measurement locations along the porous substrate, there is insufficient spatial resolution in the streamwise direction to provide further insight into the scaling behaviour of boundary layer adjustment and growth over porous substrates. Effect of pore size {#sec:poresize} ------------------- ![Mean velocity (a) and turbulence intensity (b) profiles for the smooth wall and for all the porous foams at $x/h=42$.[]{data-label="fig:ppi-u"}](Figures/Fig7-PoreSizeU.pdf) Next, we consider the effect of varying pore size on the fully-developed boundary layer profiles measured at $x/h = 42$ [[[for the foams of thickness $h = 12.7$ mm]{}]{}]{}. Figure \[fig:ppi-u\] shows the measured mean velocity and streamwise intensity for each of the foams tested, together with the smooth-wall profile measured upstream of the porous section. The mean velocity profile over the porous foams is modified in two significant ways with respect to the smooth wall profile taken upstream. First, there is a substantial slip velocity near the porous substrate. This slip velocity is approximately 30$\%$ of the external velocity across all substrates, with little dependence on pore size. The observed slip velocity is consistent with the DNS results of @breugem2006influence, who observed a slip velocity of approximately $30\%$ for their highest porosity case ($\epsilon=0.95$, $Re_k=9.35$). Note that the properties for the foams tested here (listed in table \[tab:foam\]) are similar to the properties of the porous substrate considered in the simulations. Second, there is a mean velocity deficit relative to the smooth-wall case from $0.004\leq y/\delta \leq 0.4$. This deficit generally increases with increasing pore size, though there is some non-monotonic behaviour. The maximum deficit relative to the smooth-wall profile is approximately 15$\%$ for the 100 ppi foam ($s^+ = 7$) and this increases to almost 50$\%$ for the 20 ppi foam ($s^+ = 37$). However, the deficit for the foam with the largest pore sizes (10 ppi, $s^+ = 52$) is smaller than the deficit over the 20 ppi foam. In the outer wake region ($y/\delta > 0.5$), all mean velocity profiles collapse onto the canonical smooth-wall profile again. The streamwise turbulence intensity profiles, plotted in figure \[fig:ppi-u\]b, show that the inner peak disappears for all the porous foams. Instead, there is a region of elevated but roughly constant intensity that extends from the porous interface to $y/\delta \approx 0.01$. For $y/\delta > 0.01$, the intensity profiles show a strong dependence on pore size. For the smallest pore sizes, the streamwise intensity either decreases slightly (100 ppi) or stays approximately constant until $y/\delta \approx 0.1$ (60 ppi). For foams with larger pore sizes, the intensity increases and a distinct outer peak appears near $y/\delta \approx 0.1$. All the profiles collapse towards smooth-wall values for $y/\delta \ge 0.5$. Consistent with the mean velocity measurements, the streamwise intensity profiles also show non-monotonic behaviour with pore size. While the streamwise intensity in the outer region of the flow generally increases with pore size, the magnitude of the outer peak is higher for the 20 ppi foam compared to the 10 ppi foam. Velocity spectra, presented in §\[sec:spectra\] below, provide further insight into the origin of this outer peak. ![Wall-normal profiles of skewness ($Sk$) for the smooth wall and for all the porous foams at $x/h = 42$.[]{data-label="fig:skew"}](Figures/Fig8-PoreSizeSkew.pdf) Figure \[fig:skew\] shows skewness profiles over the smooth wall and porous foams. Consistent with previous measurements at comparable Reynolds number [e.g., @mathis2011relationship], the skewness over the smooth wall is negative or close to zero across much of the boundary layer ($0.004 < y/\delta < 1$). In contrast, over the foam substrates, the sign of the skewness is positive until $y/\delta \approx 0.1$. Interestingly, the location of this change in sign for the skewness corresponds to the location of the outer peak in streamwise intensity profiles. Further, the magnitude of the skewness generally increases with pore size, which is consistent with the intensity measurements (the 10 and 20 ppi cases again show non-monotonic behaviour). These observations are particularly interesting given the intrinsic link between skewness and amplitude modulation [@schlatter2010quantifying; @mathis2011relationship], and suggest that structures responsible for the outer peak in streamwise intensity over the foams may have a modulating effect on the interfacial turbulence. This possibility is discussed in greater detail in §\[sec:amplitude-mod\] below. Effect of substrate thickness {#sec:thickness} ----------------------------- ![Mean velocity (a) and turbulence intensity (b) profiles for the smooth wall and for the 20ppi foam of varying thickness at the same physical location.[]{data-label="fig:thick"}](Figures/Fig9-ThicknessU.pdf) The thickness of the porous substrate, $h$, is another important parameter that dictates flow behaviour. Figure \[fig:thick\] shows the measured mean velocity and streamwise intensity profiles over 20 ppi foams of varying thickness, $h = 6.35, 12.7$ and $25.4$ mm. In dimensionless terms, these thicknesses correspond to $h/s = 4.3, 8.5$ and $17$, respectively, where $s = 1.5 \pm 0.2$ mm is the average pore size (Table \[tab:foam\]). Note that the measurements shown in figure \[fig:thick\] are for the same physical location, $x = 53.3$cm, which yields normalized distances increasing from $x/h = 21$ for the thickest foam to $x/h = 84$ for the thinnest foam. The foam with thickness $h=12.7$mm is considered the baseline case, since velocity measurements made over this foam have already been discussed in §\[sec:development\] and §\[sec:poresize\]. The foams of thickness $h = 6.35$mm and $h = 25.4$mm are referred to as the thin and the thick foam, respectively. As noted in §\[sec:development\], the flow was fully developed for the thin and baseline foams, but still developing for the thick foam. Mean velocity profiles for both the thick and thin foam show similar slip velocities at the interface compared to the baseline foam of thickness $h=12.7$ mm ($\approx 0.3 U_e$), though the velocity measured over the thin foam is slightly higher ($\approx 0.35U_e$). In general, the mean profile for the thin foam is consistently higher than that for the baseline foam, and collapses onto the smooth-wall profile above $y/\delta > 0.2$ (black triangles in figure \[fig:thick\]a). In contrast, the mean profile for the thicker foam (white triangles in figure \[fig:thick\]a) is closer to the baseline case in the near-wall region $y/\delta < 0.005$. However, a little farther from the wall, the mean profile for the thick foam diverges from that for the baseline foam. The thick foam mean profile shows a smaller velocity deficit in the region $y/\delta \approx 0.01$ to $y/\delta \approx 0.1$ compared to the baseline case, and verges on the smooth-wall profile for $y/\delta \ge 0.2$. The streamwise turbulence intensity profiles plotted in figure \[fig:thick\]b show that $\overline{u^2}/U_e^2$ is similar near the interface for all three foams. In fact, the baseline and thin foams show very similar intensity profiles through most of the boundary layer, barring two minor differences. First, the outer peak in streamwise intensity appears closer to the interface for the thin foam. Second, the thin foam profile shows a better collapse onto the smooth-wall profile for $y/\delta \ge 0.3$. The turbulence intensity profile for the thick foam also collapses onto the smooth-wall profile for $y/\delta \ge 0.3$. However, closer to the wall, $\overline{u^2}$ is much higher over the thick foam, and the outer peak in intensity also appears to move slightly closer to the interface. These features are also seen in the velocity spectra presented below. In summary, for the thin foam, the mean velocity and streamwise intensity measurements both collapse onto the smooth-wall profile for $y/\delta \ge 0.3$. This suggests that the outer-layer similarity hypothesis proposed by Townsend for rough-walled flows [see e.g., @schultz2007rough] also holds for thin porous media. The mean profile for the thick foam is similar to that for the baseline foam close to the interface, but moves closer to the thin foam profile further from the interface. However, the streamwise intensities are significantly higher for the thick foam compared to the thin foam for $y/\delta \le 0.1$, even in regions where the mean profiles show agreement. These discrepancies between the mean velocity and streamwise intensity profiles for the thick foam are consistent with a developing flow. Velocity spectra {#sec:spectra} ---------------- The premultiplied velocity spectra shown in figure \[fig:spectra\] provide further insight into the origin of the outer peak in streamwise intensity observed over the porous substrates. [[[As is customary in the boundary layer literature, the premultiplied spectrum is defined as $f E_{uu}$, where $f$ is the frequency and $E_{uu}$ is the power spectral density, normalized by $U_e^2$. This quantity is computed for each wall-normal location, and the results are compiled into the contour plots shown in figure \[fig:spectra\].]{}]{}]{} Due to the low data rates obtained near the interface, spectra are only shown for $y/\delta \ge 0.04$. Note that the spectra are expressed in terms of a normalized streamwise wavelength estimated using Taylor’s hypothesis, $U/(f\delta)$. ![Contour maps showing variation in premultiplied frequency spectra [[[(normalized by $U_e^2$)]{}]{}]{} for streamwise velocity as a function of wall-normal distance $y/\delta$ over the smooth wall (a), the 100 ppi foam (b), the 10 ppi foam (c), the thin 20 ppi foam (d), the baseline 20 ppi foam (e), and the thick 20 ppi foam (f). The spectra are plotted against a normalized streamwise length scale, $U/f\delta$, computed using Taylor’s hypothesis. The white box in (a) denotes the region typically associated with VLSMs while the markers ($*$) represent the nominal frequency $f_{KH}$ for structures resembling Kelvin-Helmholtz vortices. [[[The spectra refer to the same physical measurement location, corresponding to $x/h=42$ for the foams with $h = 12.7$mm and $x/h=21$ and $84$ for the thick and thin 20 ppi foams, respectively.]{}]{}]{}[]{data-label="fig:spectra"}](Figures/Fig10-Spectra.pdf) For the smooth wall case, there is evidence of weak very-large-scale motions (VLSMs), which is similar to results obtained in previous studies at comparable Reynolds number [@hutchins2007evidence]. The box in the top left panel encompasses the spectral region typically associated with VLSMs, i.e. structures of length $6\delta - 10\delta$ ($U/f\delta = 6-10$) located between $y/\delta = 0.06$ and $y^+ = 3.9 \sqrt{Re_\tau}$ [see @hutchins2007evidence; @marusic2010predictive; @smits2011high]. This box coincides with a region of elevated spectral density for the measurements. Spectra for the porous substrates are different in several ways. For all the foams, the spectra are elevated over the frequency range that corresponds to structures with streamwise length scale $1\delta-5\delta$. The spectra are most energetic [[[at, or below,]{}]{}]{} wall-normal location $y/\delta \approx 0.1$ and remain elevated until $y/\delta \approx 0.3$. There is a marked increase in the spectral energy density from the 100 ppi foam to the 20 ppi foam, and little difference between the spectra for the 20 ppi and 10 ppi foams. Together, these features suggest that the outer peak in streamwise intensity observed over the porous foams [[[in figure \[fig:ppi-u\]b]{}]{}]{} is associated with large-scale structures of length $1\delta-5\delta$ that are distinct from VLSMs. [[[Spectra for the 20ppi foam of different thickness, plotted in figure \[fig:spectra\]e-f, show a substantial increase in energy for the thickest foam, which is consistent with the elevated streamwise intensity profiles shown in figure \[fig:thick\]b. For the thick foam, the energy is also concentrated closer to the interface compared to the thin and baseline foams with identical pore sizes.]{}]{}]{} Note that the spectral features described above are consistent with previous experiments and simulations. @manes2011turbulent showed that the premultiplied frequency spectra for streamwise velocity peak at wavenumber $k_x\delta \approx 2-4$ over porous substrates, i.e. corresponding to structures of length $1.5\delta-3\delta$. Similarly, the DNS carried out by @breugem2006influence indicated the presence of spanwise rollers with streamwise length-scale comparable to the total channel height. Since these structures have been linked to a Kelvin-Helmholtz shear instability arising from the inflection point in the mean profile, it is instructive to consider the characteristic frequency associated with this mechanism. Drawing an analogy to mixing layers, @white2007shear and @manes2011turbulent suggested that the characteristic frequency for the Kelvin-Helmholtz instability can be estimated as $f_{KH} = 0.032 \overline{U}/\theta$ [@ho1984perturbed], in which $\overline{U} = (U_p + U_e)/2$ is the *average* velocity in the shear layer with $U_p$ being the velocity deep within the porous medium, and $\theta$ is the momentum thickness. Estimates for this characteristic frequency, converted to streamwise length-scale based on the mean velocity at $y/\delta = 0.1$, $\lambda_{KH}/\delta = U(0.1)/f_{KH}\delta$, are shown in figure \[fig:spectra\]. These estimates assume $U_p \approx 0$, so that $\overline{U} \approx 0.5U_e$, and that the momentum thickness $\theta$ can be approximated based on the measured velocity profile in the fluid domain, i.e. for $y \ge 0$. With these assumptions, the predicted length scales associated with the instability range from $\lambda_{KH}/\delta \approx 4.3$ for the 100 ppi foam to $\lambda_{KH}/\delta \approx 3.6$ for the 10 ppi foam, which is in reasonable agreement with the location of the peaks in the spectra. There is an evident overprediction of length scale for both the 10 ppi foam (figure \[fig:spectra\]c) and the thick 20 ppi foam (figure \[fig:spectra\]c). This overprediction can be attributed to greater flow penetration into the porous medium for thicker foams with larger pore sizes. Greater flow penetration would create higher velocities inside the porous medium, $U_p$. This would increase $\overline{U}$ and $f_{KH}$, resulting in lower $\lambda_{KH}$. Greater flow penetration into the porous medium may also result in the inflection point moving closer to the interface. In this scenario, conversion from $f_{KH}$ to $\lambda_{KH}$ should be based on a lower mean velocity from $y/\delta < 0.1$. This would also result in lower $\lambda_{KH}$ compared to the estimates shown in figure \[fig:spectra\]. To test whether the observed peaks in streamwise intensity and velocity spectra tracked the location of inflection points in the mean profile, $y_i$, finite-difference approximations of the second derivative of mean velocity, $(d^2U/dy^2)_{y_i} = 0$, were considered. This yielded inflection point locations ranging from $y/\delta \approx 0.03$ to $y/\delta \approx 0.12$ for the 10 and 20 ppi foams, which is broadly consistent with the location of energetic peaks in figure \[fig:spectra\]. However, the second derivatives estimated from experimental data were very noisy, and the inflection point locations were highly sensitive to the accuracy (i.e., order) of the finite-difference approximation and the lower limit in $y$ used for the estimates. As a result, exact inflection point locations are not presented here. Discussion {#sec:discussion} ========== Amplitude modulation {#sec:amplitude-mod} -------------------- As noted earlier, the skewness profiles shown in figure \[fig:skew\] strongly suggest that the amplitude modulation phenomenon that has received significant attention in recent smooth and rough wall literature [e.g., @mathis2009large; @marusic2010predictive; @mathis2013estimating; @pathikonda2017inner] may also be prevalent in turbulent flows over permeable walls. For smooth wall flows at high Reynolds number, it has been shown that the VLSMs prevalent in the logarithmic region of the flow can have a modulating effect on the small-scale fluctuations found in the near-wall region. Specifically, it has been suggested that the turbulent velocity field in the near-wall region can be decomposed into a large-scale (or low-frequency) component $u_L$ that represents the near-wall signature of the VLSMs, and a ‘universal’ (i.e. Reynolds number independent) small-scale component $u_S$ that is associated with the local near-wall turbulence. Analysis of the resulting signals indicates that the filtered envelope of the small-scale activity, $E_L(u_S)$, obtained via a Hilbert transform [for details, see @mathis2011relationship], is strongly correlated with the large-scale signal. In other words, the single-point correlation coefficient: $$\label{eq:R} R = \frac{\overline{u_L E_L(u_S)}}{\sqrt{\overline{u_L^2}}\sqrt{\overline{E_L(u_S)^2}}}$$ tends to be positive in the near-wall region, which suggests that the small-scale signal $u_S$ is modulated by the large-scale signal $u_L$. Since the local large-scale signal $u_L$ arises from VLSM-type structures centered further from the wall, these observations have given rise to predictive models of the form $$\label{eq:modulation} u = u^*(1+\beta u_{OL}) + \alpha u_{OL},$$ where $u(y)$ is the predicted velocity at a specified location in the near-wall region, $u^*(y)$ is a statistically universal small-scale signal at that wall-normal location, $u_{OL}$ is the large-scale velocity measured in the outer region of the flow, and $\alpha(y)$ and $\beta(y)$ are superposition and modulation coefficients, respectively [@marusic2010predictive]. Assuming that the universal small-scale signal can be obtained via detailed experiments or simulations carried out at low Reynolds numbers, equation (\[eq:modulation\]) allows for near-wall predictions at much higher Reynolds numbers based only on measurements of $u_{OL}$ in the outer region of the flow. [[[Note that the modulation coefficient $\beta$ is similar to the single-point correlation coefficient $R$, but also accounts for changes in phase and amplitude of the large-scale signal from the measurement location to the near-wall region. In other words, $\beta$ also accounts for the relationship between $u_{OL}$ and $u_L(y)$, which is thought to be Reynolds-number independent [@marusic2010predictive].]{}]{}]{} [[[Subsequent studies have shown rigorously that the correlation coefficient $R$ in (\[eq:R\]) is intrinsically linked to the skewness of the velocity [@schlatter2010quantifying; @mathis2011relationship; @duvvuri2015triadic], whereby positive correlations between scales ($R > 0$) translate into increased skewness. Thus, the increase in skewness observed in figure \[fig:skew\] near the porous interface suggests that $R>0$ in this region.]{}]{}]{} Near-wall measurements made in the present study do not have sufficient time resolution to allow for a quantitative evaluation of the amplitude modulation phenomenon over porous substrates (i.e., a decomposition into small- and large-scale components). However, the increase in the skewness in the near-wall region over the porous substrates suggests that the large-scale structures responsible for the outer peak in streamwise intensity at $y/\delta \approx 0.1$ may have a modulating effect on the turbulence near the interface. This is particularly interesting given that the large-scale structures found over porous substrates are distinct from the VLSMs found over smooth walls, and that the small-scale turbulence near the porous interface may also be modified from the near-wall cycle [@robinson1991coherent; @schoppa2002coherent] found over smooth walls. Logarithmic region {#sec:log-shift} ------------------ ![(a) Scaled velocity gradient $(y/U_e)\partial U/\partial y$ plotted as a function of $y/\delta$ for the smooth wall profile and porous foam data. Equation (\[eq:log-law-fit\]) was fitted to these data to estimate the normalized displacement height, $y_d/h$, and the friction velocity weighted by the von Karman constant, $u_\tau/(\kappa U_e)$, shown in (b) and (c), respectively. Using these estimates for $y_d$ and $u_\tau/\kappa$, the roughness height $k_0/h$, shown in (d), was evaluated from the velocity profiles using equation (\[eq:log-law\]). Dotted lines in (a) represent the upper limits of $y/\delta=0.16,0.20$ and $0.25$ employed in the fitting procedure. The dashed line represents the minimum lower limit, $y/\delta > 0.02$. Larger marker sizes in (b,c,d) denote higher values for the upper limit. The red cross in (c) represents the friction velocity estimated via a linear fit to the near-wall velocity measurements over the smooth wall. [[[Horizontal error bars in panels (b)-(d) represent uncertainty in pore size, $s$.]{}]{}]{}[]{data-label="fig:log-law"}](Figures/Fig11-loglaw.pdf) Previous studies have devoted considerable effort to testing whether a modified logarithmic region of the form shown in equation (\[eq:log-law\]) exists in turbulent flows over porous substrates. While there is no definitive consensus on how the von Karman constant, $\kappa$, displacement height, $y_d$, and equivalent roughness height, $k_0$, vary with substrate properties, experiments and simulations generally show that the displacement and roughness heights increase with increasing permeability (or pore size). @manes2011turbulent showed that empirical relationships of the form: $$\label{eq:ys} y_d^+ = 15.1 Re_k - 13.5$$ and $$\label{eq:k0} k_0^+ = 6.28 Re_k - 9.82,$$ where $Re_k = u_\tau \sqrt{k}/\nu$ is the permeability Reynolds number, led to reasonable fits for the experimental data obtained by @suga2010effects and @manes2011turbulent, but underestimated $y_d$ and $k_0$ for the simulations performed by @breugem2006influence. The von Karman constant decreased relative to smooth wall values but demonstrated a complex dependence on both the permeability Reynolds number and the ratio of displacement height to boundary layer thickness, $y_d/\delta$ [@manes2011turbulent]. Unfortunately, the present study did not involve independent measurements of the shear stress at the interface (or Reynolds’ shear stress in the near-wall region) and so it is not possible to estimate the friction velocity and von Karman constant independently. However, the velocity measurements can still be used to test whether a modified logarithmic region exists, and to estimate $u_\tau/\kappa$, $y_d$ and $k_0$. Taking the partial derivative of equation (\[eq:log-law\]) with respect to $y$ and rearranging yields: $$\label{eq:log-law-fit} (y+y_d)\frac{\partial U}{\partial y} = \frac{u_\tau}{\kappa}.$$ Following @breugem2006influence and @suga2010effects, we estimated $y_d$ as the value that forces $(y+y_d)\partial U/\partial y$ to be constant over specified ranges of $y/\delta$. Based on equation (\[eq:log-law-fit\]), the resulting constant value for the weighted velocity gradient was assumed to be the friction velocity divided by von Karman constant, $u_\tau/\kappa$. Using these estimates for displacement height and friction velocity, the roughness height was estimated directly from the velocity measurements using equation (\[eq:log-law\]). Since the fitting procedure described above relies on noisy velocity gradient data (see figure \[fig:log-law\]a), the uncertainty in the fitted values and sensitivity to fitting ranges was evaluated as follows. First, the fitting procedure was carried out over the range $0.02 < y/\delta < 0.16$. The fit was then repeated with the lower limit sequentially increased by one to four measurement points. The estimates of $y_d$, $u_\tau/\kappa$, and $k_0$ reported in figure \[fig:log-law\](b-d) represent an average of the five different values obtained via this process and the error bars represent the standard error. This entire procedure was then repeated for outer limits $y/\delta = 0.20$ and $y/\delta = 0.25$. A similar process was used to evaluate $u_\tau/\kappa$ and $k_0$ for the smooth wall case with the displacement height constrained to be zero, $y_d = 0$. Estimates for $y_d$, $k_0$, and $u_\tau/\kappa$ are reported in table \[tab:log-law\]. Note that the fitting procedure employed by @manes2011turbulent was also considered, where $y_d$ is estimated as the value that minimizes residuals between the measured velocity profile and equation (\[eq:log-law\]) over specified ranges of $y/\delta$. The resulting fitted coefficients are then used to estimate $u_\tau/\kappa$ and $k_0$. This process led to fitted values within the uncertainty ranges shown in figures \[fig:log-law\]b-d. Assuming $\kappa = 0.39$, the fitting procedure described above led to a friction velocity estimate of $u_\tau = 2.31 \pm 0.02$ cm/s for the smooth wall profiles with outer limit $y/\delta = 0.15$. This is consistent with the value obtained via a linear fit to the near-wall mean velocity measurements, $u_\tau = 2.3 \pm 0.05$ cm/s. The additive constant $B = -(1/\kappa)\ln k_0^+$ in equation (\[eq:log-law\]) was estimated to be $B = 4.8 \pm 0.2$, which is slightly higher than the value, $B = 4.3$, reported in @marusic2013logarithmic, but still broadly consistent with previous literature. As expected, the fitted log law constants for the porous substrates show a strong dependence on average pore size. Figures \[fig:log-law\](b-d) show that the displacement height, friction velocity, and roughness height generally increase with increasing pore size, though here is some evidence of saturation at the largest pore sizes. Specifically, figure \[fig:log-law\]b suggests that the displacement height levels out above $y_d/h \approx 1$ for the [[[baseline 20 ppi foam, the 10 ppi foam, and the thin 20 ppi foam, for which $s/h > 0.1$]{}]{}]{}. This saturation in $y_d/h$ as a function of [[[normalized]{}]{}]{} pore size is accompanied by saturation, or perhaps slight decreases, in normalized friction velocity $u_\tau/\kappa U_e$ (figure \[fig:log-law\]c) and roughness height $k_0/h$ (figure \[fig:log-law\]d) [[[above $s/h > 0.1$]{}]{}]{}. While the exact values of the log law constants shown in figure \[fig:log-law\] and table \[tab:log-law\] must be treated with some caution due to the uncertainty associated with the fitting procedure, the overall trends suggest the following physical interpretation. The displacement height $y_d$ represents the level at which momentum is extracted within the porous medium [@jackson1981displacement], or alternatively, the distance to which the turbulence penetrates into the medium [@luhar2008interaction] or the effective plane at which attached eddies are initiated [@poggi2004effect]. For the least permeable substrates tested in the present study (i.e. the 100 ppi and 60 ppi foams), $y_d$ increases approximately linearly with average pore size. In other words, at this low permeability or thick substrate limit with $h/s \gg 1$, the distance to which turbulence penetrates into the porous medium increases with increasing pore size or permeability. Since the flow does not interact with the entire porous medium, the foam thickness $h$ does not play a role. However, with further increases in pore size [[[or decreases in porous medium thickness]{}]{}]{}, at some point the displacement height becomes comparable to the foam thickness $y_d \approx h$. At this high permeability or thin substrate limit, turbulence penetrates the entire porous medium and the foam essentially acts as a roughness or obstruction. Results shown in figure \[fig:log-law\] suggest that [[[the baseline 20 ppi foam, the thin 20 ppi foam, and the 10 ppi foam, for which $h/s \le 10$]{}]{}]{}, may be approaching this thin substrate limit. Figures \[fig:log-law\]b-d show that the normalized friction velocity and roughness height are strongly correlated with each other as well as the displacement height. Physically, the friction velocity is a measure of momentum transfer into the porous medium while the roughness height is a measure of momentum loss, or friction increase, due to the presence of the complex substrate. As a result, correlation between $k_0$ and $u_\tau$ is unsurprising for boundary layer experiments carried out at constant free-stream velocity (n.b., for channel or pipe flow experiments, the friction velocity can be controlled independently by setting the pressure gradient). The link between the displacement and roughness heights can be explained by considering the rough-wall literature. For flows over conventional *K*-type roughness, $k_0$ has been shown to depend on both the height of the roughness elements and the solidity $\lambda$, which is defined as the total projected frontal area per unit wall-parallel area [@jimenez2004turbulent]. Similarly, for flows over porous media, $k_0$ can be expected to depend on the displacement height, which represents the thickness of porous medium that interacts with the flow, as well as the porous medium microstructure [see also @jackson1981displacement; @manes2011turbulent]. In other words, a relationship of the form $k_0/y_d = f(\lambda)$ may be appropriate for turbulent flows over porous media. Note that the physical link between the roughness and displacement heights is also evident in the empirical relationships for $y_d$ and $k_0$ shown in equations (\[eq:ys\]-\[eq:k0\]). ------------------- ------------------------ ------------ --------- --------------------- ----------- Substrate $u_\tau/\kappa$ (cm/s) $y_d$ (mm) $y_d/h$ $k_0$ (mm) $k_0/y_d$ \[0.1cm\] smooth 5.8 0 0 $5.3\times 10^{-3}$ $\infty$ \[0.1cm\] 10 ppi 27.5 14.2 1.1 7.8 0.55 \[0.1cm\] 20 ppi 32.9 15.6 1.2 9.8 0.63 20 ppi thin 22.2 8.7 1.4 3.9 0.44 20 ppi thick 26.9 10.2 0.4 5.5 0.52 \[0.1cm\] 60 ppi 16.5 5.3 0.42 1.5 0.28 \[0.1cm\] 100 ppi 11.9 3.2 0.25 0.49 0.15 ------------------- ------------------------ ------------ --------- --------------------- ----------- : Fitted values for log-law parameters in dimensional and dimensionless form. Listed values of $u_\tau/\kappa$, $y_d$, and $k_0$ are averages of the three estimates shown in figure \[fig:log-law\], which were obtained for three different outer limits in the fitting procedure. The displacement height is assumed to be zero for the smooth wall flow.[]{data-label="tab:log-law"} Non-monotonic behaviour with pore size {#sec:nonmonotonic} -------------------------------------- In many ways, the aforementioned transition from thick substrate behaviour, where turbulence penetration into the porous medium is limited and $y_d$ increases with permeability, to thin substrate behaviour, where turbulence penetrates the entire porous medium and $y_d \approx h$, is analogous to the $\lambda$-dependent transition from dense to sparse canopy behaviour proposed in the vegetated flow literature [@belcher2003adjustment; @luhar2008interaction; @nepf2012flow]. As noted in the introduction, for vegetated flows the distance to which the flow penetrates into the canopy is dependent on the drag length-scale $(C_Da)^{-1}$, where $C_D$ is a representative drag coefficient and $a$ is the frontal area per unit volume. As a result, the ratio of shear penetration to canopy height, $h$, is given by the dimensionless parameter $C_D a h = C_D \lambda$, where $\lambda$ is the solidity as before. For dense canopies with $C_D \lambda \ge O(1)$, the shear layer does not penetrate the entire canopy and so an inflection point is expected in the mean profile. This gives rise to large-scale structures resembling Kelvin-Helmholtz vortices. However, this instability mechanism is expected to weaken in sparse canopies. For $C_D \lambda < O(0.1)$, turbulence penetrates the entire canopy and there is no inflection point in the mean profile [@nepf2012flow]. The non-monotonic behaviour in mean velocity and turbulence intensity observed for the 10 ppi foam in the present experiments could be attributed to a similar weakening of the shear layer instability as the turbulence penetrates the entire porous medium, i.e. as $y_d \approx h$. Consistent with this hypothesis, the reduced magnitude of the outer peak in streamwise intensity for the 10 ppi foam relative to the 20 ppi foam (figure \[fig:ppi-u\]b) suggests that the large-scale structures resembling Kelvin-Helmholtz vortices are weaker over the 10 ppi foam. Since these structures contribute substantially to vertical momentum transfer, a reduction in their strength also translates into a smaller mean velocity deficit (figure \[fig:ppi-u\]a). Note that there is evidence of non-monotonic behaviour as a function of solidity $\lambda$ in the rough-wall literature as well. Based on a compilation of experimental data, @jimenez2004turbulent showed that the normalized roughness height (i.e. ratio of $k_0$ to roughness dimension) depends on the solidity $\lambda$, and that there are two regimes of behaviour. For sparse roughness with solidity less than $\lambda \approx 0.15$, the normalized roughness increases with increasing $\lambda$. In other words, an increase in frontal area leads to an increase in roughness drag. However, for densely packed roughness with $\lambda \gtrsim 0.15$, the normalized roughness decreases with increasing $\lambda$ because the roughness elements shelter each other. Assuming that the relevant vertical dimension for turbulent flows over porous media is the displacement height $y_d$, we may expect similar non-monotonic behaviour for the normalized roughness $k_0/y_d = f(\lambda)$. The values for $k_0/y_d$ listed in table \[tab:log-law\] provide some support for this hypothesis: the normalized roughness height increases from $k_0/y_d = 0.15$ for the 100 ppi foam to $k_0/y_d = 0.63$ for the baseline 20 ppi foam, before decreasing to $k_0/y_d = 0.55$ for the 10 ppi foam. Bear in mind that, for identical $h$, the solidity is expected to increase with decreasing pore size from the ‘sparsely packed’ 10 ppi foam to the ‘densely packed’ 100 ppi foam. Although the solidity is a difficult parameter to measure for porous media, it may be estimated for the foams employed here based on simple geometric assumptions. Consider, for instance, a cubic lattice comprising thin rectangular ligaments of cross-section $d \times d$ and length $s$ (i.e., the pore size). Each unit cell of volume $s^3$ in this lattice comprises three orthogonal filaments aligned in the $x,y,$ and $z$ directions intersecting in a three-dimensional cross. Neglecting the overlapping volume at the center of the cross, the porosity is approximately $\epsilon \approx 1 - 3d^2s/s^3 = 1-3(d/s)^2$ for this geometry. For the foams tested here, the porosity is constant, $\epsilon \approx 0.97$. So, the above equation implies that the lattice must be geometrically similar with $d/s \approx \sqrt{(1-\epsilon)/3} = 0.1$. In other words, the ligament width $d$ increases linearly with pore size $s$ to maintain constant $\epsilon$. For this assumed geometry, the frontal area per unit volume for flow in either the $x$, $y$, or $z$ directions is $a \approx 2ds/s^3 = 2d/s^2$, since there are always 2 ligaments with area $ds$ normal to the flow. This results in solidity $\lambda = ah \approx 2(d/s)(h/s) = 0.2(h/s)$. This estimate suggests that the pore size threshold above which non-monotonic behavior is observed in the log-law constants ($s/h \ge 0.12$ in figure \[fig:log-law\]) corresponds to solidity $\lambda \sim O(1)$. Of course, since the foam pore structures do not resemble a cubic lattice (figure \[fig:foam\]), the numerical factors appearing in the equations above are unlikely to be accurate. However, the linear relationship between $\lambda$ and $h/s$ is expected to hold. Finally, keep in mind that the non-monotonic behavior with solidity and pore size described above does not preclude the possibility of monotonic behavior with permeability Reynolds number $Re_k=u_\tau\sqrt{k}/\nu$. Although $y_d$ and $k_0$ are shown to decrease with increasing pore size, and hence permeability, from the 20 ppi foam to the 10 ppi foam, this decrease in the displacement and roughness heights is also accompanied by a decrease in $u_\tau/\kappa$ (Table \[tab:log-law\]). In other words, $Re_k$ may be lower for the 10 ppi foam compared to 20 ppi foam, even if $\sqrt{k}$ is higher. Unfortunately, this hypothesis cannot be tested further without independent estimates of $u_\tau$. A note on scaling {#sec:scaling} ----------------- Previous studies on turbulent flows over porous media indicate that shear penetration into the porous medium depends on the permeability length scale $\sqrt{k}$, which determines the effective flow resistance within the porous medium per the well-known Darcy-Forchheimer equation [@breugem2006influence; @suga2010effects]. This is also evident in the empirical relationship shown in equation (\[eq:ys\]), which indicates that $y_d \approx 15.1\sqrt{k}$ for sufficiently high permeability Reynolds number $Re_k \gg 1$. Although the permeability is related to geometric parameters such as pore size [e.g. $\sqrt{k}/s \approx 0.08$ for the foams tested by @suga2010effects] and frontal area per unit volume, it is essentially a dynamic parameter that is typically estimated from fitting the Darcy-Forchheimer law: $$-\frac{1}{\rho}\frac{\Delta P}{\Delta x} = \frac{\nu U_v}{k} + \frac{C_f}{\sqrt{k}}U_v^2,$$ to experimental pressure drop measurements ($\Delta P/ \Delta x$) across porous media. In the equation above, $U_v$ is the volume-averaged velocity and $C_f$ is the Forchheimer coefficient. Since such pressure drop measurements are usually carried out at low Reynolds number in steady pipe or channel flow with uniformly distributed porous media (essentially a one-dimensional system), there are inherent risks in employing the resulting permeability values for unsteady, three-dimensional, spatially varying flows at higher Reynolds number. Further, the non-linear Forchheimer term that becomes increasingly important at higher speeds requires an additional coefficient $C_f$ that is not a universal constant. To avoid these issues, the present study presents results primarily as a function of the normalized pore size, $s/h$. For example, figure \[fig:log-law\]b suggests that $y_d \propto s$ until it becomes comparable to foam thickness. Another alternative would be to use the frontal area per unit volume, $a$, and solidity, $\lambda$, as the relevant scales. [[[The frontal area per unit volume is a difficult quantity to measure for complex porous media. However, the discussion presented in the previous section suggests that the solidity $\lambda$ increases linearly with $(h/s)$ for geometrically-similar porous media with constant porosity. As a result, the use of $s/h$ for scaling purposes also allows for greater reconciliation with the canopy flow and rough wall literature.]{}]{}]{} Conclusions {#sec:conclusions} =========== The experimental results presented in this paper show that turbulent boundary layers over high-porosity foams are modified substantially compared to canonical smooth wall flows. Development data in §\[sec:development\] suggest that the boundary layer adjusts relatively quickly to the presence of the porous substrate. Specifically, [[[for most of the foams tested,]{}]{}]{} the mean velocity profile adjusts to a new equilibrium over a streamwise distance $<10\delta$, which is similar to the adjustment length observed in previous literature for the transition from smooth to rough walls.[[[However, this rough-wall analogy does not hold for the thickest foam tested, which suggests that the foam thickness may also provide a bound on development length.]{}]{}]{} Fully-developed mean velocity profiles presented in §\[sec:poresize\] show the presence of substantial slip velocity ($>0.3U_e$) that is relatively insensitive to pore size [[[for foams of constant thickness. Profiles in §\[sec:thickness\] also show a near constant slip velocity over substrates with constant pore size and varying thickness.]{}]{}]{} These observations remain to be explained fully. Profiles of streamwise intensity show the emergence of an outer peak at $y/\delta \approx 0.1$ over the porous substrates, which is associated with large-scale structures of length $2\delta - 4\delta$. Such structures have also been observed in previous simulations and experiments, and are thought to arise from a Kelvin-Helmholtz instability associated with an inflection point in the mean profile. Although the magnitude of the outer peak in streamwise intensity generally increases with pore size, there is some evidence of weakening for the foam with the largest pore size. The log-law fits presented in §\[sec:log-shift\] provide further insight into this non-monotonic behaviour. Specifically, the displacement height increases with [[[normalized pore size, $s/h$,]{}]{}]{} until it becomes comparable to the foam thickness. Further increases in pore size beyond this point do not lead to an increase in $y_d$. In other words, there is a transition from thick substrate behaviour, in which the thickness of the porous medium interacting with the flow is determined by pore size ($y_d \propto s$), to thin substrate behaviour, in which the flow penetrates the entire porous medium ($y_d \approx h$). The weakening in the outer layer structures may be attributed to this transition from thick to thin substrate behaviour. Drawing an analogy to sparse canopy behaviour for vegetated flows, at the thin substrate limit, the mean velocity profile becomes fuller with increasing pore size and ultimately loses the inflection point. This results in a weakening of the Kelvin-Helmholtz instability. [[[For canopy flows, the transition from dense-canopy behavior to sparse-canopy behavior occurs as the solidity parameter becomes small, $\lambda \ll 1$. Simple geometric arguments show that $\lambda \propto h/s$ for the foams tested here, and that the transition from thick- to thin-substrate behavior occurs around $\lambda \sim O(1)$.]{}]{}]{} Interestingly, the skewness of the near-wall velocity measurements increases substantially over the porous substrates relative to smooth wall values. Further, this increase in skewness is correlated with an increase in the magnitude of the outer peak in streamwise intensity. Given the link between skewness and the amplitude modulation phenomenon, these observations suggest that the large-scale structures that are energetic over porous media may have a modulating influence on the interfacial turbulence. This is analogous to the interaction between VLSMs and near-wall turbulence in smooth wall flows at high Reynolds number. Unfortunately, the near-wall velocity measurements collected as part of this study were not time resolved, and so did not allow for a quantitative evaluation of this effect. However, given the substantial similarities between turbulent flows over porous media and vegetation or urban canopies, further studies into such scale interactions could lead to the development of promising wall models for a variety of flows. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the Air Force Office of Scientific Research under AFOSR grant No. FA9550-17-1-0142 (Program Manager: Dr. Douglas Smith). [^1]: Email address for correspondence: [email protected]

--- abstract: 'Monads are a popular tool for the working functional programmer to structure effectful computations. This paper presents *polymonads*, a generalization of monads. Polymonads give the familiar monadic bind the more general type !forall $a,b$. L $a$ -&gt; ($a$ -&gt; M $b$) -&gt; N $b$!, to compose computations with three different kinds of effects, rather than just one. Polymonads subsume monads and parameterized monads, and can express other constructions, including precise type-and-effect systems and information flow tracking; more generally, polymonads correspond to Tate’s *productoid* semantic model. We show how to equip a core language (called [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}) with syntactic support for programming with polymonads. Type inference and elaboration in [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} allows programmers to write polymonadic code directly in an ML-like syntax—our algorithms compute principal types and produce elaborated programs wherein the binds appear explicitly. Furthermore, we prove that the elaboration is *coherent*: no matter which (type-correct) binds are chosen, the elaborated program’s semantics will be the same. Pleasingly, the inferred types are easy to read: the polymonad laws justify (sometimes dramatic) simplifications, but with no effect on a type’s generality.' author: - Michael Hicks$^1$     Gavin Bierman$^3$     Nataliya Guts$^1$     Daan Leijen$^2$     Nikhil Swamy$^2$ bibliography: - 'monadic.bib' title: - | Polymonadic Programming\ (Extended version) - Polymonadic Programming --- Introduction {#sec:intro} ============ Since the time that Moggi first connected them to effectful computation [@moggi89computational], *monads* have proven to be a surprisingly versatile computational structure. Perhaps best known as the foundation of Haskell’s support for state, I/O, and other effects, monads have also been used to structure APIs for libraries that implement a wide range of programming tasks, including parsers [@hutton98monadic], probabilistic computations [@ramsey02stochastic], and functional reactivity [@elliot97functional; @cooper06father]. Monads (and morphisms between them) are not a panacea, however, and so researchers have proposed various extensions. Examples include Wadler and Thiemann’s [@wadler:2003] indexed monad for typing effectful computations; Filli[â]{}tre’s generalized monads [@filliatre99atheory]; Atkey’s parameterized monad [@atkey09], which has been used to encode disciplines like regions [@kiselyov2008lightweight] and session types [@pucella2008haskell]; Devriese and Piessens’ [@devriese2011information] monad-like encodings for information flow controls; and many others. Oftentimes these extensions are needed to prove stronger properties about computations, for instance to prove the absence of information leaks or memory errors. Unfortunately, these extensions do not enjoy the same status as monads in terms of language support. For example, the conveniences that Haskell provides for monadic programs (e.g., the notation combined with type-class inference) do not apply to these extensions. One might imagine adding specialized support for each of these extensions on a case-by-case basis, but a unifying construction into which all of them, including normal monads, fit is clearly preferable. This paper proposes just such a unifying construction, making several contributions. Our first contribution is the definition of a *polymonad*, a new way to structure effectful computations. Polymonads give the familiar monadic bind (having type !forall $a,b$. M $a$ -&gt; ($a$ -&gt; M $b$) -&gt; M $b$!) the more general type !forall $a,b$. L $a$ -&gt; ($a$ -&gt; M $b$) -&gt; N $b$!. That is, a polymonadic bind can compose computations with three different types to a monadic bind’s one. Section \[sec:polymonads-alt\] defines polymonads formally, along with the *polymonad laws*, which we prove are a generalization of the monad and morphism laws. To precisely characterize their expressiveness, we prove that polymonads correspond to Tate’s *productoids* [@tate12productors] (Theorem \[thm:productoid\]), a recent semantic model general enough to capture most known effect systems, including all the constructions listed above.[^1] Whereas Tate’s interest is in semantically modeling sequential compositions of effectful computations, our interest is in supporting practical programming in a higher-order language. Our second contribution is the definition of [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} (Section \[sec:programming\]), an ML-like programming language well-suited to programming with polymonads. We work out several examples in [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}, including novel polymonadic constructions for stateful information flow tracking, contextual type and effect systems [@neamtiu08context], and session types. Our examples are made practical by [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}’s support for type inference and elaboration, which allows programs to be written in a familiar ML-like notation while making no mention of the bind operators. Enabling this feature, our third contribution (Section \[sec:syntactic\]) is an instantiation of Jones’ theory of qualified types [@jones1992theory] to [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}. In a manner similar to Haskell’s type class inference, we show that type inference for [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} computes *principal types* (Theorem \[thm:oml\]). Our inference algorithm is equipped with an elaboration phase, which translates source terms by inserting binds where needed. We prove that elaboration is *coherent* (Theorem \[thm:coherence\]), meaning that when inference produces constraints that could have several solutions, when these solutions are applied to the elaborated terms the results will have equivalent semantics, thanks to the polymonad laws. This property allows us to do better than Haskell, which does not take such laws into account, and so needlessly rejects programs it thinks might be ambiguous. Moreover, as we show in Section \[sec:solve\], the polymonad laws allow us to dramatically simplify types, making them far easier to read without compromising their generality. A prototype implementation of [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} is available from the first author’s web page and has been used to check all the examples in the paper. Put together, our work lays the foundation for providing practical support for advanced monadic programming idioms in typed, functional languages. Polymonads {#sec:polymonads-alt} ========== We begin by defining polymonads formally. We prove that a polymonad generalizes a collection of monads and morphisms among those monads. We also establish a correspondence between polymonads and productoids, placing our work on a semantic foundation that is known to be extremely general. \[def:polymonad\] A **polymonad** $({\mathcal{M}}, \Sigma)$ consists of (1) a collection ${\mathcal{M}}$ of unary type constructors, with a distinguished element ${\ensuremath{\sfont{Id}}}\in {\mathcal{M}}$, such that $\kw{Id}~\tau=\tau$, and (2) a collection, $\Sigma$, of ${\kw{bind}}$ operators such that the laws below hold, where $\bind{(M,N)}{P} \triangleq$!forall a b. M a -&gt; (a -&gt; N b) -&gt; P b!.\ $$\begin{array}{ll} \multicolumn{2}{l}{$For all$~\kw{M}, \kw{N}, \kw{P}, \kw{Q}, \kw{R}, \kw{S}, \kw{T}, \kw{U} \in {\mathcal{M}}.} \\ $\textbf{(Functor)}$ & \exists \kw{b}. \kw{b}@\bind{(M,{\ensuremath{\sfont{Id}}})}{M} \in \Sigma ~$and$~\kw{b}~\mbox{\ls!m!}~(\lambda \kw{y}.\kw{y}) = \mbox{\ls!m!} \\[1ex] $\textbf{(Paired morphisms)}$ & \exists \kw{b}_1@\bind{(M,{\ensuremath{\sfont{Id}}})}{N} \in \Sigma \iff \exists \kw{b}_2@\bind{({\ensuremath{\sfont{Id}}}, M)}{N} \in \Sigma~\mbox{\emph{and}} \\ & \forall \kw{b}_1@\bind{(M,{\ensuremath{\sfont{Id}}})}{N}, \kw{b}_2@\bind{({\ensuremath{\sfont{Id}}}, M)}{N}. \kw{b}_1\, \mbox{\ls!(f v)!}~(\lambda \kw{y}.\kw{y}) = \kw{b}_{2}~\mbox{\ls!v f!} \\[1ex] $\textbf{(Diamond)}$ & \exists \kw{P},\kw{b}_1,\kw{b}_2. \aset{\kw{b}_1@\bind{(M,N)}{P}, \kw{b}_2@\bind{(P,R)}{T}} \subseteq \Sigma \; \iff \\ & \exists \kw{S},\kw{b}_3,\kw{b}_4. \aset{\kw{b}_3@\bind{(N,R)}{S}, \kw{b}_4@\bind{(M,S)}{T}} \subseteq \Sigma \\[1ex] $\textbf{(Associativity)}$ & \forall \kw{b}_1,\kw{b}_2,\kw{b}_3,\kw{b}_4. $If$~\\ & \aset{\kw{b}_1@\bind{(M,N)}{P}, \kw{b}_2@\bind{(P,R)}{T}, \kw{b}_3@\bind{(N,R)}{S}, \kw{b}_4@\bind{(M,S)}{T}}\subseteq\Sigma\\ & $then$~\kw{b}_2~(\kw{b}_1~m~f)~g = \kw{b}_4~m~(\lambda x. \kw{b}_3~(f~x)~g) \\[1ex] $\textbf{(Closure)}$ & $If$~\exists \kw{b}_1,\kw{b}_2,\kw{b}_3,\kw{b}_4. \\ & \aset{\kw{b}_1@\bind{(\kw{M}, \kw{N})}{\kw{P}}, \kw{b}_2@\bind{\kw{(S,Id)}}{\kw{M}}, \kw{b}_3@\bind{\kw{(T,Id)}}{\kw{N}}, \kw{b}_4@\bind{\kw{(P,Id)}}{\kw{U}}} \subseteq \Sigma \\ & $then$~\exists \kw{b}. \kw{b}@\bind{(\kw{S}, \kw{T})}{\kw{U}} \in \Sigma \end{array}$$ Definition \[def:polymonad\] may look a little austere, but there is a simple refactoring that recovers the structure of functors and monad morphisms from a polymonad.[^2] Given $(\mathcal{M},\Sigma)$, we can easily construct the following sets: $$\begin{array}{llcl} $(Maps)$ & M & = & \aset{(\lambda f m. {\kw{bind}}~m~f)\colon \kw{(a -> b) -> M a -> M b} \mid {\kw{bind}}\colon\bind{(\mconst,{\ensuremath{\sfont{Id}}})}{\mconst} \in \Sigma}\\ $(Units)$ & U & = & \aset{(\lambda x. {\kw{bind}}~x~(\lambda y.y))\colon \kw{a -> M a} \mid {\kw{bind}}\colon\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{M} \in \Sigma}\\ $(Lifts)$ & L & = & \aset{(\lambda x. {\kw{bind}}~x~(\lambda y.y))\colon \kw{M a -> N a} \mid {\kw{bind}}\colon\bind{(\mconst,{\ensuremath{\sfont{Id}}})}{\mconst[N]} \in \Sigma}\\ \end{array}$$ It is fairly easy to show that the above structure satisfies generalizations of the familiar laws for monads and monad morphisms. For example, one can prove ${\kw{bind}}~({\kw{unit}}~e)~ f = f~e$, and ${\kw{lift}}~({\kw{unit}}_1~e) = \munit_2~e$ for all suitably typed ${\kw{unit}}_1,{\kw{unit}}_2 \in U$, ${\kw{lift}}\in L$ and ${\kw{bind}}\in \Sigma$. With these intuitions in mind, one can see that the **Functor** law ensures that each $\mconst \in \Sigma$ has a $map$ in $M$, as expected for monads. From the construction of $L$, one can see that a bind $\bind{(M,{\ensuremath{\sfont{Id}}})}{N}$ is just a morphism from $\mconst$ to $N$. Since this comes up quite often, we write $\morph{\mconst}{\kw{N}}$ as a shorthand for $\bind{(M,{\ensuremath{\sfont{Id}}})}{N}$. The **Paired morphisms** law amounts to a coherence condition that all morphisms can be re-expressed as binds. The **Associativity** law is the familiar associativity law for monads generalized for both our more liberal typing for bind operators and for the fact that we have a *collection* of binds rather than a single bind. The **Diamond** law essentially guarantees a coherence property for associativity, namely that it is always possible to complete an application of **Associativity**. The **Closure** law ensures closure under composition of monad morphisms with binds, also for coherence. It is easy to prove that every collection of monads and monad morphisms is also a polymonad. In fact, in Appendix \[sec:productoids\], we prove a stronger result that relates polymonads to Tate’s *productoids* [@tate12productors]. \[lemma:monad-is-polymonad\] If $(\kw{M}, {\kw{map}}, {\kw{unit}}, {\kw{bind}})$ is a monad then $(\{\kw{M}, \kw{Id}\}, \{b_1, b_2, b_3, b_4\})$ is a polymonad where $b_1=\lambda x\colon\kw{M a}.\lambda f\colon\kw{a->Id b}. \kw{map}~f~x$, $b_2=\lambda x\colon\kw{Id a}.\lambda f\colon\kw{a -> M b}. f~x$, $b_3={\kw{bind}}$, $b_4=\lambda x\colon\kw{Id a}.\lambda f\colon\kw{a -> Id a}. \kw{unit}~x$. \[thm:productoid\] Every polymonad gives rise to a productoid, and every productoid that contains an $Id$ element and whose joins are closed with respect to the lifts, is a polymonad. Tate developed productoids as a categorical foundation for effectful computation. He demonstrates the expressive power of productoids by showing how they subsume other proposed extensions to monads [@wadler:2003; @filinski1999representing; @atkey09]. This theorem shows polymonads can be soundly interpreted using productoids. Strictly speaking, productoids are more expressive than polymonads, since they do not, in general, need to have an $Id$ element, and only satisfy a slightly weaker form of our **Closure** condition. However, these restrictions are mild, and certainly in categories that are Cartesian closed, these conditions are trivially met for all productoids. Thus, for programming purposes, polymonads and productoids have exactly the same expressive power. The development of the rest of this paper shows, for the first time, how to harness this expressive power in a higher-order programming language, tackling the problem of type inference, elaborating a program while inserting binds, and proving elaboration coherent. Programming with polymonads {#sec:programming} =========================== $$\begin{array}{lllcl} $\textit{Signatures}$ ({\mathcal{M}},\Sigma): &k$-ary constructors$ & {\mathcal{M}}& ::= & \cdot \mid M/k, {\mathcal{M}}\\ &$ground constructor$ & \gm & ::= & M~\overline{\ty} \\ &$bind set$ & \Sigma & ::= & \cdot \mid \sfont{b}@s, \Sigma \\ &$bind specifications$ & s & ::= & \forall\bar{a}. \Phi \Rightarrow \bind{(\gm_1,\gm_2)}{\gm_3} \\ &$theory constraints $ & \Phi & \\[2ex] $\textit{Terms:}$ & $values$ & v & ::= & x \mid c \mid \slam{x}{e} \\ & $expressions$ & e & ::= & v \mid \sapp{e_1}{e_2} \mid \slet{x}{e_1}{e_2} \\ & & & \mid & \sif{e}{e_1}{e_2} \mid \sletrec{f}{v}{e} \\[2ex] $\textit{Types:}$ & $monadic types$ & m & ::= & \gm \mid \mvar\\ & $value types$ & \ty & ::= & a \mid T\, \overline{\ty} \mid \tfun{\ty_1}{\tapp{m}{\ty_2}} \\ & $type schemes$ & \sigma & ::= & \forall \bar{a}\bar\mvar. {\ensuremath{P}}=> \ty \\ & $bag of binds$ & {\ensuremath{P}}& ::= & \cdot \mid \pi, {\ensuremath{P}}\\ & $bind type$ & \pi & ::= & \tbind{(m_1,m_2)}{m_3} \end{array}$$ This section presents [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}, an ML-like language for programming with polymonads. We also present several examples that provide a flavor of programming in [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}. As such, we aim to keep our examples as simple as possible while still showcasing the broad applicability of polymonads. For a formal characterization of the expressiveness of polymonads, we appeal to Theorem \[thm:productoid\]. #### Polymonadic signatures. A [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} *polymonadic signature* $(\mathcal{M}, \Sigma)$ (Figure \[fig:lang\]) amends Definition \[def:polymonad\] in two ways. Firstly, each element $M$ of $\mathcal{M}$ may be *type-indexed*—we write $M/k$ to indicate that $M$ is a $(k+1)$-ary type constructor (we sometimes omit $k$ for brevity). For example, constructor ${\ensuremath{W}}/1$ could represent an effectful computation so that ${\ensuremath{W}}\;\epsilon\;\ty$ characterizes computations of type $\ty$ that have effect $\epsilon$. Type indexed constructors (rather than large enumerations of non-indexed constructors) are critical for writing reusable code, e.g., so we can write functions like $\sfont{app}: \forall a,b,\varepsilon. (a \rightarrow {\ensuremath{W}}\;\epsilon\;b) \rightarrow a \rightarrow {\ensuremath{W}}\;\epsilon\;b$. We write $\gm$ to denote *ground constructors*, which are monadic constructors applied to all their type indexes; e.g., ${\ensuremath{W}}\;\epsilon$ is ground. Secondly, a bind set $\Sigma$ is not specified intensionally as a set, but rather extensionally using a language of *theory constraints* $\Phi$. In particular, $\Sigma$ is a list of mappings $\sfont{b}@s$ where $s$ contains a triple $\bind{(\gm_1,\gm_2)}{\gm_3}$ along with constraints $\Phi$, which determine how the triple’s constructors may be instantiated. For example, a mapping $\sfont{sube}: \forall \varepsilon_1, \varepsilon_2.\, \varepsilon_1 \subseteq \varepsilon_2 \Rightarrow \tbind{({\ensuremath{W}}\;\varepsilon_1, {\ensuremath{\sfont{Id}}}) }{{\ensuremath{W}}\,\varepsilon_2}$ specifies the set of binds involving type indexes $\varepsilon_1, \varepsilon_2$ such that the theory constraint $\varepsilon_1 \subseteq \varepsilon_2$ is satisfied. [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}’s type system is parametric in the choice of theory constraints $\Phi$, which allows us to encode a variety of prior monad-like systems with [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}. To interpret a particular set of constraints, [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} requires a theory entailment relation [$\vDash$]{}. Elements of this relation, written $\Sigma {\ensuremath{\vDash}}\pi \rew \sfont{b}; \theta$, state that there exists $\sfont{b}@\forall\bar{a}. \Phi \Rightarrow \bind{(\gm_1,\gm_2)}{\gm_3}$ in $\Sigma$ and a substitution $\theta'$ such that $\theta\pi = \theta'\bind{(\gm_1,\gm_2)}{\gm_3}$, and the constraints $\theta'\Phi$ are satisfiable. Here, $\theta$ is a substitution for the free (non-constant) variables in $\pi$, while $\theta'$ is an instantiation of the abstracted variables in the bind specification. Thus, the interpretation of $\Sigma$ is the following set of binds: $\aset{\sfont{b}@\pi \mid \Sigma {\ensuremath{\vDash}}\pi \rew \sfont{b}; \cdot}$. Signature $(\mathcal{M}, \Sigma)$ is a polymonad if this set satisfies the polymonad laws (where each ground constructor is treated distinctly). Our intention is that type indices are *phantom*, meaning that they are used as a type-level representation of some property of the polymonad’s current state, but a polymonadic bind’s implementation does not depend on them. For example, we would expect that binds treat objects of type ${\ensuremath{W}}\,\varepsilon\,\tau$ uniformly, for all $\varepsilon$; different values of $\varepsilon$ could statically prevent unsafe operations like double-frees or dangling pointer dereferences. Of course, a polymonad may include other constructors distinct from ${\ensuremath{W}}$ whose bind operators could have a completely different semantics. For example, if an object has different states that would affect the semantics of binds, or if other effectful features like exceptions were to be modeled, the programmer can use a different constructor $M$ for each such feature. As such, our requirement that the type indices are phantom does not curtail expressiveness. #### Terms and types. [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}’s term language is standard. [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} programs do not explicitly reference binds, but are written in *direct style*, with implicit conversions between computations of type $m\;\ty$ and their $\ty$-typed results. Type inference determines the bind operations to insert (or abstract) to type check a program. To make inference feasible, we rely crucially on [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}’s call-by-value structure. Following our prior work on monadic programming for ML [@swamy11monadICFP], we restrict the shape of types assignable to a [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} program by separating value types $\ty$ from the types of polymonadic computations $m~\ty$. Here, metavariable $m$ may be either a ground constructor $\gm$ or a polymonadic type variable $\mvar$. The co-domain of every function is required to be a computation type $m~\ty$, although pure functions can be typed $\ty -> \ty'$, which is a synonym for $\ty -> {\ensuremath{\sfont{Id}}}~\ty'$. We also include types $T~\bar\ty$ for fully applied type constructors, e.g., $\sfont{list}~\tint$. Programs can also be given type schemes $\sigma$ that are polymorphic in their polymonads, e.g., $\forall a,b,\mvar.$ $(a -> \mvar\,b) -> a -> \mvar\,b$. Here, the variable $a$ ranges over value types $\tau$, while $\mvar$ ranges over ground constructors $\gm$. Type schemes may also be qualified by a set $P$ of bind constraints $\pi$. For example, $\forall \mvar. \bind{(\mvar,{\ensuremath{\sfont{Id}}})}{\gm} \Rightarrow (\tint -> \mvar~\tint) -> \gm~\tint$ is the type of a function that abstracts over a bind having shape $\bind{(\mvar,{\ensuremath{\sfont{Id}}})}{\gm}$. Notice that $\pi$ triples may contain polymonadic type variables $\mvar$ while specification triples $s \in \Sigma$ may not. Moreover, $\Phi$ constraints never appear in $\sigma$, which is thus entirely independent of the choice of the theory. Polymonadic information flow controls {#sec:ist-example} ------------------------------------- Polymonads are appealing because they can express many interesting constructions as we now show. Figure \[fig:ist\] presents a polymonad $\mIST$, which implements *stateful* information flow tracking [@devriese2011information; @russo08lightweight; @li2006encoding; @crary2005monadic; @abadi1999dcc]. The idea is that some program values are secret and some are public, and no information about the former should be learned by observing the latter—a property called noninterference [@goguen1982security]. In the setting of $\mIST$, we are worried about leaks via the heap. Heap-resident storage cells are given type ${\ensuremath{\mathit{intref}}}\;l$ where $l$ is the secrecy label of the referenced cell. Labels $l \in \aset{\Lo,\Hi}$ form a lattice with order $L \sqsubset H$. A program is acceptable if data labeled $H$ cannot flow, directly or indirectly, to computations or storage cells labeled $L$. In our polymonad implementation, $\Lo$ and $\Hi$ are just types $T$ (but only ever serve as indexes), and the lattice ordering is implemented by theory constraints $l_1 \sqsubseteq l_2$ for $l_1,l_2 \in \aset{\Lo,\Hi}$. ---------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- $\nquad\begin{array}{l@{~}c@{~}ll} ----------------------------------------------------------------------------------------------------------------- \multicolumn{4}{l}{\!\!$\textit{Signature:}$} \\ $\nquad\begin{array}{ll} {\mathcal{M}}& = & \mIST/2 \\ \multicolumn{2}{l}{\!\!$\textit{Types and auxiliary functions:}$} \\ \Phi & ::= & \multicolumn{2}{l}{l_1 \sqsubseteq l_2 \mid \Phi_1,\Phi_2} \\ \tau : & ... \mid {\ensuremath{\mathit{intref}}}~\tau \mid \Lo \mid \Hi \\ \Sigma & = & \sfont{bId} : & \morph{{\ensuremath{\sfont{Id}}}}{{\ensuremath{\sfont{Id}}}}, \\ \code{read} : & \forall l.\, {\ensuremath{\mathit{intref}}}~l \rightarrow \mIST\; H\; l\; \tint \\ & & \sfont{unitIST}: & \forall p,l.\,\morph{{\ensuremath{\sfont{Id}}}}{\mIST\;p\;l},\\ \code{write} : & \forall l.\, {\ensuremath{\mathit{intref}}}~l \rightarrow \tint \rightarrow \mIST\; l\; L\; () & & \sfont{mapIST}: &\forall p_1,l_1,p_2,l_2.\, p_2 \end{array}$ \sqsubseteq p_1, l_1\sqsubseteq l_2 \Rightarrow \\ & & & \morph{\mIST\;p_1\;l_1}{\mIST\;p_2\;l_2},\\   & & \sfont{appIST}: & \forall p_1,l_1,p_2,l_2.\, p_2 \sqsubseteq p_1, l_1\sqsubseteq l_2 \Rightarrow \\ $\nquad\begin{array}{ll} & & & \bind{({\ensuremath{\sfont{Id}}},\mIST\;p_1\;l_1)}{\mIST\;p_2\;l_2}, \\ \multicolumn{2}{l}{\!\!$\textit{Example program:}$} \\ & & \sfont{bIST}: &\forall p_1,l_1,p_2,l_2,p_3,l_3. \\ \multicolumn{2}{l}{\kw{let add_interest = lam savings. lam interest.}} \\ & & & l_1 \sqsubseteq p_2, l_1 \sqsubseteq l_3, l_2 & \kw{let currinterest = read interest in} \\ \sqsubseteq l_3, \\ & \kw{if currinterest > 0 then} \\ & & & p_3 \sqsubseteq p_1, p_3 \sqsubseteq p_2 & \quad\kw{let currbalance = read savings in}\\ \Rightarrow \\ & \quad\kw{let newbalance =}\\ & & & \bind{(\mIST\; p_1\; l_1,\mIST\; p_2\; l_2)}{\mIST\; p_3\; l_3} & \quad \quad \kw{currbalance + currinterest in}\\ \end{array}$ & \quad\kw{write savings newbalance} \\ & \kw{else ()} $improved type$ & \forall \mu. \morph{(\mIST\_H\_H,\mIST\_H\_L)}{\mu}, \\ \end{array}$ ----------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- The polymonadic constructor $\mIST/2$ uses secrecy labels for its type indexes. A computation with type $\mIST\;p\;l\;\ty$ potentially writes to references labeled $p$ and returns a $\ty$-result that has security label $l$; we call $p$ the *write label* and $l$ the *output label*. Function $read$ reads a storage cell, producing a $\mIST\; H\; l\; \tint$ computation—the second type index $l$ matches that of $l$-labeled storage cell. Function $write$ writes a storage cell, producing a $\mIST\; l\; L\; ()$ computation—the first type index $l$ matches the label of the written-to storage cell. $\Hi$ is the most permissive write label and so is used for the first index of $read$, while $\Lo$ is the most permissive output label and so is used for the second index of $write$. Aside from the identity bind $\sfont{bId}$, implemented as reverse apply, there are four kinds of binds. Unit $\sfont{unitIST}\;p\;l$ lifts a normal term into an $\mIST$ computation. Bind $\sfont{mapIST}\;p\;l$ lifts a computation into a more permissive context (i.e., $p_2$ and $l_2$ are at least as permissive as $l_1$ and $l_2$), and $\sfont{appIST}\;p\;l$ does likewise, and are implemented using $\sfont{mapIST}$ as follows: $\sfont{appIST}\;p\;l = \lambda x. \lambda f. \sfont{mapIST}\;p\;l\; (f\;x)\;(\lambda x.x)$. Finally, bind $\sfont{bIST}$ composes a computation $\mIST\;p_1\;l_1~\alpha$ with a function $\alpha -> \mIST\;p_2\;l_2~\beta$. The constraints ensure safe information flow: $l_1 \sqsubseteq p_2$ prevents the second computation from leaking information about its $l_1$-secure $\alpha$-typed argument into a reference cell that is less than $l_1$-secure. Dually, the constraints $l_1 \sqsubseteq l_3$ and $l_2 \sqsubseteq l_3$ ensure that the $\beta$-typed result of the composed computation is at least as secure as the results of each component. The final constraints $p_3 \sqsubseteq p_1$ and $p_3 \sqsubseteq p_2$ ensure that the write label of the composed computation is a lower bound of the labels of each component. Proving $(\mathcal{M},\Sigma)$ satisfies the polymonad laws is straightforward. The functor and paired morphism laws are easy to prove. The diamond law is more tedious: we must consider all possible pairs of binds that compose. This reasoning involves consideration of the theory constraints as implementing a lattice, and so would work for any lattice of labels, not just $\Hi$ and $\Lo$. In all, there were ten cases to consider. We prove the associativity law for the same ten cases. This proof is straightforward as the implementation of $\mIST$ ignores the indexes: , and various binds are just as in a normal state monad, while the indexes serve only to prevent illegal flows. Finally, proving closure is relatively straightforward—we start with each possible bind shape and then consider correctly-shaped flows into its components; in all there were eleven cases. #### Example. The lower right of Figure \[fig:ist\] shows an example use of $\mIST$. The $\kw{add_interest}$ function takes two reference cells, $\kw{savings}$ and $\kw{interest}$, and modifies the former by adding to it the latter if it is non-negative.[^3] Notice that expressions of type $\mIST\;p\;l\;\ty$ are used as if they merely had type $\ty$—see the branch on |currinterest|, for example. The program is rewritten during type inference to insert, or abstract, the necessary binds so that the program type checks. This process results in the following type for $add_interest$:[^4] $$\small\begin{array}{l@{~}l} \multicolumn{2}{l}{\forall \mvar_6,\mvar_{27}, a_1, a_2. {\ensuremath{P}}=> {{\tiny \mathit{{\ensuremath{\mathit{intref}}}\;a_1 \rightarrow {\ensuremath{\mathit{intref}}}\;a_2 \rightarrow \mvar_{27}\;()}}}} \\ \text{where } {\ensuremath{P}}= & \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{6}}, \bind{(\mIST\; \Hi\; a_1, \mIST\; a_1\; \Lo)}{\mvar_{6}}, \bind{(\mIST\; \Hi\; a_2,\mvar_{6})}{\mvar_{27}} \end{array}$$ The rewritten version of $add_interest$ starts with a sequence of $\lambda$ abstractions, one for each of the bind constraints in ${\ensuremath{P}}$. If we imagine these are numbered $\sfont{b1}$ ... $\sfont{b3}$, e.g., where $\sfont{b1}$ is a bind with type $\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{6}}$, then the term looks as follows (notation $\kw{...}$ denotes code elided for simplicity): lam savings. lam interest. b3 (read interest) (lam currinterest. if currinterest > 0 then (b2 ...) else (b1 () (lam z. z))) In a program that calls |add\_interest|, the bind constraints will be solved, and actual implementations of these binds will be passed in for each of $\sfont{b}_i$ (using a kind of dictionary-passing style as with Haskell type classes). Looking at the type of |add\_interest| we can see how the constraints prevent improper information flows. In particular, if we tried to call with $a_1 = \Lo$ and $a_2 = \Hi$, then the last two constraints become $\bind{(\mIST\; \Hi\; \Lo, \mIST\; \Lo\; \Lo)}{\mvar_{6}}, \bind{(\mIST\; \Hi\; \Hi,\mvar_{6})}{\mvar_{27}}$, and so we must instantiate $\mvar_6$ and $\mvar_{27}$ in a way allowed by the signature in Figure \[fig:ist\]. While we can legally instantiate $\mvar_6 = \mIST\;\Lo\;l_3$ for any $l_3$ to solve the second constraint, there is then no possible instantiation of $\mvar_{27}$ that can solve the third constraint. After substituting for $\mvar_6$, this constraint has the form $\tbind{(\mIST\;\Hi\; \Hi, \mIST\;\Lo\;l_3)}{\mvar_{27}}$, but this form is unacceptable because the $\Hi$ output of the first computation could be leaked by the $\Lo$ side effect of the second computation. On the other hand, all other instantiations of $a_1$ and $a_2$ (e.g., $a_1 = \Hi$ and $a_2 = \Lo$ to correspond to a secret savings account but a public interest rate) do have solutions and do not leak information. Having just discussed the latter two constraints, consider the first, $\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{6}}$. This constraint is important because it says that $\mvar_6$ must have a unit, which is needed to properly type the else branch; units are not required of a polymonad in general. The type given above for |add\_interest| is not its principal type, but an *improved* one. As it turns out, the principal type is basically unreadable, with 19 bind constraints! Fortunately, Section \[sec:solve\] shows how some basic rules can greatly simplify types without reducing their applicability, as has been done above. Moreover, our coherence result (given in the next section) assures that the corresponding changes to the elaborated term do not depend on the particular simplifications: the polymonad laws ensure all such elaborations will have the same semantics. Contextual type and effect systems ---------------------------------- Wadler and Thiemann [@wadler:2003] showed how a monadic-style construct can be used to model type and effect systems. Polymonads can model standard effect systems, but more interestingly can be used to model *contextual effects* [@neamtiu08context], which augment traditional effects with the notion of *prior* and *future* effects of an expression within a broader context. As an example, suppose we are using a language that partitions memory into *regions* $R_1, ..., R_n$ and reads/writes of references into region $R$ have effect $\aset{R}$. Then in the context of the program $\kw{read}\;r_1; \kw{read}\;r_2$, where $r_1$ points into region $R_1$ and $r_2$ points into region $R_2$, the contextual effect of the subexpression $\kw{read}\;r_1$ would be the triple $[ \emptyset; \aset{R_1}; \aset{R_2} ]$: the prior effect is empty, the present effect is $\aset{R_1}$, and the future effect is $\aset{R_2}$. ---------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\begin{array}{lcl} $\begin{array}{ll} {\mathcal{M}}& = & {\ensuremath{\mbox{\textit{CE}}}}/3 \\ \multicolumn{2}{l}{\!\!$\textit{Types and theory constraints:}$} \\ \Sigma & = & \sfont{bId} : \tbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{{\ensuremath{\sfont{Id}}}}, \\ \ty & ::= ... \mid \{A_1\} ... \{A_n\} \mid \emptyset \mid \top \mid \ty_1 \cup \ty_2 \\ &&\sfont{unitce}: \tbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})} \Phi & ::= \ty \subseteq \ty' \mid \ty = \ty' \mid \Phi,\Phi \\ {{\ensuremath{\mbox{\textit{CE}}}}\,\top\,\emptyset\,\top}\\ \\ && \sfont{appce}: \forall \multicolumn{2}{l}{\!\!$\textit{Auxiliary functions:}$} \\ \alpha_1,\alpha_2,\epsilon_1,\epsilon_2,\omega_1,\omega_2. \\ \code{read} : & \forall \alpha,\omega,r.\, {\ensuremath{\mathit{intref}}}~r \rightarrow {\ensuremath{\mbox{\textit{CE}}}}\; \alpha\; r\; \omega\; \tint \\ && \quad (\alpha_2 \subseteq \alpha_1, \epsilon_1 \subseteq \epsilon_2, \omega_2 \subseteq \omega_1) \Rightarrow\\ \code{write} : & \forall \alpha,\omega,r.\, {\ensuremath{\mathit{intref}}}~r \rightarrow \tint \rightarrow {\ensuremath{\mbox{\textit{CE}}}}\; \alpha\;r\; \omega\; ()\\ &&\quad \tbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\mbox{\textit{CE}}}}\;\alpha_1\;\epsilon_1\,\omega_1) \\ }{{\ensuremath{\mbox{\textit{CE}}}}\,\alpha_2\;\epsilon_2\, \\ \omega_2} \\ \\ && \sfont{mapce}: \forall \\ \alpha_1,\alpha_2,\epsilon_1,\epsilon_2,\omega_1,\omega_2. \\ \\ && \quad (\alpha_2 \subseteq \alpha_1, \epsilon_1 \subseteq \epsilon_2, \omega_2 \subseteq \omega_1) \Rightarrow\\ \end{array}$ &&\quad \tbind{({\ensuremath{\mbox{\textit{CE}}}}\;\alpha_1\;\epsilon_1\,\omega_1, {\ensuremath{\sfont{Id}}}) }{{\ensuremath{\mbox{\textit{CE}}}}\,\alpha_2\;\epsilon_2\, \omega_2} \\ &&\sfont{bindce}: \forall \alpha_1,\epsilon_1,\omega_1,\alpha_2,\epsilon_2,\omega_2,\epsilon_3. \\ && \quad\epsilon_2 \cup \omega_2 = \omega_1, \epsilon_1\cup \alpha_1 = \alpha_2, \epsilon_1 \cup \epsilon_2 = \epsilon_3) \Rightarrow \\ &&\quad \tbind{({\ensuremath{\mbox{\textit{CE}}}}\;\alpha_1\;\epsilon_1\,\omega_1, {\ensuremath{\mbox{\textit{CE}}}}\, \alpha_2\;\epsilon_2\,\omega_2)}{{\ensuremath{\mbox{\textit{CE}}}}\,\alpha_1\;\epsilon_3\,\omega_2} \end{array}$ ---------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Figure \[fig:ctxeff\] models contextual effects as the polymonad ${\ensuremath{\mbox{\textit{CE}}}}~\alpha~\epsilon~\omega~\ty$, for the type of a computation with prior, present, and future effects $\alpha$, $\epsilon$, and $\omega$, respectively. Indices are sets of atomic effects $\{A_1\} ... \{A_n\}$, with $\emptyset$ the empty effect, $\top$ the effect set that includes all other effects, and $\cup$ the union of two effects. We also introduce theory constraints for subset relations and extensional equality on sets, with the obvious interpretation. As an example source of effects, we include and functions on references into region sets $r$. The bind $\sfont{unitce}$ ascribes a pure computation as having an empty effect and any prior and future effects. The binds $\sfont{appce}$ and $\sfont{mapce}$ express that it is safe to consider an additional effect for the current computation (the $\epsilon$s are covariant), and fewer effects for the prior and future computations ($\alpha$s and $\omega$s are contravariant). Finally, $\sfont{bindce}$ composes two computations such that the future effect of the first computation includes the effect of the second one, provided that the prior effect of the second computation includes the first computation; the effect of the composition includes both effects, while the prior effect is the same as before the first computation, and the future effect is the same as after the second computation. Parameterized monads, and session types --------------------------------------- Finally, we show ${\ensuremath{\lambda{\mbox{\textsc{\underline{pm}}}}}}$ can express Atkey’s parameterized monad [@atkey09], which has been used to encode disciplines like regions [@kiselyov2008lightweight] and session types [@pucella2008haskell]. The type constructor ${\ensuremath{A}}~p~q~\ty$ can be thought of (informally) as the type of a computation producing a $\ty$-typed result, with a pre-condition $p$ and a post-condition $q$. --------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- $\begin{array}{lcl} ------------------------------------------------------------------------------------------------------------------------------ {\mathcal{M}}& = & {\ensuremath{\sfont{Id}}},{\ensuremath{A}}/2 \\ $\nquad\begin{array}{ll} \Sigma & = & \sfont{bId} : \tbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{{\ensuremath{\sfont{Id}}}}, \\ \multicolumn{2}{l}{\!\!$\textit{Types:}$} \\ & & \sfont{mapA}: \forall p,r.\; \tbind{({\ensuremath{A}}\;p\;r,{\ensuremath{\sfont{Id}}})}{{\ensuremath{A}}\;p\;r}, \\ \multicolumn{2}{l}{\ty ::= \dots \mid {\tfont{send}\,\ty_1\,\ty_2} \mid {\tfont{recv}\,\ty_1\,\ty_2} \mid {\tfont{end}}} \\ & & \sfont{appA}: \forall p,r.\; \tbind{({\ensuremath{\sfont{Id}}},{\ensuremath{A}}\;p\;r)}{{\ensuremath{A}}\;p\;r}, \\ \\ & & \sfont{unitA}: \forall p.\; \tbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{{\ensuremath{A}}\;p\;p}, \\ \multicolumn{2}{l}{\!\!$\textit{Auxiliary functions:}$} \\ & & \sfont{bindA}:\forall p,q,r.\; \tbind{({\ensuremath{A}}\,p\,q,\; {\ensuremath{A}}\,q\,r)}{{\ensuremath{A}}\,p\,r} \\ \sfont{send}\, : & \forall a,q.\,a\,\xrightarrow\, {\ensuremath{A}}\, ({\tfont{send}\,a\,q})\,q\, () \\ \end{array}$ \sfont{recv}\, : & \forall a,q.\,()\,\xrightarrow\, {\ensuremath{A}}\, ({\tfont{recv}\,a\,q})\,q\, a \end{array}$ ------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- As a concrete example, Figure \[fig:session\] gives a polymonadic expression of Pucella and Tov’s notion of session types [@pucella2008haskell]. The type ${\ensuremath{A}}\,{p}\,{q}\, \ty$ represents a computation involved in a two-party session which starts in protocol state $p$ and completes in state $q$, returning a value of type $\ty$. The key element of the signature $\Sigma$ is the $\sfont{bindA}$, which permits composing two computations where the first’s post-condition matches the second’s precondition. We use the type index ${\tfont{send}\,\ty\,q}$ to denote a protocol state that requires a message of type $\ty$ to be sent, and then transitions to $q$. Similarly, the type index ${\tfont{recv}\,\ty\,r}$ denotes the protocol state in which once a message of type $\ty$ is received, the protocol transitions to $r$. We also use the index ${\tfont{end}}$ to denote the protocol end state. The signatures of two primitive operations for sending and receiving messages capture this behavior. As an example, the following [$\lambda{\mbox{\textsc{\underline{pm}}}}$]{} program implements one side of a simple protocol that sends a message $x$, waits for an integer reply $y$, and returns $y+1$. $$\begin{array}{c} \kw{let go = lam x. let _ = send x in incr (recv ())} \\ $Simplified type: $\forall a,b,q,\mvar.\, \tbind{({\ensuremath{A}}\,({\tfont{send}\,\,\,a}\,b)\, b,\;{\ensuremath{A}}\,({\tfont{recv}\,\,\,\tint}\;q)\, q))}{\mvar} \Rightarrow \,(a\rightarrow\mvar\;\tint) \\ \end{array}$$ There are no specific theory constraints for session types: constraints simply arise by unification and are solved as usual when instantiating the final program (e.g., to call $go 0$). Coherent type inference for [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} {#sec:syntactic} ============================================================================= $$\begin{array}{l} \fbox{${\ensuremath{P}}|= {\ensuremath{P}}'$} \qquad \inference{\forall \pi \in {\ensuremath{P}}'. \pi \in {\ensuremath{P}}\vee \pi \in \Sigma} {{\ensuremath{P}}|= {\ensuremath{P}}'}[(TS-Entail)] \\\\ \fbox{${\ensuremath{P}}|= \sigma \tygt \ty\;\leadsto\mathsf{f}$} \qquad \inference{ \theta = [\bar \tau/\bar{a}][\bar{m}/\bar{\mvar}] & {\ensuremath{P}}|= \theta{\ensuremath{P}}_1} {{\ensuremath{P}}|= (\tscheme{\bar{a}\bar{\mvar}}{{\ensuremath{P}}_1}{\ty}) \,\tygt\, {\theta\ty} \;\leadsto {\mathsf{app}(\theta{\ensuremath{P}}_1)}}[(TS-Inst)] \\\\ \fbox{$\prefix{\ensuremath{P}}\Gamma v : \ty \;\leadsto\mathsf{e}$} \qquad \inference{v\in\aset{x,c} & {\ensuremath{P}}|= \Gamma(v) \tygt \ty \;\leadsto \mathsf{f}} {\prefix{\ensuremath{P}}\Gamma v : \ty \;\leadsto \mathsf{f}\,v}[(TS-XC)] \\\\ \inference{\prefix{\ensuremath{P}}{\Gamma,x@\ty_1} e : \tapp{m}{\ty_2} \;\leadsto \mathsf{e}} {\prefix{\ensuremath{P}}\Gamma \slam{x}{e} : \tfun{\ty_1}{\tapp{m}{\ty_2}} \;\leadsto \slam{x}{\mathsf{e}}}[(TS-Lam)] \\\\ \fbox{$\prefix{\ensuremath{P}}\Gamma e : \tapp{m}\ty \;\leadsto\mathsf{e}$}\quad \inference{\prefix{\ensuremath{P}}\Gamma v : \ty \;\leadsto \mathsf{e}} {\prefix{{\ensuremath{P}},\morph{{\ensuremath{\sfont{Id}}}}{m}}\Gamma v : \tapp{m}{\ty} \;\leadsto {\mathsf{b}_{{\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}},m}\,}{\mathsf{e}}\;(\lambda x.x)}[(TS-V)] \\\\ \inference{\prefix{{\ensuremath{P}}_1}{\Gamma,x@\ty} v : \ty \;\leadsto \mathsf{e}_1 & (\sigma,\mathsf{e_2}) = \Gen{\Gamma}{{\ensuremath{P}}_1 => \ty,\,\mathsf{e_1}} \\ \prefix{\ensuremath{P}}{\Gamma,x@\sigma} e : \tapp{m}{\ty'} \;\leadsto \mathsf{e}_3} {\prefix{\ensuremath{P}}\Gamma \sletrec{x}{v}{e} : \tapp{m}{\ty'} \;\leadsto \sletrec{x}{\mathsf{e}_2}{\mathsf{e}_3} }[(TS-Rec)] \\\\ \inference{\prefix{{\ensuremath{P}}_1}\Gamma v : \ty \;\leadsto \mathsf{e}_1 & (\sigma,\mathsf{e}_2) = \Gen{\Gamma}{{\ensuremath{P}}_1 => \ty,\,\mathsf{e}_1} \\ \prefix{\ensuremath{P}}{\Gamma,x@\sigma} e : \tapp{m}\ty' \;\leadsto\mathsf{e}_3} {\prefix{\ensuremath{P}}\Gamma \slet{x}{v}{e} : \tapp{m}\ty' \;\leadsto \slet{x}{\mathsf{e}_2}{\mathsf{e}_3} }[(TS-Let)] \\\\ \inference{ \prefix{\ensuremath{P}}\Gamma e_1 : \tapp{m_1}{\ty_1} \;\leadsto \mathsf{e}_1 & \prefix{\ensuremath{P}}{\Gamma,x@\ty_1} e_2 : \tapp{m_2}{\ty_2} \;\leadsto \mathsf{e}_2 \\ e_1 \neq v &{\ensuremath{P}}|= (m_1,m_2) \rhd {m_3}} {\prefix{\ensuremath{P}}\Gamma \slet{x}{e_1}{e_2} : \tapp{m_3}{\ty_2} \;\leadsto {\mathsf{b}_{m_1,m_2,m_3}\,}{\mathsf{e}_1}\,{(\lambda x.\,\mathsf{e}_2)} }[(TS-Do)] \\\\ \inference{\prefix{\ensuremath{P}}\Gamma e_1 : \tapp{m_1}{(\tfun{\ty_2}{\tapp{m_3}{\ty}})} \;\leadsto \mathsf{e}_1 & \prefix{\ensuremath{P}}\Gamma e_2 : \tapp{m_2}{\ty_2} \;\leadsto \mathsf{e}_2 \\ {\ensuremath{P}}|= {(m_1,m_4)}\rhd{m_5} & {\ensuremath{P}}|= {(m_2,m_3)}\rhd{m_4} } {\prefix{\ensuremath{P}}\Gamma \eapp{e_1}{e_2} : \tapp{m_5}{\ty} \;\leadsto {\mathsf{b}_{m_1,m_4,m_5}\,}{\mathsf{e}_1}\;{({\mathsf{b}_{m_2,m_3,m_4}\,}{\mathsf{e}_2}})}[(TS-App)] \\\\ \inference{\prefix{\ensuremath{P}}\Gamma e_1 : \tapp{m_1}\kw{bool} \;\leadsto \mathsf{e}_1 & \prefix{\ensuremath{P}}\Gamma e_2 : \tapp{m_2}{\ty} \;\leadsto \mathsf{e}_2 \\ \prefix{\ensuremath{P}}\Gamma e_3 : \tapp{m_3}{\ty} \;\leadsto \mathsf{e}_3 & {\ensuremath{P}}|= \morph{m_2}{m}, \morph{m_3}{m}, {(m_1,m)}\rhd{m'}} {\prefix{\ensuremath{P}}\Gamma \sif{e_1}{e_2}{e_3} : \tapp{m'}{\ty} }[(TS-If)] \\{ \;\leadsto {\mathsf{b}_{m_1,m,m'}\,}{\mathsf{e}_1}\,{(\lambda b.\,\mathsf{if}\;b\;\mathsf{then}\;{\mathsf{b}_{m_2,{\ensuremath{\sfont{Id}}},m}\,}{\mathsf{e}_2}\;(\lambda x.x)\;\mathsf{else}\; {\mathsf{b}_{m_3,{\ensuremath{\sfont{Id}}},m}\,}{\mathsf{e}_3}\; (\lambda x. x))} } \iffull \else \\\\ \begin{array}{ll} \Gen{\Gamma}{{\ensuremath{P}}=> \ty, \mathsf{e}} & = (\forall (\ftv{{\ensuremath{P}}=> \ty} \setminus \ftv{\Gamma}). {\ensuremath{P}}=> \ty,\;\mathsf{abs}({\ensuremath{P}},\mathsf{e}))\\ \mathsf{abs}((\tbind{(m_1,m_2)}{m_3},P),\mathsf{e}) &= \lambda{\mathsf{b}_{m_1,m_2,m_3}\,}.\,\mathsf{abs}(P,\mathsf{e})\\ \mathsf{abs}(\cdot,\mathsf{e}) &= \mathsf{e} \\ {\mathsf{app}(P,\tbind{(m_1,m_2)}{m_3}))} &= \lambda f.\,{\mathsf{app}(P)}\,(f\;{\mathsf{b}_{m_1,m_2,m_3}\,})\\ {\mathsf{app}(\cdot)} &= \lambda x.\,x \end{array} \fi \end{array}$$ $$\begin{array}{ll} \Gen{\Gamma}{{\ensuremath{P}}=> \ty, \mathsf{e}} & = (\forall (\ftv{{\ensuremath{P}}=> \ty} \setminus \ftv{\Gamma}). {\ensuremath{P}}=> \ty,\;\mathsf{abs}({\ensuremath{P}},\mathsf{e}))\\ \\ \mathsf{abs}((\tbind{(m_1,m_2)}{m_3},P),\mathsf{e}) &= \lambda{\mathsf{b}_{m_1,m_2,m_3}\,}.\,\mathsf{abs}(P,\mathsf{e})\\ \mathsf{abs}(\cdot,\mathsf{e}) &= \mathsf{e} \\ \\ {\mathsf{app}(P,\tbind{(m_1,m_2)}{m_3}))} &= \lambda f.\,{\mathsf{app}(P)}\,(f\;{\mathsf{b}_{m_1,m_2,m_3}\,})\\ {\mathsf{app}(P,\cdot)} &= \lambda x.\,x\\ \end{array}$$ This section defines our declarative type system for [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{} and proves that type inference produces principal types, and that elaborated programs are coherent. Figure \[fig:ssyntaxrules\] gives and Figure \[fig:extraops\] give a syntax-directed type system, organized into two main judgments. The value-typing judgment $\prefix{{\ensuremath{P}}}{\Gamma} v : \ty \;\leadsto\mathsf{e}$ types a value $v$ in an environment $\Gamma$ (binding variables $x$ and constants $c$ to type schemes) at the type $\ty$, provided the constraints ${\ensuremath{P}}$ are satisfiable. Moreover, it *elaborates* the value $v$ into a lambda term $\mathsf{e}$ that explicitly contains binds, lifts, and evidence passing (as shown in Section \[sec:ist-example\]). However, note that the elaboration is independent and we can read just the typing rules by igoring the elaborated terms. The expression-typing judgment $\prefix{{\ensuremath{P}}}{\Gamma} e : \tapp{m}{\ty}\;\leadsto\mathsf{e}$ is similar, except that it yields a computation type. Constraint satisfiability ${\ensuremath{P}}|= {\ensuremath{P}}'$, defined in the figure, states that ${\ensuremath{P}}'$ is satisfiable under the hypothesis ${\ensuremath{P}}$ if ${\ensuremath{P}}' \subseteq {\ensuremath{P}}\cup \Sigma$ where we consider $\pi \in \Sigma$ if and only if $\Sigma {\ensuremath{\vDash}}\pi \rew \sfont{b}; \cdot$ (for some $\sfont{b}$). The rule (TS-XC) types a variable or constant at an instance of its type scheme in the environment. The instance relation for type schemes ${\ensuremath{P}}|= \sigma \geq \ty\;\leadsto\mathsf{f}$ is standard: it instantiates the bound variables, and checks that the abstracted constraints are entailed by the hypothesis ${\ensuremath{P}}$. The elaborated $\mathsf{f}$ term supplies the instantiated evidence using the form. The rule (TS-Lam) is straightforward where the bound variable is given a value type and the body a computation type. The rule (TS-V) allows a value $v:\ty$ to be used as an expression by lifting it to a computation type $\tapp{m}{\ty}$, so long as there exists a morphism (or unit) from the $Id$ functor to $m$. The elaborated term uses ${\mathsf{b}_{{\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}},m}\,}$ to lift explicitly to monad $m$. Note that for evidence we make up names (${\mathsf{b}_{{\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}},m}\,}$) based on the constraint ($\morph{{\ensuremath{\sfont{Id}}}}{m}$). This simplifies our presentation but an implementation would name each constraint explicitly [@jones1994improvement]. We use the name ${\mathsf{b}_{m_1,{\ensuremath{\sfont{Id}}},m_2}\,}$ for morphism constraints $\morph{m_1}{m_2}$, and use ${\mathsf{b}_{m_1,m_2,m_3}\,}$ for general bind constraints ${(m_1,m_2)}\rhd{m_3}$. (TS-Rec) types a recursive let-binding by typing the definition $v$ at the same (mono-)type as the $letrec$-bound variable $f$. When typing the body $e$, we generalize the type of $f$ using a standard generalization function $\Gen{\Gamma}{{\ensuremath{P}}=> \ty,\;\mathsf{e}}$, which closes the type relative to $\Gamma$ by generalizing over its free type variables. However, in constrast to regular generalization, we return both a generalized type, as well as an elaboration of $\mathsf{e}$ that takes all generalized constraints as explicit evidence parameters (as defined by rule $\mathsf{abs}$). (TS-Let) is similar, although somewhat simpler since there is no recursion involved. (TS-Do) is best understood by looking at its elaboration: since we are in a call-by-value setting, we interpret a $let$-binding as forcing and sequencing two computations using a single bind where $e_1$ is typed monomorphically. (TS-App) is similar to (TS-Do), where, again, since we use call-by-value, in the elaboration we sequence the function and its argument using two bind operators, and then apply the function. (TS-If) is also similar, since we sequence the expression $e$ in the guard with the branches. As usual, we require the branches to have the same type. This is achieved by generating morphism constraints, $\morph{m_2}{m}$ and $\morph{m_3}{m}$ to coerce the type of each branch to a functor $m$ before sequencing it with the guard expression. Principal types --------------- $$\begin{array}{ll} {\llbracket x\rrbracket}^\kstar &= x \\ {\llbracket c\rrbracket}^\kstar &= c \\ {\llbracket \lambda x.e\rrbracket}^\kstar &= \lambda x. {\llbracket e\rrbracket} \\ \\ {\llbracket v\rrbracket} &= \mathtt{ret}\;{\llbracket v\rrbracket}^\kstar \\ {\llbracket e_1\; e_2\rrbracket} &= \mathtt{app}\; {\llbracket e_1\rrbracket}\; {\llbracket e_2\rrbracket} \\ {\llbracket \slet{x}{v}{e}\rrbracket} &= \slet{x}{{\llbracket v\rrbracket}^\kstar}{{\llbracket e\rrbracket}} \\ {\llbracket \slet{x}{e_1}{e_2}\rrbracket} &= \mathtt{do}\;{\llbracket e_1\rrbracket}\; {\llbracket \slam{x}{e_2}\rrbracket}^\kstar \qquad\textrm{(when $e_1 \neq v$)}\\ {\llbracket \sif{e_1}{e_2}{e_3}\rrbracket} &= \mathtt{cond}\; {\llbracket e_1\rrbracket}\; \slam{()}{{\llbracket e_2\rrbracket}} \; \slam{()}{{\llbracket e_3\rrbracket}} \\ {\llbracket \sletrec{f}{v}{e}\rrbracket} &= \mathtt{letrec}\; {f = {\llbracket v\rrbracket}^\kstar}~\mathtt{in}~{{\llbracket e\rrbracket}} \end{array}$$ $$\begin{array}{ll} {\textsf{ret}}&: \forall\alpha \mvar.\,(\morph{{\ensuremath{\sfont{Id}}}}{\mvar}) => \tfun{\alpha}{\tapp{\mvar}{\alpha}} \\ {\textsf{do}}&: \forall\alpha \beta \mvar_1 \mvar_2 \mvar.\,(\bind{(\mvar_1,\mvar_2)}{\mvar}) => \tfun{\tapp{\mvar_1}{\alpha}}{(\tfun{\tfun{\alpha}{\tapp{\mvar_2}{\beta}})}{\tapp{\mvar}{\beta}}} \\ {\textsf{app}}&: \forall\alpha \beta \mvar_1 \mvar_2 \mvar_3 \mvar_4 \mvar.\, (\bind{(\mvar_1,\mvar_4)}{\mvar}, \bind{(\mvar_2,\mvar_3)}{\mvar_4}) => \tfun{\tapp{\mvar_1}{(\tfun{\alpha}{\tapp{\mvar_3}{\beta}})}}{\tfun{\tapp{\mvar_2}{\alpha}}{\tapp{\mvar}{\beta}}} \\ {\textsf{cond}}&: \forall\alpha\mvar_1\mvar_2\mvar_3\mvar\mvar'.\, (\morph{\mvar_2}{\mvar}, \morph{\mvar_3}{\mvar}, \bind{(\mvar_1,\mvar)}{\mvar'}) \\ & \qquad=> \tfun{\tapp{\mvar_1}{\kw{bool}}} {\tfun{(\tfun{()}{\tapp{\mvar_2}{\alpha}})} {\tfun{(\tfun{()}{\tapp{\mvar_3}{\alpha}})} {\tapp{\mvar'}{\alpha}}}} \end{array}$$ The type rules admit principal types, and there exists an efficient type inference algorithm that finds such types. The way we show this is by a translation of polymonadic terms (and types) to terms (and types) in Overloaded ML (OML) [@jones1992theory] and prove this translation is sound and complete: a polymonadic term is well-typed if and only if its translated OML term has an equivalent type. OML’s type inference algorithm is known to enjoy principal types, so a corollary of our translation is that principal types exist for our system too. We encode terms in our language into OML as shown in Figure \[fig:xlate-oml\]. We rely on four primitive OML terms that force the typing of the terms to generate the same constraints as our type system does: ${\textsf{ret}}$ for lifting a pure term, ${\textsf{do}}$ for typing a do-binding, ${\textsf{app}}$ for typing an application, and ${\textsf{cond}}$ for conditionals. Using these primitives, we encode values and expressions of our system into OML. We write ${\prefix{{\ensuremath{P}}}{\Gamma}_{\textsc{\tiny OML}}} e : \ty$ for a derivation in the syntax directed inference system of OML (cf. Jones [@jones1992theory], Fig. 4). \[thm:oml\] \ **Soundness**: Whenever $\prefix{{\ensuremath{P}}}{\Gamma} v : \tau$ we have ${\prefix{{\ensuremath{P}}}{\Gamma}_{\textsc{\tiny OML}}} {\llbracket v\rrbracket}^\kstar : \tau$. Similarly, whenever $\prefix{\ensuremath{P}}\Gamma e : \tapp{m}{\ty}$ then we have ${\prefix{{\ensuremath{P}}}{\Gamma}_{\textsc{\tiny OML}}} {\llbracket e\rrbracket} : \tapp{m}{\ty}$. **Completeness**: Whenever ${\prefix{{\ensuremath{P}}}{\Gamma}_{\textsc{\tiny OML}}} {\llbracket v\rrbracket}^\kstar : \tau$, then we have $\prefix{\ensuremath{P}}\Gamma v : \tau$. Similarly, whenever ${\prefix{{\ensuremath{P}}}{\Gamma}_{\textsc{\tiny OML}}} {\llbracket e\rrbracket} : \tapp{m}{\ty}$, then we have $\prefix{\ensuremath{P}}\Gamma e : \tapp{m}{\ty}$. The proof is by straightforward induction on the typing derivation of the term. It is important to note that our system uses the same instantiation and generalization relations as OML which is required for the induction argument. Moreover, the constraint entailment over bind constraints also satisfies the monotonicity, transitivity and closure under substitution properties required by OML. As a corollary of the above properties, our system admits principal types via the general-purpose OML type inference algorithm. Ambiguity --------- Seeing the previous OML translation, one might think we could directly translate our programs into Haskell since Haskell uses OML style type inference. Unfortunately, in practice, Haskell would reject many useful programs. In particular, Haskell rejects as ambiguous any term whose type $\forall \bar{\alpha}. {\ensuremath{P}}=> \ty$ includes a variable $\alpha$ that occurs free in ${\ensuremath{P}}$ but not in $\ty$;[^5] we call such type variables *open*. Haskell, in its generality, must reject such terms since the instantiation of an open variable can have operational effect, while at the same time, since the variable does not appear in $\ty$, the instantiation for it can never be uniquely determined by the context in which the term is used. A common example is the term `show . read` with the type `(Show a, Read a) => String -> String`, where `a` is open. Depending on the instantiation of `a`, the term may parse and show integers, or doubles, etc. Rejecting all types that contain open variables works well for type classes, but it would be unacceptable for [[$\lambda{\mbox{\textsc{\underline{pm}}}}$]{}]{}. Many simple terms have principal types with open variables. For example, the term $\slam{f}{\slam{x}{\sapp{f}{x}}}$ has type $\forall a b \mvar_1 \mvar_2 \mvar_3.$ $((\mathsf{Id},\mvar_1)\rhd\;\mvar_2, (\mathsf{Id},\mvar_2)\rhd\;\mvar_3)$ $\Rightarrow\;(a \rightarrow \mvar_1\;b) \rightarrow \alpha \rightarrow \mvar_3\;b$ where type variable $\mvar_2$ is open. In the special case where there is only one polymonadic constructor available when typing the program, the coherence problem is moot, e.g., say, if the whole program were to only be typed using only the polymonad of Section \[sec:ist-example\]. However, recall that polymonads generalize monads and morphisms, for which there can be coherence issues (as is well known), so polymonads must address them. As an example, imagine combining our $\mIST$ polymonad (which generalizes the state monad) with an exception monad $\sfont{Exn}$, resulting in an $\sfont{ISTExn}$ polymonad. Then, an improperly coded bind that composed $\mIST$ with $\sfont{Exn}$ could sometimes reset the heap, and sometimes not (a similar example is provided by Filinski [@filinski94representing]). A major contribution of this paper is that for binds that satisfy the polymonad laws, we need not reject all types with open variables. In particular, by appealing to the polymonadic laws, we can prove that programs with open type variables in bind constraints are indeed unambiguous. Even if there are many possible instantiations, the semantics of each instantiation is equivalent, enabling us to solve polymonadic constraints much more aggressively. This coherence result is at the essence of making programming with polymonads practical. Coherence {#sec:coherence} --------- The main result of this section (Theorem \[thm:coherence\]) establishes that for a certain class of polymonads, the ambiguity check of OML can be weakened to accept more programs while still ensuring that programs are coherent. Thus, for this class of polymonads, programmers can reliably view our syntax-directed system as a specification without being concerned with the details of how the type inference algorithm is implemented or how programs are elaborated. The proof of Theorem \[thm:coherence\] is a little technical—the following roadmap summarizes the structure of the development. - We define the class of *principal* polymonads for which unambiguous typing derivations are coherent. All polymonads that we know of are principal. - Given $\prefix{{\ensuremath{P}}}{\Gamma} e : t \rew \tgte$ (with $t \in \aset{\ty, \tapp{m}{\ty}}$), the predicate ${\ensuremath{\mathsf{unambiguous}({\ensuremath{P}},\Gamma,t)}}$ characterizes when the derivation is unambiguous. This notion requires interpreting ${\ensuremath{P}}$ as a graph $G_{\ensuremath{P}}$, and ensuring (roughly) that all open variables in ${\ensuremath{P}}$ have non-zero in/out-degree in $G_{\ensuremath{P}}$. - A *solution* $S$ to a constraint graph with respect to a polymonad $({\mathcal{M}}, \Sigma)$ is an assignment of ground polymonad constructors ${\ensuremath{\sfont{M}}}\in{\mathcal{M}}$ to the variables in the graph such that each instantiated constraint is present in $\Sigma$. We give an equivalence relation on solutions such that $S_1 \cong S_2$ if they differ only on the assignment to open variables in a manner where the composition of binds still computes the same function according to the polymonad laws. - Finally, given $\prefix{{\ensuremath{P}}}{\Gamma} e : t \rew \tgte$ and ${\ensuremath{\mathsf{unambiguous}({\ensuremath{P}},\Gamma,t)}}$, we prove that all solutions to ${\ensuremath{P}}$ that agree on the free variables of $\Gamma$ and $t$ are in the same equivalence class. While Theorem \[thm:coherence\] enables our type system to be used in practice, this result is not the most powerful theorem one can imagine. Ideally, one might like a theorem of the form $\prefix{{\ensuremath{P}}}{\Gamma} e : t \rew \tgte$ and $\prefix{{\ensuremath{P}}'}{\Gamma} e : t \rew \tgte'$ implies $\tgte$ is extensionally equal to $\tgte'$, given that both ${\ensuremath{P}}$ and ${\ensuremath{P}}'$ are satisfiable. While we conjecture that this result is true, a proof of this property out of our reach, at present. There are at least two difficulties. First, a coherence result of this form is unknown for qualified type systems in a call-by-value setting. In an unpublished paper, Jones [@jones93coherencefor] proves a coherence result for OML, but his techique only applies to call-by-name programs. Jones also does not consider reasoning about coherence based on an equational theory for the evidence functions (these functions correspond to our binds). So, proving the ideal coherence theorem would require both generalizing Jones’ approach to call-by-value and then extending it with support for equational reasoning about evidence. In the meantime, Theorem \[thm:coherence\] provides good assurance and lays the foundation for future work in this direction. #### Defining and analyzing principality. {#defining-and-analyzing-principality. .unnumbered} We introduce a notion of principal polymonads that corresponds to Tate’s “principalled productoids.” Informally, in a principal polymonad, if there is more than one way to combine pairs of computations in the set $F$ (e.g., $\bind{(M,M')}{M_1}$ and $\bind{(M,M')}{M_2}$), then there must be a “best” way to combine them. This best way is called the principal join of $F$, and all other ways to combine the functors are related to the principal join by morphisms. All the polymonadic libraries we have encountered so far are principal polymonads. It is worth emphasizing that principality does not correspond to functional dependency—it is perfectly reasonable to combine ${\ensuremath{\sfont{M}}}$ and ${\ensuremath{\sfont{M}}}'$ in multiple ways, and indeed, for applications like sub-effecting, this expressiveness is important. We only require that there be an ordering among the choices. In the definition below, we take $\downarrow\!\!\mathcal{M}$ to be set of ground instances of all constructors in $\mathcal{M}$. A polymonad $(\mathcal{M}, \Sigma)$ is a *principal polymonad* if and only if for any set $F \subseteq \downarrow\!\!\mathcal{M}^2$, and any $\aset{{\ensuremath{\sfont{M}}}_1, {\ensuremath{\sfont{M}}}_2}\subseteq\downarrow\!\!\mathcal{M}$ such $\aset{\tbind{({\ensuremath{\sfont{M}}}, {\ensuremath{\sfont{M}}}')}{{\ensuremath{\sfont{M}}}_1} \mid ({\ensuremath{\sfont{M}}},{\ensuremath{\sfont{M}}}') \in F} \subseteq \Sigma$ and $\aset{\tbind{({\ensuremath{\sfont{M}}}, {\ensuremath{\sfont{M}}}')}{{\ensuremath{\sfont{M}}}_2} \mid ({\ensuremath{\sfont{M}}},{\ensuremath{\sfont{M}}}') \in F} \subseteq \Sigma$, then there exists $\hat{\ensuremath{\sfont{M}}}\in \downarrow\!\!\mathcal{M}$ such that $\aset{\morph{\hat{\ensuremath{\sfont{M}}}}{{\ensuremath{\sfont{M}}}_1}, \morph{\hat{\ensuremath{\sfont{M}}}}{{\ensuremath{\sfont{M}}}_2}} \subseteq \Sigma$, and $\aset{\tbind{({\ensuremath{\sfont{M}}},{\ensuremath{\sfont{M}}}')}{\hat{\ensuremath{\sfont{M}}}} \mid ({\ensuremath{\sfont{M}}},{\ensuremath{\sfont{M}}}') \in F} \subseteq \Sigma$. We call $\hat{\ensuremath{\sfont{M}}}$ the principal join of $F$ and write it as $\bigsqcup F$ A graph-view $G_{\ensuremath{P}}=(V,A,E_\rhd,E_{eq})$ of a constraint-bag ${\ensuremath{P}}$ is a graph consisting of a set of vertices $V$, a vertex assignment $A: V -> m$, a set of directed edges $E_\rhd$, and a set of undirected edges $E_{eq}$, where: - $V = \aset{\pi.0, \pi.1, \pi.2 \mid \pi \in {\ensuremath{P}}}$, i.e., each constraint contributes three vertices. - $A(\pi.i) = m_i$ when $\pi = \tbind{(m_0,m_1)}{m_2}$, for all $\pi.i \in V$ - $E_\rhd = \aset{(\pi.0,\pi.2), (\pi.1, \pi.2) \mid \pi \in {\ensuremath{P}}}$ - $E_{eq} = \aset{(v,v') \mid v,v' \in V ~\wedge~v\neq v'\wedge \exists \mvar.\mvar=A(v)=A(v')}$ **Notation** We use $v$ in this section to stand for a graph vertex, rather than a value in a program. We also make use of a pictorial notation for graph views, distinguishing the two flavors of edges in a graph. Each constraint $\pi \in {\ensuremath{P}}$ induces two edges in $E_\rhd$. These edges are drawn with solid lines, with a triangle for orientation. Unification constraints arise from correlated variable occurrences in multiple constraints—we [r]{}[3.5cm]{} $$\nqquad \xymatrix@C=1em@R=0.5em{ m_1 \ar@{-}[dr] & & & m_2\ar@{-}[dr] \\ \mvar \ar@{-}[r] & \rhd\ar@{-}[r] & \mvar'\ar@{:}[r] & \mvar'\ar@{-}[r] & \rhd\ar@{-}[r] & \mvar\ar@/^/@{:}[lllll] }$$ depict these with double dotted lines. For example, the pair of constraints $\tbind{(m_1, \mvar)}{\mvar'}, \tbind{(m_2,\mvar')}{\mvar}$ contributes four unification edges, two for $\mvar$ and two for $\mvar'$. We show its graph view alongside. Unification constraints reflect the dataflow in a program. Referring back to Figure \[fig:ssyntaxrules\], in a principal derivation using (TS-App), correlated occurrences of unification variables for $m_4$ in the constraints indicate how the two binds operators compose. The following definition captures this dataflow and shows how to interpret the composition of bind constraints using unification edges as a lambda term (in the expected way).[^6] Given a constraint graph $G = (V, A, E_\rhd, E_{eq})$, an edge $\eta=(\pi.2, \pi'.i) \in E_{eq}$, where $i \in \aset{0,1}$ and $\pi\neq\pi'$ is called a *flow edge*. The flow edge $\eta$ has a functional interpretation $F_G(\eta)$ defined as follows:\ $$\begin{array}{lcl} $If$~~i=0,~~ F_G(\eta) & = & \lambda (x@A(\pi.0)~a)~(y@a->A(\pi.1)~b)~(z@b->A(\pi'.1)~c).\\ & & ~~\sfont{bind}_{A(\pi'.0),A(\pi'.1),A(\pi'.2)}(\sfont{bind}_{A(\pi.0),A(\pi.1),A(\pi.2)}~x~y)~z\\ $If$~~i=1,~~ F_G(\eta) & = & \lambda (x@A(\pi'.0)~a)~(y@a -> A(\pi.0)~b)~(z@b->A(\pi.1)~c).\\ & & ~~\sfont{bind}_{A(\pi'.0),A(\pi'.1),A(\pi'.2)}~x~(\lambda a.\sfont{bind}_{A(\pi.0),A(\pi.1),A(\pi.2)}~(y~a)~z) \end{array}$$ We can now define our ambiguity check—a graph is unambiguous if it contains a sub-graph that has no cyclic dataflows, and where open variables only occur as intermediate variables in a sequence of binds. \[def:unambiguous\] Given $G_{\ensuremath{P}}=(V,A,E_\rhd,E_{eq})$, the predicate ${\ensuremath{\mathsf{unambiguous}({\ensuremath{P}},\Gamma,t)}}$ holds if and only if there exists $E_{eq}' \subseteq E_{eq}$, such that in the graph $G'=(V,A,E_\rhd,E_{eq}')$ all of the following are true. 1. For all $\pi \in {\ensuremath{P}}$, there is no path from $\pi.2$ to $\pi.0$ or $\pi.1$. 2. For all $v \in V$, if $A(v)\in \ftv{{\ensuremath{P}}} \setminus \ftv{\Gamma,t}$, then there exists a flow edge that connects to $v$. We call $G'$ a *core* of $G_{\ensuremath{P}}$. For a polymonadic signature $(\mathcal{M}, \Sigma)$, a solution to a constraint graph $G=(V, A, E_\rhd, E_{eq})$, is a vertex assignment $S : V -> \mathcal{M}$ such that all of the following are true. 1. For all $v \in V$, if $A(v) \in \mathcal{M}$ then $S(v)=A(v)$ 2. For all $(v_1,v_2) \in E_{eq}$, $S(v_1) = S(v_2)$. 3. For all $\aset{(\pi.0,\pi.2), (\pi.1,\pi.2)} \subseteq E_\rhd$, $\tbind{(S(\pi.0),S(\pi.1))}{S(\pi.2)} \in \Sigma$. We say that two solutions $S_1$ and $S_2$ to $G$ *agree on* $\mvar$ if for all vertices $v \in V$ such that $A(v) = \mvar$, $S_1(v) = S_2(v)$. Now we define $\cong_R$, a notion of equivalence of two solutions which captures the idea that the differences in the solutions are only to the internal open variables while not impacting the overall function computed by the binds in a constraint. It is easy to check that $\cong_R$ is an equivalence relation. Given a polymonad $(\mathcal{M},\Sigma)$ and constraint graph $G=(V,A,$ $E_\rhd,E_{eq})$, two solutions $S_1$ and $S_2$ to $G$ are equivalent with respect to a set of variables $R$ (denoted $S_1 \cong_R S_2$) if and only if $S_1$ and $S_2$ agree on all $\mvar \in R$ and for each vertex $v \in V$ such that $S_1(v) \neq S_2(v)$ for all flow edges $\eta$ incident on $v$, $F_{G_1}(\eta) = F_{G_2}(\eta)$, where $G_i=(V, S_i, E_\rhd, E_{eq})$. \[thm:coherence\] For all principal polymonads, derivations ${\ensuremath{P}}|\Gamma |- e : t \rew \tgte$ such that\ ${\ensuremath{\mathsf{unambiguous}({\ensuremath{P}},\Gamma,t)}}$, and for any two solutions $S$ and $S'$ to $G_{\ensuremath{P}}$ that agree on $R=ftv(\Gamma,t)$, we have $S \cong_R S'$. (Sketch; full version in appendix) The main idea is to show that all solutions in the core of $G_{\ensuremath{P}}$ are in the same equivalence class (the solutions to the core include $S$ and $S'$). The proof proceeds by induction on the number of vertices at which $S$ and $S'$ differ. For the main induction step, we take vertices in [r]{}[5cm]{} $$\begin{tiny}\nquad \begin{array}{c} \underline{S/S'} \\ \xymatrix@C=.1em@R=0.75em{ & {\ensuremath{\sfont{M}}}_1/{\ensuremath{\sfont{M}}}_1' & \ldots & {\ensuremath{\sfont{M}}}_2/{\ensuremath{\sfont{M}}}_2' \\ & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] & \ldots{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u]\ar@{-}[d]\ldots &{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] \\ {\ensuremath{\sfont{M}}}_3/{\ensuremath{\sfont{M}}}_3'\ar@{-}[ru] & {\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{-}[u]\ar@{:}[r]\ar@{:}[dr] & \ldots & \ar@{:}[dl]\ar@{:}[l]{\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{-}[u] & {\ensuremath{\sfont{M}}}_4/{\ensuremath{\sfont{M}}}_4'\ar@{-}[ul] \\ & \eta_1\ar@{:}[u] & \ldots\eta\ldots & \eta_k\ar@{:}[u] \\ & {\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{:}[u]\ar@{:}[r]\ar@{:}[ur] & \ldots & \ar@{:}[ul]\ar@{:}[l]{\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{:}[u] \\ & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] & \ldots{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u]\ar@{-}[d]\ldots & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] \\ {\ensuremath{\sfont{M}}}_5\ar@{-}[ur] & {\ensuremath{\sfont{M}}}_6\ar@{-}[u] & \ldots & {\ensuremath{\sfont{M}}}_7\ar@{-}[u] & {\ensuremath{\sfont{M}}}_8\ar@{-}[ul] \\ } \end{array} \end{tiny}$$ topological order, considering the least (in the order) set of vertices $Q$, all related by unification constraints, and whose assignment in $S$ is ${\ensuremath{\sfont{A}}}$ and in $S'$ is ${\ensuremath{\sfont{B}}}$, for some ${\ensuremath{\sfont{A}}}\neq{\ensuremath{\sfont{B}}}$. The vertices in $Q$ are shown in the graph alongside, all connected to each other by double dotted lines (unification constraints), and their neighborhood is shown as well. Since vertices are considered in topological order, all the vertices below $Q$ in the graph have the same assignment in $S$ and in $S'$. We build solutions $S_1$ and $S_1'$ from $S$ and $S'$ respectively, that instead assign the principal join ${\ensuremath{\sfont{J}}}=\bigsqcup \aset{({\ensuremath{\sfont{M}}}_5,{\ensuremath{\sfont{M}}}_6),\ldots,({\ensuremath{\sfont{M}}}_7,{\ensuremath{\sfont{M}}}_8)}$ to the vertices in $Q$, where $S_1 \cong_R S_1'$ by the induction hypothesis. Finally, we prove $S \cong_R S_1$ and $S' \cong_R S_1'$ by showing that the functional interpretation of each of the flow edges $\eta_i$ are equal according to the polymonad laws, and conclude $S \cong_R S'$ by transitivity. Simplification and solving {#sec:solve} ========================== Before running a program, we must solve the constraints produced during type inference, and apply the appropriate evidence for these constraints in the elaborated program. We also perform *simplification* on constraints prior to generalization to make types easier to read, but without compromising their utility. A simple syntactic transformation on constraints can make inferred types easier to read. For example, we can hide duplicate constraints, identity morphisms (which are trivially satisfiable), and constraints that are entailed by the signature. Formally, we can define the function [$\mbox{\textit{Hide}}(P)$]{} to do this, as follows: $$\nquad\begin{array}{lclr} {\ensuremath{\mbox{\textit{Hide}}(P,\pi,P')}} & = & {\ensuremath{\mbox{\textit{Hide}}(P,P')}} & \quad\mbox{if}~\pi\in P,P'~\vee~\pi=\morph{m}{m}~\vee~|= \pi\\ {\ensuremath{\mbox{\textit{Hide}}(P)}} & = & P & \quad\mbox{otherwise} \end{array}$$ Syntactically, given a scheme $\forall \bar\nu.{\ensuremath{P}}=> \ty$, we can simply show the type $\forall\bar\nu. {\ensuremath{\mbox{\textit{Hide}}({\ensuremath{P}})}} => \ty$ to the programmer. Formally, however, the type scheme is unchanged since simply removing constraints from the type scheme changes our evidence-passing elaboration. More substantially, we can find instantiations for open variables in a constraint set before generalizing a type (and at the top-level, before running a program). To do this, we introduce below a modified version of (TS-Let) (from Figure \[fig:ssyntaxrules\]); a similar modification is possible for (TS-Rec). $$\inference{\prefix{{\ensuremath{P}}_1}\Gamma v : \ty \rew \tgte_1 & \bar\mvar,\bar{a} = \ftv{{\ensuremath{P}}_1 => \ty} \setminus \ftv{\Gamma} \\ {\ensuremath{P}}_1 {\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar \setminus \ftv{\ty})}} \theta & (\sigma,\tgte_2) = \Gen{\Gamma}{\theta{\ensuremath{P}}_1 => \ty, \tgte_1} & \prefix{{\ensuremath{P}}}{\Gamma,x@\sigma} e : \tapp{m}\ty' \rew \tgte_3} {\prefix{{\ensuremath{P}}}\Gamma \slet{x}{v}{e} : \tapp{m}\ty' \rew \slet{x}{\tgte_2}{\tgte_3}}$$ This rule employs the judgment ${\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\theta$, defined in Figure \[fig:decl-solving\], to simplify constraints by eliminating some open variables in ${\ensuremath{P}}$ (via the substitution $\theta$) before type generalization. There are three main rules in the judgment, (S-$\Uparrow$), (S-$\Downarrow$) and (S-$\sqcup$), while the last two simply take the transitive closure. $$\small \begin{array}{c} \inference[S-$\Uparrow$] {\pi = \bind{({\ensuremath{\sfont{Id}}},m)}{\mvar} \;\vee\; \pi = \bind{(m,{\ensuremath{\sfont{Id}}})}{\mvar} \\ \mvar \in \bar\mvar & {\textit{flowsFrom}_{{\ensuremath{P}},{\ensuremath{P}}'}{~\mvar}} \neq \{\} \\ {\textit{flowsTo}_{{\ensuremath{P}},{\ensuremath{P}}'}{~\mvar}} =\{\} } {{\ensuremath{P}},\pi,{\ensuremath{P}}' {\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\mvar \mapsto m} \qquad \inference[S-$\Downarrow$] {\pi = \bind{({\ensuremath{\sfont{Id}}},\mvar)}{m} \;\vee\; \pi = \bind{(\mvar,{\ensuremath{\sfont{Id}}})}{m} \\ \mvar \in \bar\mvar & {\textit{flowsFrom}_{{\ensuremath{P}},{\ensuremath{P}}'}{~\mvar}} = \{\} \\ {\textit{flowsTo}_{{\ensuremath{P}},{\ensuremath{P}}'}{~\mvar}} \neq \{\} } {{\ensuremath{P}},\pi,{\ensuremath{P}}' {\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\mvar \mapsto m} \\\\ \inference[S-$\sqcup$] { F = {\textit{flowsTo}_{{\ensuremath{P}}}{~\mvar}} \\ m \in F \Rightarrow m = \gm\\ \text{for some $\gm$} } {{\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\mvar \mapsto \bigsqcup F} \qquad \inference{{\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\theta \\ \theta{\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\theta'} {{\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\theta'\theta} \qquad \inference{} {{\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\cdot} \end{array}$$ $$\small \text{where~~} \begin{array}{lcl} {\textit{flowsTo}_{{\ensuremath{P}}}{~\mvar}} & = & \aset{\,(m_1,m_2) \mid \bind{(m_1,m_2)}{\mvar} \in {\ensuremath{P}}\,} \\ {\textit{flowsFrom}_{{\ensuremath{P}}}{~\mvar}} & = & \aset{\,m \mid \exists m'.\;~ \pi \in {\ensuremath{P}}~ \wedge~ (\pi = \bind{(\mvar,m')}{m}~ \vee~ \pi = \bind{(m',\mvar)}{m})\,} \\ \end{array}$$ Rule (S-$\Uparrow$) solves monad variable $\mvar$ with monad $m$ if we have a constraint $\pi =\bind{({\ensuremath{\sfont{Id}}}, m)}{\mvar}$, where the only edges directed inwards to $\mvar$ are from ${\ensuremath{\sfont{Id}}}$ and $m$, although there may be many out-edges from $\mvar$. (The case where $\pi=\bind{(m,{\ensuremath{\sfont{Id}}})}{\mvar}$ is symmetric.) Such a constraint can always be solved without loss of generality using an identity morphism, which, by the polymonad laws is guaranteed to exist. Moreover, by the closure law, any solution that chooses $\mvar=m'$, for some $m'\neq m$ could just as well have chosen $\mvar=m$. Thus, this rule does not impact solvability of the costraints. Rule S-$\Downarrow$ follows similar reasoning in the reverse direction. Finally, we the rule (S-$\sqcup$) exploits the properties of a principal polymonad. Here we have a variable $\mvar$ such that all its in-edges are from pairs of ground constructors $\gm_i$, so we can simply apply the join function to compute a solution for $\mvar$. For a principal polymonad, if such a solution exists, this simplification does not impact solvability of the rest of the constraint graph. #### Example. {#example.-1 .unnumbered} Recall the information flow example we gave in Section \[sec:ist-example\], in Figure \[fig:ist\]. Its principal type is the following, which is hardly readable: $$\small\begin{array}{l@{~}l} \multicolumn{2}{l}{\forall \bar\mvar_i, a_1, a_2. {\ensuremath{P}}_0 => {{\tiny \mathit{{\ensuremath{\mathit{intref}}}\;a_1 \rightarrow {\ensuremath{\mathit{intref}}}\;a_2 \rightarrow \mvar_{27}\;()}}}} \\ \text{where } {\ensuremath{P}}_0 = & \bind{({\ensuremath{\sfont{Id}}},\mvar_{3})}{\mvar_{2}}, \bind{({\ensuremath{\sfont{Id}}},\mIST\; \Hi\; a_2)}{\mvar_{3}}, \bind{(\mvar_{26},{\ensuremath{\sfont{Id}}})}{\mvar_{4}}, \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{4}}, \\ & \bind{(\mvar_{8},\mvar_{4})}{\mvar_{6}}, \bind{({\ensuremath{\sfont{Id}}},\mvar_{9})}{\mvar_{8}}, \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{9}}, \bind{(\mvar_{11},\mvar_{25})}{\mvar_{26}}, \\ & \bind{({\ensuremath{\sfont{Id}}},\mvar_{12})}{\mvar_{11}}, \bind{({\ensuremath{\sfont{Id}}},\mIST\; \Hi\; a_1)}{\mvar_{12}}, \bind{(\mvar_{17},\mvar_{23})}{\mvar_{25}}, \bind{(\mvar_{14},\mvar_{18})}{\mvar_{17}}, \\ & \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{18}}, \bind{({\ensuremath{\sfont{Id}}},\mvar_{15})}{\mvar_{14}}, \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{15}}, \bind{(\mvar_{20},\mvar_{24})}{\mvar_{23}}, \\ & \bind{({\ensuremath{\sfont{Id}}},\mIST\; a_1\; \Lo)}{\mvar_{24}}, \bind{({\ensuremath{\sfont{Id}}},\mvar_{21})}{\mvar_{20}}, \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_{21}}. \end{array}$$ After applying (S-$\Uparrow$) and (S-$\Downarrow$) several times, and then hiding redundant constraints, we simplify ${\ensuremath{P}}_0$ to ${\ensuremath{P}}$ which contains only three constraints. If we had fixed $a_1$ and $a_2$ (the labels of the function parameters) to $\Hi$ and $\Lo$, respectively, we could do even better. The three constraints would be $\bind{(\mIST\,\Hi\,\Lo,\mvar_{6})}{\mvar_{27}}, \bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{\mvar_6},\bind{(\mIST\,\Hi\,\Hi,\mIST\,\Hi\,\Lo)}{\mvar_{6}}$. Then, applying (S-$\sqcup$) to $\mvar_6$ we would get $\mvar_{6} \mapsto \mIST\,\Hi\,\Hi$, which when applied to the other constraints leaves only $\bind{(\mIST\,\Hi\,\Lo,\mIST\,\Hi\,\Hi)}{\mvar_{27}}$, which cannot be simplified further, since $\mvar_{27}$ appears in the result type. Pleasingly, this process yields a simpler type that can be used in the same contexts as the original principal type, so we are not compromising the generality of the code by simplifying its type. \[lem:simplification\] For a principal polymonad, given $\sigma$ and $\sigma'$ where $\sigma$ is $\forall \bar{a}\bar{\mvar}. {\ensuremath{P}}=> \ty$ and $\sigma'$ is an *improvement* of $\sigma$, having form $\forall \bar{a'}\bar{\mvar'}. \theta{\ensuremath{P}}=> \ty$ where ${\ensuremath{P}}{\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}\theta$ and $\bar{a'}\bar{\mvar'} = (\bar{a}\bar{\mvar}) - dom(\theta)$. Then for all ${\ensuremath{P}}'', \Gamma, x, e, m, \tau$, if $\prefix{{\ensuremath{P}}''}{\Gamma,x@\sigma} e : m\,\tau$ such that $|= {\ensuremath{P}}''$ then there exists some ${\ensuremath{P}}'''$ such that $\prefix{{\ensuremath{P}}'''}{\Gamma,x@\sigma'} e : m\,\tau$ and $|= {\ensuremath{P}}'''$. The proof is by induction on the derivation $\prefix{{\ensuremath{P}}''}{\Gamma,x@\sigma} e : m\,\tau$. Most cases are by assumption or induction, with the interesting one being (TS-Var) where the variable in question is $x$, and we know that all of the constraints are solvable according to the reasoning we used to justify the simplifications, above. Note that our ${\xrightarrow{\mbox{\tiny{simplify}}(\bar\mvar)}}$ relation is non-deterministic in the way it picks constraints to analyze, and also in the order in which rules are applied. In practice, for an acyclic constraint graph, one could consider nodes in the graph in topological order and, say, apply (S-$\sqcup$) first, since, if it succeeds, it eliminates a variable. For principal polymonads and acyclic constraint graphs, this process would always terminate. However, if unification constraints induce cycles in the constraint graph, simply computing joins as solutions to internal variables may not work. This should not come as a surprise. In general, finding solutions to arbitrary polymonadic constraints is undecidable, since, in the limit, they can be used to encode the correctness of programs with general recursion. Nevertheless, simple heuristics such as unrolling cycles in the constraint graph a few times may provide good mileage, as would the use of domain-specific solvers for particular polymonads, and such approaches are justified by our coherence proof. Related work and conclusions ============================ This paper has presented *polymonads*, a generalization of monads and morphisms, which, by virtue of their relationship to Tate’s *productoids*, are extremely powerful, subsuming monads, parameterized monads, and several other interesting constructions. Thanks to supporting algorithms for (principal) type inference, (provably coherent) elaboration, and (generality-preserving) simplification (none of which Tate considers), this power comes with strong supports for the programmer. Like monads before them, we believe polymonads can become a useful and important element in the functional programmer’s toolkit. Constructions resembling polymonads have already begun to creep into languages like Haskell. Notably, Kmett’s `Control.Monad.Parameterized` Haskell package [@kmett] provides a type class for bind-like operators that have a signature resembling our $\tbind{(m_1,m_2)}{m_3}$. One key limitation is that Kmett’s binds must be *functionally dependent*: $m_3$ must be functionally determined from $m_1$ and $m_2$. As such, it is not possible to program morphisms between different constructors, i.e., the pair of binds $\tbind{(m_1,{\ensuremath{\sfont{Id}}})}{m_2}$ and $\tbind{(m_1,{\ensuremath{\sfont{Id}}})}{m_3}$ would be forbidden, so there would be no way to convert from $m_1$ to $m_2$ and from $m_1$ to $m_3$ in the same program. Kmett also requires units into ${\ensuremath{\sfont{Id}}}$, which may later be lifted, but such lifting only works for first-order code before running afoul of Haskell’s ambiguity restriction. Polymonads do not have either limitation. Kmett does not discuss laws that should govern the proper use of non-uniform binds. As such, our work provides the formal basis to design and reason about libraries that functional programmers have already begun developing. While polymonads subsume a wide range of prior monad-like constructions, and indeed can express any system of *producer effects* [@tate12productors], as might be expected, other researchers have explored generalizing monadic effects along other dimensions that are incomparable to polymonads. For example, Altenkirch et al. [@Altenkirch10relative] consider *relative monads* that are not endofunctors; each polymonad constructor must be an endofunctor. Uustalu and Vene [@Uustalu08comonad] suggest structuring computations comonadically, particularly to work with context-dependent computations. This suggests a loose connection with our encoding of contextual effects as a polymonad, and raises the possibility of a “co-polymonad”, something we leave for the future. Still other generalizations include reasoning about effects equationally using Lawvere theories [@plotkin01semantic] or with arrows [@Hughes00arrows]—while each of these generalize monadic constructions, they appear incomparable in expressiveness to polymonads. A common framework to unify all these treatments of effects remains an active area of research—polymonads are a useful addition to the discourse, covering at least one large area of the vast design space. == **Appendix** Polymonads are productoids and vice versa {#sec:productoids} ========================================= Given a polymonad $(\mathcal{M},\Sigma)$, we can construct a 4-tuple $({\mathcal{M}}, U, L, B)$ as follows: (Units) : $U = \aset{(\lambda x. {\kw{bind}}~x~(\lambda y.y))\colon \kw{a -> M a} \mid {\kw{bind}}\colon\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{M} \in \Sigma}$, (Lifts) : $L = \aset{(\lambda x. {\kw{bind}}~x~(\lambda y.y))\colon \kw{M a -> N a} \mid {\kw{bind}}\colon\morph{{\ensuremath{\sfont{M}}}}{{\ensuremath{\sfont{N}}}} \in \Sigma}$, (Binds) : The set $B = \Sigma- \aset{{\kw{bind}}\mid {\kw{bind}}\colon\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{M}~\mbox{or}~{\kw{bind}}\colon\bind{(M,{\ensuremath{\sfont{Id}}})}{N} \in \Sigma}$. It is fairly easy to show that the above structure satisfies generalizations of the familiar laws for monads and monad morphisms. \[thm:bind-as-mubl\] Given a polymonad $(\mathcal{M},\Sigma)$, the induced 4-tuple $({\mathcal{M}}, U, L, B)$ satisfies the following properties. (Left unit) : $\forall{\kw{unit}}\in U, {\kw{bind}}\in B$. if $\kw{unit: forall a. a -> M a}$ and $\kw{bind:} \bind{(M,N)}{N}$ then $ {\kw{bind}}~({\kw{unit}}~e)~ f = f(e) $ where $e\colon\tau$ and $f\colon\tau~\kw{-> N}~\tau'$. (Right unit) : $\forall{\kw{unit}}\in U, {\kw{bind}}\in B$. if $\kw{unit: forall a. a -> N a}$ and $\kw{bind:} \bind{(M,N)}{M}$ then ${\kw{bind}}~m~({\kw{unit}}) = m$ where $m\colon \kw{M}~\tau$. (Associativity) : $\forall{\kw{bind}}_1, {\kw{bind}}_2, {\kw{bind}}_3, {\kw{bind}}_4 \in B$. if ${\kw{bind}}_1: \bind{(M,N)}{P}$, ${\kw{bind}}_2: \bind{(P,R)}{T}$, ${\kw{bind}}_3: \bind{(M,S)}{T}$, and ${\kw{bind}}_4: \bind{(N,R)}{S}$ then ${\kw{bind}}_2~({\kw{bind}}_1~m~f)~g = {\kw{bind}}_3~m~(\lambda x. {\kw{bind}}_4~(f~x)~g)$ where $m\colon \kw{M}~\tau$, $f\colon \tau~\kw{-> N}~\tau'$ and $g\colon\tau'~\kw{-> R}~\tau''$ (Morphism 1) : $\forall{\kw{unit}}_{1}, {\kw{unit}}_{2}\in U, {\kw{lift}}\in L$. if ${\kw{unit}}_{1}\kw{: forall a. a -> M a}$, ${\kw{unit}}_2\kw{: forall a. a -> N a}$ and $\kw{lift: forall a. M a -> N a}$ then ${\kw{lift}}~({\kw{unit}}_1~e) = \munit_2~e$ where $e\colon\tau$. (Morphism 2) : $\forall{\kw{bind}}_1, {\kw{bind}}_2\in B, {\kw{lift}}_1, {\kw{lift}}_2, {\kw{lift}}_3\in L$. if ${\kw{bind}}_1: \bind{(M,P)}{S}$, ${\kw{bind}}_2: \bind{(N,Q)}{T}$, ${\kw{lift}}_1\kw{: forall a. M a -> N a}$, ${\kw{lift}}_2\kw{: forall a. P a -> Q a}$ and ${\kw{lift}}_3\kw{: forall a. S a -> T a}$ then ${\kw{lift}}_3~({\kw{bind}}_1~m~f) = {\kw{bind}}_2~({\kw{lift}}_1~m)~(\lambda x.{\kw{lift}}_2~(f~x))$ where $m\colon \kw{M}~\tau$ and $f\colon\tau~\kw{-> P}~\tau'$. Now we show how this definition can be used to relate polymonads to Tate’s *productoids* [@tate12productors]. The definition of a productoid is driven by an underlying algebraic structure: the effectoid [@tate12productors Theorem 1]. \[def:effectoid\] An **effectoid** $(E, U, \leq, \mapsto)$ is a set $E$, with an identified subset $U\subseteq E$ and relations $\mathord{\leq} \subseteq E\times E$ and $(\_;\_)\mapsto \_\subseteq E\times E\times E$, that satisfies the following conditions: 1. $\forall e_1, e_2\in E$. $e_1\leq e_2$ iff $\exists u\in U. u;e_1\mapsto e_2$ 2. $\forall e_1, e_2\in E$. $e_1\leq e_2$ iff $\exists u\in U. e_1;u\mapsto e_2$ 3. $\forall e_1, e_2, e_3, e_4\in E. (\exists e. e_1;e_2\mapsto e$ and $e;e_3\mapsto e_4$) iff $(\exists e'. e_2;e_3\mapsto e'$ and $e_1;e'\mapsto e_4$) 4. $\forall e\in E$. $e\leq e$ 5. $\forall e\in E, u\in U$. if $u\leq e$ then $e\in U$ 6. $\forall e_1,e_2,e_3,e_4\in E$. if $e_1;e_2\mapsto e_3$ and $e_3\leq e_4$ then $e_1;e_2\mapsto e_4$ It is fairly simple to see that a polymonad directly induces an effectoid structure. The definition of a productoid is driven by an underlying algebraic structure: the effectoid [@tate12productors Theorem 1]. An effectoid $(E, U, \leq, \mapsto)$ is a set $E$, with an identified subset $U\subseteq E$ and relations $\mathord{\leq}\subseteq E\times E$ and $(\_;\_)\mapsto \_\subseteq E\times E\times E$, that satisfies six monoid-like conditions. It is possible to show that a polymonad directly induces an effectoid structure and hence a productoid. Given a polymonad $({\mathcal{M}}, U, L, B)$ we can define an effectoid [$({E}, {U}, \leq, {\ensuremath{(\_\mathbb{;}\_) \mapsto \_}})$]{} as follows. $$\begin{array}{ll} {E}= {\mathcal{M}}& {U} \; = \{ \kw{M} \mid \kw{unit: a -> M a} \in U\}\\ \leq\ = \{ (\kw{M},\kw{N}) \mid \kw{lift: M a -> N a} \in L\}~~~~ & {\ensuremath{(\_\mathbb{;}\_) \mapsto \_}}\ = \{ (\kw{M}, \kw{N}, \kw{P}) \mid \bind{(M,N)}{P} \in B\} \end{array}$$ Identity (1): Need to show: $$\exists L. {U} \wedge {\ensuremath{(L\mathbb{;}M) \mapsto M'}} \;\iff\; M \leq M'$$ Case $(=>)$: From [$(L\mathbb{;}M) \mapsto M'$]{}, we know that $\bind{(L,M)}{M'} \in B$. (1) From [U]{}, we know that $L$ has a unit, and hence $\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{L} \in B$. (2.1) From the \[Functor\] law, we have $\bind{(M,{\ensuremath{\sfont{Id}}})}{M}$ and $\bind{(M',{\ensuremath{\sfont{Id}}})}{M'} \in B$. (2.2) From \[Closure\] applied to (1) and (2.1, 2.2), we have $\bind{({\ensuremath{\sfont{Id}}},M)}{M'} \in B$ (3) From \[Paired morphisms\] and (3), have $\bind{(M,{\ensuremath{\sfont{Id}}})}{M'} \in B$ (4) By construction, we have $\morph{M}{M'} \in J$ and hence $M\leq M'$. Case $(<=)$: From $M\leq M'$, we have $\morph{M}{M'} \in J$, or $\bind{(M,{\ensuremath{\sfont{Id}}})}{M'} \in B$. (1) Pick ${\ensuremath{\sfont{Id}}}$ as a witness for the existentially bound $L$. To show that ${U}$, we use the \[Functor\] law, which requires $\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{{\ensuremath{\sfont{Id}}}} \in B$. Hence, $a -> {\ensuremath{\sfont{Id}}}~a \in U$, and ${U}$ is derivable. To show that $\joineff{({\ensuremath{\sfont{Id}}},M)}{M'}$, we use \[Paired morphisms\] applied to (1), deriving $\bind{({\ensuremath{\sfont{Id}}}, M)}{M'} in B$, as required Identity (2): Need to show: $$M \leq M' \;\iff\; \exists L. {U} \wedge {\ensuremath{(M\mathbb{;}L) \mapsto M'}}$$ Proof is similar to the previous case. Associativity: Immediate from polymonad Associativity (1) Reflexive congruence: Case (1) $$\forall M. M \leq M$$ Proof: easy from \[Functor\] Case (2) $$\forall M,M'. {U} \;\wedge\; M \leq M' => {U}$$ Proof: From the hypothesis, we have $\aset{\mbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{M}, \mbind{(M,{\ensuremath{\sfont{Id}}})}{M'}} \subseteq B$. (1) From \[Functor\], we get $\mbind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{{\ensuremath{\sfont{Id}}}}$ and $\mbind{(M',{\ensuremath{\sfont{Id}}})}{M'}$ $\in B$. (2) From \[Closure\] applied to (1) and (2), we have $\bind{({\ensuremath{\sfont{Id}}},{\ensuremath{\sfont{Id}}})}{M'} \in B$, or $a -> M' a \in U$, or ${U}$. Case (3) $$\forall M_1,M_2,M, M'. {\ensuremath{(M_1\mathbb{;}M_2) \mapsto M}} \;\wedge\; M\leq M' => {\ensuremath{(M_1\mathbb{;}M_2) \mapsto M'}}$$ Proof: immediate from \[Closure\] A polymonad $({\mathcal{M}}, U, L, B)$ induces a productoid $({\mathcal{M}}, F, U, L, J)$, where - $F=\aset{map_M:(a -> b) -> M a -> M b \mid \kw{b}_M : \bind{(M,{\ensuremath{\sfont{Id}}})}{M} \in B}$, where @map$_M$ f m = b$_M$ m (Val $\circ$ f)@; - and $J = \aset{j_{M,N,P}:M (N a) -> P a \mid \kw{b}_{M,N,P}@\bind{(M,N)}{P} \in B}$, where @j$_{M,N,P}$ mn = b$_{M,N,P}$ mn $\lambda$x.x@. As the previous lemma shows, $({\mathcal{M}}, U, L, B)$ induces an effectoid. What remains to be shown is that $({\mathcal{M}}, F, U, L, J)$ obeys the five productoid laws. Note, we write: $I$ for ${\ensuremath{\sfont{Id}}}$. $j_{m_1m_2m_3}$ for $j@m_1 (m_2 a) -> m_3 a \in J$. $l_{m_1m_2}$ for $l@m_1 a -> m_2 a \in L$. $map_{m}$ for $map@(a -> b) -> m a -> m b \in F$. $unit_{m}$ for $unit@a -> m a \in U$. 1. $j_{ONP}~\circ~j_{LMO} = j_{LO'P}~\circ~(map_{L}~j_{MNO'})$ For any $x : LMN a$: $j_{LO'P}~(map_L~j_{MNO'}~x)$ $\nquad$$=$(def) $b_{LO'P}~(b_{LIL}~x~(\lambda x. \sfont{Val}~(j_{MNO'} x)))~\lambda x.x$ $\nquad$$=$(Associativity (2)) $b_{LO'P}~x~(\lambda x.~b_{IO'O'}~(\sfont{Val}~(j_{MNO'} x))~\lambda x.x)$ $\nquad$$=$(Paired morphisms) $b_{LO'P}~x~(\lambda x.~b_{O'IO'}~(j_{MNO'}~x)~\sfont{Val})$ $\nquad$$=$(Identity) $b_{LO'P}~x~j_{MNO'}$ $\nquad$$=$(def) $b_{LO'P}~x~(\lambda x.b_{MNO'}~x~\lambda y.y)$ $\nquad$$=$(Associativity (2)) $b_{ONP}~(b_{LMO}~x~\lambda x.x)~\lambda y.y$ $\nquad$$=$(def) $(j_{ONP}~\circ~j_{LMO})~x$  \ 2. @j$_{MNP}$ $\circ$ (map$_M$ unit$_N$) = l$_{MP}$@ For any $x:M a$ $j_{MNP}~(map_M~unit_N~x)$ $\nquad=$(def) $b_{MNP}~(b_{MIM}~x~(\lambda x. \sfont{Val}~(unit_N~x)))~\lambda y.y$ $\nquad=$(Associativity 2) $b_{MNP}~x~(\lambda x.~b_{INN}~(\sfont{Val}~(unit_N~x))~\lambda y.y)$ $\nquad=$(Paired morphisms) $b_{MNP}~x~(\lambda x.~b_{NIN}~(unit_N~x)~\sfont{Val})$ $\nquad=$(Identity) $b_{MNP}~x~unit_N$ $\nquad=$(def) $b_{MNP}~(b_{IMM}~(\sfont{Val}~x)~\lambda x.x)~unit_N$ $\nquad=$(Associativity 1 and 2) $\exists Q. b_{IQP}~(\sfont{Val}~x)~(\lambda x. b_{MNQ} x unit_N)$ $\nquad=$(def) $b_{IQP}~(\sfont{Val}~x)~(\lambda x. b_{MNQ} x (\lambda y. b_{IIN}~(\sfont{Val}~y)~\sfont{Val}))$ $\nquad=$(Associativity 1 and 2) $\exists R. b_{IQP}~(\sfont{Val}~x)~(\lambda x. b_{RIQ}~(b_{MIR}~x~\sfont{Val})~\sfont{Val})$ $\nquad=$(Lemma \[lem:bind-coherence\] and given that $b_{IMP},b_{MIM}$ exist from hyp. and Functor) $b_{IMP}~(\sfont{Val}~x)~(\lambda x. b_{MIM}~(b_{MIM}~x~\sfont{Val})~\sfont{Val})$ $\nquad=$(Identity) $b_{IMP}~(\sfont{Val}~x)~(\lambda x. x)$ $\nquad=$(Paired morphisms) $b_{MIP}~x~\sfont{Val}$ $\nquad=$(def) $l_{MP}~x$\ 3. $l_{MM'} \circ j_{M_1M_2M} = j_{M_1M_2M'}$ For any $x:M_1M_2 a$ $(l_{MM'}~\circ~j_{M_1M_2M})~x$ $\nquad=$(def) $b_{MIM'}~(b_{M_1M_2M}~x~\lambda x.x)~\sfont{Val}$ $\nquad=$(Associativity 2) $b_{M_1M_2M'}~x~(\lambda y. b_{M_2IM_2}~y~\sfont{Val})$ $\nquad=$(Identity) $b_{M_1M_2M'}~x~(\lambda y. y)$ $\nquad=$(def) $j_{M_1M_2M'}$ 4. @j$_{NMP}$ $\circ$ unit$_N$ = l$_{MP}$@ Similar to the case (2) 5. @l$_{MM'}$ $\circ$ unit$_M$ = unit$_{M'}$@ This is identical to \[Morphism (1)\] Every polymonad gives rise to a productoid. We have shown that a polymonad gives rise to an effectoid. Given an effectoid [$({E}, {U}, \leq, {\ensuremath{(\_\mathbb{;}\_) \mapsto \_}})$]{}  a productoid is defined as a collection of functors indexed by the collection $E$, and three collections of natural transformations indexed by the three relations. These functors and natural transformations are required to satisfy five addition properties [@tate12productors Theorem 2]. The five properties are the five properties of Theorem \[thm:bind-as-mubl\], so the proof is immediate. Interestingly, we can identify conditions where the opposite direction also holds. A productoid $(\mathbf{C}, \{F_e\colon\mathbf{C}\to\mathbf{C}\}_{e\in E}, \{\eta\colon 1\Rightarrow F_e\}_{e\in U}, \{\mu\colon F_{e_1}\circ F_{e_2}\Rightarrow F_{e_{3}}\}_{(e_1;e_2)\mapsto e_3}, \{\sigma\colon F_{e_1}\Rightarrow F_{e_{2}}\}_{e_{1}\leq e_{2}})$ that in addition satisfies the following conditions gives rise to a polymonad. ----------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- 1\. ${\ensuremath{\sfont{Id}}}\in E$ and $F_{\ensuremath{\sfont{Id}}}= 1$ 4\. For all $e\in E, \mu\colon F_e\circ 1\Rightarrow F_e={\ensuremath{\sfont{Id}}}$ 2\. ${\ensuremath{\sfont{Id}}}\in U$ 5\. ${\ensuremath{(e_1\mathbb{;}e_2) \mapsto e}} \;\wedge\; e_1' \leq e_1 \;=>\; {\ensuremath{(e_1'\mathbb{;}e_2) \mapsto e}}$ 3\. For all $e\in E$, ${\ensuremath{(e\mathbb{;}{\ensuremath{\sfont{Id}}}) \mapsto e}}$ 6\. ${\ensuremath{(e_1\mathbb{;}e_2) \mapsto e}} \;\wedge\; e_2' \leq e_2 \;=>\; {\ensuremath{(e_1\mathbb{;}e_2') \mapsto e}}$ ----------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- These additional conditions are fairly mild: (1)-(4) simply ensure that the ${\ensuremath{\sfont{Id}}}$ element is interpreted as the identity functor. Conditions (5)-(6) are also quite straightforward; certainly if the category is cartesian closed then the extra natural transformations are always defined. Coherence of solutions ====================== \[lem:acycliccore\] For a polymonad $(\mathcal{M},\Sigma)$, and a constraint graph $G$ with a core $G'$, the set of all solutions $\mathcal{S}'$ to $G'$ includes all the solutions $\mathcal{S}$ of $G$. (Sketch) This is easy to see, since $G'$ differs from $G$ only in that it includes fewer unification constraints. So, all solutions to $G$ are also solutions to $G'$. $$\begin{tiny}\nquad \begin{array}{c|c} \underline{S/S'} & \underline{S_1/S_1'} \\ \xymatrix@C=.1em@R=0.75em{ & {\ensuremath{\sfont{M}}}_1/{\ensuremath{\sfont{M}}}_1' & \ldots & {\ensuremath{\sfont{M}}}_2/{\ensuremath{\sfont{M}}}_2' \\ & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] & \ldots{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u]\ar@{-}[d]\ldots &{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] \\ {\ensuremath{\sfont{M}}}_3/{\ensuremath{\sfont{M}}}_3'\ar@{-}[ru] & {\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{-}[u]\ar@{:}[r]\ar@{:}[dr] & \ldots & \ar@{:}[dl]\ar@{:}[l]{\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{-}[u] & {\ensuremath{\sfont{M}}}_4/{\ensuremath{\sfont{M}}}_4'\ar@{-}[ul] \\ & \eta_1\ar@{:}[u] & \ldots\eta\ldots & \eta_k\ar@{:}[u] \\ & {\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{:}[u]\ar@{:}[r]\ar@{:}[ur] & \ldots & \ar@{:}[ul]\ar@{:}[l]{\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}\ar@{:}[u] \\ & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] & \ldots{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u]\ar@{-}[d]\ldots & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] \\ {\ensuremath{\sfont{M}}}_5\ar@{-}[ur] & {\ensuremath{\sfont{M}}}_6\ar@{-}[u] & \ldots & {\ensuremath{\sfont{M}}}_7\ar@{-}[u] & {\ensuremath{\sfont{M}}}_8\ar@{-}[ul] \\ } \qquad&\qquad \xymatrix@C=.1em@R=0.75em{ & {\ensuremath{\sfont{M}}}_1/{\ensuremath{\sfont{M}}}_1' & \ldots & {\ensuremath{\sfont{M}}}_2/{\ensuremath{\sfont{M}}}_2' \\ & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] & \ldots{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u]\ar@{-}[d]\ldots &{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] \\ {\ensuremath{\sfont{M}}}_3/{\ensuremath{\sfont{M}}}_3'\ar@{-}[ru] & {\ensuremath{\sfont{J}}}\ar@{-}[u]\ar@{:}[r]\ar@{:}[dr] & \ldots & \ar@{:}[dl]\ar@{:}[l]{\ensuremath{\sfont{J}}}\ar@{-}[u] & {\ensuremath{\sfont{M}}}_4/{\ensuremath{\sfont{M}}}_4'\ar@{-}[ul] \\ & \eta_1\ar@{:}[u] & \ldots\eta\ldots & \eta_k\ar@{:}[u] \\ & {\ensuremath{\sfont{J}}}\ar@{:}[u]\ar@{:}[r]\ar@{:}[ur] & \ldots & \ar@{:}[ul]\ar@{:}[l]{\ensuremath{\sfont{J}}}\ar@{:}[u] \\ & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] & \ldots{{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u]\ar@{-}[d]\ldots & {{\rotatebox[origin=c]{90}{$\rhd$}}}\ar[u] \\ {\ensuremath{\sfont{M}}}_5\ar@{-}[ur] & {\ensuremath{\sfont{M}}}_6\ar@{-}[u] & \ldots & {\ensuremath{\sfont{M}}}_7\ar@{-}[u] & {\ensuremath{\sfont{M}}}_8\ar@{-}[ul] \\ } \end{array} \end{tiny}$$ For all principal polymonads, derivations ${\ensuremath{P}}|\Gamma |- e : t \rew \tgte$ such that\ ${\ensuremath{\mathsf{unambiguous}({\ensuremath{P}},\Gamma,t)}}$, and for any two solutions $S$ and $S'$ to $G_{\ensuremath{P}}$ that agree on $R=ftv(\Gamma,t)$, we have $S \cong_R S'$. We consider the set $\mathcal{S}$ of all solutions to the core of $G_{\ensuremath{P}}$ that agree on $ftv(\Gamma,t)$, and prove that all these solutions are in the same equivalence class. By Lemma \[lem:acycliccore\], $\aset{S,S'} \subseteq \mathcal{S}$, establishing our goal. Let $G=(V,A,E_\rhd,E_{eq})$ be a core of $G_{\ensuremath{P}}$ and let $S$ and $S'$ be arbitrary elements of $\mathcal{S}$. $S$ and $S'$ may only differ on the open variables of ${\ensuremath{P}}$. Since $G$ is unambiguous, the nodes associated with these variables all have non-zero in- and out-degree. Let $U_{S,S'} = \aset{v \mid v \in V \;\wedge\; S(v) \neq S'(v)}$; the proof proceeds by induction on the the size of $U$. **Base case $|U_{S,S'}|=0$:** Trivial, since we have $S(v) = S'(v)$, for all $v$. **Induction step $|U_{S,S'}=i|$:** From the induction hypothesis: All solutions $S_1$ and $S_1'$ such that $|U_{S_1,S_1'}| < i$, we have $S_1 \cong S_1'$. Topologically sort $G$, such that all vertices in the same connected component following edges in $E_{eq}$ have the same index, and each vertex $v$ is assigned an index greater than the index of all vertices $v'$ such that $(v,v')$ is an edge in $E_\rhd$. That is, “leaf” nodes have the highest indices. Pick a vertex $v$ with the maximal index, such that $S(v) = {\ensuremath{\sfont{A}}}$ and $S'(v) = {\ensuremath{\sfont{B}}}$, for ${\ensuremath{\sfont{A}}} \neq {\ensuremath{\sfont{B}}}$, and let $I$ be the set of vertices reachable from $v$ via unification edges. Since both $S$ and $S'$ are solutions, there must exist an open variable $\mvar$ such that $A(v) =\mvar$, and since $G$ is a core, there must be some non-empty set of flow edges incident on $v$. Thus, the neighborhood of $v$ in the graphs $G$, under assignment $S$ and $S'$ has a shape as shown in graph at left in Figure \[fig:graphs\]. All the nodes in $I$ are shown connected by double dotted lines—they each have assignment ${\ensuremath{\sfont{A}}}/{\ensuremath{\sfont{B}}}$ in $S/S'$. Since all the nodes in $I$ have an index greater than the index of any variable that differs among $S$ and $S'$, all their immediate predecessors have identical assignments in the two solutions (i.e,. ${\ensuremath{\sfont{M}}}_5, \ldots, {\ensuremath{\sfont{M}}}_8$). However, the other assignments may differ, (e.g., the top-left node could be assigned ${\ensuremath{\sfont{M}}}_1$ in $S$ and ${\ensuremath{\sfont{M}}}_1'$ in $S'$, etc.) Each flow-edge $\aset{\eta_1, \ldots, \eta_k}$ incident upon one of the nodes with the same index as $v$ is also labeled. Now, since we have a principal polymonad, there exists a principal join of $\aset{({\ensuremath{\sfont{M}}}_5, {\ensuremath{\sfont{M}}}_6), \ldots, ({\ensuremath{\sfont{M}}}_7, {\ensuremath{\sfont{M}}}_8)}$—call it ${\ensuremath{\sfont{J}}}$. Consider the assigment $S_1$ (resp. $S_1'$) that differs from $S$ (resp. $S'$) only by assigning ${\ensuremath{\sfont{J}}}$ to each vertex in $I$ instead of ${\ensuremath{\sfont{A}}}$ (resp. ${\ensuremath{\sfont{B}}}$). We first show that $S_1$ (resp. $S_1'$) is a solution and that $S \cong S_1$ (resp. $S' \cong S_1'$). Then, we note that since $S_1$ and $S_1'$ agree on all the vertices in $I$, $|U_{S_1,S_1'}| < i$, so we apply the induction hypothesis to show that $S_1 \cong S_1'$ and conclude with transitivity of $\cong$. To show that $S_1$ (resp $S_1'$) is a solution, since ${\ensuremath{\sfont{J}}}$ is a join of ${\ensuremath{\sfont{M}}}_5,{\ensuremath{\sfont{M}}}_6, \ldots$, then $\aset{\bind{({\ensuremath{\sfont{M}}}_5, {\ensuremath{\sfont{M}}}_6)}{{\ensuremath{\sfont{J}}}}, \ldots, \bind{({\ensuremath{\sfont{M}}}_7, {\ensuremath{\sfont{M}}}_8)}{{\ensuremath{\sfont{J}}}}}$ all exist, as well as $\morph{{\ensuremath{\sfont{J}}}}{{\ensuremath{\sfont{A}}}}$ (resp. $\morph{{\ensuremath{\sfont{J}}}}{{\ensuremath{\sfont{B}}}}$). By the Closure property, for every $\bind{({\ensuremath{\sfont{M}}},{\ensuremath{\sfont{A}}})}{{\ensuremath{\sfont{M'}}}}$ (resp. ${\ensuremath{\sfont{B}}}$) there also exists $\bind{({\ensuremath{\sfont{M}}},{\ensuremath{\sfont{J}}})}{{\ensuremath{\sfont{M'}}}}$. Thus, the assignment of ${\ensuremath{\sfont{J}}}$ to $I$ is valid for a solution. To show that $S \cong S_1$ (resp. $S' \cong S_1'$), we have to show that $F_S(\eta_i) = F_{S_1}(\eta_i)$ (resp. $F_{S'}(\eta_i) = F_{S_1'}(\eta_i)$), for all $i$. Taking $\eta_k$ as a representative case (the other cases are similar), we need to show the identity below, which is an immediate corollary of Associativity 1 and 2 (resp. for ${\ensuremath{\sfont{B}}}, {\ensuremath{\sfont{M}}}_4',{\ensuremath{\sfont{M}}}_2'$). $$\begin{array}{l} \sfont{bind}_{{\ensuremath{\sfont{A}}},{\ensuremath{\sfont{M}}}_4,{\ensuremath{\sfont{M}}}_2}(\sfont{bind}_{{\ensuremath{\sfont{M}}}_7,{\ensuremath{\sfont{M}}}_8,{\ensuremath{\sfont{A}}}}~x~y)~z = \sfont{bind}_{{\ensuremath{\sfont{J}}},{\ensuremath{\sfont{M}}}_4,{\ensuremath{\sfont{M}}}_2}(\sfont{bind}_{{\ensuremath{\sfont{M}}}_7,{\ensuremath{\sfont{M}}}_8,{\ensuremath{\sfont{J}}}}~x~y)~z \end{array}$$ The following lemma is useful for proving coherence. \[lem:bind-coherence\] Given a polymonad $({\mathcal{M}}, \Sigma)$; 1. For all $\aset{b_{PQR}@\bind{(P,Q)}{R}, b_{RST}@\bind{(R,S)}{T}, b_{PQR'}@\bind{(P,Q)}{R'}, b_{R'ST}@\bind{(R',S)}{T}} \subseteq \Sigma$\ and all $x@P a, f@a -> Q b, g@b -> S c$\ $b_{RST}~(b_{PQR}~x~f)~g = b_{R'ST}~(b_{PQR'}~x~f)~g$ 2. For all $\aset{b_{PQR}@\bind{(P,Q)}{R}, b_{STQ}@\bind{(S,T)}{Q}, b_{PQ'R}@\bind{(P,Q')}{R}, b_{STQ}@\bind{(S,T)}{Q'}} \subseteq \Sigma$\ and all $x:P a, f@a -> S b, g@b -> T c$\ $b_{PQR}~x(\lambda x.b_{STQ}~(f x)~g) = b_{PQ'R}~x(\lambda x.b_{STQ'}~(f x)~g)$ From Associativity (1) and (2), and transitivity. Foundations =========== This section formally defines a polymonad using category theory, and then shows how this definition can be interpreted in a programming language. This section can be safely skipped on a first read. A categorical account of polymonads {#sec:polymonads} ----------------------------------- A monad on a category consists of a endofunctor and a pair of natural transformations involving that endofunctor satisfying a number of conditions [@maclane71]. As mentioned in the introduction, we have come across a number of monad-like programming patterns that, in contrast, involve a number of endofunctors and natural transformations between them. This leads to the following rather compact definition. \[defn:polymonad-category\] A polymonad over a category ${\mathbb{C}}$ is a pair of a set $\functors$ of endofunctors on ${\mathbb{C}}$ and a set $\joins$ of natural transformations: - $\functors = \aset{M ~|~ M\colon{\mathbb{C}}\to{\mathbb{C}}}$ - $\joins = \aset{\mu\colon MN\To P ~|~ M,N,P\in\functors}$ These must satisfy the following conditions (where we write $MN \To P \in \joins$ to mean $\exists \mu. \mu\colon MN \To P \in \joins$) 1. [***Identity***]{}: \(i) $I \in \functors$ is the identity functor and $MI = IM = M$ for all $M \in \functors$. \(ii) for all $M \in \functors$, $MI \To M \in \joins$ is an identity map (equivalently, $M \To M \in \joins$). 2. [******]{}: For all $\aset{R, S, T, W}\subseteq \functors$, $\exists U \in \functors$ such that $\aset{RS \To U, UT \To W} \subseteq \joins$ if and only if $\exists V \in \functors$ such that $\aset{ST \To V, RV \To W} \subseteq \joins$. 3. [***Associativity***]{}: For all $\aset{R, S, T, U, V, W}\subseteq \functors$,\ if $\aset{\mu_1,\mu_2,\mu_3,\mu_4} \subseteq \joins$, then the following diagram commutes: $$\xymatrix{ RST \ar[r]^{\mu_1T} \ar[d]_{R\mu_3} & UT \ar[d]^{\mu_2} \\ RV \ar[r]_{\mu_4} & W }$$ It is important to observe that this definition is not equivalent to simply a collection of monads (connected, perhaps, by monad morphisms). A functor $M\in\functors$ may, or may not, be pointed (meaning $II\To M\in\joins$) and may, or may not, have a multiplication (meaning $MM\To M\in\joins$). The definition of a polymonad is considerably more general. However, let us reassure the reader immediately with the following theorem. Every monad, $\triple{R}{\eta_{R}}{\mu_{R}}$, is a polymonad. We take as our collection of endofunctors the set $\{ R, I \}$ and as our collection of natural transformations the set $\{ \eta_R\colon II\To R, \mu_{R}\colon RR\To R, \mathit{id}_R\colon RI\To R, \mathit{id}_I\colon II\To I\}$. It is simple to see that the three conditions of Definition \[defn:polymonad-category\] hold. Interestingly, we can prove the opposite direction. A polymonad $(\{ R, I \}, \{ \eta_R\colon II\To R, \mu_{R}\colon RR\To R, \mathit{id}_R\colon RI\To R, \mathit{id}_I\colon II\To I\})$ gives rise to a monad $\triple{R}{\eta_R}{\mu_R}$. The left unit law for a monad is given by the following diagram that commutes by the associativity requirement of the polymonad. $$\xymatrix{ IIR = IR \ar[r]^{\eta R} \ar[d]_{I\mathit{id}} & RR \ar[d]^{\mu}\\ IR \ar[r]_{\mathit{id}} & R }$$ The right unit law holds similarly. The associativity law of the monad is clearly given by the associativity requirement of the polymonad. In fact, polymonads support more general versions of the familiar monad laws. Given a polymonad $\PM$, we refer to a map $\eta_R: II \To R \in \joins$ as a *unit* for $R$. We also refer to a map $\delta: RI \To S \in \joins$ as a *morphism* from $R$ to $S$. We can show that polymonads support generalizations of the monad morphism and monad unit laws. Let  be a polymonad. ---------------------------------------------------------------------------------- ------------------------------------------------------------ 1\. For all $\delta\colon RI\To S, \eta_R\colon II\To R, \eta_S\colon II\To S\in $$\xymatrix{ \joins$ the following diagram commutes: I \ar[r]^{\eta_R} \ar[dr]_{\eta_S} & R\ar[d]^{\delta} \\ & S }$$ ---------------------------------------------------------------------------------- ------------------------------------------------------------ ------------------------------------------------------------------------------- ----------------------------------------------------------------------- 2\. For all $\delta_1\colon MI\To M', \delta_2\colon NI\To $$\xymatrix{ N', \delta_3\colon LI\To L', \mu_1\colon MN\To L, \mu_2\colon M'N'\To L'\in MN \ar[r]^{\delta_1 \delta_2}\ar[d]_{\mu_1} & M'N'\ar[d]^{\mu_2} \\ \joins$, the following diagram commutes: L \ar[r]_{\delta_3} & L' }$$ ------------------------------------------------------------------------------- ----------------------------------------------------------------------- ----------------------------------------------------------------------------- ----------------------------------------------------------- 3\. For all $\eta\colon II\To R, \mu\colon RS\To T, \delta\colon SI\To T\in $$\xymatrix{ \joins$ the following diagram commutes: S \ar[r]^{\eta S} \ar[rd]_{\delta} & RS \ar[d]^{\mu} \\ & T }$$ ----------------------------------------------------------------------------- ----------------------------------------------------------- ----------------------------------------------------------------------------- ---------------------------------------------------------- 4\. For all $\eta\colon II\To S, \mu\colon RS\To T, \delta\colon RI\To T\in $$\xymatrix{ \joins$ the following diagram commutes: R \ar[r]^{R\eta} \ar[rd]_{\delta} & RS \ar[d]^{\mu} \\ & T }$$ ----------------------------------------------------------------------------- ---------------------------------------------------------- 1. The following diagram commutes, from Associativity, and from the Identity law, is equivalent to our goal. $$\xymatrix{ I = III \ar[r]^{I \mathit{id}_I}\ar[d]_{\eta_RI} & II \ar[d]^{\eta_S} \\ R = RI \ar[r]_\delta & S }$$ 2. From , the map $\mu_2'$ exists in $\joins$, and that the diagram below commutes. Using the identity laws, the diagram simplifies to our goal. $$\xymatrix{ MN\!\!=\!\!MNI \ar[r]^{M\delta_2}\ar[d]_{\mu_1.I} & MN' {\!\!=\!\!}MIN' \ar@{..>}[d]^{\mu_2'}\ar[rr]^{\delta_1N'} & & M'N'\ar[d]^{\mu_2} \\ L=LI \ar[r]_{\delta_3} & L' {\!\!=\!\!}L'I \ar[rr]_{id_{L'}} & & L' }$$ 3. We have from the identity law and associativity, that the diagram below commutes. $$\xymatrix{ SII {\!\!=\!\!}IIS \ar[r]^{~~~~\eta_R S}\ar[d]_{S\mathit{id}_{I}} & RS\ar[d]^\mu \\ SI \ar[r]_\delta & T }$$ 4. We have from the identity law and associativity that the following diagram commutes: $$\xymatrix{ RII{\!\!=\!\!}IRI \ar[d]_{R \eta_S}\ar[r]^{~~~~I\mathit{id}_R} & IR\ar[d]^\delta \\ RS \ar[r]_{\mu} & T }$$ Interestingly, we can show polymonads are not quite as liberal as one might think. In particular, an important uniqueness property holds, which has ramifications in our programming model. For any polymonad $\PM$, if $\aset{\mu_1\colon MN\To P, \mu_2\colon MN\To P} \subseteq \joins$ then $\mu_1 = \mu_2$. The following diagram commutes from the Associativity property of a polymonad, where $\mathit{id}_M\colon IM\To M$, $\mathit{id}_P\colon IP\To P$. $$\xymatrix{ IMN \ar[r]^{\mathit{id}_M N} \ar[d]_{I\mu_1} & MN \ar[d]^{\mu_2} \\ IP \ar[r]_{\mathit{id}_P} & P }$$ Alternatively, we can define polymonads where the unit and monad morphisms are distinguished from the multiplication natural transformations. Appendix \[app:more-categories\] sketches this definition but we leave further category theory to future work. Alternatively, we can define polymonads where the unit and monad morphisms are distinguished from the multiplication natural transformations. We sketch this definition in our supplementary document and leave further category theory to future work. The following theorem shows that the polymonads are quite foundational. The first shows that polymonads subsume Filinski’s layered monads [@filinski1999representing], which Filinski showed are equivalent to monads connected by monad morphisms. The second gives an example of a weak categorical structure that can not be represented by layered monads or other models that require all functors to have full monadic structure. We conclude by giving some further examples of polymonads. 1. Any two monads, $\triple{R}{\eta_{R}}{\mu_{R}}$ and $\triple{S}{\eta_{S}}{\mu_{S}}$ on a category ${\mathbb{C}}$ and a monad morphism, $\delta\colon R\To S$, between them give rise to a polymonad. 2. Let $R$ be an endofunctor over ${\mathbb{C}}$, $(S,\eta_S,\mu_S)$ a monad over ${\mathbb{C}}$, and $\delta\colon R\To S$ be a natural transformation. These give rise to a polymonad. **Tate’s productors and productoids** A polymonad is a productoid if it satisfies the following condition: - [***Functor closure***]{} $\aset{M,N}\subseteq \functors => MN \in\functors$ - [***Decomposition:***]{} For all $\aset{R, S, T, W} \subseteq \functors$, $RST -> W \in \joins$, if and only if, $\exists \aset{U,W} \subseteq \functors$ such that $\aset{RS -> U, UT -> W} \subseteq \joins$ For any polymonadic productoid $\PM$, for arbitrary $\bar{Q}, P \in \functors$, and any $\aset{\mu@\bar{Q} -> P, \mu'@\bar{Q} -> P} \subseteq \joins$, $\mu = \mu'$. First, observe that $\bar{Q}$ can always be viewed as the product of precisely $3$ functors in $\functors$. If $\abs{\bar{Q}} < 2$, it can be padded with as many identity functors as needed. If $\bar{Q} = Q_0Q_1Q_2Q_3\ldots Q_n$ or more, then, by noting that $\functors$ is closed under composition, we can take $Q_2Q_3\ldots Q_n$ as a single element of $\functors$. Thus, without loss of generality, we have $\bar{Q} = Q_0Q_1Q_2$, for $\aset{Q_0,Q_1,Q_2}\subseteq \functors$. Since we have $\aset{\mu@\bar{Q} -> P, \mu'@\bar{Q} -> P} \in \joins$, by (ii), we have $\exists Q. \aset{\mu_1@Q_0Q_1 -> Q, \mu_2@QQ_2 -> P} \subseteq \joins$ and from (i), we have $\exists Q'.\aset{\mu_3@Q_1Q_2 -> Q', \mu_4@Q_0Q' -> P} \subseteq \joins$. From Associativity, we have that $\mu_4 \circ Q_0.\mu_3 = \mu_2\circ\mu_1.Q_2 = \mu = \mu'$. For any polymonadic productoid $\PM$, given $\bar{Q} = Q_0\ldots Q_{n-1} \in \functors$, we write $\bar{Q} {\longrightarrow}P$, for $P \in \functors$, if either 1. $n=1$ and $\mu@Q_0 -> P \in \joins$; 2. $n=2$ and $\mu@Q_0Q_1 -> P \in \joins$; or 3. there exists a partition $\bar{Q}_0, \ldots, \bar{Q}_m$ of $\bar{Q}$ such that $\exists i. \abs{\bar{Q}_i} \geq 2$, and $\forall i. \exists P_i. \bar{Q}_i {\longrightarrow}P_i$ and $P_0\ldots P_m {\longrightarrow}P$. For any polymonadic productoid $\PM$, for arbitrary $\bar{Q}, P \in \functors$, and any two derivations of $\bar{Q} {\longrightarrow}P$ are equivalent and $\bar{Q} -> P \in \joins$. By strong induction on $n=\abs{\bar{Q}}$. We reduce this to case $n=3$ by padding with $\bar{Q}$ with $I$.\ We have $\bar{Q} = Q_0Q_1Q_2$ and we have two sub-cases to consider: $$\begin{array}{cc} \xymatrix{ \bar{Q} \ar[d]_{Q_0.\mu_1} & \\ Q_0P_1'\ar[r]_{\mu_2} & P } & \xymatrix{ \bar{Q} \ar[r]^{\mu_1'.Q_2} & P_0'Q_2 \ar[d]^{\mu_2'} \\ & P } \end{array}$$ Taken together, the diagrams commute because of Associativity, and from Coherence (ii), we have $\bar{Q} -> P \in \joins$.\ We show one case below (the other is symmetric): $$\begin{array}{cc} \xymatrix{ \bar{Q} \ar[r]^{\mu_1.Q_2} & P_0Q_2 \ar[d]^{\mu_2} \\ & P } & \xymatrix{ \bar{Q} \ar[r]^{\mu_1'.Q_2} & P_0'Q_2 \ar[d]^{\mu_2'} \\ & P } \end{array}$$ From Coherence (i), we know that there exists $\mu_3,\mu_4, P_1$ such that the following diagram commutes; from Coherence (ii), we know that $\mu@\bar{Q} -> P \in \joins$; and we conclude with transitivity. $$\xymatrix{ \bar{Q} \ar[rr]^{\mu_1.Q_2/\mu_1'.Q_2}\ar[d]_{Q_0.\mu_3} & & P_0Q_2 \ar[d]^{\mu_2/\mu_2'} \\ Q_0P_1\ar[rr]_{\mu_4} & & P }$$ From the induction hypothesis, we have that all pairwise joins on $\bar{Q}$, for $\abs{\bar{Q}} < n$, are coherent. Now, given $\bar{Q}$ with $\abs{\bar{Q}} = n > 3$, take any two splits of $\bar{Q}$ into $\bar{Q}_0, \bar{Q}_1$ and $\bar{Q}_0', \bar{Q}_1'$. From the induction hypothesis, we have $\aset{\mu_0, \mu_1, \mu, \mu_0', \mu_1',\mu'} \subseteq \joins$, according the the diagram below, (where, since $\functors$ is closed under composition, each $\aset{\bar{Q}_0,\bar{Q}_1,\bar{Q}_0',\bar{Q}_1'} \subseteq \functors$). $$\xymatrix{ \bar{Q}=\bar{Q}_0'\bar{Q}_1' \ar[d]_{\bar{Q}_0'.\mu_1'}\ar@{..>}[r] & \_\ar@{..>}[d] & \_\ar@{..>}[d] & \bar{Q}_0\bar{Q}_1=\bar{Q}\ar[d]^{\bar{Q}_0.\mu_1}\ar@{..>}[l] \\ \bar{Q}_0'P_1' \ar[d]_{\mu_0'.P_1'}\ar@{..>}[r] & P'\ar@{..>}[d] \ar@{=}[r] & P''\ar@{..>}[d] & \ar@{..>}[l]\bar{Q}_0P_1 \ar[d]^{\mu_0.P_1} \\ P_0'P_1' \ar[r]_{\mu'} & P \ar@{=}[r] & P & P_0P_1 \ar[l]^{\mu} }$$ From Associativity on the bottom-left and bottom-right squares, we know that that $P'$ and the dotted arrows exist. From Associativity applied to the top-left and top-right squares, we know that $P'=P''$ and hence the whole diagram commutes. Finally, to show that we have a map from $\bar{Q} -> P \in \joins$, we from Coherence (ii) applied to the top-left square, we have a join from $I\bar{Q}_0'\bar{Q}_1' -> IP'$; we also have a join from $I\bar{P'} -> P$, so applying Coherence (ii) to these two joins, we get a join $\bar{Q} -> P$. Polymonads in System [F$\omega$]{} {#sec:pmfomega} ---------------------------------- A polymonad $\PM$ can be interpreted in the category of System [F$\omega$]{}, thus establishing a foundation for its use in programming. The interpretation follows a familiar route. #### Interpreting functors and joins {#interpreting-functors-and-joins .unnumbered} Each endofunctor $M \in \functors$ is represented by a unary type constructor $M$ and a function $mapM: forall a b. (a -> b) -> M a -> M b$. These are expected to satisfy the functor laws below. (1) $\Lambda$a. mapM a a ($\lambda$x:a.x) = $\Lambda$a. $\lambda$x:M a. x (2) $\Lambda$a b c. $\lambda$ (f:b -> c) (g:a -> b). mapM a c (f $\circ$ g) = $\Lambda$a b c. $\lambda$ (f:b -> c) (g:a -> b). mapM b c f $\circ$ mapM a b g Each $\mu@PQ -> R\in \joins$ is represented as an [F$\omega$]{} function $joinPQR:$ $forall a. P (Q a) -> R a$, and satisfy the natural transformation law below. (3) $\Lambda$a b.$\lambda$f:a -> b. mapR a b f $\circ$ joinPQR a = $\Lambda$a b.$\lambda$f:a -> b. joinPQR b $\circ$ mapP (Q a) (Q b) (mapQ a b f) #### Interpreting the polymonad laws. {#interpreting-the-polymonad-laws. .unnumbered} The Identity law requires that the identity functor $I \in \functors$ be represented as the [F$\omega$]{} type function $Id a = a$ together with the map $mapId a b f = f$. The law ensures that that the appropriate $join$s exist in the [F$\omega$]{} interpretation, while the Associativity law translates to the following equation: (4) $\Lambda$a. joinRVW a $\circ$ mapR (S (T a)) (V a) (joinSTV a) = $\Lambda$a. joinUTW a $\circ$ joinRSU (T a) #### Programming with binds. {#programming-with-binds. .unnumbered} With this interpretation in place, one can program with $map$s and $join$s. However, we observe that in the context of monads it is often more convenient to program with a Kleisli-style ‘$bind$’ operator, defined in terms of $map$s and $join$s, both because composing computations using $bind$ is syntactically lightweight, and because the monad laws require that bind associates, ensuring common program transformations (e.g., inlining) are semantics preserving. Pleasingly, as we now show, polymonadic binds can be similarly defined and provide the same convenience as their monadic counterparts. For each $joinPQR$ in the interpretation, we can define an operator $bindPQR$ as follows: let bindPQR : forall a b. P a -> (a -> Q b) -> R b = $\Lambda$a b.$\lambda$p q. joinPQR b (mapP a (Q b) q p) To see this, consider the following function, which composes three computations , , and , where the latter two are each parameterized by the result of the previous computation: let f : forall a b c. R a -> (a -> S b) -> (b -> T c) -> W c = $\Lambda$a b c. $\lambda$r s t. bindUTW b c (bindRSU a b r s) t Inlining the $bind$s, and appealing to the laws (in particular, that the $join$s are natural transformations), one can show that $f$ is equivalent to the code below: let f' = $\Lambda$a b c. $\lambda$r s t. (joinUTW c $\circ$ joinRSU (T c)) (mapR (S b) (S (T c)) (mapS b (T c) t) (mapR a (S b) s r)) If $f'$ is definable, then the law ensures that there exists a type constructor $U$ such that the function $g'$ below is also definable, and the Associativity law (cast as equation (4) above), ensures that it is equivalent to $f'$. let g' = $\Lambda$a b c. $\lambda$r s t. (joinRVW c $\circ$ mapR (S (T c)) (U c) (joinSTV c)) (mapR (S b) (S (T c)) (mapS b (T c) t) (mapR a (S b) s r)) Fortunately, appealing to the laws again (this time using the fact that $map$s distribute over function composition), we can prove that $g'$ is equivalent to the function $g$ below, which, like $f$, uses $bind$s for sequencing, although in a different order. However, our reasoning shows that the order does not influence the semantics. let g = $\Lambda$a b c. $\lambda$r s t. bindRVW a c r $\lambda$x:a. bindSTV b c (s x) t Thus, pleasantly, the familiar Kleisli-style $bind$ operators work for polymonads as well. As this form of operator is dominant in the world of monadic programming, we will adopt it for the rest of this paper. let f' = $\Lambda$a b c. $\lambda$(r:R a) (s:a -> S b) (t:b -> T c). bindUTW b c (bindRSU a b r s) t = $\eqannot{definition of bindRSU and bindUTW}$ joinUTW c (mapU b (T c) t (joinRSU b (mapR a (S b) s r))) = $\eqannot{function composition}$ joinUTW c ((mapU b (T c) t $\circ$ joinRSU b) (mapR a (S b) s r)) = $\eqannot{joinRSU is a natural transformation}$ joinUTW c ((joinRSU b $\circ$ mapRS b (T c) t) (mapR a (S b) s r)) = $\eqannot{associativity of function composition}$ (joinUTW c $\circ$ joinRSU b) (mapRS b (T c) t (mapR a (S b) s r)) = $\eqannot{composition of functors}$ (joinUTW c $\circ$ joinRSU b) (mapR (S b) (S (T c)) (mapS b (T c) t) (mapR a (S b) s r)) = $\eqannot{definition}$ f let g' = $\Lambda$a b c. $\lambda$(r:R a) (s:a -> S b) (t:b -> T c). bindRVW a c r $\lambda$x:a. bindSTV b c (s x) t = $\eqannot{definition of bindRVW and bindSTR}$ joinRVW c (mapR a (V c) ($\lambda$x:a. joinSTV c (mapS b (T c) t (s x))) r) = $\eqannot{function composition}$ joinRVW c (mapR a (V c) (joinSTV c $\circ$ mapS b (T c) t $\circ$ s) r) = $\eqannot{distributivity of mapR over function composition}$ joinRVW c (mapR (S (T c)) (V c) (joinSTV c) (mapR a (S (T c)) (mapS b (T c) t $\circ$ s) r)) = $\eqannot{distributivity of mapR over function composition}$ joinRVW c (mapR (S (T c)) (V c) (joinSTV c) (mapR (S b) (S (T c)) (mapS b (T c) t) (mapR a (S b) s r))) = $\eqannot{definition}$ g Alternative categorical model {#app:more-categories} ============================= As mentioned in §\[sec:polymonads\], it is possible to give a less compact definition of a polymonad, where the unit and multiplication natural transformations are distinguished. A polymonad over a category ${\mathbb{C}}$ is a triple, $\triple{\tee}{\ee{\tee}}{\emm{\tee}}$, where - $\tee = \{T_i ~|~ i\in\{1..m\}, T_i\colon{\mathbb{C}}\to{\mathbb{C}}\}$ - $\ee{\tee} = \{ \eta_j\colon I\To T ~|~ j\in\{0..n\}, T\in\tee\}$ - $\emm{\tee} = \{ \mu_k\colon RS\To T ~|~ k\in\{0..p\}, R,S,T\in\tee\}$ that satisfies the following conditions: 1. **(Associativity)** $\forall \mu_1, \mu_2,\mu_3,\mu_4\in\emm{\tee}$, $\exists U\in\tee$, $\mu_1\colon RS\To U$ and $\mu_2\colon UT\To W$ if and only if $\exists V\in\tee$, such that $\mu_3\colon ST\To V$ and $\mu_4\colon RV\To W$, and moreover the following commutes: $$\xymatrix{ RST \ar[r]^-{\mu_1 T} \ar[d]_-{R\mu_3} & UT \ar[d]^-{\mu_2}\\ RV \ar[r]_-{\mu_4} & W }$$ 2. **(Left unit)** $\forall \eta\colon I\To R\in\ee{\tee}, \mu\in\emm{\tee}$ if $\mu\colon RS\To S$ then the following commutes: $$\xymatrix{ I S \ar[r]^-{\eta S} \ar@{=}[d] & RS \ar[d]^-{\mu}\\ S \ar@{=}[r] & S }$$ 3. **(Right unit)** $\forall \eta\colon I\To S\in\ee{\tee}, \mu\in\emm{\tee}$ if $\mu\colon RS\To R$ then the following commutes: $$\xymatrix{ R I \ar[r]^-{R \eta} \ar@{=}[d] & RS \ar[d]^-{\mu}\\ R \ar@{=}[r] & R }$$ We can extend this definition with natural transformations between the functors, i.e. the generalization of monad morphisms, in the following way. A polymonad system over a category ${\mathbb{C}}$ is a four-tuple, $\fourtuple{\tee}{\ee{\tee}}{\emm{\tee}}{\dee{\tee}}$, where $\triple{\tee}{\ee{\tee}}{\emm{\tee}}$ is a polymonad and - $\dee{\tee} = \{ \delta_i\colon R\To S ~|~ i\in\{1..q\}, R,S\in\tee\}$ that satisfies the following conditions: 1. $\forall \delta\in\dee{\tee}$, $\eta_R, \eta_S\in\ee{\tee}$ if $\delta\colon R\To S$, $\eta_R\colon I\To R$, and $\eta_S\colon I\To S$ then the following commutes: $$\xymatrix{ R \ar[r]^-{\delta} & S\\ I \ar[u]^-{\eta_R} \ar@{=}[r] & I \ar[u]_-{\eta_S} }$$ 2. $\forall \delta_1, \delta_2, \delta_3\in\dee{\tee}$, $\mu_1, \mu_2\in\ee{\tee}$ if $\delta_1\colon R\To S$, $\delta_2\colon T\To U$, $\delta_3\colon V\To W$, $\mu_1\colon RT\To U$, and $\mu_2\colon SU\To W$ then the following commutes: $$\xymatrix{ RT \ar[r]^-{\delta_{1}\delta_{2}} \ar[d]_-{\mu_1} & SU \ar[d]^-{\mu_2}\\ V \ar[r]_-{\delta_3} & W }$$ [^1]: We discovered the same model concurrently with Tate and independently of him, though we have additionally developed supporting algorithms for (principal) type inference, (provably coherent) elaboration, and (generality-preserving) simplification. Nevertheless, our presentation here has benefited from conversations with him. [^2]: An online version of this paper provides an equivalent formulation of Definition \[def:polymonad\] in terms of join operators instead of binds. It can be found here: <http://research.microsoft.com/en-us/um/people/nswamy/papers/polymonads.pdf>. The join-based definition is perhaps more natural for a reader with some familiarity with category theory; the bind-based version shown here is perhaps more familiar for a functional programmer. [^3]: For ease of presentation, the program in Figure \[fig:ist\] uses $let$ to sequence computations. This is not essential, e.g., we need not have $let$-bound $currbalance$. [^4]: This and other example types were generated by our prototype implementation. [^5]: The actual ambiguity rule in Haskell is more involved due to functional dependencies and type families but that does not affect our results. [^6]: Note, for the purposes of our coherence argument, unification constraints between value-type variables $a$ are irrelevant. Such variables may occur in two kinds of contexts. First, they may constrain some value type in the program, but these do not depend on the solutions to polymonadic constraints. Second, they may constrain some index of a polymonadic constructor; but, as mentioned previously, these indices are phantom and do not influence the semantics of elaborated terms.

--- abstract: 'The gravitational-wave candidate GW151216 is a proposed binary black hole event from the first observing run of Advanced LIGO–Virgo. Not identified as a bona fide signal by LIGO–Virgo, there is disagreement as to its authenticity, which is quantified by $p_\text{astro}$, the probability that the event is astrophysical in origin. Previous estimates of $p_\text{astro}$ from different groups range from 0.18 to 0.71, making it unclear whether this event should be included in population analyses, which typically require $p_\text{astro}>0.5$. Whether GW151216 is an astrophysical signal or not has implications for the population properties of stellar-mass black holes and hence the evolution of massive stars. Using the astrophysical odds, a Bayesian method based on coherence, we find that $p_\text{astro}={\ensuremath{0.03}}{}$, suggesting that GW151216 is unlikely to be a genuine signal. We also analyse GW150914 (the first gravitational-wave detection) and GW151012 (initially considered to be an ambiguous detection) and find $p_\text{astro}$ values of 1 and 0.997 respectively. We argue that the astrophysical odds presented here improve upon traditional methods for distinguishing signals from noise.' author: - | Gregory Ashton$^{1,2,}$[^1], Eric Thrane$^{1,2,}$\ $^{1}$School of Physics and Astronomy, Monash University, Vic 3800, Australia,\ $^{2}$OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery, Clayton VIC 3800, Australia bibliography: - 'bibliography.bib' title: The astrophysical odds of GW151216 --- [email protected] \[firstpage\] gravitational waves – black hole mergers Introduction {#sec:introduction} ============ Transient gravitational wave-astronomy has opened a new window with which to study black holes and neutron stars. The LIGO [@LIGO] and Virgo [@virgo] collaborations have now completed three observing runs and announced 13 binary coalescence signals [@GWTC1; @GW190425; @GW190412]. The data collected by these observatories is public allowing independent groups to reaffirm observations and identify new candidates [@Zackay2019; @Nitz2019; @Venumadhav2018; @Venumadhav2019]. In addition to astrophysical signals, gravitational-wave detector data contains transient non-Gaussian noise artefacts, often referred to as glitches [@2008CQGra..25r4004B; @transient_noise; @nutall2015; @cabero2019; @powell2018]. Glitches degrade our ability to identify signals, i.e. the sensitivity of the detector; when the cause of the glitch is fully understood, the optimal solution is to remove the data containing the glitches which improves the sensitivity of the detector [@2018CQGra..35f5010A]. However, the cause of many glitches is not understood and hence they cannot be removed from the data, but must be treated as part of the background noise of the detector. Traditional search methods (see, e.g. @cannon2013 [@usman2016] and @capano2017 for a review of the methods) deal with this by estimating the background using bootstrap methods [@Efron]. Bootstrap methods are defined by the use of an empirical distribution to estimate a quantity of interest. Subsequently, candidates are assigned an astrophysical probability, [$p_{\rm astro}$]{}, based on the empirical output of the search pipeline; see @2016ApJ...833L...1A [@2016ApJS..227...14A; @GWTC1] for details. For loud events such as the GW150914 @GW150914, the first observed binary black hole coalescence, ${\ensuremath{p_{\rm astro}}}\approx 1$. Meanwhile, for marginal candidates, ${\ensuremath{p_{\rm astro}}}\in [0.5, 0.99]$. Different search pipelines produce different values of ${\ensuremath{p_{\rm astro}}}$ due to differing assumptions. For loud events, this is of little consequence, but as we will see, understanding these assumptions can be crucial for marginal candidates. GW151216 was reported as a significant trigger in O1, the first LIGO observing run, with ${\ensuremath{p_{\rm astro}}}=0.71$ by @Zackay2019. The event was not included in the first LIGO–Virgo gravitational-wave transient catalogue covering the O1 and O2 observing runs [@GWTC1]. The candidate was also identified in @nitz2020, but with ${\ensuremath{p_{\rm astro}}}=0.18$, less than the 0.5 threshold used to determine inclusion in the catalogue. In the original analysis, @Zackay2019 noted the large effective spin of the candidate, which led to a range of implications; e.g. [@piran2020; @fragione2020; @luca2020]). However, in a systematic study [@Huang2020], it was shown that support for the effective spin was sensitive to the choice of prior. We summarise the various significance estimates for all events analysed in this work in Table \[tab:overview\]. In this work, we study the significance of GW151216 using the astrophysical odds [@bcr2]. This method is different from traditional methods in that it eschews bootstrap noise estimation. Instead, it directly models and fits for the population properties of glitches as they appear projected onto the parameter space of compact binary coalescence signals. By combining the notion of glitches as incoherent signals [@veitch2010; @isi2018], and using contextual data to measure the population properties of glitches, the astrophysical odds can elevate the significance of marginal candidates based on their coherence between detectors and their properties in the context of typical glitches. The odds is a Bayesian ratio of probabilities comparing a signal and noise hypothesis complete with prior probability; it can be used to directly weight posteriors in the context of a population analysis, disposing of the need for arbitrary thresholds for inclusion [@Galaudage2019; @gaebel2019] and can be employed directly to the analysis of multi-messenger events using the framework laid out in @ashton2018. In order to give the results for GW151216 context, and to validate our method, we also analyse two other binary black hole signals: GW150914, the first and most significant signal in O1 and GW151012, first reported as a “trigger” [@2016PhRvD..93l2003A] and subsequently upgraded in significance to a candidate [@GWTC1; @1OGC]. In the future, we expect more candidates to be identified in the open data by independent pipelines (see, e.g. @Venumadhav2019). While we focus here on GW151216, our broader goal is to establish a unified catalogue, sourced from multiple groups, each event with a single, reliable value of $p_\text{astro}$. The $p_\text{astro}$ in this unified catalogue will not depend on the search pipeline used to first identify each trigger. Method {#sec:method} ====== Following @bcr2, we use a Bayesian framework to calculate the astrophysical odds, ${\mathcal{O}}$. The odds answers the question: what is the ratio of probability that a $\Delta t= \unit[0.2]{s}$-duration data segment $d_i$ spans the coalescence time[^2] of an astrophysical signal versus the probability that it contains noise? The noise can be either Gaussian or it can include glitch. The odds for a signal in data segment $d_i$ in a larger dataset $d$ are $${\mathcal{O}}_{{\mathcal{S}}_i/{\mathcal{N}}_i}(d) \approx \frac{ \langle\xi\rangle {\mathcal{L}}(d_i| {\mathcal{S}}_i)}{ \int {\mathcal{L}}(d_i | {\mathcal{N}}_i, \Lambda_{\mathcal{N}}) \pi(\Lambda_{\mathcal{N}}| d_{i \neq k}, I) \, d\Lambda_{{\mathcal{N}}}} \,. \label{eqn:bcr2}$$ Here, $\xi$ is the probability of a signal in $d_i$ and its expectation value $\langle\xi\rangle$ is the prior odds; we discuss this more below. The term ${\mathcal{L}}(d_i|{\mathcal{S}}_i)$ is the Bayesian evidence (marginal likelihood) for $d_i$ given the signal hypothesis. This is the likelihood function commonly used to estimate the parameters of merging binaries, [@veitch15]. Meanwhile, ${\mathcal{L}}(d_i|{\mathcal{N}}_i, \Lambda_{\mathcal{N}})$ is the likelihood of the data given the noise hypothesis. The noise hypothesis is that the data contain either Gaussian noise or non-Gaussian glitches, modelled by uncorrelated (between detectors) binary mergers [@veitch2010; @isi2018; @bcr2]. The noise likelihood is marginalized over $\Lambda_{\mathcal{N}}$, a set of hyper-parameters that describe the distributions of glitch parameters. Finally, $\pi(\Lambda_{\mathcal{N}}| d_{i \neq k}, I)$ is the noise parameter prior informed by conditional data $d_{i \neq k}$ and any other cogent information $I$; the importance of this will be made clear later on. We refer to this distribution as the glitch population properties. Our present purpose is to describe how we calculate ${\mathcal{O}}$ for the three events considered in this paper, so we take Eq. \[eqn:bcr2\] as given and refer readers to [@bcr2] for more information including a derivation of ${\mathcal{O}}$ and a discussion of the motivation for our noise model. Equation \[eqn:bcr2\] differs slightly from the expression in [@bcr2] because of two simplifying assumptions. First, we assume that the prior signal probability $\xi$ is independent of the glitch hyper-parameters $\Lambda_{\mathcal{N}}$. Second we assume that the prior signal probability $\xi \ll 1$ (as expected for astrophysical signals). This allows us to factorise the prior-odds $\pi_{{\mathcal{S}}_i/{\mathcal{N}}_i}(d_{i\neq k}, I)$ and approximate them by $$\begin{aligned} \pi_{{\mathcal{S}}_i/{\mathcal{N}}_i}(d_{i\neq k}, I) = & \frac{\int \xi \pi(\xi | d_{i\neq k}, I)\, d\xi}{\int (1 - \xi) \pi(\xi| d_{i\neq k}, I)\, d\xi} \approx \langle \xi \rangle \,, \label{eqn:prior_odds}\end{aligned}$$ where $\langle \xi \rangle$ is the expectation of value of $\xi$. We omitted $I$ in @bcr2 as all inferences were made from the contextual data alone. In this work, we will make good use of cogent prior information and hence re-introduce it in order to show where this information is important. With this formalism out of the way, we turn our attention to the evaluation of ${\mathcal{O}}$ using data from O1. The first step is to define the contextual data. The contextual data is drawn from a span of time near to the candidate of interest. Ideally, one would like to include as much contextual data as possible, though, not so much that the detector performance is likely to have changed. A comprehensive study of transient noise in O1 was performed by @transient_noise. Using the single-detector burst identification algorithm [<span style="font-variant:small-caps;">Omicron</span>]{}[@transient_noise; @omicron2], the rate of all glitches with  $>5$ during O1 was found to vary by epoch, but typically was less than $\SI{0.5}{\per\second}$. However, louder glitches with  $>10$ have a typical rate (excluding vetoed epochs) between $0.01$ and $\SI{0.001}{\per\second}$. Given this rate, a coincident-observing period will contain several thousand quiet glitches and a few hundred loud glitches: a sufficient number to estimate typical population properties. Thus, we use $\unit[24]{h}$ of contextual data, which is long enough to provide adequate estimates of the population properties of glitches, but short enough to control computational costs. We define $d_{i \neq k}$ to be the set of [<span style="font-variant:small-caps;">Omicron</span>]{}triggers in the contextual data. The next step is to calculate the expectation value of $\xi$. This is often referred to as the “duty cycle” [@smith18]. It is the expectation value for the fraction of segments containing the coalescence time of a gravitational-wave signal. The duty cycle is straightforwardly related to the local merger rate $R$ and the average time between mergers in the Universe $\tau$: $\xi \sim R \sim \tau^{-1}$. By assuming a plausible cosmological model,[^3] @GW170817_stochastic obtained $\tau=223^{+352}_{-115} \si{\second}$ based on a local merger rate of $R=103.2^{+110}_{-115}$. Since then, @O1O2RAP updated the estimated local merger rate to be $R=53.2^{+58.5}_{-28.8}$. Combining these results, we obtain a point estimate of $$\widehat\xi \approx 4.5\times10^{-4} \left(\frac{\Delta T}{\unit[0.2]{s}}\right) \left(\frac{R}{\unit[59]{Gpc^{-3}yr^{-1}}}\right) .$$ We approximate the posterior for merger rate as a log-normal distributions, centred on $\widehat\xi$, with shape parameters estimated by fitting the 90% credible intervals given above. Using these fits, we Monte-Carlo sample the distribution $\pi(\xi | I)$ in Eq. \[eqn:prior\_odds\] (we are dropping the contextual data $d_{i\neq k}$ in deference to the information $I$ used above) and find that $\langle \xi \rangle=7.4\times10^{-4}$. Thus, roughly one in $1/\langle\xi\rangle\approx1400$ segments contains a coalescence time. For the odds to favour a signal hypothesis, the astrophysical Bayes factor (i.e. all the terms in Eq.  *except* $\langle \xi \rangle$) must be larger than these prior odds. The next step is to estimate the glitch population properties $\pi(\Lambda_{\mathcal{N}}| d_{i\neq k}, I)$. We write the set of glitch hyper-parameters as $\Lambda_{\mathcal{N}}\equiv\{{\xi_{g}^{\textsc{h}}}, {\xi_{g}^{\textsc{l}}}, \lambda_{{\mathcal{N}}}\}$ where ${\xi_{g}^{\textsc{h}}}$ and ${\xi_{g}^{\textsc{l}}}$ are the prior probability for a glitch in the LIGO Hanford and Livingston detectors and $\lambda_{{\mathcal{N}}}$ is the remaining set of hyper-parameters describing the glitch population properties. Making the simplifying assumption that these are independent, we can write $\pi(\Lambda_{\mathcal{N}}| d_{i\neq k}, I)=\pi({\xi_{g}^{\textsc{h}}}| d_{i\neq k}, I) \pi({\xi_{g}^{\textsc{l}}}| d_{i\neq k}, I)\pi(\lambda_{\mathcal{N}}| d_{i\neq k}, I)$. A computationally efficient means to infer $\lambda_{\mathcal{N}}$ is to use the &gt;5 [<span style="font-variant:small-caps;">Omicron</span>]{}triggers present in the $\unit[24]{h}$ span of contextual data as a representative sample of glitches (we pre-filter this list to only include triggers with frequencies between 20 and ). By using only these [<span style="font-variant:small-caps;">Omicron</span>]{}triggers, we can save the time that would otherwise be spent analysing data segments consistent with Gaussian noise; they do not teach us about the properties of glitches. The inferred distribution of ${\xi_{g}^{\textsc{h}}}$ and ${\xi_{g}^{\textsc{l}}}$ given this contextual data is consistent with unity. This is not surprising since the [<span style="font-variant:small-caps;">Omicron</span>]{}pipeline is designed to identify non-Gaussian noise. For calculations of the astrophysical odds, we approximate the distribution of ${\xi_{g}^{\textsc{h}}}$ and ${\xi_{g}^{\textsc{l}}}$ using a point estimates ${\hat{\xi}_{g}^{\textsc{h}}}$ and ${\hat{\xi}_{g}^{\textsc{l}}}$ given by the ratio of the number of [<span style="font-variant:small-caps;">Omicron</span>]{}triggers, for each detector, to the available data span. That is, we assume $\pi(\xi_{g} | d_{i\neq k}, I) = \delta\left(\xi_{g} - \hat{\xi}_{g}\right)$. The values of these point estimates are reported in Table \[tab:overview\]. To verify that these point estimates are appropriate, we additionally analyse an auxiliary set of conditional data: 1000 randomly selected times near to GW151216. This contextual data has too few glitches to give reasonable inferences about $\lambda_{\mathcal{N}}$, but gives a good measure of the glitch probability with medians and 90% credible intervals ${\xi_{g}^{\textsc{h}}}=0.013^{+0.02}_{-0.01}$ and ${\xi_{g}^{\textsc{l}}}=0.0034^{+0.01}_{-0.003}$. The [<span style="font-variant:small-caps;">Omicron</span>]{}rate estimates (Table \[tab:overview\]) lie at the 80% and 96% percentiles for the Hanford and Livingston detectors respectively. We conclude that the [<span style="font-variant:small-caps;">Omicron</span>]{}triggers provide reliable point estimates, but that they are slightly conservative; by slightly overestimating $\xi_G$, there is a modest bias against the astrophysical hypothesis. In Sec. \[sec:discusion\] we show that the results are robust to this conservative choice. When writing out the prior previously, each term was conditional on both the contextual data as well as $I$. However, by using the [<span style="font-variant:small-caps;">Omicron</span>]{}triggers to infer $\lambda_{\mathcal{N}}$, but point estimates to infer ${\xi_{g}^{\textsc{h}}}$ and ${\xi_{g}^{\textsc{l}}}$ we see that we are calculating $\pi(\Lambda_{\mathcal{N}}| d_{i\neq k}, I)=\pi({\xi_{g}^{\textsc{h}}}| I) \pi ({\xi_{g}^{\textsc{l}}}| I)\pi\left(\lambda_{\mathcal{N}}| d_{i\neq k}\right)$. Having described details of our calculation, we now recap the procedure from start to finish. There are three steps. First, we identify a period of data passing the standard data-quality vetoes and absent of injected signals and the analysis segment itself. Second, we filter the available data against [<span style="font-variant:small-caps;">Omicron</span>]{}triggers to produce a list of contextual data segments known to contain glitches. Third, we analyse the loudest $N$ of these triggers and estimate the glitch hyper-parameters $\lambda_{\mathcal{N}}$. In this step we vary $N$ by a factor of two and check that the resulting glitch population posteriors are invariant: this demonstrates that we have captured the typical glitch population properties without analysing the entire available data set. Finally, we calculate the astrophysical odds, Eq. , using the distribution of hyper-parameters found in the second step, the prior odds $\langle \xi \rangle=7.4\times10^{-4}$, and the point estimates ${\hat{\xi}_{g}^{\textsc{h}}}$ and ${\hat{\xi}_{g}^{\textsc{l}}}$. Waveform models, priors, and noise uncertainty {#sec:details} ============================================== We use the aligned-spin waveform model `IMRPhenomD` [@PhysRevD.93.044006; @Khan:2015jqa] for the signal model and for the incoherent-between-detectors glitch model. In the future, it is desirable to extend this analysis to use more sophisticated waveforms including precession of the orbital plane and marginalization over systematic waveform uncertainties [@ashton2020]. However, we elect to use `IMRPhenomD` because it is fast and no published candidate events exhibit strong evidence of precession. We use data from the Gravitational Wave Open Science Centre [@gwosc] spanning $\unit[20-512]{Hz}$. We estimate the noise properties, the , from the median average of 31 non-overlapping s periodograms using `gwpy` [@2019ascl.soft12016M; @duncan_macleod_2020_3598469]. The data used for estimating the is off-source and immediately before the analysis segment in each instance. We do not include the effects of calibration uncertainty [@Cahillane:2017jb]. For signals, we use uniform priors in the chirp mass and mass ratio over the ranges $[13, 100$ ${\textrm{M}_{\sun}}$ and $[0.125, 1]$ respectively; for the component spin prior we use the “$z$-prior” (see Eq. (A7) of @lange2018) which places much of the prior support at small spins; this is equivalent to the aligned-spin prior (Config. B) used in @Huang2020. For the remaining parameters we use standard priors (see @bilbyO1O2), which are informed by the astrophysical nature of expected signals. In the future, it is worth employing more realistic population models for mass and spin, though, this is outside our present scope; see [@Fishbach; @Galaudage2019]. The informative prior distributions used for signals are not necessarily appropriate for the glitch model in which we project glitches into the compact binary coalescence signal parameter space. The astrophysical odds framework is designed to use knowledge about typical glitches by marginalizing over the contextual data. It does so by “recycling” posteriors obtained with an initial prior (see Appendix B of @bcr2). This process is inefficient if the glitch posteriors strongly disagree with the initial prior. We find, in agreement with @Davis2020, that glitches tend to have posterior support in regions of parameter space unusual for typical astrophysical signals, e.g., large negative spins and extreme mass ratios. To counter this inefficiency, we apply, a glitch prior uniform in the component spin $\chi_1\in[-1, 1]$ and $\chi_2\in[-1, 1]$. In testing, we find this improved the efficiency of the astrophysical odds in properly classifying glitches. One might worry that, by applying a different prior for glitches and signals, we are biasing the odds. However, posterior samples are ultimately recycled using hierarchical inference, and so these prior choices do not affect our results except to improve computational efficiency. We also find that the astrophysically motivated comoving volumetric prior [@bilbyO1O2] for luminosity distance can also decrease the efficiency of recycling as most glitches tend to occur around $\sim 100$ Mpc. To be clear, glitches have no physical distance; we refer here to the effective distance obtained by fitting glitches to binary merger waveforms. We therefore employ a uniform-in-luminosity distance prior for both signals and glitches, which ensures efficient recycling. In testing, we find that it is important to include uncertainty in our estimate of the estimation. Failing to take this into account yields false-positive signals (${\mathcal{O}}>1$) in time-slide checks in which the H1 data is offset from L1 to destroy the coherence of real gravitational-wave signals in the data. The solution is to marginalise over uncertainty in the noise as in [@Talbot2020; @Banagiri2020]. Using the median Student-$t$ method from [@Talbot2020]), the astrophysical odds calculated for the set of time-slid [<span style="font-variant:small-caps;">Omicron</span>]{}triggers behaves properly: all triggers result in an odds disfavouring an astrophysical interpretation (see Fig. \[fig:plot\]). We conclude that marginalizing over uncertainty in the is necessary for a reliable odds, and so we apply this to all the results discussed below. The glitch population {#sec:glitches} ===================== We infer the properties of the glitch population by analysing the top 100 [<span style="font-variant:small-caps;">Omicron</span>]{}triggers from Hanford and Livingston in a period around each of the events. Our glitch model consists of compact binary coalescence signals with uncorrelated parameters in each detector [@bcr2]. As such, we are projecting the properties of glitches (for which we do not have a first-principle model) onto the parameter space of astrophysical signals. We find broadly consistent features in the glitch populations surrounding each of the three events. Namely, glitches tend to have large anti-aligned spins $\chi \sim -1$ , large masses, and large mass ratios. These finding are consistent with the impulse-like glitches studied by @Davis2020 for the <span style="font-variant:small-caps;">PyCBC</span> search algorithm [@Nitz2017]. For this analysis, we apply a simple glitch hyper-model to capture salient features of the triggers in our study. For each component spin, we apply a hyper-model consisting of a mixture distribution between a uniform and normal distribution. We verify the predictive power of the model by generating posterior predictive distributions and comparing these with the data. In future work, we will look to study a larger population of glitches and develop a more sophisticated glitch hyper model, potentially improving the ability of this method to distinguish astrophysical signals. In this analysis, we study a limited set of contextual data to infer the properties of typical glitches. The conclusions are robust to this choice of contextual data and we find that doubling the number of points does not significantly change the inferred glitch population distribution. In future work, we will extend the analysis of glitches to a broader population and develop a more sophisticated glitch hyper-models. This will likely improve the ability of the method to distinguish astrophysical signals. Event GstLAL PyCBC 1-OGC 2-OGC IAS $\langle \xi \rangle$ ${\hat{\xi}_{g}^{\textsc{h}}}$ ${\hat{\xi}_{g}^{\textsc{l}}}$ $\ln B^G_{\rm S/N}$ $\ln B_{\rm S/N}$ $\ln B_{\rm coh,inc}$ $\ln$ BCR $\ln {\mathcal{O}}$ $1 - {\ensuremath{p_{\rm astro}}}$ ---------- -------------- -------------- --------------------- -------------- ------ ----------------------- -------------------------------- -------------------------------- --------------------- ------------------- ----------------------- ----------- --------------------- ------------------------------------ GW150914 $ <10^{-3} $ $ <10^{-3} $ $<8{\times}10^{-4}$ $ <10^{-3} $ – $7.4{\times}10^{-4}$ 0.0094 0.013 307 205 12.5 14.3 16.2 $9\times10^{-8}$ GW151012 $0.001$ 0.04 $0.024$ $ <10^{-3} $ – $7.4{\times}10^{-4}$ 0.031 0.021 28.2 13.2 9.63 5.64 5.74 0.003 GW151216 – – 0.997 0.82 0.29 $7.4{\times}10^{-4}$ 0.022 0.016 12.7 3.70 3.10 -3.53 -3.50 $0.97$ \[tab:overview\] Results & Discussion {#sec:discusion} ==================== We present the astrophysical odds for the three events analysed in this work in Table \[tab:overview\]. For GW151216, we find ${\ensuremath{p_{\rm astro}}}={\ensuremath{0.03}}{}$ suggesting it is likely of terrestrial origin. While the astrophysical odds disfavour an astrophysical origin, the posterior odds, $\mathcal{O}={\ensuremath{0.03}}{}$, are larger than the prior odds $7.4\times10^{-4}$, showing that this segment is 40 times more likely than average to contain a signal. Our result suggests that the astrophysical implications inferred from GW151216 may be premised on a terrestrial event [@piran2020; @fragione2020; @luca2020]. In Table \[tab:overview\], we also provide several Bayesian estimates of significance. These are the signal vs. Gaussian noise Bayes factor as measured directly $B_{\rm S/N}^{G}$, the signal vs. Gaussian noise Bayes factor after marginalizing over uncertainty in the ; $B_{\rm S/N}$, the coherent vs. incoherent Bayes factor $B_{\rm coh,inc}$ [@veitch15], and the Bayesian coherence ratio (BCR) [@isi2018]. The astrophysical odds builds on each of these concepts, as such, we can see each as a special case. $B_{\rm coh,inc}$, does not include the prior-odds and gives only the evidence for a signal vs. a glitch (i.e. the non-Gaussian component of the noise in our noise model). The BCR is an odds comparing the signal and noise hypotheses used in this work, but does not include the glitch hyper model marginalization. In [@isi2018] the prior-odds and glitch probabilities are used to tune the statistic to maximise the detection power of the statistic in a bootstrap framework. In this work, we instead apply the usual direct interpretation for the tuning parameters as prior probabilities. In this sense, the astrophysical odds in the absence of a glitch hyper-model are equivalent to the BCR up to the choice of tuning parameters/prior probabilities. The difference between $\ln{\rm BCR}$ and $\ln {\mathcal{O}}$ in Tab. \[tab:overview\] quantifies the effect of the glitch hyper model. The astrophysical odds, as with any significance estimate, depends on the choice of priors and on the noise model. It is therefore useful to consider which aspects of the analysis are most important for the conclusion that GW151216 is not astrophysical in origin. From Table \[tab:overview\], it is clear that GW151216 is a less significant trigger than the other two candidates from $B_{\rm S/N}^{G}$ alone; it has a lower signal-to-noise ratio. However, the critical factor in our analysis responsible for reducing the significance of this event is the marginalization over the uncertainty in the (see Sec. \[sec:details\]). In Table \[tab:overview\], we see that the signal/noise Bayes factor falls from $B_{\rm S/N}^{G}=12.7$ for Gaussian noise to $B_{\rm S/N}=3.70$ using the Student-$t$ likelihood. Naively combining this Bayes factor (ignoring the effect of glitches) with a prior odds of $\ln \langle \xi \rangle = -7.2$ the resulting odds is less than unity, providing evidence against an astrophysical origin. The subsequent BCR and astrophysical odds (which include the effect of this prior odds) make minor corrections, but retain the overall conclusion. As discussed in Section \[sec:details\], we cannot neglect this marginalization if we want reliable odds. This underlines the importance of estimation (for further discussion, see also @Chatziioannou2019). In the future, with improved methods for evaluating the uncertainty in the (for example, building on the work of @Biscoveanu2020 or developing a joint and model method building on @2015PhRvD..91h4034L), we can reassess GW151216. We now discuss the prior sensitivity of our results. The dominant prior choice is that of the $\xi$-distribution. In this work, we use an astrophysical prior based on the rate of binary black hole events in the O1 and O2 observing runs. The factorisation of the prior odds in Eq.  allows us to update the odds based on differing prior assumptions. In order to change the conclusions for GW151216, one would need to increase $\langle \xi \rangle$ by a factor of $\sim 36$ Translating this into an updated merger rate, this would require a merger rate of $R'\sim 1600$ , much larger than the current uncertainty on the merger rate [@GWTC1]. Similarly, a merger rate which would make GW151012 not of astrophysical origin (based on an updated prior odds) would also require a merger rate well outside of the current uncertainty. This demonstrates that our results are not sensitive to the choice of prior odds, given the current uncertainty. Technically, the odds for GW151012 and GW150914 are biased because the data from these events is used to estimate the rate. However, we expect the error from this double-counting to be negligible. The other potential bias from our prior assumptions is the choice of point estimates for ${\xi_{g}^{\textsc{h}}}$ and ${\xi_{g}^{\textsc{l}}}$. To check how sensitive our results are to this choice, we rerun the analysis of GW151216 using ${\xi_{g}^{\textsc{h}}}= {\xi_{g}^{\textsc{l}}}=0$ and find that $\ln {\mathcal{O}}= -3.5$. This small shift from our calculated value confirms that our conclusion, that ${\ensuremath{p_{\rm astro}}}={\ensuremath{0.03}}{}$, is robust to the choice of glitch hyper prior. For GW150914 and GW151012, the astrophysical odds provide unequivocal evidence that these events are of astrophysical origin. Comparing the BCR and the ${\mathcal{O}}$ in Table \[tab:overview\] allows to assess the effect of the glitch hyper-model. For GW151012 and GW151216, only a small effect is observed, but for GW150914 the astrophysical odds is larger than the BCR by a factor of $\approx 7$. This demonstrates the ability of the astrophysical odds to increase our confidence in a signal based on how unlike the glitch population it is. We can also compare the BCR values derived in this work with that of @isi2018. For GW150914 and GW151012, they find $\ln{\rm BCR}$ values of 19.6 and 8.5 respectively; larger than the values found in this work (see Tab. \[tab:overview\]). This difference is caused by an unknown combination of the choice of tuning parameters, the narrower source parameter priors, the use of a precessing waveform, or the marginalization over the applied in this work. Given the significant impact of the marginalization over the , we suspect this is likely to dominate, but we cannot determine this without further investigation. ![Visualisation of candidates and triggers considered in this work for three Bayesian significance estimates: $B_{\rm coh,inc}$, the coherent vs. incoherent Bayes factor [@veitch2010]; the “Bayes Coherence Ratio” (BCR) an odds comparing a signal with both incoherent glitches and Gaussian noise [@isi2018]; and finally the astrophysical odds, Eq. . We label the three events analysed in this work in the figure and provide the numerical values for each in Tab. \[tab:overview\]. Blue circles and connecting curves are drawn for each of the [<span style="font-variant:small-caps;">Omicron</span>]{}triggers used to characterise the background. Pink crosses and dashed connecting curves mark the values for time-shifted [<span style="font-variant:small-caps;">Omicron</span>]{}trigger results—a background where we can be sure there are no coherent signals.[]{data-label="fig:plot"}](Significance_lines.png) To visualise our results for the three candidates and various realisations of a background, in Fig. \[fig:plot\], we show the evolution of candidates through three stages of Bayesian significance estimates. Individual candidates are labeled by their ID. In blue, are the [<span style="font-variant:small-caps;">Omicron</span>]{}triggers identified for each of the three epochs around each event; we show these together as no differences in behaviour per-epoch were found. All the significance estimates use evidence obtained by marginalizing over the uncertainty in the . For the [<span style="font-variant:small-caps;">Omicron</span>]{}trigger candidates, we see two distinct clusters: those with $\ln(B_{\rm coh, inc})\sim 0$ and those with $\ln(B_{\rm coh, inc}) < -1$. These can be understood as a cluster of candidates where the data is reasonably Gaussian in both detectors (thus tricking the coherent Bayes factor which only compares signal evidence against glitch evidence) and a cluster of candidates with a strong glitch in one detector resulting in a Bayes factor favouring the glitch hypothesis. When subsequently analysed with the BCR metric [@isi2018], the Gaussian cluster is weighted down because the BCR includes Gaussian noise in its alternative hypothesis. Finally, when applying the glitch hyper-prior a small correction is applied based on the likeness of the candidates to the glitch population. For the candidates initially in the $\ln(B_{\rm coh, inc}) < -1$ cluster, this results in a modest down-weighting: i.e. the odds having marginalized over the glitch population are slightly better at distinguishing glitches. In future work, we expect that a more detailed glitch model will yield further improvement in the ability of the odds to distinguish glitches. In pink, we also show the evolution of a set of [<span style="font-variant:small-caps;">Omicron</span>]{}triggers analysed with a time-slide. That is, we take the set of triggers and apply a shift between the Hanford and Livingston data. This ensures that the set of triggers do not contain coherent astrophysical signals. The figure demonstrates that the three Bayesian significance estimates perform equivalently for the [<span style="font-variant:small-caps;">Omicron</span>]{}triggers under a time-slide as they do without. Conclusion ========== We find that the marginal gravitational wave candidate GW151216 is not of astrophysical origin, ${\ensuremath{p_{\rm astro}}}={\ensuremath{0.03}}{}$. Our $p_\text{astro}$ estimate is smaller than that of the original detection claim ${\ensuremath{p_{\rm astro}}}=0.71$ [@Zackay2019], or the PyCBC analysis ${\ensuremath{p_{\rm astro}}}=0.18$ [@nitz2020]. Taken together with [@Huang2020], we urge the community to use caution when considering the astrophysical implications of this event. We also analyse GW150914, the loudest signal in the first advanced-LIGO observing run, and GW151012, a candidate first marked as marginal, but subsequently upgraded. We find overwhelming support that these are astrophysical signals. This work lays out the framework for applying the astrophysical odds [@bcr2] to a growing catalogue of gravitational-wave transients. In doing so, we seek to provide a single $p_\text{astro}$ for candidate events from multiple groups. Our results do not rely on the output of a search pipeline, and it is easy to see the assumptions that go into our calculations. It is also straightforward to update our significance estimates to keep pace with advances in noise modelling. Unlike traditional search methods, it does not use bootstrap realisations of the noise, but models the noise as incoherent-between-detector signals. In future work, we anticipate a number of improvements including: adding additional alternative models, for example, sine-Gaussians; improved waveforms; improved methods of estimating the noise ; and the addition of calibration uncertainty. Acknowledgements ================ The authors are grateful to Jess McIver, Will Farr, Laura Nutall, Sebastian Khan, Max Isi, Thomas Massinger, and Thomas Dent for useful comments during the development of this work. We acknowledge the support of the Australian Research Council through grants CE170100004, FT150100281, and DP180103155. This research has made use of data, and web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. The authors are grateful for computational resources provided by the LIGO Laboratory and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459. We use the `bilby` [@bilby] inference package and the `dynesty` [@dynesty] Nested Sampling algorithm. \[lastpage\] [^1]: E-mail: [email protected] [^2]: The coalescence time is defined differently for different waveforms, but it is approximately synonymous with time of peak gravitational-wave amplitude. While a gravitational waveform can span several segments, the coalescence time always falls in just one segment. [^3]: These error bars don’t include systematic uncertainty associated with the cosmological model, which might increase the uncertainty by a factor of $\sim2$.

--- abstract: 'Finite–temperature local dynamical spin correlations $S_{nn}(\omega)$ are studied numerically within the random spin–$1/2$ antiferromagnetic Heisenberg chain. The aim is to explain measured NMR spin–lattice relaxation times in , which is the realization of a random spin chain. In agreement with experiments we find that the distribution of relaxation times within the model shows a very large span similar to the stretched–exponential form. The distribution is strongly reduced with increasing $T$, but stays finite also in the high–$T$ limit. Anomalous dynamical correlations can be associated to the random singlet concept but not directly to static quantities. Our results also reveal the crucial role of the spin anisotropy (interaction), since the behavior is in contrast with the ones for XX model, where we do not find any significant $T$ dependence of the distribution.' author: - 'J. Herbrych$^{1}$' - 'J. Kokalj$^{1}$' - 'P. Prelovšek$^{1,2}$' title: Local spin relaxation within the random Heisenberg chain --- One–dimensional (1D) quantum spin systems with random exchange couplings reveal interesting phenomena fundamentally different from the behavior of ordered chains. Since the seminal studies of antiferromagnetic (AFM) random Heisenberg chains (RHC) by Dasgupta and Ma [@ma1979; @dasgupta1980] using the renormalization–group approach and further development by Fisher [@fisher1994], it has been recognized that the quenched disorder of exchange couplings $J$ leads at lowest energies to the formation of random singlets with vanishing effective $\tilde J$ at large distances. The consequence for the uniform static susceptibility $\chi^0$ is the singular Curie–type temperature ($T$) dependence, dominated by nearly uncoupled spins at low–$T$ and confirmed by numerical studies of model systems [@hirsch1980], as well by measurements of $\chi^0(T)$ on the class of materials being the realizations of RHC physics, in particular the mixed system [@zheludev2007; @shiroka2011; @shiroka2012]. Recent measurements of NMR spin–lattice relaxation times $T_1$ in [@shiroka2011] reveal a broad distribution of different $T_1$ resulting in a nonexponential magnetization decay being rather of a stretched–exponential form. In connection to this the most remarkable is the strong $T$ dependence of the $T_1$ span becoming progressively large and the corresponding distribution non–Gaussian at low–$T$. It is evident that in a random system $T_1$, which is predominantly testing the local spin correlation function $S_{nn}(\omega \to 0)$, becomes site $n$ dependent and we are therefore dealing with the distribution of $T_{1n}$ leading to a nonexponential magnetization decay. Theoretically the behavior of dynamical spin correlations in RHC has not been adequately addressed so far. There is (to our knowledge) no established model result and moreover no clear prediction for the behavior of dynamical ($\omega \ne 0$) spin correlations at $T>0$ in RHC. It seems plausible that the low–$T$ behavior should follow from the random–singlet concept and its scaling properties, discussed within the framework of the renormalization–group approaches [@dasgupta1980; @fisher1994; @westerberg1997; @motrunich2001]. Still, the relation to singular static correlations as evidenced, e.g., by $\chi^0(T )$ diverging at $T\to 0$, and low- $\omega$ dynamical correlations is far from clear. One open question is also the qualitative similarity to the behavior of the random anisotropic XX chain invoked in several studies [@bulaevskii1972; @hirsch1980; @westerberg1997; @motrunich2001]. The latter system is equivalent to more elaborated problem of noninteracting (NI) spinless fermions with the off–diagonal (hopping) disorder [@theodorou1976; @eggarter1978]. In the following we present results for the dynamical local spin correlation function $S_{nn}(\omega)$, in particular for its limit $s=S_{nn}(\omega \to 0)$ relevant for the NMR $T_1$, within the AFM RHC model for $T>0$, obtained using the numerical method based on the density–matrix renormalization group (DMRG) approach [@kokalj2009]. At high $T \geq J$, distribution of $s$ reveals a modest but finite width qualitatively similar both for the isotropic and the XX chain. On the other hand, the low–$T$ variation established numerically is essentially different. While for the XX chain there is no significant $T$ dependence, results for the isotropic case reveal at low $T \ll J$ a very large span of $s$ values and corresponding $T_{1n}$, qualitatively and even quantitatively consistent with NMR experiments [@shiroka2011]. We study in the following the 1D spin–$1/2$ model representing the AFM RHC, $$H=\sum_{i} J_i\left(S^x_i S^x_{i+1}+S^y_i S^y_{i+1}+\Delta S^z_i S^z_{i+1} \right)\,, \label{ham}$$ where $J_i$ are random and we will assume their distribution as uncorrelated and uniform in the interval $J-\delta J\le J_i\le J+\delta J$, with the width $\delta J< J$ as the parameter. In the following we will consider predominantly the isotropic case $\Delta=1$, but as well the anisotropic XX case with $\Delta=0$. The chain is of the length $L$ with open boundary conditions (o.b.c.) as useful for the DMRG method. We further on use $J=1$ as the unit of energy as well as $\hbar=k_B=1$. Our aim is to analyse the local spin dynamics in connection with the NMR spin–lattice relaxation [@shiroka2011]. In a homogeneous system the corresponding relaxation rate $1/T_1$ is expressed in terms of the $q$–dependent spin correlation function, $$\frac{1}{T_{1}}=\sum_{q \alpha } A^2_\alpha(q) S^{\alpha \alpha}(q,\omega\to 0)\,, \label{st1}$$ where $A^2_\alpha(q)$ involve hyperfine interactions and NMR form factors [@shiroka2011]. In the Supplement [@supp] we show that the dominant dynamical $\omega \to 0$ contribution at low–$T$ is coming from the regime $q \sim \pi$. Therefore the variation $A^2_\alpha(q)$ is not essential and the rate depends only on the local spin correlation function $1/T_{1} \propto S^{zz}_{\mathrm{loc}}(\omega \to 0)$. In a system with quenched disorder the relaxation time becomes site dependent, i.e. $T_{1n}$, hence we study in the following the local correlations $S_{nn}(\omega)$ and the distribution of local limits $s=S_{nn}(\omega\to 0)$ and related relaxation times $\tau=1/s$ where $$S_{nn}(\omega)=\frac{1}{\pi}\mathrm{Re}\int\limits_{0}^{\infty}\mathrm{d}t\, {\rm e}^{\imath \omega t}\langle S^z_n(t)S^z_n(0)\rangle\,. \label{auco}$$ In order to reduce finite–size effects we study large systems employing the finite–temperature dynamical DMRG (FTD–DMRG) [@kokalj2009; @schollwock2005; @prelovsek2013] method to evaluate the dynamical $S_{nn}(\omega)$, Eq. . To reduce edge effects we choose the local site $n$ to be in the middle of the chain, $n=L/2$. The distribution of $s$ is then calculated with $N_r \sim 10^3$ different realizations of the system with random $J_i$. More technical detail on the calculation can be found in the Supplement [@supp]. We start the presentation of results with typical examples of $S_{nn}(\omega)$. In Fig. \[spect\] we show calculated spectra for system with $L=80$ sites, $T=0.5$, $\Delta=1$, and three different realizations of $J_i$, i.e. the homogeneous system with $J_i=1$ and two configurations with $\delta J= 0.7$. Spectra for the uniform system are broad and regular at $\omega\sim 0$ agreeing with those obtained with other methods [@naef1999], while $S_{nn}(\omega)$ for random case strongly depend apart from $\delta J$ also on the local $J_i, i \sim n$. In particular, spectra with both $J_{n-1}$ and $J_{n}$ small have large amplitude at the relevant $\omega \sim 0$, while spectra with one large $J_{n-1}$ or $J_{n}$ have most of the weight at high–$\omega$ and small amplitude at $\omega \sim 0$ (elaborated further in the conclusions). For the following analysis it is important that $s=S_{nn}(\omega \to 0)$ can be extracted reliably. ![(Color online) Dynamical local spin correlations $S_{nn}(\omega)$ for different configurations of $J_i$. Shown are spectra for the homogeneous case $\delta J=0$ and two configurations with $\delta J=0.7$, calculated for $T=0.5$ and $L=80$ sites.[]{data-label="spect"}](spec.eps){width="1.0\columnwidth"} [*Results for $T \geq J$:*]{} Before displaying results for most interesting $T<J$ regime, we note that even at $T \gg J$ one cannot expect a well defined $\tau=\tau_0$ but rather a distribution of values. One can understand this by studying analytically local frequency moments within the high–$T$ expansion and using the Mori’s continued fraction representation [@mori1965] with the Gaussian–type truncation at the level of $l>3$ [@tommet1975; @oitmaa1984] (see [@supp] for more details). In the inset of Fig. \[momex\] we present the high–$T$ result for $\mathrm{PDF}(s)$ and compare it with the numerical results evaluated for $T=1$. Several conclusions can be drawn from results presented on Fig. \[momex\]: (a) The agreement of PDF$(s)$ obtained via the analytical approach and numerical FTD–DMRG method is satisfactory having the origin in quite broad and featureless spectra $S_{nn}(\omega)$ at $T \geq J$. Still we note that median value of $s$ ($s_{\mathrm{med}}$) differ between both approaches and that for $T\gg J$ (unlike $T \leq J$) contribution of $q\to 0$ can become essential [@supp; @sandvik1997]. (b) PDF$(s)$ becomes quite asymmetric and broad for $\delta J\ge 0.5$. (c) Consequently, also the distribution of local relaxation times PDF$(\tau)$ has finite but modest width for $T \to \infty$. This seems in a qualitative agreement with NMR data for , where the width was hardly detected at high–$T$ [@shiroka2011]. ![(Color online) Probability distribution function of local relaxation rates PDF$(s)$ at $T\gg 1$ evaluated using the moment expansion for different $\delta J$. Inset: Comparison of analytical and FTD–DMRG result for $\delta J=0.5$, $L=20$ with full basis and averaged over $N_{r}=10^3$ realizations.[]{data-label="momex"}](momex.eps){width="1.0\columnwidth"} [*Results for $T < J$:*]{} More challenging is the low–$T$ regime which we study using the FTD–DMRG method for typically $L=80$ and $N_r\sim 10^3$. Besides the isotropic case $(\Delta =1 )$, we investigate for comparison also the XX model ($\Delta=0$). As the model of NI fermions with the off–diagonal disorder [@bulaevskii1972; @theodorou1976] it can be easily studied via full diagonalization on much longer chains with $L \sim 16000$. PDF for $T<J$ can become very broad and asymmetric. Hence, we rather present results as the cumulative distribution $\mathrm{CDF}(x)=\int_{0}^{x}\mathrm{d}y\,\mathrm{PDF}(y)$. Further we rescale $x$ values to the median defined as $\mathrm{CDF}(x_{\mathrm{med}})=0.5$. Results for CDF$(s)$ are presented in Fig. \[cdf\]. Note that $\mathrm{PDF}(\tau)=\mathrm{PDF}(s)/\tau^{2}$. Panels in Fig. \[cdf\] represent results for the isotropic case $\Delta=1$ with (a) fixed $T=0.2$ and varying $\delta J=0.1-0.9$, while in (b) $\delta J=0.7$ is fixed and $T=0.1 - 0.5$. Inset of Fig. \[cdf\]b displays the $T$ dependence (for fixed $\delta J=0.7$) of CDF for the XX chain. ![(Color online) Cumulative distribution function of $s$. Shown are FTD–DMRG results for $\Delta=1$: (a) for fixed $T=0.2$ and various $\delta J$, (b) for fixed $\delta J=0.7$ and various $T \leq 0.5 $. Inset of (b):full diagonalization results for $\Delta=0$, $\delta J=0.7$ and various $T$.[]{data-label="cdf"}](cdf.eps){width="1.0\columnwidth"} We first note that within the XX chain CDF$(s)$ are essentially $T$ independent. This appears as quite a contrast to, e.g., static $\chi^0(T)$ which exhibits a divergence at $T \to 0$ [@hirsch1980; @supp]. Results for the isotropic case $\Delta=1$ in Figs. \[cdf\]a,b are evidently different. The span in CDF becomes very large (note the logarithmic scale) either by increasing $\delta J$ at fixed $T$ or even more by decreasing $T$ at fixed $\delta J$. From the corresponding PDF one can calculate the relaxation function $R(t)=\int \mathrm{d}s\,\mathrm{PDF}(s){\rm e}^{-ts}$, which is in fact the quantity measured in the NMR as a time–dependent magnetization recovery [@shiroka2011]. As in experiment the large span in our results for low–$T$ can be captured by a stretched exponential form, $R(t)\approx\exp[-\left(t/\tau_0 \right)^{\Gamma}]$, where $\Gamma$ and $\tau_0$ are parameters to be fitted for particular PDF$(s)$ and corresponding $R(t)$. It is evident that $\Gamma \ll 1$ means large deviations from the Gaussian–like form, and in particular very pronounced tails in PDF$(s)$, both for $s \gg s_{\mathrm{med}}$ as well as a singular variation for $s \to 0$. In the latter regime $1/\tau_0$ can deviate substantially from average of local $1/\tau$. It should be also noted that stretched exponential form, is the simplest one capturing the large span of $s$ values. It is also used in the experimental analysis [@shiroka2011], but the corresponding PDF$(s)$ reveal somewhat enhanced tails for $s > s_{\mathrm{med}}$ relative to calculated ones in Fig. \[cdf\]a,b, and the opposite trend for $s < s_{\mathrm{med}}$. This suggests possible improvements and description beyond stretched exponential form, which we leave as a future challenge. More details can be found in the Supplement [@supp]. Results for the fitted exponent $\Gamma(T)$ for $\Delta=1$ as extracted from numerical PDF$(s)$ for various $\delta J$ are shown in Fig. \[fit\]a. They confirm experimental observation [@shiroka2011] of increasing deviations from simple exponential variation ($\Gamma=1$) for $T \ll J$. While for $T > J$, $\Gamma \lesssim 1$ for modest $\delta J < 0.7$, low–$T$ values can reach even $\Gamma < 0.5$ at lowest reachable $T < 0.1$. Note that in such a case values of $s$ are distributed over several orders of magnitude. Of interest for the comparison with experiment is also the $T$ variation of fitted $1/\tau_0$. Results are again essentially different for $\Delta=0$ and $\Delta=1$. $\tau_0$ (as well $s_{\mathrm{med}}$) for $\Delta=0$ follows well the Korringa law $1/\tau_0 \propto T$ for $T<0.5$, as usual for the system of NI fermions with a constant density of states (DOS) (divergent DOS at $E \to 0$ could induce a logarithmic correction). On the other hand, for the isotropic ($\Delta=1$) chain with no randomness $\tau_0=\tau$ it should follow $1/\tau \sim \mathrm{const.}$ for $T <J$ [@sandvik1995; @naef1999]. Similar behavior is observed for weak disorder $\delta J=0.1$ as shown in Fig. \[fit\]b. However, with increasing randomness $\delta J$, $1/\tau_0$ becomes more $T$ dependent and increases with $T$. Such $T$ dependence in the RHC of $1/\tau_0$ is, although in agreement with experiment, in apparent contrast with diverging $\chi^0(T\to 0)$. This remarkable dichotomy between static and dynamical $\omega \to 0$ behaviour can be reconciled by the observation that in a random system $S(q,\omega\sim 0)$ reveals besides the regular part also a delta peak at $\omega=0$ (not entering $1/T_1$), which can be traced back to diagonal matrix elements [@supp] being an indication of a nonergodic behaviour (at least at low–$T$). Note that more frequently studied static $S(q)$ (equal-time correlation) [@hoyos2007] represents a sum rule containing both parts. Also, the relation $\chi^0(T)=S(q=0)/T$ in spite of divergent $\chi(T \to 0)$ leads to vanishing $S(q=0)$ at $T\to 0$ only slower than linearly [@supp; @hoyos2007]. ![(Color online) (a) Exponent $\Gamma$ vs. $T$ obtained from PDF$(s)$ data for different $\delta J$ and isotropic case $\Delta=1$. (b) $T$ dependence of fitted $1/\tau_0$ for $\Delta=1$ and different $\delta J$.[]{data-label="fit"}](fit.eps){width="1.0\columnwidth"} As a partial summary of our results, we comment on the relation to the experiment on [@zheludev2007; @shiroka2011]. The spin chain is in this case assumed to be random mixture of two different values $J_i=280$ K, $580$ K, which correspond roughly to our $\delta J\simeq0.6$ (fixing the same effective width) and $J=430$ K. Taking these values, our results for $\Gamma(T)$ as well as $1/\tau_0(T)$ agree well with experiment. In particular we note that at lowest $T \ll J$ our calculated $\Gamma \sim 0.5$ for $\delta J=0.6$ matches the measured one. Some discrepancy appears to be a steeper increase of measured $\Gamma(T)$ towards the limiting $\Gamma=1$ coinciding with observed very narrow PDF$(\tau)$ which remains of finite width in our results even for $T \to \infty$ as seen in Fig. \[momex\]. As far as calculated $1/\tau_0(T)$ vs. NMR experiment is concerned we note that taken into account the normalization of average $J$ disordered system reveals at $T \to 0$ smaller $1/\tau_0$ than a pure one consistent with the experiment [@shiroka2011]. In agreement with the experimental analysis is also strong $T$ variation of $1/\tau_0$ at low–$T$ in disordered system in contrast to a pure one. Our results on the local spin relaxation $S_{nn}(\omega)$ and in particular its $T$ dependence cannot be directly explained within the framework of existing theoretical studies and scaling approaches to RHC [@dasgupta1980; @fisher1994; @motrunich2001]. Our study clearly shows the qualitative difference in the behavior of the XX chain and the isotropic RHC. While in the former model mapped on NI electrons, $T$ does not play any significant role on PDF$(s)$ as seen in inset of Fig. \[cdf\]b, $\Delta=1$ case shows strong variation with $T \ll J$. It is plausible that the difference comes from the interaction and many–body character involved in the isotropic RHC. To account for that we design in the following a simple qualitative argument. The behavior of $S_{nn}(\omega\sim 0)$ at low–$T$ is dominated by transitions between low–lying singlet and triplet states which become in a RHC nearly degenerate following the scaling arguments with effective coupling $\tilde J \to 0$ for more distant spins and reflected in diverging $\chi^0(T \to 0)$ [@supp; @dasgupta1980; @hirsch1980; @fisher1994]. Such transitions are relevant at $\omega \to 0$ behavior as presented in Fig. \[spect\]. Moreover, local $S_{nn}(\omega \sim 0)$ exhibit large spread due to the variations in the local environment. Let us for simplicity consider the symmetric Heisenberg model on four sites (with o.b.c.) with a stronger central bond $J_2 \gg J_1=J_3$ and $J=(J_1+J_2+J_3)/3$. It is then straightforward to show that the lowest singlet–triplet splitting is strongly reduced, i.e. $\Delta E \propto \eta^2 J$ where $\eta=J_1/J_2 $. Within the same model one can evaluate also the ratio between two different amplitudes of $S_{nn}(\omega \sim \Delta E)=A_{nn}\delta(\omega-\Delta E)$, on sites $n=1,2$ neighboring the weak and strong bond, $$\frac{1}{W}= \frac{A_{22}}{A_{11}}= \frac{ |\langle \Psi_t |S^z_2|\Psi_s\rangle |^2} {|\langle \Psi_t |S^z_1|\Psi_s\rangle |^2}\sim \eta^2\,. \label{3s}$$ The relation shows that the span between largest and smallest amplitudes increases as $W\propto 1/\eta^2 \propto 1/\Delta E$. Continuing in the same manner the scaling procedure for AFM RHC [@dasgupta1980; @fisher1994] for a long chain the smallest effective coupling between further spins $\tilde J$ vanishes at $T=0$ and $\Delta E \propto \tilde J \to 0$, so that one expects $W \to \infty$ for $T \to 0$. On the other hand, for $T>0$ the scaling should be cut off at $\tilde J \sim T$ at least for $\Delta=1$, finally leading to the strong $W(T)$ dependence ($W \propto 1/T$). In the summary, we have reproduced qualitatively main experimental NMR results on mixed system including anomalously wide distribution of relaxation rates, together with $T$ dependencies of experimental parameters ($1/\tau_0$, $\Gamma$) and provide microscopic explanation with the help of the random-singlet framework. Our qualitative conclusions on the RHC do not change by changing $S^z_{\mathrm{tot}}$ (adding finite field in the fermionic language) or even reducing $\Delta <1$ provided that $\Delta > 0$ (see Supplement [@supp]). We also comment on striking difference between static and dynamic quantities and observed deviations from stretched exponential phenomenology. [23]{} S.-K. Ma, C. Dasgupta, and C.-k. Hu, Phys. Rev. Lett. **43**, 1434 (1979). C. Dasgupta and S.-k. Ma, Phys. Rev. B **22**, 1305 (1980). D. S. Fisher, Phys. Rev. B **50**, 3799 (1994). J. E. Hirsch, Phys. Rev. B **22**, 5355 (1980). A. Zheludev, T. Masuda, G. Dhalenne, A. Revcolevschi, C. Frost, and T. Perring, Phys. Rev. B **75**, 054409 (2007). T. Shiroka, F. Casola, V. Glazkov, A. Zheludev, K. Prša, H.–R. Ott, and J. Mesot, Phys. Rev. Lett. **106**, 137202 (2011). F. Casola, T. Shiroka, V. Glazkov, A. Feiguin, G. Dhalenne, A. Revcolevschi, A. Zheludev, H.-R. Ott, and J. Mesot, Phys. Rev. B **86**, 165111 (2012). E. Westerberg, A. Furusaki, M. Sigrist, and P. A. Lee, Phys. Rev. B **55**, 12578 (1997). O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B **63**, 134424 (2001). L. N. Bulaevskii, Zh. Eksp. Teor. Fiz. **62**, 725 (1972). G. Theodorou and M. H. Cohen, Phys. Rev. B **13**, 4597 (1976). T. P. Eggarter and R. Riedinger, Phys. Rev. B **18**, 569 (1978). J. Kokalj and P. Prelovšek, Phys. Rev. B **80**, 205117 (2009). See Suplemetary material for more details. For a recent review, see P. Prelovšek and J. Bonča, in [*Strongly Correlated Systems - Numerical Methods*]{}, edited by A. Avella and F. Mancini (Springer Series in Solid–State Sciences 176, Berlin, 2013), pp. 1–29. U. Schollwöck, Rev. Mod. Phys. **77**, 259 (2005). F. Naef, X. Wang, X. Zotos, and W. von der Linden, Phys. Rev B **60**, 359 (1999). H. Mori, Prog. Theor. Phys. **34**, 399 (1965). J. Oitmaa, M. Plischke, and T. A. Winchester, Phys. Rev. B **29**, 1321 (1984). T. N. Tommet and D. L. Huber, Phys. Rev. B **11**, 1971 (1975). O. A. Starykh, A. W. Sandvik, and R. R. P. Singh, Phys. Rev. B **55**, 14953 (1997). A. W. Sandvik, Phys. Rev. B **52**, R9831 (1995). J. A. Hoyos, A. P. Vieira, N. Laflorencie and E. Miranda, Phys. Rev. B **76**, 174425 (2007). **[ for “Local spin relaxation within random Heisenberg chain”]{}** = In this section we present in more detail the numerical method, finite–temperature dynamical DMRG (FTD–DMRG). The method is a variation of a zero temperature ($T=0$) DMRG [@s_white1992; @s_schollwock2005], with targeting of the ground state or ground state density matrix $\rho_0=|0\rangle\langle 0|$ generalized to targeting of the finite–$T$ density matrix $\rho^\beta= \frac{1}{Z}\sum_n |n\rangle \textrm{e}^{-\beta H}\langle n|$, [@s_kokalj2009; @s_kokalj2010; @s_prelovsek2011]. Similar generalization is applied to targeting of the operator on the ground state. From such targets, the reduced density matrix is calculated and then truncated in the standard DMRG like manner for basis optimization. All quantities, that need to be evaluated at finite–$T$, are calculated with the use of finite–temperature Lanczos method (FTLM) [@s_jaklic2000; @s_prelovsek2011], which in FTD–DMRG replaces $T=0$ Lanczos method used in the standard DMRG algorithm. The method is most efficient at low–$T$ and for low frequencies, where basis can be efficiently truncated and only small portion ($M$ basis states) of the whole basis for block can be kept. In this regime large system sizes can be reached. The truncation error becomes larger at higher–$T$, and one needs to either use larger $M$, or reduce system size, which is legitimate approach, since finite size effects are smaller at higher–$T$ due to reduced correlation lengths. We typically keep $M \sim 200$ basis states in the DMRG block and use systems with length $L \sim 80$ at low $T < J/2$, while for $T > J/2$ smaller systems are employed down to $L \sim 20$, for which full basis can be used. We stress that randomness of $J_i$ reduces the truncation error since some larger values of $J_i$ induce strong tension for formation of a local singlet and therefore in turn reduces the entanglement on larger distances. Also the local operator, acting on the middle of the chain, where the local one site basis is not truncated, helps in this respect. The quenched random $J_i$ are introduced into the DMRG procedure at the beginning of [*finite*]{} algorithm. [*Infinite*]{} algorithm is preformed for homogeneous system $J_i=J$ and the randomness of $J_i$ is introduced in the first sweep (see Fig. \[sche\] for schematic presentation). In this way the preparation of the basis in the [ *infinite*]{} algorithm is performed just once and for all realizations of $J_i$–s, while larger number of sweeps (usually $\sim 5$) is needed to converge the basis within the [*finite*]{} algorithm for random $J_i$. After [*finite*]{} algorithm local dynamical spin structure factor $S_{nn}(\omega)$ at desired $T$ is calculated for the site in the middle of the chain within [*measurements*]{} part of DMRG procedure. ![(Color online) Schematic ($L=6$) representation of the beginning of the sweeping method in [*finite*]{}–DMRG algorithm, in which randomness is introduced. Open circle represents site of a local operator used to calculated local spin correlation function in the [*measurement*]{} part of the DMRG method.[]{data-label="sche"}](fin_rand.eps){width="0.75\columnwidth"} Any spectra on finite system consists of separate $\delta$–peaks, which we broaden by changing them into Gaussian with small broadening $\delta=0.05$ and in this way obtain a smooth spectra. Since NMR relaxation rate is related to $S_{nn}(\omega \to 0)$, we are interested in the limit $\omega \to 0$, which should be contrasted with the singular $S_{nn}(\omega=0)$. In order to avoid the problem of diagonal elements and keeping $\omega \neq 0$ we perform the evaluation of $S_{nn}(\omega \to 0)$ in the magnetization sector $S^z_{\mathrm{tot}}=0$, which in terms of spinless fermions corresponds to the canonical ensemble (in the thermodynamic limit canonical and grand canonical give the same result) and we remove $\delta(\omega=0)$ peak (see also Section ). Diagonal elements are however essential when evaluating static uniform susceptibility $\chi^0(T)=\langle(S^z_{\mathrm{tot}})^2 \rangle/(LT)$, which we show in Fig. \[chi\]. It has been argued [@s_hirsch1980; @s_fisher1994; @s_zheludev2007], that in 1D random Heisenberg chain the density of low lying excitation is strongly increased, which is observed in diverging $\chi^0(T)$ for $T\to 0$. Our numerical results show similar behavior (see Fig. \[chi\]), which agrees also with experiment [@s_shiroka2011; @s_masuda2004]. ![(Color online) Static spin susceptibility $\chi^0(T)$ vs. temperature $T$ for various randomness $\delta J$. For random system ($\delta J \neq 0$) $\chi^0(T)$ is strongly increased at low–$T$ and agrees with the random singlet [@s_fisher1994] prediction (black dotted line). Sudden drop of $\chi^0(T)$ at low–$T$ shown for $\delta J=0$ and $\delta J=0.4$ represents opening of finite–size gap, which is strongly reduced in the random (shown for $\delta J=0.4$) system. Results are obtained with the finite–temperature Lanczos method [@s_jaklic2000; @s_prelovsek2011] on $L=24$ sites.[]{data-label="chi"}](chi_ftlm.eps){width="1.0\columnwidth"} Increased number of low–lying excitations (see Fig. \[chi\]) also reduces the finite size effect, since, e.g., finite size gap is reduced, and in this way also the temperature $T_\textrm{fs}$, below which the finite size effects become important. Therefore, smaller $T$ can be numerically reached in a random system. = The local spin correlation function can be related to the (local) dynamical spin susceptibility by relation $$S_{nn}(\omega)\left[1-\exp(-\beta\omega)\right]=\chi_{nn}''(\omega)\,, \label{fdt}$$ with $$\chi_{nn}(\omega)=\imath\int\limits_{0}^{\infty}\mathrm{d}t\, \mathrm{e}^{\imath\omega t}\langle[S^z_n(t),S^z_n(0)]\rangle\,.$$ Taking the high–$T$ limit ($\beta\to0$) of Eq.  one gets $\beta S_{nn}(\omega)=\chi_{nn}''(\omega)/\omega$, which is so–called relaxation function - symmetric with respect to $\omega=0$, non–negative function. Note that due to symmetric form of relaxation function all odd frequency moments, $m_{ln}$, are equal to zero. The local spin correlation function can by expressed by the Mori’s continued fraction representation [@s_mori1965]: $$\hat{S}_{nn}(z=\imath\omega)= \cfrac{\delta_{0n}}{z+ \cfrac{\delta_{1n}}{z+ \cfrac{\delta_{2n}}{z+ \cdots}}}\,, \label{moricf}$$ where coefficient $\delta_{ln}$ are cumulants of $S_{nn}(\omega)$, i.e. $\delta_{0n}=m_{0n}$, $\delta_{1n}=m_{2n}/m_{0n}$, $\delta_{2n}=m_{4n}/m_{2n}-m_{2n}/m_{0n}$. $m_{ln}$ are frequency moments of the local spectra, $m_{ln}=\int\mathrm{d}\omega\,\omega^{l}S_{nn}(\omega)$. For $l>3$ we chose a truncation $\zeta_n=\delta_{3n}/(z+\dots)$, which assumes [@s_tommet1975; @s_oitmaa1984] a Gaussian–like decay of correlation function, i.e. $\zeta_n=\sqrt{2/\pi}(\delta_{1n}+\delta_{2n})/\delta_{2n}^{3/2}$. The $S_{nn}(\omega)$ can be recovered from Eq.  by the relation $S_{nn}(\omega)=\mathrm{Re}[\hat{S}_{nn}(z=\imath\omega)]/\pi$, leading to $$S_{nn}(\omega)=\frac{1}{\pi}\frac{\zeta_{n}\delta_{0n}\delta_{1n}\delta_{2n}} {\left[\omega\zeta_{n}\left(\omega^{2}-\delta_{1n}-\delta_{2n}\right)\right]^2 +\left(\omega^{2}-\delta_{1n}\right)^2}\,. \label{hislo}$$ Note that Eq.  gives the first three nonzero ($l=0,2,4$) frequency moments $m_{ln}$ correctly, independent of a choice of $\zeta_n$. Frequency moments $m_{ln}$ of $S_{nn}(\omega)$ can be evaluated analytically for $T=\infty$, e.g., $m_{0n}=\langle S^{z}_{n}S^{z}_{n}\rangle$, $m_{2n}=\langle [H,S^{z}_{n}][H,S^{z}_{n}]\rangle$, etc. For zero magnetization, $S^{z}_{\mathrm{tot}}=(L_{\uparrow}-L_{\downarrow})/2L=0$, where $L_{\uparrow}$ ($L_{\downarrow}$) is number of up (down) spins, the first three nonzero moments of the order of ${\cal O}(\beta)$ are: $$\begin{aligned} m_{0n}=\frac{1}{4}\,,\qquad m_{2n}=\frac{J^2_{n-1}+J^2_{n}}{8}\,, \nonumber\\ m_{4n}=\frac{1+\Delta^2}{32}\left(J^{2}_{n-2}J^{2}_{n-1}+J^{2}_{n}J^{2}_{n+1}\right) \nonumber\\ +\frac{3+2\Delta^2}{32}\left(J^{4}_{n-1}+J^{4}_{n}\right) +\frac{7+2\Delta^2}{32}J^{2}_{n-1}J^{2}_{n}\,.\end{aligned}$$ ![(Color online) Comparison of $S_{nn}(\omega)$ between analytical high–$T$ expansion result and numerical FTD–DMRG result ($L=20$, $T=100$, full basis) for three realization of $J_i$.[]{data-label="spechT"}](spec_highT.eps){width="1.0\columnwidth"} In Fig. \[spechT\] we present comparison of high–$T$ expansion result and FTD–DMRG result ($L=20$, $T=100$, full basis) for $S_{nn}(\omega)$ and three realizations of $J_i$. One can see that the agreement is good for actual finite size system. It should be, however, noted that $q \to 0$ contribution (leading to finite size corrections) can become essential for $T\gg J$ [@s_sandvik1997]. As a final remark of this section we comment on the probability distribution function (PDF) of $s=S_{nn}(\omega\to 0)$ presented in Fig. 2 in the main text. Assuming the uniform distribution of $J_i$, $i=n-2,\cdots,n+1$ the PDF$(s)$ can be can be generated from expression $$s=S_{nn}(\omega\to0)= \frac{1}{\sqrt{8\pi^3}}\frac{\delta_{1n}+\delta_{2n}}{\delta_{1n}\delta_{2n}^{1/2}}\,.$$ The PDF-s presented in Fig. 2 (main text) where obtained from $N_r=10^6$ realizations of $J_i$. = ![(Color online) Cumulative distribution function of relaxation rates $s$ and times $\tau=1/s$. Shown are FTD–DMRG results for various $T \leq 0.5$ and: (a) for $\Delta=0.5$, $S^z_{\mathrm{tot}}=0$ and $\delta J=0.7$, (b) for $\Delta=1$, $S^z_{\mathrm{tot}}=1/4$ and $\delta J=0.7$, while (c) are diagonalization results for $\Delta=0$, $\delta J=0.7$ and $h/J=0.5$.[]{data-label="cdf_supp"}](cdf_supp.eps){width="1.0\columnwidth"} In Fig. \[cdf\_supp\] we show that our main conclusions stay valid also in a more general case, such as for $\Delta=0.5<1$ and for finite magnetization ($S_{tot}^z=1/4$), where considerable $T$ dependence of distribution with large spread is observed. In the last panel of Fig. \[cdf\_supp\] we show that the distribution for noninteracting case (XX model) stays $T$ independent even in a finite magnetic field $h$ or for finite magnetization. Wavevector resolved spin structure factor ========================================== Looking at the diverging uniform ($q=0$) susceptibility $\chi^0(T)$ as $T\to 0$ (Fig. \[chi\]) intuitively suggests large low–$q$ response and in turn increasing contribution of $S(q\sim0,\omega)$ to the spin relaxation rate $1/T_1$ as $T\to 0$. This is not what is observed, since we see no increase of $1/\tau_0$ (Fig. 4b in the main text) as $T\to 0$, but instead $1/\tau_0$ decreases with decreasing $T$, which is in agreement also with experimental data (Ref. [@s_shiroka2011], Fig. 3a). This dichotomy can be partly understood by exploring the connection between static uniform spin susceptibility $\chi^0(T)$ with the static spin structure factor (equal-time correlation) $S(q)$, representing also the frequency integral of dynamical spin structure factor $S(q)=\int _{-\infty}^{\infty}\mathrm{d}\omega\,S(q,\omega)$. The connection $\chi^0(T)= S(q=0)/T$ together with the low–$T$ RG results (see Fig. \[chi\]) $\chi^0(T)=A/[T \ln^2(J_\textrm{max}/T)]$ leads to $S(q=0)=A/[ \ln^2(J_\textrm{max}/T)]$. This shows that $S(q=0)$ goes to 0 as $T\to 0$ and is not diverging, rather its slow logarithmic approach to $0$ (in contrast to linear in $T$ decrease for homogeneous system). This is in agreement with results in Fig. 15 in Ref. [@s_hoyos2007], which show that $S(q)$ at $T=0$ goes to 0 as $q\to 0$ and is only slightly increased by randomness for $q\sim 0$. Another remarkable property of RHC can be seen in the difference between dynamic ($\omega > 0$) properties, e.g. $S(q,\omega\to 0)$, and strictly $\omega=0$ contribution. This is shown in the lower panel of Fig. \[s\_fig\_sqw\], which shows that $S(q,\omega)$ is singular at $\omega=0$ since it consists besides the continuous background (regular part) also of a distinct delta peak at $\omega=0$ (see Fig. \[s\_fig\_sqw\]b). This peak is non–dispersive and is the signature of non–ergodicity in the random system (absence of diffusion) at least for low–$T$. Similar peak is observed even in a random non–interacting electron system at finite–$T$ (not presented). In our analysis of spin relaxation for which $\omega \to 0$ is relevant, the $\omega=0$ peak was excluded (see Fig. \[s\_fig\_sqw\]c and also Section above). Distinction between strictly $\omega=0$ and $\omega>0$ properties can be traced back to the difference between diagonal elements, e.g. $\langle (S^z_{tot})^2\rangle$ determining $\chi^0$ with only total spin $S>0$ (triplet) states contributing, and non–diagonal elements describing the transitions between, e.g. singlet and triplet states, which are relevant for dynamical ($\omega>0$) properties. ![(Color online) (a) Wavevector resolved $S(q,\omega)$ for $\delta J= 0.8$ and $T=0.1$ showing much larger weight at low–$\omega$ at $q\sim \pi$. Panel (b) show $S(q,\omega)$ for low $\omega$ where the non–dispersive $\omega=0$ delta peak is seen. The width of the peak is due to thew broadening used in the presentation. Panel (c) show low–$\omega$ $S(q,\omega)$ with removed $\omega=0$ delta peak.[]{data-label="s_fig_sqw"}](sq_3D.eps){width="1.0\columnwidth"} a. Dominance of wavevectors $q\sim \pi$ in relaxation rate ---------------------------------------------------------- By approximating NMR relaxation rate $1/T_1$ with local $S_\textrm{loc}(\omega\to 0)$, we neglected the effect of form factors ($A_\alpha^2(q)$, Eq. 2 in the main text), which is a good approximation since in the regime of our calculations the main contribution comes only from $q\sim \pi$. To show this, we present $S(q,\omega)$ in Fig. \[s\_fig\_sqw\], where it is evident that the main contribution at $\omega\to0$ comes from $q\sim \pi$. This stays valid even in the low–$T$ regime, where $\chi^0(T)$ is already increased due to renormalization of $J$-s as observed in the RG flow. Randomness does in fact slightly reduces the contribution of $q\sim \pi$ and slightly increases the contribution of $q\sim 0$, but the transfer of weight is much too small to make $q\sim 0$ dominant. This is in agreement with finding for $S(q)$ shown in Fig. 15 in Ref. [@s_hoyos2007], where even for very large randomness and $T=0$ the main contribution stays at $q\sim \pi$ similarly to the homogeneous system [@s_sandvik1995]. It also agrees with experimental observation and direct statement of the authors [@s_shiroka2011], that there in no indication of important $q\sim 0$ contributions as, e.g., the d.c. field dependence of $1/T_1$, being indication of the absence of (anomalous) low–$q$ (diffusion) contribution. b. Long–wavelength contributions -------------------------------- Our analysis of local $s$ was based on assumption that there is no singular contribution emerging from long–wavelength $q \to 0$ physics. Indeed, all our available data for $S^{zz}(q,\omega \to 0)$ for AFM RHC confirm that the dominant regime at low–$T$ is $q \sim \pi$. Still, $q \to 0$ regime needs further attention since it can lead at $T>J$ to a divergent $S_{nn}(\omega \to 0) \propto 1/\omega^\alpha$ either from the propagation (prevented by randomness in the RHC) in the homogeneous XX chain [@s_naef1999] (with $\alpha \to 0$) or even more as the consequence of the spin diffusion [@s_sirker2009] ($\alpha =1/2$). The latter can be realized at $T>0$ but vanishes at $T \to 0$ within the RHC [@s_theodorou1976; @s_motrunich2001]. Our results so far indicate that in spite of possible $T>0$ diffusion its contribution to $S_{nn}(\omega \to 0)$ is unresolvable for reachable systems, as follows also from NMR experiments [@s_shiroka2011] where it can be directly tested via the magnetic field dependence of $T_1$. Binary disorder distribution ============================= Concrete realization of a random system in Ref. [@s_shiroka2011], namely , has binary disorder distribution with two exchange couplings, $J_{\text{Si}}=280\,[K]$ and $J_{\text{Ge}}= 580\,[K]$, which is in contrast with our continuous disorder distribution model, motivated by a usual theoretical reference. Therefore the question arises, how different are the results for the binary distribution from our results. Here we argue that the results are qualitatively and even quantitatively very similar for both distributions. This was realized already by J. E. Hirsch [@s_hirsch1980], who showed that arbitrary disorder distribution lead to the similar low–$T$ behaviour. ![(Color online) Comparison of CDF-s for continuous and binary disorder distribution. Both distributions give almost the same CDF for small disorder, while for larger disorder they show only small quantitative difference. Note that the $x$–axis is not rescaled and both distributions would therefore give almost the same $1/\tau_0$. Plots are for $L=80$, $T=0.3$ and obtained with $N_r=10^3$ realizations.[]{data-label="s_fig_bin"}](cdf_bin.eps){width="1.0\columnwidth"} To demonstrate the effect of binary distribution we show in Fig. \[s\_fig\_bin\] the comparison of relaxation rate CDF-s for continuous disorder distribution with the ones for binary disorder distribution with the same effective width. It is seen that the difference is small and largest for strongest disorder, where it still remains only quantitative, while for low disorder CDF-s are essentially the same for both disorder distributions. Therefore our results obtained with continuous distribution can easily be compared with measurements and they indeed agree qualitatively and to some extend even quantitatively with them (see main text). Comments on stretched exponential ================================== Using phenomenological stretched exponential form to fit experimental data on magnetization relaxation seems to be a common practice, which can be attributed to the fact that stretched exponential form can capture anomalously long tails in the distribution of the relaxation rates (normal distribution can not) and is at the same time very convenient for the fitting procedure. This immediately raises the question, how good this form really is for the description of experimental data and can it be motivated by some microscopic picture, e.g. model Hamiltonian. In Fig. \[s\_fig\_pdf\] we show our RHC model results for relaxation time distributions (PDF-s), which shows several important features when compared to the experimentally suggested stretched exponential forms (see Fig. 4 in Ref. [@s_shiroka2011]). First one can see that the $T$ evolution is similar to the experimental one and more importantly at low–$T$ anomalously long tails (or large spread) in PDF appear, which can be captured with stretched exponential form and not with, e.g., normal (Gaussian) distribution. This could be the reason for the success of the stretched exponential form in the fitting procedures and its phenomenological description of experimental data. ![(Color online) Probability distribution function PDF($\tau$) for several temperatures and one randomness $\delta J=0.3$. These results should be compared with the shape of experimental ones in Fig. 4 in Ref. [@s_shiroka2011]. Temperature evolution of the distribution agrees with the experiment, as well as appearance of anomalously long tails of the distribution at low–$T$. Points correspond to fitted $\tau_0$ which shows similar $T$–dependence as experiment, while dashed vertical line corresponds to homogeneous ($\delta J=0$) model.[]{data-label="s_fig_pdf"}](pdf.eps){width="1.0\columnwidth"} However, the description of our RHC model results with stretched exponential is not perfect as can be expected, and the most obvious deviations can be found in the long tails. E.g., for few specific value of $\Gamma$ analytical form of PDF-s is know [@s_lindsey1980; @s_johnston2006] and for $\Gamma=0.5$ has a form $$\mathrm{PDF}_{\Gamma=0.5}(s)=\frac{\exp\left[-1/\left(4\tau_0s\right)\right]}{\sqrt{4\pi\tau_0s^3}}\,.\label{pdfan}$$ On Fig. \[s\_fig\_sea\] we compare our numerical result with $\Gamma=0.49$ (for $T=0.1$ and $\delta J=0.5$) with stretched exponential CDF obtained from Eq. , $\mathrm{CDF}_{\Gamma=0.5}(s)=\mathrm{Erf}[1/(2\sqrt{s\,\tau_0})]$. We observe that the RHC model predicts longer (shorter) tails in the PDF for smaller (larger) $s$ than stretched exponential form. This is in turn reflected in the corresponding time dependent magnetization relaxation function (being Laplace transform of PDF) directly probed by experiment. One could, for example, from our PDF-s propose a new form (instead of stretched exponential) by approximating PDF-s with some function and performing its Laplace transform. This is however not trivial and we leave it as a motivation for future work. In this way obtained form is expected to describe experimental data better than stretched exponential, although differences might be small and experimental data with higher resolution might be needed. ![(Color online) One of more critical comparisons of our calculated CDF (blue line) to approximated stretched exponential form (red line) for $\Gamma \simeq 0.5$, for which stretched exponential distribution has particularly simple analytical form [@s_lindsey1980; @s_johnston2006].[]{data-label="s_fig_sea"}](sea.eps){width="1.0\columnwidth"} [24]{} S. R. White, Phys. Rev. Lett. **69**, 2863 (1992). U. Schollwöck, Rev. Mod. Phys. **77**, 259 (2005). J. Kokalj and P. Prelovšek, Phys. Rev. B **80**, 205117 (2009). J. Kokalj and P. Prelovšek, Phys. Rev. B **82**, 060406(R) (2010). For a recent review, see P. Prelovšek and J. Bonča, in [*Strongly Correlated Systems - Numerical Methods*]{}, edited by A. Avella and F. Mancini (Springer Series in Solid–State Sciences 176, Berlin, 2013), pp. 1–29. J. Jaklič and P. Prelovšek, Adv. Phys. **49**, 1 (2000). D. Fisher, Phys. Rev. B **50**, 3799 (1994). J. E. Hirsch, Phys. Rev. B **22**, 5355 (1980). A. Zheludev, T. Masuda, G. Dhalenne, A. Revcolevschi, C. Frost, and T. Perring, Phys. Rev. B **75**, 054409 (2007). T. Shiroka, F. Casola, V. Glazkov, A. Zheludev, K. Prša, H.–R. Ott, and J. Mesot, Phys. Rev. Lett. **106**, 137202 (2011). T. Masuda, A. Zheludev, K. Uchinokura, J.-H. Chung, and S. Park, Phys. Rev. Lett. **93**, 077206 (2004). H. Mori, Prog. Theor. Phys. **34**, 399 (1965). T. N. Tommet and D. L. Huber, Phys. Rev. B **11**, 1971 (1975). J. Oitmaa, M. Plischke, and T. A. Winchester, Phys. Rev. B **29**, 1321 (1984). O. A. Starykh, A. W. Sandvik, and R. R. P. Singh, Phys. Rev. B **55**, 14953 (1997). J.A. Hoyos, A.P. Vieira, N. Laflorencie and E. Miranda, Phys. Rev. B **76**, 174425 (2007). A. W. Sandvik, Phys. Rev. B **52**, R9831 (1995). F. Naef, X. Wang, X. Zotos, and W. von der Linden, Phys. Rev B **60**, 359 (1999). J. Sirker, R. G. Pereira, and I. Affleck, Phys. Rev. Lett. **103**, 216602 (2009). G. Theodorou and M. H. Cohen, Phys. Rev. B **13**, 4597 (1976). O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B **63**, 134424 (2001). O. Motrunich, K. Damle, and D. A. Huse, Phys. Rev. B **63**, 134424 (2001). C. P. Lindsey and G. D. Patterson, J. Chem. Phys. **73**, 3348 (1980). D. Johnston, Phys. Rev B **74**, 184430 (2006).

--- abstract: | It is known that every $R$-module has a flat precover. We show in the paper that every $R$-module has a Gorenstein flat precover.\ address: - 'School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou [730070]{}, P.R. China' - 'School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou [730070]{}, P.R. China' author: - Gang Yang - Li Liang title: All modules have Gorenstein flat precovers --- **Introduction** ================ A class $\mathcal{L}$ of objects of an abelian category $\mathcal{C}$ is called a precovering class [@Enoc81] if every object of $\mathcal{C}$ has an $\mathcal{L}$-precover (see Definition 2.2). In the language of [@AR] this means that $\mathcal{L}$ is a contravariantly finite subcategory. Precovering classes (or contravariantly finite subcategories) play a great important role in homological algebra. One of the reasons is that one can construct proper $\mathcal{L}$-resolutions using a precovering class $\mathcal{L}$ to compute homology and cohomology (see [@EJ00] for details). For any ring $R$, recall from [@EJT93] that an $R$-module $G$ is Gorenstein flat if there exists an exact sequence $\cdots\rightarrow F^{-2}\rightarrow F^{-1}\rightarrow F^0\rightarrow F^1\rightarrow F^2\rightarrow\cdots$ of flat $R$-modules with $G=\text{Ker}(F^0\rightarrow F^1)$ such that $I\otimes_R-$ leaves the sequence exact whenever $I$ is an injective right $R$-module. Obviously, flat $R$-modules are Gorenstein flat. Further studies on Gorenstein flat $R$-modules can be found in [@Ben09; @EJ00; @EJLR04; @EJT93; @Holm04a]. Bican, El Bashir and Enochs [@BBE01] proved that the class of flat $R$-modules is a precovering class. On the other hand, Enochs, Jenda and López-Ramos [@EJLR04] proved that the class of Gorenstein flat $R$-modules is a precovering class over a right coherent ring. Furthermore, it was shown in [@YL] that the result holds over a left GF-closed ring (that is, a ring over which the class of Gorenstein flat $R$-modules is closed under extensions). In this paper, we prove that the class of Gorenstein flat $R$-modules is a precovering class over any ring as follows. **Theorem A.** *Let $R$ be any ring. Then every $R$-module has a Gorenstein flat precover.* We prove the above result by constructing a perfect cotorsion pair in the category of complexes of $R$-modules. **Preliminaries** ================= Throughout the paper, we assume all rings have an identity and all modules are unitary. Unless stated otherwise, an $R$-module will be understood to be a left $R$-module. To every complex $\xymatrix@C=0.6cm{C= \cdots \ar[r]^{} & C^{m-1} \ar[r]^{d^{m-1}} & C^m \ar[r]^{d^{m}} & C^{m+1} \ar[r]^{d^{m+1}} & \cdots },$ the $m$th cycle is defined as ${\mbox{\rm Ker}}(d^m)$ and is denoted $\text{Z}^m(C)$. The $m$th boundary is $\text{Im}(d^{m-1})$ and is denoted $\text{B}^m(C)$. The $m$th homology of $C$ is the module $$\text{H}^m(C)=\text{Z}^m(C)/\text{B}^m(C).$$ A complex $C$ is exact if $\text{H}^m(C)=0$ for all $m\in\mathbb{Z}$. For an integer $n$, $C[n]$ denotes the complex such that $C[n]^m=C^{m+n}$ and whose boundary operators are $(-1)^nd^{m+n}$. Given an $R$-module $M$, we denote by $\overline{M}$ the complex $$\xymatrix@C=0.6cm{ \cdots \ar[r]^{ } & 0 \ar[r]^{ } & M \ar[r]^{id} & M\ar[r]^{ } & 0 \ar[r]^{ } & \cdots }$$ with $M$ in the $-1$ and 0th degrees and $\underline{M}$ the complex $$\xymatrix@C=0.6cm{ \cdots \ar[r]^{ } & 0 \ar[r]^{ } & M \ar[r]^{ } & 0 \ar[r]^{ } & \cdots }$$ with $M$ in the $0$th degree. A complex $C$ is finitely presented (generated) if only finitely many components are nonzero and each $C^m$ is finitely presented (generated). Clearly, both $\overline{R}$ and $\underline{R}$ are finitely presented. Recall that a complex $P$ is projective if it is exact and $\text{Z}^m(P)$ is a projective $R$-module for each $m\in \mathbb{Z}$, so it is easy to see that $P$ is a direct sum of the form $\overline{Q}[m]$ for some projective $R$-modules $Q$. Given two complexes $X$ and $Y$, we let ${\mbox{\rm Hom}}^\bullet(X, Y)$ denote a complex of $\mathbb{Z}$-modules with $m$th component $${\mbox{\rm Hom}}^\bullet(X, Y)^m=\prod_{t\in \mathbb{Z}}{\mbox{\rm Hom}}(X^t, Y^{m+t})$$ and such that if $f\in{\mbox{\rm Hom}}^\bullet(X, Y)^m$ then $$(d^m(f))^n=d_Y^{n+m}\circ f^n-(-1)^{m}f^{n+1}\circ d_X^n.$$ We say $f:X\rightarrow Y$ a morphism of complexes if $d_Y^{n}\circ f^n=f^{n+1}\circ d_X^n$ for all $n\in \mathbb{Z}$. ${\mbox{\rm Hom}}(X, Y)$ denotes the set of morphisms of complexes from $X$ to $Y$ and ${\mbox{\rm Ext}}^i(X, Y)$ $(i\geq1)$ are right derived functors of ${\mbox{\rm Hom}}$. Obviously, ${\mbox{\rm Hom}}(X, Y)=\text{Z}^0({\mbox{\rm Hom}}^\bullet(X, Y))$. We let $\underline{{\mbox{\rm Hom}}}(X, Y)$ denote a complex with $\underline{{\mbox{\rm Hom}}}(X, Y)^m$ the abelian group of morphisms from $X$ to $Y[m]$ and with a boundary operator given by: $f\in\underline{{\mbox{\rm Hom}}}(X, Y)^m$, then $d^m(f): X\rightarrow Y[m+1]$ with $d^m(f)^n=(-1)^md_Y\circ f^n$, $\forall n\in \mathbb{Z}$. We note that the new functor $\underline{{\mbox{\rm Hom}}}(X, Y)$ has right derived functors whose values will be complexes. These values should certainly be denoted $\underline{{\mbox{\rm Ext}}}^i(X, Y)$. It is not hard to see that $\underline{{\mbox{\rm Ext}}}^i(X, Y)$ is the complex $$\cdots\rightarrow{{\mbox{\rm Ext}}}^i(X, Y[n-1])\rightarrow {{\mbox{\rm Ext}}}^i(X, Y[n])\rightarrow{{\mbox{\rm Ext}}}^i(X, Y[n+1])\rightarrow\cdots$$ with boundary operator induced by the boundary operator of $Y$. If $X$ is a complex of right $R$-modules and $Y$ is a complex of left $R$-modules, let $X\otimes^\bullet Y$ be the usual tensor product of complexes. I.e., $X\otimes^\bullet Y$ is the complex of abelian groups with $$(X\otimes^\bullet Y)^m=\bigoplus_{t\in \mathbb{Z}}X^t\otimes_R Y^{m-t}$$ and $$d(x\otimes y)=d_X^t(x)\otimes y+(-1)^{t}x\otimes d_Y^{m-t}(y)$$ for $x\in X^t$ and $y\in Y^{m-t}$. Obviously, $\underline{M}\otimes^\bullet Y=M\otimes_R Y= \cdots\rightarrow M\otimes_R Y^{-1}\rightarrow M\otimes_R Y^0\rightarrow M\otimes_R Y^1\rightarrow\cdots$ for a right $R$-module $M$. We define $X\otimes Y$ to be $\frac{(X\otimes^\bullet Y)}{\text{B}(X\otimes^\bullet Y)}$. Then with the maps $$\frac{(X\otimes^\bullet Y)^n}{\text{B}^n(X\otimes^\bullet Y)}\rightarrow \frac{(X\otimes^\bullet Y)^{n+1}}{\text{B}^{n+1}(X\otimes^\bullet Y)}, \quad x\otimes y\mapsto d_X(x)\otimes y,$$ where $x\otimes y$ is used to denote the coset in $\frac{(X\otimes^\bullet Y)^n}{\text{B}^n(X\otimes^\bullet Y)}$, we get a complex of abelian groups. One can found the next result in [@Garc99 Proposition 4.2.1]. \[l2.1\] Let $X$, $Y$, $Z$ be complexes. Then we have the following natural isomorphisms: 1. $X\otimes(Y\otimes Z)\cong (X\otimes Y)\otimes Z$; 2. For a right $R$-module $M$, $\overline{M}[n]\otimes Y\cong M\otimes_R Y[n]$; 3. $X\otimes (\varinjlim Y_i)\cong \varinjlim (X\otimes Y_i)$ for a directed family $(Y_i)_{i\in I}$ of complexes. Let $\mathcal{L}$ be a class of objects of an abelian category $\mathcal{C}$ and $X$ an object. A homomorphism $f: L\rightarrow X$ is called an $\mathcal{L}$-precover if $L\in\mathcal{L}$ and the abelian group homomorphism $\text{Hom}(L', f): \text{Hom}(L', L)\rightarrow \text{Hom}(L', X)$ is surjective for each $L'\in\mathcal{L}$. An $\mathcal{L}$-precover $f: L\rightarrow X$ is called an $\mathcal{L}$-cover if every endomorphism $g: L\rightarrow L$ such that $fg=f$ is an isomorphism. Dually we have the definitions of an $\mathcal{L}$-preenvelope and an $\mathcal{L}$-envelope. A pair $(\mathcal{A}, \mathcal{B})$ in an abelian category $\mathcal{C}$ is called a cotorsion pair if the following conditions hold: 1. ${\mbox{\rm Ext}}^1_\mathcal{C}(A, B)=0$ for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$; 2. If ${\mbox{\rm Ext}}^1_\mathcal{C}(A, X)=0$ for all $A\in\mathcal{A}$ then $X\in\mathcal{B}$; 3. If ${\mbox{\rm Ext}}^1_\mathcal{C}(X, B)=0$ for all $B\in\mathcal{B}$ then $X\in\mathcal{A}$. We think of a cotorsion pair $(\mathcal{A}, \mathcal{B})$ as being $\lq\lq$orthogonal with respect to ${\mbox{\rm Ext}}^1_\mathcal{C}$". This is often expressed with the notation $\mathcal{A}={^\perp\mathcal{B}}$ and $\mathcal{B}=\mathcal{A}^\perp$. The notion of a cotorsion pair was first introduced by Salce in [@S79] and rediscovered by Enochs and coauthors in 1990’s. Its importance in homological algebra has been shown by its use in the proof of the existence of flat covers of modules over any ring [@BBE01]. A cotorsion pair $(\mathcal{A}, \mathcal{B})$ is said to be complete if for any object $X$ there are exact sequences $0\rightarrow X\rightarrow B\rightarrow A\rightarrow 0$ and $0\rightarrow B'\rightarrow A'\rightarrow X\rightarrow 0$ with $A, A'\in \mathcal{A}$ and $B, B'\in \mathcal{B}$. A cotorsion pair $(\mathcal{A}, \mathcal{B})$ is said to be cogenerated by a set if there is a set $\mathcal{S}\subset \mathcal{A}$ such that $\mathcal{S}^\bot=\mathcal{B}$. By a well-known theorem of Eklof and Trlifaj [@ET01], a cotorsion pair $(\mathcal{A}, \mathcal{B})$ is complete if it is cogenerated by a set (see [@BBE01]). A cotorsion pair $(\mathcal{A}, \mathcal{B})$ is said to be perfect if every object has an $\mathcal{A}$-cover and a $\mathcal{B}$-envelope. **All modules have Gorenstein flat precovers** {#ns} ============================================== Recall from [@Garc99] that an exact sequence $0\rightarrow P\rightarrow X\rightarrow X/P\rightarrow 0$ of complexes is *pure* if for any complex $Y$ of right $R$-modules, the sequence $0\rightarrow Y\otimes P\rightarrow Y\otimes X\rightarrow Y\otimes X/P\rightarrow 0$ is exact. We state here the characterizations of purity that can be found in [@Garc99 Theorem 5.1.3]. \[p2.1\] Let $0\rightarrow P\rightarrow X\rightarrow X/P\rightarrow 0$ be an exact sequence of complexes. Then the following statements are equivalent. 1. $0\rightarrow P\rightarrow X\rightarrow X/P\rightarrow 0$ is pure; 2. $0 \rightarrow\underline{{\mbox{\rm Hom}}}(U, P)\rightarrow\underline{{\mbox{\rm Hom}}}(U, X) \rightarrow\underline{{\mbox{\rm Hom}}}(U, X/P)\rightarrow0$ is exact for any finitely presented complex $U$. Recall from [@AF91] that a complex $Q$ is DG-projective, if each $R$-module $Q^m$ is projective and ${\mbox{\rm Hom}}^\bullet(Q, E)$ is exact for any exact complex $E$. By [@Garc99 Proposition 2.3.5], a complex $Q$ is DG-projective if and only if ${\mbox{\rm Ext}}^1(Q, E)=0$ for every exact complex $E$. \[l2.2\] Let $ 0 \rightarrow P \rightarrow X\rightarrow X/P\rightarrow 0$ be a pure exact sequence of complexes. If $X$ is exact then both $P$ and $X/P$ are also exact. By Lemma \[p2.1\], the sequence $\underline{{\mbox{\rm Hom}}}(D,X)\rightarrow\underline{{\mbox{\rm Hom}}}(D,X/P)\rightarrow0$ is exact for all finitely presented complex $D$, and so the sequence $$\underline{{\mbox{\rm Hom}}} (\underline{R},X)\rightarrow\underline{{\mbox{\rm Hom}}}(\underline{R},X/P)\rightarrow0$$ is exact since $\underline{R}$ is finitely presented. On the other hand, the sequence $$\underline{{\mbox{\rm Hom}}}(\underline{R},X)\rightarrow\underline{{\mbox{\rm Hom}}}(\underline{R},X/P)\rightarrow \underline{{\mbox{\rm Ext}}}^1(\underline{R},P)\rightarrow \underline{{\mbox{\rm Ext}}}^1(\underline{R},X)$$ is exact, where $\underline{{\mbox{\rm Ext}}}^1(\underline{R},X)=0$ since $\underline{R}$ is DG-projective and $X$ is exact. Thus we get that $\underline{{\mbox{\rm Ext}}}^1(\underline{R},P)=0$, and so $\text{H}^{-n+1}(P)\cong {\mbox{\rm Ext}}^1(\underline{R},P[-n])=0$ for all $n\in \mathbb{Z}$. This means that $P$ is an exact complex, and now it is easily seen that $X/P$ is also exact. Let $R$ be a ring, we denote by $\mathbf{E}(R)$ the class of exact complexes of flat $R$-modules such that they remain exact after applying $I\otimes_R-$ for any injective right $R$-module $I$. Recall that a complex $F$ is flat if $F$ is exact and each $\text{Z}^n(F)$ is a flat $R$-module for each $n\in \mathbb{Z}$. Clearly, $\mathbf{E}(R)$ contains all flat complexes. As characterized in [@Gill04] and [@Garc99], there are initiate connections between the purity and the flatness of complexes. Inspired by this fact we give the following result. \[l3.3\] Let $R$ be any ring and $E\in \mathbf{E}(R)$. If $S\subseteq E$ is pure, then $S$ and $E/S$ are both in $\mathbf{E}(R)$. Let $M$ be any right $R$-module. Then $$0\rightarrow\overline{M}[n]\otimes S\rightarrow \overline{M}[n]\otimes E\rightarrow \overline{M}[n]\otimes E/S\rightarrow0$$ is exact. By Lemma \[l2.1\](2), the sequence $$0\rightarrow M\otimes_R S[n]\rightarrow M\otimes_R E[n]\rightarrow M\otimes_R (E/S)[n]\rightarrow0$$ is exact. Therefore $S^n\subseteq E^n$ is pure for each $n\in\mathbb{Z}$. Since each $E^n$ is flat, we get that $S^n$ and $E^n/S^n$ are flat for each $n\in\mathbb{Z}$. By Lemma \[l2.2\], we get that $S$ and $E/S$ are exact. It remains to show that for any injective right $R$-module $I$, $I\otimes_RS$ and $I\otimes_RE/S$ are exact. Since the exact sequence $0\rightarrow S\rightarrow E\rightarrow E/S\rightarrow0$ is pure, we get that the sequence $$0\rightarrow\overline{I}\otimes S\rightarrow \overline{I}\otimes E\rightarrow \overline{I}\otimes E/S\rightarrow0$$ is exact and pure by Lemma \[l2.1\](1). Note that $\overline{I}\otimes E\cong I\otimes_R E$ is exact by Lemma \[l2.1\](2), then $\overline{I}\otimes S$ and $\overline{I}\otimes E/S$ are exact by Lemma \[l2.2\], and so $I\otimes_RS$ and $I\otimes_RE/S$ are exact by Lemma \[l2.1\](2). \[l2.4\] Let ${\rm Card}(R)\leq \kappa$, where $\kappa$ is some infinite cardinal. Then for any $F\in\mathbf{E}(R)$ and any element $x\in F$ (by this we mean $x\in F^n$ for some $n$), there exists a subcomplex $L\subseteq F$ with $x\in L$, $L, F/L\in \mathbf{E}(R)$ and ${\rm Card}(L)\leq \kappa$. By [@Gill04 Lemma 4.6], there exists a pure subcomplex $L\subseteq F$ with $x\in L$ and ${\rm Card}(L)\leq \kappa$, then, by Lemma \[l3.3\], we get that $L$ and $F/L$ are contained in $\mathbf{E}(R)$. \[l3.5\] For any ring $R$ the pair $(\mathbf{E}(R), \mathbf{E}(R)^\bot)$ is a perfect cotorsion pair. By Lemma \[l2.1\](3) the class $\mathbf{E}(R)$ is closed under direct limits. Clearly, $\mathbf{E}(R)$ is closed under direct sums, direct summands and extensions. Using Lemma \[l2.4\] and a similar method as proved in [@AE01 Remark 3.2], we get that the pair $(\mathbf{E}(R), \mathbf{E}(R)^\bot)$ is cogenerated by a set. On the other hand, the class $\mathbf{E}(R)$ contains all projective complexes. Thus, by [@AE01 Corollaris 2.11, 2.12 and 2.13], the pair $(\mathbf{E}(R), \mathbf{E}(R)^\bot)$ is a perfect cotorsion pair. *Proof of Theorem A.* Let $M$ be any $R$-module and $g: E\rightarrow \underline{M}[1]$ be an $\mathbf{E}(R)$-precover which exists by Lemma \[l3.5\]. This gives the following commutative diagram: $$\xymatrix@C=15pt@R=30pt{ E=: \ \cdots \ar[r]^{} &E^{-2} \ar[rr]^{}\ar[dd]^{} & & E^{-1} \ar[dr]_{\pi}\ar[rr]^{} \ar[dd]^{g^{-1}} & & \ E^0 \ar[rr]^{}\ar[dd]^{} && E^1 \ar[rr]^{}\ar[dd]^{} & & \cdots \\ & \ \ & & \ & G \ar[dd]^{\widetilde{g}} \ar@{.>}[ur]^{} \\ \underline{M}[1]=: \ \cdots \ar[r]^{}&0 \ar[rr]^{} & &M \ar[dr]_{=}\ar[rr]_{} & & 0 \ar[rr]^{} && 0 \ar[rr]^{} & & \cdots \\ & \ \ & &\ & M \ar[ur]^{} }$$ where $G=\text{Z}^0(E)$ is Gorenstein flat. In the following we show that $\widetilde{g}: G\rightarrow M$ is a Gorenstein flat precover of $M$. Let $\widetilde{f}: H\rightarrow M$ be a homomorphism with $H$ Gorenstein flat. Then there exists a complex $F$ in $\mathbf{E}(R)$ such that $H=\text{Z}^0(F)$. Now one can extend $\widetilde{f}$ to a morphism $f: F\rightarrow \underline{M}[1]$ of complexes as follows: $$\xymatrix@C=15pt@R=30pt{ F=: \ \cdots \ar[r]^{} &F^{-2} \ar[rr]^{}\ar[dd]^{} & & F^{-1} \ar[dr]_{\sigma}\ar[rr]^{} \ar[dd]^{f^{-1}} & & \ F^0 \ar[rr]^{}\ar[dd]^{} && F^1 \ar[rr]^{}\ar[dd]^{} & & \cdots \\ & \ \ & & \ & H \ar[dd]^{\widetilde{f}} \ar@{.>}[ur]^{} \\ \underline{M}[1]=: \ \cdots \ar[r]^{}&0 \ar[rr]^{} & &M \ar[dr]_{=}\ar[rr]_{} & & 0 \ar[rr]^{} && 0 \ar[rr]^{} & & \cdots \\ & \ \ & &\ & M \ar[ur]^{} }$$ Since $g: E\rightarrow \underline{M}[1]$ is an $\mathbf{E}(R)$-precover, there exists a morphism $h: F\rightarrow E$ of complexes such that the diagram $$\xymatrix{ & E \ar[d]^{g} \\ F \ar[ur]^{h} \ar[r]_{f} & \underline{M}[1] }$$ is commutative. The morphism $h$ induces a homomorphism $\widetilde{h}: H\rightarrow G$ such that the following diagram $$\xymatrix@C=15pt@R=30pt{ F=: \ \cdots \ar[r]^{} &F^{-2} \ar[rr]^{}\ar[dd]^{h^{-2}} & & F^{-1} \ar[dr]_{\sigma}\ar[rr]^{} \ar[dd]^{h^{-1}} & & \ F^0 \ar[rr]^{}\ar[dd]^{h^0} && F^1 \ar[rr]^{}\ar[dd]^{h^1} & & \cdots \\ & \ \ & & \ & H \ar[dd]^{\widetilde{h}} \ar@{.>}[ur]^{} \\ E=: \ \cdots \ar[r]^{}&E^{-2} \ar[rr]^{} & &E^{-1} \ar[dr]_{\pi}\ar[rr]_{} & & E^0 \ar[rr]^{} && E^1 \ar[rr]^{} && \cdots \\ & \ \ & &\ & G \ar[ur]^{} }$$ is commutative. Note that $\widetilde{f}\sigma=f^{-1}=g^{-1}h^{-1}=\widetilde{g}\pi h^{-1}=\widetilde{g}\widetilde{h}\sigma,$ then $\widetilde{f}=\widetilde{g}\widetilde{h}$ since $\sigma$ is an epimorphism. This implies that $\widetilde{g}: G\rightarrow M$ is a Gorenstein flat precover of $M$. [10]{} S. T. Aldrich, E. E. Enochs, J. R. García Rozas and L. Oyonarte, COvers and envelopes in Grothendieck categories: flat covers of complexes with applications. J. Algebra **243** (2001), 615-630. M. Auslander and I. Reiten, Applications of contravariantly finite subcategories. Adv. Math. **86** (1991), 111-152. L. L. Avramov and H.-B. Foxby, Homological dimensions of unbounded complexes. J. Pure Appl. Algebra **71** (1991), 129-155. D. Bennis, Rings over which the class of Gorenstein flat modules is closed under extensions. Comm. Algebra **37** (2009), 855-868. L. Bican, R. El Bashir, E. E. Enochs, All modules have flat covers. Bull. Lond. Math. Soc. **33** (2001), 385-390. P. C. Eklof and J. Trlifaj, How to make Ext vanish. Bull. London Math. Soc. **33** (2001), 41-51. E. E. Enochs, Injective and flat covers, envelopes and resolvents. Israel J. Math. (3) **39** (1981), 189-209. E. E. Enochs and O. M. G. Jenda, *Relative Homological Algebra*. De Gruyter Expositions in Mathematics no. 30, Walter De Gruyter, Berlin-New York, 2000. E. E. Enochs and O. M. G. Jenda, J. A. López-Ramos, The existence of Gorenstein flat covers. Math. Scand. **94** (2004), 46-62. E. E. Enochs and O. M. G. Jenda, B. Torrecillas, Gorenstein flat modules. Nanjing Daxue Xuebao Shuxue Bannian Kan **10** (1993), 1-9. J. Gillespie, The flat model structure on Ch($R$). Trans. Amer. Math. Soc. **356** (2004), 3369-3390. J. R. García Rozas, *Covers and Envelope in the Category of Complexes of Modules*. CRC Press, Boca Raton-London-New York-Washington, D.C., 1999. H. Holm, Gorenstein homological dimensions. J. Pure Appl. Algebra **189** (2004), 167-193. L. Salce, Cotorsion theories for abelian groups. Symposia Math. 23, 1979. G. Yang and Z. K. Liu, Gorenstein Flat Covers over GF-Closed Rings, Comm. Algebra **40** (2012), 1632-1640.

--- author: - 'Y. C. Joshi, A. K. Pandey, D. Narasimha, Y. Giraud-Héraud, R. Sagar and J. Kaplan' date: 'Received 26 May 2003 /accepted 15 October 2003' title: Photometric study of two novae in M31 --- Introduction ============ Cataclysmic variables (CVs) are close binary systems consisting of a white dwarf primary and a late-type main sequence secondary star. Novae are a sub-class of cataclysmic variables characterized by the presence of a sudden increase of brightness, called outbursts, due to thermonuclear runway in the envelope of the primary, causing the system brightness to increase typically by 10-20 mag. These are bright objects which reach up to $M_{V} \sim$ -9.0 mag at maximum and their rate of decline is tightly correlated with their absolute magnitude at maximum (McLaughlin 1945). The study of novae in external galaxies is important to infer their distances as these objects are one of the brightest standard candles up to the Virgo cluster (cf. Jacoby et al. 1992 for a review) and tracers of differences in the stellar content among galaxies (cf. Van den Bergh 1988 for a review). M31, our nearest large galaxy, has been a target of searches for novae since the pioneering work of Hubble (1929). Later Arp (1956), Rosino (1964, 1973), Rosino et al. (1989), Ciardullo et al. (1987), Sharov & Alksnis (1991), Tomaney & Shafter (1992), Rector et al. (1999) and Shafter & Irby (2001) have extended the systematic search for novae in M31. In collaboration with the AGAPE (Andromeda Gravitational Amplification Pixel Experiment) group, we started Cousins $R$ and $I$ photometric observations of M31 in 1998 to search for microlensing events. Based on the 4 year observations, we have already reported the discovery of new Cepheids and other variable stars (Joshi et al. 2003a). As a microlensing survey program is ideally suited to monitor the flux and temperature variation during transient events, we have extended our search to detect nova outbursts. Here we report photometric light curves of two novae detected in the target field, one each in 2000 and 2001 observing seasons. We show that an increase in flux at longer wavelengths a few weeks after the initial rapid decline is a common phenomenon in the novae light curves. Observations ============ We have undertaken a program called the [**Nainital Microlensing Survey**]{} to detect microlensing events in the direction of M31, at the State Observatory, Nainital, India, since 1998. Cousins $R$ and $I$ broad band CCD observations of M31 were carried out for an $\sim6'\times 6'$ field in 1998 and an $\sim13'\times 13'$ field during 1999 to 2001 observing seasons using the 104-cm Sampurnanand Telescope at the f/13 Cassegrain focus. The total integrated observing time devoted to the survey ranges from $\sim$ 30 minutes to 2 hours each night. The $13'\times 13'$ target field ($\alpha _{2000}$ = $0^{h} 43^{m} 38^{s}$ and $\delta_{2000}$ = $+41^{\circ}09^{\prime}.1$) is centered at a distance of $\sim$ 15 arcmin away from the center of M31. The average seeing during the 141 observed nights spanning 4 years was $\sim$ 2 arcsec. An overview of the observational detail has been given by Joshi et al. (2003a). {height="6.0cm" width="6.0cm"} {height="6.0cm" width="6.0cm"} A large database collected during the observing period was planned as: \(a) To search for microlensing events. \(b) To search for variable stars, particularly Cepheids. \(c) To search for other transient events e.g. novae, where a decrease in brightness of 2 to 5 mag within the first two months of the nova evolution can easily be detected in our observations. We have already published a catalog of variable stars which contains data for 26 Cepheids and 333 red variable stars (Joshi et al. 2003a). We are in the process of analysing some of the possible microlensing candidates found in our data set (Joshi et al. 2003b, 2003c). Here we report the photometry of two novae detected in the target field. Data reduction and photometry ============================= Standard techniques were used for data reduction using MIDAS and IRAF software. The dark current correction was not applied due to its negligible contribution during the maximum exposure time of a frame. Cosmic rays (CRs) were removed from each frame independently. We added all the frames of a particular filter taken on a single night and made one frame per filter per night to increase the signal to noise ratio. The CCD frames were processed using the DAOPHOT photometry routine (Stetson 1987). To obtain $R$ and $I$ standard magnitudes, the photometric calibration was done using Landolt’s (1992) standard field SA98, on a good photometric night of 25/26 October, 2000. A total of 13 secondary stars having $0.09 \le (R-I) \le 1.0$ were observed over a wide range of airmasses. The typical error in magnitudes at $\sim$20.0 mag level is $\sim$ 0.10 and 0.15 mag in $R$ and $I$ bands respectively. In order to use observations in non-photometric conditions, differential photometry was done assuming that the errors introduced due to colour difference between nova and comparison stars were much smaller than the zero point errors. Further details of the photometric calibration has been given in Joshi et al. (2003a). Identification of the novae =========================== We identified two novae in the target field while searching for microlensing events using the pixel technique which was initially proposed to detect microlensing events by Baillon et al. (1993). The implementation of the pixel technique in our data is described in detail by Joshi et al. (2001, 2003c). The two novae detected in our survey, one in 2000 and the other in 2001 observing seasons, are named nova NMS-1 and nova NMS-2 where NMS is an acronym for our project ‘Nainital Microlensing Survey’. The two novae are individually discussed in the following subsections. Nova NMS-1 ---------- The nova NMS-1 having celestial coordinates $\alpha _{2000}$ = $00^{h} 42^{m} 57^{s}.1$ and $\delta_{2000}$ = $+41^{\circ}07^{\prime}15^{''}.7$ was reported in the IAU circular by Donato et al. (2001). When we started observations at 18:54 UT on October 18, 2000, the nova was still brightening. We followed it through the 2000-2001 observing season. Fig. 1 shows two images where the nova is unresolved and at maximum brightness. {height="12.0cm" width="9.0cm"} ------------ ---------------- ------------ ---------------- ------------ ---------------- ---------------- J.D. $R$ J.D. $I$ J.D. $R$ $I$ (+2450000) (mag) (+2450000) (mag) (+2450000) (mag) (mag) 1836.287 18.69$\pm$0.02 1836.316 18.38$\pm$0.03 1869 19.64$\pm$0.03 19.32$\pm$0.06 1836.292 18.64$\pm$0.02 1836.321 18.38$\pm$0.02 1870 19.67$\pm$0.02 19.23$\pm$0.05 1836.296 18.60$\pm$0.02 1836.327 18.36$\pm$0.02 1872 19.69$\pm$0.03 19.39$\pm$0.05 1836.299 18.57$\pm$0.02 1836.333 18.36$\pm$0.02 1874 19.80$\pm$0.02 19.42$\pm$0.05 1836.303 18.59$\pm$0.02 1836.338 18.15$\pm$0.05 1877 19.84$\pm$0.03 19.34$\pm$0.07 1838.213 17.19$\pm$0.01 1838.236 16.97$\pm$0.01 1878 19.93$\pm$0.02 19.24$\pm$0.06 1838.218 17.16$\pm$0.01 1838.245 16.95$\pm$0.01 1879 20.00$\pm$0.03 19.44$\pm$0.08 1838.224 17.19$\pm$0.01 1838.250 16.94$\pm$0.01 1881 19.97$\pm$0.02 19.70$\pm$0.06 - - 1838.255 16.96$\pm$0.01 1884 20.13$\pm$0.02 19.55$\pm$0.05 - - 1838.259 16.94$\pm$0.01 1888 19.97$\pm$0.07 19.35$\pm$0.08 1843 17.74$\pm$0.01 17.27$\pm$0.01 1890 20.18$\pm$0.06 19.37$\pm$0.05 1845 18.09$\pm$0.02 17.67$\pm$0.02 1905 20.88$\pm$0.06 19.72$\pm$0.05 1851 18.69$\pm$0.02 18.44$\pm$0.03 1915 20.60$\pm$0.12 19.40$\pm$0.12 1853 18.84$\pm$0.02 18.54$\pm$0.05 1918 - 19.12$\pm$0.08 1855 19.04$\pm$0.02 18.71$\pm$0.04 1921 20.82$\pm$0.08 19.40$\pm$0.06 1862 19.20$\pm$0.03 18.89$\pm$0.04 1924 20.96$\pm$0.07 19.56$\pm$0.04 1863 19.35$\pm$0.03 19.03$\pm$0.04 1925 20.81$\pm$0.06 19.52$\pm$0.04 1866 19.57$\pm$0.02 19.29$\pm$0.07 1926 20.85$\pm$0.08 - 1867 19.61$\pm$0.03 19.09$\pm$0.04 1927 21.28$\pm$0.09 19.50$\pm$0.07 1868 19.68$\pm$0.02 19.19$\pm$0.06 1929 21.63$\pm$0.11 - ------------ ---------------- ------------ ---------------- ------------ ---------------- ---------------- The photometry of nova NMS-1 has been carried out using the PSF profile fitting technique. Since it was sufficiently bright during the early phase of its eruption and gradually increased its brightness, we carried out photometry on each individual frame of the observations of 18 and 20 October, 2000 (see Table 1). The light curve of nova NMS-1 is shown in Fig. 2. The brightness of the nova increased by $\sim$ 0.10 mag in the $R$ band from 18:54 UT to 19:17 UT and by $\sim$ 0.23 mag in $I$ band from 19:35 UT to 20:07 UT. From a comparison of individual frames taken on that day, it appears that we have captured nova NMS-1 during its peak brightness. The total brightness of the nova increased by $\sim$ 1.5 mag during October 18-20, 2000 followed by an almost exponential decay. Its rate of decline, $v_d$, defined as the rate in mag day$^{-1}$ at which a nova drops to two magnitudes below maximum brightness, is estimated to be: $$v_{d,R} \sim 0.11 ~ \tt{mag~day}^{-1}$$ $$v_{d,I} \sim 0.11 ~ \tt{mag~day}^{-1}$$ for $R$ and $I$ bands respectively. The observed values of $v_d$ suggest that the nova NMS-1 was a fast nova. The maximum magnitude of the nova NMS-1 at peak brightness is $M_R$(max) $\sim -7.96$ and $M_I$(max) $\sim -8.02$ mag in $R$ and $I$ bands respectively. Here we consider a true distance modulus of 24.49 mag for M31 and a total extinction of 0.63 mag and 0.47 mag towards our observed direction in the $R$ and $I$ bands respectively (cf. Joshi et al. 2003a). The correlation between observed $R_{max}$ and rate of decline is consistent with that illustrated by Capaccioli et al. (1989) for novae in M31. {height="6.0cm" width="6.0cm"} {height="6.0cm" width="6.0cm"} Nova NMS-2 ---------- The nova NMS-2 was already reported in the IAU circular by Li (2001). Fig. 3 shows two images where the nova is at unresolved and maximum brightness phase. The celestial coordinates of the nova NMS-2 are $\alpha _{2000}$ = $00^{h} 43^{m} 03^{s}.3$ and $\delta_{2000}$ = $+41^{\circ}12^{\prime}10^{''}.8$. Since this nova was situated at the edge of our target field, it could not be observed in all the images. When we started our observation on October 12, 2001, it was already in a descending phase of brightness. Photometry of the nova was carried out during the brighter phase of outburst. We give its $R$ and $I$ magnitudes in Table 2 and the light curve is shown in Fig. 4. A variation of more than 1.5 mag in $R$ band is seen during the first 35 days of its observations. For the nova NMS-2, our photometry is insufficient to characterize its speed class since we do not have any information of its maximum brightness. The brightest magnitude of the nova NMS-2 in our observations is estimated to be $\sim$ 17.7 mag in both $R$ and $I$ bands indicating that it must be a star brighter than $M_R,M_I\sim -7.4$ mag during its peak brightness. The decay rate for the nova NMS-2 during our observation period is estimated to be: $$\frac {dM(R)}{dt} \sim 0.03\pm0.003 ~ \tt{mag~day}^{-1}$$ $$\frac {dM(I)}{dt} \sim 0.05\pm0.004 ~ \tt{mag~day}^{-1}$$ for $R$ and $I$ bands respectively. The small value of the decay rate for the nova NMS-2 suggests that either it was a slow nova or, most likely, we observed it very late after it reached its peak brightness. A re-brightening in the light curve of the nova NMS-2 from JD $\sim$ 2452210 to 2452220 is evident in both $R$ and $I$ bands. This type of re-brightening profile in nova evolution is quite rare but not unique (Bonifacio et al. 2000). However, due to insufficient data points, we are not able to draw any firm conclusion about its exact behaviour. The evolution of $(R-I)$ colour for the nova NMS-2 is shown in the lower panel of Fig. 4. It becomes bluer with time which is consistent with normal nova behaviour. However, scattering in the data due to poor flat fielding at the edge of frames prevents us from drawing any conclusion about the dust formation in the ejecta. {height="11.0cm" width="9.0cm"} ------------ ---------------- ---------------- ------------ ---------------- ---------------- J.D. $R$ $I$ J.D. $R$ $I$ (+2450000) (mag) (mag) (+2450000) (mag) (mag) 2198 17.98$\pm$0.01 17.86$\pm$0.04 2221 - 18.99$\pm$0.08 2202 17.74$\pm$0.02 17.72$\pm$0.02 2222 18.53$\pm$0.03 18.75$\pm$0.05 2203 18.04$\pm$0.02 17.98$\pm$0.04 2225 18.93$\pm$0.03 19.11$\pm$0.06 2206 18.26$\pm$0.03 18.23$\pm$0.03 2226 18.67$\pm$0.03 - 2207 18.40$\pm$0.02 - 2229 - 19.28$\pm$0.06 2208 18.11$\pm$0.02 - 2230 19.03$\pm$0.03 19.41$\pm$0.07 2213 18.53$\pm$0.03 - 2231 18.83$\pm$0.04 19.00$\pm$0.07 2216 18.43$\pm$0.03 18.48$\pm$0.05 2232 19.05$\pm$0.03 19.30$\pm$0.08 2217 18.42$\pm$0.02 18.61$\pm$0.07 2236 19.18$\pm$0.03 - 2218 18.16$\pm$0.03 18.20$\pm$0.04 2237 19.16$\pm$0.04 19.48$\pm$0.15 ------------ ---------------- ---------------- ------------ ---------------- ---------------- The detection rate of novae in M31 is $\sim$ 30 novae per year (Arp 1956, Capaccioli et al. 1989, Shafter & Irby 2001) and 2 novae detected in our survey, keeping our field of view and total observing time in mind, is consistent with these estimates. Discussion ========== The $R$ and $I$ magnitudes of the nova NMS-1 at the brightest phase are estimated to be -7.96 and -8.02 mag respectively with a rate of decline of $\sim$ 0.11 mag day$^{-1}$. A correlation of the rate of decline with the observed peak flux is in good agreement with the maximum magnitude versus rate of decline (MMRD) curve given by Capaccioli et al. (1989) for M31 novae. It is instructive to analyse the light curves and colour index, to investigate the evolution of the remnant. From the observed light curve of the nova NMS-1 (Fig. 2), it is evident that the flux in $R$ and $I$ bands decline in a similar way for the first three weeks but the late time evolution is qualitatively different. A difference in the profile of the light curves in two bands is a valuable diagnostic of the physical processes operating in the ejecta. We have neither spectroscopic information nor infra-red photometry, nevertheless, we might estimate the possible evolution of the expanding shell from the $R, I$ light curves. To study the photometric behavior of the nova NMS-1, we divided the declining part of the light curve into three sections as shown in Fig. 5(a1). The initial rapid decline in the flux follows a similar pattern in $R$ and $I$ bands and the colour remains practically unchanged; essentially this is the phase where emission from a region of progressively lower effective radius is received. However, after about three weeks, the ($R-I$) colour shows a gradual increase indicating that the region is cooling. After about 40 days, the $R$ band flux is still progressively declining similar to that observed for the nova Herculis 1991 (Harrison & Stringfellow 1994) in the $V$ band. The $I$ band, however, probably shows a slight gradual increase in flux during this period; a rapid rise in the colour index of the nova is seen in Fig. 5(a2). A similar behaviour has been observed in the case of nova Herculis 1991 where an almost constant plateau is seen in $J$, $H$, $K$ and $L$ bands (see Fig. 5(b1) and 5(b2)). Although, we have neither sufficient observations in the later stage nor supporting spectroscopy, we still believe that probably there is a signature of the formation of neutral hydrogen. The decrease in $R$ band flux but the reverse trend in $I$ band flux is likely to be due to the $H_\alpha$ absorption and re-radiation in longer wavelengths. There is some reason to speculate this because: a) we see a nearly constant peak Balmer flux of $10^{37}$ ergs/sec declining rapidly, b) moderately large velocity gradient in the shell is shown by Balmer lines and c) A Balmer jump in the spectra (cf. Downes et al. 2001) are seen. If such a scenario can be confirmed, it could provide a diagnostic of dust formation during the expansion of wind in the nova. {height="11.0cm" width="14.0cm"} For the nova NMS-2, we do not have light curve measurements in the initial phase and hence the epoch of peak brightness cannot be estimated. Unlike the nova NMS-1 where we had observations for about three months, the nova NMS-2 could be monitored only for about 40 days and during this period, $R$ and $I$ flux show a similar trend, except that $I$ declines faster. This is consistent with emission coming from a region of lower radius but higher temperature. Probably, the flux in both bands shows a bump similar to that observed in some novae. What does our light curves imply about the formation of dust in the nova wind? Novae are believed to be a source of dust enrichment in the interstellar medium. However, there are two conflicting views (cf. Bode & Evans, chap9): (1) Graphite or silicate grains can be formed in the cooling wind of nova depending on the chemical composition of white dwarf. (2) The nova environment could be hot and might not be the conducive of dust formation, in which case dust grains already present in the shell could cause an infra-red excess. If the expanding cool wind in the nova can produce grains, we should be able to see the reduction in temperature followed by an infra-red excess. A good monitoring program that tracks the temperature, luminosity as well as possible spectroscopy will be helpful in settling the question. Our limited $R, I$ observations suggest that the possible neutral hydrogen formation, when ($R-I$) increases to about 1 mag from near zero, could act to shield dust grains against evaporation. This might be the reason for the slow increase in $I$ band flux despite the drop in $R$ band flux which is contributed in part by regions bluewards of the Balmer alpha line. Conclusion ========== We found two novae in our survey carried out during 1998 to 2001 for an $\sim 13'\times13'$ field belonging to the M31 disk population. The rate of decline of nova NMS-1 suggests that it was a fast nova while the decay rate for nova NMS-2 suggests that either it was a slow nova or we observed it at a very late stage after the maximum brightness phase. The nova NMS-1 becomes cooler with time about six weeks after the maximum brightness phase accompanied by a slow increase in $I$ band flux, although the $R$ band flux still continues to decrease. This is in contrast with the normal behaviour of the novae. We suspect a secondary bump in $R, I$ light curves of the nova NMS-2, which is not unusual for normal nova evolution.   [*Acknowledgments*]{} We are grateful to Stéphane Paulin-Henriksson for his help. We thank the anonymous referee for the useful comments. This study is a part of the project 2404-3 supported by Indo-French center for the Promotion of Advanced Research, New Delhi. Arp, H., 1956, AJ, 61, 15 Baillon, P., Bouquet, A., Kaplan J. & Giraud-Héraud Y., 1993, A&A, 277, 1. Bode, M.F. & Evans, A., 1989, Classical Novae, Chap. 9, (Publisher, John Willy & Sons Ltd) Bonifacio, P., Selvelli, P.L. & Caffau, E., 2000, A&A, 356, L53 Ciardullo, R., Ford, H.C., Neil, J.D., Jacoby, G.H. & Shafter, A.W., 1987, ApJ, 318, 520 Capaccioli, M., Della Valle, M., D’Onofrio, M. & Rosino, L., 1989, AJ, 97, 1622 Donato, L., Garzvia, S., Conano, V., Sostero, G. & Korlevic, K., 2001, IAUC 7516 Downes, R.A., Duerbeck, H.W. & Delahodde, C.E., 2001, JAD, 7, 6 Harrison, T.E. & Stringfellow, G.S., 1994, ApJ, 437, 827 Hubble, E., 1929, ApJ, 69, 103 Jacoby, C.H., Branch, D.G., Ciardullo, R., et al., 1992, PASP, 104, 599 Joshi, Y.C., Pandey, A.K., Narasimha, D. & Sagar, R., 2001, BASI, 29, 531 Joshi, Y.C., Pandey, A.K., Narasimha, D., Sagar, R. & Giraud-Héraud, Y., 2003a, A&A, 402, 113 Joshi, Y.C., Pandey, A.K., Narasimha, D. & Sagar, R., 2003b, BASI, presented in XXII ASI Meeting, Feb. 13-15, 2003 Joshi, Y.C., et al., 2003c, under preparation Landolt, A.U., 1992, AJ, 104, 340 Li, W.D., 2001, IAUC 7729 McLaughlin, D.B., 1945, PASP, 57, 69 Rector, T.A., Jacoby, G.H., Corbett, D.L., Denham, M., RBSE Nova Search Team 1999, BAAS, 195, 36.08 Rosino, L., 1964, A&A, 27, 498 Rosino, L., 1973, A&ASS, 3, 347 Rosino, L., Capacciol, M., D’Onofrio, M., & Della Valle, M., 1989, AJ, 97, 83 Sharov, A.S. & Alksnis, A., 1991, ApSS, 180, 273 Shafter, A.W. & Irby, B.K., 2001, ApJ, 563, 749 Stetson, P.B., 1987, PASP, 99, 191 Tomaney, A.B. & Shafter, A.W., 1992, ApJS, 81, 683 Van den Bergh, S., 1988, PASP, 100, 1486

--- abstract: | The recent excitement about Dirac fermion systems has renewed interest in magnetotransport properties of multi-carrier systems. However, the complexity of their analysis, even in the simplest two-carrier case, has hampered a good understanding of the underlying phenomena. Here we propose a new analysis scheme that strongly reduces the numerical uncertainty of previous studies and therefore allows to draw robust conclusions. This is demonstrated explicitly for the example of three-dimensional topological insulators. Their temperature and gate voltage-dependent Hall coefficient and transverse magnetoresistance behavior, including the phenomenon of huge linear transverse magnetoresistance, is fully reproduced by the scheme, allowing for an unambiguous identification of the carrier numbers and mobilities. Also the Fermi level can be determined from the analysis. We derive an upper limit for the transverse magnetoresistance as functions of mobility and field. Violation of this limit is a strong indication for field-dependences in the electronic band structure or scattering processes, that are not captured by our model, or for more than two effective carrier types. Remarkably, none of the three-dimensional topological insulators with particularly large transverse magnetoresistance violate the limit. [Keywords: Dirac fermion systems, topological insulators, Hall effect, magnetoresistance]{} author: - 'G. Eguchi' - 'S. Paschen' title: New scheme for magnetotransport analysis in topological insulators --- Topological insulators continue to be of tremendous interest to condensed matter physicists [@RevModPhys.82.3045; @RevModPhys.83.1057; @JPSJ.82.102001], both to advance the fundamental understanding of topological matter and to pave the way for new applications. Topologically protected states with linear band dispersion have been observed for numerous systems, including (Bi,Sb)$_2$(Se,Te)$_3$, TlBi(Se,Te)$_2$, Cd$_3$As$_2$, and (Ta,Nb)(As,P), by angle-resolved photoemission spectroscopy (ARPES) [@nphys1270; @PhysRevLett.105.146801; @PhysRevLett.105.136802; @nmat3990; @Xue1501092]. However, (magneto)transport evidence for electrical conduction by Dirac fermions – relativistic quasiparticle in the solid associated with these bands – has remained more circumstantial [@PhysRevB.81.241301; @PhysRevB.82.241306; @PhysRevB.91.041203; @srep04859; @nphys3372]. This is largely due to finite conductivity contributions from topologically trivial (e.g., bulk) bands. To disentangle the different effects, transport experiments are usually being analyzed with two-carrier models [@PhysRevB.82.241306; @science.1189792; @acsnano.5b00102; @nmat4143; @nphys3372]. Because of the large number of open parameters in these models, however, this has frequently lead to large uncertainties in the extracted information. As a consequence, unequivocal transport evidence for the expected ultrahigh mobilities of Dirac particles, a cornerstone for “topotronic” devices, remains to be found. Here we propose a new scheme for such analyses, that largely eliminates previous problems. It clarifies the physical meaning of both the $R_{\rm{H}}$ sign inversion and the huge linear transverse magnetoresistance (TrMR) [@srep04859; @nphys3372], phenomena deemed characteristic of Dirac fermion systems. It also allows to determine the Fermi level as well as the upper limit for TrMR as functions of the Hall mobility and field. We start by describing the differences between the common two-carrier analysis and our new scheme. In the former, the resistance $R_{xx}(B)$ and the Hall resistance $R_{xy}(B)$, where $B$ is the magnetic field, are characterized by four free parameters (the charge carrier numbers $n_1$ and $n_2$, and the mobilities $\mu_1$ and $\mu_2$) and two constant parameters (the charge carrier types $q_1$, $q_2 = \pm e$, where $e$ is the elementary charge) that need to be anticipated. Following the usual notation, $n_i$ is positive for both electrons and holes, whereas $\mu_i$ is negative for electrons and positive for holes. In the new scheme, only two free parameters are used, namely the relative charge carrier number difference $$N \equiv \frac{n_1-n_2}{n_1+n_2} \label{eq0a}$$ and the relative mobility difference $$M \equiv \frac{\mu_1-\mu_2}{\mu_1+\mu_2}\quad . \label{eq0b}$$ Two further parameters, the effective Hall coefficient $$R_{\rm{H}} \equiv \lim_{B \to 0} \frac{R_{xy}}{B} \label{eq0c}$$ and the effective Hall mobility $$\mu_{\rm{H}} \equiv \lim_{B \to 0}\frac{R_{xy}}{R_{xx}B} \label{eq0d}$$ can be directly read off the data. No [*ad hoc*]{} assumption has to be made on the charge carrier type. $N$ and $M$ can thus be determined with minimal ambiguity. $q_1$ and $q_2$ are determined as functions of $N$, $M$, and $\mu_{\rm{H}}$ (see supplemental material for the complete description). ![(Color online) (a,left) Sign inversion of $R_{xy}(B)$ traces typically observed in a multi-carrier system as a function of temperature $T$ or gate voltage $V_{\rm{G}}$ (1-4). The slope of $R_{xy}(B)$ varies continuously from 1 to 4. (a, right) Electronic band dispersion of a single Dirac cone, where $E$ is the energy and $k$ is the wave number (left), and corresponding $R_{xy}(B)$ traces at $T=0$ (right) as $E_{\rm{F}}$ is varied across the Dirac point (5-8). The sign change of $R_{xy}$ is accompanied by a discontinuity in the slope of $R_{xy}(B)$. (b) Contour plots of the Hall factor $\alpha$ for $q_2/q_1=-1$ as functions of $N$ and $M$, for $|\mu_1|>|\mu_2|$ (left) and $|\mu_1|<|\mu_2|$ (right). The $R_{\rm{H}}$ sign inversion occurs only for $|\mu_1|<|\mu_2|$, the situation of higher mobility minority carriers. The four situations (1-4) depicted in (a) could arise, for instance, along the arrow, with sign inversion between 2 and 3. (c) Contour plots of the transverse magnetoresistance TrMR for $q_2/q_1=-1$ and $|\mu_{\rm{H}}B|=1$ as functions of $N$ and $M$, for $|\mu_1|>|\mu_2|$ (left) and $|\mu_1|<|\mu_2|$ (right). The same arrow depicted in (c) indicates a monotonic increase between 2 and 3. The upper limits for the TrMR are 5.96 and 12.5 for $|\mu_1|>|\mu_2|$ and $|\mu_1|<|\mu_2|$, respectively.[]{data-label="fig1"}](fig1.pdf){width="\columnwidth"} Next we show how to understand the sign inversion in $R_{\rm{H}}$, observed in many Dirac fermion systems as a function of temperature $T$ or gate voltage $V_{\rm{G}}$ [@srep04859; @nphys3372]. Figure\[fig1\](a, left) depicts the typically observed signature in consecutive (1-4) $R_{xy}(B)$ traces: the slope varies continuously from 1 to 4. This is to be contrasted with the expectation for $R_{xy}(B)$ traces resulting from a single Dirac cone as the Fermi level $E_{\rm{F}}$ is varied across the Dirac point (5-8 in Fig.\[fig1\] (a, right)) at zero temperature ($T=0$) [@nature04235]. In this case the sign change of $R_{xy}$ is accompanied by a discontinuity in the slope of $R_{xy}(B)$. If $n_1$, $\mu_1$, and $q_1$ denote the majority carrier type ($n_1 > n_2$) and if charge carriers of opposite sign, i.e., both electrons and holes, are present ($q_2/q_1=-1$), $R_{\rm{H}}$ can be expressed as $$R_{\rm{H}}= \frac{1}{n_+q_1} \alpha\quad , \label{eq1a}$$ where $$n_+=n_1+n_2 \label{eq1b}$$ is the total charge carrier number and $$\alpha= \frac{N+2M+NM^2}{(N+M)^2} \label{eq1c}$$ is the Hall factor. The values of the parameters $N$ and $M$ can assume in this case follow from their definitions (Eqns.\[eq0a\] and \[eq0b\]) as $0<N<1$ and $1<|M|<\infty$. For any ($N$,$M$) combination within these limits $\alpha$ directly follows from Eqn.\[eq1c\]. Figure\[fig1\](b) shows contour plots of $\alpha$ in the full $N$ range and for $1<M<5$ (left) and $-5<M<-1$ (right). These describe the situations $|\mu_1|>1.5|\mu_2|$ and $|\mu_1|<0.667|\mu_2|$, respectively. Smaller mobility differences would be captured by plots to larger values of $|M|$. These are, however, less relevant here because we aim at separating contributions of highly mobile Dirac fermions from those of topologically trivial fermions with much lower mobility. Our first key result follows directly from these contour plots. A sign change of $\alpha$ occurs only in the right panel. Therefore, within the two-carrier picture, also $R_{\rm{H}}$ can show sign inversion only for $|\mu_1|<|\mu_2|$. In experiments on putative three-dimensional topological insulators (3D-TIs) where the observed sign inversion in $R_{\rm{H}}$ was taken as evidence for the presence of Dirac surface states [@PhysRevB.82.241306; @science.1189792; @acsnano.5b00102; @srep04859], Dirac fermions were thus the minority carriers, and transport was dominated by topologically trivial charge carriers of lower mobility, most likely associated with residual bulk states. To illustrate this further, two concrete examples of temperature and gate voltage tuning are given in what follows. {width="\textwidth"} ![(Color online) (a) Sketch of the electronic band dispersion of a 3D-TI around the Dirac point. An increase of $E_{\rm{F}}$ by $V_{\rm{G}}$ (5: lowest $E_{\rm{F}}$, 8: highest $E_{\rm{F}}$; left) and consecutive $R_{xy}(B)$ (center) and ${\rm{TrMR}}(B)$ (right) traces are also presented. (b) Contour plots of $\alpha(N, M)$ ($q_2/q_1=1$, left) around $|\mu_2/\mu_1|=5 (M=-0.667)$ and $\alpha(N, M)$ ($q_2/q_1=-1$, right) around $|\mu_2/\mu_1|=5 (M=-1.5)$. No sign inversion occurs at the Dirac point ($6\rightarrow 7$). (c) Contour plot of ${\rm{TrMR}}(N,\,M)$ for $|\mu_{\rm{H}}B|=1$, $q_2/q_1=1$ (left) and $q_2/q_1=-1$ (right). The minimum TrMR is observed at the Dirac point ($6\rightarrow 7$).[]{data-label="fig3"}](fig3.pdf){width="\columnwidth"} The experimentally observed $R_{\rm{H}}$ sign inversion as a function of temperature can be understood by taking the temperature dependence of the Fermi distribution function $f(E,T)$ into account. Figure \[fig2\](a) shows a sketch of the electronic band structure of a 3D-TI, for the situation where $E_{\rm{F}}$ lies above the Dirac point. The Dirac fermions are thus electron like (n-type carriers). The temperature dependence of $f(E,T)$ is also sketched, for different temperatures decreasing from 1 to 4. At high temperatures (1), the majority carriers ($n_1$) are thermally excited bulk holes (p-type carriers). With decreasing temperature ($1\rightarrow 4$), $n_1$ decreases exponentially, resulting in a decrease of $n_+$ and $N$. Minor variations are also expected for $n_2$, $\mu_1$, and $\mu_2$, but they are neglected here for simplicity. To extract information on the system at the sign inversion of $R_{\rm{H}}$, we replot a section of the $\alpha(N,\,M)$ contour plot of Fig.\[fig1\](b,right) around $|\mu_2/\mu_1|=5$ ($M=-1.5$), a situation considered realistic for experimentally studied 3D-TIs, in Fig.\[fig2\](b). Upon lowering the temperature ($1\rightarrow 4$), sign inversion ($\alpha = 0$) occurs at $N=0.923$, corresponding to only 4% of surface carriers ($n_2=0.04 n_1$). Larger mobility differences (smaller negative $M$ values, towards top of Fig.\[fig2\](b)) correspond to even smaller fractions of surface carriers. Gate voltage tuning can be mimicked by a variation of $E_{\rm{F}}$ around the Dirac point (Fig.\[fig3\](a,left), $E_{\rm{F}}$ increases from 5 to 8). At not too low temperatures, the above situation with minority surface carriers ($n_2$) is still relevant here because of the very small density of states of Dirac particles near the Dirac point. The corresponding $R_{xy}(B)$ traces (Fig.\[fig3\](a,center)) and $\alpha(N,\,M)$ contour plots for $q_2/q_1=1$ ($E_{\rm{F}}$ below Dirac point, 5 and 6 in Fig.\[fig3\](b,left)) and $q_2/q_1=-1$ ($E_{\rm{F}}$ above Dirac point, 7 and 8 in Fig.\[fig3\](b,right)) reveal the situation at $R_{xy}(B)$ sign inversion (details on $\alpha$ for $q_2/q_1=1$ are provided in the supplemental material): For the typical mobility ratio $|\mu_2/\mu_1|=5$ ($M=-0.667$ for $q_2/q_1=1$ and $M=-1.5$ for $q_2/q_1=-1$) also considered above, it occurs at $N=0.923$, corresponding to 4% of surface carriers, similar to the temperature tuning case. Interesting conclusions can, however, also be drawn if $R_{\rm{H}}(T)$ reveals no sign inversion. Such a situation may arise in a truly bulk-insulating 3D-TI where surface carriers are the majority carriers ($n_1$). Here, the exponential decrease of the number of minority bulk holes ($n_2$) with decreasing temperature ($1\rightarrow 4$) results in only a small decrease of $n_+$ (Fig. \[fig2\](d,center)) and an increase of the $N$ (Fig. \[fig2\](e), arrow assumes again a mobility ratio $|\mu_1/\mu_2|=5$ ($M=1.5$)). Such a minor effect on the $R_{xy}(B)$ isotherms can, on its own, hardly be taken as strong evidence for the detection of Dirac fermions. However, in conjecture with transverse magnetoresistance measurements, strong conclusions can be drawn, as detailed in what follows. The transverse magnetoresistance ${\rm{TrMR}}\equiv [R_{xx}(B)-R_{xx}(0)]/R_{xx}(0)$ of a two-carrier system with $q_2/q_1=-1$ $${\rm{TrMR}} = \frac{(N^2-1)M^2(1-M^2)(\mu_{\rm{H}}B)^2}{(2M+N+NM^2)^2+N^2(1-M^2)^2(\mu_{\rm{H}}B)^2} \label{eq2}$$ depends on the parameters $N$, $M$, and $\mu_{\rm{H}}$ defined in Eqns.\[eq0a\],\[eq0b\], and \[eq0d\]. As stated above, $\mu_{\rm{H}}$ can be directly read off from $R_{xy}(B)$ and $R_{xx}(B)$ data. Then, for a given $\mu_{\rm{H}}B$, TrMR follows for each ($N$,$M$) from Eqn.\[eq2\]. Thus, as for $\alpha$, contour plots of ${\rm{TrMR}}(N,\,M)$ can be determined and analyzed for the different situations of interest (Fig.\[fig1\](c), Fig.\[fig2\](c,f), Fig.\[fig3\](c)). The sign inversion in $R_{\rm{H}}(T)$ in a system with surface minority carriers (Figs.\[fig1\] and \[fig2\](a-c)) is, for the exemplary case of $|\mu_{\rm{H}}B|=1$, accompanied by a monotonic increase of TrMR$(T)$ with decreasing temperature, in agreement with recent experiments [@PhysRevB.90.201307]. By contrast, if surface carriers are the majority carriers (Fig.\[fig2\](d-f)), TrMR$(T)$ should decrease with decreasing temperature. The sign inversion in $R_{\rm{H}}(V_{\rm{G}})$ is accompanied by a distinct feature in TrMR: a non-monotonic variation of TrMR$(V_{\rm{G}})$ with a minimum of TrMR at the Dirac point (Fig.\[fig3\](c)). This insight establishes a new technique to determine the position of the Fermi level [@PhysRevB.91.235117; @PhysRevB.93.174428], complementary to that from Shubnikov-de Haas experiments [@PhysRevB.82.241306; @science.1189792] which are only possible in materials of extremely high quality and/or at very low temperatures. Thus, the simultaneous analysis of both $R_{xy}$ and TrMR within our two-carrier scheme is particularly rewarding and allows to draw robust conclusions. The phenomenon of TrMR has puzzled researchers since long. A single-carrier Drude model predicts zero TrMR. However, large TrMR values are reported even for the simplest metals such as potassium or copper. This inconsistency was recognized early on [@Ashcroft_Mermin; @Colin; @Pippard], and was highlighted again more recently in conjecture with the huge linear TrMR observed in Dirac fermion systems [@PhysRevB.81.241301; @PhysRevB.82.241306; @PhysRevB.91.041203; @srep04859; @nphys3372]. Our two-carrier analysis scheme advances the understanding of this phenomenon by revealing that, for any given $|\mu_{\rm{H}}B|$ value, there is an upper limit to the TrMR. {width="\textwidth"} Figure \[fig4\] shows this TrMR limit for the two cases $q_2/q_1=\pm 1$ and more mobile minority carriers ($|\mu_1| < |\mu_2|$). Interestingly, the TrMR limit increases linearly with $|\mu_{\rm{H}}B|$ for $|\mu_{\rm{H}}B| > 10^3$ and $|\mu_{\rm{H}}B| > 10^{-1}$ for $q_2/q_1=-1$ and $q_2/q_1=1$, respectively. The corresponding $N$ and $M$ values are also plotted (middle and lower panel). They suggest that linear increase of the TrMR limit with $|\mu_{\rm{H}}B|$ occurs if $|\mu_1| \ll |\mu_2|$. The largest TrMR values for various Dirac fermion materials reported in the literature, together with the corresponding $\mu_{\rm{H}}$ values at the same field, are also summarized in Fig. \[fig4\]. None of them overshoots the limit. This shows that, surprisingly, all TrMR observed to date are consistent with a two-carrier model. Let us not conclude without mentioning limitations of our considerations. The two-carrier model assumes that the charge carrier numbers and mobilities, and thus $N$ and $M$ are independent of $B$. Thus, systems with a strongly $B$-dependent electronic structure or scattering processes cannot be expected to be described. Whether or not a certain materials class obeys the TrMR limit discussed above is therefore an indication of the validity of these conditions. The fact that a large number of Dirac fermion systems all conform with the TrMR limit (Fig.\[fig4\]) underpins the validity of this analysis for this materials class. Strongly field-dependent parameters have for instance been observed in strongly correlated electron systems [@nphys3555; @PhysRevLett.111.056601], which indeed break the limit [@tobepublished]. Finally, we mention several possible applications of the new two-carrier analysis scheme. As the two-carrier model does not specify the origin of the carriers, the analysis is valid not only for intrinsic two-band transport, but can also describe extrinsic carriers arising from spatial inhomogeneity or by multi-layer films. In summary, we have put forward a new two-carrier analysis scheme that clarifies the physical meaning of the $R_{\rm{H}}$ sign inversion and the huge linear TrMR. Transport behavior observed in 3D-TIs is discussed. Several features, such as the absence of a sign inversion of $R_{\rm{H}}$ in the case where surface carriers are the majority carriers and a minimum of the TrMR at the Dirac point are suggested. Upper limit of the TrMR is determined as a function of $|\mu_{\rm{H}}B|$. All huge linear TrMR values observed to date in Dirac fermion systems are consistently explained within the analysis, implying the absence of $B$-dependent parameters in the two-carrier model. We acknowledge fruitful discussions with Kenta Kuroda, Akio Kimura, Yuuichiro Ando, and Masashi Shiraishi and financial support from the U.S. Army Research Office (grant W911NF-14-0496) and the Austrian Science Fund (project FWF I2535-N27). 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--- abstract: 'We have examined a method of direct extraction of accurate tight-binding parameters from an [*ab-initio*]{} band-structure calculation. The linear muffin-tin potential method, in its full-potential implementation, has been used to provide the hamiltonian and overlap matrix elements in the momentum space. These matrix elements are Fourier transformed to real space to produce the tight-binding parameters. The feasibility of this method has been tested on the intermetallic alloy NiAl, using [*spd*]{} orbitals for each atom. The parameters generated for this alloy have been used as input to a real-space calculation of the local density of states using the recursion method.' address: ' Department of Physics, West Virginia University, P.O. Box 6315, Morgantown, WV 26506-6315' author: - 'David Djajaputra and Bernard R. Cooper' title: 'Tight-binding parameters from the full-potential linear muffin-tin orbital method: A feasibility study on NiAl' --- 0[[**0**]{}]{} Introduction ============ There has been a growing interest in recent years to construct methods of electronic-structure calculation in which the required computational time scales linearly with the size of the system, the so-called $O(N)$ method.[@goedecker1999] The proposed approaches abandon the -space method used in convential band-structure calculations in favor of working directly in real space. [@yang1991; @galli1992; @galli2000; @goedecker1994; @kohn1996; @yang1997; @jayanthi1998] Instead of calculating the energy bands, one instead focuses on local properties, e.g., the local density of states. Extensive properties, like the total energy, are obtained by integrating over space instead of over the Brillouin zone (hence the linear scaling with the size of the system). This real-space approach is particularly relevant to cases in which lack of perfect crystalline symmetry is an essential part of the problem (e.g., in surfaces, impurities, defects, and amorphous materials) since in those cases Brillouin zone and energy bands are simply non-existent.[@heine1980] The tight-binding (TB) method is among the most popular implementations of this real-space strategy. The tight-binding approach to electronic structure has a long history which dates back to the classic work of Slater and Koster half a century ago. [@slater1954; @slater1965; @sharma1979; @harrison1989] It has been found especially useful in the modeling of the important technological material silicon.[@menon1997; @bernstein1997; @cohen1997; @lenosky1997] The conventional means to obtain the TB parameters is to follow the original procedure of Slater and Koster namely to fit the TB energy bands to the ones obtained from accurate self-consistent calculations. The parameters for elemental solids obtained in this way have been compiled by Papaconstantopoulos.[@papa1986] This database has also been actively enlarged to include parameters for various alloys.[@papa1996; @mehl1998; @yang1998; @papa1998] Recent interest in real-space electronic structure methods has supplied a renewed vigor to this field of TB parametrization.[@turchi1998; @goringe1997; @bowler1997; @ohno1999; @raabe1998] Despite its seemingly simple principle, the fitting process that one uses in obtaining the TB parameters is not always straightforward in practice. Starting from a guess set of TB parameters one calculates the TB energy bands which are then compared with the accurate bands from a self-consistent calculation. One can proceed to minimize the merit function by using standard nonlinear optimization procedures.[@press1999] This, in general, is quite a straightforward procedure for simple systems with one atom per unit cell. However, for more complicated multiatom unit cells with more intricate “spaghetti” of bands, the number of independent parameters on which the merit function depends grows rapidly. In the optimization process for this case, the merit function can get trapped easily in a local minimum and the output TB bands have little resemblance to the bands that they are intended to fit. Getting around this problem by choosing a good initial set of parameters, unfortunately, still comprises more art than science. Another drawback of the TB parameters that are obtained from fitting is that one normally fits only the bands below the Fermi energy and a few bands above it. In general the parameters obtained depend on the number of bands that are used in the fitting. This feature makes it difficult to attach a physical meaning to the parameters. Andersen and coworkers have proposed a method to obtain TB parameters directly from his linear muffin-tin orbital (LMTO) method.[@andersen1984; @andersen1985; @andersen1987; @andersen1994; @tank2000] This method has also been extended by other groups.[@xie1997] The method outputs the [*orthogonal*]{} hamiltonian (hopping) parameters between highly-localized (screened) muffin-tin orbitals in the two-center approximation. Since each hopping parameter effectively vanishes beyond second-nearest neighbors, TB calculations using these parameters can be highly efficient. Moreover, the obtained parameters are also independent of structure (transferable), a property which is highly desirable in TB applications. Most of these nice properties, however, depend on the atomic sphere approximation (ASA) which is used to derive these results.[@andersen1984a] This approximation has been known to work rather well for close-packed structures but is not very accurate for open structures, although this can be remedied to some extent by using the so-called empty spheres.[@tank2000] The TB-LMTO method assigns a minimal base (at most one [*s*]{}, three [*p*]{}, and five [*d*]{} orbitals per atom) to each atom in the crystal. In the LMTO-ASA method, the muffin-tin orbitals are set to have zero kinetic energy ($\kappa^2 = 0$) in the interstitial region. Full-potential (FP) implementations of linear electronic-structure methods, e.g., the FPLMTO method or the FP Linear Augmented Plane Wave (FPLAPW) method, are among the most accurate electronic-structure methods.[@andersen1975; @skriver1984; @nemoshkalenko1998; @loucks1967; @singh1994] It is therefore desirable to have a direct method of extracting practical TB parameters from these methods. The basis functions in the FPLAPW method are solutions to equation inside the MT sphere and plane waves in the interstitial region. Since the basis functions are not localized, its matrix elements therefore cannot be mapped into the TB form. (It should be noted, however, that it is possible to project a FPLAPW calculation into localized basis.[@portal1995]) The basis functions in the FPLMTO method, on the other hand, can be [*chosen*]{} (by setting the value of the kinetic energy $\kappa^2$) to have a decaying tail outside its MT sphere. Direct mapping to TB parameters is therefore possible for the FPLMTO method. To our knowledge this direct approach has been seriously studied only by McMahan and Klepeis who used it to study Si and Si/B phases. [@mcmahan1997; @mcmahan1998; @manh2000] In this paper we use this direct approach and extract TB (hamiltonian and overlap) parameters for the intermetallic alloy NiAl from a FPLMTO method. These parameters are then used as input to a real-space calculation of the local density of states using the recursion method. To our knowledge, this is the first test of this direct method to a transition-metal alloy. Our choice of NiAl is determined by our previous experience with it[@djajaputra2001] and by the fact that NiAl has a simple B2 structure in which each atom in the unit cell has the full cubic point group. The latter is of practical relevance since the symmetry allows us to reduce the number of independent TB parameters that need to be calculated and saved in the database. Since the TB parameters that we extract from the FPLMTO method are to be used in a recursion calculation, we do not need to extract the (two-center) Slater-Koster parameters as was done by McMahan and Klepeis. [@mcmahan1997] Our motivation in performing this calculation is twofold. First, one hopes to obtain a general idea of the level of numerical accuracy that can be expected to result from a combination method like this. Second, eventually one would like to use the TB parameters extracted from this method in applications where it is too expensive computationally to use direct [*ab-initio*]{} methods, e.g., in studying the effects of a small concentration of impurity atoms to static and dynamic properties of alloys. Since the TB method is a [*non*]{}-self-consistent method, the plausibility of its result can only be judged by how well it reproduces certain “benchmark” results. In this paper the “benchmarks” are the local density of states that can also be obtained from the FPLMTO method. Section \[fplmto\_section\] of this paper discusses the specific implementation of the FPLMTO method that we use for this work. The choice of basis is of the utmost importance in a TB calculation. Since the FPLMTO basis functions are well-known for being complicated we have chosen to discuss it in some details in Sec. \[basis\_functions\]. Practical points regarding the transformation to real space are listed in Sec. \[fourier\_transform\]. The recursion calculation and the results that we obtained from it are discussed in Sec. \[recursion\]; and major points of the paper are discussed in a summary section at the end of the paper. FPLMTO Method {#fplmto_section} ============= In this paper we have used an implementation of the FPLMTO method that is developed by Wills and Price.[@price1989; @price1992; @wills2000]. Several other implementations are available in the literature and, although the main formalism is largely the same, each implementation is different in its details.[@nemoshkalenko1998; @savrasov1992; @methfessel1988; @methfessel1989] The FPLMTO uses a fixed basis set in calculating its hamiltonian and overlap matrix elements; it is therefore similar to the linear combination of atomic orbitals (LCAO) method.[@ziman1971] However, while in the LCAO method the basis set is determined, and fixed, at the beginning, the basis set in the FPLMTO method is determined self-consistently. The FPLMTO method is very similar to the Augmented Spherical Wave (ASW) of Williams, Kübler, and Gelatt. [@williams1979] One of the strengths of the FPLMTO method is that it allows unlimited number of basis functions to be assigned to each atom in the unit cell. This provides unlimited variational flexibility in finding the ground state configuration. The practical requirement of TB, however, requires us to use a minimal basis set. Furthermore, although FPLMTO method allows the use of localized ($\kappa^2 < 0$) or unlocalized ($\kappa^2 >0$) basis,[@springborg1987] only the localized basis should be used for TB purposes. Williams [*et al.*]{} have used $\kappa^2 \approx -0.2$ Ry exclusively in their ASW paper.[@williams1979] They have also warned that the use of a single fixed $\kappa$ is rather restrictive and cited the work of Gunnarsson [*et al.*]{} [@gunnarsson1976] that shows the variation of eigenenergies with $\kappa$ in a molecular calculation. Basis Functions {#basis_functions} --------------- In the FPLMTO method the crystal is divided into non-overlapping muffin-tin (MT) spheres surrounding each atomic sites. The radius of the MT sphere that is centered at $\boldtau_\alpha$, the position of the $\alpha$-th atom in the unit cell, is denoted by $s_\alpha$. On this muffin-tin geometry we construct a muffin-tin potential. This potential is spherically symmetric within each muffin tin and equal to a constant ${\rm V_0}$, the muffin-tin zero, at the interstitial region. The potential inside the MT, $v_{\rm MT}(r)$, does [*not*]{} have to join continuously to the interstitial potential ${\rm V_0}$ at the MT radius. We emphasize that in the FPLMTO method the MT potential is used [*solely*]{} to construct the basis functions. The final output charge density and potential, which are obtained from a self-consistent calculation, do not necessarily have the form of a MT potential. During each iteration in the self-consistency loop, the LMTO basis functions are used to solve the (or Dirac) equation in a variational manner similar to the LCAO method. The resulting eigenfunctions are then used to calculate the new charge density and potential. The important difference from the LCAO method is that here, in addition to the charge density and potential, one also updates the basis functions on every step. This update is performed by constructing a new MT potential from the output potential. The constant potential ${\rm V_0}$ is given the value of the average potential at the interstitial, while the spherically-symmetric potential $v_{\rm MT}(r)$ is obtained from the angular average of the potential inside the muffin tin.[@loucks1967; @springborg2000; @ham1961; @mattheiss1968] Note that we average the potential rather than the charge density as used in an alternative scheme.[@liberman1967] The final output MT orbitals therefore provide the “best” set of basis functions (within the variational freedom imposed by the choice of the LMTO parameters) for performing an LCAO-like calculation on the system under consideration. This view of the MT orbitals as basis functions for an LCAO-type calculation has appeared from time to time in the literature, under the name of the linear combination of muffin-tin orbitals (LCMTO) method. It has been particularly useful for calculations of the electronic structure of systems that do not possess lattice periodicity, e.g., surfaces, impurities, and atomic clusters.[@andersen1971; @andersen1973; @kasowski1976; @harris1980] Unfortunately, the difficulty in performing the necessary three-dimensional integrations in this method has made it less popular relative to other methods that utilize orbitals with simpler analytical properties like the Slater- or Gaussian-type orbitals.[@springborg2000; @kasowski1976; @porezag1995] Each basis function in the FPLMTO method is centered around an atomic site that we will call its host site. With respect to this site, all other atomic sites will be referred to as remote sites. The basis function is constructed as a continuous patchwork of three different parts: (1) inside the host MT sphere; (2) at the interstitial region; (3) inside all remote MT spheres. Thus the basis function centered at atomic site $\boldtau_\alpha$ inside the unit cell with lattice vector $\boldR_i$ has the form: $$\phi_{i \alpha}({\bf r}) = \phi_{i \alpha}(i \alpha | {\bf r}) + \phi_{i \alpha}(I | {\bf r}) + {\sum_{i' \alpha'}}' \phi_{i \alpha}(i' \alpha' | {\bf r}).$$ Here the primed summation means that it should be carried out over all $(i'\alpha') \neq (i \alpha).$ The MTO also carries other indices: $L \equiv (lm)$ which are the angular-momentum quantum numbers of the part of the MTO inside the host sphere $\phi_{i \alpha}(i \alpha | {\bf r})$; the tail parameter $\kappa$ which controls the behavior of the orbital at the interstitial region; and possibly also the principal quantum number $n$ which controls the number of radial nodes of $\phi_{i \alpha}(i \alpha | {\bf r})$. The last is usually not necessary since one normally assigns only one principal quantum number for each $(lm)$. In any case, for clarity we will suppress these indices unless their presence is really necessary. An MTO is essentially an augmented spherical wave (ASW).[@williams1979] This means that it is constructed by defining it to be a spherical wave (spherical Bessel, Neumann, or Hankel function, i.e. solution of Helmholtz equation in spherical coordinates) in the interstitial region: $$\phi_{i\alpha}(I | {\bf r}) = K_l(\kappa,r_{i \alpha}) \cdot i^l Y_{lm}(\hat{\bf r}_{i \alpha}) \cdot \Theta(I|\boldr), \label{interstitial}$$ with a radial part:[@wills2000; @springborg1987] $$K_l(\kappa,r) = - \kappa^{l+1} \times \cases{n_l(\kappa r) - i j_l(\kappa r), \ &$\kappa^2 < 0$, \cr n_l(\kappa r), \ &$\kappa^2 > 0$.} \label{interstitial_radial}$$ Here ${\bf r}_{i \alpha} \equiv ({\bf r} - \boldR_{i \alpha})$, $\boldR_{i \alpha} \equiv (\boldR_i + \boldtau_\alpha)$, $r_{i \alpha} \equiv |\boldr_{i \alpha}|$, and $\hat{\bf r}_{i \alpha} \equiv \boldr_{i \alpha} / r_{i \alpha}$. The masking function $\Theta(I|\boldr)$ is equal to 1 if $\boldr$ is in the interstitial region and 0 otherwise. For $\kappa^2 < 0$, $\kappa = i |\kappa|$, this spherical wave is usually expressed in terms of the spherical Hankel function of the first kind:[@springborg1987] $$K_l(\kappa,r) = i \kappa^{l+1} h_l^{(+)}(\kappa r) = {e^{-z} \over r} \times \cases{1, &$l=0$, \cr |\kappa| (1 + z^{-1}), &$l=1$, \cr |\kappa|^2 (1 + 3 z^{-1} + 3 z^{-2}), &$l=2$,}$$ where $z \equiv |\kappa| r$. Since this spherical wave is the envelope of the full MTO we see that, for negative $\kappa^2$, the MTO has a tail that decays exponentially with distance from its host site. The decay rate is controlled directly by the magnitude of $\kappa$. This feature makes it suitable for tight-binding calculations. Inside the MT spheres, the envelope spherical wave is augmented (i.e. replaced) with a solution of the equation for the MT potential. The augmentation process is similar to the augmentation of the plane wave in the Augmented Plane Wave (APW) method.[@loucks1967; @singh1994] The difference, of course, is that here we are augmenting a spherical wave instead of a plane wave. Note that the basis functions in the APW method are not suitable for tight binding since their plane-wave envelopes are not localized in space. As we approach the host MT sphere from the outside, the basis function is determined by Eq.(\[interstitial\]). Note that, relative to the origin at the host site, this function is separable, i.e. it is a product of a radial and an angular part. Inside the host MT sphere, this function is augmented with a solution of the equation for the MT potential. Since this potential is spherically symmetric inside the MT, the basis function inside the host MT is also separable: $$\phi_{i \alpha}(i \alpha | {\bf r}) = \big[ A \varphi_{i \alpha l} (e_{nl},r_{i \alpha}) + B \dot{\varphi}_{i \alpha l}(e_{nl},r_{i \alpha}) \big] \cdot i^l Y_{lm}(\hat{\bf r}_{i \alpha}) \cdot \Theta(i \alpha | \boldr ), \label{host_sphere}$$ where the MT masking function $\Theta(i \alpha | \boldr )$ is equal to 1 if $\boldr$ is inside the MT sphere $(i \alpha)$ and 0 otherwise. The energy dependence of the radial solution has been approximated with a linear combination of the radial solution $\varphi_{i \alpha l}$ and its energy derivative $\dot{\varphi}_{i \alpha l}$ ($\equiv \partial \varphi_{i \alpha l} / \partial \varepsilon$) calculated at a fixed chosen energy $e_{nl}$.[@andersen1975] The coefficients are obtained by requiring the function to match continuously and smoothly with the interstitial function at the MT radius: $$\left[ \matrix{ \varphi_{i \alpha l} (s_{i \alpha}) & \dot{\varphi}_{i \alpha l} (s_{i \alpha}) \cr \varphi_{i \alpha l}' (s_{i \alpha}) & \dot{\varphi}_{i \alpha l}' (s_{i \alpha})} \right] \cdot \left[ \matrix{ A \cr B} \right] = \left[ \matrix{ K_l(\kappa, s_{i \alpha}) \cr K_l'(\kappa, s_{i \alpha})} \right].$$ Here a prime denotes differentiation with respect to the radial coordinate: $\varphi' \equiv \partial \varphi / \partial r$. The augmentation process at the remote spheres proceeds rather differently from the one for the host sphere. As we approach a remote MT sphere from the interstitial region, the basis function is again determined by Eq.(\[interstitial\]). The important difference from the host-sphere case is that relative to the origin at the remote site the interstitial function is [*not*]{} separable. We can, however, express the spherical Hankel function at the interstitial region as a sum of separable functions in the neighbourhood of the remote site:[@danos1965; @nozawa1966; @talman1968; @andersen1971a; @gonis2000] $$K_l(\kappa,r_{i \alpha}) \cdot i^l Y_L(\hat{\bf r}_{i \alpha}) = \sum_{L'} J_{l'}(\kappa,r_{i' \alpha'}) \cdot i^{l'} Y_{L'}(\hat{\bf r}_{i' \alpha'}) \cdot S_{L'L}(\kappa,\boldR_{i' \alpha'} - \boldR_{i \alpha}), \qquad r_{i'\alpha'} < | \boldR_{i'\alpha'} - \boldR_{i \alpha} |, \label{addition_theorem}$$ with the spherical Bessel function as the radial part in each term: $$J_l(\kappa,r) = \kappa^{-l} j_l(\kappa r) = {1 \over z} \times \cases{\sinh z, &$l=0$, \cr |\kappa|^{-1} (\cosh z - z^{-1} \sinh z), &$l=1$, \cr |\kappa|^{-2} [(3z^{-2} + 1)\sinh z - 3 z^{-1} \cosh z], &$l=2$.}$$ The addition theorem for the Bessel functions, Eq.(\[addition\_theorem\]), is a consequence of the fact that both sides of the identity are solution of the translationally-invariant Helmholtz equation. The spherical Hankel function, $K_l(\kappa,r)$, is regular everywhere except at the origin, while the spherical Bessel function, $J_l(\kappa,r)$, is regular everywhere except at infinity. The domain of validity of Eq.(\[addition\_theorem\]) is just the domain where both sides of the identity are regular. The radial part of each term in the one-center expansion, Eq.(\[addition\_theorem\]), is centered at the remote site $\boldR_{i' \alpha'}$. Thus the $(\varphi,\dot{\varphi})$ augmentation of the interstitial function, Eq.(\[interstitial\]), at this remote site can be performed by augmenting each $J_l(\kappa,r_{i'\alpha'})$ with the radial part of the solution of the equation for the spherically-symmetric MT potential centered at $\boldR_{i'\alpha'}$. Thus we replace: $$J_l(\kappa,r_{i'\alpha'}) \rightarrow \big[ C_l \varphi_{i'\alpha'l}(e_{nl},r_{i'\alpha'}) + D_l \dot{\varphi}_{i'\alpha'l}(e_{nl},r_{i'\alpha'}) \big].$$ The coefficients are again obtained by requiring continuity in the value of the function and its first derivative with those of the interstitial envelope function at the MT radius: $$\left[ \matrix{ \varphi_{i' \alpha' l} (s_{i' \alpha'}) & \dot{\varphi}_{i' \alpha' l} (s_{i' \alpha'}) \cr \varphi_{i' \alpha' l}' (s_{i' \alpha'}) & \dot{\varphi}_{i' \alpha' l}' (s_{i' \alpha'})} \right] \cdot \left[ \matrix{ C_l \cr D_l} \right] = \left[ \matrix{ J_l(\kappa, s_{i' \alpha'}) \cr J_l'(\kappa, s_{i' \alpha'})} \right].$$ The part of basis function inside a remote MT sphere is thus given by: $$\phi_{i \alpha}(i' \alpha' | {\bf r}) = \Theta(i' \alpha' | \boldr ) \cdot \sum_{L'}^{l \leq l_m} \big[ C_{l'} \varphi_{i'\alpha'l'}(r_{i'\alpha'}) + D_{l'} \dot{\varphi}_{i'\alpha'l'}(r_{i'\alpha'}) \big] \cdot i^{l'} Y_{L'}(\hat{\bf r}_{i' \alpha'}) \cdot S_{L'L}(\kappa,\boldR_{i' \alpha'} - \boldR_{i \alpha}). \label{remote_sphere}$$ The summation on the r.h.s. of the equation ideally runs to infinity. Practically, a converged total energy is achieved using $l \leq l_m \sim 6-8$ in most cases.[@wills2000] Note that, inside the remote sphere, an angular-momentum expansion is necessary since the resulting function needs to match the envelope interstitial function at its MT sphere. Relative to this remote site, the envelope function contains multiple harmonics since it is centered at a different site. The MT orbital, constructed from Eqs.(\[interstitial\]), (\[host\_sphere\]), and (\[remote\_sphere\]), is continuous and smooth everywhere. Note, however, that it is not normalized and it does [*not*]{} have a separable form: $$\phi_{i \alpha}({\bf r}) \neq u(r_{i \alpha}) Y_{lm}(\hat{\bf r}_{i \alpha}),$$ in other words, the MTO does not have good angular-momentum quantum numbers. Although the parts of the MTO inside the host MT and at the interstitials are separable, the presence of the remote sites breaks the perfect spherical symmetry around the host site, and therefore also removes the separability. The MTO, however, does obey exactly the point symmetry of its host site. Moreover, since the non-separable contributions come from the remote sites, where the amplitude of the exponentially-decaying envelope function is small, to a good approximation the MTO does transform as if it has the quantum numbers $(lm)$ of the part of the orbital inside the host sphere. The fact that the MTO is not fully separable should be noted especially if one intends to map the FPLMTO result into a TB form using the Slater-Koster parametrization method. In the SK-LCAO parametrization method the basis functions are assumed to be atomic orbitals; these orbitals are separable since they are derived from a single-atom spherically-symmetric potential.[@slater1954; @slater1965; @sharma1979; @harrison1989] Fourier Transform {#fourier_transform} ----------------- In applications of the FPLMTO method to ideal crystalline systems, instead of using the basis functions in real space, one works with the Bloch functions: $$\psi_\alpha (\boldk, \boldr) = {1 \over \sqrt{N}} \sum_i \exp(i \boldk \cdot \boldR_i ) \ \phi_{i \alpha}(\boldr). \label{bloch_function}$$ The energy bands are obtained by solving the eigenvalue equation: $$\det | H_{\alpha \beta} (\boldk) - E S_{\alpha \beta}(\boldk) | = 0. \label{nonorthogonal_eigenvalue}$$ The rank of the matrices is equal to the total number of the orbitals used for all the atoms within the unit cell. The hamiltonian matrix is given by integration over an arbitrarily chosen unit cell $\boldR_n$: $$H_{\alpha \beta} (\boldk) = \langle \psi_\alpha (\boldk, \boldr) | H | \psi_\beta(\boldk,\boldr) \rangle = {1 \over N} \sum_{ij} e^{i \boldk \cdot (\boldR_j - \boldR_i)} \sum_n \Big[ \langle \phi_{i \alpha} ( I_n) | \ H \ | \phi_{j \beta} ( I_n ) \rangle + \sum_{\gamma} \langle \phi_{i \alpha} ( n \gamma ) | \ H \ | \phi_{j \beta} (n \gamma ) \rangle \Big], \label{hamiltonian_matrix}$$ and similar expression is obtained for the overlap matrix by replacing the hamiltonian operator $H$ with identity. The vectors have been defined by: $$\langle \boldr | \phi_{i \alpha} (n \gamma) \rangle = \phi_{i \alpha} (n \gamma | \boldr),$$ which is the part of the basis function, centered at the host site $i \alpha$, inside the MT sphere $n \gamma$ or the interstitial region $I_n$, which is defined to be the interstitial region within the unit cell $\boldR_n$. Cross terms are absent from the square bracket in Eq.(\[hamiltonian\_matrix\]) since the overlap of their masking functions is zero. It should be mentioned that in the actual FPLMTO formalism, the matrix elements are not computed using a multicenter expansion (over all lattice sites) as shown in Eq.(\[bloch\_function\]). Rather, using the addition theorem for the tail of the MT orbital, Eq.(\[addition\_theorem\]), the multicenter expansion is first transformed into a one-center expansion (over angular momenta at a single site), and this sum is the one that is actually computed. [@skriver1984; @wills2000] This, however, is merely a computational contrivance, albeit a pivotal one in the FPLMTO method, and should not mask the true structure of the basis function in [**k**]{}-space as a simple Bloch sum of the MT orbitals. In the FPLMTO method of Wills [*et al.*]{}[@wills2000], the fact that each term of the matrix elements is to be calculated only over a certain region, either inside the MT sphere or the interstitial region as seen in Eq.(\[hamiltonian\_matrix\]), is used to facilitate its computation. Instead of using the true MTO, which is complicated, one uses a pseudo basis orbital.[@weinert1980] This orbital is equal to the true MTO inside the relevant integration region but outside the region it is replaced with a smooth function which is chosen to facilitate the computation. The integration can then be performed efficiently by working in the momentum space. Using the translational properties of the Bloch functions, the matrix element can be written as: $$H_{\alpha \beta} (\boldk) = \sum_j \exp(i \boldk \cdot \boldR_j ) \ H_{\alpha \beta} (\boldR_j), \label{k_from_r}$$ with the matrix elements in real space: $$H_{\alpha \beta}(\boldR_j) = \sum_n \Big[ \langle \phi_{0 \alpha} ( I_n) | \ H \ | \phi_{j \beta} ( I_n ) \rangle + \sum_{\gamma} \langle \phi_{0 \alpha} ( n \gamma ) | \ H \ | \phi_{j \beta} (n \gamma ) \rangle \Big].$$ To calculate the inverse transform of Eq.(\[k\_from\_r\]) we use the fact that the real-space matrix element $H_{\alpha \beta}(\boldR)$ decays exponentially with the distance $|\boldR|$. It is thus reasonable to ignore matrix elements between orbitals separated by a distance greater than a certain cutoff radius $R_c$. In practice this can be performed by working with a finite crystal which contains $N_x$ unit cells along the $x$-direction such that $N_x a_x > 2 R_c$, with $a_x$ being the lattice constant along the $x$-direction, and similar ranges for the $y$ and $z$ directions. The real-space matrix elements can then be obtained from a discrete Fourier transform: $$H_{\alpha \beta} (\boldR_j) = {1 \over N} \sum_\boldk \exp(-i \boldk \cdot \boldR_j ) \ H_{\alpha \beta} (\boldk). \label{r_from_k}$$ Here $N = N_x N_y N_z$ and $k_i = \pi n_i/(N_i a_i)$ with $-N_i < n_i \leq N_i$ for $i \in \{ x, y, z \}$. Some practical remarks should be mentioned regarding this calculation of the inverse Fourier transform. First, the output matrix elements from the FPLMTO are calculated using the spherical harmonic $i^l Y_{lm}(\unitr)$ as the angular part of the basis function inside its host sphere. Usually it is easier to work with real basis functions and this can be achieved by making linear combinations of the spherical harmonics.[@weissbluth1978] Second, the basis functions are not normalized. The normalized matrix elements are obtained by using the overlap matrix elements: $$H_{\alpha \beta}'(\boldR) = S_{\alpha \alpha} (\bold0)^{-{1 \over 2}} \cdot H_{\alpha \beta} (\boldR) \cdot S_{\beta \beta} (\bold0)^{-{1 \over 2}}.$$ Third, the actual displacement vector connecting the centers of the two orbitals in $H_{\alpha \beta} (\boldR)$ is not $\boldR$ but is instead $(\boldR + \boldtau_\beta - \boldtau_\alpha)$. Instead of the original hamiltonian matrix in Eq.(\[k\_from\_r\]), it is preferable to work with a modified matrix that is related to the original by a unitary transformation: $$H_{\alpha \beta}''(\boldk) = e^{i \boldk \cdot (\boldtau_\beta - \boldtau_\alpha)} H_{\alpha \beta}(\boldk) = \sum_j e^{i \boldk \cdot (\boldR_j + \boldtau_\beta - \boldtau_\alpha)} \ H_{\alpha \beta} (\boldR_j). \label{hopping}$$ This positions the origin at the center of orbital $(0\alpha)$ and allows the matrix elements to obey the point symmetry of the corresponding site. It may therefore be used to reduce the number of matrix elements that need to be stored. Note that the eigenvalues are unaffected by this unitary transformation. Löwdin Orthogonalization {#lowdin_transform} ------------------------ The real-space hamiltonian matrix elements in Eq.(\[r\_from\_k\]) (together with the overlap matrix elements $S_{\alpha \beta}$ which can be obtained similarly) can be used directly in a nonorthogonal recursion calculation.[@riedinger1989; @ballentine1986; @mckinnon1995] Unfortunately, this requires us to obtain the inverse of the overlap matrix in real space, which is a non-trivial computational task. In this paper we instead use an orthogonal hamiltonian which is obtained by Löwdin orthogonalization of the original hamiltonian and overlap matrices. [@weissbluth1978] An alternative scheme is to use the chemical pseudopotential approach and work with a non-hermitian matrix $S^{-1}H$.[@bullett1980; @foulkes1993] The advantage of Löwdin orthogonalization is that it is a symmetry transformation: angular symmetry is preserved, i.e., orbitals with angular momentum $l$ are not mixed by the transformation with orbitals with $l' \neq l$. [@altmann1995] This means that the real-space matrix elements can still be parametrized by Slater-Koster parametrization, e.g., using the procedure proposed by McMahan and Klepeis.[@mcmahan1997] Since we are working with a perfect crystal, the Löwdin orthogonalization process can be applied to the momentum-space matrix elements. One first diagonalize the overlap matrix: $$D=U^\dagger \cdot S \cdot U,$$ where $U$ is the matrix containing the eigenvectors of $S$ as its columns, and $D$ is a diagonal matrix containing the eigenvalues of $S$. Using these matrices, we define a new matrix: $$A = U \cdot {D^{-{1 \over 2}}} \cdot U^\dagger.$$ Since the overlap matrix is a positive-definite hermitian matrix, its eigenvalues are all positive real numbers and the matrix $D^{-{1 \over 2}}$ is well defined. In practice, however, small negative eigenvalues of $S$ may sometimes appear due to the finite machine precision used in the computation. This problem is especially relevant if the diagonalization of the overlap matrix is performed in real space and when one uses an overlap matrix which is necessarily [*approximate*]{} due to the finite cutoff imposed on the range of the matrix elements or due to the Slater-Koster two-center approximation used in parametrizing them. One commonly-used fix to this problem is to add a diagonal matrix with small elements to make the overlap matrix positive definite.[@roder1997] The orthogonal Löwdin hamiltonian matrix is obtained by sandwiching the original hamiltonian matrix with $A$ matrices: $$H^{(L)} = A \cdot H \cdot A.$$ The nonorthogonal eigenvalue problem, Eq.(\[nonorthogonal\_eigenvalue\]), is then transformed to an equivalent orthogonal one: $$\det | H_{\alpha \beta}^{(L)} (\boldk) - E \delta_{\alpha \beta}(\boldk) | = 0. \label{orthogonal_eigenvalue}$$ The orthogonal hopping parameters are obtained by substituting $H_{\alpha \beta}^{(L)} (\boldk)$ in Eq.(\[r\_from\_k\]). These are the parameters that we use in the recursion calculation which is described in the rest of this paper. Recursion Calculation {#recursion} ===================== The recursion method, initially introduced to electronic-structure calculation by Haydock, is one of the most efficient and stable ways of calculating the electronic Green’s function from the TB hamiltonian describing the system:[@haydock1980; @pettifor1985] $$G_{uv}(E) = \langle u | (E- H)^{-1} | v \rangle,$$ where $|u \rangle$ and $|v \rangle$ are two arbitrary vectors in the Hilbert space for the problem. The local density of states (LDOS) is a projection of the total density of states to a local orbital: $$\rho_\alpha(E) = \sum_n | \langle \alpha | \Psi_n \rangle |^2 \ \delta(E - E_n),$$ where $| \Psi_n \rangle$ is an eigenvector with eigenvalue $E_n$. This can be obtained from the Green’s function: $$\rho_\alpha(E) = - {1 \over \pi} \ {\rm Im} \ \big[ G_{\alpha \alpha}(E + i \epsilon) \big],$$ with $\epsilon \rightarrow 0^+$. The token Im$[\cdots]$ denotes the operation of taking the imaginary part of its argument. The recursion method outputs the Green’s function as a continued fraction: $$G(E) = {1 \over (E - a_0) - {\textstyle b_1^2 \over {\textstyle (E - a_1) - {\textstyle b_2^2 \over {\textstyle \hphantom{E} \cdots \hphantom{E} } }}}}. \label{greens_function}$$ Note that there are two conventions used in the literature regarding the indexing of the $a$-parameters in the above continued fraction. One convention is to assign $a_0$ as the first $a$-parameter,[@haydock1980; @ballentine1986] while the other use $a_1$ as the first parameter.[@turchi1982] Here we follow the first convention. The calculation of the LDOS is therefore reduced to the computation of the strings of recursion coefficients $\{ a_n \}$ and $\{ b_n \}$. The procedure for performing this has been described extensively in the literature and will not be repeated here.[@haydock1980; @pettifor1985] We have used the recursion method to calculate the LDOS for NiAl crystal. This alloy crystallizes in B2 structure with two atoms per (cubic) unit cell. One component (Ni or Al) sits at $(0,0,0)$, while the other is at the body-center of the cube, $( {1 \over 2}, {1 \over 2}, {1 \over 2})$. We obtained the real-space orthogonal hopping parameters by direct extraction and Löwdin transformation as described in the preceding section. The parameters were extracted from a single-$\kappa$ FPLMTO calculation using 9 $spd$ orbitals and $\kappa^2 = -0.04$ Ry for each Ni or Al atom. This rather small value of $\kappa$ produces basis functions with relatively long range (the envelope of the basis function decays roughly as $e^{-|\kappa| r}$). Using a larger negative value of $\kappa$, however, produces a greater discrepancy between the output bands from this single-$\kappa$ calculation and the corresponding accurate bands from a multiple-$\kappa$ calculation. The chosen value of $\kappa$ that we used is obtained by compromising the need to have a short-ranged TB basis functions with the requirement to have a precision comparable to accurate multiple-$\kappa$ result. All of the extracted parameters are saved in a database for use in the recursion calculation (crystal symmetry was used to reduce the size of the database). They were calculated for separation distance of up to 8 lattice constants. The obtained TB parameters fall roughly one order of magnitude for each increase in separation distance of one lattice constant. The recursion calculation was performed using a cubic cluster of $(16)^3$ unit cells and the LDOS’s were computed for Ni and Al orbitals within the unit cell at the center of the cluster. Since the TB parameters connecting the orbitals at the central unit cell to the ones at the surface of the cluster are practically zero, the calculated LDOS should not depend on the size of the cluster if we increase it further. Figs. \[Ni\_d\_parameters\] and \[Al\_p\_parameters\] show the calculated recursion parameters for Ni-$d$ and Al-$p$ states, respectively. Here the $d$-state is defined by: $$| d \rangle = {1 \over \sqrt{5}} \sum_{m=-2}^2 |l=2,m \rangle,$$ with a corresponding definition for the $p$-state. It is seen that the parameters tend to settle around certain constant values. Indeed it is known that as $n \rightarrow \infty$: [@turchi1982] $$a_n \rightarrow (E_t + E_b)/2, \quad b_n \rightarrow W/4,$$ where $E_t$ and $E_b$ are the values of energy at the top and bottom of the spectrum, respectively, and $W = (E_t - E_b)$ is the total bandwidth. Furthermore, asymptotically the parameters should oscillate around their limit values with a decaying amplitude as $n \rightarrow \infty$.[@gaspard1973] The frequency and the rate of decay are determined by specific features (the Van Hove singularities) of the spectrum.[@hodges1977] The recursion parameters shown in Figs. \[Ni\_d\_parameters\] and \[Al\_p\_parameters\] are seen to roughly follow this decayed oscillation prediction. However, for large values of $n$ (greater than $n_m \sim 40$), the amplitudes of the deviation tend to increase rather than decrease. We interpret this as the limit at which the machine precision noise start to interfere with the calculation. This interpretation is supported by our experience that incorporating the recursion parameters with $n > n_m$ in the reconstruction of LDOS in general does not help improve the agreement, and indeed it tends to reduce the agreement, with the accurate LDOS obtained from the FPLMTO method. Note that the main workhorse in the recursion method involves a matrix-vector multiplication with the dimension of the vector being the total number of orbitals in the cluster. In our case this dimension is equal to 73728 (= $16^3$ unit cells $\times$ 2 atoms/cell $\times$ 9 orbitals/atom). Compounded with this large dimension is the fact that the elements of the matrix and the vector in general have widely different orders of magnitude which makes the calculation rather prone to rounding errors. Figs. \[alldos\] display the main results in this paper: the [*s,p,d*]{} LDOS of Ni and Al in NiAl as calculated using the TB recursion method. These LDOS’s are compared with the corresponding accurate spectra obtained from Brillouin zone integration using atom and angular-momentum projection of the eigenvalues obtained directly from the FPLMTO method. The agreement in general is good although the TB recursion method is unable to precisely reproduce the sharp peaks in the FPLMTO spectra. Not very surprisingly, the best fit is obtained for the tightly-bound Ni-$d$ state. The limited number ($n_m \sim 40$) of reliable recursion parameters that we can use here is generally not sufficient for acceptable reproduction of the FPLMTO LDOS. Furthermore, construction of LDOS by directly taking the imaginary part of the Green’s function, Eq.(\[greens\_function\]), normally introduces an undesirable feature in the output LDOS in the form of spurious rapid oscillations near the edges of the spectrum, as can still be seen in, e.g., Fig. \[alldos\]a. The simplest extrapolation method for the recursion parameters is the square-root terminator method:[@haydock1980] one simply replaces all $a_n$ and $b_n$ for $n > n_m$ with constants $a_\infty$, and $b_\infty$. The tail of the continued fraction for the Green’s function can then be summed exactly as a square-root expression. The best values for the terminating coefficients $a_\infty$ and $b_\infty$ can be obtained using the Beer-Pettifor method.[@beer1984]. Other ways to construct a terminator for the continued fraction have been suggested,[@haydock1985] but for our calculation we have chosen to use the Beer-Pettifor method. Finally, further smoothing of the spectra shown in Fig. \[alldos\] have been performed using a Chebyshev-polynomial method.[@vargas1994] Summary ======= In this paper, we have described a method for directly extracting real-space TB parameters from a FPLMTO method. The basis functions used in the FPLMTO calculation and their construction are described in considerable details. Special emphasis has been placed on the fact that these basis functions do not have a well-defined angular-momentum quantum number, although to a good approximation they do. We believe this fact should be kept in mind in transcribing FPLMTO matrix elements to the Slater-Koster two-center form, since there it is implicitly assumed that the basis functions do have a good angular-momentum quantum number. This direct extraction method has been applied to an intermetallic alloy NiAl. The TB parameters extracted have been used as input to a real-space calculation of local densities of states using the recursion method. We believe this is the first application of the direct extraction method to a real-space calculation of the electronic structure of an intermetallic alloy. The good agreement between the LDOS obtained using the TB extracted from the FPLMTO method and the accurate LDOS obtained directly from the FPLMTO method shows the feasibility of using this method to study more complicated cases, e.g., alloys with small concentration of impurity atoms. This work was supported by AF-OSR Grant No. F49620-99-1-0274. DD would like to thank Dr. M.J. Mehl of the Naval Research Laboratory, Washington D.C., for useful discussions and insightful inputs regarding the NRL Tight-Binding code. [99]{} S. Goedecker, Rev. Mod. Phys. [**71**]{}, 1085 (1999). W. Yang, Phys. Rev. Lett. [**66**]{}, 1438 (1991). G. Galli and M. Parrinello, Phys. Rev. Lett. [**69**]{}, 3547 (1992). G. Galli, Phys. Stat. Sol. (b) [**217**]{}, 231 (2000). S. Goedecker and L. Colombo, Phys. Rev. Lett. [**73**]{}, 122 (1994). W. Kohn, Phys. Rev. Lett. [**76**]{}, 3168 (1996). W. Yang, Phys. Rev. B [**56**]{}, 9294 (1997). C.S. Jayanthi, S.Y. Wu, J. Cocks, N.S. Luo, Z.L. Xie, M. Menon, and G. Yang, Phys. Rev. B[**57**]{}, 3799 (1998). V. Heine, [*Solid State Physics*]{} [**35**]{}, 1980, pp. 1–127. J.C. Slater and G.F. Koster, Phys. Rev. [**94**]{}, 1498 (1954). J.C. Slater, [*Quantum Theory of Molecules and Solids Vol. 2*]{}, McGraw-Hill, New York, 1965. R.R. Sharma, Phys. Rev. B [**19**]{}, 2813 (1979). W.A. Harrison, [*Electronic Structure and the Properties of Solids*]{}, Dover Publications, Mineola, New York, 1989. M. Menon and K.R. Subbaswamy, Phys. Rev. B [**55**]{}, 9231 (1997). N. Bernstein and E. Kaxiras, Phys. Rev. B [**56**]{}, 10488 (1997). R.E. Cohen, L. Stixrude, E. Wasserman, Phys. Rev. B [**56**]{}, 8575 (1997). T.J. Lenosky, J.D. Kress, I. Kwon, A.F. Voter, B. Edwards, D.F. Richards, S. Yang, and J.B. Adams, Phys. Rev. B [**55**]{}, 1528 (1997). D.A. Papaconstantopoulos, [*Handbook of the Band Structure of Elemental Solids*]{}, Plenum, New York, 1986. D.A. Papaconstantopoulos and M.J. Mehl, Phys. Rev. B [**54**]{}, 4519 (1996). M.J. Mehl and D.A. Papaconstantopoulos, in [*Topics in Computational Materials Science*]{}, edited by C.Y. Fong (World Scientific, Singapore, 1998), pp. 169–213. S.H. Yang, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B [**57**]{}, R2013 (1998). D.A. Papaconstantopoulos, M.J. Mehl, S.C. Erwin, and M.R. Pederson, in Ref. \[turchi1998\], pp. 221–230. , edited by P.E.A. Turchi, A. Gonis, and L. Colombo, MRS Symposium Proceedings, Vol. [**491**]{}, Materials Research Society, Warrendale, PA, 1998. \[turchi1998\] C.M. Goringe, D.R. Bowler, and E. Hernandez, Rep. Prog. Phys. [**60**]{}, 1447 (1997). D.R. Bowler, M. Aoki, C.M. Goringe, A.P. Horsfield, and D.G. Pettifor, Modelling Simul. Mater. Sci. Eng. [**5**]{}, 199 (1997). K. Ohno, K. Esfarjani, Y. Kawazoe, [*Computational Materials Science*]{}, Springer-Verlag, Berlin, 1999. D. Raabe, [*Computational Materials Science*]{}, Wiley-VCH, Weinheim, Germany, 1998. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, [*Numerical Recipes in Fortran 77*]{}, Cambridge University Press, New York, 1999. O.K. Andersen and O. Jepsen, Phys. Rev. Lett. [**53**]{}, 2571 (1984). O.K. Andersen, O. Jepsen, and D. Glötzel, in [*Highlights of Condensed Matter Theory*]{}, edited by F. Bassani, F. Fumi, and M.P. Tosi (North-Holland, Amsterdam, 1985), pp. 59–176. O.K. Andersen, O. Jepsen, and M. Sob, in [*Electronic Band Structure and Its Applications*]{} (Springer Lecture Notes in Physics), edited by M. Yussouff (Springer-Verlag, Berlin, 1987), pp. 1–57. O.K. Andersen, O. Jepsen, and G. Krier, in [*Lectures on Methods of Electronic Structure Calculations*]{}, edited by V. Kumar, O.K. Andersen, and A. Mookerjee (World Scientific, Singapore, 1994), pp. 63–124. R.W. Tank and C. Arcangeli, Phys. Stat. Sol. (b) [**217**]{}, 89 (2000). O.K. Andersen, in [*The Electronic Structure of Complex Systems*]{}, edited by P. Phariseau and W.M. Temmerman, Plenum, New York, 1984, pp. 11–66. Z.L. Xie, K.S. Dy, and S.Y. Wu, Phys. Rev. B [**55**]{}, 1748 (1997). O.K. Andersen, Phys. Rev. B [**12**]{}, 3060 (1975). H.L. Skriver, [*The LMTO Method*]{}, Springer-Verlag, Berlin, 1984. V.V. Nemoshkalenko and V.N. Antonov, [*Computational Methods in Solid State Physics*]{}, Gordon and Breach, Amsterdam, Netherlands, 1998. T.L. Loucks, [*Augmented Plane Wave Method*]{}, Benjamin/Cummings, New York, 1967. D.J. Singh, [*Planewaves, Pseudopotentials, and the LAPW Method*]{}, Kluwer, Boston, 1994. D. Sanchez-Portal, E. Artacho, and Jose M. Soler, Solid State Commun. [**95**]{}, 685 (1995). A.K. McMahan and J.E. Klepeis, Phys. Rev. B [**56**]{}, 12250 (1997). A.K. McMahan and J.E. Klepeis, in Ref. \[turchi1998\], pp. 199–210. D. Nguyen-Manh, D.G. Pettifor, and V. Vitek, Phys. Rev. Lett. [**85**]{}, 4136 (2000). D. Djajaputra and B.R. Cooper, Phys. Rev. B [**64**]{}, 085121 (2001). D.L. Price and B.R. Cooper, Phys. Rev. B [**39**]{}, 4945 (1989). D.L. Price, J.M. Wills, and B.R. Cooper, Phys. Rev. B [**46**]{}, 11368 (1992). J.M. Wills, O. Eriksson, M. Alouani, D.L. Price, in [*Electronic Structure and Physical Properties of Solids*]{}, edited by H. Dreyssé, Springer-Verlag, Berlin, 2000, pp. 148–167. S.Yu. Savrasov and D.Yu. Savrasov, Phys. Rev. B [**46**]{}, 12181 (1992). M. Methfessel, Phys. Rev. B [**38**]{}, 1537 (1988). M. Methfessel, C.O. Rodriguez, and O.K. Andersen, Phys. Rev. B [**40**]{}, 2009 (1989). J.M. Ziman, in [*Solid State Physics*]{}, Vol [**26**]{}, edited by H. Ehrenreich, F. Seitz, and D. Turnbull, Academic Press, New York, 1971, pp. 1–101. A.R. Williams, J. Kübler, and C.D. Gelatt, Jr., Phys. Rev. B [**19**]{}, 6094 (1979). \[williams1979\] M. Springborg and O.K. Andersen, J. Chem. Phys. [**87**]{}, 7125 (1987). O. Gunnarsson, J. Harris, and R.O. Jones, J. Phys. C [**9**]{}, 2739 (1976). M. Springborg, [*Methods of Electronic-Structure Calculations*]{}, John Wiley and Sons, Chichester, England, 2000. F.S. Ham and B. Segall, Phys. Rev. [**124**]{}, 1786 (1961). L.F. Mattheiss, J.H. Wood, and A.C. Switendick, in [*Methods in Computational Physics Vol. 8*]{}, edited by B. Alder, S. Fernbach, and M. Rotenberg, Academic Press, New York, 1968, pp. 63–147. D.A. Liberman, Phys. Rev. [**153**]{}, 704 (1967). O.K. Andersen and R.V. Kasowski, Phys. Rev. B [**4**]{}, 1064 (1971). O.K. Andersen and R.G. Woolley, Mol. Phys. [**26**]{}, 905 (1973). R.V. Kasowski, Phys. Rev. B [**14**]{}, 3398 (1976). J. Harris and G.S. Painter, Phys. Rev. B [**22**]{}, 2614 (1980). D. Porezag, Th. Frauenheim, Th. Köhler, G. Seifert, and R. Kaschner, Phys. Rev. B [**51**]{}, 12947 (1995). M. Danos and L.C. Maximon, J. Math. Phys. [**6**]{}, 766 (1965). R. Nozawa, J. Math. Phys. [**7**]{}, 1841 (1966). J.D. Talman, [*Special Functions*]{}, Benjamin, New York, 1968. O.K. Andersen, in [*Computational Methods in Band Theory*]{}, edited by P.M. Marcus, J.F. Janak, and A.R. Williams, Plenum, New York, 1971, pp. 178–182. A. Gonis and W.H. Butler, [*Multiple Scattering in Solids*]{}, Springer-Verlag, New York, 2000. M. Weinert, J. Math. Phys. [**22**]{}, 2433 (1980). M. Weissbluth, [*Atoms and Molecules*]{}, Academic Press, New York, 1978, Table 1.2. R. Riedinger, M. Habar, L. Stauffer, H. Dreyssé, P. Léonard, and M.N. Mukherjee, Phys. Rev. B [**39**]{}, 13175 (1989). L.E. Ballentine and M. Kolá$\check{\rm r}$, J. Phys. C [**19**]{}, 981 (1986). B.A. McKinnon and T.C. Choy, Phys. Rev. B [**52**]{}, 14531 (1995). D.W. Bullett, [*Solid State Physics*]{} [**35**]{}, 1980, pp. 129–214. W.M.C. Foulkes and D.M. Edwards, J. Phys. Cond. Matt. [**5**]{}, 7987 (1993). S.L. Altmann, [*Band Theory of Solids: An Introduction from the Point of View of Symmetry*]{}, Clarendon Press, Oxford, UK, 1995. H. Röder, R.N. Silver, D.A. Drabold, and J.J. Dong, Phys. Rev. B [**55**]{}, 15382 (1997). R. Haydock, [*Solid State Physics*]{} [**35**]{}, 1980, pp. 215–294. D.G. Pettifor and D.L. Wearie, [*The Recursion Method and Its Applications*]{}, Springer-Verlag, Berlin, 1985. P. Turchi, F. Ducastelle, and G. Tréglia, J. Phys. C [**15**]{}, 2891 (1982). J.P. Gaspard and F. Cyrot-Lackmann, J. Phys. C [**6**]{}, 3077 (1973). C.H. Hodges, J. de Physique, [**38**]{}, L-187 (1977). N. Beer and D.G. Pettifor, in [*The Electronic Structure of Complex Systems*]{}, edited by P. Phariseau and W.M. Temmerman, Plenum, New York, 1984, pp. 769–777. R. Haydock and C.M.M. Nex, J. Phys. C [**18**]{}, 2235 (1985). P.C. Vargas, in [*Lectures on Methods of Electronic Structure Calculations*]{}, edited by V. Kumar, O.K. Andersen, and A. Mookerjee (World Scientific, Singapore, 1994), pp. 147–191.

--- date: | Tapobrata Sarkar[^1], Gautam Sengupta, [^2], Bhupendra Nath Tiwari [^3]\ Department of Physics,\ Indian Institute of Technology,\ Kanpur 208016, India. title: On the Thermodynamic Geometry of BTZ Black Holes --- Introduction {#one} ============ Over the last few decades, black hole thermodynamics has been one of the most intense topics of research in theoretical physics (for a comprehensive review, see [@wald1]). It is by now well known that black holes are thermodynamical systems, which posess a Bekenstein-Hawking (BH) entropy, and a characteristic Hawking temperature, related to the surface gravity on the event horizon. Indeed, these quantities satisfy the four laws of black hole thermodynamics. However an understanding the microscopic statistical origin of the black hole entropy has been an outstanding theoretical question. Although considerable progress has been made in the recent past, a clear comprehension of the statistical microstates of black holes still remain elusive. It is well known that equilibrium thermodynamic systems posess interesting geometric features [@callen]. An interesting inner product on the equilibrium thermodynamic state space in the energy representation was provided by Weinhold [@wein] as the Hessian matrix of the internal energy with resect to the extensive thermodynamic variables. However there was no physical interpretation asociated with this metric structure. The Weinhold inner product was later formulated in the entropy representation by Ruppeiner [@rupen] into a Riemannian metric in the thermodynamic state space. The Ruppeiner geometry was however meaningful in the context of equlibrium thermodynamic fluctuations of the system. The curvature scalar obtained from this geometry signified interactions and was proportional to the correlation volume which diverges at the critical points of phase transitions. The [*Ruppeiner metric*]{} on the thermodynamic state space, is defined as the Hessian of the entropy with respect to the extensive variables and is given by $$g_{i,j}=-\partial_i\partial_j S(U,N^a).$$ Here $U$ denotes the internal energy of the system, and $N^a$ are the other extensive thermodynamic variables in the entropy representation. Here, $i,j$ runs over all the extensive variables. It is to be noted that here the volume $V$ is held fixed to provide a physical scale. The Ruppeiner metric is conformaly related to the Weinhold metric with the inverse temperature as the conformal factor. Although, isolated asymptoticaly flat black holes do not follow the usual precepts of extensive thermodynamic systems it is possible to consider the black hole entropy as an extensive thermodynamic quantity provided the black hole is a part of a larger system with which it is in equilibrium. From this perspective the geometric notions of thermodynamics may be applied to investigate the nature of the black hole entropy. In particular the investigation of the covariant thermodynamic geometry of Ruppeiner for black holes have elucidated interesting aspects of black hole phase transitions and moduli spaces. This was first explored in the context of charged black hole configurations of $N=2$ supergravity [@fgk], and since then, several authors have attempted to understand this connection [@cai1],[@aman1; @aman2; @aman3] both for supersymmetric as well as non-supersymmetric black holes. and five dimensional rotating black rings. The simplest black hole system for which it is possible to analyze the thermodynamic geometry is the three dimensional rotating BTZ black hole. Here, one may construct the Ruppeiner metric in terms of the black hole mass and its angular momentum (the non-rotating BTZ black hole is trivial), and it turns out that this metric is flat [^4] [@aman3]. [^5] Recently Kraus and Larsen [@krauslarsen] and Solodukhin [@solo] have shown how various properties of BTZ black holes are affected by the addition of the gravitational Chern-Simons term to the three dimensional Einstein-Hilbert action. In particular, they show that the black hole entropy is modified by the presence of this term and obtain an explicit expression for this modified entropy. In this context, it is imperative to re-examine the Ruppeiner geometry of the BTZ black hole, in the presence of the Chern-Simons term. Although the statistical origin of black hole entropy is still elusive it stands to reason that a black hole in equilibrium with the thermal Hawking radiation at a fixed Hawking temperature is described by a canonical ensemble. The thermodynamic geometry of the black hole entropy function has been determined with reference to the canonical ensemble. However its well known that thermal fluctuations in the canonical ensemble generates logarithmic corrections to the entropy. These corrections vanish in the thermodynamic limit where the canonical and the microcanonical entropy are identical. For black holes such logarithmic corrections to the canonical entropy have been obtained in [@dasetal]. It is a natural question as to whether the thermodynamic geometry of black holes are sensitive to these fluctuations. As we will show in the next few sections, thermal fluctuations indeed modify the Ruppeiner geometry of the BTZ black-holes with and without the Chern-Simons term. In another interesting development, Sahoo and Sen [@sahoosen] have computed the BTZ black hole entropy in the presence of the Chern-Simons and higher derivative terms [@senentropy]. A variant of the [*attractor mechanism*]{} involving the use of the Sen entropy function was applied to an effective two-dimensional theory that results upon making the angular coordinate of the BTZ solution as a compact direction for this analysis. It is indeed natural to examine the thermodynamic geometry of BTZ Chern Simons black holes with higher derivative terms and investigate the effect of thermal fluctuations to this geometry. It is to be emphasized here that the thermal fluctuations in the canonical ensemble may be analysed through purely thermodynamic considerations. In contrast the corrections to the black hole entropy from the $\alpha'$ corrections of higher derivative terms in the effective action needs to be analysed through gravitational considerations. Although the structures of the corrections are simmilar they may enter with opposing signs leading to a cancellation. In addition there should be quantum corrections following from purely quantum gravitational effects. We note that it is not meaningful to analyse corrections due to thermal fluctuations over and above corrections due to quantum effects. It is the above considerations, that we set out to explore in this paper. Our main result is that the thermodynamic geometry is flat for the rotating BTZ black hole in the presence of the Chern-Simons and higher derivative terms. We show that inclusion of thermal fluctuations non-trivially modify the thermodynamic geometry of the BTZ black hole both with and without the Chern-Simons and the higher derivative corrections. As a by-product of our results, we show that the leading order correction to the canonical entropy of the BTZ black hole due to thermal fluctuations are reproduced in the presence of Chern-Simons terms also illustrating further the universality of these corrections. The article is organized as follows. In section \[two\], we first review some known facts about the thermodynamic geometry for BTZ black holes, mainly to set the notations and conventions used in this paper, and then examine the thermodynamic geometry of BTZ black holes including small thermodynamic fluctuations. In section \[three\], we examine the Ruppeiner geometry of the BTZ black hole in the presence of the Chern-Simons term [@solo] and show that including small fluctuations in the analysis, the leading order correction to the entropy turns out to be the same as that of [@carlip]. We then calculate the Ruppeiner curvature scalar and verify the bound on the Chern-Simons coupling, as predicted by Solodukhin. Section \[four\] contains some comments on higher derivative corrections to the BTZ black hole entropy, and discussions and directions for future investigations. Some of the calculations are unfortunately too long to reproduce here, and whereever necessary, we have used numerical techniques to highlight and illustrate our results. Thermodynamic Geometry of BTZ black holes {#two} ========================================= In this section, we study certain aspects of the thermodynamic (Ruppeiner) geometry of BTZ black holes. We will use the units $8G_N=\hbar=c=1$ and start by reviewing the results for the rotating BTZ black hole and then examine the role of small thermal fluctuations. Rotating BTZ black holes ------------------------ The purpose of this subsection is mainly to set the notations and conventions that will be followed in the rest of the paper. We start with the BTZ metric $$ds^2 = -N(r)dt^2 + \frac{1}{N(r)}dr^2 + r^2\left(N^{\phi}dt + d\phi \right)^2 \label{btz1}$$ where $N$ and $N^{\phi}$ are the (squared) lapse and shift functions defined by $$N(r) = \frac{J^2}{4r^2}+\frac{r^2}{l^2} - M;~~~~ N^{\phi}=-\frac{J}{2r^2}$$ with $M$ and $J$ being the mass and the angular momentum of the black hole, and $l^2$ represents the Cosmological constant term. The BTZ black hole has two horizons, located at $$r_{\pm}=\sqrt{\frac{1}{2}Ml^2\left(1\pm \Delta\right)} \label{rpm}$$ where $$\Delta = \sqrt{1-\frac{J^2}{M^2l^2}} \label{delta}$$ The mass and angular momentum of the black hole may be expressed in terms of $r_{\pm}$ of eq. (\[rpm\]) as; $$M = \frac{r_+^2 + r_-^2}{l^2};~~~~J=\frac{2r_+ r_-}{l} \label{mj}$$ The BH entropy of the ordinary BTZ black hole is given by $$S=4\pi r_+ \label{entropy}$$ The Ruppeiner metric is two dimensional, and is a function of the black hole mass $M$ and angular momentum $J$. Explicitly, the metric is given by $$g_{ij}=-\pmatrix{\frac{\partial^2 S}{\partial J^2}& \frac{\partial^2 S}{\partial J\partial M}\cr \frac{\partial^2 S}{\partial J\partial M}& \frac{\partial^2 S}{\partial M^2}} \label{rupenmatrix}$$ with $i,j\equiv J,M$. We will use this general form of the Ruppeiner metric throughout this paper. A simple calculation shows that the Christoffel symbols are given by [^6] $$\begin{aligned} \Gamma_{JJJ}&=&-\frac{1}{2}\frac{\partial^3 S}{\partial J^3}~~~ \Gamma_{MMM}=-\frac{1}{2}\frac{\partial^3 S}{\partial M^3}~~~ \Gamma_{JJM}=-\frac{1}{2}\frac{\partial^3 S}{\partial M \partial J^2} \nonumber\\ \Gamma_{JMJ}&=&-\frac{1}{2}\frac{\partial^3 S}{\partial M \partial J^2} ~~~\Gamma_{JMM}=-\frac{1}{2}\frac{\partial^3 S}{\partial J \partial M^2} ~~~\Gamma_{MMJ}=-\frac{1}{2}\frac{\partial^3 S}{\partial M^2\partial J}\end{aligned}$$ with the symmetries relating the other components. The only non-vanishing component of the Riemann-Christoffel curvature tensor is $R_{JMJM}=N/D$, where $$\begin{aligned} N=&~&\left(\frac{\partial^2S}{\partial J^2}\right)\left[ \left(\frac{\partial^3S}{\partial M\partial J^2}\right) \left(\frac{\partial^3S}{\partial M^3}\right) - \left(\frac{\partial^3 S}{\partial J\partial M^2}\right)^2\right] \nonumber\\ &+& \left(\frac{\partial^2S}{\partial M^2}\right)\left[ \left(\frac{\partial^3S}{\partial J\partial M^2}\right) \left(\frac{\partial^3S}{\partial J^3}\right) - \left(\frac{\partial^3 S}{\partial M\partial J^2}\right)^2\right] \nonumber\\ &+& \left(\frac{\partial^2S}{\partial J\partial M}\right)\left[ \left(\frac{\partial^3S}{\partial M\partial J^2}\right) \left(\frac{\partial^3S}{\partial J\partial M^2}\right) - \left(\frac{\partial^3S}{\partial J^3}\right) \left(\frac{\partial^3S}{\partial M^3}\right)\right] \label{numerator}\end{aligned}$$ and $$D= 4\left[ \left(\frac{\partial ^{2}S}{\partial {J}^{2}}\right) \left(\frac{\partial^{2}S}{\partial{M}^{2}}\right)- \left(\frac {\partial ^{2}S}{\partial J\partial M}\right)^{2} \right] \label{denominator}$$ The Ricci scalar is $$R=\frac{2}{{\mbox{det}}g}R_{JMJM} \label{ricciscalar}$$ It is easy to compute the Ricci scalar by using $$r_+=\frac{1}{2}\left[\sqrt{l\left(Ml+J\right)} + \sqrt{l\left(Ml-J\right)}\right] \label{ricci1}$$ Using eq. (\[ricci1\]) in eqs. (\[entropy\]), (\[ricciscalar\]), (\[numerator\]) and (\[denominator\]), it can be easily shown that the Ricci scalar vanishes identically [@aman3]. We might point out here that in [@cai1], from considerations of the laws of black hole thermodynamics, the authors have argued that the internal energy of a charged or rotating black hole might not always be equal to its mass. Although we are not in full agreement with the arguments of [@cai1], we have checked nevertheless that a modification of the internal energy of the rotating BTZ black hole in lines with [@cai1] does not change the observation above. Inclusion of thermal fluctuations --------------------------------- We will now discuss the Ruppeiner geometry of BTZ black holes including thermal fluctuations about the equilibrium. As is well known, any thermodynamical system, considered as a canonical ensemble has logarithmic and polynomial corrections to the entropy [@huang]. These considerations apply to black holes as well (considered as a canonical ensemble), and the specific forms of the logarithmic and polynomial corrections has been calculated for a wide class of black holes in [@dasetal]. It is to be noted that the applicability of this analysis presupposes that the canonical ensemble is thermodynamicaly stable. This requires a positive specific heat or correspondingly the Hessian of the entropy function must be negative definite. The microcanonical entropy for any thermodynamical system, incorporating such corrections, is [@huang] $$S = S_0 - \frac{1}{2}{\mbox{ln}}\left(CT^2\right) \label{correctedbtz}$$ where $S_0$ is the entropy calculated in the canonical ensemble, and $S$ is the corrected microcanonical entropy. $C$ is the specific heat, and it is understood that appropriate factors of the Boltzmann’s constant are included to make the logarithm dimensionless. The approximation is valid only in the regime where thermal fluctuations are much larger than quantum fluctuations. In [@dasetal], the BTZ black hole was analysed in this framework and eq. (\[correctedbtz\]) reproduces the leading order correction to the entropy as obtained in [@carlip]. It is then a natural question as to how the Ruppeiner geometry for the BTZ black hole is modified due to the thermal fluctuations in the canonical ensemble and this is what we will analyse in the rest of this section. The Ruppeiner metric for the corrected entropy for the BTZ black hole of eq. (\[correctedbtz\]) can be calculated using the equations (\[numerator\]), (\[denominator\]) and (\[ricciscalar\]). Since the expressions involved are lengthy, we will set the cosmological constant $l=1$ . The Hawking temperature of the BTZ black hole is given by $$T_H=\frac{1}{2\pi}\left[\frac{r_+^2 - r_-^2}{r_+}\right]$$ which can be readily expressed in terms of the entropy of eq. (\[entropy\]) as $$T_H=\frac{S}{8\pi^2} - \frac{8\pi^2J^2}{S^3} \label{hawkingbtz}$$ The specific heat is $$C = \left(\frac{\partial M}{\partial T}\right)_J = \frac{S\left(S^4 - 64\pi^4J^2\right)}{S^4 + 192\pi^4J^2} \label{spheatbtz}$$ The specific heat is positive and this ensures that the stability of the corresponding canonical ensemble. Alternatively the Hessian of the internal energy ( ADM mass) with respect to the extensive variables in the energy representation is given as $$\mid\mid \frac {\partial^2 M}{\partial X_i \partial X_j}\mid\mid =\frac {1}{S^2l^2}- \frac {64 \pi^4 J^2}{S^6}$$. This is positive provided $\frac {J}{S^2}\ < \ 1$ ensuring the thermodynamic stability of the corresponding BTZ black hole. It is to be noted that this condition also governs the situation away from extremality. Substituting the expressions of (\[hawkingbtz\]) and (\[spheatbtz\]) in (\[correctedbtz\]), we obtain the corrected entropy of the BTZ black hole, and the Ruppeiner metric for this entropy. The expression of the curvature scalar of this metric is far too complicated to present here, so we present the results numerically. First, we consider the Ruppeiner metric with just the leading logarithmic correction of [@carlip]. In this case, the analysis is simplified and (\[correctedbtz\]) reduces to $$S = S_0 - \frac{3}{2}{\mbox{ln}}S_0 \label{correctedbtzlog}$$ Figure (\[fig1\]) shows the curvature scalar of the Ruppeiner metric, $R$, plotted against the angular momentum $J$ for $M=100$, where we have taken only the logarithmic correction of eq. (\[correctedbtzlog\]) to the entropy into account. We have restricted to small values of $J$, so that we are far from extremality, i.e in the regime where these results are valid. Indeed, for near extremal BTZ black holes (i.e for very low temperatures), our analysis is not valid [@dasetal]. From fig. (\[fig1\]), we see that in this case, the curvature scalar is not positive definite, and indeed, by extending the values of $J$, it is seen that the curvature scalar goes to zero at $J$ increases towards its extremal value. However, we must point out that our calculations that lead to this result can only be trusted when the black hole is far from extremality. Also note that even at zero angular momentum, there is a small but finite value of the curvature scalar. This indicates that even at zero angular momentum, the statistical system is interacting, once small fluctuations are included. This should be contrasted with the non-rotating BTZ black hole which is a non-interacting system even when small fluctuations are included. We have checked that increasing the value of $M$, the value of the curvature scalar becomes smaller, while preserving the shape of the graph. Figure (\[fig2\]) shows the Ruppeiner curvature scalar plotted against the angular momentum, calculated using eq. (\[correctedbtz\]). Interestingly, in this case, the Ruppeiner scalar is positive definite. Again, we have restricted ourselves to values of $J$ small compared to $M$ (i.e far from extremality) where our results can be trusted. BTZ black holes with the Chern-Simons term {#three} ========================================== Recently, Kraus and Larsen [@krauslarsen] and Solodukhin [@solo] have studied gravitational anomalies for three-dimensional gravity in the presence of the Chern-Simons term. Indeed, the BTZ black hole is a bonafide solution to the gravitational action that included both the Einstein-Hilbert and the Chern-Simons term. We will henceforth refer to the BTZ black hole with the Chern-Simons term as the BTZ-CS black hole.In [@krauslarsen],[@solo], the entropy of BTZ-CS black holes have been analysed, and these authors have derived an expression for the entropy, which differes from the entropy of the “usual” BTZ black hole, eq. (\[entropy\]). The modified entropy for the BTZ-CS black hole is $$S=4\pi\left(r_+ - \frac{K}{l}r_-\right) \label{btzcs}$$ where $K$ is the Chern-Simons coupling. The extra term in eq. (\[btzcs\]) as compared to eq. (\[entropy\]) is the contribution from the Chern-Simons term and has very interesting properties. In particular, [@solo] predicts a stability bound $$|K| \leq l \label{stability}$$ on the Chern-Simons coupling, from physical considerations. In view of the above, it is natural to ask what type of Ruppeiner geometry is seen by the BTZ black hole in the presence of the Chern-Simons term and it is this issue that we address in this section. It is important to remember here that the usual mass and angular momentum of the BTZ black hole is modified in the presence of the Chern-Simons term. This may be calculated by integrating the modified stress tensor of the theory using the Fefferman-Graham expansion of the BTZ metric and reads [@solo] $$M = M_0 - \frac{K}{l^2}J_0; ~~~~ J= J_0 - KM_0 \label{mjcs}$$ where $M_0$ and $J_0$ are the the mass and angular momentum of the usual BTZ black hole of eqn. (\[mj\]). We have calculated the Ruppeiner metric for the BTZ black hole (with the thermodynamic coordinates now being $M$ and $J$, rather than $M_0$ and $J_0$) in the presence of the Chern-Simons term, taking into account the modifications of the mass and angular momentum as in eq. (\[mjcs\]). [^7] Writing the entropy as $$S=2\pi\left[\sqrt{\left(1-K\right)\left(M+J\right)} +\sqrt{\left(1+K\right)\left(M-J\right)}\right] \label{btzcsentropy}$$ it is easy to calculate the geometric quantities. The expressions leading to the calculation of the Ricci scalar are not important, and we simply point out that the curvature scalar for this geometry turns out to be zero, i.e, the Ruppeiner geometry of the BTZ-CS black hole is flat showing that it is a non interacting statistical system. This is the main result of this subsection. BTZ-CS black holes with small fluctuations ------------------------------------------ We will now discuss some thermodynamic properties of the BTZ-CS black holes, treating the system as a canonical ensemble. We allow for small thermal fluctuations of the system considered as a canonical ensemble, and study the thermodynamic geometry of the BTZ-CS black hole in lines with our treatment of the usual rotating BTZ black hole described earlier As before, we would like to analyse the Ruppeiner metric for the BTZ-CS black hole, with the entropy now being given by eq. (\[btzcs\]). Again, for ease of notation, we set the cosmological constant $l=1$. We begin by expressing the outer and inner horizons of the BTZ-CS black hole as $$\begin{aligned} r_+=\frac{1}{2}\left[\sqrt{M_0 + J_0} + \sqrt{M_0 - J_0}\right] \nonumber\\ r_-=\frac{1}{2}\left[\sqrt{M_0 + J_0} - \sqrt{M_0 - J_0}\right]\end{aligned}$$ When expressed in terms of the corrected mass and angular momentum of eq. (\[mjcs\]), these expressions become $$\begin{aligned} r_+=\frac{1}{2}\left[\sqrt{\frac{M+J}{1-K}} + \sqrt{\frac{M-J}{1+K}} \right]\nonumber\\ r_-=\frac{1}{2}\left[\sqrt{\frac{M+J}{1-K}} - \sqrt{\frac{M-J}{1+K}} \right]\end{aligned}$$ The equation for the entropy, given by (\[btzcs\]), can now be solved to obtain the mass $M$ in terms of $S$ and $J$, and gives $$M=\frac{1}{2K^2}\left[\left(2KJ+\frac{S^2}{4\pi^2}\right)+ \left[\left(2KJ+\frac{S^2}{4\pi^2}\right)^2-4K^2\left(\frac{S^4}{64\pi^4} +\frac{S^2KJ}{4\pi^2}+J^2\right)\right]^\frac{1}{2}\right]$$ The temperature of the BTZ-CS black hole, given by $\left(\frac {\partial M}{\partial S}\right)_J$ may be obtained from the expression for $M$, and is given by $$T=\frac{SK^2\left[S^2\left(1-K^2\right)+8JK\pi^2\left(1-K^2\right) +\left[S^2\left(1-K^2\right)\left(S^2+16\pi^2JK\right)\right]^{1\over 2} \right]}{4\pi^2\left[S^2\left(1-K^2\right)\left(16\pi^2KJ+S^2\right)\right]}$$ The specific heat may be calculated from the expression $$C=\left(\frac{\partial M}{\partial T}\right)_J= \frac{T}{\left(\frac{\partial T}{\partial S}\right)_J}$$ and is evaluated as $$C=\frac{S\alpha\left[\beta + \left(8KJ\pi^2 + S^2\right) \left(1-K^2\right)\right]}{\alpha\beta + S^2\left(1-K^2\right) \left(S^2 + 24KJ\pi^2\right)} \label{spheatbtzcs}$$ where $\alpha = 16\pi^2KJ + S^2$ and $\beta = \left(S^2\left(1-K^2\right)\alpha\right)^{\frac{1}{2}}$. It may be checked that the secific heat is positive ensuring local thermodynamic stability. Using eq. (\[spheatbtzcs\]), we calculate the correction to the canonical entropy including small thermal fluctuations of the statistical system and this leads to, $$S=S_0 - \frac{1}{2}{\mbox{ln}}CT^2 \label{btzcsfull}$$ where $S_0$ is the entropy (\[btzcs\]) of the BTZ-CS in the canonical ensemble. We approximate (\[btzcsfull\]) in the limit of large entropy, following [@dasetal]. It may be easily examined that in the limit of $S \gg J^2$ which is the stability bound, the above formula reduces to $$S=S_0 - \frac{3}{2}{\mbox{ln}}S_0 \label{btzcslog}$$ It is interesting to note that the factor of $\frac{3}{2}$, first calculated in [@carlip] is reproduced for the BTZ-CS black hole as well. Illustrating the seeming universality of this factor. This is one of the main result of this subsection. We now calculate the Ruppeiner geometry correspoding to the modified entropy of the BTZ-CS black hole with thermal fluctuations. As in the last section, we first present the numerical result for the Ricci scalar using the leading order correction of eq. (\[btzcslog\]). This is depicted in fig. (\[fig3\]). In this analysis, we have set $M=100$ and $J=1$, to ensure that we are far from extremality. The Ricci scalar is positive definite in this case. The Ricci scalar diverges for $|K| = 1$. More appropriately, since we had set the cosmological constant to unity, it is not difficult to see that the bound on $K$ from the Ruppeiner geometry is $|K| \leq l$ where $l$ is the cosmological constant. This is of course as expected, since the entropy (\[btzcsentropy\]) becomes unphysical beyond this limit. In fig. (\[fig4\]), we present the result for the Ricci scalar of the Ruppeiner geometry taking into account the full correction of eq. (\[btzcsfull\]). Again, as a function of $K$, the curvature scalar is positive definite and the graph has the same qualitative features as in fig. (\[fig3\]). For the sake of completeness, we have also numerically evaluated the Ricci scalar of the Ruppeiner metric for the BTZ-CS black hole as a function of the angular momentum, and studied its behaviour. The plots in this case are qualitatively the same as in fig. (\[fig1\]) and fig. (\[fig2\]) and we do not discuss them further. Discussions and Conclusions {#four} =========================== In this article, we have mainly investigated the thermodynamic geometry of a class of BTZ black holes, both with and without the Chern-Simons term. We have shown that the Ruppeiner geometry remains flat even with the introduction of the Chern-Simons term, as it was without this term. However, introducing small thermal fluctuations in the analysis produces a non-zero Ricci scalar for the thermodynamic geometry, and we have calculated this quantity for some special cases. As a byproduct of our calculations, we have shown that the leading logarithmic correction to the canonical entropy of the BTZ-CS black hole retains the same form as for the ordinary rotating BTZ black hole thus illustrating the universality of this correction. We should mention here that the validity of this analysis depends on the local thermodynamic stability which is ensured by a positive specific heat for the BTZ and the BTZ-CS black holes. This is also generaly true for charged and rotating charged black holes. It would be intetesting to extend our analysis to other black holes and investigate the subtle interplay between the corrections due to thermal fluctutaions and $\alpha"$ corrections resulting from higher derivative terms. It is expeted that the corresponding thermodynamic geomteries would be sensitive to these corrections. Furthermore thermodynamic geometries provide a direct way to analyse critical points of black hole phase transitions which is an area of current interest. This may have important implications for black holes in string theory and the geomtry of moduli spaces. Some of these issues will be investigated in future. A few comments are in order here. It is clear that our analysis will be similar for BTZ black holes with higher derivative corrections. As shown in [@krauslarsen] and [@sahoosen], the form of the entropy for the BTZ-CS black hole remains the same in the presence of the higher derivative corrections, and it is the central charge of the underlying conformal field theory that is modified. Hence, we expect qualitatively similar results for the thermodynamic geometry of BTZ-CS black holes with higher derivative corrections. We have explicitly verified this. As we have pointed out earlier, leading logarithmic correction to the black hole entropy arises from various sources. The black hole considered as a canonical ensemble admits such corrections to the entropy due to standard thermal fluctuations. The effect of such fluctuations may be analysed from purely thermodynamic considerations. It is to be understood that these fluctuations vanish in the thermodynamic limit of large systems where the canonical and the microcanonical entropy becomes identical. Apart from these the black hole entropy also admits logarithmic corrections due to presence of higher derivative terms to the gravitational action from the perspective of low energy effective field theories resulting from some underlying theory of quantum gravity. These higher derivative corrections are accessible to analysis through purely gravtitational considerations like Walds formula or through gravitational anomalies. It is a meaningful exercise to analyse these two corrections simultaneously and in certain cases leads to a cancellation. However corrections due to purely quantum effects must be considered separately. Lacking a viable fundmaental theory of quantum gravity these quantum corrections still need to be elucidated. We should also remark here that as pointed out in [@solo] that the modification of the entropy due to the gravitational Chern-Simons term being dependent on the radius of the inner horizon seems to probe the black hole interior. This is in contarst to the higher derivative $\alpha'$ corrections which are only dependent on the radii of the outer horizon. This seems to indicate that contrary to the existing point of view certain degrees of freedom may be associated with the black hole interior. This may have implications for space time holography and is an important issue for future investigations. Acknowledgements {#six} ================ We are grateful to Ron-Gen Cai for helpful email correspondence. All of us would like to thank Geetanjali Sarkar for computational support and V. Subrahmanyam for discussions. GS would like to thank Sutapa Mukherji for a reference. BNT acknowldges CSIR, India, for financial support under the research grant CSIR-SRF-9/92(343)/2004-EMR-I. [99]{} R. M. Wald, “The Thermodynamics of Black Holes,” Living Rev. Rel. [**4**]{} (2001) 6. H. B. Callen, [*Thermodynamics and an Introduction to Thermostatistics*]{}, Pub. Wiley, New York (1985); L.Tisza [*Generalized Thermodynamics*]{}, Pub. MIT Press, Cambridge, MA (1966). F. Weinhold, J.Chem. Phys. [**63**]{}, 2479 (1975), [*ibid*]{} J. Chem. Phys [**63**]{} , 2484 ( 1975). G. Ruppeiner, Rev. Mod. Phys [**67**]{} (1995) 605, Erratum [**68**]{} (1996) 313. S. Ferrara, G. W. Gibbons, R. Kallosh, “Black Holes and Critical Points in Moduli Space,” Nucl. Phys. [**B500**]{} (1997) 75, [hep-th 9702103]{}. J. Y. Shen, R. G. Cai, B. Wang, R. K. Su, “Thermodynamic geometry and critical behavior of black holes,” [gr-qc/0512035]{}. K. Huang, “Statistical Mechanics,” Publishers John Wiley, 1963,\ L. D. Landau, E. M. Lifshitz, “Statistical Physics,” Publishers Pergamon, 1969 J. E. Aman, I. Bengtsson, N. Pidokrajt, “Flat Information Geometries in Black Hole Thermodynamics,” [gr-qc/0601119]{} J. E. Aman, N. Pidokrajt, “Geometry of Higher Dimensional Black Hole Thermodynamics,” Phys. Rev. [**D73**]{}(2006) 024017 [hep-th/0510139]{} J. E. Aman, I. Bengtsson, N. Pidokrajt, “Geometry of Black Hole Thermodynamics,” Gen. Rel. Grav. [**35**]{} (2003) 1733, [gr-qc/0304015]{}, G. Arcioni, E. Lozano-Tellechea, Phys.Rev. D72 (2005) 104021, [hep-th/ 0412118]{}. S. Das, P. Majumdar, R. K. Bhaduri, “General logarithmic corrections to black hole entropy,” Class. Quant. Grav. [**19**]{} (2002) 2355, [hep-th/0111001]{}, S. S. More, Class. Quant. Grav. 22 (2005) 4129-4140 [gr-qc/0410071]{}, D. Grumiller, [hep-th/0506175]{}. (2005). S. Carlip, “ Logarithmic corrections to black hole entropy from the Cardy formula,” Class. Quant. Grav. [**17**]{} (2000) 4175, [gr-qc/0005105]{} F. Weinhold, J. Chem. Phys [**63**]{} (1975) 2479. P. Kraus, F. Larsen, “Holographic Gravitational Anomalies,” [hep-th/0508218]{} S. Solodukhin, “Holography with Gravitational Chern-Simons Term,” [hep-th/0509148]{} B. Sahoo, A. Sen, “BTZ Black Hole with Chern-Simons and Higher Derivative Terms,” [hep-th/0601228]{}. H. Saida, J.Soda, Phys.Lett. B471 (2000) 358-366 [gr-qc/9909061]{} A. Sen, “Black hole entropy function and the attractor mechanism in higher derivative gravity,” JHEP [**0509**]{} (2005) 038. [^1]: [email protected] [^2]: [email protected] [^3]: [email protected] [^4]: Ricci flatness in two dimensions implies a flat space. [^5]: See Ref. [@aman2] and [@aman3] for a classification of the nature of the Ruppeiner metrics for black holes in various dimensions. [^6]: Our notation is $\Gamma_{ijk}=g_{ij,k}+g_{ik,j} -g_{jk,i}$ [^7]: We have set the cosmological constant $l=1$ for simplicity.

--- abstract: 'We used a renormalisation group based smoothing to address two questions related to Abelian dominance. Smoothing enabled us to extract the Abelian heavy-quark potential from time-like Wilson loops on Polyakov gauge projected configurations. We obtained a very small string tension which is inconsistent with the string tension extracted from Polyakov loop correlators. This shows that the Polyakov gauge projected Abelian configurations do not have a consistent physical meaning. We also applied the smoothing on SU(2) configurations to test how sensitive Abelian dominance in the maximal Abelian gauge is to the short distance fluctuations. We found that on smoothed SU(2) configurations the Abelian string tension was about 30% smaller than the SU(2) string tension which was unaffected by smoothing. This suggests that the approximate Abelian dominance found with the Wilson action is probably an accident and it has no fundamental physical relevance.' address: - | Department of Physics, University of Colorado,\ Boulder CO 80309-390, USA - | Department of Theoretical Physics, Kossuth Lajos University,\ Debrecen H-4010, Hungary author: - 'Tamás G. Kovács and Zsolt Schram' title: 'Some remarks on Abelian dominance[^1]' --- INTRODUCTION ============ It is an old idea to try to understand non-Abelian gauge theories in terms of an effective Abelian model with a smaller symmetry group. One possible way of doing this on the lattice is to isolate $U(1)^{N-1}$ link variables belonging to a maximal torus of SU(N). This is called Abelian projection. The hope is that non-Abelian confinement might be explained as a condensation of monopoles in the resulting Abelian projected model (see e.g. [@Polikarpov] for a recent review). If one wants to explain the non-Abelian physics in the Abelian projected system, a necessary condition is that the Abelian model has to reproduce the physical features of the non-Abelian system. This property is referred to as Abelian dominance. The projection procedure necessarily involves some gauge fixing. In principle the physical properties of the projected system can depend on the gauge choice. Up to now the only gauge in which the Abelian projected system seems to capture the physics of the non-Abelian model is the maximal Abelian gauge [@MAG]. Here in the SU(2) case the Abelian and non-Abelian string tensions at Wilson $\beta=2.51$ agree to within 8% [@Bornyakov]. In other gauges, most notably in the Polyakov gauge (where Polyakov loops are diagonalised) the situation is more controversial. Since all the Polyakov loops can be exactly diagonalised at the same time, in this case “Abelian dominance” exactly and trivially holds if the string tension is measured with Polyakov loop correlators. On the other hand due to the high level of noise on the projected configurations, it is impossible to extract the string tension from Wilson loops [@Suzuki]. In this talk we discuss some related issues. The first question we address is that of the gauge choice. We use a recently proposed smoothing technique based on renormalisation group ideas [@DeGrand]. We can drastically reduce the short-distance fluctuations while preserving the long-distance physical properties of our configurations, most importantly the SU(2) string tension. This allows us to extract the heavy quark potential from Wilson loops on Polyakov gauge projected configurations. The resulting Abelian string tension turns out to be practically zero. This result is inconsistent with the string tension measured from Polyakov loop correlators. It shows that the physical meaning of Polyakov gauge projected configurations is questionable. The only gauge known to us in which approximate Abelian dominance has been found (with the Wilson action) is the maximal Abelian one. Therefore in the second part of the talk we shall concentrate only on this gauge. We study the question, how Abelian dominance depends on the details of the short-distance fluctuations in this particular gauge. Using the above mentioned smoothing on Monte Carlo generated SU(2) gauge configurations we can produce smoothed configurations with the same long-distance properties but reduced short-distance fluctuations. Comparing the Abelian string tension on the original and the smoothed configurations we can gain insight into its dependence on the short-distance details. For a more detailed account of this work the reader is referred to Ref.  [@KS]. THE GAUGE CHOICE ================ The very idea of Abelian dominance is that the diagonal Abelian degrees of freedom can account for the physical properties of the full non-Abelian configurations. The issue of gauge fixing is definitely important here since the part of the system that we retain/discard with the Abelian projection very strongly depends on it. Let us consider the Polyakov gauge first. On any given SU(2) configuration all the links belonging to the Polyakov loops can be diagonalised simultaneously by a suitable gauge transformation. Therefore any physical quantity derived from Polyakov loops will be trivially and exactly reproduced after Abelian projection in this gauge. In particular there is exact Abelian dominance for the string tension measured with Polyakov loop correlators [@Ejiri]. A good test of whether the Polyakov gauge projected Abelian configurations capture some genuine physics would be to measure the string tension using time-like Wilson loops and compare this to the string tension obtained with Polyakov loop correlators. Unfortunately this cannot be done directly because the gauge fixing introduces so much noise that one would need a huge number of configurations to get enough statistics. We can however use an ensemble of smoothed configurations and do all the measurements on them. We generated an ensemble of 20 $12^4$ configurations with the fixed point action of Ref. [@DeGrand] at $\beta=1.5$ which corresponds to a physical lattice spacing of 0.144 fm. After one smoothing step we measured both the full SU(2) and the Polyakov gauge projected U(1) heavy quark potential on them using time-like Wilson loops. We used the method and computer code of Heller et al. [@Heller]. Our results are shown in Figure \[fig:pot\_pg\]. In the SU(2) case we have a good plateau at $T=3$ (this has also been confirmed on another ensemble of larger statistics) but in the U(1) case the potential decreases considerably with increasing $T$ even at this point. One can conclude that in the $T \rightarrow \infty$ limit the U(1) string tension is probably very close to zero. The discrepancy is striking. We would also like to note that the static quark potential measured by Polyakov-loop correlators is exactly the same as the full non-Abelian potential. We also note that the string tension obtained from Polyakov loop correlators and timelike Wilson loops should be the same (up to some small finite size effects). This means that two different but physically equivalent measurements of the same physical quantity give absolutely different results on the Polyakov gauge projected configurations. Our result for the Polyakov gauge strongly suggests that the physics of the Abelian projection is not only very strongly gauge dependent but in most of the arbitrarily chosen gauges the Abelian projected configurations do not even have a consistent physical meaning. The maximal Abelian gauge (MAG) is special as it minimises the off-diagonal components of the link degrees of freedom, the ones that are discarded in the projection [@MAG]. For this reason the MAG is a priori a better choice than the gauges that diagonalise an arbitrarily selected set of operators like the Polyakov loops. ABELIAN DOMINANCE AND SHORT RANGE FLUCTUATIONS ============================================== In this section we study how Abelian dominance in the maximal Abelian gauge depends on the precise nature of short distance fluctuations. We generated 100 $8^3 \times 12$ lattices with the fixed point action of Ref. [@DeGrand] at $\beta=1.5$ (lattice spacing $a=0.144$ fm). At first as a check we verified that Abelian dominance holds for this ensemble. We transformed the configurations into the maximal Abelian gauge. This was done using the usual overrelaxation procedure iterated until the change in the gauge fixing action became less than $10^{-8}$ per link. After Abelian projecting these configurations the heavy quark potential was extracted from time-like Wilson loops in the same way as in the previous section. From the heavy-quark potential we obtained $\sigma_{na}=0.123(7)$ for the non-Abelian and $\sigma_{ab}=0.119(5)$ for the Abelian string tension in lattice units. After this check we applied one step of smoothing to the same ensemble of SU(2) configuration and repeated the measurement of the Abelian and non-Abelian potential on the smoothed configurations. It gave $\sigma_{na}=0.115(9)$ and $\sigma_{ab}=0.080(10)$ for the SU(2) and the U(1) string tension respectively. The SU(2) string tension on the smoothed configurations is essentially the same as on the unsmoothed ones, reflecting the fact that smoothing does not change the long-distance features. On the other hand, as a result of smoothing, the Abelian string tension dropped by about 30%. This shows that the Abelian string tension is very sensitive to the details of the short-distance fluctuations on the SU(2) configurations. A similar result has been found for the monopole string tension using cooling with the Wilson action [@Hart]. It seems to us quite impossible to reconcile this fact with the expectation that the Abelian string tension is a genuine long-distance physical observable which is in some sense equivalent to the SU(2) string tension. In view of this, the approximate Abelian dominance found with Wilson action in the maximal Abelian gauge seems to be an accident rather than a fundamental physical phenomenon. [9]{} M.I. Polikarpov, Nucl. Phys. B (Proc. Suppl.) [**53**]{} (1997) 134. A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Phys. Lett. (1987) 516; A.S. Kronfeld, G. Schierholz and U.-J. Wiese, Nucl. Phys. [**B293**]{} (1987) 461. G. Bali, V. Bornyakov, M. Müller-Preussker and K. Schilling, Phys. Rev. [**D54**]{} (1996) 2863. T. Suzuki, S. Ilyar, Y. Matsubara, T. Okude and K. Yotsuji, Phys. Lett. (1995) 375. T. DeGrand, A. Hasenfratz and T.G. Kovács, hep-lat/9705009. Tamás G. Kovács and Zsolt Schram, hep-lat/9706012. S. Ejiri et al., Nucl. Phys. [**B47**]{} (Proc. Suppl.) (1996) 322. U.M. Heller, K.M. Bitar, R.G. Edwards and A.D. Kennedy, Phys. Lett. (1994) 71. A. Hart and M. Teper, hep-lat/9707009. [^1]: Research supported in part by U.S. Department of Energy grant DE-FG02-92ER-40672, OTKA Hungarian Science Foundation T 017311 and the Physics Research Group of the Hungarian Academy of Sciences, Debrecen.

--- author: - 'M. Castro' - 'F. D’Amico' - 'J. Braga' - 'T. Maiolino' - 'K. Pottschmidt' - 'J. Wilms' bibliography: - 'our\_refs.bib' date: title: 'Confirming the thermal Comptonization model for black hole X-ray emission in the low-hard state' --- Introduction {#intro} ============ Since its discovery with the Einstein Observatory (), the putative black-hole system 1E1740.7$-$2942 has been extensively studied. Historical observations () have established 1E1740.7$-$2942 as the brightest hard X-ray source ($E$$>$$20$keV) in the direction of the Galactic Center. From the years of observation with SIGMA/Granat, the source was discovered to have 3 spectral states resembling those of CygX-1 (). Another heritage from the SIGMA years was a detection of 1E1740.7$-$2942 up to $\sim$$500$keV (). The source was dubbed a [ *microquasar*]{} after radio jets were observed (). The intense optical extinction toward the Galactic Center still prevents the identification of the counterpart of 1E1740.7$-$2942 at optical and infrared wavelengths, despite all efforts (). ASCA () as well as Chandra () that were helpful in determining the column density of hydrogen ($N_{\mathrm{H}}$). Recent INTEGRAL observations have shown that the source can be clearly detected up to $\sim$$600$keV (). A broadband study with Suzaku with a small spectral gap ($10$–$12$keV) has shown the presence for the first time of an iron K-edge absorption in the 1E1740.7$-$2942 spectra (). In general, the combined soft/hard X-ray spectra of 1E1740.7$-$2942 are fitted by a combination of a thermal component and a non-thermal powerlaw. These fits are useful in constraining the spectral state of 1E1740.7$-$2942 through the powerlaw index (see, e.g., [). Alternatively, a combination of two thermal models can be used to explain the broadband spectrum of this source: a soft component, which is associated with the accretion disk, and a hard Comptonized component coming from a corona-type region which has $kT$]{}$\sim$50keV and an optical depth around 1 (e.g., ). In this study, we show evidence that modeling the spectra of 1E1740.7$-$2942 in the latter way provides a consistent picture. We also report a detection here (by XMM) of an iron K-edge absorption feature, which confirms the Suzaku results (). Recently, a NuSTAR + INTEGRAL study on 1E1740.7$-$2942 was published based on data of 2012 (). The spectrum, starting at $\sim$3keV and extending up to 250keV, was fitted by a combination of a Comptonization model ([compTT]{}) and a soft component ([diskbb]{}), which enables us to make useful comparisons with our results. Finally, we highlight that this present study is an extended and improved version of a previous work, where we analyzed data for only two epochs (2003 and 2005 ) and only from PN/XMM and IBIS/INTEGRAL (). Data selection and analysis {#data} =========================== To test the thermal Comptonization model in the low/hard state of 1E1740.7$-$2942, we carried out a search in the databases of XMM and INTEGRAL and looked for nearly simultaneous observations. The best matches satisfying our criteria are presented in Table[\[tab1\]]{}, which results in three observations. The 2003 observation was performed by the two satellites almost simultaneously. Some hours of delay can be seen in the 2005 data set, and in 2012 the data from the two satellites are not simultaneous with days of delay in between the observations. PN MOS1 JEMX ISGRI -- -------------- --------------------- --------------------- --------------------- --------------------- TStart(UTC) 2003-09-11 23:05:36 2003-09-11 23:00:21 2003-09-11 22:49:29 2003-09-11 22:50:31 Exposure(ks) 1.6 8.0 3.0 9.0 TStart(UTC) 2005-10-02 01:21:18 2005-10-02 01:16:05 2005-10-02 11:07:19 2005-10-02 02:48:43 Exposure(ks) 16.0 7.7 2.2 23.4 TStart(UTC) 2012-04-03 08:32:12 2012-04-03 08:27:03 2012-04-07 09:01:18 2012-04-07 05:59:52 Exposure(ks) 82.7 34.1 1.6 7.7 Data from XMM-Newton () cameras PN (), and MOS1 () were reduced using standard procedures with SAS (V. 12.0.1) (<http://xmm.esa.int/sas/>). INTEGRAL () data from IBIS () and JEM-X () telescopes were treated using the recipes described in the OSA10.0 documentation (<http://www.isdc.unige.ch/integral/>). We have also made use of [*XSPEC*]{} (V. 12.8.0) in performing our spectral fits. Data from PN were constrained to the $\sim$$2$ to $12$keV region, and MOS1 data were limited to the $\sim$$2$–$10$keV band. The lower energy threshold is due to the low count rate and signal-to-noise ratio, S/N below $\sim$2keV for both cameras (PN and MOS1) according to the pileup analysis. We also note that $\sim$$2$keV was the lower limit in energy used by ASCA () and Chandra on two occasions (; ). With our spectra also starting around 2keV we left the hydrogen column density as a free parameter in our fits. The energy range from $\sim$10 up to $\sim$$20$keV was filled with the use of the JEM-X/INTEGRAL telescope, even though the 1E1740.7$-$2942 count rates are quite low in this band for this instrument. The data presented here from $20$ up to $200$keV were collected by the IBIS telescope onboard INTEGRAL, completing our broadband coverage. We have made use only of the ISGRI/IBIS data (). Fits to our spectra included the multiplicative [const]{} and [phabs]{} components in [*XSPEC*]{} when one accounts for the difference in the counts for the four instruments and the other for absorption by neutral material. The normalization factors were relative to the PN instrument, which has the highest count rate among the four instruments. While the ASCA () and Chandra () studies on 1E1740.7$-$2942 have made use of an absorbed powerlaw to fit the $2$–$20$keV spectra, the broadband ($2$–$200$keV) Suzaku study () used a combination of models. On the other hand, the INTEGRAL hard X-ray spectrum of 1E1740.7$-$2942 has been fitted with a Comptonization model (). We have thus fitted our XMM and INTEGRAL broadband ($2$–$200$keV) spectra of 1E1740.7$-$2942 with two components: a soft standard extended blackbody ([diskbb]{}), which comes from the accretion disk and a Comptonized component. We have made use of thermal Comptonization and other variations. This component is probably coming from a larger region (a [*corona*]{}). In this two-component model, we tied (as usual) the temperature of the seed soft photons of the Comptonized component to the disk average temperature (modeled as [diskbb]{}). To help us in classifying the 1E1740.7$-$2942 spectral state, we have also made use of a classical exponential folded powerlaw to fit our spectra (see details in ). We note that this model had also been used before for 1E1740.7$-$2942 (e.g., ). Results =======  The lower (XMM) parts of our spectra can be fitted very well as an accretion disk consisting of multiple blackbody components, as the [diskbb]{} model. It is noteworthy, however, that [*another*]{} (soft) component is necessary (F$_{\hbox{\scriptsize{test}}}$ of the order of 10$^{-7}$) to adequately describe the soft part or our 1E1740.7$-$2942 data in 2003 and 2012. This component is only marginally needed in the 2005 spectrum (F$_{\hbox{\scriptsize{test}}}$ of 10$^{-4}$). Nevertheless, to maintain uniformity in our comparisons between the three spectra, we kept this second component in the XMM spectrum of 2005. This second component is very well fitted by any Comptonization model. With the addition of this Comptonization component, our broadband $2$–$200$keV spectra can be very well modeled. As we have already stated, instead of using Comptonization models, an exponential folded powerlaw can also be used to fit (the INTEGRAL) part of our spectra. We also performed those fits accordingly, which help us to conclude (see details in ) that 1E1740.7$-$2942 was in its (canonical) low/hard state (LHS) in the three observations reported in this study. It is interesting to note that we found no evidence in any of our XMM data sets of the emission lines due to the soft X-ray background in the direction of this source, as reported, for example, in [@Reynolds2010]{}. A feature near 7keV was found, however. Even though it was apparent from the PN spectrum that the feature seems to be an edge, the feature is very well fitted by a [Gauss]{} model in [*XSPEC*]{}. The centroid energy is 7.11keV, which is exactly the energy of the iron absorption K-edge reported by Suzaku. We proceed as in the Suzaku study, modeling this edge using the [zvfeabs]{} model in [*XSPEC*]{}. The fit, in this case, returns an iron absorption K-edge of 7.19keV. Freezing the edge energy to 7.11 returned a $\chi^2_{\hbox{\tiny{red}}}$ value of 1.4, which is not as good as the value obtained with the [gauss]{} model (1.2). Notwithstanding this difference in goodness of fit, it is our interpretation on physical grounds that this is the iron absorption K-edge observed before by Suzaku (). To our knowledge, this is the first detection by XMM of this feature in 1E1740.7$-$2942. For the Comptonization component, our first attempts were with the simplest thermal [compTT]{} () model in [*XSPEC*]{}, since this form of modeling was widely used in past spectra modeling of 1E1740.7$-$2942, and for others black hole binaries and in a previous version of this study (). The spectra resulting of those fits are shown in Figure([\[fig1\]]{}). All the fits with this thermal Comptonization provided a very adequate description of the spectrum. In our fits we kept the geometry parameter equal to 1, which corresponds to a disk geometry in [compTT]{}. Fits parameters can be found in Table([\[tab2\]]{}). Motivated by the evidence of Compton reflection found by some authors in 1E1740.7$-$2942 (), and by the study of @Natalucci2013, which found no evidence of it, we also used the convolutive reflect component in our fits (). Presence of reflection is very common in the low/hard spectra of black hole binaries (see, e.g., ). If the Comptonizing plasma is surrounding the disk, then the presence of Compton reflection is unavoidable. The results of our first attempt in modeling such reflection, with the reflect model (acting on the [compTT]{} component) is shown in Table([\[tab2\]]{}). We found indications of the possible presence of the Compton reflection in the 2003 and 2005 spectra but not in the 2012 spectrum. It is interesting to note that the absence of reflection in 2012 and the presence of an Fe-edge agrees with the results of @Reynolds2010. As can be see in Table([\[tab2\]]{}), the presence or absence of the Compton reflection does not alter the parameters of the fit with respect to the [compTT]{} fit alone (i.e., without the convolutive reflect component). We have also made an attempt to fit our spectra with the [compPS]{} model () in [*XSPEC*]{} to check what has been tried before (). [CompPS]{} provides a numerical solution of the radiative transfer equation, and it comprises Compton reflection as one of its parameters. However, we noted that the 2012 spectrum in our fits did not fit very well by this model in the sense that it returns an unrealistic value for the plasma temperature when using a slab geometry with an optical depth around the value of one, as is the case for our 2003 and 2005 fits with [compTT]{}. Therefore, we decided not to use the [compPS]{} model in this study. In Figure([\[fig2\]]{}), we show the spectral variation between the 2003, 2005, and 2012 spectrum. We also show the parameters of our fits to a exponential folded powerlaw model ([cutoffpl]{} in [*XSPEC*]{}) in Table([\[tab2\]]{}). In Table([\[tab3\]]{}), we show the measured fluxes for the observations in this study. It is interesting to compare our broadband fluxes with those obtained by Suzaku (). The average flux measured by Suzaku in two observations (one in $2006$ and another in $2008$) is $2.2 \times 10^{-9}$ergcm$^{-2}$s$^{-1}$ in the $2$–$300$keV range. For our observations, the average between $2003$ and $2005$ observations is $2.5 \pm 0.4 \times 10^{-9}$ergcm$^{-2}$s$^{-1}$ in the same energy range ($2.97 \pm 0.1\times10^{-9}$ in 2003 and $1.97 \pm 0.2\times10^{-9}$ in 2005, in ergcm$^{-2}$s$^{-1}$), whereas our 2012 value is $1.7 \pm 0.1 \times 10^{-9}$ergcm$^{-2}$s$^{-1}$ (unabsorbed fluxes), characterizing a significant decrease in the $2$–$300$keV flux from 1E1740.7$-$2942 from 2003 up to 2012. It is also interesting to compare our results for the hydrogen column density ($N_{\mathrm{H}}$) with other studies. Observations with ASCA [@1999ApJ...520..316S] found an average value of $9.7 \pm 0.4$ in eight observations. A Chandra/HETGS study found $11.8 \pm 0.6$ (), whereas a Chandra/ACIS-I found $10.5 \pm 0.6$ (). The average in our three observations is $13.6 \pm 0.1$. The average of all of these results is $13.3 \pm 0.1$. Without our results, the average is $10.4 \pm 0.3$ with all quoted values in units of $10^{22}$cm$^{-2}$. We caution the reader that we performed our fits leaving $N_{\mathrm{H}}$ as a free parameter. We then carefully verified that neither our conclusions nor the quality of our fits is changed noticeably by adopting (and freezing) $N_\mathrm{H}$ to $10.4$. [ccccc]{}\ & & 2003 & 2005 & 2012\ & $N_{\mathrm{H}}$ ($10^{22}$ cm$^{-1}$) & 14.1$^{+0.3}_{-0.3}$ & 12.5$^{+0.1}_{-0.1}$ & 14.1$^{+0.1}_{-0.1}$\ & $T_{in}$ (keV) & 0.24$^{+0.02}_{-0.02}$ & 0.17$^{+0.03}_{-0.03}$ & 0.19$^{+0.01}_{-0.01}$\ & $T_{0}$ (keV) & = $T_{in}$ & = $T_{in}$ & = $T_{in}$\ [compTT]{} & $kT$ (keV) & 65.6$^{+2.2}_{-1.9}$ & 65.7$^{+1.5}_{-1.9}$ & 20.1$^{+0.1}_{-0.1}$\ &[$\tau$]{} & 0.90$^{+0.08}_{-0.02}$ & 0.85$^{+0.02}_{-0.02}$ & 3.56$^{+0.03}_{-0.03}$\ &[LineE]{} (keV) & – & – & 7.11$^{+0.01}_{-0.01}$\ ${\chi}^2$/dof& & 308/277 & 400/294 & 352/298\ \ & $N_\mathrm{H}$ ($10^{22}$ cm$^{-2}$) & 13.4$^{+0.4}_{-0.3}$ & 12.5$^{+0.1}_{-0.1}$ & 14.0$^{+0.1}_{-0.1}$\ & $T_{in}$(keV) & 0.21$^{+0.03}_{-0.03}$ & 0.16$^{+0.01}_{-0.01}$ & 0.17$^{+0.01}_{-0.01}$\ &${\Omega}$/$2{\pi}$ & 0.74$^{+0.29}_{-0.28}$ & 0.32$^{+0.15}_{-0.14}$ & $\leq$0.01\ &$T_{0}$ & =$T_{in}$ & =$T_{in}$ & =$T_{in}$\ &$kT$(keV) & 64.7$^{+2.6}_{-2.4}$ & 65.5$^{+2.1}_{-2.0}$ & 20.8$^{+0.4}_{-0.4}$\ &${\tau}$ & 0.88$^{+0.05}_{-0.05}$ & 0.80$^{+0.03}_{-0.03}$ & 3.52$^{+0.03}_{-0.03}$\ &$norm$ & 44$^{+3}_{-3}\times10^{-4}$ & 43$^{+7}_{-5}\times10^{-4}$ & 76$^{+2}_{-2}\times10^{-4}$\ & [LineE]{} (keV) & – & – & 7.11$^{+0.01}_{-0.01}$\ $\chi^2$/dof & & 296/275 & 394/292 & 347/295\ \ & $N_\mathrm{H}$ ($10^{22}$ cm$^{-2}$) & 13.0$^{+0.4}_{-0.3}$ & 12.3$^{+0.1}_{-0.1}$ & 13.8$^{+0.1}_{-0.1}$\ & $T_{in}$(keV) & 0.21$^{+0.03}_{-0.03}$ & 0.15$^{+0.04}_{-0.04}$ & 0.17$^{+0.01}_{-0.01}$\ & $\Gamma$ & 1.42$^{+0.04}_{-0.04}$ & 1.52$^{+0.02}_{-0.02}$ & 1.24$^{+0.02}_{-0.02}$\ & $E_{cut}$(keV) & 89.7$^{+15.7}_{-12.1}$ & 87.1$^{+14.5}_{-11.4}$ & 138$^{+51.7}_{-30.1}$\ & $norm$ & 85$^{+7}_{-6}\times10^{-3}$ & 81$^{+3}_{-3}\times10^{-3}$ & 333$^{+8}_{-8}\times10^{-4}$\ & [LineE]{} (keV) & – & – & 7.11$^{+0.01}_{-0.01}$\ $\chi^2$/dof & & 286/276 & 379/293 & 350/296\ ------ ------------ ----------------------------------------- ----------------------------------------- ----------------------------------------- ----------------------------------------- 20–50 keV 50–200 keV [diskbb]{} 1.15$^{+0.06}_{-0.06}$ $\times10^{-12}$ 0 0 0 2003 [compTT]{} 2.74$^{+0.14}_{-0.14}$ $\times10^{-10}$ 3.72$^{+0.19}_{-0.19}$ $\times10^{-10}$ 5.65$^{+0.28}_{_0.28}$ $\times10^{-10}$ 1.02$^{+0.05}_{-0.05}$ $\times10^{-9}$ total 2.76$^{+0.14}_{-0.14}$ $\times10^{-10}$ 3.72$^{+0.19}_{-0.19}$ $\times10^{-10}$ 5.65$^{+0.28}_{-0.28}$ $\times10^{-10}$ 1.02$^{+0.05}_{-0.05}$ $\times10^{-9}$ [diskbb]{} 1.18$^{+0.20}_{-0.20}$ $\times10^{-13}$ 0 0 0 2005 [compTT]{} 2.57$^{+0.28}_{-0.26}$ $\times10^{-10}$ 2.56$^{+0.28}_{-0.26}$ $\times10^{-10}$ 3.77$^{+0.41}_{-0.38}$ $\times10^{-10}$ 6.44$^{+0.71}_{-0.64}$ $\times10^{-10}$ total 2.57$^{+0.28}_{-0.26}$ $\times10^{-10}$ 2.56$^{+0.28}_{-0.26}$ $\times10^{-10}$ 3.77$^{+0.41}_{-0.38}$ $\times10^{-10}$ 6.44$^{+0.71}_{-0.64}$ $\times10^{-10}$ [diskbb]{} 3.16$^{+0.02}_{-0.01}$ $\times10^{-13}$ 0 0 0 2012 [compTT]{} 1.86$^{+0.02}_{-0.01}$ $\times10^{-10}$ 2.76$^{+0.01}_{-0.01}$ $\times10^{-10}$ 3.89$^{+0.02}_{-0.01}$ $\times10^{-10}$ 6.11$^{+0.01}_{-0.01}$ $\times10^{-10}$ total 1.86$^{+0.02}_{-0.01}$ $\times10^{-10}$ 2.76$^{+0.01}_{-0.01}$ $\times10^{-10}$ 3.89$^{+0.02}_{-0.01}$ $\times10^{-10}$ 6.11$^{+0.01}_{-0.01}$ $\times10^{-10}$ ------ ------------ ----------------------------------------- ----------------------------------------- ----------------------------------------- ----------------------------------------- Discussion ========== The observations we describe in this study are one of the few with broadband ($2$–$200$keV) coverage of 1E1740.7$-$2942 spectra with no data gaps, which is provided only by imaging instruments, avoiding any source confusion and, thus, flux contamination. In our study, we tested the thermal–Compton paradigm for the source spectra by following, for example, other studies of 1E1740.7$-$2942 spectra. Such a paradigm was first tested in CygX$-$1 spectral analysis ().\ For the first time to our knowledge, we reported here evidence of a 7.11keV Fe-edge detected by XMM, which agrees with previous reports for such a feature derived from Suzaku studies ().\ Our results for the magnitude of $N_{\mathrm{H}}$ for 1E1740.7$-$2942 show a higher value than the average. However, we do not claim here a variation of $N_{\mathrm{H}}$ in the line of sight of 1E1740.7$-$2942, as well as any intrinsic change at or nearby the source, since our fits are not very sensitive to the value of $N_{\mathrm{H}}$.\ Our model of 1E1740.7$-$2942 spectra shows a variation in the optical depth of the plasma between 2012 when compared to 2003 or 2005. This variation implies a change in the so-called $y$ parameter of inverse Compton scattering (see, e.g., ) on the order of 3. From our database, it is not clear what causes this variation. More simultaneous observations by XMM and INTEGRAL will be very important in shedding light in this issue.\ Compared to the Suzaku study of 1E1740.7$-$2942 (), we verified a possible decrease in the 2–300keV unabsorbed flux of the source. Our average value found for 2003 and 2005 is $2.5{\pm}0.4{\times}10^{-9}$ergcm$^{-2}$s$^{-1}$, and our measured value for 2012 is $1.7{\pm}0.1{\times}10^{-9}$ergcm$^{-2}$s$^{-1}$, while the averaged between 2006 and 2008 value measured by Suzaku is $2.2{\times}10^{-9}$ergcm$^{-2}$s$^{-1}$. It is noteworthy that recent monitoring by the INTEGRAL/IBIS program in the Galactic Bulge reported that 1E1740.7$-$2942 is below the detection sensitivity limit ().\ From the results in Table([\[tab3\]]{}), we can clearly see a decrease in the $50$–$200$keV flux. The absorbed flux in the $2$–$10$keV range remained within errors constant in 2003 and 2005 and then decreased in 2012. In trying to associate the change in the $50$–$200$keV flux with some parameter of our models, no clear evidence is found, i.e., our decrease in flux is not evidently correlated with any parameter. This is contrary to what was observed in GX339$-$4, where a decrease in the luminosity is associated with an increase in $kT_{\hbox{\tiny{e}}}$ (). The high energy $50$–$200$keV flux must be associated with the accretion disk (or, for example, with a corona surrounding it), and, similarly to other Galactic black holes, it is the brightest component of the X-ray spectrum when the source is in the LHS.\ The comparison of our derived plasma temperatures with other broadband hard X-ray studies of 1E1740.7$-$2942 must be considered with caution. For example, our measured temperatures are different from the ones reported by @DelSanto05, but this is probably due to the fact that these authors have used the physical assumptions of the [compPS]{} model, which was not used here. Similarly, a different modeling was used by @Bouchet2009 and @Reynolds2010. On the other hand, low temperatures of ${\sim}$$20$keV, as in our models of the 2012 spectrum, were already reported, for instance, by the NuSTAR recent study (), which has made use of the same models we applied (i.e., [diskbb]{} and [compTT]{}). We caution the reader that the NuSTAR reported value for the optical depth is of the order of 1.4, while it varies in our observations (see Table[\[tab2\]]{}).\ It is also noteworthy that our fits with the [cutoffpl]{} model also provided acceptable fits, implying that non–thermal process may also explain the broadband 1E1740.7$-$2942 behavior. We note that a model based on a jet emission for explaining the hard X-ray spectrum of 1E1740.7$-$2942 was ruled out (). A model consisting of two thermal Comptonization components was already used () as an alternative to non–thermal processes.\ In our spectral analysis for 2003 and 2005, we found the presence of Compton reflection, which agrees with previous reports for 1E1740.7$-$2942 spectra (e.g., ). Our 2012 spectral modeling, however, found no evidence for such a component, which is also compatible with other results (e.g., ). It is very interesting to note that in 2012 the observation of a Fe-edge in our spectrum without the presence of Compton reflection is also agrees with previous spectral analysis of 1E1740.7$-$2942 (). Since the Compton reflection seems not to be permanent in 1E1740.7$-$2942 spectra, it is tempting to associate the vanishing of such a feature with physical changes in possible corona surrounding the accretion disk.\ Our study has revealed a rich spectral variability in 1E1740.7$-$2942 by highlighting the importance of broadband coverage by XMM and INTEGRAL in future simultaneous observations.\ Conclusions {#conclusion} =========== We have shown a simultaneous broadband study of 1E1740.7$-$2942 in three different epochs with the use of ESA’s XMM and INTEGRAL satellites here that covers the band from $\sim$2 up to 200keV with no data gaps. The imaging instruments onboard XMM and INTEGRAL prevent any kind of source-confusion/flux contamination. To our knowledge for the first time, we reported here a XMM/PN observation of the iron absorption K-edge at $7.11$keV, a value reported previously in Suzaku studies. We derived an historical decrease in the $2$–$300$keV flux of 1E1740.7$-$2942. Our study has revealed a rich spectral variability in 1E1740.7$-$2942. We have shown that the plasma temperature has varied between the 2003–2005 and the 2012 spectra, but it is unclear from our analysis what the causes of this are. We note that this variation is accompanied by a huge increase in the optical depth from 2003 to 2012. We believe that only more broadband observations of 1E1740.7$-$2942, for example, with simultaneous XMM and INTEGRAL campaigns as the ones discussed in this study, will be able to provide a data base from where tight constrains can be derived to the Comptonization model for the emission in the LHS state of 1E1740.7$-$2942. MC gratefully acknowledges CAPES/Brazil for support. MC gratefully acknowledges Mariano Méndez at the Kapteyn Astronomical Institute (Groningen-The Netherlands) host in a visit in the period of September - October/2012, sponsored by the COSPAR Program for Capacity Building Fellowship, for helpful discussions in the subject of this study. MC and FD acknowledges Raimundo Lopes de Oliveira Filho for early guidance in XMM data analysis issues. FD acknowledges Tomaso Belloni for helpful discussions. We deeply acknowledge Andrzej Zdziarski, our referee, for superb comments and for helping us in improving the quality of this study.

--- abstract: | Let ${{\mathfrak X}}_d$ be the $p$-adic analytic space classifying the semisimple continuous representations ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}}_p) \rightarrow {\mathrm{GL}}_d({\overline{\mathbb{Q}}_p})$. We show that the crystalline representations are Zarski-dense in many irreducible components of ${{\mathfrak X}}_d$, including the components made of residually irreducible representations. This extends to any dimension $d$ previous results of Colmez and Kisin for $d = 2$. For this we construct an analogue of the infinite fern of Gouvêa-Mazur in this context, based on a study of analytic families of trianguline ${(\varphi,\Gamma)}$-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline ${(\varphi,\Gamma)}$-modules, as well as the density of the crystalline ${(\varphi,\Gamma)}$-modules in this family. These results may be viewed as a local analogue of the theory of $p$-adic families of finite slope automorphic forms, they are new already in dimension $2$. The technical heart of the paper is a collection of results about the Fontaine-Herr cohomology of families of trianguline ${(\varphi,\Gamma)}$-modules. address: | Gaëtan Chenevier\ C.N.R.S, Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex\ France. [email protected] author: - Gaëtan Chenevier title: | Sur la densité des representations cristallines de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$ --- Introduction {#introduction .unnumbered} ============ Soient[^1] ${{\mathbb{F}}}_q$ un corps fini de caractéristique $p>0$, $W$ l’anneau des vecteurs de Witt de ${{\mathbb{F}}}_q$, et $F=W[1/p]$. Si $K$ est une extension finie de ${\mathbb{Q}}_p$ on désigne par $G_K={\rm Gal}(\overline{K}/K)$ son groupe de Galois absolu. Fixons $${{\bar\rho}}: G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d({{\mathbb{F}}}_q)$$ une représentation continue absolument irréductible[^2] et désignons par $R$ la $W$-algèbre de déformation universelle de ${{\bar\rho}}$ au sens de Mazur [@mazurdef]. Supposons enfin que ${{\bar\rho}}\not\simeq {{\bar\rho}}(1)$ (condition automatiquement satisfaite si $p-1$ ne divise pas $d$). D’après Tate [@tate][^3], cela entraîne que $R \simeq W[[T_0,\dots,T_{d^2}]]$. En particulier, l’espace rigide analytique $X$ associé par Berthelot à $R[1/p]$ est la boule unité ouverte $\{ (T_0,\dots,T_d), |T_i|<1\}$ de dimension $d^2+1$ sur $F$. PS. Rappelons que si $L$ est une extension finie de $F$, $X(L)$ paramètre les classes de $L$-isomorphie de représentations continues $G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d({\mathcal{O}}_L)$ dont la réduction modulo $\pi_L$ est isomorphe à ${{\bar\rho}}\otimes_{{{\mathbb F}}_q} k_L$. Bien sûr, ${\mathcal{O}}_L$ désigne ici l’anneau des entiers de $L$, $\pi_L$ en est une uniformisante, et $k_L={\mathcal{O}}_L/\pi_L{\mathcal{O}}_L$. Enfin, on dira que $x \in X(L)$ a une certaine propriété si la représentation associée $\rho_x : G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d(L)$ a cette propriété. Le résultat principal de cet article est le suivant. PS. PS. [**Théorème A :**]{} [*Il existe une extension finie $L$ de $F$ telle que les points cristallins de $X(L)$ sont Zariski-denses et d’accumulation dans $X$*]{}.PS. PS. Autrement dit $X(L)$ contient au moins un point cristallin et pour tout $x \in X(L)$ cristallin et tout ouvert affinoïde connexe $U \subset X$ contenant $x$ alors l’ensemble des points cristallins de $U(L)$ est Zariski-dense dans $U$ : toute fonction rigide analytique sur $U$ s’annulant sur les points cristallins de $U(L)$ est identiquement nulle. En fait, on verra que tout $L$ contenant une extension explicite de $F$ (de degré $\leq d^2$) convient.PS. PS. Ce résultat est facile quand $n=1$ et démontré indépendamment par Colmez et Kisin quand $n=2$ dans [@colmeztri §5] et [@kisinfern §1]. Leurs preuves sont inspirées de la fougère infinie de Gouvêa et Mazur [@gm], qui concerne un analogue global du problème qui nous intéresse, et qui est elle-même issue des travaux fondateurs de Coleman [@coleman] sur les familles $p$-adiques de formes modulaires. Bien que les techniques employées par Colmez et Kisin soient différentes, leurs approches sont très similaires ; nous nous bornerons ci-dessous à décrire celle de Colmez car c’est son point de vue que nous étendrons par la suite. PS. Quand $n=2$ l’espace $X$ est une boule de dimension $5$. Considérons l’ensemble $S$ des paires $(x,t)$ où $x \in X$ est [*trianguline*]{}[^4] et où $t$ est la donnée d’une triangulation de $x$. D’après Colmez, $S$ admet une structure “naturelle” d’espace analytique $p$-adique, qui est lisse et de dimension $4$. Notons $$\pi : S \rightarrow X$$ l’application ensembliste $(x,t) \mapsto x$. Son image $\pi(S) \subset X$ est appelée la [*fougère infinie*]{}. Colmez démontre que : - $\pi(S)$ contient tous les points cristallins de $X$. De plus, chaque $x \in X$ cristallin non exceptionnel[^5] admet exactement deux antécédents dans $S$. PS. - Les $(x,t) \in S$ avec $x$ cristallin non exceptionnel sont Zariski-denses et d’accumulation dans $S$.PS. - $\pi$ est localement analytique : pour (presque[^6]) tout $s \in S$ il existe un voisinage affinoïde de $s$ dans $S$ (une boule fermée de dimension $4$) surlequel $\pi$ est une immersion analytique. PS. - si $x \in X$ est cristallin non exceptionnel, alors les deux sous-boules[^7] de $X$ passant par $x$ qui sont données par (a) et (c) ne sont pas confondues. Comme dans chacune des deux sous-boules mentionnées dans le (d), les points cristallins non exceptionnels sont Zariski-denses par le (b), on obtient une idée de la structure fractale de la fougère infinie dans $X$ (et une justification pour son nom!). Le théorème ci-dessus s’en déduit aisément. Dans l’approche de Kisin, disons simplement que l’espace $S$ est remplacé dans cet argument par sa version $X_{fs}$ construite dans [@kisin]. La différence essentielle est que Kisin découpe $X_{fs}$ dans l’espace $X \times \mathbb{G}_m$ (il doit donc le “majorer”, ce qui est assez délicat) alors que Colmez le construit explicitement, ce qui est peut-être plus naturel par rapport à notre problème. Quand $n=2$, mentionnons que ce résultat de densité des cristallines joue un rôle important dans la preuve par Colmez de la correspondance de Langlands locale $p$-adique [@colmezgros]. Ajoutons enfin que toujours quand $n=2$, Nakamura a étendu dans [@nakamura] l’approche de Kisin et le théorème ci-dessus au cas où $G_{{\mathbb{Q}}_p}$ est remplacé par $G_K$ avec $K/{\mathbb{Q}}_p$ finie arbitraire. PS. Notre démonstration reprend pour $d$ quelconque les étapes de la preuve esquissée ci-dessus. L’ensemble $S$, ainsi que la fougère infinie, sont définis de la même manière. Son étude a été amorcée par Bellaïche et l’auteur dans [@bch §2]. Une conséquence simple des travaux de Berger est que les triangulations d’une représentation cristalline $V$ de dimension $d$ quelconque sont en bijection avec les drapeaux $\varphi$-stables de $D_{\rm cris}(V)$, de sorte que génériquement (au sens naïf) il y en a $d!$. C’est la généralisation du (a) dont nous aurons besoin ici. De plus, mentionnons que bien que l’on n’y munisse pas $S$ d’une structure analytique, une étude du voisinage infinitésimal de chaque point est aussi menée [*loc. cit.*]{}, ce qui est un premier pas en direction (c), tout en étant insuffisant[^8] pour le théorème $A$. Les généralisations de (b) et (c) à la dimension $d$ formeront le coeur technique et nouveau de cet article, et nous y reviendrons plus loin en détail. Un point assez délicat concerne la généralisation du (d) : en un point cristallin $x \in X$ assez général, la fougère infinie admet $d!$ feuilles chacune étant de dimension $\frac{d(d+1)}{2}+1$, et le problème est de comprendre leurs positions relatives. Ce problème est facile à résoudre pour des raisons de dimension quand $d=2$, car une analyse ad-hoc des paramètres triangulins sur chacune des deux feuilles de dimension $4$ montre que les deux feuilles ne sont pas confondues. Le cas général est nettement plus délicat, mais a déjà été résolu par l’auteur dans [@chU3] (en fait dans [@chpeccot]). Le résultat clé est que si $x$ cristallin est assez générique en un sens précis rappelé plus bas, alors la somme des espaces tangents en $x$ des $d!$ feuilles de la fougère en $x$ est l’espace tangent de $X$ en $x$ tout entier. L’argument d’adhérence Zariski employé dans [@chU3] permet alors de conclure : nous renvoyons à la section \[reductions\] pour l’argument complet.PS. PS. Revenons sur la structure analytique de l’espace $S$. Nous raisonnerons entièrement dans le monde des ${(\varphi,\Gamma)}$-modules sur l’anneau de Robba (non nécéssairement étales), ce qui est essentiellement permis par un théorème de Kedlaya et Liu [@kedliu]. Si $A$ est une ${\mathbb{Q}}_p$-algèbre affinoïde, on note ${\mathcal{R}}_A$ l’anneau de Robba à coefficients dans $A$ (voir § \[defRoA\] pour la définition précise). Soient ${{\rm Aff}}$ la catégorie des ${\mathbb{Q}}_p$-algèbres affinoïdes et $$F_d^\square : {{\rm Aff}}\longrightarrow {\rm Ens}$$ le foncteur associant à $A$ l’ensemble des classes d’équivalence de ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ qui sont triangulins, réguliers et rigidifiés : nous renvoyons à la section \[San\] pour les définitions de ces termes.[^9] PS. PS. [**Théorème B :**]{} [*Le foncteur $F_d^{\square}$ est représentable par un espace analytique $p$-adique $S_d^\square$ qui est irréductible et lisse sur ${\mathbb{Q}}_p$, de dimension $\frac{d(d+3)}{2}$. Les ${(\varphi,\Gamma)}$-modules cristallins sont Zariski-denses et d’accumulation dans $S_d^\square$.*]{} PS. PS. Ce théorème peut être vu comme un analogue local de la théorie des variétés de Hecke (ou “[*eigenvarieties*]{}”).[^10] Notons que sa première partie est nouvelle même pour $d=2$, où elle complète des résultats de Colmez et de Kisin. Toujours dans le cas $d=2$, nous construisons en fait une famille “tautologique” des ${(\varphi,\Gamma)}$-modules non nécessairement réguliers (Théorème \[thmdegal2\]). Mentionnons que la notion de rigidification utilisée fait que l’espace $S_d^\square$ admet moralement $d-1$ dimensions de plus que sa définition naïve, en revanche elle permet de traiter sur un pied d’égalité tous les ${(\varphi,\Gamma)}$-triangulins (y compris ceux qui sont scindés par exemple). Le théorème $B$ a des conséquences intéressantes concernant la construction de représentations cristallines ayant certaines propriétés. Par exemple il permet de montrer que [*si ${{\bar\rho}}: G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d({{\mathbb F}}_q)$ est une représentation semi-simple continue quelconque, il existe une extension finie $L/{\mathbb{Q}}_p$ et une représentation $G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d(L)$ cristalline absolument irréductible dont la représentation résiduelle est ${{\bar\rho}}$*]{} (Prop. \[liftgeneral\]).PS. Le coeur technique du théorème $B$, et de cet article, est un ensemble de résultats sur les ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$, notamment sur leur cohomologie à la Fontaine-Herr ([@herr1],[@herr2]). Ceci fait l’objet de la section \[cohomologie\]. Une étape de la démonstration est la vérification dans ce contexte que les complexes $C_{\varphi,\gamma}$ et $C_{\psi,\gamma}$ sont quasi-isomorphes comme dans la théorie classique de Herr, ce qui avait notamment été conjecturé par Kedlaya dans [@kedlayaseul §2.6]. Concrètement, il s’agit d’étudier la structure de $D^{\psi=0}$ comme $\Gamma$-module quand $D$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ ou sur ${\mathcal{R}}_A^+$. Ce point assez technique repose sur une étude préliminaire des familles de $\Gamma$-modules effectuée en section \[prelimfamilles\]. Notre preuve est directement inspirée de [@colmezgros §V] qui en démontre le cas particulier où $A$ est un corps. PS. Nous calculons enfin la cohomologie de ${\mathcal{R}}_A(\delta)$ quand $\delta : {\mathbb{Q}}_p^\ast \rightarrow A^\ast$ est un caractère continu, étendant des résultats de Colmez (en degrés $0$ et $1$, [@colmeztri])) et de Liu (en degré $2$, [@liu]) dans le cas particulier où $A$ est un corps. Notre preuve, bien qu’inspirée de celle de Colmez, est en fait un peu plus simple que celle dans [@colmeztri] : l’idée nouvelle essentielle est de remplacer son dévissage pour se ramener au cas où $v(\delta(p)) < 0$ par un argument direct utilisant la transformée de ... Colmez ! qui est un dévissage de l’anneau de Robba. Nous décrivons de plus la structure de ${\mathcal{R}}_A(\delta)^{\psi=1}$ comme module sur une certaine complétion de $A[\Gamma]$ notée ${\mathcal{R}}_A^+(\Gamma)$. Nous étendons enfin ces résultats à tous les ${(\varphi,\Gamma)}$-modules triangulins sur ${\mathcal{R}}_A$. Mentionnons qu’une des spécificités de cette théorie en famille est l’absence de dualité. De plus, le coeur (au sens de Fontaine) d’une famille de ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ n’est pas nécéssairement libre sur ${\mathcal{R}}_A^+(\Gamma)$. Voici un échantillon des résultats obtenus.PS. [**Théorème C :**]{} [*Si $D$ est un ${(\varphi,\Gamma)}$-module triangulin de rang $d$ sur ${\mathcal{R}}_A$, alors $H^i(D)$ est de type fini sur $A$ pour tout $i$ et dans le groupe de Grothendieck des $A$-modules de type fini on a la relation $[H^0(D)]-[H^1(D)]+[H^2(D)]=-[A^d]$. De plus, la formation des $H^i(D)$ commute à tout changement de base plat affinoïde. Enfin, $D^{\psi=1}$ contient un ${\mathcal{R}}_A^+(\Gamma)$-module libre de rang $d$, le quotient étant de type fini sur $A$. PS. Si $D$ est régulier, alors $H^0(D)=H^2(D)=0$ et $H^1(D)$ est libre de rang $d$ sur $A$. La formation des $H^i(D)$ commute alors à tout changement de base affinoïde. Si $D$ est $p$-régulier, alors $D^{\psi=1}$ est libre de rang $d$ sur ${\mathcal{R}}_A^+(\Gamma)$.*]{} PS. PS. L’auteur remercie chaleureusement Pierre Colmez pour les explications précieuses de ses résultats, dont les § \[checkgamma\] et § \[cohoRoA\] sont très largement inspirés, ainsi que Laurent Fargues et Olivier Taïbi pour des discussions utiles. ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ {#prelimfamilles} ================================================== Quelques anneaux de fonctions analytiques {#defRoA} ----------------------------------------- Nous renvoyons à [@bgr] pour les généralités sur les espaces analytiques $p$-adiques au sens de Tate. Si $X$ est un tel espace, nous noterons ${\mathcal{O}}(X)$ la ${\mathbb{Q}}_p$-algèbre de ses fonctions globales. Quand $X$ est affinoïde, c’est une algèbre de Banach noethérienne, et on note aussi $X={\rm Sp}({\mathcal{O}}(X))$. En général, on munit ${\mathcal{O}}(X)$ de la topologie de la convergence uniforme sur tout ouvert affinoïde, c’est une algèbre de Fréchet si $X$ admet un recouvrement dénombrable admissible par des affinoïdes, ce qui sera le cas pour tous les espaces ci-dessous. Si $X$ est réunion admissible d’ouverts affinoïdes $X_n$ ($n\geq 0$) avec $X_n \subset X_{n+1}$, c’est une algèbre de Fréchet-Stein au sens de [@schneiderteitelbaum §3].[^11] Soit $A$ une ${\mathbb{Q}}_p$-algèbre affinoïde. Un [*modèle*]{} de $A$ est une sous-${\mathbb{Z}}_p$-algèbre $\mathcal{A} \subset A$ topologiquement de type fini, i.e. quotient de ${\mathbb{Z}}_p\langle t_1,\dots,t_m\rangle$ pour un certain $m\geq 1$, et telle que $\mathcal{A}[1/p]=A$. Un modèle est ouvert, borné, sans $p$-torsion, et complet séparé pour la topologie $p$-adique. Pour toute famille finie $x_1$,…, $x_r$ d’éléments de $A$ à puissances (positives) bornées, il existe toujours un modèle $\mathcal{A}$ de $A$ contenant les $x_i$. Si $A$ est réduit, l’ensemble de ses éléments à puissances bornées est le plus grand modèle de $A$ (Tate). Si $|.|$ est une norme sur $A$ (sous-entendu, sous-multiplicative et pour laquelle $A$ est complet), alors sa boule unité $A^0$ est un modèle de $A$. PS. Soit $I \subset [0,1[$ un intervalle d’extrémités dans $p^{\mathbb{Q}}$, on note $B_I$ l’ouvert admissible de la droite affine rigide $\mathbb{A}^1$ (de paramètre $T$) défini par $|T| \in I$. C’est un disque si $0 \in I$ et une couronne sinon, il est affinoïde si $I$ est un segment. On notera encore $T \in {\mathcal{O}}(B_I)$ le paramètre tautologique. Soit $A$ une ${\mathbb{Q}}_p$-algèbre affinoïde. Si $I \subset [0,1[$ est un intervalle on pose $${{\mathcal{E}}}_A^I:={\mathcal{O}}( {\rm Sp}(A) \times B_I).$$ - Si $0 \in I$, ${{\mathcal{E}}}_A^I \subset A[[T]]$ est aussi la sous-algèbre des séries $f=\sum_{n \in {\mathbb{N}}} a_n T^n$ telles que pour tout $r \in I$ on ait $|a_n|r^n \rightarrow 0$ quand $n\rightarrow \infty$. C’est une $A$-algèbre de Fréchet pour les normes $|f|_{[0,r]}:=\sup_{n \in {\mathbb{N}}} |a_n|r^n$, si $|.|$ est une norme fixée sur $A$. - Si $0 \notin I$, ${{\mathcal{E}}}_A^I$ est aussi la $A$-algèbre des séries de Laurent $f=\sum_{n \in {\mathbb{Z}}} a_n T^n$ telles que pour tout $[r,s] \subset I$ on ait $|a_n|s^n \rightarrow 0$ et $|a_{-n}| r^{-n} \rightarrow 0$ quand $n\rightarrow \infty$. C’est une $A$-algèbre de Fréchet pour les normes $|f|_{[r,s]}:=\sup(\sup_{n \in {\mathbb{N}}} |a_n|s^n, \sup_{n \in {\mathbb{N}}} |a_{-n}|r^{-n})$, où $[r,s] \subset I$. Notons que les descriptions ci-dessus sont classiques quand $I$ est un segment, et s’en déduisent en général en considérant le recouvrement admissible de ${\rm Sp}(A) \times B_I$ par les ${\rm Sp}(A) \times B_J$ pour $J \subset I$ un segment. De plus, ${{\mathcal{E}}}_A^I$ est une algèbre affinoïde si $I$ est un segment et de Fréchet-Stein en général (considérer le recouvrement par les ${\rm Sp}(A) \times B_{I_n}$ où $I_n$ est une suite croissante de segments recouvrant $I$). PS. On pose encore $${\mathcal{R}}_{A,r}={{\mathcal{E}}}_A^{[r,1[}, \, \, \, {\mathcal{R}}_A=\bigcup_{0<r<1} {\mathcal{R}}_{A,r},\, \, \, {\rm et}\, \, \, \, {\mathcal{R}}^+_A={\mathcal{R}}_{A,0}.$$ Lorsque $A={\mathbb{Q}}_p$ on omettra souvent de le mentionner en indice. Par exemple ${\mathcal{R}}:={\mathcal{R}}_{{\mathbb{Q}}_p}$ et ${{\mathcal{E}}}^I:={{\mathcal{E}}}_{{\mathbb{Q}}_p}^I$. De plus, si $X={\rm Sp}(A)$ on remplacera parfois $A$ par $X$ dans les notations ci-dessus, de sorte que par exemple ${{\mathcal{E}}}_X^I:={{\mathcal{E}}}_{{\mathcal{O}}(X)}^I$. PS. Il sera commode par la suite d’introduire certains modèles sur ${\mathbb{Z}}_p$ des anneaux ci-dessus.PS. \[lemmmodel\] Soit $I=[p^{-\frac{a}{n}},p^{-\frac{b}{m}}]$ avec $a,b,m,n$ entiers tels que $0\leq \frac{a}{n} \leq \frac{b}{m}$ (on suppose les dénominateurs non nuls et le fractions réduites). Alors ${\mathcal{O}}^I:={\mathbb{Z}}_p\langle \frac{T^m}{p^b},\frac{p^a}{T^n},T \rangle$ est un modèle de ${{\mathcal{E}}}^I$. De plus, le morphisme surjectif naturel ${\mathbb{Z}}_p \langle T,U,V \rangle /(p^bU-T^m,T^nV-p^a) \rightarrow {\mathcal{O}}^I$ a pour noyau la $p^\infty$-torsion de ${\mathbb{Z}}_p \langle T,U,V \rangle /(p^bU-T^m,T^nV-p^a)$ En effet, par définition de $B_I$ on a ${{\mathcal{E}}}^I={\mathbb{Q}}_p\langle T, U, V\rangle/(p^b U-T^m, T^n V-p^a)$, donc les éléments $\frac{T^m}{p^b}, \frac{p^a}{T^n} \in {{\mathcal{E}}}^I$ sont à puissances bornées et ${\mathcal{O}}^I$ est un modèle de ${{\mathcal{E}}}^I$ car image de ${\mathbb{Z}}_p\langle T, U, V\rangle$. Le dernière assertion suit car le morphisme de l’énoncé est un isomorphisme après avoir inversé $p$. Si $I$ est un segment on définit ${\mathcal{O}}^I$ comme dans le lemme ci-dessus. Si $\mathcal{A}$ est un modèle de $A$, alors ${\mathcal{O}}_{{\mathcal A}}^I:={{\mathcal A}}\widehat{\otimes}_{{\mathbb{Z}}_p} {\mathcal{O}}^I$ est un modèle de ${{\mathcal{E}}}_A^I$. PS. Enfin, on pose ${{\mathcal{E}}}_{{\mathcal A}}^{\dag, 0}=:{{\mathcal A}}[[T]]$, et si $n\geq 1$, on définit ${{\mathcal{E}}}_{\mathcal{A}}^{\dag,n}$ comme étant le complété de $\mathcal{A}[[T]][\frac{p}{T^n}]$ pour la topologie $p$-adique. C’est aussi l’anneau des séries de Laurent de la forme $\sum_{k\in {\mathbb{Z}}} a_k T^k$ telles que $a_k \in \mathcal{A}$ pour tout $k \in {\mathbb{Z}}$, $a_{-k} \in p^{\alpha_k}\mathcal{A}$ pour $k \geq 0$, où $\alpha_k$ est une suite d’entiers $\geq [\frac{k}{n}]$ et tendant vers l’infini avec $k$. En particulier, ${{\mathcal{E}}}_{\mathcal{A}}^{\dag,n} \subset {\mathcal{R}}_{A,p^{-\frac{1}{n}}}$. On le munit de sa [*topologie faible*]{} : une base de voisinages de $0$ est l’ensemble des $p^\alpha {{\mathcal{E}}}_{\mathcal{A}}^{\dag, n} + T^\beta \mathcal{A}[[T]]$, $\alpha,\beta\geq 0$ entiers. Il est aussi complet pour cette topologie. Nous renvoyons à [@schneider §17] pour les généralités sur les produits tensoriels complétés. Le (i) ci-dessous fait notamment le lien entre les définitions employées ici et celles de [@kedliu] et de [@bergercolmez].PS. \[topo\] - Pour tout $I$, l’application naturelle $A \widehat{\otimes}_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I \rightarrow {{\mathcal{E}}}^I_A$ est un isomorphisme de Fréchets. PS. - Pour tout $0<r<1$, la norme $|.|_{[0,r]}$ sur ${\mathcal{R}}_A^+$ induit sur ${{\mathcal A}}[[T]]$ la topologie définie par l’idéal $(p,T)\mathcal{A}[[T]]$, ou ce qui revient au même, la [*topologie faible*]{} dont une base de voisinages de $0$ est donnée par les $p^\alpha {{\mathcal A}}[[T]] + T^\beta {{\mathcal A}}[[T]]$ avec $\alpha,\beta \geq 0$. L’injection naturelle $\mathcal{A}[[T]] \rightarrow {\mathcal{R}}_A^+$ est d’image fermée. PS. - Pour tout $n\geq 1$ et $p^{-1/n} \leq s <1$, la norme $|.|_{[p^{-1/n},s]}$ induit sur ${{\mathcal{E}}}_{\mathcal{A}}^{\dag, n}\subset {\mathcal{R}}_{A,p^{-1/n}}$ la topologie faible. En particulier, ${{\mathcal{E}}}_{\mathcal{A}}^{\dag, n}$ est fermé dans ${\mathcal{R}}_{A,p^{-1/n}}$. PS. - Si $J$ est un ideal de $A$ et $I$ est un intervalle quelconque de $[0,1[$, alors l’application naturelle ${{\mathcal{E}}}_A^I/J{{\mathcal{E}}}_A^I \rightarrow {{\mathcal{E}}}_{A/J}^I$ est un isomorphisme.PS. - Pour tout intervalle $I$ de $[0,1[$, le $A$-module ${{\mathcal{E}}}_A^I$ est plat. En particulier, ${\mathcal{R}}_A$ est plat sur $A$. Quand $I$ est un segment, $B_I$ est affinoïde et le (i) est évident. En général, l’injectivité de $A \otimes_{{\mathbb{Q}}_p} {\mathbb{Q}}_p^{{\mathbb{Z}}} \rightarrow A^ {\mathbb{Z}}$ entraîne celle de $\iota_I : A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I \rightarrow {{\mathcal{E}}}^I_A$. Si $J \subset I$ est un segment, on a une semi-norme naturelle produit-tensoriel $|.|_J:=|.|\otimes |.|_J$ sur $A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I$ associée : $|x|_J$ est l’infimum sur toutes les écritures $x=\sum_i a_i \otimes f_i$ des ${\rm sup}_i |a_i||f_i|_{J}$. En particulier, $|\iota_I(x)|_J\leq |x|_J$ pour tout $x \in A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I$. Réciproquement, considérons le diagramme commutatif évident $$\xymatrix{ A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I \ar@{->}[rr] ^{\iota_I}\ar@{->}[d]_\mu & & {{\mathcal{E}}}_A^I \ar@{->}[d] \\ A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^J \ar@{->}[rr]^{\iota_J} & & {{\mathcal{E}}}_A^J}.$$ D’après [@schneider Prop. 17.4 (iii)], $|\mu(x)|_J=|x|_J$ pour tout $x \in A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I$. De plus, le cas d’un segment entraîne qu’il existe une constante $C_J>0$ telle que $|x|_J \leq C_J |\iota(x)|_J$ pour tout $x$ dans $A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^J$. Ainsi, $|.|_J$ et $|\iota_I(.)|_J$ sont équivalentes sur $A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I$. On conclut car ${{\mathcal{E}}}_A^I$ est un Fréchet contenant $A \otimes_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I$ comme sous-espace dense.PS. Le premier point du (ii) est un exercice classique sans difficulté laissé au lecteur. Comme $\mathcal{A}[[T]]$ est complet pour la topologie $(p,T)$-adique, c’est un sous-espace fermé de ${\mathcal{R}}_A^+$ et le (ii) suit. Pour le (iii), on constate sur la description donnée plus haut de ${{\mathcal{E}}}_{{\mathcal A}}^{\dag, n}$ que ${{\mathcal{E}}}_{{\mathcal A}}^{\dag,n}=T{{\mathcal A}}[[T]] \oplus (\oplus_{i=0}^{n-1} T^i {\mathbb{Z}}_p \langle \frac{p}{T^n} \rangle)$ est une somme directe topologique, les termes de droite étant respectivement munis de la topologie $(p,T)$-adique pour le premier et de la topologie $p$-adique pour le second. Par définition des $|.|_J$ sur ${\mathcal{R}}_{A,r}$, cette somme directe est une isométrie pour chaque $|.|_J$ (les termes de droites étant munis de la norme induite). Le premier point du (iii) se déduit alors de celui du (ii). Le (iii) suit car ${{\mathcal{E}}}_{{\mathcal A}}^{\dag,n}$ est complet pour la topologie faible. Comme il est complet pour cette topologie, cela conclut.PS. Pour démontrer le (iv) il faut voir si $f \in {{\mathcal{E}}}_A^I$ a tous ses coefficients dans $J$ (vue comme série de Laurent), alors $f \in J{{\mathcal{E}}}_A^I$. Soient $e_1,\dots,e_g$ une famille finie de générateurs de $J$ comme $A$-module, d’après Tate la surjection ($A$-linéaire) $A^g \rightarrow I$ qui s’en déduit est nécessairement ouverte. En particulier il existe une constante $C>0$ telle que tout élément $x \in J$ s’écrive sous la forme $\sum_i x_i e_i$ avec $|x_i| \leq C |x|$ pour tout $i$. Ainsi, appliquant ceci à tous les coefficients d’un $f \in {{\mathcal{E}}}_A^I$ à coefficients dans $J$, il vient que $f \in J {{\mathcal{E}}}_A^I$.PS. Prouvons enfin (v). Si $I$ est un segment, alors ${{\mathcal{E}}}_A^I=A \widehat{\otimes}_{{\mathbb{Q}}_p} {{\mathcal{E}}}^I$ est isomorphe à $A \langle t \rangle$ comme $A$-module, et il est bien connu que ce dernier est plat sur $A$. En effet, si ${{\mathcal A}}\subset A$ est un modèle de $A$, alors ${{\mathcal A}}\langle t \rangle$ est plat sur ${{\mathcal A}}[T]$ comme complété de ce dernier (qui est un anneau noethérien) pour la topologie $p$-adique, et donc plat sur ${{\mathcal A}}$. On conclut en inversant $p$. En général, on écrit ${{\mathcal{E}}}_A^I$ est comme limite projective de ${{\mathcal{E}}}_A^{I_n}$ pour une suite arbitraire croissante $I_n$ de segments recouvrant $I$. Soit $0 \longrightarrow P \longrightarrow Q \longrightarrow R \longrightarrow 0$ une suite exacte de $A$-modules de type fini. Par platitude de ${{\mathcal{E}}}_A^{I_n}$ sur $A$ pour tout $n\geq 0$ (cas précédent), on dispose d’un système projectif de suites exactes $0 \longrightarrow P_n \longrightarrow Q_n \longrightarrow R_n \longrightarrow 0$ où $X_n:=X \otimes_A {{\mathcal{E}}}_A^{I_n}$ (la famille $(X_n)$ définit donc un ${{\mathcal{E}}}_A^I$-module cohérent au sens de Schneider-Teitelbaum). D’après [@schneiderteitelbaum §3, Thm.], c’est un fait général que cette suite reste exacte après passage à la limite projective sur $n$. Pour conclure, il suffit de voir que si $X$ est un $A$-module de type fini, alors l’application naturelle $$X \otimes_A {{\mathcal{E}}}_A^I \longrightarrow \projlim_n X_n$$ est un isomorphisme. C’est clair si $X$ est libre et cela suit en général si l’on choisit une présentation $L \rightarrow L' \rightarrow X \rightarrow 0$ avec $L,L'$ libres de type fini sur $A$, et considère le diagramme commutatif $$\xymatrix{ L \otimes_A {{\mathcal{E}}}_A^I \ar@{->}[d] \ar@{->}[rr] & & L' \otimes_A {{\mathcal{E}}}_A^I \ar@{->}[d] \ar@{->}[rr] & & X\otimes_A {{\mathcal{E}}}_A^I \ar@{->}[d] \rightarrow 0 \\ \projlim_n L_n \ar@{->}[rr] & & \projlim_n L'_n \ar@{->}[rr] & & \projlim_n X_n \rightarrow 0}$$ dont les suites horizontales sont exactes (par exactitude à droite du produit tensoriel pourcelle du haut et par  [@schneiderteitelbaum §3, Thm.] pour celle du bas) et les deux verticales de gauche sont des isomorphismes. L’assertion sur ${\mathcal{R}}_A$ suit car c’est une limite inductive filtrante de ${{\mathcal{E}}}_A^I$. Le groupe $\Gamma:={\mathbb{Z}}_p^\ast$ agit par automorphismes des $B_I$ par la formule $$\label{formgamma} \gamma(T)=(1+T)^\gamma-1=\sum_{n\geq 1} {\gamma \choose {n}} T^n \in {\mathbb{Z}}_p[[T]],$$ de sorte que $|\gamma(t)|=|t|$ si $|t|<1$. Si $A$ est une algèbre affinoïde sur ${\mathbb{Q}}_p$, cette action de $\Gamma$ s’étend donc en une action $A$-linéaire sur ${{\mathcal{E}}}_A^I$, et en fait sur tous les anneaux introduits ci-dessus. \[estimeegamma\] Soient $A$ une algèbre affinoïde sur ${\mathbb{Q}}_p$ et ${{\mathcal A}}\subset A$ un modèle. - Si $I$ est un segment, l’application induite $\Gamma \rightarrow {\rm End}_A ({{\mathcal{E}}}_A^I)$ est continue.PS. - Si $1\leq n \leq m$, $I=[p^{-1/n},p^{-1/m}]$ et $\gamma \in 1+2p^M{\mathbb{Z}}_p$ alors $(\gamma-1){\mathcal{O}}^I_{{\mathcal A}}\subset T^M {\mathcal{O}}^I_{{\mathcal A}}$.PS. (Le terme à droite dans (i) est l’algèbre des endomorphismes $A$-linéaires continus du $A$-module de Banach ${{\mathcal{E}}}_A^I$, c’est une $A$-algèbre de Banach pour la norme d’opérateurs).PS. L’action de $\Gamma$ étant $A$-linéaire, on peut supposer $A={\mathbb{Q}}_p$ et $\mathcal{A}={\mathbb{Z}}_p$ dans (i) et (ii). Vérifions (i), il suffit par multiplicativité de montrer la continuité en $1 \in \Gamma$. Écrivons $I=[p^{-a/n},p^{-b/m}]$ (resp. $I=[0,p^{-b/m}]$). Soient $U=\frac{T^m}{p^b}$ et $V=\frac{p^a}{T^n}$, ainsi que ${\mathcal{O}}^I={\mathbb{Z}}_p\langle U, V,T \rangle \subset {{\mathcal{E}}}^I$ (resp. ${\mathbb{Z}}_p\langle U,T \rangle$) le modèle de ${{\mathcal{E}}}^I$ défini au lemme \[lemmmodel\]. Si $\gamma \in \Gamma$ alors $$\gamma(U)=\gamma(T)^m/p^b=\gamma^m U(1+\sum_{k\geq 2} \frac{{ \gamma \choose k}}{\gamma} T^{k-1})^m \in U{\mathbb{Z}}_p\langle U \rangle [T]$$ $$\gamma(V)=\frac{p^a}{\gamma(T)^n}=\gamma^{-n} V (1+\sum_{k\geq 2} {\frac{{\gamma \choose k}}{\gamma}} T^{k-1})^{-n} \in V{\mathbb{Z}}_p\langle U \rangle[T].$$ (car ${\mathbb{Z}}_p[[T]] \subset {\mathbb{Z}}_p\langle U \rangle [T]$). En particulier, $\Gamma$ préserve ${\mathcal{O}}^I$. Notons que $T^m \in p{\mathcal{O}}^I$. Ainsi, si $M\geq 1$ est un entier suffisamment grand pour que ${\gamma \choose i} \in p{\mathbb{Z}}_p$ si $i=2,\dots,m$ et $\gamma \in 1+p^M{\mathbb{Z}}_p$, les formules ci-dessus entraînent que $1+p^M{\mathbb{Z}}_p$ agit trivialement sur ${\mathcal{O}}^I/p{\mathcal{O}}^I$. L’identité $$\label{identite} W^p-1 \equiv (W-1)^p \bmod p(W-1){\mathbb{Z}}[W]$$ montre alors que $(\gamma-1){\mathcal{O}}^I \subset p^N{\mathcal{O}}^I$ si $\gamma \in 1+2p^{M+N-1}{\mathbb{Z}}_p$, ce qui termine la preuve du (i).PS. Vérifions donc (ii). Les formules pour $\gamma(T)$, $\gamma(U)$ et $\gamma(V)$ données ci-dessus montrent alors que ${\mathcal{O}}^I$ et $T{\mathcal{O}}^I$ sont $\Gamma$-stables, puis que $\gamma$ agit trivialement sur ${\mathcal{O}}^I/T{\mathcal{O}}^I$ (car sur les images de $U$ et $V$) dès que $\gamma \in 1+p{\mathbb{Z}}_p$. Cela démontre le (ii) pour $M=1$. On en déduit que pour tout $i\geq 0$ et tout $\gamma \in 1+p{\mathbb{Z}}_p$, alors $(\gamma-1)T^i{\mathcal{O}}^I \subset T^{i+1}{\mathcal{O}}^I$. En effet, cela vient par récurrence sur $i$ de la formule $(\gamma-1)(ab)=(\gamma-1)(a)\gamma(b)+ (\gamma-1)(b)a$ et de ce que $$(\gamma-1)(T)\in (T^2,pT){\mathbb{Z}}_p[[T]] \subset T^2 {\mathcal{O}}^I$$ (car $p=VT^n \in T {\mathcal{O}}^I$). En particulier, $(\gamma-1)^jT^i{\mathcal{O}}^I \subset T^{i+j}{\mathcal{O}}^I$ pour tout $i,j\geq 0$ et $\gamma \in 1+p{\mathbb{Z}}_p$. Le (ii) suit alors pour tout $M$, encore par récurrence sur $M$, en utilisant cette fois-ci l’identité (\[identite\]) et encore le fait que $p=VT^n \in T {\mathcal{O}}^I$. Préliminaire sur les familles de $\Gamma$-modules {#checkgamma} ------------------------------------------------- Soit $A$ une ${\mathbb{Q}}_p$-algèbre affinoïde et $N\geq 0$ un entier. On définit un [*$\Gamma$-module sur ${{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$*]{} comme étant un ${{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$-module $D$ libre de rang fini muni d’une action semi-linéaire de $\Gamma$ qui soit continue, [*i.e.*]{} telle qu’il existe une base $e_1,\dots,e_d$ de $D$ sur ${{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$ pour laquelle l’application $\gamma \in \Gamma \mapsto {{\rm Mat}}(\gamma) \in M_d({{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N})$ associant à $\gamma$ sa matrice dans la base $(e_i)$ soit continue coefficient par coefficient. On vérifirait aisément à l’aide des lemmes \[estimeegamma\] (i) et \[topo\] (ii) et (iii) que si cela vaut pour une base alors cela vaut pour toutes. PS. Pour tout segment $I \subset [p^{-\frac{1}{N}},1[$, on a un morphisme naturel ${{\mathcal{E}}}_{{\mathcal A}}^{\dag,N} \rightarrow {\mathcal{O}}_{{\mathcal A}}^I$, de sorte qu’il y a un sens à considérer $D^I=D \otimes_{{{\mathcal{E}}}_{{\mathcal A}}^{\dag,N}} {\mathcal{O}}_{{\mathcal A}}^I$. \[estimeeD\] Soit $s\geq 0$ un entier. Il existe un entier $M\geq 1$ tel que $\forall \gamma \in 1+p^M{\mathbb{Z}}_p$ : PS. - $(\gamma-1)D^I \subset T^s D^I$ pour tout $I=[p^{-1/m},p^{-1/m'}]$ avec $1\leq m \leq m'$ et $m\geq N$,PS. PS. - et de plus $(\gamma-1)D^{[0,p^{-1/2}]} \subset p^s D^{[0,p^{-1/2}]}$ si $N=0$. Supposons d’abord $N\geq 1$. Par continuité de $\Gamma$, on peut choisir $M\geq s$ tel que pour tout $\gamma \in 1+p^{M}{\mathbb{Z}}_p$ on ait ${{\rm Mat}}(\gamma)-{\rm id} \in p^sM_d({{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,n})+T^sM_d({{\mathcal A}}[[T]])$. Mais $p \in T^N{{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$, donc ${{\rm Mat}}(\gamma)-{\rm id} \in T^sM_d({{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N})$. En particulier, si $1\leq m \leq m'$, $m\geq N$ et $I=[p^{-1/m},p^{-1/m'}]$, alors $$(\gamma-1)(e_i) \in T^s D^I,\,\, \,\,\,\forall \gamma \in 1+p^M{\mathbb{Z}}_p, \,\,\forall i=1,\dots,d.$$ On conclut alors le (a) par l’identité $$(\gamma-1)(\sum_i x_i e_i)=\sum_i (\gamma-1)(x_i)\gamma(e_i)+x_i(\gamma-1)(e_i),$$ et le lemme \[estimeegamma\] (ii). Si $N=0$, le fait que l’on puisse choisir $M$ de sorte que (a) soit satisfait découle du cas précédent en considérant le $\Gamma$-module sur ${{\mathcal{E}}}_{{\mathcal A}}^{\dag,1}$ obtenu par extension des scalaires. Pour vérifier (b) on procède de même que ci-dessus en remarquant par exemple que $T^2 \in p{\mathcal{O}}_I$ si $I=[0,p^{-1/2}]$ et en utilisant le lemme \[estimeegamma\] (i). Terminons par une proposition qui jouera un rôle important dans la suite. Fixons $D$ un $\Gamma$-module sur ${{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$. Si $\gamma \in \Gamma$, considérons l’application $$G_\gamma : D \rightarrow D, \, \, x \mapsto Tx+(1+T) \cdot (\gamma-1)(x),$$ c’est un endomorphisme $A$-linéaire de $D$. On considère le morphisme de $A$-algèbres $A[G_\gamma] \rightarrow {\mathcal{R}}_{A,p^{-1/n}}$ envoyant $G_\gamma$ sur $T$. PS. \[propcle\] Il existe un entier $M\geq 1$ tel que pour tout $\gamma \in 1+p^M{\mathbb{Z}}_p$, et pour tout $n\geq N$, le $A[G_\gamma]$-module $D^{(n)} = D \otimes_{{{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}} {\mathcal{R}}_{A,p^{-1/n}}$ s’étend de manière unique en un ${\mathcal{R}}_{A,p^{-1/n}}$-module noté[^12] $D^{'(n)}$ tel que l’application structurelle ${\mathcal{R}}_{A,p^{-1/n}} \times D^{'(n)} \rightarrow D^{'(n)}$ soit continue. De plus, si $(e_i)$ est une base de $D$ sur ${{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$ alors $(e_i \otimes 1)$ est une base de $D^{'(n)}$ sur ${\mathcal{R}}_{A,p^{-1/n}}$. La topologie sous-entendue sur $D^{'(n)}$ dans cet énoncé est celle de ${\mathcal{R}}_{A,p^{-1/n}}$-module de Fréchet $D^{(n)}$ sous-jacent (qui est libre de rang fini par définition). Appliquons le lemme précédent pour $s=2$. Fixons une fois pour toutes $\gamma \in 1+p^M {\mathbb{Z}}_p$, où $M$ est donné par ce lemme. Soit $I=[p^{-1/m},p^{-1/m'}]$ avec $m\geq N$ et $m\geq 1$, et considérons $D^I$. C’est un ${\mathcal{O}}_{{\mathcal A}}^I$-module libre de rang $d$, il est en particulier complet pour la topologie $p$-adique, et aussi pour la topologie $T$-adique (car ${\mathcal{O}}_{{\mathcal A}}^I$ l’est, puisque $T^{m'}{\mathcal{O}}_{{\mathcal A}}^I \subset p{\mathcal{O}}_{{\mathcal A}}^I$). L’anneau ${\mathcal}{B}^I={\mathrm{End}}_{{\mathcal A}}({\mathcal{O}}_{{\mathcal A}}^I)$ est donc lui aussi complet pour les topologies $p$-adique et $T$-adique. En particulier, si on pose $$\psi(x):=\frac{1+T}{T}(\gamma-1)(x)$$ alors $\psi(D^I) \subset T D^I$ par choix de $M$, en particulier $\psi \in {\mathcal}{B}^I$, mais aussi $\sum_{k\geq 0} (-1)^k\psi^k$ converge dans ${\mathcal}{B}^I$, vers un inverse de $1+\psi$. De plus, on a évidemment $\psi(TD^I) \subset TD^I$, donc $\psi$ induit l’endomorphisme nul sur $D^I/TD^I$. Si on note $m_u$ la multiplication par $u$, nous avons donc montré que : - $G_\gamma=m_T \cdot (1+\psi)$ dans ${\mathcal}{B}^I$ et $G_\gamma \equiv m_T$ dans ${\mathrm{End}}_{{\mathcal{O}}^I/(T)}(D^I/TD^I)$. - $\frac{G_\gamma^{m'}}{p}=m_{\frac{T^{m'}}{p}}\cdot (1+\psi)^{m'} \in {\mathcal}{B}^I$ et $\frac{G_\gamma^{m'}}{p} \equiv m_{\frac{T^{m'}}{p}}$ dans ${\mathrm{End}}_{{\mathcal{O}}^I/(T)}(D^I/TD^I)$.PS. - $\frac{p}{G_\gamma^m}=m_{\frac{p}{T^m}} \cdot (\sum_{k\geq 0} (-1)^k \psi^k)^m \in {\mathcal}{B}^I$ et $\frac{p}{G_\gamma^{m}} \equiv m_{\frac{p}{T^m}}$ dans ${\mathrm{End}}_{{\mathcal{O}}^I/(T)}(D^I/TD^I)$. Comme ${\mathcal}{B}^I$ est complet pour la topologie $p$-adique, et comme ${\mathcal{O}}^I$ est par définition de quotient de ${\mathbb{Z}}_p\langle T,U,V\rangle/(pU-T^m,T^{m'}V-p)$ par sa $p$-torsion (lemme \[lemmmodel\]), il découle de (i), (ii) et (iii) que le morphisme naturel ${{\mathcal A}}[{\mathrm{G}}_\gamma] \rightarrow {\mathcal}{B}^I$ s’étend en un morphisme continu ${{\mathcal{E}}}_{{\mathcal A}}^I \rightarrow {\mathcal}{B}^I$. On note $D^{'I}$ le groupe abélien $D^I$ muni de cette structure de ${{\mathcal{E}}}_{{\mathcal A}}^I$-module. Comme $1+\psi$ est inversible dans ${\mathcal}{B}^I$, notons que $T^kD^{'I}=G_\gamma^k D^I=T^k D^I$ pour tout $k\geq 0$, puis que $D^{'I}$ est complet pour la topologie $T$-adique et sans $T$-torsion. Considérons l’application identité $$D^I/TD^I \rightarrow D^{'I}/TD^{'I}.$$ Le (i), (ii) et (iii) ci-dessus assurent que cette application est un morphisme de ${\mathcal{O}}_{{\mathcal A}}^I$-modules. En particulier, $D^{'I}/(T)$ est libre comme ${{\mathcal{E}}}_{{\mathcal A}}^I/(T)$-module. Comme $D^{'I}$ et ${{\mathcal{E}}}_{{\mathcal A}}^I$ sont complets pour la topologie $T$-adique et sans $T$-torsion, un raisonnement standard montre que $D^{'I}$ est libre sur ${{\mathcal{E}}}_{{\mathcal A}}^I$, et qu’une famille $ (e_i)$ est une base de $D^I$ si et seulement si c’est une base de $D^{'I}$. Quand $n=0$, on traite le cas de $I=[0,p^{-1/2}]$ par un raisonnement entièrement analogue en ne considérant que la topologie $p$-adique (et non pas $T$-adique, c’est en fait seulement plus simple il n’y a pas de condition de type (iii) à vérifier). PS. Pour terminer, il ne reste qu’à “recoller” les $D^I[1/p]$. Pour cela, fixons $n\geq N$ et considérons l’ensemble $\mathcal{I}$ des intervalles de $[p^{-1/n},1[$ de la forme $[p^{-1/m},p^{-1/m'}]$ avec $m\leq m'$, ou de la forme $[0,p^{-1/2}]$ (ce qui ne se produit que si $n=0$). Si $J \subset I$ sont dans $\mathcal{I}$ on a bien sûr un morphisme de restriction $$r_{I,J}: D^I[1/p] \rightarrow D^J[1/p].$$ Si $I,J \in \mathcal{I}$, alors $I\cap J=\emptyset$ ou $I\cap J \in \mathcal{I}$. Comme les $B_I$ avec $I\in \mathcal{I}$ recouvrent admissiblement $B_{[p^{-1/n},1[}$, il vient que $D^{(n)}$ (resp. ${\mathcal{R}}_{A,p^{-1/n}}$) s’identifie à la limite projective sur $\mathcal{I}$ des ${{\mathcal{E}}}_A^I$-modules $D^I[1/p]$ (resp. des ${{\mathcal{E}}}_A^I$). Il est de plus immédiat sur la construction ci-dessus que si $f \in {{\mathcal{E}}}_A^I$ et $v \in D^{I}[1/p]$, alors $r_{I,J}(f \ast v)=r_{I,J}(f) \ast r_{I,J}(v)$ où $\ast$ désigne ici la structure de module de $D^{'I}[1/p]$ et $D^{'J}[1/p]$ sur ${{\mathcal{E}}}_A^I$ et ${{\mathcal{E}}}_A^J$. Cela nous permet d’une part de munir $D^{(n)}$ d’une structure de ${\mathcal{R}}_{A,p^{-1/n}}$-module, disons $D^{'(n)}$, en posant $(f_I)\ast (v_I):=(f_I\ast v_I)$. D’autre part, si $(e_i)$ est une base de $D$ sur ${{\mathcal{E}}}_{{{\mathcal A}}}^{\dag,N}$ nous avons vu que $e_i \otimes 1$ est une base de $D^{'I}[1/p]$ sur ${{\mathcal{E}}}_A^I$, on en déduit que $D^{'(n)}$ est libre sur ${\mathcal{R}}_{A,p^{-1/n}}$ de base $e_i \otimes 1$.PS. Le choix d’une base de $D$ munit $D^{(n)} = \oplus_i {\mathcal{R}}_{A,p^{-1/n}} e_i$ d’une structure d’espace de Fréchet qui ne dépend pas du choix de la base $(e_i)$. La continuité de ${\mathcal{R}}_{A,p^{-1/n}} \times D^{'(n)} \rightarrow D^{'(n)}$ se déduit alors de celle des ${{\mathcal{E}}}_A^I \times D^{'I}[1/p] \rightarrow D^{'I}[1/p]$. L’assertion d’unicité vient plus précisément de ce qu’il existe au plus une structure de ${\mathcal{R}}_{A,p^{-1/n}}$-module telle que pour tout $v \in D^{(n)}$, l’application $f \mapsto f\ast v$ soit continue, car $A[T,T^{-1}]$ (resp. $A[T]$) est dense dans ${\mathcal{R}}_{A,p^{-1/n}}$ si $n\geq 1$ (resp. si $n=N=0$). Familles de ${(\varphi,\Gamma)}$-modules sur l’anneau de Robba -------------------------------------------------------------- Soit $A$ une ${\mathbb{Q}}_p$-algèbre affinoïde. Les anneaux ${\mathcal{R}}^+_A$ et ${\mathcal{R}}_A$ sont munis d’un endomorphisme d’anneaux $\varphi$ défini par $$\varphi(f)(T)=f((1+T)^p-1)$$ et qui commute à l’action de $\Gamma$. Plus précisément, supposons $r=0$ ou $r>p^{-\frac{1}{(p-1)}}$, on dispose d’un morphisme analytique $\varphi_\ast : B_{[r,1[} \rightarrow B_{[r^p,1[}$ défini par $t \mapsto (1+t)^p-1$ et $\varphi$ est par définition le morphisme ${\mathcal{R}}_{A,r^p} \rightarrow {\mathcal{R}}_{A,r}$ qui s’en déduit. Il est évident sur la formule pour $\varphi_\ast$ que $\varphi$ commute à l’action de $\Gamma$. PS. Un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ est un ${\mathcal{R}}_A$-module $D$ libre et de rang fini muni d’actions semi-linéaires de $\varphi$ et $\Gamma$ qui commutent et satisfaisant les axiomes suivants. D’une part on demande que $\varphi$ envoie une ${\mathcal{R}}_A$-base de $D$ sur une ${\mathcal{R}}_A$-base de $D$, i.e. ${\mathcal{R}}\varphi(D)= D$. D’autre part, on demande que l’action de $\Gamma$ soit continue au sens suivant : il existe une ${\mathcal{R}}_A$-base $e_1,\dots,e_d$ de $D$ et $r \in [0,1[$ tels que si $\gamma \mapsto {{\rm Mat}}(\gamma)$ désigne la matrice de $\gamma \in \Gamma$ dans cette base, alors $M(\gamma) \in M_d({\mathcal{R}}_{A,r})$ pour tout $\gamma \in \Gamma$ et l’application $\Gamma \rightarrow M_d({\mathcal{R}}_{A,r})$, $\gamma \mapsto M(\gamma)$, est continue (coefficient par coefficient). On dira que $D$ est [*$\Gamma$-borné*]{} si on peut trouver un modèle $\mathcal{A} \subset A$, un entier $n\geq 0$, et une ${\mathcal{R}}_A$-base $e_i$ de $D$ dans laquelle ${{\rm Mat}}(\gamma) \in M_d({{\mathcal{E}}}_{{{\mathcal A}}}^n)$ pour tout $\gamma \in \Gamma$. Il résulte du lemme \[topo\] (iii) que le $\Gamma$-module $\oplus_i {{\mathcal{E}}}_{{{\mathcal A}}}^n e_i$ qui s’en déduit est bien un $\Gamma$-module sur ${{\mathcal{E}}}_{{{\mathcal A}}}^n$ au sens du § \[checkgamma\]. PS. Les ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ forment un catégorie $A$-linéaire ${{\rm FG}}_A$ s’il on considère pour ${\mathrm{Hom}}_{{{\rm FG}}_A}(D_1,D_2)$ les applications ${\mathcal{R}}_A$-linéaires qui commutent à $\Gamma$ et à $\varphi$. PS. L’intérêt des ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ réside dans leurs liens avec les représentations continues ${\mathrm{Gal}}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p}) \rightarrow {\mathrm{GL}}_d(A)$ (Fontaine, Cherbonnier-Colmez, Kedlaya, Berger-Colmez, Kedlaya-Liu) pour lequel nous renvoyons à Berger-Colmez [@bergercolmez] et Kedlaya-Liu [@kedliu]. La condition $\Gamma$-bornée introduite ci-dessus n’est pas standard, et apparaît ici pour des raisons techniques. Bien que nous n’utiliserons pas ce résultat, mentionnons que la méthode de Berger-Colmez [@bergercolmez], généralisant un résultat de Colmez-Cherbonnier, assure que si $M$ est un ${{\mathcal A}}$-module libre muni d’une action ${{\mathcal A}}$-linéaire continue de[^13] ${\mathrm{Gal}}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$, alors on peut lui associer un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ qui est $\Gamma$-borné. PS. Enfin, définissons un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A^+$ comme étant un ${\mathcal{R}}^+_A$-module $D$ libre de rang fini muni d’actions semi-linéaires de $\varphi$ et $\Gamma$ qui commutent, telles que $\varphi(D)$ contienne une ${\mathcal{R}}^+_A$-base de $D$, et telles qu’il existe une ${\mathcal{R}}^+_A$-base $e_1,\dots,e_d$ de $D$ dans laquelle $\gamma \mapsto {{\rm Mat}}(\gamma)$ (la matrice de $\gamma \in \Gamma$ dans cette base) soit continue. On dira encore que $D$ est [*$\Gamma$-borné*]{} si on peut trouver un modèle $\mathcal{A} \subset A$ et une ${\mathcal{R}}^+_A$-base $e_i$ de $D$ dans laquelle ${{\rm Mat}}(\gamma) \in M_d({{\mathcal A}}[[T]])$ pour tout $\gamma \in \Gamma$. Il résulte du lemme \[topo\] (ii) que le $\Gamma$-module $\oplus_i {{\mathcal A}}[[T]] e_i$ qui s’en déduit est bien un $\Gamma$-module sur ${{\mathcal A}}[[T]]$ au sens du § \[checkgamma\]. PS. PS. PS. Les ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ qui nous intéressent principalement dans cet article sont les ${(\varphi,\Gamma)}$-modules triangulins, variante en famille d’une notion introduite par Colmez dans [@colmeztri]. Soit ${{\mathcal T}}$ l’espace analytique $p$-adique paramétrant les caractères continus de ${\mathbb{Q}}_p^\ast$ : pour toute algèbre affinoïde $A$, ${{\mathcal T}}(A)$ est l’ensemble des morphismes continus ${\mathbb{Q}_p}^\ast \rightarrow A^\ast$. Il est bien connu que cet espace est isomorphe au produit de $\mathbb{G}_m$ par l’espace ${\mathcal{W}}$ des caractères continus $p$-adiques de ${\mathbb{Z}}_p^\ast$, lui-même étant une réunion disjointe finie de boules unités ouvertes. Si $\delta \in {{\mathcal T}}(A)$, i.e. si $\delta : {\mathbb{Q}}_p^\ast \rightarrow A^\ast$ est un morphisme continu de groupes, on définit un ${(\varphi,\Gamma)}$-module ${\mathcal{R}}_A(\delta)$ de rang $1$ sur ${\mathcal{R}}_A$, disons ${\mathcal{R}}_A(\delta)={\mathcal{R}}_A e$, par la formule $\gamma(e)=\delta(\gamma)e$ pour tout $\gamma \in \Gamma$ et $\varphi(e)=\delta(p)e$. On définit de même ${\mathcal{R}}_A^+(\delta)$ de manière évidente. Un ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_A$ est la donnée d’un ${(\varphi,\Gamma)}$-module $D$ sur ${\mathcal{R}}_A$ et d’une suite croissante $({{\rm Fil}}_i(D))_{i=0,\dots,d}$, $d={\rm rg}_{{\mathcal{R}}_A}(D)$, de sous ${\mathcal{R}}_A$-modules de $D$ stables par $\varphi$ et $\Gamma$ telle que ${{\rm Fil}}_0(D)=0$, ${{\rm Fil}}_d(D)=D$, et telle que pour chaque $i=1,\dots,d-1$, ${{\rm Fil}}_i(D)/{{\rm Fil}}_{i-1}(D) \simeq {\mathcal{R}}_A(\delta_i)$ pour un certain $\delta_i \in {{\mathcal T}}(A)$. Nous verrons plus bas (Lemme \[classrg1\]) que la suite des $\delta_i$ est uniquement déterminée par $({{\rm Fil}}_i(D))$, nous l’appelons le [*paramètre*]{} de $D$. Notons aussi que ${{\rm Fil}}_i(D)$ est libre de rang $i$ sur ${\mathcal{R}}_A$, et facteur direct dans $D$ comme ${\mathcal{R}}_A$-module. PS. Terminons ce paragraphe par un sorite sur l’extension des scalaires. Soit $B$ une $A$-algèbre affinoïde. On dispose pour chaque $0<r<1$ d’un morphisme continu d’anneaux ${\mathcal{R}}_{A,r} \rightarrow {\mathcal{R}}_{B,r}$ induisant à la limite un morphisme ${\mathcal{R}}_A \rightarrow {\mathcal{R}}_B$. Si $D$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ on note $$D \widehat{\otimes}_A B$$ le ${\mathcal{R}}_B$-module $D \otimes_{{\mathcal{R}}_A} {\mathcal{R}}_B$ : c’est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_B$ de manière naturelle. On pourrait justifier cette notation en le voyant comme un produit tensoriel complété mais cela ne sera pas nécessaire. On a défini ainsi un foncteur $ -\,\widehat{\otimes}_A B : {(\varphi,\Gamma)}/A \rightarrow {(\varphi,\Gamma)}/B$. Si $I$ est un idéal de $A$, le lemme \[topo\] (iv) entraîne que ${\mathcal{R}}_A/I{\mathcal{R}}_A={\mathcal{R}}_{A/I}$ et donc que $D \widehat{\otimes}_A A/I = D/ID$ pour tout ${(\varphi,\Gamma)}$-module $D$ sur ${\mathcal{R}}_A$. Si $I=m_x$ est l’idéal maximal correspondant à $x \in {\rm Sp}(A)$, on posera aussi $$D_x:=D/m_xD,$$ c’est un ${(\varphi,\Gamma)}$-module sur $k(x):=A/m_x$ (une extension finie de ${\mathbb{Q}}_p$). PS. PS. Par exemple, si $\delta \in {{\mathcal T}}(A)$ alors ${\mathcal{R}}_A(\delta) \widehat{\otimes}_A B = {\mathcal{R}}_B(\delta')$ où $\delta' \in {{\mathcal T}}(B)$ est le caractère $\delta$ composé par $A^\ast \rightarrow B^\ast$. Notons que si $0 \rightarrow D \rightarrow D' \rightarrow D'' \rightarrow 0$ est une suite exacte de ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$, elle est scindée comme suite de ${\mathcal{R}}_A$-modules, et induit donc une suite exacte de ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_B$ après extension des scalaires. En particulier, si $(D,{{\rm Fil}}_\bullet(D))$ est ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_A$, alors $D\widehat{\otimes}_A B$ est triangulin sur ${\mathcal{R}}_B$ pour la filtration ${{\rm Fil}}_i(D) \widehat{\otimes} B$.PS. PS. \[classrg1\] Si $\delta,\delta' \in {{\mathcal T}}(A)$, alors ${\mathcal{R}}_A(\delta) \simeq {\mathcal{R}}_A(\delta')$ si et seulement si $\delta=\delta'$. En effet, si $A$ est artinien c’est [@bch Prop. 2.3.1]. En général, on remarque que si $I$ est un idéal de $A$ alors ${\mathcal{R}}_{A}(\delta) \otimes_A A/I = {\mathcal{R}}_{A/I}(\delta \bmod I)$. On conclut car si $A$ est une ${\mathbb{Q}}_p$-algèbre affinoïde alors l’intersection de ses idéaux de codimension finie est nulle par le théorème d’intersection de Krull, et donc $A$ se plonge dans le produit des $A/I$ avec $I$ de codimension finie. Terminons par une question, dont une réponse (affirmative) ne semble connue que lorsque $A$ est artinien :PS. [**Question:**]{} Est-ce que tout ${(\varphi,\Gamma)}$-module de rang $1$ sur ${\mathcal{R}}_A$ est isomorphe à un ${\mathcal{R}}_A(\delta)$ ? PS. PS. Cohomologie des ${(\varphi,\Gamma)}$-modules triangulins sur ${\mathcal{R}}_A$ {#cohomologie} ============================================================================== L’objectif de cette partie est de calculer la cohomologie des ${(\varphi,\Gamma)}$-modules triangulins sur ${\mathcal{R}}_A$. PS. PS. Généralités sur la cohomologie des ${(\varphi,\Gamma)}$-modules {#gencoho} --------------------------------------------------------------- Si $D$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$, et si $\gamma \in \Gamma$ est un générateur topologique[^14] on rappelle que suivant Fontaine et Herr [@herr1] on dispose du complexe $C_{\varphi,\gamma}(D)^\bullet$ : $$0 \rightarrow D \overset{x\mapsto (\varphi-1)x+(\gamma-1)x}{\longrightarrow} D \times D \overset{(x,y)\mapsto (\gamma-1)x-(\varphi-1)y}{\longrightarrow} D \rightarrow 0,$$ le premier $D$ étant placé en degré $0$. On désigne par $H^i(D)$ la cohomologie de ce complexe, ce sont donc des $A$-modules nuls en degré $i \notin \{0,1,2\}$. Par définition, $H^0(D)={\mathrm{Hom}}_{{{\rm FG}}_A}({\mathcal{R}}_A,D)$. De plus, PS. PS. $H^1(D)$ est canoniquement isomorphe à ${\rm Ext}_{{{\rm FG}}_A}({\mathcal{R}}_A,D)$. Donner une action de $\varphi$ et $\gamma$ sur $D\oplus {\mathcal{R}}_A$ étendant la structure de ${(\varphi,\Gamma)}$-module de $D$ est équivalent à donner $x:=(\varphi-1)(e) \in D$ et $y:=(\gamma-1)(e) \in D$, $\varphi$ et $\gamma$ commutant si et seulement si $(x,y)$ est dans $Z^1(C_{\varphi,\gamma}(D))$. L’action de $\gamma$ s’étend automatiquement en une action continue de $\Gamma$. En effet, il découle du Lemme \[estimeegamma\] (i) que si $f \in {\mathcal{R}}_{A,r}$ alors la suite $(1+\gamma+\cdots+\gamma^{p^n-1})f$ tend vers $0$. Cela vaut donc aussi si $f \in M_d({\mathcal{R}}_{A,r})$, et donc tout $1$-cocycle $\gamma^{\mathbb{Z}}\rightarrow M_d({\mathcal{R}}_{A,r})$ s’étend en un cocycle continu $\Gamma \rightarrow M_d({\mathcal{R}}_{A,r})$. On vérifie immédiatement que deux $1$-cocycles donnent des extensions isomorphes si et seulement si ils diffèrent d’un cobord. Il se trouve que toujours suivant Fontaine et Herr, un autre complexe est relié à la cohomologie de $D$. Supposons $r=0$ ou $r>p^{-\frac{1}{p(p-1)}}$. L’application $({\mathcal{R}}_{A,r^p})^p \rightarrow {\mathcal{R}}_{A,r}$ définie par $$(f_0,\dots,f_{p-1}) \mapsto \sum_{i=0}^{p-1} (1+T)^i\varphi(f_i)$$ est alors un isomorphisme topologique. En effet, c’est un résultat standard quand $A={\mathbb{Q}}_p$ et le lemme \[topo\] (i) nous y ramène en général. En particulier, $\varphi$ est fini et plat de degré $p$, continu et injectif. On définit alors $\psi : {\mathcal{R}}_{A,r} \rightarrow {\mathcal{R}}_{A,r}$ par la formule $\varphi \psi = \frac{1}{p} {\rm trace}_{{\mathcal{R}}_{A,r}/\varphi({\mathcal{R}}_{A,r^p})}(\varphi)$. Sur $A\otimes_{{\mathbb{Q}}_p}{\mathbb{Q}}_p(\mu_p)$ on a donc la formule $\varphi\psi(f)= \frac{1}{p}\sum_{\zeta^p=1}f(\zeta (1+T)-1)$. Le calcul de la trace des $(1+T)^i$ assure que si $f=\sum_{i=0}^{p-1} (1+T)^i\varphi(f_i)$ alors $\psi(f)=f_0$. En particulier, $\psi : {\mathcal{R}}_{A,r} \rightarrow {\mathcal{R}}_{A,r^p}$ est continu. PS. Soit $D$ un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ ou ${\mathcal{R}}_A^+$. Comme $D$ a par définition une ${\mathcal{R}}_A$ ou ${\mathcal{R}}_A^+$-base dans $\varphi(D)$, on a encore une décomposition $$D=\oplus_{i=0}^{p-1} (1+T)^i \varphi(D).$$ On peut donc définir un opérateur $\psi: D \rightarrow D$ par la formule $$\psi(\sum_{i=0}^{p-1} (1+T)^i \varphi(x_i))=x_0.$$ Il est $A$-linéaire surjectif, commute à l’action de $\Gamma$, et satisfait $\psi \varphi = {\rm id}$. Enfin, si $u \in {\mathrm{Hom}}_{{{\rm FG}}_A}(D_1,D_2)$, alors $u \cdot \psi = \psi \cdot u$. Le complexe $C_{\psi,\gamma}(D)^\bullet$ est alors définit de la même manière que $C_{\varphi,\gamma}(D)^\bullet$ à ceci près que $\varphi$ est remplacé par $\psi$. Un calcul immédiat montre que l’on dispose d’un morphisme $$\eta : C_{\varphi,\gamma}(D)^\bullet \rightarrow C_{\psi,\gamma}(D)^\bullet$$ qui vaut l’identité en degré $0$, $(x,y)\mapsto (-\psi(x),y)$ en degré $1$, et $-\psi$ en degré $2$. Le morphisme $\eta$ est surjectif car $\psi$ l’est, son noyau étant le complexe $$0 \longrightarrow 0 \longrightarrow D^{\psi=0} \overset{\gamma-1}{\longrightarrow} D^{\psi=0} \longrightarrow 0.$$ En particulier, $C_{\psi,\gamma}(D)^\bullet$ et $C_{\varphi,\gamma}(D)^\bullet$ sont quasi-isomorphes si $\gamma-1$ est bijectif sur $D^{\psi=0}$. La proposition suivante est immédiate. \[devissagecoho\] Soit $D$ un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$. Si $\gamma-1$ est bijectif sur $D^{\psi=0}$ alors on a des identifications naturelles $H^0(D)=D^{\psi=1,\gamma=1}$, $H^2(D)=D/(\psi-1,\gamma-1)$, ainsi qu’une suite exacte naturelle $$0 \rightarrow D^{\psi=1}/(\gamma-1) \overset{y \mapsto (0,y)}{\longrightarrow} H^1(D) \overset{(x,y)\mapsto \overline{x}}{\longrightarrow} (D/(\psi-1))^{\gamma=1} \rightarrow 0.$$ Enfin, si on pose $C(D)=(\varphi-1)D^{\psi=1} \subset D^{\psi=0}$, on a une suite exacte naturelle $$0 \rightarrow D^{\varphi=1}/(\gamma-1)\rightarrow D^{\psi=1}/(\gamma-1) \rightarrow C(D)/(\gamma-1) \rightarrow 0.$$ En théorie des ${(\varphi,\Gamma)}$-modules de Fontaine classique, un résultat de Herr [@herr1 Thm. 3.8] assure que $\gamma-1$ est toujours bijectif sur $D^{\psi=0}$. Dans le cadre ci-dessus, un résultat de Colmez assure aussi que c’est toujours le cas si $A$ est un corps [@colmezgros Prop. 5.1.19]. Nous allons démontrer que c’est aussi le cas en général sous une hypothèse assez faible sur $D$. PS. \[invgamma\] Soit $D$ un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ qui est $\Gamma$-borné. Alors $\gamma-1$ est bijectif sur $D^{\psi=0}$. Plus généralement, supposons que $D$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ possédant une suite croissante $D_1 \subset D_2 \subset \cdots \subset D_s$ de sous-${\mathcal{R}}_A$-modules qui sont facteurs directs comme ${\mathcal{R}}_A$-modules, et de plus stables par $\varphi$ et $\Gamma$. Supposons enfin que les $D_{i+1}/D_i$ sont $\Gamma$-bornés. Alors $\gamma-1$ est bijectif sur $D^{\psi=0}$. En effet, si $0 \rightarrow D_1 \rightarrow D_2 \rightarrow D_3 \rightarrow 0$ alors la surjectivité de $\psi$ (sur $D_1$) entraîne que la suite associée $$0 \rightarrow D_1^{\psi=0} \rightarrow D_2^{\psi=0} \rightarrow D_3^{\psi=0} \rightarrow 0$$ est exacte. Comme elle est $\gamma$-équivariante, il vient que si $\gamma-1$ est bijectif sur $D_i^{\psi=0}$ pour $i=1,3$ alors il l’est aussi pour $i=2$, de sorte que le second cas suit du premier, que nous considérons maintenant. Soient $e_1,\dots,e_d$ une ${\mathcal{R}}_A$-base de $D$, ${{\mathcal A}}\subset A$ un modèle, et $N\geq 0$, tels que ${\mathcal}{D}:=\oplus_i {{\mathcal{E}}}_{{\mathcal A}}^{\dag,N} e_i$ soit stable par $\Gamma$. Choisissons un entier $M$ comme dans la proposition \[propcle\]. Comme $\gamma-1$ divise $\gamma^{(p-1)p^{M-1}}-1$ dans ${\mathbb{Z}}[\gamma]$, il suffit de montrer que $\gamma-1$ est bijectif sur $D^{\psi=0}$ si $\gamma \in 1+p^M{\mathbb{Z}}_p^\ast$. Posons $\gamma_0=1+p^M \in \Gamma$. La relation $\gamma_0(1+T)=(1+T)\varphi^M(1+T)$ entraîne pour tout $x$ dans $D$ $$(\gamma_0-1)((1+T)\varphi^M(x))=(1+T)\varphi^M((1+T)\gamma_0(x)-x)=(1+T)\varphi^M(G_{\gamma_0}(x)).$$ Mais $D$ est la réunion des ${\mathcal}{D} \otimes_{{{\mathcal{E}}}_{{\mathcal A}}^{\dag,n}} {\mathcal{R}}_{A,p^{-1/n}}$ pour $n\geq N$. La proposition \[propcle\] assure que le $A[G_{\gamma_0}]$-module ${\mathcal}{D} \otimes_{{{\mathcal{E}}}_{{\mathcal A}}^{\dag,n}} {\mathcal{R}}_{A,p^{-1/n}}$ s’étend en un ${\mathcal{R}}_{A,p^{-1/n}}$-module via $G_{\gamma_0} \mapsto T$. Comme $T$ est inversible dans ${\mathcal{R}}_{A,p^{-1/n}}$ si $n>0$ il vient que $G_{\gamma_0}$ est inversible sur ${\mathcal}{D} \otimes_{{{\mathcal{E}}}_{{\mathcal A}}^{\dag,n}} {\mathcal{R}}_{A,p^{-1/n}}$. Pour conclure, il reste à remarquer deux choses. Premièrement, si $u \in {\mathbb{Z}}_p^\ast$ alors $\gamma_0^u-1$ agit sur ${\mathcal}{D} \otimes_{{{\mathcal{E}}}_{{\mathcal A}}^{\dag,n}} {\mathcal{R}}_{A,p^{-1/n}}$ via l’élément $u(G_{\gamma_0}) \in {\mathcal{R}}_{A,p^{-1/n}}^\ast$, qui est aussi inversible. Cela montre que si $\gamma' \in 1+p^M{\mathbb{Z}}_p^\ast$ alors $\gamma'-1$ est bijectif sur $(1+T)\varphi^M(D)$. Deuxièmement, cela vaut encore si on remplace $(1+T)$ par $(1+T)^a$ pour $a \in {\mathbb{Z}}_p^\ast$. En effet, pour tout $a \in {\mathbb{Z}}_p^\ast$ et $\gamma' \in 1+p^M{\mathbb{Z}}_p^\ast$ l’action de $a$ sur $D$ induit un isomorphisme $\gamma'$-équivariant $$(1+T)\varphi^M(D) {\overset{\sim}{\rightarrow}}(1+T)^a\varphi^M(D).$$ On conclut car $D^{\psi=0}=\bigoplus_{1\leq i \leq (p-1)p^{M-1}, (p,i)=1} (1+T)^i\varphi^M(D)$. \[cortri1\] Si $D$ est triangulin sur ${\mathcal{R}}_A$ alors $\gamma-1$ est bijectif sur $D^{\psi=0}$. Il suffit de vérifier que ${\mathcal{R}}_A(\delta)$ est $\Gamma$-borné si $\delta \in {{\mathcal T}}(A)$. Soit $e$ une base de ${\mathcal{R}}_A(\delta)$ telle que $\gamma(e)=\delta(\gamma)e$ pour tout $\gamma \in \Gamma$. Comme $\Gamma$ est compact et $\delta$ est continu, les éléments $\delta(\gamma)$ et $\delta(\gamma)^{-1}$ de $A$ sont à puissances positives bornées pour tout $\gamma \in \Gamma$. Comme $\Gamma$ est topologiquement de type fini, on peut donc trouver un modèle ${{\mathcal A}}\subset A$ tel que $\delta(\Gamma) \subset {{\mathcal A}}^\ast$. À fortiori, $\Gamma.e \subset {{\mathcal A}}[[T]]e$, ce qui conclut. Cohomologie de ${\mathcal{R}}_A(\delta)$ partie I : calcul de $H^0$ et $H^2$ {#cohoRoA} ---------------------------------------------------------------------------- Nous allons maintenant calculer les $H^i({\mathcal{R}}_A(\delta))$. Rappelons que lorsque $A$ est un corps, ce calcul est dû à Colmez pour $i=0,1$ [@colmeztri], et Liu pour $i=2$ [@liu]. Dans le cas général, nous allons procéder de manière légèrement différente à celle de Colmez en utilisant un dévissage de l’anneau de Robba (aussi dû à Colmez) que nous rappelons maintenant. Si $f=\sum_{n\in {\mathbb{Z}}} a_n T^n \in {\mathcal{R}}_A$, on note ${{\mathrm{Res}}}(f) \in A$ le résidu en $0$ de la forme différentielle $ f(T) \frac{dT}{1+T}$, c’est à dire l’élément $a_{-1}$ dans l’écriture $\frac{f(T)}{1+T}=\sum_{n \in {\mathbb{Z}}}a_n T^n$. \[transfocolmez\] Si $f \in {\mathcal{R}}_{A}$, la transformée de Colmez de $f$ est la fonction $C(f) : {\mathbb{Z}}_p \rightarrow A$ définie par la formule $$C(f)(x)={\rm Res}(f(1+T)^x)\,\,\,\, \forall x \in {\mathbb{Z}}_p.$$ Si $h\geq 0$ est un entier, notons ${{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$ le $A$-module des fonctions $A$-valuées et $h$-analytiques sur ${\mathbb{Z}}_p$, i.e. telles que pour tout $x \in {\mathbb{Z}}_p$, la fonction $f_{x,h}(t):=f(x+p^ht)$ est dans $A\langle t \rangle $. C’est un espace de Banach pour la norme $|f|_h={\rm sup}_{x \in {\mathbb{Z}}_p}|f_{x,h}|$ où $A\langle t\rangle$ est muni de la norme du sup. des coefficients. On a de plus $|af|_h\leq |a| |f|_h$ si $a \in A$ et $f \in {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$. On munit ${{\mathrm{LA}}}_h({\mathbb{Z}}_p)$ d’actions de $\psi$ et $\Gamma$ par les formules ($f \in {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$) $$\forall \gamma \in \Gamma \, \, \, \, \, \gamma(f)(x)=\gamma f(\gamma^{-1}x), \, \, \psi(f)(x)=f(px).$$ On prendra garde que l’action de $\Gamma$ définie ci-dessus n’est pas l’action naïve. De plus, ${{\mathrm{LA}}}({\mathbb{Z}}_p,A)=\cup_{h\geq 0} {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$ est muni d’une action de $\varphi$ si l’on pose $\varphi(f)(x)=0$ si $x \in {\mathbb{Z}}_p^\ast$, $\varphi(f)(x)=f(x/p)$ si $x \in p{\mathbb{Z}}_p$. PS. \[transfcolmez\] $C$ induit une suite exacte commutant aux actions de $\varphi,\psi$ et $\Gamma$ : $$0 \longrightarrow {\mathcal{R}}_A^+ \longrightarrow {\mathcal{R}}_A \overset{C}{\longrightarrow} {{\mathrm{LA}}}({\mathbb{Z}}_p,A) \longrightarrow 0.$$ Quand $A$ est un corps, c’est le théorème I.1.3 de [@colmezgros], l’argument est similaire en général. En effet, si $x \in {\mathbb{Z}}_p$ on a par définition $(1+T)^x=\sum_{n\geq 0} {{x}\choose{n}} T^n \in {\mathbb{Z}}_p[[T]]$. L’application $f \mapsto {\rm Res}(f)$ étant clairement continue sur chaque ${{\mathcal{E}}}^I_A$, on a la formule $$C(f)(x+1)=\sum_{n\geq 0} a_{-1-n} {{x}\choose{n}} \in A$$ où $f=\sum_{n \in {\mathbb{Z}}} a_n T^n$. Appliquant ceci à $x=0,1,2,\dots$, il vient que ${\rm Ker}\,\,C(f)={\mathcal{R}}_A^+$. De plus, on obtient que ${\rm Im}\,\,C(f)$ est exactement l’ensemble des fonctions continues ${\mathbb{Z}}_p \rightarrow A$ de la forme $\sum_{n\geq 0} c_n {{x}\choose{n}}$ telles qu’il existe $r > 1$ tel que $|c_n| r^n \rightarrow 0$ quand $n\rightarrow \infty$. D’autre part, un théorème d’Amice assure que ${{\mathrm{LA}}}_h({\mathbb{Z}}_p,{\mathbb{Q}}_p)$ pour $h\geq 0$ est un espace de Banach sur ${\mathbb{Q}}_p$ ayant pour base orthonormée $[\frac{n}{p^h}]!{{x}\choose{n}}$. Via l’isométrie naturelle $A\langle t \rangle = {\mathbb{Q}}_p \langle t \rangle \widehat{\otimes}_{{\mathbb{Q}_p}} A$, on dispose d’un isomorphisme naturel ${{\mathrm{LA}}}_h({\mathbb{Z}}_p,{\mathbb{Q}}_p)\widehat{\otimes}_{{\mathbb{Q}}_p} A {\overset{\sim}{\rightarrow}}{{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$, de sorte que $[\frac{n}{p^h}]!{{x}\choose{n}}$ est aussi une base orthonormée du $A$-module de Banach ${{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$. Pour conclure la surjectivité de $C$, il suffit donc de voir que pour une suite $(c_n) \in A^{\mathbb{N}}$, il y a équivalence entre satisfaire $|c_n|r^n \rightarrow 0$ pour un certain $r>1$ et satisfaire $\frac{|c_n|}{|[\frac{n}{p^h}]!|} \rightarrow 0$ pour un entier $h$ assez grand. Mais $v_p([\frac{n}{p^h}]!)=\frac{n}{p^h(p-1)}+O({\rm log}(n))$ et $v_p([\frac{n}{p^h}]!)\geq \frac{n}{p^h(p-1)}$, donc si on pose $r_h=p^{-\frac{1}{p^h(p-1)}}$, alors pour $h\geq 0$ fixé et tout $n$ assez grand on a $r_h^n \leq |[\frac{n}{p^h}]!| \leq r_{h+1}^n$. PS. Il ne reste qu’à voir que $C$ est équivariant pour $\varphi,\psi$ et $\Gamma$. Remarquons pour cela que les estimées ci-dessus montrent que pour tout entier $h\geq 0$ on a $C({\mathcal{R}}_{A,r_h}) \subset {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$ et $$C_{|{\mathcal{R}}_{A,r_h}} : {\mathcal{R}}_{A,r_h} \rightarrow {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$$ est continue. La commutation à $\varphi,\psi$ et $\Gamma$ se vérifie alors sur la partie dense $A.{\mathcal{R}}_{{\mathbb{Q}}_p,r_h}$, soit encore dans le cas $A={\mathbb{Q}}_p$, où elle est démontrée dans [@colmezgros Prop. I.2.2]. Notons $x \in {{\mathrm{LA}}}_0({\mathbb{Z}}_p,{\mathbb{Z}}_p)$ la fonction identité. Si $N\geq 0$, notons ${\rm Pol}_{\leq N}({\mathbb{Z}}_p,A) \subset {{\mathrm{LA}}}_0({\mathbb{Z}}_p,A)$ le sous-$A$-module (libre de rang $N+1$) des fonctions polynomiales de degré $\leq N$, et ${\rm Pol}({\mathbb{Z}}_p,A)=A[x]=\cup_N {\rm Pol}_{\leq N}({\mathbb{Z}}_p,A)$. On a pour tout entier $h\geq 0$ (et pour $h=\emptyset$) $${\rm Pol}_{\leq N}({\mathbb{Z}}_p,A)\oplus x^{N+1}{{\mathrm{LA}}}_h({\mathbb{Z}}_p,A) = {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A),$$ les deux sous-espaces étant stables par $\psi$ et $\Gamma$. La $A$-base des monômes $x^i$ de ${\rm Pol}_{\leq N}({\mathbb{Z}}_p,A)$ est propre pour $\psi$ et $\Gamma$ : pour $i \leq N$ on a $\psi(x^i)=p^ix^i$ et $\gamma(x^i)=\gamma^{1-i} x^i$ pour tout $\Gamma$.PS. On pose enfin $t=\log(1+T)=\sum_{k\geq 1} (-1)^{k+1} \frac{T^k}{k} \in {\mathcal{R}}^+$. On a $\varphi(t)=pt$, donc $\psi(t)=p^{-1} t$, et $\gamma(t)=\gamma t$ pour tout $\gamma \in \Gamma$. \[mainlemme\] Soient $\lambda \in A^\ast$ et $N\geq 0$ un entier.PS. - 0n a une décomposition $\varphi$-stable ${\mathcal{R}}_A^+=( \bigoplus_{0\leq i < N} A t^i ) \oplus T^N{\mathcal{R}}_A^+$. PS. - $1-\lambda\varphi$ est injectif sur ${{\mathrm{LA}}}({\mathbb{Z}}_p,A)$, ${\mathcal{R}}_A^{\lambda\varphi=1}=({\mathcal{R}}_A^{+})^{\lambda\varphi=1}$, et si $|\lambda p^N|<1$ alors $1-\lambda\varphi$ est bijectif sur $T^N{\mathcal{R}}_A^+$, d’inverse continu.PS. - Si $|p^{N+1}\lambda|<1$, alors $1-\lambda \psi$ est bijectif sur $x^{N+1}{{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$ pour tout $h\geq 0$. PS. - $\oplus_{i=0}^N A.T^{-(i+1)} \subset {\mathcal{R}}_{A}$ est un sous-$A$-module $\psi$-stable sur lequel $C$ induit un isomorphisme avec ${\rm Pol}_{\leq N}(A,{\mathbb{Z}}_p)$. PS. - $(1-\lambda \psi){\mathcal{R}}_A^+={\mathcal{R}}_A^+$ et la transformée de Colmez induit une suite exacte $$0 \longrightarrow ({\mathcal{R}}_A^+)^{\lambda \psi=1} \longrightarrow {\mathcal{R}}_A^{\lambda\psi=1} \overset{C}{\longrightarrow} {{\mathrm{LA}}}({\mathbb{Z}}_p,A)^{\lambda\psi=1} \longrightarrow 0$$ ainsi qu’un isomorphisme $$C: {\mathcal{R}}_A/(\lambda\psi-1) {\overset{\sim}{\rightarrow}}{{\mathrm{LA}}}({\mathbb{Z}}_p,A)/(\lambda\psi-1).$$ - Si $|\lambda p^N|<1$ alors $1-\lambda\varphi$ induit une bijection $\Gamma$-équivariante $$(T^N{\mathcal{R}}_A^+)^{\lambda^{-1}\psi=1} {\overset{\sim}{\rightarrow}}({\mathcal{R}}_A^+)^{\psi=0}\cap T^N{\mathcal{R}}_A^+.$$ PS. Le (i) découle de ce que $t \in T+T^2{\mathcal{R}}_A^+$ et $\varphi(T) \in T{\mathcal{R}}_A^+$. Vérifions le premier point du (ii). Tout d’abord, l’injectivité de $1-\lambda\varphi$ sur ${{\mathrm{LA}}}({\mathbb{Z}}_p,A)$ entraîne que ${\mathcal{R}}_A^{\lambda\varphi=1}=({\mathcal{R}}_A^{+})^{\lambda\varphi=1}$ via la suite exacte de la proposition \[transfcolmez\]. Soit donc $f \in {{\mathrm{LA}}}({\mathbb{Z}}_p,A)$ telle que $\lambda \varphi(f)=f$. Il vient que $\lambda^n \varphi^n(f)= f$ pour tout $n\geq 1$. En particulier, $f$ est nulle sur $p^{n-1}{\mathbb{Z}}_p^\ast$ pour tout $n\geq 1$ : $f=0$, ce que l’on voulait démontrer. Le second point du (ii) est l’argument de Colmez [@colmeztri Lemme A.1], que l’on rappelle par commodité pour le lecteur. D’une part, pour tout $0<r<1$ et tout $f \in {\mathcal{R}}_A^+$, $|\varphi(f)|_{[0,r]}\leq |f|_{[0,r]}$ (se ramener à $A={\mathbb{Q}}_p$ auquel cas cela découle de l’interpretation de $|.|_r$ comme norme sup. sur $B_{[0,r]}$). D’autre part, pour tout $0<r<1$ il existe $C_r>0$ tel que pour tout $i\geq 0$, $|\varphi^i(T^N)|_{[0,r]}\leq \frac{C_r}{p^{Ni}}$ (idem). Ainsi, si $|\lambda p^N|<1$, alors pour tout $f \in T^N{\mathcal{R}}_A^+$, on a $|\lambda^k \varphi^k(f)|_{[0,r]} \leq C_r |f|_{[0,r]}|\lambda p^N|^k$ et $$|\sum_{k\geq 0}\lambda^k \varphi^k(f)|_{[0,r]}\leq \frac{C_r |f|_{[0,r]}}{1-|\lambda p^N|}.$$ On a donc construit un inverse continu de $1-\lambda \varphi$ sur $T^N{\mathcal{R}}_A^+$. PS. Montrons le (iii). Remarquons que si $f \in {{\mathrm{LA}}}_h({\mathbb{Z}}_p,A)$ et $h\geq 1$ alors $\psi(f) \in {{\mathrm{LA}}}_{h-1}({\mathbb{Z}}_p,A)$. Ainsi, $\psi(x^i{{\mathrm{LA}}}_{h}({\mathbb{Z}}_p,A)) \subset x^i{{\mathrm{LA}}}_{h-1}({\mathbb{Z}}_p,A)$ pour tout $i\geq 0$. Enfin, si $f \in x^{N+1}{{\mathrm{LA}}}_{0}({\mathbb{Z}}_p,A)$ alors $|\psi(f)|_0\leq \frac{|f|_0}{p^{N+1}}$. Ainsi, pour tout $m\geq h$ on a $$|\psi^m(f)|_0\leq \frac{1}{p^{(N+1)(m-h)}} |\psi^h(f)|_0.$$ Sous l’hypothèse sur $N$, la série $\sum_{m\geq 0} \lambda^m\psi^m$ converge donc dans les endomorphismes de $x^{N+1}{{\mathrm{LA}}}_{h}({\mathbb{Z}}_p,A)$, vers un inverse continu de ${\rm id}-\lambda\psi$.PS. Pour le (iv), un calcul sans difficulté montre que $\psi(T^{-i-1})=Q(T)T^{-i-1}$ où $Q(T) \in {\mathbb{Q}}[T]$ est un polynôme de degré $<i+1$ (et tel que $Q(0)=p^{-i}$). On conclut car $C(T^{-i-1})(x+1)={{x}\choose{i}}$. PS. Montrons maintenant le (v). Si $N$ est suffisament grand de sorte que $|\lambda^{-1} p^N|<1$, on a vu au (ii) que $\sum_{k\geq 0} \lambda^{-k} \varphi^k$ converge normalement sur $T^N{\mathcal{R}}_A^+$ vers un inverse de $1-\lambda^{-1}\varphi$. L’opérateur $\psi$ étant continu sur ${\mathcal{R}}_A^+$, la relation formelle $(1-\lambda\psi)(\sum_{k\geq 0}\lambda^{-k}\varphi^k)=-\lambda \psi$ a donc un sens sur $T^N{\mathcal{R}}_A^+$, puis $$(1-\lambda\psi) T^N {\mathcal{R}}_A^+=\psi(T^N{\mathcal{R}}_A^+) \supset \psi(\varphi(T^N){\mathcal{R}}_A^+)=T^N\psi({\mathcal{R}}_A^+)=T^N{\mathcal{R}}_A^+.$$ Enfin on a $\psi(T)=-1$, donc si $i \geq 0$ on a l’identité $$(\lambda\psi-1)(T\varphi(T^i))=-\lambda T^i-T\varphi(T)^i.$$ Comme $T\varphi(T)^i \in p^iT^{i+1}+T^{i+2}{\mathcal{R}}_A^+$ une récurrence descendante montre que $T^i \in (\lambda\psi-1){\mathcal{R}}_A^+$ pour tout $i\leq N$. Le (v) suit en appliquant $\lambda \psi = 1$ à la suite exacte de la transformée de Colmez, et d’après le (i).PS. La dernière assertion découle de (i) et (ii), et de ce que si $(1-\lambda \varphi)x=y$ alors $\lambda(\lambda^{-1}\psi-1)x=\psi(y)$, donc $\psi(y)=0$ si et seulement si $(1-\lambda^{-1}\psi)x=0$. Une conséquence de (i), (ii), (iii) et (v) du lemme ci-dessus est la proposition suivante, qui donne une description complète de $H^i({\mathcal{R}}_A(\delta))$ pour $i=0$ et $i=2$. PS. \[corH0H2\] Soit $\delta \in {{\mathcal T}}(A)$. On a $${\mathcal{R}}_A(\delta)^{\varphi=1}={\mathcal{R}}_A^{\delta(p)\varphi=1}=A[t]^{\delta(p)\varphi=1}$$ En particulier, $H^0({\mathcal{R}}_A(\delta))=A[t]^{\delta(p)\varphi=1, \delta(\gamma)\gamma=1}$. De plus, la transformée de Colmez induit un isomorphisme $\Gamma$-équivariant $${\mathcal{R}}_A(\delta)/(\psi-1)={\mathcal{R}}_A/(\delta(p)^{-1}\psi-1) {\overset{\sim}{\rightarrow}}{\rm Pol}({\mathbb{Z}}_p,A) /(\delta(p)^{-1}\psi-1).$$ En particulier, $H^2({\mathcal{R}}_A(\delta))={\rm Pol}({\mathbb{Z}}_p,A) /(\delta(p)^{-1}\psi-1,\delta(\gamma)\gamma-1)$. Suivant Colmez, désignons par $x : {\mathbb{Q}_p}^\ast \rightarrow {\mathbb{Q}_p}^\ast$ le caractère identité et par $\chi :{\mathbb{Q}_p}^\ast \rightarrow {\mathbb{Z}}_p^\ast$ le caractère cyclotomique, c’est à dire $\chi=x|x|$. Un corollaire immédiat de la proposition ci-dessus est le suivant.PS. \[corsuiteH0H2\] Soit $\delta \in {{\mathcal T}}(A)$. - $H^0({\mathcal{R}}_A(\delta)) \neq 0$ si et seulement si il existe $i \geq 0$ et $f \neq 0 \in A$ tels que $f\cdot(\delta-x^{-i})$ est identiquement nul sur ${\mathbb{Q}}_p^\ast$.PS. - $H^2({\mathcal{R}}_A(\delta)) \neq 0$ si et seulement si il existe $i\geq 0$ tel que le fermé de ${\rm Sp}(A)$ sur lequel $\delta = \chi(x)x^i$ soit non vide.PS. Il reste à décrire $H^1({\mathcal{R}}_A(\delta))$. D’après le dévissage de la cohomologie démontré plus haut il nous faut nous intéresser au $A[\Gamma]$-module ${\mathcal{R}}_A(\delta)^{\psi=1}$. Une étape clef sera alors la structure du $A[\Gamma]$-module ${\mathcal{R}}_A^+(\delta)^{\psi=0}$. PS. Structures sur ${\mathcal{R}}_A^+(\Gamma)$ ------------------------------------------ Soit $C$ un groupe profini isomorphe à ${\mathbb{Z}}_p$ et $c$ un générateur topologique de $C$. Si $\mathcal{A}$ est complet séparé pour la topologie $p$-adique, et en particulier si c’est un modèle d’une algèbre affinoïde $A$, il est connu depuis Iwasawa que l’application $${{\mathcal A}}[[T]] \rightarrow {{\mathcal A}}[[C]]:=\projlim_n {{\mathcal A}}[C/p^nC]$$ envoyant $T$ sur $[c]-1$ est un isomorphisme. Cela permet de définir un anneau ${\mathcal{R}}_A^+(C)$ en remplaçant simplement la variable $T$ dans la définition de ${\mathcal{R}}_A^+$ par $[c]-1$. Cette définition ne dépend pas du choix de $c$ car $\Gamma$ agit par automorphismes sur ${\mathcal{R}}_A^+$. On remarque de plus que ${\mathcal{R}}_A^+(pC)=\varphi({\mathcal{R}}_A(C))$, et donc que ${\mathcal{R}}_A^+(pC) \otimes_{A[pC]}A[C]={\mathcal{R}}_A^+(C)$, car les $(1+T)^i$ pour $0\leq i \leq p-1$ forment une base ${\mathcal{R}}_A^+$ sur $\varphi({\mathcal{R}}_A)^+$. On pose enfin $${\mathcal{R}}_A^+(\Gamma):={\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p) \otimes_{A[1+p^M{\mathbb{Z}}_p]}A[{\mathbb{Z}}_p^*]$$ qui ne dépend par du choix de $M\geq 1$ par ce que l’on vient de dire. On dispose d’inclusions naturelles denses $$A[\Gamma] \subset ({{\mathcal A}}[[\Gamma]])[1/p] \rightarrow {\mathcal{R}}_A^+(\Gamma).$$ On posera aussi $A[[\Gamma]]_b=({{\mathcal A}}[[T]])[1/p] \subset {\mathcal{R}}_A^+(\Gamma)$. Il ne dépend pas du choix de ${{\mathcal A}}$. Quand $A$ est un corps le résultat suivant est essentiellement dû à Berger ([@berger1 §5]). Soit $D$ un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ ou ${\mathcal{R}}_A^+$. L’action $A$-linéaire de $\Gamma$ sur $D$ s’étend de manière unique en une structure de ${\mathcal{R}}_A^+(\Gamma)$-module sur $D$ qui est continue au sens suivant : si $(e_i)$ est une ${\mathcal{R}}_A$-base de $D$ telle que $\Gamma(D_r) \subset D_r$ où $D_r=\oplus_i {\mathcal{R}}_{A,r}e_i$, alors pour tout $f \in D_r$ l’application orbite ${\mathcal{R}}_A^+(\Gamma) \rightarrow D_r={\mathcal{R}}_{A,r}^d$, $u \mapsto u(f)$, est continue. Cette action de ${\mathcal{R}}_A^+(\Gamma)$ commute à $\varphi$ et $\psi$. PS. En effet, il suffit de voir que si $I$ est un intervalle fermé de $[0,1[$ et si $D=({{\mathcal{E}}}^I_A)^n$ est muni d’une action semi-linéaire de $\Gamma$ qui soit continue dans le sens que la matrice $M_\gamma \in M_n({{\mathcal{E}}}^I_A)$ de $\gamma \in \Gamma$ dans la base canonique $(e_i)$ dépende continûment de $\gamma$, alors l’application naturelle $$A[\Gamma] \rightarrow {\mathrm{End}}_A(D)$$ se prolonge en une application continue ${\mathcal{R}}_A^+(\Gamma) \rightarrow {\mathrm{End}}_A(D)$ (un tel prolongement étant nécessairement unique s’il existe). Fixons un tel $I$ ainsi qu’un modèle ${{\mathcal A}}\subset A$. Le sous-espace ${\mathcal}{D}=({\mathcal{O}}^I_{{\mathcal A}})^n \subset D$ est un ouvert (pour la topologie de module de Banach sur ${{\mathcal{E}}}_A^I$ de ce dernier), ainsi que $p{\mathcal}{D}$, de sorte qu’il existe un entier $M\geq 1$ tel que pour tout $\gamma \in 1+p^M{\mathbb{Z}}_p$ et tout entier $i$ on ait $(M_\gamma-1)(e_i) \in pM_n({\mathcal{O}}^I_{{\mathcal A}})$. Le lemme \[estimeegamma\] (i) permet de supposer de plus que $(\gamma-1){{\mathcal{E}}}_{{\mathcal A}}^I \subset p{{\mathcal{E}}}_{{\mathcal A}}^I$ pour tout $\gamma \in 1+p^M{\mathbb{Z}}_p$. La relation $$(\gamma-1)(\sum_i a_i e_i)=\sum_i (\gamma-1)(a_i)\gamma(e_i)+a_i(\gamma-1)(e_i)$$ assurent alors que $(\gamma-1){\mathcal}{D} \subset p{\mathcal}{D}$ pour tout $\gamma \in 1+p^M{\mathbb{Z}}_p$. On en déduit que le morphisme $A[1+p^M{\mathbb{Z}}_p] \rightarrow {\mathrm{End}}_A(D)$ s’étend continûment en un morphisme $${\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p) \rightarrow {\mathrm{End}}_A(D)$$ ainsi donc qu’à ${\mathcal{R}}_A^+(\Gamma)={\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p) \otimes_{A[1+p^M{\mathbb{Z}}_p]} A[{\mathbb{Z}}_p^\times]$. Remarquons que jusqu’ici nous n’avons pas utilise la structure de $\varphi$-module sur $D$. La commutation de ${\mathcal{R}}_A^+(\Gamma)$ à $\varphi$ et $\psi$ vient de leur commutation à $\Gamma$ et de ce que pour tout $r$ assez grand, $\varphi : D_{r^p} \rightarrow D_r$ et $\psi : D_r \rightarrow D_r$ sont continues (ceci découlant du cas $D={\mathcal{R}}_A$ considéré au § \[gencoho\]). Un point crucial est que la structure de $D^{\psi=0}$ comme ${\mathcal{R}}_A^+(\Gamma)$-module est particulièrement simple. \[propRplus\] Si $D$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}^+_A$ de rang $d$ qui est $\Gamma$-borné alors $D^{\psi=0}$ est libre de rang $d$ sur ${\mathcal{R}}_A^+(\Gamma)$. PS. Plus précisément, si $(e_i)$ est une ${\mathcal{R}}_A^+$-base de $D$ telle que $\oplus {{\mathcal A}}[[T]] e_i$ est stable par $\Gamma$ pour un certain modèle ${{\mathcal A}}\subset A$, alors $(1+T)\varphi(e_i)$ est une base de $D^{\psi=0}$ sur ${\mathcal{R}}_A^+(\Gamma)$. Avant de démontrer ce résultat nous devons dire un mot sur la topologie considérée sur $D^{\psi=0}$. Tout d’abord, le choix d’une base $(e_i)$ fournit une écriture $D=\oplus_i {\mathcal{R}}_A^+ e_i$ et donc une structure d’espace de Fréchet sur $D$. Cette structure ne dépend pas du choix de la base et l’application structurale ${\mathcal{R}}^+_A \times D \rightarrow D$ est continue. Dans la base $\varphi(e_i)$, on a $\psi(\sum_i x_i \varphi(e_i))=\sum_i \psi(x_i)e_i$ donc la continuité de $\psi$ sur ${\mathcal{R}}_A^+$ entraîne sa continuité sur $D$; celle de $\varphi$ est immédiate. En particulier, $D^{\psi=0}$ est fermé, ainsi que $(1+T)\varphi^M(D)$ pour tout $M\geq 1$, et l’application $x \mapsto (1+T)\varphi^M(x)$ est un homéomorphisme de $D$ sur $(1+T)\varphi^M(D)$ d’inverse $y \mapsto \psi^M(\frac{y}{1+T})$. Soit $(e_i)$ comme dans l’énoncé, ${\mathcal}{D}=\oplus_i {{\mathcal A}}[[T]]e_i$ et $M$ associé à ${\mathcal}{D}$ comme dans la proposition \[propcle\]. Soit $\gamma_0=1+p^M \in \Gamma$ et $x \in D$. Comme on l’a déjà vu, on a la relation $$(\gamma_0-1)((1+T)\varphi^M(x))=(1+T)\varphi^M(G_\gamma(x)).$$ La proposition \[propcle\] assure donc que l’action de $1+p^M{\mathbb{Z}}_p$ sur $(1+T)\varphi^M(D)$ s’étend en une structure de ${\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p)$-module, qui est de plus libre de base $(1+T)\varphi^M(e_i)$ d’après la proposition \[propcle\], et telle que ${\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p) \times (1+T)\varphi^M(D) \rightarrow (1+T)\varphi^M(D)$ soit continue. Cette structure est nécessairement celle de la proposition précédente par unicité de cette dernière. Les identités $$D^{\psi=0}=\bigoplus_{1\leq i \leq p^{M-1}(p-1), (p,i)=1}(1+T)^i\varphi^M(D),$$ ${\mathcal{R}}_A^+(\Gamma)={\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p)\otimes_{A[1+p^M{\mathbb{Z}}_p]}A[\Gamma]$, et pour $a \in {\mathbb{Z}}_p^\ast$, $a((1+T)\varphi^M(D))=(1+T)^a\varphi^M(D)$, concluent la démonstration. Le dernier point vient de ce que $${\mathcal{R}}_A^+(\Gamma)/(\gamma-1)={\mathcal{R}}_A^+(1+p{\mathbb{Z}}_p)/(\gamma^{p-1}-1)={\mathcal{R}}_A^+/(T)=A.$$ [Dans le même genre, on déduirait aisément de la proposition \[propcle\] que si $D$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ qui est $\Gamma$-borné alors l’action de $\Gamma$ sur $D^{\psi=0}$ s’étend en une structure de ${\mathcal{R}}_A(\Gamma)$-module (voir [@colmezgros V §3] pour la définition), qui est libre sur ${\mathcal{R}}_A(\Gamma)$ de rang le rang de $D$ sur ${\mathcal{R}}_A$.]{} Cohomologie de ${\mathcal{R}}_A(\delta)$, partie II : structure de ${\mathcal{R}}_A(\delta)^{\psi=1}$. ------------------------------------------------------------------------------------------------------ Retournons au calcul de $H^1({\mathcal{R}}_A(\delta))$. Étant donné le dévissage donné par la proposition \[devissagecoho\], il convient d’étudier tout d’abord ${\mathcal{R}}_A(\delta)^{\psi=1}$. Si $X$ est un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_A$ ou ${\mathcal{R}}_A^+$, on pose suivant Fontaine $C(X)=X^{\psi=0}\cap (1-\varphi)X=(1-\varphi)X^{\psi=1}$, de sorte que l’on dispose d’une suite ${\mathcal{R}}_A^+(\Gamma)$-équivariante tautologique $$0 \longrightarrow X^{\varphi=1} \longrightarrow X^{\psi=1} \overset{1-\varphi}{\longrightarrow} C(X) \longrightarrow 0.$$ \[devissedplus\] Soient $\delta \in {{\mathcal T}}(A)$, $D={\mathcal{R}}_A(\delta)$ et $D^+={\mathcal{R}}_A^+(\delta)$. On a des suites exactes naturelles ${\mathcal{R}}_A^+(\Gamma)$-équivariantes : - $0 \longrightarrow (D^+)^{\psi=1} \longrightarrow D^{\psi=1} \overset{C}{\longrightarrow} {\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1} \rightarrow 0$,PS. - $0 \longrightarrow C(D^+) \longrightarrow C(D) \overset{(1-\delta(p)\varphi)^{-1} C}{\longrightarrow} {\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1} \longrightarrow 0$. De plus, $(A[t])^{\delta(p)\varphi=1}=(D^+)^{\varphi=1}=D^{\varphi=1}$ et $D/(\psi-1)={\rm Pol}({\mathbb{Z}}_p,A)/(\delta(p)^{-1}\psi-1)$. Pour le (i) on applique $\delta^{-1}(p)\psi=1$ à la suite exacte définie par la transformée de Colmez et on note que $(D^+)/(\psi-1)=0$ et ${{\mathrm{LA}}}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1}={\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1}$ d’après le Lemme \[mainlemme\] (iv) et (iii). Le (ii) découle du (i) et de l’injectivité $1-\delta(p)\varphi$ sur ${{\mathrm{LA}}}({\mathbb{Z}}_p,A)$ (lemme \[mainlemme\] (ii)). En effet, cette injectivité entraîne d’une part que $C(D^+)=C(D)\cap D^+$, puis que la dernière flèche de l’énoncé est bien définie ; elle est surjective par le (i). La dernière assertion a déjà été démontrée (corollaire \[corH0H2\]). La structure de $C(D)^+$ s’avère intéressante, avant de la décrire nous avons besoin d’un lemme sur ${\mathcal{R}}_A^+(\Gamma)$. \[lemmeker\] - Si $\delta : {\mathbb{Z}}_p^\ast \rightarrow A^\ast$ est un caractère continu, il s’étend de manière unique en un morphisme de $A$-algèbres continu ${\widetilde{\delta}}: {\mathcal{R}}_A^+(\Gamma) \rightarrow A$. Tout morphisme de $A$-algèbres continu ${\mathcal{R}}_A^+(\Gamma) \rightarrow A$ est de cette forme.PS. - Soit $\gamma_0 \in \Gamma$ un générateur topologique de $\Gamma$ (resp. de $1+4{\mathbb{Z}}_2$ si $p=2$), soit $T_\delta:=[\gamma_0]-\delta(\gamma_0) \in {\mathcal{R}}_A^+(\Gamma)$. On a ${\rm Ker}({\widetilde{\delta}})=(T_\delta)$ où $(T_\delta,[-1]-\delta(-1))$ selon que $p>2$ ou non. PS. - La multiplication par $T_\delta$ est injective sur ${\mathcal{R}}_A^+(\Gamma)$ et ${\mathcal{R}}_A^+(\Gamma)=T_\delta{\mathcal{R}}_A^+(\Gamma) \oplus A$ (resp. ${\mathcal{R}}_A^+(\Gamma)=T_\delta{\mathcal{R}}_A^+(\Gamma) \oplus A[\{\pm 1\}]$ si $p=2$). Soit ${{\mathcal A}}$ un modèle de $A$ contenant $\delta(\Gamma)$. Pour $M\geq 1$ assez grand, on a $\delta(1+p^M{\mathbb{Z}}_p) \subset 1+p{{\mathcal A}}$. Rappelons que ${\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p)$ s’identifie à ${\mathcal{R}}_A^+$ si l’on envoit $[c]-1$ vers $T$, $c$ étant un générateur quelconque de $1+p^M{\mathbb{Z}}_p$. Il est alors immédiat que $\delta: A[1+p^M{\mathbb{Z}}_p] \rightarrow A$ s’étend à ${\mathcal{R}}_A^+(1+p^M{\mathbb{Z}}_p)$, ainsi donc qu’à ${\mathcal{R}}_A^+(\Gamma)$ par extension des scalaires. La réciproque découle aisément de ce que l’application naturelle ${\mathbb{Z}}_p[[\Gamma]] \rightarrow {\mathcal{R}}_A^+(\Gamma)$ est continue (lemme \[topo\] (ii)) : cela démontre le (i). Pour le (ii), il est clair que $T_\delta$ et $[-1]-\delta(-1)$ sont dans ${\rm Ker}({\widetilde{\delta}})$, et que $[\gamma_0^n]-\delta(\gamma_0)^n \in (T_\delta)$ pour tout $n \in {\mathbb{Z}}$. Soit $n \in {\mathbb{Z}}$ tel que $c=\gamma_0^n$ engendre topologiquement $1+2p{\mathbb{Z}}_p$. ${\mathcal{R}}_A^+(1+2p{\mathbb{Z}}_p)$ s’identifie à l’anneau de Robba $A$-valué positif sur la variable $U=[c]-1$, et il vient que $(T_\delta) \supset (U-(\delta(c)-1)){\mathcal{R}}_A^+(1+2p{\mathbb{Z}}_p)$. Comme $(\delta(c)-1)^m$ tend vers $0$ quand $m$ tend vers l’infini, on a ${\mathcal{R}}_A^+(1+2p{\mathbb{Z}}_p)=A \oplus ([c]-1)-(\delta(c)-1)){\mathcal{R}}_A^+(1+2p{\mathbb{Z}}_p)$. Le (ii) suit car l’application canonique $A[\Gamma_{\rm tors}] \rightarrow {\mathcal{R}}_A^+(\Gamma)/{\mathcal{R}}_A^+(1+2p{\mathbb{Z}}_p)=A[\Gamma/(1+2p{\mathbb{Z}}_p)]$ est un isomorphisme. Le premier point du (iii) suit de l’analyse ci-dessus. En effet, ${\mathcal{R}}_A^+(\Gamma)$ est libre de rang fini sur ${\mathcal{R}}_A^+(1+2p{\mathbb{Z}}_p)$ et la multiplication par $U-(\delta(c)-1) \in (T_\delta)$ est injective sur l’anneau de Robba positif en $U$. Le second point découle du (ii). Considérons pour tout $k\geq 0$ l’application $$J_k : {\mathcal{R}}_A^+ \rightarrow {\mathcal{R}}_A^+/T^k{\mathcal{R}}_A^+=A[T]/T^k.$$ Notons que $T^k{\mathcal{R}}_A^+$ est fermé dans ${\mathcal{R}}_A^+$ et qu’il est stable par $\Gamma$ (et donc ${\mathcal{R}}_A^+(\Gamma)$) et $\varphi$. Ainsi, $J_k$ est équivariante sous ${\mathcal{R}}_A^+(\Gamma)$ et $\varphi$. De plus, les images de $1,t,\dots,t^{k-1}$ forment une $A$-base de ${\mathcal{R}}_A^+/T^k{\mathcal{R}}_A^+$ propre pour $\varphi$ et $\Gamma$ de valeurs propres évidentes. \[surjjk\] $J_k$ induit une surjection $({\mathcal{R}}_A^+)^{\psi=0} \rightarrow {\mathcal{R}}_A^+/T^k{\mathcal{R}}_A^+$. En effet, l’écriture formelle “$1+T={\rm exp}(t)$” assure que $J_k(1+T)=\sum_{i=0}^{k-1} \frac{t^i}{i!}$, l’important pour ce qui suit étant que le coefficient de chaque $t^i$ est non nul. Comme pour $i=0,\dots,k-1$ les caractères $\gamma \mapsto \gamma^i$, $\Gamma \rightarrow {\mathbb{Q}}_p^\ast$, sont linéairement indépendants sur ${\mathbb{Q}}_p$ (car distincts), il vient que $J_k(A[\Gamma](1+T))={\mathcal{R}}_A^+/T^k{\mathcal{R}}_A^+$. Cela conclut car $1+T \in ({\mathcal{R}}_A^+)^{\psi=0}$. \[struccplus\] Le ${\mathcal{R}}_A^+(\Gamma)$-module $C(D^+)$ est isomorphe à l’idéal $$\cap_{i\geq 0}(1-\delta(p)p^i,{\mathrm{Ker}}(\widetilde{x^i\delta}))$$ de ${\mathcal{R}}_A^+(\Gamma)$. Plus précisément, pour tout $k$ assez grand $J_k$ induit une suite exacte ${\mathcal{R}}_A^+(\Gamma)$-équivariante $$0 \rightarrow C(D^+) \longrightarrow {\mathcal{R}}_A^+(\Gamma)\cdot (1+T) \overset{J_k}{\longrightarrow} \oplus_{i=0}^{k-1} A/(1-\delta(p)p^i) \,\,\cdot \widetilde{\delta x^i} \rightarrow 0.$$ (Comme $1-\delta(p)p^i$ est inversible dans $A$ pour $i$ assez grand, l’intersection ci-dessus est finie. De plus, le terme centrale est libre de rang $1$ sur ${\mathcal{R}}_A^+(\Gamma)$.) En effet, soit $k$ suffisament grand de sorte que $|\delta(p)p^k|<1$. Le lemme \[mainlemme\] (ii) assure que $$(1-\delta(p)\varphi){\mathcal{R}}_A^+=J_k^{-1}((1-\delta(p)\varphi){\mathcal{R}}_A^+/T^k{\mathcal{R}}_A^+).$$ Comme ${\mathcal{R}}_A^+(\Gamma)$-module on a $$(1-\varphi){\mathcal{R}}_A^+(\delta)/T^k{\mathcal{R}}_A^+(\delta)=\oplus_{i=0}^{k-1} (1-\delta(p)p^i)A( \widetilde{\delta x^i}).$$ Mais ${\mathcal{R}}_A^+(\delta)^{\psi=0}$ est libre de rang $1$ sur ${\mathcal{R}}_A^+(\Gamma)$ engendré par $1+T$ d’après la proposition \[propRplus\]. La proposition suit alors du lemme \[surjjk\]. Fixons un générateur topologique $\gamma_0$ de $\Gamma$ (resp. de $1+4{\mathbb{Z}}_2$ si $p=2$). On rappelle l’élément $T_\delta \in A[\Gamma]$ défini dans le lemme \[lemmeker\] (ii). Pour $i\in {\mathbb{Z}}$ on note $T_i \in A[\Gamma]$ l’élément $T_\delta$ où $\delta(\gamma)=\gamma^i$ pour tout $\gamma \in \Gamma$. \[defpresque\] On dira qu’un ${\mathcal{R}}_A^+(\Gamma)$-module $D$ est presque libre de rang $d$ si il existe une suite exacte de ${\mathcal{R}}_A^+(\Gamma)$-modules de la forme $$0 \rightarrow {\mathcal{R}}_A^+(\Gamma)^d \rightarrow D \rightarrow Q \rightarrow 0$$ telle que $Q$ est de type fini sur $A$ et annulé par un monôme en des $T_i$ pour $i \in {\mathbb{Z}}$. On dira que $D$ est presque nul si il est presque libre de rang $0$, c’est à dire de type fini sur $A$ et annulé par un mônome en des $T_i$ pour $i \in {\mathbb{Z}}$. \[lemmepl\] - Les ${\mathcal{R}}_A^+(\Gamma)$-modules presque libres sont de type fini. PS. - Soit $0 \rightarrow D' \rightarrow D \rightarrow D'' \rightarrow 0$ une suite exacte de ${\mathcal{R}}_A^+(\Gamma)$-modules. Si $D''$ (resp. $D'$) est presque nul et $D$ est presque libre de rang $d$, alors $D'$ (resp. $D''$) est presque libre de rang $d$. PS. - Enfin, si on a une suite exacte longue de ${\mathcal{R}}_A^+(\Gamma)$-modules $$D_1 \longrightarrow D_2 \longrightarrow D_3 \longrightarrow D_4 \longrightarrow D_5$$ avec $D_1$ et $D_5$ presque nuls, $D_2$ et $D_4$ presque libres de rang respecifs $d_2$ et $d_4$, alors $D_3$ est presque libre de rang $d_2+d_4$. Le premier point est évident. Pour le second, soient $L \subset D$ libre de rang $d$ et $M$ un monôme en les $T_i$ tel que $MD \subset L$. Si $M'$ est un autre tel monôme tel que $M'D \subset D'$ alors le lemme \[lemmeker\] (ii) assure que $M'L$ est libre de rang $d$ et que $L/M'L$ est de type fini sur $A$. Il vient que $D/M'L$ est de type fini sur $A$ annulé par $MM'$, ainsi donc que son sous-module $D'/M'L$ par noethérianité de $A$, et donc $D'$ est presque libre de rang $d$ ce qui prouve le (ii) dans le premier cas. Dans le second cas $D'$ est presque nul, donc $D' \cap L =0$ par le lemme \[lemmeker\] (ii). Ainsi, la projection $\pi : D \rightarrow D''$ est injective sur $L$ et $D''/\pi(L)$ est un quotient du $A$-module presque nul $D/L$, ce qui conclut le (ii). Pour le (iii), on peut supposer $D_1=D_5=0$ par le (ii). Si $D_3$ est libre sur ${\mathcal{R}}_A^+(\Gamma)$ alors la suite est scindée et le résultat suit. En général, quitte à remplacer $D_2$ par l’image inverse dans $D_2$ d’un sous- ${\mathcal{R}}_A^+(\Gamma)$-module libre de $D_3$ on peut donc supposer que $D_3$ est presque nul, auquel cas l’affirmation est évidente. \[mainiwa\] Soit $D$ un ${(\varphi,\Gamma)}$-module triangulin de rang $d$ sur ${\mathcal{R}}_A$. Alors les ${\mathcal{R}}_A^+(\Gamma)$-modules $D^{\varphi=1}$ et $D/(\psi-1)$ sont presque nuls, et $D^{\psi=1}$ et $C(D)$ sont presque libres de rang $d$. PS. Soit $(\delta_i) \in {{\mathcal T}}^d(A)$ le paramètre de $D$ et supposons de plus que pour tout $i=1,\dots,d$ et pour tout $j\in {\mathbb{N}}$ alors $1-\delta_i(p)p^j$ est non diviseur de zéro dans $A$ et $1-\delta_i(p) p^{-j} \in A^\times$. Alors $D^{\varphi=1}=D/(\psi-1)=0$ et $D^{\psi=1} {\overset{\sim}{\rightarrow}}C(D)$ est libre de rang $d$ sur ${\mathcal{R}}_A^+(\Gamma)$. On procède par récurrence sur $d\geq 1$. On a une suite exacte dans ${(\varphi,\Gamma)}/A$ $$0 \longrightarrow D' \rightarrow D \rightarrow {\mathcal{R}}_A(\delta_d) \rightarrow 0$$ avec $D'$ triangulin de rang $d-1$. Par la suite exacte longue de cohomologie associée et par le lemme \[lemmepl\] (iii), on peut supposer $d=1$, i.e. $D={\mathcal{R}}_A(\delta)$. Dans ce cas, il découle de la proposition \[corH0H2\] que $D^{\varphi=1}$ et $D/(\psi-1)$ sont presque nuls car $1-\delta(p)^{\pm 1}p^i$ est inversible dans $A$ pour tout entier $i$ assez grand, c’est aussi évidemment le cas de ${\rm Pol}(A,{\mathbb{Z}}_p)^{\delta(p)^{-1}\psi=1}$. De plus, le premier de ces trois modules est nul si et seulement si $1-\delta(p)p^i$ est non diviseur de $0$ dans $A$ pour tout $i\geq 0$, et les deux autres le sont si et seulement si $1-\delta(p)p^{-i} \in A^\times$ pour tout $i\geq 0$. Sous ces hypothèses, le lemme \[devissedplus\] assure aussi que $D^{\psi=1} {\overset{\sim}{\rightarrow}}C(D)=C(D^+)$ et la proposition \[struccplus\] montre que $C(D^+)=(D^+)^{\psi=0}={\mathcal{R}}_A^+(\Gamma)\cdot (1+T)$ est libre de rang $1$. Mentionnons qu’en général, $C(D)$ n’est pas libre sur ${\mathcal{R}}_A^+(\Gamma)$, et ce même si $D={\mathcal{R}}_A(\delta)$. En effet, considérons le cas particulier où $\delta \in {{\mathcal T}}(A)$ est tel que $1-\delta(p)p^{-i}$ est non diviseur de $0$ dans $A$ pour tout $i\geq 0$, auquel cas $C(D)=C(D^+)$. On conclut par le résultat général suivant. \[critlibre\] Soit $\delta \in {{\mathcal T}}(A)$ tel que $1-\delta(p)p^i$ est non diviseur de $0$ dans $A$ pour tout $i\geq 0$. Alors $C({\mathcal{R}}_A^+(\Gamma))$ est projectif comme ${\mathcal{R}}_A^+(\Gamma)$-module si et seulement si $1-\delta(p)p^i \in A^\times$ pour tout $i\geq 0$, auquel cas il est en fait libre de rang $1$. Remarquons que si deux idéaux de type fini $I$ et $J$ d’un anneau commutatif $B$ sont tels que l’idéal $IJ$ est projectif comme $B$-module, et si de plus $I+J=B$, alors $I$ et $J$ sont des $B$-modules projectifs. En effet, être projectif de type fini est une propriété locale sur ${\rm Spec}(B)$, mais si $x \in {\rm Spec}(B)\backslash V(I)$ alors $I_x=B_x$, et si $x \in V(I)$ alors $x \notin V(J)$ et donc $J_x=B_x$ puis $(IJ)_x=I_xJ_x=I_x$. D’après le lemme \[comax\] ci-dessous et la proposition \[struccplus\], il s’agit de voir que si $I_\delta$ est projectif de rang fini, avec $1-\delta(p)$ non diviseur de $0$ dans $A$, alors $1-\delta(p) \in A^\times$ (et donc $I_\delta={\mathcal{R}}_A^+(\Gamma)$). Fixons donc $\delta \in {{\mathcal T}}(A)$ avec $1-\delta(p)$ non diviseur de $0$ et supposons $p>2$ pour simplifier, de sorte que ${\rm Ker}({\widetilde{\delta}})=(T_\delta)$. La suite de ${\mathcal{R}}_A^+(\Gamma)$-modules $$0 \longrightarrow {\mathcal{R}}_A^+(\Gamma) \overset{u \mapsto (T_\delta u,(1-\delta(p))u)}{\longrightarrow} {\mathcal{R}}_A^+(\Gamma)^2 \overset{(x,y) \mapsto (1-\delta(p))x-T_\delta y}{\longrightarrow} I_{\delta} \longrightarrow 0$$ est exacte. En effet, si $x,y \in {\mathcal{R}}_A^+(\Gamma)$ satisfont $x T_\delta = y (1-\delta(p))$ alors en appliquant ${\widetilde{\delta}}$ il vient que $0={\widetilde{\delta}}(y)(1-\delta(p)) \in A$ donc $y=T_\delta u$ avec $u \in {\mathcal{R}}_A^+(\Gamma)$ unique par le lemme \[lemmeker\], puis $x=u (1-\delta(p))$. Comme la multiplication par chaque $T_{\delta'}$ est injective sur $I_{\delta} \subset {\mathcal{R}}_A^+(\Gamma)$ on en déduit que la suite ci-dessus est reste exacte modulo $T_{\delta'}$ pour tout $\delta'$. L’isomorphisme naturel ${\widetilde{\delta}}' : {\mathcal{R}}_A^+(\Gamma)/(T_{\delta'}) {\overset{\sim}{\rightarrow}}A$ envoie tout $\gamma \in \Gamma$ sur $\delta'(\gamma)$. Si $\gamma$ engendre topologiquement $\Gamma$ on obtient donc une suite exacte $$0 \longrightarrow A \overset{u \mapsto ((\delta'(\gamma)-\delta(\gamma))u, (1-\delta(p))u)}{\longrightarrow} A \overset{(x,y) \mapsto (1-\delta(p))x-(\delta'(\gamma)-\delta(\gamma))y}{\longrightarrow} I_\delta/T_{\delta'}I_\delta \longrightarrow 0,$$ et donc $$I_\delta/T_{\delta'}I_\delta \simeq A^2/A(1-\delta(p),\delta'(\gamma)-\delta(\gamma)).$$ On conclut en appliquant ceci à $\delta'=\delta$ : si le ${\mathcal{R}}_A^+(\Gamma)$-module $I_\delta$ est projectif il en va de même du $A$-module $A/(1-\delta(p)) \times A$. Comme $1-\delta(p)$ est non diviseur de $0$ dans $A$, cela entraîne qu’il est inversible, ce qui conclut. L’argument est similaire pour $p=2$. \[comax\] Pour $\delta \in {{\mathcal T}}(A)$, notons $I_\delta$ l’idéal $(1-\delta(p),{\rm Ker}({\widetilde{\delta}}))$ de ${\mathcal{R}}_A^+(\Gamma)$. Alors pour tout $i \in {\mathbb{Z}}\backslash \{0\}$ on a $I_\delta+I_{\delta x^i}={\mathcal{R}}_A^+(\Gamma)$. En particulier, $C({\mathcal{R}}_A^+(\delta))=\prod_{i\geq 0} I_{\delta x^i}$. En effet, $(1-\delta(p)p^i)-(1-\delta(p))=\delta(p)(1-p^i) \in A^\times$ si $i\neq 0$. De la discussion précédent la proposition \[critlibre\] on déduit le : Soit $\delta \in {{\mathcal T}}(A)$ tel que $1-\delta(p)p^i$ est non diviseur de $0$ dans $A$ pour tout $i \in {\mathbb{Z}}$. Alors le ${\mathcal{R}}_A^+(\Gamma)$-module $C({\mathcal{R}}_A(\delta))$ est projectif si et seulement si $1-\delta(p)p^i \in A^\times$ pour tout $i\geq 0$, auquel cas il est libre de rang $1$. Cohomologie de ${\mathcal{R}}_A(\delta)$ partie III : calcul du $H^1$ --------------------------------------------------------------------- Le calcul de $H^1({\mathcal{R}}_A(\delta))$ est maintenant une formalité. On rappelle que $x \in {{\mathcal T}}({\mathbb{Q}}_p)$ désigne le caractère tautologique identité et que $\chi \in {{\mathcal T}}({\mathbb{Q}}_p)$ est le caractère tel que $\chi(p)=1$ et $\chi_{|{\mathbb{Z}}_p^\times}=x_{|{\mathbb{Z}}_p^\times}$ (“caractère cyclotomique”). On désigne par ${{\mathcal T}}^{\rm reg} \subset {{\mathcal T}}$ l’ouvert complémentaire de l’ensemble discret[^15] des points ${\mathbb{Q}}_p$-rationnels de la forme $x^{-i}$ ou $\chi x^i$ pour $i\geq 0$. Un caractère $\delta \in {{\mathcal T}}(A)$ est dit régulier si il est dans ${{\mathcal T}}^{\rm reg}(A)$, ce qui revient à dire que pour tout $z \in {\rm Sp}(A)$ le caractère $\delta_z : {\mathbb{Q}_p}^\times \rightarrow k(z)^\times$ obtenu par évaluation en $z$ n’est pas de la forme $x^i$ ou $\chi x^{-i}$ pour $i\geq 0$ entier. On note $K(A)$ le groupe de Grothendieck des $A$-modules de type fini, et si $M$ est un tel $A$-module on note $[M]$ sa classe dans $K(A)$. On rappelle qu’une algèbre affinoïde est noethérienne. \[thmH1\] Soit $D$ un ${(\varphi,\Gamma)}$-module triangulin de rang $d$ sur ${\mathcal{R}}_A$. Alors $H^i(D)$ est de type fini sur $A$ pour tout entier $i$ et on a la relation dans $K(A)$ $$[H^0(D)]-[H^1(D)]+[H^2(D)]=-[A^d].$$PS. Si de plus le paramètre $(\delta_i)$ de $D$ est dans ${{\mathcal T}}^{\rm reg}(A)^d$, alors $H^0(D)=H^2(D)=0$ et $H^1(D)$ est libre de rang $d$ sur $A$. Enfin, si $D={\mathcal{R}}_A(\delta)$ avec $\delta \in {{\mathcal T}}^{\rm reg}(A)$, les quatre morphismes naturels $$(D^+)^{\psi=0}/(\gamma-1) \leftarrow C(D^+)/(\gamma-1) \rightarrow C(D)/(\gamma-1) \leftarrow D^{\psi=1}/(\gamma-1) \rightarrow H^1(D)$$ sont des isomorphismes, un générateur de $(D^+)^{\psi=0}/(\gamma-1)$ étant donné par la classe de $1+T$. Par récurrence sur $d$ et en utilisant la suite longue de cohomologie, on peut supposer que $d=1$, i.e. $D={\mathcal{R}}_A(\delta)$. On a déjà vu que $D^{\varphi=1}$ et $D/(\psi-1)$ sont de type fini sur $A$, et donc à plus forte raison que $H^0(D)$ et $H^2(D)$ le sont aussi. Pour vérifier le premier point du théorème, et compte tenu du dévissage donné par le théorème \[devissagecoho\], il suffit donc de démontrer que $C(D)/(\gamma-1)$ est de type fini sur $A$ et de classe $[A^d]$ dans $K(A)$. (On rappelle que si $X$ est un $A$-module de type fini et $u \in {\mathrm{End}}_A(X)$ alors la multiplication par $u$ sur $X$ implique l’identité $[X^{u=0}]=[X/u(X)]$ dans $K(A)$.) Les deux suites exactes données par le lemme \[devissedplus\] (ii) et la proposition \[struccplus\] induisent des suites exactes $$\label{sexacte1} \begin{split} 0 \longrightarrow {\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1,\gamma=1} \longrightarrow C(D^+)/(\gamma-1) \longrightarrow \\ C(D)/(\gamma-1) \overset{(1-\varphi)^{-1} C}{\longrightarrow} {\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1}/(\gamma-1) \longrightarrow 0. \end{split}$$ $$\label{sexacte2}\begin{split} 0 \longrightarrow (\oplus_{i=0}^{k-1} A/(1-\delta(p)p^i) \,\,\cdot \widetilde{\delta x^i})^{\gamma=1} \longrightarrow C(D)^+/(\gamma-1) \longrightarrow \\ {\mathcal{R}}_A^+(\Gamma)\cdot (1+T)/(\gamma-1) \overset{J_k}{\longrightarrow} (\oplus_{i=0}^{k-1} A/(1-\delta(p)p^i) \,\,\cdot \widetilde{\delta x^i})/(\gamma-1) \longrightarrow 0 \end{split}$$ car $\gamma-1$ est injectif sur $C(D)$ et $({\mathcal{R}}_A^+)^{\psi=0}$ (théorème \[invgamma\]). On conclut car ${\mathcal{R}}_A^+(\Gamma)/([\gamma]-1)=A$ (lemme \[lemmeker\] pour $\delta=1$). Supposons maintenant $\delta$ régulier. Dans ce cas, les noyaux et conoyaux de $\gamma-1$ sur $D^{\varphi=1}$, $D/(\psi-1)$, ${\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1}$ et $\oplus_{i\geq 0} A/(1-\delta(p)p^i) \widetilde{\delta x^i}$ sont tous nuls par hypothèse. En effet, si $a, b \in A$ sont tels que $(a,b)=A$, alors la multiplication par $a$ est bijective sur $A/bA$ et $A[b]$ (la $b$-torsion dans $A$): si $au+bv=1$ la multiplication par $u$ en est un inverse. Cela montre que $H^0(D)=H^2(D)=0$ puis que les quatre flèches de l’énoncé sont des isomorphismes. En particulier, $H^1(D)$ est isomorphe à ${\mathcal{R}}_A^+(\Gamma)/(\gamma-1)=A$. \[changementdebase\] Soient $A \rightarrow B$ un morphisme de ${\mathbb{Q}}_p$-algèbres affinoïdes et $D$ un ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_A$. Supposons que $A \rightarrow B$ est plat ou que le paramètre de $D$ est régulier. Alors pour tout entier $i$ l’application naturelle $$H^i(D) \otimes_A B \longrightarrow H^i(D \widehat{\otimes}_A B)$$ est un isomorphisme. L’application de l’énoncé est celle déduite du $A$-morphisme de complexes $C_{\varphi,\gamma}(D)^\bullet \rightarrow C_{\varphi,\gamma}(D \widehat{\otimes}_A B)^\bullet$. En particulier, si on a une suite exacte de ${(\varphi,\Gamma)}$-modules sur ${\mathcal{R}}_A$ disons $0 \rightarrow D_1 \rightarrow D_2 \rightarrow D_3 \rightarrow 0$, et donc une suite exacte $0 \rightarrow D_1 \widehat{\otimes}_A B \rightarrow D_2 \widehat{\otimes}_A B \rightarrow D_3 \widehat{\otimes}_A B \rightarrow 0$, on dispose d’une morphisme $A$-linéaire naturel entre les suites exactes longues de cohomologie. Il vient que par dévissage et par le lemme des $5$, on peut supposer $D={\mathcal{R}}_A(\delta)$ est de rang $1$. Soit $\delta \in {{\mathcal T}}(A)$, on note $\delta_B$ l’image de $\delta$ dans ${{\mathcal T}}(B)$. Remarquons que si $F(\delta)$ désigne le $A[\varphi,\Gamma]$-module $A[t](\delta)$ et ${\rm Pol}({\mathbb{Z}}_p,A)(\delta)$, alors l’application naturelle $F(\delta) \otimes_A B \rightarrow F(\delta_B)$ est un isomorphisme. En particulier, pour $\ast \in \{\varphi,\psi\}$ et si $C_{\ast,\gamma}(F(\delta))$ désigne le complexe à trois termes évident, dont on désignera par $H^i_\ast(F(\delta))$ le $A$-module de cohomologie, et si $A \rightarrow B$ est plat, alors $$H^i_\ast(F(\delta)) \otimes_A B {\overset{\sim}{\rightarrow}}H^i_\ast(F(\delta)\otimes_A B) {\overset{\sim}{\rightarrow}}H^i_\ast(F(\delta_B)).$$ Mais on a défini (prop. \[corH0H2\]) des isomorphismes naturels $H^0_\varphi(F(\delta)) {\overset{\sim}{\rightarrow}}H^0({\mathcal{R}}_A(\delta))$ et $H^2({\mathcal{R}}_A(\delta)) {\overset{\sim}{\rightarrow}}H^2_\psi(F(\delta))$ ($F$ valant respectivement $A[t]$ dans le premier cas et ${\rm Pol}$ dans le second), le théorème en découle pour $i=0,2$. Supposons toujours $D={\mathcal{R}}_A(\delta)$. Par un argument similaire à celui ci-dessus utilisant les suites exactes  et , et le fait que la formation des modules $(D^+)^{\psi=0}$, $D^{\psi=1}$, $C(D)$ et $C(D^+)$ est fonctorielle en $A$, ainsi donc que leurs quotients par $\gamma-1$, le théorème suit dans le cas $i=1$ si l’on montre que pour tout morphisme $A \rightarrow B$ l’application naturelle $$f : ({\mathcal{R}}_A^+(\delta))^{\psi=0}/(\gamma-1) \otimes_A B \rightarrow ({\mathcal{R}}_B^+(\delta_B))^{\psi=0}/(\gamma-1)$$ est un isomorphisme. Mais on a déjà vu que ${\mathcal{R}}_A^+(\delta)^{\psi=0}$ est libre sur ${\mathcal{R}}_A^+(\Gamma)$ engendré par $1+T$, de sorte que son quotient par $\gamma-1$ est libre sur $A$ engendré par la classe $\overline{1+T}$ de $1+T$. Mais par construction $f$ provient par quotient du morphisme évident ${\mathcal{R}}_A^+(\delta)^{\psi=0} \rightarrow {\mathcal{R}}_B^+(\delta_B)^{\psi=0}$ qui envoie $1+T$ sur $1+T$. Remarquons que jusqu’ici nous n’avons pas donné d’énoncé exact sur la structure de $H^1({\mathcal{R}}_A(\delta))$ quand $\delta$ n’est pas régulier. Si l’on concatène les suites exactes données par la proposition \[devissagecoho\] et les formules  et  nous en obtenons un dévissage explicite, bien que peu ragoûtant en général. Ce dévissage se simplifie dans le cas utile suivant. On dit que $\delta \in {{\mathcal T}}(A)$ est bien plaçé si : - pour tout $i \in {\mathbb{Z}}$, alors $1-\delta(p)p^i$ est non diviseur de $0$ dans $A$,PS. - pour tout $i \geq 0$, l’image de $1-\delta(\gamma)\gamma^{1-i}$ dans[^16] $A/(1-\delta(p)p^{-i})$ est non diviseur de $0$. \[h1bienplace\] Si $\delta \in {{\mathcal T}}(A)$ est bien plaçé et $D={\mathcal{R}}_A(\delta)$, alors $H^0(D)=0$ et les morphismes naturels $$C(D^+)/(\gamma-1) \rightarrow C(D)/(\gamma-1) \leftarrow D^{\psi=1}/(\gamma-1) \rightarrow H^1(D)$$ sont des isomorphismes. On a de plus une suite exacte naturelle de $A$-modules $$0 \longrightarrow \prod_{i\geq 0} (A/(1-\delta(p)p^i))[1-\delta(\gamma)\gamma^i] \longrightarrow H^1(D) \longrightarrow \bigcap_{i\geq 0} (1-\delta(p)p^i,1-\delta(\gamma)\gamma^i) \longrightarrow 0.$$ L’intersection dans le terme de droite de cette dernière suite est sous-entendue à l’intérieur de $A$ (c’est donc un idéal de $A$). En effet, l’annulation de $D^{\varphi=1}$, $D/(\psi-1))^{\gamma=1}$ et ${\rm Pol}({\mathbb{Z}}_p,A)^{\delta(p)^{-1}\psi=1}$ équivaut au caractère bien placé de $\delta$, et on conclut la première assertion par la proposition \[devissagecoho\] et la suite exacte . D’après la proposition \[struccplus\], on a une suite exacte de ${\mathcal{R}}_A^+(\Gamma)$-modules $$0 \longrightarrow C(D^+) \longrightarrow {\mathcal{R}}_A^+(\Gamma) \longrightarrow \prod_{i\geq 0} A/(1-\delta(p)p^i)(\delta) \longrightarrow 0$$ dont la dernière assertion se déduit en appliquant $\gamma=1$, en utilisant que $(\gamma-1)$ est injectif sur ${\mathcal{R}}_A^+(\Gamma)$ (lemme \[lemmeker\] pour $\delta=1$). (Quand $p=2$ l’argument ci-dessus et l’énoncé ne sont évidemment pas tout à fait corrects) Terminons par un cas particulier important concernant le caractère universel. Si $U \subset {{\mathcal T}}$ est un ouvert affinoïde, on désigne par $\delta_U \in {{\mathcal T}}(U)$ le caractère tautologique. \[h1universel\] Pour tout ouvert affinoïde $U \subset {{\mathcal T}}$, $H^0({\mathcal{R}}_U(\delta_U))=0$ et le ${\mathcal{O}}(U)$-module $H^1({\mathcal{R}}_U(\delta_U))$ s’identifie naturellement à l’idéal de ${\mathcal{O}}(U)$ constitué des fonctions qui s’annulent en tous les points de $U$ paramétrant les caractères de la forme $x^{-i}$ pour $i\geq 0$. De plus, pour tout $z \in U$, de caractère associé $\delta_z$, l’application naturelle $$H^1({\mathcal{R}}_U(\delta_U))\otimes_{{\mathcal{O}}(U)}k(z) \longrightarrow H^1({\mathcal{R}}_{k(z)}(\delta_z))$$ est un isomorphisme, à moins que $\delta_z$ ne soit de la forme $\chi x^i$ avec $i\geq 0$, auquel cas cette application est nulle. Plus précisément, supposons que le seul point non régulier de $U$, disons $u \in U$, paramètre un caractère de la forme $\chi x^i$ avec $i\geq 0$ et soit $m=m_u \subset {\mathcal{O}}(U)$ l’idéal maximal des fonctions s’annulant en ce point. On a $k(u)={\mathbb{Q}}_p$ et on considère l’espace tangent $T_u={\mathrm{Hom}}_{{\mathbb{Q}_p}}(m/m^2,{\mathbb{Q}}_p)$. Alors : - Le morphisme naturel $H^1(m {\mathcal{R}}_U(\delta)) \rightarrow H^1({\mathcal{R}}_U(\delta_U))$ est un isomorphisme entre ${\mathcal{O}}(U)$-module libres de rang $1$, - Le ${\mathcal{O}}(U)$-morphisme canonique $m \rightarrow m/m^2$ induit une injection $$H^1(m {\mathcal{R}}_U(\delta_U)) \otimes_{{\mathcal{O}}(U)} k(u) \longrightarrow {\mathrm{Hom}}_{{\mathbb{Q}}_p}(T_u,H^1({\mathcal{R}}(\chi x^i)))$$ dont l’image est une droite constituée d’isomorphismes. En particulier, cette droite induit une isomorphisme canonique $ \mathbb{P}(T_u) {\overset{\sim}{\rightarrow}}\mathbb{P}(H^1({\mathcal{R}}(\chi x^i))$ entre espaces projectifs sur ${\mathbb{Q}}_p$ de dimension $1$. On notera le rôle non symétrique joué ici par les points singuliers de la forme $x^{-i}$ et ceux de la forme $\chi x^i$. Nous préciserons un peu plus loin ce théorème en introduisant l’éclaté de $U$ aux points non réguliers, ce qui nous permettra notamment de comprendre complètement la structure analytique de l’espace des ${(\varphi,\Gamma)}$-modules triangulins de rang $2$, y compris au voisinage des points de paramètre singulier : c’est exactement la structure suggérée par Colmez dans sa définition de l’espace des triangulines en rang $2$. Le caractère $\delta_U$ est évidemment bien placé. De même, pour tout $i \geq 0$ l’élément $1-\delta(\gamma)\gamma^i$ n’est pas diviseur de zéro sur l’anneau (localement intègre) ${\mathcal{O}}(U)/(1-\delta(p)p^i)$. La première assertion découle donc de la proposition \[h1bienplace\]. Pour vérifier la seconde assertion, il résulte de la commutativité de $H^1$ au changement de base plat (ici une immersion ouverte) que l’on peut supposer que $U$ ne contient qu’un seul point singulier, disons $z$. Si $m \subset {\mathcal{O}}(U)$ désigne l’idéal maximal des fonctions qui s’annulent en $z$, alors il est classique que si $a=\delta_U(p)-\delta_z(p)$ et $b=\delta_U(\gamma)-\delta_z(\gamma)$ alors $m=(a,b)$ et on a des suites exactes $$0 \longrightarrow {\mathcal{O}}(U) \overset{f\mapsto (af,bf)}{\longrightarrow} {\mathcal{O}}(U)^2 \overset{(f,g) \mapsto af-bg}{\longrightarrow} m \longrightarrow 0,$$ et $$0 \longrightarrow m \longrightarrow {\mathcal{O}}(U) \longrightarrow k(z) \longrightarrow 0.$$ Comme ${\mathcal{R}}_U$ est plat sur ${\mathcal{O}}(U)$ (lemme \[topo\] (vi)), ces suites restent exactes après $- \otimes_{{\mathcal{O}}(U)} D$, $D={\mathcal{R}}_U(\delta_U)$, de sorte qu’en prenant la suite longue de cohomologie on obtienne des suites exactes de ${\mathcal{O}}(U)$-modules $$0 \longrightarrow H^0(mD) \longrightarrow H^1(D) \overset{(a,b)}{\longrightarrow} H^1(D)^2 \longrightarrow H^1(mD) \longrightarrow H^2(D) \overset{(a,b)}{\longrightarrow} H^2(D)^2 \longrightarrow H^2(mD) \longrightarrow 0,$$ $$0 \longrightarrow H^0(D_z) \longrightarrow H^1(mD) \longrightarrow H^1(D) \longrightarrow H^1(D_z) \longrightarrow H^2(mD) \longrightarrow H^2(D) \longrightarrow H^2(D_z) \longrightarrow 0.$$ On a utilisé que $D \otimes_{{\mathcal{O}}(U)} m = mD$ (platitude de ${\mathcal{R}}_U$ sur ${\mathcal{O}}(U)$). Si $\delta_z=x^{-i}$, alors la proposition \[corH0H2\] assure que $H^2(D)=H^2(D_z)=0$. On déduit des suites ci-dessus qu’alors $H^2(mD)=0$ puis que $H^1(D) \rightarrow H^1(D_z)$ est surjectif. Comme $H^1(D) \simeq m$ et que $m/m^2$ et $H^1(D_z)$ sont de dimension $2$ sur $k(z)$, l’application $H^1(D) \otimes_{{\mathcal{O}}(U)}k(z) \rightarrow H^1(D_z)$ est un isomorphisme. Si $\delta_z=\chi x^i$, la proposition \[corH0H2\] assure que $H^2(D) \simeq k(z)$ est tué par $m$, de sorte que la flèche $H^2(D) \rightarrow H^2(D)^2$ ci-dessus est nulle. Ainsi, $H^2(D)^2 {\overset{\sim}{\rightarrow}}H^2(mD)$ et ce dernier est isomorphe à $k(z)^2$. Comme d’autre part $H^2(D) \rightarrow H^2(D_z)$ est un isomorphisme (car surjectif), on en déduit que $H^1(D_z) \longrightarrow H^2(mD)$ est surjective : c’est donc un isomorphisme pour des raisons de dimension. Ainsi, $H^1(D) \rightarrow H^1(D_z)$ est nul. Comme $H^0(D_z)=0$ on en déduit enfin $H^1(mD)=H^1(D)$ : il ne reste qu’à prouver le (ii) du théorème. Comme $m/m^2$ est annulé par $m$, il vient que $D \otimes_{{\mathcal{O}}(U)}(m/m^2)= D_z \otimes_{{\mathbb{Q}}_p} (m/m^2)$ (avec action de $\varphi$ et $\Gamma$). On en déduit pour tout $i$ un isomorphisme canonique de ${\mathcal{O}}(U)$-modules $$H^i(D \otimes_{{\mathcal{O}}(U)}(m/m^2)) {\overset{\sim}{\rightarrow}}H^i(D_z) \otimes_{{\mathbb{Q}}_p} (m/m^2) {\overset{\sim}{\rightarrow}}{\mathrm{Hom}}_{{\mathbb{Q}}_p}(T_z,H^i(D_z)).$$ Remarquons que $T_z$ et $H^1(D_z)$ sont des ${\mathbb{Q}}_p$-espace vectoriel de dimension $2$. Considérons le morphisme $$\mu : H^1(mD) \rightarrow H^1(D\otimes_{{\mathcal{O}}(U)}m/m^2) = {\mathrm{Hom}}_{{\mathbb{Q}}_p}(T_z,H^1(D_z))$$ déduit du morphisme naturel $mD \rightarrow mD/m^2D=(m/m^2)\otimes_{{\mathcal{O}}(U)}D$ (on rappelle que ${\mathcal{R}}_U$ est plat sur ${\mathcal{O}}(U)$). Comme $H^1(mD)=H^1(D)$ est libre de rang $1$, son image dans ${\mathrm{Hom}}_{{\mathbb{Q}}_p}(T_z,H^1(D_z))$ est soit nulle soit une ${\mathbb{Q}}_p$-droite. Si $L \neq 0 \subset T_z$, $\mu(L)$ est par définition l’élément de $H^1(D_z)$ obtenu comme composé de $\mu$ et de l’applicatio naturelle $(L \otimes 1) : (m/m^2) \otimes_{{\mathbb{Q}}_p} D_z \rightarrow D_z$, ou ce qui revient au même comme image de l’application ${\mathcal{O}}(U)$-linéaire naturelle $mD \rightarrow D_z$ après passage au $H^1$. Soit $m^2 \subset J \subset m$ le noyau de $L$. La suite exacte de ${\mathcal{O}}(U)$-modules $0 \rightarrow J \rightarrow m \rightarrow m/J=k(z) \rightarrow 0$ reste exacte après extension des scalaires à ${\mathcal{R}}_U$, de sorte que l’on dispose d’une suite longue de ${\mathcal{O}}(U)$-modules $$H^1(mD) \rightarrow H^1(D_z) \rightarrow H^2(JD) \rightarrow H^2(mD) \rightarrow H^2(D_z) \longrightarrow 0.$$ L’image de la première flèche est le ${\mathbb{Q}}_p$-module engendré par $\mu(L)$. Pour conclure il suffit donc de voir que le noyau de la flèche $H^1(D_z) \rightarrow H^2(JD)$ est une doite. On a déjà vu que $H^1(D_z) \simeq H^2(mD)$ et $H^2(D_z)$ sont de ${\mathbb{Q}}_p$-dimensions respectives $2$ et $1$, il suffit donc de voir que $H^2(JD)$ est de dimension $2$ sur ${\mathbb{Q}}_p$. Mais tout comme $m$, l’idéal $J$ a deux générateurs et admet une présentation de la forme $$0 \longrightarrow {\mathcal{O}}(U) \overset{g}{\longrightarrow} {\mathcal{O}}(U)^2 \longrightarrow J \longrightarrow 0$$ avec $g \otimes_{{\mathcal{O}}(U)} k(u) = 0$, de sorte qu’un argument déjà donné plus haut montre que $H^2(D)^2 \simeq H^2(JD)$, ce qui conclut la preuve du théorème. L’espace des ${(\varphi,\Gamma)}$-modules triangulins sur ${\mathcal{R}}_A$ {#San} =========================================================================== ${(\varphi,\Gamma)}$-modules triangulins réguliers rigidifiés. -------------------------------------------------------------- Fixons $d\geq 1$ un entier. L’espace ${{\mathcal T}}^d$ est muni d’une famille universelle $(\widetilde{\delta}_i)$ de caractères ${\widetilde{\delta}}_i : {\mathbb{Q}}_p^\ast \rightarrow {\mathcal{O}}({{\mathcal T}}^d)^\ast$. On notera $${{\mathcal T}}^{\rm reg}_d \subset {{\mathcal T}}^d$$ l’ouvert admissible (en fait, de Zariski) défini par les relations $\widetilde{\delta}_i/\widetilde{\delta}_j \in {{\mathcal T}}^{\rm reg}$ pour tout $1 \leq i < j \leq d$. Par définition, si $A$ est une algèbre affinoïde alors ${{\mathcal T}}_d^{\rm reg}(A)$ est donc le sous-ensemble des $(\delta_i) \in {{\mathcal T}}(A)^d$ tels que $\delta_i/\delta_j \in {{\mathcal T}}^{\rm reg}(A)$ pour tout $1\leq i < j\leq d$. PS. Un ${(\varphi,\Gamma)}$-module triangulin [*régulier rigidifié*]{} est un triplet $(D,{{\rm Fil}}_\bullet(D),\nu)$ sur ${\mathcal{R}}_A$ où : - $(D,{{\rm Fil}}_\bullet(D))$ est un ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_A$ dont le paramètre $(\delta_i)$ est dans ${{\mathcal T}}_d^{\rm reg}(A)$ (condition de régularité), PS. - $\nu=(\nu_i)$ est une famille d’isomorphismes $\nu_i : {{\rm Fil}}_{i+1}(D)/{{\rm Fil}}_i (D) {\overset{\sim}{\rightarrow}}{\mathcal{R}}_A(\delta_i)$ dans ${(\varphi,\Gamma)}/A$ pour $i=0,\dots,{\rm rang}_{{\mathcal{R}}_A}(D)-1$ (rigidification). Deux tels triplets $(D,{{\rm Fil}}_\bullet(D),\nu)$ et $(D',{{\rm Fil}}_\bullet(D'),\nu')$ seront dit [*équivalents*]{} si il existe un isomorphisme $f: D \rightarrow D'$ dans ${(\varphi,\Gamma)}/A$ envoyant ${{\rm Fil}}_i(D)$ sur ${{\rm Fil}}_i(D')$ pour tout $i$ (auquel cas $D$ et $D'$ ont mème paramètre) et tel que $\nu'_i \,\,{\rm o} \,\, f = \nu_i$ pour tout $i=0,\dots,{\rm rang}_{{\mathcal{R}}_A}(D)-1$. Considérons le foncteur $$F_d^\square : {{\rm Aff}}\longrightarrow {\rm Ens}$$ de la catégorie ${{\rm Aff}}$ des ${\mathbb{Q}}_p$-algèbres affinoïdes vers celle ${\rm Ens}$ des ensembles associant à chaque objet $A$ l’ensemble[^17] des classes d’équivalence de ${(\varphi,\Gamma)}$-modules triangulins $p$-réguliers rigidifiés sur ${\mathcal{R}}_A$. Si $u=(D,{{\rm Fil}}_\bullet(D),\nu) \in F_d^\square(A)$ et si $f: A \rightarrow B$ est un morphisme dans ${{\rm Aff}}$, on pose bien entendu $$F_d^\square(f)(u)=(D\widehat{\otimes}_A B,({{\rm Fil}}_i(D) \widehat{\otimes}_A B), (\nu_i \otimes_{{\mathcal{R}}_A} {\mathcal{R}}_B)) \in F_d^\square(B),$$ ce qui fait bien de $F_d^\square$ un foncteur covariant. Le paramètre fournit un morphisme de foncteurs $\delta : F_d^\square \rightarrow {{\mathcal T}}_d^{\rm reg}$. \[repfdsquare\] Le foncteur $F_d^\square$ est représentable par un espace analytique $p$-adique ${\mathcal{S}}_d^\square$. Le morphisme $\delta: {\mathcal{S}}_d^\square \rightarrow {{\mathcal T}}_d^{\rm reg}$ est lisse de dimension relative $\frac{d(d-1)}{2}$. L’espace ${\mathcal{S}}_d^\square$ est irréductible, régulier, et équidimensionnel de dimension $\frac{d(d+3)}{2}$. Nous aurons besoin du lemme suivant. (Rigidité) Soient $(D,{{\rm Fil}}_\bullet(D),\nu)$ et $(D',{{\rm Fil}}_\bullet(D'),\nu')$ des ${(\varphi,\Gamma)}$-modules triangulins réguliers rigidifiés sur ${\mathcal{R}}_A$. S’ils sont équivalents, alors il existe une unique équivalence entre eux. On peut supposer $(D,{{\rm Fil}}_\bullet(D),\nu)=(D',{{\rm Fil}}_\bullet(D'),\nu')$ et il s’agit de voir que ce triplet a pour seule auto-équivalence l’identité. Si $f : D \rightarrow D$ en est une, alors par hypothèse $f({{\rm Fil}}_i(D))={{\rm Fil}}_i(D)$ et $f$ induit l’identité sur chaque ${{\rm Fil}}_{i+1}(D)/{{\rm Fil}}_i(D)$. Ainsi, $u:=f-{\rm id} \in {\mathrm{End}}_{{(\varphi,\Gamma)}/A}(D)$ a la propriété que $u({{\rm Fil}}_{i+1}(D)) \subset {{\rm Fil}}_i(D)$ pour tout $i<{\rm rang}_{{\mathcal{R}}_A}(D)$. Pour voir que $u=0$ il suffit donc de voir que $${\mathrm{Hom}}_{{(\varphi,\Gamma)}/A}({\mathcal{R}}_A(\delta_j),{\mathcal{R}}_A(\delta_i))=0$$ dès que $j>i$, soit encore que $H^0({\mathcal{R}}_A(\delta_i\delta_j^{-1})=0$ sous cette hypothèse. Mais ceci vient de ce que $\delta_i\delta_j^{-1}$ est régulier et du théorème \[thmH1\]. Démontrons maintenant le théorème. Quand $d=1$, $F_d^\square={{\mathcal T}}$ et le résultat est évident. Pour $d\geq 2$ on procède par récurrence sur $d$. On dispose d’un morphisme de foncteurs évident $F_d^\square \rightarrow F_{d-1}^\square \times F_1^\square = {\mathcal{S}}_{d-1}^\square \times {{\mathcal T}}$, associant à la classe de $(D,{{\rm Fil}}_\bullet(D),\nu) \in F_d^\square(A)$ la paire formée de la classe de $({{\rm Fil}}_{d-1}(D),({{\rm Fil}}_i(D))_{i\leq d-1},(\nu_i)_{1\leq i \leq d-1})$ et de $\delta_d$. Il se factorise par l’ouvert Zariski $$U_d \subset {\mathcal{S}}_{d-1}^\square \times {{\mathcal T}}$$ qui est l’image inverse de l’ouvert ${{\mathcal T}}_d^{\rm reg} \subset {{\mathcal T}}_{d-1}^{\rm reg} \times {{\mathcal T}}$ par le morphisme paramètre $\pi_d: {\mathcal{S}}_{d-1}^\square \times {{\mathcal T}}\rightarrow {{\mathcal T}}_{d-1}^{\rm reg} \times {{\mathcal T}}$. Nous allons démontrer que le morphisme ci-dessus $F_d^\square\rightarrow U_{d}$ est relativement représentable par un fibré vectoriel de rang $d-1$, ce qui provera le théorème. Le fibré en question sera trivial au dessus de tout ouvert affinoïde de $U_d$. PS. Nous aurons besoin d’un sorite préliminaire. Soient $A$ une ${\mathbb{Q}}_p$-algèbre affinoïde et $u=(c_A,\delta_d) \in U_d(A)$ et $x=(D_A,{{\rm Fil}}_\bullet,\nu)$ un représentant de la classe $c_A$. Considérons le $A$-module $M(u)=H^1(D_A(\delta_d^{-1})$. Si $y=(D'_A,{{\rm Fil}}_\bullet,\nu')$ est équivalent à $x$, l’unique équivalence $y \rightarrow x$ identifie donc canoniquement $M(y)$ et $M(x)$. Ainsi, il y a un sens à définir le $A$-module $M(u)$ associé à un élément $u \in U_d(A)$, comme étant par exemple la limite inductive des $M(x)$ pour $x$ parcourant l’ensemble[^18] des représentants de $c_A$ : le choix d’un représentant $x$ de $c_A$ fournit alors un isomorphisme canonique $M(c_A) {\overset{\sim}{\rightarrow}}M(x)$. On vérifie de suite que si $A \rightarrow B$ est un morphisme entre algèbres affinoïdes, alors pour tout $u_A \in U_d(A)$, d’image $u_B \in U_d(B)$, on dispose d’un morphisme canonique $$M(u_A) \otimes_A B \longrightarrow M(u_B)$$ défini de manière évidente sur les représentants. Les théorèmes \[changementdebase\] et \[thmH1\] assurent que c’est un isomorphisme entre modules libres de rang $d-1$. En particulier $\Omega \mapsto M(\Omega)$, pour $\Omega \subset U_d$ ouvert affinoïde, définit un faisceau cohérent sur $U_d$ tel qu’en fait $M(\Omega)$ est libre de rang $d-1$ sur ${\mathcal{O}}(\Omega)$ pour tout $\Omega$. On note encore ${{\mathcal M}}$ ce faisceau cohérent sur $U_d$. D’après ce que nous venons de voir, si $u \in U_d(A)$, alors on a une identification canonique $$\label{idcan} u^*({{\mathcal M}})(A)=M(u).$$ Considérons alors $$\eta : {\mathcal{S}}_d^\square : = {\rm Spec}_{U_d}^{\rm an}({\rm Symm}\,\,{{\mathcal M}}^\vee) \rightarrow U_d$$ le spec relatif analytique de la ${\mathcal{O}}_{U_d}$-algèbre quasi-cohérente ${\rm Symm}\,\,{{\mathcal M}}^\vee$ (voir [@conradample §2.2]) : c’est le fibré vectoriel sur $U_d$ associé à ${{\mathcal M}}^\vee$. Par la propriété universelle de cette construction, pour toute algèbre affinoïde $A$ et tout $u \in U_d(A)$, disons $u=([(D_A,{{\rm Fil}}_\bullet,\nu)],\delta_d)$, on a des identifications canoniques $$\{ v \in {\mathcal{S}}_d^\square(A), \eta(v)=u\}={\mathrm{Hom}}_A(u^*({{\mathcal M}}^\vee)(A),A)=u^*({{\mathcal M}})(A)=H^1(u) {\overset{\sim}{\rightarrow}}{\rm Ext}({\mathcal{R}}_A(\delta_d),D_A).$$ Cela définit un morphisme de foncteurs ${\mathcal{S}}_d^\square \rightarrow F_d^\square$ au dessus de $U_d$ : on associe à une paire formée d’un élément $([(D_A,{{\rm Fil}}_\bullet,\nu)],\delta_d) \in U_d(A)$ et d’une classe $E \in {\rm Ext}({\mathcal{R}}_A(\delta_d),D_A)$, que l’on voit comme la donnée d’une suite exacte $$0 \longrightarrow D_A \overset{\iota}{\longrightarrow} E \overset{\pi}{\longrightarrow} {\mathcal{R}}_A(\delta) \longrightarrow 0,$$ la classe du ${(\varphi,\Gamma)}$-module triangulin $E$ avec ${{\rm Fil}}_i(E):=\iota({{\rm Fil}}_i(D_A))$ et $\nu_i {\rm \cdot} \iota^{-1}=:\nu_i$ pour $i\leq d-1$, et ${\rm Fil}_{d}(E)=E$ et $\nu_{d}=\pi$. Il est immédiat que ${\mathcal{S}}_d^\square \rightarrow F_d^\square$ est un isomorphisme de foncteurs. $\square$ Notons $\mathbb{D}^r$ la boule unité affinoïde fermée de rayon $1$ sur ${\mathbb{Q}}_p$, d’algèbre ${\mathbb{Q}}_p\langle t_1,\dots,t_r\rangle$. \[corfamcoleman\] Si $x \in S_d^\square$, il existe un voisinage ouvert affinoïde $U$ de $x$ dans $S_d^\square$, un voisinage ouvert affinoïde $\Omega$ de $\delta(x)$ dans ${{\mathcal T}}_d^{\rm reg}$, et un isomorphisme $$\iota : U {\overset{\sim}{\rightarrow}}\Omega \times {\mathbb{D}}^{\frac{d(d-1)}{2}}$$ tels que ${\rm pr}_2 \cdot \iota = \delta$. Par définition, si $E$ est un fibré vectoriel sur un espace rigide $Y$ alors pour tout $y \in Y$ on peut trouver un voisinage ouvert $W$ de $y$ dans $Y$ tel que $W \times_Y E {\overset{\sim}{\rightarrow}}W \times {\mathbb{A}}^m$ comme fibré. Le corollaire découle alors de la construction inductive de ${\mathcal{S}}_d^\square$ établie ci-dessus. ${(\varphi,\Gamma)}$-modules triangulins réguliers non rigidifiés {#diverstrucs} ----------------------------------------------------------------- Bien que ce ne soit pas nécessaire pour le théorème principal de cet article, il est naturel de considérer le foncteur $$F_d : {{\rm Aff}}\rightarrow {\rm Ens}$$ où $F_d(A)$ est l’ensemble des classes d’équivalence de ${(\varphi,\Gamma)}$-modules triangulins $(D,{{\rm Fil}}_\bullet(D))$ sur ${\mathcal{R}}_A$ dont le paramètre est dans ${{\mathcal T}}_d^{\rm reg}(A)$ (sans rigidification). La notion d’équivalence utilisée ici est celle dans ${(\varphi,\Gamma)}/A$ avec préservation de la filtration. Pour éliminer les auto-équivalences des objets paramétrés par $F_d$ il est nécessaire de se restreindre à un sous-foncteur adéquat. Si $L$ est une extension finie de ${\mathbb{Q}}_p$ et si $(D,{{\rm Fil}}_\bullet(D))$ est un ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_L$, on dira que $D$ est [*non scindé*]{} si pour tout $0 \leq i < {\rm rg}_{{\mathcal{R}}_L}(D)$, l’extension $$0 \longrightarrow {{\rm Fil}}_i(D) \longrightarrow {{\rm Fil}}_{i+1}(D) \rightarrow {{\rm Fil}}_{i+1}(D)/{{\rm Fil}}_i(D) \longrightarrow 0$$ n’est pas scindée dans ${(\varphi,\Gamma)}/L$. Si $(D,{{\rm Fil}}_\bullet(D))$ est un ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_A$, on dira que $D$ est [*partout non scindé*]{} si pour tout $x \in {\rm Sp}(A)$, $(D_x,{{\rm Fil}}_\bullet(D)_x)$ est non scindée. Si un ${(\varphi,\Gamma)}$-module $D$ triangulin sur ${\mathcal{R}}_A$ est partout non scindé et si $B$ est une $A$-algèbre affinoïde, alors $D \widehat{\otimes}_A B$ est aussi partout non scindé vu comme ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_B$. On dispose donc d’un sous-foncteur $$F_d^{\rm ns} \subset F_d$$ paramétrant les ${(\varphi,\Gamma)}$-modules triangulins réguliers partout non scindés, et idem pour $F_d^{\square,{\rm ns}} \subset F_d^\square$. L’oubli de la rigidification définit un morphisme de foncteurs $$\eta : F_d^\square \rightarrow F_d$$ qui est surjectif sur les points. Soit $G_d=\mathbb{G}_m^d/\mathbb{G}_m$ (plongé diagonalement) vu comme tore rigide analytique sur ${\mathbb{Q}}_p$. On dispose enfin d’une action de $G_d$ sur $F_d^\square$ agissant sur les rigidifications : $(a_i)\cdot [(D,{{\rm Fil}}_\bullet(D),(\nu_i))]:=[(D,{{\rm Fil}}_\bullet(D),(a_i \cdot \nu_i))]$. Cette action se factorise bien par $G$ car si $a \in A^\ast$ et $x$ est un représentant d’une classe de $F_d^\square(A)$ alors $(a,a,\dots,a)\cdot x$ est équivalent à $x$ via la multiplication par $a$. Elle préserve $F_d^{\square,{\rm ns}}$. \[rigidite2\] - Si $(D_A,{{\rm Fil}}_\bullet)$ est un ${(\varphi,\Gamma)}$-module triangulin régulier sur ${\mathcal{R}}_A$ qui est partout non-scindé, alors ses auto-équivalences sont les homothéties $A^\times$.PS. - $F_d^{\square,{\rm ns}}$ est représenté par un ouvert Zariski de $F_d^\square$.PS. - Pour toute algèbre affinoïde $A$, $G_d(A)$ agit librement sur $F_d^{\square,{\rm ns}}(A)$ et l’application $\eta(A): F_d^{\square,{\rm ns}}(A)/G_d(A) \longrightarrow F_d^{\rm ns}(A)$ est bijective. Vérifions le (i). Nous allons montrer plus généralement que $${\mathrm{End}}_{{(\varphi,\Gamma)}_A}((D_A,{{\rm Fil}}_\bullet))=A.$$ Quitte à remplacer $A$ par $A/I$ pour un idéal $I$ de codimension finie, on peut supposer que $A$ est artinien d’après le théorème d’intersection de Krull. Dans ce cas, une récurrence sur la longueur de $A$ permet de supposer que $A=L$ est un corps. Dans ce cas, on procède par récurrence sur $d$. Quand $d=1$ cela vient de ce que $H^0({\mathcal{R}}_L)=L$. Pour $d\geq 1$, remarquons que si $(D,{{\rm Fil}}_\bullet)$ est non scindé, il en va de même de $({{\rm Fil}}_{d-1}(D),{{\rm Fil}}_\bullet)$. Ainsi, un endomorphisme de $D$ préservant sa filtration agit par une homothétie sur ${{\rm Fil}}_{d-1}$ et sur le quotient $D/{{\rm Fil}}_{d-1}$. Notons qu’il agit par $0$ sur ces deux ${(\varphi,\Gamma)}$-modules si et seulement si il provient d’un morphisme $D/{{\rm Fil}}_{d-1}(D) \rightarrow {{\rm Fil}}_{d-1}(D)$ dans ${(\varphi,\Gamma)}/L$, auquel cas il est en fait nul car si $\delta_i$ est le paramètre de $D$ alors $H^0({{\rm Fil}}_{d-1}(D)(\delta_d^{-1}))=0$ par l’hypothèse de régularité. Nous avons donc montré que le morphisme de $L$-algèbres $$\alpha : {\mathrm{End}}_{{(\varphi,\Gamma)}/L}((D,{{\rm Fil}}_\bullet)) \longrightarrow {\mathrm{End}}_{{(\varphi,\Gamma)}/L}(({{\rm Fil}}_{d-1}(D),{{\rm Fil}}_\bullet)) \times {\mathrm{End}}(D/{{\rm Fil}}_{d-1}(D)) {\overset{\sim}{\rightarrow}}L \times L$$ est injectif. En particulier, si ${\mathrm{End}}_{{(\varphi,\Gamma)}/L}((D,{{\rm Fil}}_\bullet))$ n’est pas réduit aux homothéties alors $\alpha$ est bijectif : il existe donc un endomorphisme idempotent de $D$ valant l’identité sur ${{\rm Fil}}_{d-1}$ et $0$ sur $D/{{\rm Fil}}_{d-1}$, ce qui contredit le fait que $0 \rightarrow {{\rm Fil}}_{d-1}(D) \rightarrow D \rightarrow D/{{\rm Fil}}_{d-1}(D) \rightarrow 0$ est non scindée.PS. Vérifions le (ii) par récurrence sur $d$. Si $d=1$, $F_d^{\square,{\rm ns}}=F_d^\square$ et il n’y a rien à démontrer. En général nous avons vu que $S_d^\square$ est un certain fibré vectoriel de rang $d-1$ sur un ouvert $U_d \subset S_{d-1}^\square \times {{\mathcal T}}$. Ce point de vue fait apparaître $F_d^{\square,{\rm ns}}$ comme l’ouvert du complémentaire de la section nulle de ce fibré pris au dessus de $F_{d-1}^{\square,{\rm ns}} \times {{\mathcal T}}$, d’où le résultat.PS. Vérifions le (iii). Si $(a_i)(D,{{\rm Fil}}_\bullet,\nu)$ est équivalent à $(D,{{\rm Fil}}_\bullet,\nu)$ alors $(D,{{\rm Fil}}_\bullet)$ admet un automorphisme agissant sur chaque ${{\rm Fil}}_i(D)/{{\rm Fil}}_{i-1}(D)$ par le scalaire $a_i$. Comme les seuls automorphismes sont des homothéties par le (i) il vient que tous les $a_i$ sont égaux : l’action de l’énoncé est libre. Le second point du (iii) vient de ce que les automorphismes dans ${(\varphi,\Gamma)}/A$ de ${\mathcal{R}}_A(\delta)$ sont les homothéties $A^\times$. Nous n’avons pas trouvé de références pour l’existence d’un quotient pour une action libre d’un tore sur un espace analytique. Ceci, combiné au lemme ci-dessus, entraînerait que le faisceau pour la topologie de Tate ${{\mathcal{F}}}_d^{\rm ns}$ associé à $F_d^{\rm ns}$ est représentable. Concrètement, ${{\mathcal{F}}}_d^{\rm ns}(A)$ est la limite inductive (qui est filtrante injective) sur tous les recouvrements finis de ${\rm Sp}(A)$ par des ouverts affinoïdes $U_i$ des noyaux des $\prod_i F_d^{\rm ns}({\mathcal{O}}(U_i)) \rightrightarrows \prod_{i,j} F_d^{\rm ns}({\mathcal{O}}(U_i\cap U_j))$. Nous nous contenterons ici de traiter le cas $d=2$, pour lequel le problème se résoud aisément. $F_2^{\rm ns}$ est représenté par ${{\mathcal T}}_2^{\rm reg}$. En effet, le morphisme paramètre $F_2^{\rm ns} \rightarrow {{\mathcal T}}_2^{\rm reg}$ est un isomorphisme : si $\delta=(\delta_1,\delta_2) \in {{\mathcal T}}_2^{\rm reg}(A)$ est donné, alors on a vu que $H^1({\mathcal{R}}_A(\delta_1\delta_2^{-1}))$ est libre de rang $1$ sur $A$, donc il existe un et un seul élement de $F_2^{\rm ns}(A)$ de paramètre $\delta$. Il se trouve que dans ce cas nous pouvons décrire aussi ce qui se passe au voisinage des points non réguliers. Pour tout $i \geq 0$ entier, notons $F_i,F'_i \subset {{\mathcal T}}^2$ les fermés définis respectivement par les équations $\delta_1\delta_2^{-1}=\chi x^i$ et $\delta_1 \delta_2^{-1} = x^{-i}$. Tous ces fermés sont deux à deux disjoints et chaque ouvert affinoïde de ${{\mathcal T}}^2$ ne rencontre qu’un nombre fini d’entre eux. On désigne par $F$ la réunion des $F_i$ et $F'$ celle des $F'_i$ : ce sont encore des fermés de ${{\mathcal T}}^2$, et on a ${{\mathcal T}}_2^{\rm reg} \coprod F \coprod F' = {{\mathcal T}}^2$. Soit $\pi : {{\widetilde{{{\mathcal T}}}}}_2 \rightarrow {{\mathcal T}}^2\backslash F'$ l’éclaté de ${{\mathcal T}}^2\backslash F'$ le long de $F$. Le résultat suivant confirme l’intuition de Colmez dans [@colmeztri] selon laquelle ${{\widetilde{{{\mathcal T}}}}}_2$ est l’espace de module grossier des ${(\varphi,\Gamma)}$-modules triangulins partout non scindés de rang $2$ de paramètre dans ${{\mathcal T}}^2\backslash F'$ (sur l’ouvert ${{\mathcal T}}_2^{\rm reg}$ c’est même un espace de module fin par le résultat précédent). \[thmdegal2\]Pour toute extension finie $L/{\mathbb{Q}}_p$, il existe une bijection canonique entre ${{\widetilde{{{\mathcal T}}}}}_2(L)$ et l’ensemble des classes d’isomorphie de ${(\varphi,\Gamma)}$-modules triangulins sur ${\mathcal{R}}_L$ qui sont de rang $2$, non scindé, et de paramètre dans ${{\mathcal T}}^2(L)\backslash F'(L)$, l’application $\pi$ donnant le paramètre associé. De plus, il existe un recouvrement affinoïde admissible $(U_i)$ de ${{\widetilde{{{\mathcal T}}}}}_2$, et pour chaque $i$ un ${(\varphi,\Gamma)}$-module triangulin $D_i$ de rang $2$ sur ${\mathcal{R}}_{U_i}$ et de paramètre $\pi_{|U_i}$, tels que pour toute extension $L/{\mathbb{Q}}_p$ finie et tout $x \in U_i(L)$, $(D_i)_x$ est isomorphe au ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_L$ associé à $x$ par la bijection précédente. La première partie du théorème est dûe à Colmez : c’est son calcul de $H^1({\mathcal{R}}_L(\delta))$ pour l’ouvert ${{\mathcal T}}_2^{\rm reg}$, combiné à sa formule pour l’invariant $L$ ([@colmezinvL]) au voisinage des points de la forme $x \mapsto \chi x^i$ avec $i\geq 1$. Colmez a aussi démontré une version faible de la seconde partie au voisinage de tout $x \in {{\mathcal T}}_2^{\rm reg}$ qui est de plus [*$p$-régulier*]{} au sens que $\delta_1\delta_2^{-1}(p) \notin p^{\mathbb{Z}}$. Au dessus de ${{\mathcal T}}_2^{\rm reg}$, le théorème est un cas particulier de la proposition précédente. Quitte à tordre par une famille de ${(\varphi,\Gamma)}$-modules de rang $1$, on peut donc supposer que l’on se place dans un voisinage ouvert affinoïde $U \subset {{\mathcal T}}$ contenant un unique point $u \in U({\mathbb{Q}}_p)$ tel que $\delta(u)=\chi x^i$ est non régulier, et que l’on s’intéresse à l’éclaté $\pi : \widetilde{U} \rightarrow U$ en $u$. Dans ce cas, on dispose d’une identification naturelle donnée par le théorème \[h1universel\] $$\mu : \pi^{-1}(u) {\overset{\sim}{\rightarrow}}{\mathbb P}(H^1({\mathcal{R}}_{{\mathbb{Q}}_p}(\chi x^i))).$$ Ce théorème assure aussi que $H^1({\mathcal{R}}_U(\delta_U))$ est canoniquement isomorphe à ${\mathcal{O}}(U)$, et que si $m \subset {\mathcal{O}}(U)$ désigne l’idéal des fonctions s’annulant en $u$, alors l’inclusion induit une égalité $$H^1(m{\mathcal{R}}_U(\delta_U))=H^1({\mathcal{R}}_U(\delta_U))={\mathcal{O}}(U).$$ Soit $D_U=m{\mathcal{R}}_U \oplus {\mathcal{R}}_U$ un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_U$ dont la classe dans $H^1(m{\mathcal{R}}_U(\delta_U))$ en est un ${\mathcal{O}}(U)$-générateur. Soit $U_i$ un recouvrement fini de $\widetilde{U}$ par des ouverts affinoïdes sur chacun desquels $m{\mathcal{O}}(U_i) \subset {\mathcal{O}}(U_i)$ est libre de rang $1$, disons engendré par l’élément $f_i$. Le transformé strict de $D_U$ sur ${\mathcal{R}}_{U_i}$, qui est aussi le quotient de $D_U \otimes_{{\mathcal{R}}_U} {\mathcal{R}}_{U_i}$ par sa $f_i$-torsion, est un ${(\varphi,\Gamma)}$- module sur ${\mathcal{R}}_{U_i}$ de ${\mathcal{R}}_{U_i}$-module sous-jacent $m{\mathcal{R}}_{U_i}\oplus {\mathcal{R}}_{U_i}=f_i{\mathcal{R}}_{U_i}\oplus {\mathcal{R}}_{U_i}$ : il est bien libre de rang $2$. Par construction de l’éclaté, le choix d’un $z \in (\pi^{-1}(u)\cap U_i)(L)$ définit un ${\mathcal{O}}(U)$-morphisme surjectif $$(m/m^2)\otimes_{{\mathbb{Q}}_p} L \rightarrow f_i{\mathcal{O}}(U_i)\otimes_{{\mathcal{O}}(U_i)} L,$$ soit encore à un vecteur tangent $v_z \in T_u \otimes_{{\mathbb{Q}}_p} L$, tout vecteur tangent s’obtenant ainsi pour un certain $i$. Un tel choix définit donc un ${\mathcal{O}}(U)$-morphisme surjectif $$D_U\otimes_{{\mathcal{O}}(U)}L \rightarrow (D_i)_z.$$ La preuve du (ii) du théorème \[h1universel\] dit exactement que l’image de ce morphisme a pour classe dans $H^1({\mathcal{R}}_{L}(\chi x^{-i}))$ l’élement $\mu(v_z)$ associé à l’isomorphisme $\mu(L) : {\mathbb P}(T_u)(L) {\overset{\sim}{\rightarrow}}{\mathbb P}(H^1({\mathcal{R}}_L(\chi x^i)))$. [*La situation est différente au voisinage des points dans $F'$. En effet, négligeons les torsions en nous plaçant dans un voisinage ouvert affinoïde $U \subset {{\mathcal T}}$ du point $\delta=x^{-i}$. D’après le théorème \[h1universel\], pour tout élément $E \in H^1({\mathcal{R}}(\delta))$ on peut trouver un ${(\varphi,\Gamma)}$-module triangulin $D$ sur ${\mathcal{R}}_U$ de paramètre $(\delta_U,1)$ tel que pour tout $z \in {{\mathcal T}}^{\rm reg}\cap U$ son évaluation $D_z$ est non scindée, et dont l’évaluation en $\delta$ est exactement $E$. On pourrait penser aller plus loin en introduisant ici aussi l’éclaté de $U$ en $\delta$. Il n’y cependant pas de manière naturelle de construire de famille de ${(\varphi,\Gamma)}$-modules sur cet éclaté. Disons simplement qu’un indice de ceci est que le théorème \[h1universel\] identifie canoniquement $T_\delta$ avec le [dual]{} de l’espace vectoriel $H^1({\mathcal{R}}_{{\mathbb{Q}}_p}(x^{-i}))$, plutôt qu’avec ce dernier.* ]{} Ainsi qu’il l’est expliqué dans [@berch], notons que la proposition \[thmdegal2\] entraïne la : La conjecture 5.1 de [@berch] est vraie : les représentations potentiellement triangulines de dimension $2$ forment une partie fine de la variété des caractères $p$-adiques de dimension $2$ de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$. Terminons ce paragraphe par une comparaison entre les foncteurs définis ici et les “foncteurs de déformations triangulines” considérés dans [@bch §2.5] et [@chU3 §3]. Soit $F : {{\rm Aff}}\rightarrow {\rm Ens}$ un foncteur quelconque, $L/{\mathbb{Q}}_p$ une extension finie et $x \in F(L)$. Soit $\mathcal{C}$ la catégorie des $L$-algèbres locales artiniennes de corps résiduel $L$ : c’est une sous-catégorie pleine de celle des affinoïdes sur $L$. Le [*complété formel*]{} de $F$ en $x$ est le foncteur $\widehat{F}_x : \mathcal{C} \rightarrow {\rm Ens}$ défini comme suit : pour tout objet $A$ de $\mathcal{C}$, $\widehat{F}_x(A) \subset F(A)$ est le sous-ensemble des éléments dont l’image dans $F(L)$ par l’unique morphisme $A \rightarrow L$ est l’élément $x$ (c’est donc un sous-foncteur de $F_{|\mathcal{C}}$). Quand $F$ est représenté par un espace analytique $Z$ sur ${\mathbb{Q}}_p$, $\widehat{F}_x$ est pro-représenté par $\widehat{{\mathcal{O}}_{Z,x}}\otimes_{k(x)} L$. \[sousfonct\] Soit $x=[(D,{{\rm Fil}}_\bullet)] \in F_d(L)$. Si $D$ est non-scindé, et plus généralement si ${\mathrm{End}}_{{(\varphi,\Gamma)}/L}((D,{{\rm Fil}}_\bullet))=L$, alors $\widehat{(F_d)}_x$ est canoniquement isomorphe au foncteur des déformations triangulines de $(D,{{\rm Fil}}_\bullet)$ au sens de [@bch §2.5]. De plus, $\widehat{(F_d)}_x$ est pro-représentable. En effet, le foncteur ${{\mathfrak X}}_{D,{{\rm Fil}}_\bullet}$ des déformations triangulines de $(D,{{\rm Fil}}_\bullet)$ défini [*loc. cit.*]{} paramètre les classes d’isomorphismes de triplets $(D_A,{{\rm Fil}}_\bullet,\pi)$ où $\pi : D_A \otimes_A L \rightarrow D$ est un isomorphisme dans ${(\varphi,\Gamma)}/A$ envoyant ${{\rm Fil}}_i(D_A)$ sur ${{\rm Fil}}_i(D)$. L’association $(D_A,{{\rm Fil}}_\bullet,\pi) \mapsto [(D_A,{{\rm Fil}}_\bullet)]$ définit un morphisme de foncteurs ${{\mathfrak X}}_{D,{{\rm Fil}}_\bullet} \rightarrow \widehat{(F_d)}_x$ qui est surjectif sur les points. Pour l’injectivité, il faut remarquer que si ${\rm Aut}_{{(\varphi,\Gamma)}/L}((D,{{\rm Fil}}_\bullet))=L^\times$ (les homothéties), alors pour tout $(D_A,{{\rm Fil}}_\bullet)$ dont la classe est dans $\widehat{(F_d)}_x$, alors l’application naturelle $${\rm Aut}_{{(\varphi,\Gamma)}/A}(D_A,{{\rm Fil}}_\bullet) \rightarrow {\rm Aut}_{{(\varphi,\Gamma)}/L}(D,{{\rm Fil}}_\bullet)$$ est surjective, car le terme de gauche contient $A^\times$. L’assertion de pro-représentabilité est [@chU3 Prop. 3.4]. Densité des ${(\varphi,\Gamma)}$-modules cristallins dans $S_d^\square$. ------------------------------------------------------------------------ Si $L$ est une extension finie de ${\mathbb{Q}}_p$ et $D$ un ${(\varphi,\Gamma)}$-module sur ${\mathcal{R}}_L$, on dit que $D$ est [*cristallin*]{} si le $L$-espace vectoriel $${\mathcal{D}_{\rm cris}}(D):=(D[1/t])^\Gamma$$ est de dimension ${\rm rg}_{{\mathcal{R}}_L}(D)$. Nous renvoyons à [@bch §2.2.7] pour une discussion de cette définition, principalement motivée par des travaux de Berger : si $D$ est étale alors il est cristallin si et seulement si sa représentation galoisienne $V$ associée l’est, auquel cas ${\mathcal{D}_{\rm cris}}(D)$ est canoniquement isomorphe à ${D_{\rm cris}}(V)$ comme $L[\varphi]$-module filtré (nous ne donnerons pas ici la recette de la filtration naturelle sur ${\mathcal{D}_{\rm cris}}(D)$). Bien entendu, un point $x \in {\mathcal{S}}_d^\square(L)$ est dit cristallin si le ${(\varphi,\Gamma)}$-module triangulin $D_x$ sur ${\mathcal{R}}_L$ qui lui est associé l’est. Dans ce cas, le paramètre $(\delta_i)$ de $D_x$ est [*algébrique*]{} : pour tout $i$, il existe $k_i \in {\mathbb{Z}}$ tel que $\delta_i(\gamma)={\gamma}^{k_i}$ pour tout $\gamma \in \Gamma$ (voir par exemple [@bch prop. 2.4.1]). PS. PS. On rappelle qu’une partie $A$ d’un espace rigide $Y$ est dite Zariski-dense si le seul fermé analytique global réduit de $Y$ contenant $A$ est la nilréduction de $Y$. Si $Y$ est affinoïde il est équivalent de demander que $A$ est Zariski-dense dans ${\rm Spec}({\mathcal{O}}(Y))$. On dit de plus que la partie $A$ s’accumule en une partie $B \subset Y$ si tout élément de $B$ admet une base de voisinages ouverts affinoïdes $U_i$ tels que $A\cap U_i$ est Zariski-dense dans $U_i$. On dit que $A$ est d’accumulation si $A$ s’accumule en $A$. On étend ces définitions à des parties de $Y({\overline{\mathbb{Q}}_p}):=\bigcup_L Y(L)$ (la réunion portant sur les sous-extensions finies) en considérant l’ensemble des poins fermés sous-jacents. Comme exemple typique, remarquons que ${\mathbb{N}}^d$ est Zariski-dense et d’accumulation dans ${\mathbb{A}}^d$.PS. Nous renvoyons à [@conradirr] pour les généralités sur les composantes irréductibles des espaces rigides analytiques $p$-adiques. \[densecrisdanstri\] Pour chaque extension finie $L/{\mathbb{Q}}_p$, l’ensemble des points cristallins de ${\mathcal{S}}_d^\square(L)$ est Zariski-dense et s’acculume en chaque point de ${\mathcal{S}}_d^\square(L)$ de paramètre algébrique. Remarquons que ${{\mathcal T}}_d^{\rm reg}$ est irréductible, comme ouvert Zariski de l’espace irréductible ${{\mathcal T}}^d$. Comme un fibré vectoriel sur une base irréductible lisse est irréductible lisse, ${\mathcal{S}}_d^{\square}$ est irréductible lisse par construction. Pour démontrer la densité Zariski de ${\mathcal{S}}_d^\square({\mathbb{Q}}_p)$, il suffit donc de démontrer qu’il est non vide ainsi que la propriété d’accumulation. PS. Soit $L$ une extension finie de ${\mathbb{Q}}_p$. Considérons $A_d(L) \subset {{\mathcal T}}_d^{\rm reg}(L)$ l’ensemble des points paramétrant les $(\delta_i) \in {{\mathcal T}}(L)^d$ tels que : \(a) $\delta_i(p)/\delta_j(p) \neq p^{\pm 1}$ pour tout $i<j$, \(b) il existe une suite d’entiers $k_i \in {\mathbb{Z}}$ vérifiant : $\delta_i(\gamma)=\gamma^{-k_i}$ pour tout $\gamma \in \Gamma$ et tout $i=1,\dots,d$, \(c) la suite $k_i$ est strictement croissante : $k_1 < k_2 < \dots < k_d$. On note aussi $B_d(L) \subset {{\mathcal T}}_d^{\rm reg}(L)$ l’ensemble des points satisfaisant uniquement la condition (b), c’est à dire les paramètres algébriques. Il est évident que $A_d(L)$ est non-vide, que $A_d(L) \subset B_d(L)$, et que $A_d(L)$ s’accumule en $B_d(L)$ dans ${{\mathcal T}}_d^{\rm reg}$. \[critcris\] Un ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_L$ qui a son paramètre dans $A_d(L)$ est cristallin. C’est un cas particulier de [@bch Prop. 2.3.4], largement précisée dans [@benois], qui repose de manière essentielle sur des résultats de Berger [@berger1] [@berger2]. En effet, si $D$ est comme dans l’énoncé alors il est de De Rham par cette proposition (à cause de la condition (c)), et donc potentiellement semi-stable par le théorème de Berger. Comme $D$ est extension successive de ${(\varphi,\Gamma)}$-modules cristallins par hypothèse, et comme la formation du ${\mathcal D}_{\rm pst}$ est exacte sur les ${(\varphi,\Gamma)}$-modules potentiellement semi-stables d’après [@benois Prop. 1.2.9], il vient que $D$ est semi-stable. La condition (a) force alors $D$ à être cristallin. Retournons à la preuve du théorème \[densecrisdanstri\]. Supposons donc $x \in S_d^\square(L)$ cristallin, ou plus généralement tel que $\delta(x) \in B_d(L)$. Soit $U$ un voisinage ouvert affinoïde de $x$ dans $S_d^\square$, ainsi que $\Omega$ et $\iota$, comme dans le corollaire \[corfamcoleman\]. Soit $(\Omega_i)_{i\in I}$ une base de voisinages ouverts affinoïdes de $\delta(x)$ dans ${{\mathcal T}}_d^{\rm reg}$ dans lesquels $A_d(L)$ est Zariski-dense. Quand $i$ parcourt $I$ et $V$ les ouverts affinoïdes de ${\mathbb{D}}^{\frac{d(d-1)}{2}}$, les $U_{i,V}:=\iota^{-1}(\Omega_i \times V)$ forment une base de voisinages ouverts affinoïdes de $x$ dans ${\mathcal{S}}_d^\square$. Comme $A_d(L)$ est Zariski-dense dans chaque $\Omega_i$, il en va de même de[^19] $\iota^{-1}((A_d(L)\cap \Omega_i) \times V)$ dans $U_{i,V}$, qui est constitué de points cristallins d’après le lemme \[critcris\]. La famille de représentations galoisiennes sur le lieu étale ------------------------------------------------------------ Pour terminer ce chapitre, considérons le [*sous-ensemble*]{} $$S_d^{\square,0} \subset S_d^\square$$ constitué des points $x \in S_d^{\square}$ tels que le ${(\varphi,\Gamma)}$-module $D_x$ sur ${\mathcal{R}}_{k(x)}$ associé est étale. Si $x \in S_d^{\square, 0}$ on désigne par $V_x$ la représentation de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$ telle que ${D_{\rm rig}}(V_x) \simeq D_x$. \[conskliu\] Pour chaque $x \in S_d^\square$, il existe un voisinage ouvert affinoïde $\Omega$ de $x$ dans $S_d^\square$, un modèle ${{\mathcal A}}\subset {\mathcal{O}}(\Omega)$, et un ${{\mathcal A}}$-module libre $M$ de rang $d$ muni d’une application ${{\mathcal A}}$-linéaire continue de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$ tels que : - $\Omega \subset S_d^{\square,0}$, - pour tout morphisme $Z \rightarrow \Omega$ avec $Z$ affinoïde, le ${(\varphi,\Gamma)}$-module ${D_{\rm rig}}(M \otimes_{{{\mathcal A}}} {\mathcal{O}}(Z))$ sur ${\mathcal{R}}_{Z}$ défini par Berger et Colmez est isomorphe au ${(\varphi,\Gamma)}$-module triangulin rigidifié universel déduit du morphisme donné $Z \rightarrow S_d^\square$. Le ${\mathcal{O}}(\Omega)[{\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})]$-module $M \otimes_{{{\mathcal A}}} {\mathcal{O}}(\Omega)$ est unique à isomorphisme près pour la propriété (ii). L’existence de ${{\mathcal A}}$ et $M$ satisfaisant (i) et (ii) pour $Z=\Omega$, ainsi que l’assertion d’unicité, est un cas particulier du théorème [@kedliu Thm. 0.2] de Kedlaya-Liu appliqué à la famille universelle de ${(\varphi,\Gamma)}$-modules sur $S_d^\square$. Vérifions le (ii) pour $Z \rightarrow \Omega$ quelconque. Il s’agit de voir qu’avec la définition du ${D_{\rm rig}}$ d’une famille de Berger-Colmez l’application naturelle $$\label{isombcol}{D_{\rm rig}}(M \otimes_{{{\mathcal A}}} {\mathcal{O}}(\Omega)) \otimes_{{\mathcal{R}}_\Omega} {\mathcal{R}}_Z \rightarrow {D_{\rm rig}}(M \otimes_{{{\mathcal A}}} {\mathcal{O}}(Z))$$ est un isomorphisme. Le terme de droite a bien un sens car la représentation $M \otimes_{{{\mathcal A}}} {\mathcal{O}}(Z)$ admet un ${{\mathcal B}}$-réseau libre stable pour tout modèle ${{\mathcal B}}$ de ${\mathcal{O}}(Z)$ contenant l’image de ${{\mathcal A}}$ (de tels modèles existent et on en fixe un). L’isomorphisme ci-dessus découle alors de l’assertion d’unicité de la proposition [@bergercolmez prop. 4.2.8] comme dans la preuve de leur théorème 4.2.9. : les détails sont sans difficulté et laissés au lecteur. En particulier, $S_d^{\square,0} \subset S_d^\square$ est un ouvert pour la topologie naïve. Notons qu’à priori, ce résultat ne confère pas à $S_d^{\square,0}$ de structure naturelle d’espace rigide analytique. (On pourrait cependant choisir une écriture de $S_d^{\square,0}$ comme réunion disjointe d’ouverts affinoïdes $\mathcal{U}= \{\Omega_i, i \in I\}$ de $S_d^{\square}$, ce qui est loisible (et l’on peut même demander que chaque $\Omega_i$ satisfasse les conclusions de la proposition \[conskliu\]), et décrêter que $\mathcal{U}$ est un recouvrement admissible de $S_d^{\square,0}$. Pour chaque telle structure l’inclusion $S_d^{\square,0} \rightarrow S_d^{\square}$ est une immersion ouverte.) On a en revanche une notion canonique de fonction [*localement analytique $f: S_d^{\square,0} \rightarrow Y$*]{} vers un espace analytique $Y$ quelconque : nous entendrons par là une application telle que pour chaque $x \in S_d^{\square,0}$ la restriction de $f$ a un voisinage affinoïde suffisament petit de $x$ est une fonction rigide analytique (noter que $S_d^\square$ est réduit). Il résulte de la proposition ci-dessus que les applications $x \mapsto {\rm trace}(g | V_x)$, pour $g \in {\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$, sont localement analytiques sur $S_d^{\square,0}$, et simultanément analytiques sur un voisinage de chaque $x$, ce qui implique le : Soit $\mathcal{X}_d$ la variété des caractères $p$-adiques de dimension $d$ de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$. Il existe une unique application localement analytique $$f : S_d^{\square, 0} \rightarrow \mathcal{X}_d$$ associant à tout $x \in S_d^{\square,0}$ la semi-simplification de la représentation $V_x$. L’image de cette application est la fougère infinie régulière. Nous renvoyons à [@chdet] pour la définition de $\mathcal{X}_d$. Pour le lecteur peu familier avec cette théorie, donnons une version plus concrète concernant la ${{\bar\rho}}$-composante connexe $\mathcal{X}_d({{\bar\rho}})\simeq X \subset \mathcal{X}_d$ où ${{\bar\rho}}$ est fixée comme dans l’introduction (sans supposer nécessairement ${{\bar\rho}}\not\simeq {{\bar\rho}}(1)$). Pour cela nous allons définir indépendamment l’ensemble $$S_d^{\square}({{\bar\rho}}) \subset S_d^{\square,0}$$ pull-back par l’application $f$ du corrollaire ci-dessus de l’ouvert ${{\mathfrak X}}_d({{\bar\rho}}) \subset {{\mathfrak X}}_d$. Cela nous oblige à quelques sorites et rappels préliminaires sur la notion de [*représentation résiduelle associée à une famille de représentations*]{} et sur la propriété universelle de $X$ (qui rappelons-le est défini comme l’espace analytique associé par Berthelot à la fibre générique de l’anneau de déformation universelle de ${{\bar\rho}}$). Nous renvoyons à [@chdet §3] pour plus de détails. PS. Si $Y$ est un affinoïde, une [*famille de représentations de $G:={\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}}_p)$ paramétrée par $Y$*]{} est la donnée d’un ${\mathcal{O}}(Y)$-module libre de rang fini muni d’une action ${\mathcal{O}}(Y)$-linéaire continue de $G$. Pour $y \in Y$, on note $M_y:=M\otimes_{{\mathcal{O}}(Y)} k(y)$ l’évaluation de $M$ en $y$ et $\overline{M}_y$ la semi-simplifié de la réduction modulo $\pi_{k(y)}$ d’un ${\mathcal{O}}_{k(y)}$-réseau stable par $G$ dans $M_y$ (elle est donc à coefficients dans le corps fini $k_y$ résiduel de $k(y)$). Si $Y$ est connexe, on peut montrer ([@chdet Def. 3.11]) qu’il existe une représentation semi-simple continue $r : G \rightarrow {\mathrm{GL}}_m(\overline{\mathbb{F}}_p)$, et un morphisme d’algèbres $W(\mathbb{F}) \rightarrow {\mathcal{O}}(Y)$ où $\mathbb{F} \subset \overline{\mathbb{F}}_p$ désigne le corps (fini) engendré par les coefficients des polynômes caractéristiques des $r(g)$ pour $g \in G$, tels que pour tout $y \in Y$ alors $$\forall g \in G, \, \, \det(1-T g | \overline{M}_y) = \det(1-T r(g)) \in \mathbb{F}[T].$$ Cette identité a un sens car $\mathbb{F} \subset k_y$ pour tout $y$. Notons que modifier le plongement $W(\mathbb{F}) \rightarrow {\mathcal{O}}(Y)$ par le Frobenius de $W(\mathbb{F})$ revient à appliquer à $r$ le Frobenius sur les coefficients (ou son inverse). À cette indétermination près, un résultat standard dû à Brauer-Nesbitt implique que la classe d’isomorphisme de $r$ est uniquement déterminée par $M$. On dit alors que $r$ est la représentation résiduelle de $M$. C’est un fait relativement formel mais important que [*l’espace analytique $X$ sur ${\mathbb{Q}}_p$ (oubliant la $F$-structure) représente le foncteur de la catégorie des affinoïdes vers les ensembles associant à $Y$ l’ensemble des classes de ${\mathcal{O}}(Y)[G]$-isomorphie de familles de représentations de $G$ paramétrées par $Y$ dont la représentation résiduelle est ${{\bar\rho}}$*]{} ([@chdet §3]).PS. Si $r : G \rightarrow {\mathrm{GL}}_d(\overline{\mathbb{F}}_p)$ est une représentation semi-simple continue quelconque, on définit enfin $$S_d^{\square}(r) \subset S_d^{\square,0}$$ comme l’ensemble des $x \in S_d^{\square,0}$ tels que $V_x$ a pour réduction $r$. \[scholieuniv\] ([et scholie]{}) Pour tout $r$, $S_d^{\square}(r)$ est un ouvert naïf[^20] de $S_d^{\square}$. Si $r={{\bar\rho}}$ et $X$ sont comme dans l’introduction[^21], il existe une unique application localement analytique $$f : S_d^{\square}({{\bar\rho}}) \rightarrow X$$ associant à tout $x \in S_d^{\square}({{\bar\rho}})$ la représentation $V_x$. Plus précisément, elle a la propriété que pour tout $x \in S_d^{\square}({{\bar\rho}})$, il existe $\Omega \subset S_d^{\square}({{\bar\rho}})$ un voisinage ouvert affinoïde assez petit de $x$ dans $S_d^{\square}$ tel que : - $f$ est analytique sur $\Omega$, PS. - Pour tout morphisme $Z \rightarrow \Omega$ avec $Z$ affinoïde, le ${(\varphi,\Gamma)}$-module associé par Berger-Colmez à la famille déduite du morphisme $f : \Omega \rightarrow X$ soit isomorphe au ${(\varphi,\Gamma)}$-module associé au morphisme $Z \rightarrow S_d^\square$ par la propriété universelle de ce dernier. En effet, le premier point résulte de la discussion ci-dessus et de la proposition \[conskliu\] en considérant la composante connexe $\Omega_x$ contenant $x$ du $\Omega$ donné par la proposition. La propriété universelle de $X$ définit alors un unique morphisme analytique $\Omega_x \rightarrow X$ qui n’est autre que l’application de l’énoncé au niveau des points et le corollaire tout entier résulte de la proposition \[conskliu\] (ii). Par définition, $S_d^{\square,0}$ est la réunion disjointe des $S_d^{\square}(r)$ sur l’ensemble des $r : G \rightarrow {\mathrm{GL}}_d(\overline{\mathbb{F}}_p)$ semi-simples continus considérés modulo isomorphisme et action du Frobenius sur les coefficients. En fait, $S_d^{\square}(r)$ est non vide pour tout $r$ : quand $r$ est irréductible cela découle de la proposition \[crislift\] (i) ci-dessous, le cas général s’en déduit en considérant des sommes directes. On verra même plus tard que $S_d^{\square}(r)$ contient des points cristallins absolument irréductibles. Démonstration du théorème $A$ ============================= Reprenons les notations de l’introduction. Premières réductions {#reductions} -------------------- Commençons la démonstration du théorème $A$. Si $L$ est une extension finie de ${\mathbb{Q}}_p$ et si $V$ est une $L$-représentation cristalline de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$, on dira que $V$ est [*générique*]{} si les conditions suivantes sont satisfaites ([@chU3 §3.18]) : - $V$ a $d:=\dim_L(V)$ poids de Hodge-Tate[^22] distincts. PS. - Les valeurs propres $\varphi_1,\dots,\varphi_d$ du Frobenius cristallin de ${D_{\rm cris}}(V)$ dans $\overline{L}$ satisfont $\varphi_i\varphi_j^{-1} \notin p^{\mathbb{Z}}$ pour tout $i\neq j$.PS. - Si $S \subset {D_{\rm cris}}(V)$ est un sous-$L$-espace vectoriel $\varphi$-stable, alors $S$ est en somme directe avec le sous-espace de la filtration de Hodge de ${D_{\rm cris}}(V)$ dont la dimension est $d-\dim_L(S)$. On dira aussi que [*$V$ est déployée sur $L$*]{} si les valeurs propres du Frobenius cristallin de ${D_{\rm cris}}(V)$ dans $\overline{L}$ sont toutes dans $L$.PS. L’ingrédient principal de la démonstration est alors le suivant. Notons $T_y(Y)$ l’espace tangent de Zariski au point $y$ de l’espace rigide $Y$ (c’est un espace vectoriel de dimension finie sur le corps résiduel $k(y)$). Pour une extension finie $L$ de ${\mathbb{Q}}_p$, on note aussi $T_y(Y)$ l’espace $T_{|y|}(Y) \otimes_{k(y)} L$, $|y| \in Y$ désignant le point fermé sous-jacent à $y$ et $k(y) \rightarrow L$ l’application structurale de $y$. \[tggen\] Soient $x \in X(L)$ cristallin générique déployé sur $L$, $U \subset X$ un voisinage ouvert affinoïde de $x$ dans $X$, et $W \subset U$ l’adhérence Zariski des points cristallins de $U(L)$. Alors $\dim_L T_x(W)=d^2+1$. Nous reportons la preuve de cette proposition au § \[secfin\]. Le second ingrédient est le suivant. \[existcrisgen\]\[crislift\] - Il existe une extension $L/F$ de degré $\leq d^2$ telle que $X(L)$ contienne des points cristallins déployés sur $L$ et dont tous les poids de Hodge-Tate sont disctincts.PS. - Si $x \in X(L)$ est cristallin déployé sur $L$, et si $\Omega \subset X$ est un ouvert affinoïde contenant $x$, alors $\Omega(L)$ contient un point cristallin générique et déployé sur $L$. Montrons le (i). Il est bien connu que si $F_d=W({{\mathbb{F}}}_{p^d})[1/p]$ est l’extension non ramifiée de degrée $d$ de ${\mathbb{Q}}_p$, il existe un caractère continu $\eta : G_{F_d} \rightarrow {\overline{\mathbb{F}}_p}^\ast$ tel que $${{\bar\rho}}\simeq {\rm Ind}_{G_{F_d}}^{G_{{\mathbb{Q}}_p}} \eta,$$ par irréductibilité absolue de ${{\bar\rho}}$. Soit $\Sigma:={\rm Hom}_{\rm corps}({{\mathbb{F}}}_{p^d},{\overline{\mathbb{F}}_p})$, cet ensemble à $d$ éléments s’identifie canoniquement par réduction modulo $p$ à ${\rm Hom}_{\rm corps}(F_d,W({\overline{\mathbb{F}}_p})[1/p])={\rm Hom}_{\rm ann}({\mathcal{O}}_{F_d},W({\overline{\mathbb{F}}_p}))$. Si ${\rm rec} : \widehat{F_d^\ast} \rightarrow G_{F_d}^{\rm ab}$ est l’isomorphisme du corps de classe local, alors $\eta \,\,{\rm o}\,\, {\rm rec}$ est nécessairement (1) trivial sur $1+p{\mathcal{O}}_{F_d}$, (2) de la forme $x \mapsto \prod_{\sigma \in \Sigma} \sigma(x)^{a_\sigma}$ pour certains entiers uniques $a_\sigma \in \{0,1,\dots,p-1\}$ sur ${\mathcal{O}}_{F_d}^\ast$, et (3) envoie $p$ sur un certain element $\overline{\lambda} \in {\overline{\mathbb{F}}_p}^\ast$. Notons aussi que $$\det(X-{{\bar\rho}}_{|G_{F_d}^{\rm ab}}(p))=(X-\overline{\lambda})^d \in \mathbb{F}[X]$$ assure que $\overline{\lambda} \in \mathbb{F}$.PS. Il vient que pour toute collection d’entiers $\{b_\sigma, \sigma \in \Sigma\}$ telle que $b_\sigma \equiv a_\sigma \bmod p^d$ pour tout $\sigma$, et pour tout $\lambda \in {\mathcal{O}}_F^\ast$ tel que $\overline{\lambda}\equiv \lambda \bmod p$, le caractère $\widetilde{\eta} : G_{F_d} \rightarrow (F_d \cdot F)^\ast$ défini par la formule $$\forall u \in {\mathcal{O}}_{F_d}^\ast, \widetilde{\eta} \,\,{\rm o}\,\, {\rm rec}(p u)=\lambda \prod_{\sigma \in \Sigma} \sigma(u)^{b_\sigma}$$ relève $\eta$, et donc que $V:={\rm Ind}_{G_{F_d}}^{G_{{\mathbb{Q}}_p}} \widetilde{\eta}$ relève ${{\bar\rho}}$. Enfin, $V$ est cristalline car $F_d$ est non ramifiée et $\eta$ est cristallin d’après un résultat de Fontaine : les poids de Hodge-Tate de $V$ sont les $b_\sigma$ et le polynôme caractéristique du Frobenius cristallin sur ${D_{\rm cris}}(V)$ est $X^d-\lambda p^{\sum_\sigma a_\sigma}$. Le (i) s’en déduit. PS. Le (ii) est une conséquence des résultats de Kisin dans [@kisinJAMS §3], expliquons comment. Soit $x \in X(L)$ un point cristallin à poids de Hodge-Tate distincts ${{\bf k}}=(k_1 < \dots < k_d)$. Si $V_R$ désigne la déformation universelle de ${{\bar\rho}}$ (l’anneau $R$ étant comme dans l’introduction), Kisin démontre l’existence d’un quotient $$R[1/p] \rightarrow R_{{{\bf k}}}$$ dont les points dans toute $F$-algèbre de dimension finie $B$ paramètrent exactement les morphismes $R[1/p] \rightarrow B$ tels que $V_R \otimes_R B$ est cristalline de poids de Hodge-Tate $k_1 < \dots < k_d$ ([@kisinJAMS Thm. 2.5.5]). Notons $X_{{\bf k}}\subset X$ le fermé analytique de $X$ défini par l’idéal noyau de $R[1/p] \rightarrow R_{{{\bf k}}}$. D’après Kisin [@kisinJAMS Thm. 3.3.4], $X_{{\bf k}}$ est lisse de dimension $\frac{d(d-1)}{2}+1$. Rappelons que c’est un fait général que l’application naturelle $R_{{\bf k}}\rightarrow {\mathcal{O}}(X_{{\bf k}})$ induit une bijection $\psi : X_{{\bf k}}\longrightarrow {\rm Specmax}(R_{{{\bf k}}})$ et des isomorphismes $\widehat{(R_{{\bf k}})}_{\psi(x)} \rightarrow {\mathcal{O}}_{X_{{\bf k}},x}$ sur les annaux locaux complétés pour tout $x \in X_{{\bf k}}$. Le (ii) découle du lemme suivant. L’ensemble des points $x \in X_{{{\bf k}}}(L)$ tels que $\rho_x$ est générique est dense dans $X_{{\bf k}}(L)$. De plus, si $x \in X_{{\bf k}}(L)$ est déployé sur $L$, il existe un ouvert affinoïde $O$ de $x$ dans $X_{{\bf k}}$ tel que $O(L)$ est constitué de points déployés sur $L$. En effet, Kisin démontre [*loc. cit.*]{} l’existence d’un $\varphi$-module filtré $D$ localement libre de rang $d$ sur $R_{{{\bf k}}}$ tel que pour tout quotient artinien $R_{{{\bf k}}} \rightarrow B$, $D_{\rm cris}(V_R \otimes_R B)$ est isomorphe à $D \otimes_{R_{{{\bf k}}}} B$, ce qui détermine donc $V_B$ par l’équivalence de Fontaine. PS. Soient $P_\varphi \in R_{{\bf k}}[T]$ le poynôme caractéristique de $\varphi$ sur $D$ et ${{\mathcal{F}}}$ la variété des drapeaux complets de $F^d$, disons vue comme variété $F$-analytique. Remarquons déjà que l’existence même de $P_\varphi$, ainsi que le lemme de Krasner, impliquent la seconde partie du lemme, concentrons nous donc sur la première.PS. Notons que ${\mathrm{GL}}_d \times {{\mathcal{F}}}$ est muni d’une action de ${{\mathrm{PGL}}}_d$ définie sur les points affinoïdes par $g\cdot (\varphi,{\bf F})=(g \varphi g^{-1},g({\bf F}))$ (strictement il faudrait insérer des recouvrements pffp ou même Zariski). Soient $x \in X_{{\bf k}}(L)$ et $\Omega$ un voisinage ouvert affinoïde de $x$ dans $X_{{\bf k}}$ assez petit de sorte que $D \otimes_{R_{{\bf k}}} {\mathcal{O}}(\Omega)$ soit libre de rang $d$ sur ${\mathcal{O}}(\Omega)$. Le choix d’une base de ce dernier définit alors un morphisme analytique $\Omega \rightarrow {\mathrm{GL}}_d \times {{\mathcal{F}}}$ associé à la matrice de $\varphi$ dans cette base et à la donnée de la filtration de $D$. Il sera commode de le modifier un peu en $$\mu : {{\mathrm{PGL}}}_d \times \Omega \rightarrow {\mathrm{GL}}_d \times {{\mathcal{F}}}$$ défini sur les points par $(g,x) \mapsto g\cdot (\varphi(x),{\bf F}(x))$. La propriété de $D$ vis-à-vis des $F$-algèbres artiniennes locales $B$ énoncée plus haut a plusieurs conséquences. Appliquée aux $B$ qui sont des corps elle entraîne que $\mu$ est injective. Si de plus un point $z \in {{\mathrm{PGL}}}_d \times \Omega$ est fixé, et appliquée aux épaississement infinitésimaux de $z$, elle entraîne que $$\mu_z^{\ast} : \widehat{{\mathcal{O}}}_{{\mathrm{GL}}_d \times {{\mathcal{F}}},\mu(z)} \rightarrow \widehat{{\mathcal{O}}}_{{{\mathrm{PGL}}}_d\times\Omega,z}$$ est un isomorphisme. En effet, c’est un fait général que si $\theta : A_1 \rightarrow A_2$ est un $F$-morphisme local entre des $F$-algèbres locales noethériennes complètes telles que pour toute $F$-algèbre locale de dimension finie $B$ l’application naturelle $A_2(B) \rightarrow A_1(B)$ est bijective, alors $\theta$ est un isomorphisme. Ce critère se vérifie bien ici car d’une part pour tout tel $B$-point de $X_{{\bf k}}$, les automorphismes de $V_R \otimes B$, ou ce qui revient au même de $D \otimes_{R_{{{\bf k}}}}B$, sont réduits aux scalaires $B^\times$, et d’autre part ${{\mathrm{PGL}}}_d(B)={\mathrm{GL}}_d(B)/B^\ast$ car ${\rm Pic}(B)=0$. Ainsi, $\mu$ est injectif et un isomorphisme local formel en tout point de sa source. En particulier, le morphisme $\mu$ est plat. Comme sa source et son but sont quasi-compacts, un résultat de Bosch-Lütckebohmert (existence de modèles formels plats) assure que $\mu(\Omega) \subset {\mathrm{GL}}_d \times {{\mathcal{F}}}$ est un ouvert admissible (quasi-compact). Comme $\mu$ induit des isomorphismes en chaque point sur les corps résiduels, suffit de voir pour conclure que si $$(\varphi,{\bf F}) \in {\mathrm{GL}}_d(L) \times {{\mathcal{F}}}(L)$$ est donné, et si $U$ en est un voisinage ouvert pour la topologie $p$-adique, alors $U$ contient une paire $(\varphi',{\bf F}')$ où $\varphi'$ est semi-simple à valeurs propres $\lambda_i \in \overline{L}$ distinctes, telles que $\lambda_i/\lambda_j \notin p^{\mathbb{Z}}$ pour $i\neq j$, et dont les espaces propres dans $\overline{L}^d$ sont en position générale par rapport à ${\bf F'}$. Choisir un $\varphi'$ proche de $\varphi$ satisfaisant les deux premières conditions est immédiat, fixons le. Soit $V \subset {{\mathcal{F}}}(L)$ un voisinage ouvert assez petit de ${\bf F}$ de sorte que que $\{\varphi'\} \times V \subset U$. Notons que $V$ est dense dans ${{\mathcal{F}}}(\overline{L})$ pour la topologie de Zariski de ce dernier (qui est irréductible). Il suffit pour conclure de remarquer que l’ensemble des drapeaux complets de $\overline{L}^d$ qui sont en position générale avec un nombre fini fixé de drapeaux complets forment un ouvert Zariski : c’est standard pour un drapeau (“grosse cellule de Bruhat”), et le cas général s’en déduit par intersection finie. Remarquons que les relevements cristallins exhibés au (i) ci-dessus ne sont pas Zariski-denses dans $X$, car ils restent dans le sous-espace fermé des représentations induites d’un caractère de $F_d$, dont on vérifie facilement qu’il est de dimension $d < d^2+1$. PS. Démontrons enfin le théorème $A$ de l’introduction. Fixons une fois pour toutes un point cristallin $x \in X(L)$ déployé sur $L$ donné par la proposition \[existcrisgen\] (i), ainsi qu’un ouvert affinoïde $U \subset X$ connexe et contenant $x$. Notons que $X$ étant normal, $U$ est irréductible. Soit $W \subset U$ le fermé analytique réduit obtenu en prenant l’adhérence Zariski des points cristallins de $U(L)$. Les affinoïdes étant excellents et de Jacobson (voir [@conradirr §1]), le lieu régulier de $W$ est un ouvert Zariski qui est dense dans $W$. En particulier, il existe des points cristallins $y \in U(L)$ qui sont réguliers dans $W$. Quitte à remplacer $x$ par un tel point, nous pouvons donc supposer que $x$ est régulier dans $W$. Appliquant la proposition \[existcrisgen\] (ii) à un voisinage ouvert affinoïde $\Omega$ de $x$ dans $U$ qui soit assez petit de sorte que $\Omega \cap W$ soit régulier, on peut finalement supposer que $x \in W(L)$ est cristallin générique, déployé sur $L$, et régulier dans $W$. La proposition \[tggen\] assure alors que $\dim W = \dim_F T_x(W) = d^2+1 = \dim X$, et donc que $W$ contient un ouvert de $X$ contenant $x$. Comme $U$ est irréductible, cet ouvert est Zariski-dense dans $U$, ce qui conclut la preuve du théorème.PS. Preuve de la proposition \[tggen\]. {#secfin} ----------------------------------- Fixons $L$ une extension finie de ${\mathbb{Q}}_p$ et $x \in X(L)$ tels que $$V_0:=\rho_x$$ est cristalline générique et déployée sur $L$. On notera $k_1<\dots<k_d$ les poids de Hodge-Tate de $V_0$ rangés par ordre croissant. (Nous prenons pour convention que ${\mathbb{Q}}_p(1)$ a pour poids de Hodge-Tate $-1$.) On rappelle que le ${(\varphi,\Gamma)}$-module $$D_0:={D_{\rm rig}}(V_0)$$ est triangulin sur ${\mathcal{R}}_L$ d’exactement $d!$ manières différentes. Plus précisément, fixons ${{\mathcal}{F}}$ un [*raffinement*]{} de $V_0$, c’est à dire un drapeau complet $L[\varphi]$-stable dans ${D_{\rm cris}}(V_0)$ $${{\mathcal}{F}}=\left({{\mathcal}{F}}_0=\{0\} \subsetneq {{\mathcal}{F}}_1 \subsetneq \cdots \subsetneq {{\mathcal}{F}}_d={D_{\rm cris}}(V_0)\right).$$ Il est équivalent ici de se donner un ordre $\varphi_1,\dots,\varphi_d$ sur les valeurs propres de $\varphi$ sur ${D_{\rm cris}}(V_0)$, $\varphi$ agissant sur ${{\mathcal}{F}}_i/{{\mathcal}{F}}_{i-1}$ par la multiplication par $\varphi_i$. À chaque tel ${{\mathcal}{F}}$ est associée une triangulation $$\mathcal{T}=({{\rm Fil}}_i(D_0))$$ de $D_0$ définie comme suit. Nous avons déjà dit que d’après Berger le $L[\varphi]$-module $(D_0[1/t])^{\Gamma}$ est canoniquement isomorphe à ${D_{\rm cris}}(V_0)$, on pose alors $${{\rm Fil}}_i(D_0)=({\mathcal{R}}_L[1/t].{{\mathcal}{F}}_i)\cap D_0.$$ L’application ${{\mathcal}{F}}\mapsto {\mathcal{T}}$ induit alors la bijection cherchée entre raffinements de $V_0$ et triangulations de $D_0$, d’après [@bch §2.4]. Dans cette bijection, le paramètre $(\delta_i)$ de $\mathcal{T}$ est relié à ${{\mathcal}{F}}$ par les formules $$\delta_i(p)=\varphi_i p^{-k_i}, \, \, \, \delta_i(\gamma)=\gamma^{-k_i} \, \,\, \,\forall \gamma \in {\mathbb{Z}}_p^\ast,$$ (cela découle de [@bch prop. 2.4.1] et de l’hypothèse (G3) de généricité). PS. Fixons ${{\mathcal}{F}}$ un raffinement de $V_0$ et considérons le ${(\varphi,\Gamma)}$-module triangulin sur ${\mathcal{R}}_L$ associé $(D_0,{{\mathcal T}})$. L’hypothèse (G2) de généricité entraîne qu’il est régulier. Fixons enfin une rigidification $\nu_0$ de $(D_0,{{\mathcal T}})$, ce qui nous fournit donc un point $$x_0 =(D_0,{{\mathcal T}},\nu_0) \in S_d^\square(L).$$ Soient $\Omega \subset S_d^{\square}$ un voisinage ouvert affinoïde de $x_0$ et $$f : \Omega \rightarrow X$$ comme dans le corollaire \[scholieuniv\]. Quitte à remplacer $\Omega$ par un voisinage plus petit, on peut supposer d’une part que $f(\Omega) \subset U$ (l’ouvert $U$ étant celui de l’énoncé de la proposition \[tggen\]), et d’autre part que les points cristallins sont Zariski-dense dans $\Omega(L)$ d’après le théorème \[densecrisdanstri\]. Si $W$ est l’adhérence Zariski des points cristallins de $U(L)$, l’analyticité de $f$ entraîne que $$\label{inclfond} f(\Omega) \subset W.$$ D’autre part, si $L[\varepsilon]$ désigne les nombres duaux sur $L$ (de sorte que $\varepsilon^2=0$) alors la fonction analytique $f$ induit une application tangente (voir la fin du § \[diverstrucs\]) $$df_{x_0} : \widehat{(F_d^\square)}_{x_0}(L[\varepsilon]) \longrightarrow T_x(X)$$ qui n’est autre, d’après le corollaire \[scholieuniv\] appliqué aux morphismes ${\rm Sp}(L[\varepsilon]) \rightarrow \Omega$ d’image $x_0$, où ce qui revient au même à tous les morphismes ${\rm Sp}(L[\varepsilon]) \rightarrow S_d^\square$ d’image $x_0$ car $\Omega \subset S_d^\square$ est ouvert, que l’application associant à un $c=[(D,{{\rm Fil}}_\bullet,\nu)]$ dans $F_d^\square(L[\varepsilon])$ et d’image $[x_0]$ dans $F_d^\square(L)$ l’unique déformation de $V_0$ sur $L[\varepsilon]$ dont le ${D_{\rm rig}}$ est isomorphe à $D$. Il vient que $$T_{V_0,{{\mathcal}{F}}}:={\rm Im}(df_{x_0}) \subset T_x(X)$$ est exactement le sous-espace des déformations ${{\mathcal}{F}}$-triangulines de $V_0$ au sens de [@bch §2.5] et [@chU3 §3]. En effet, $df_{x_0}$ se factorise évidemment par le morphisme d’oubli $$\widehat{(F_d^\square)}_{x_0}(L[\varepsilon]) \rightarrow \widehat{(F_d)}_{[(D_0,{{\mathcal T}})]}(L[\varepsilon]),$$ et $V_0$ étant générique l’algèbre ${\mathrm{End}}_{{(\varphi,\Gamma)}/L}(D_0)$ est réduite aux homothéties d’après [@chU3 lemma 3.21], de sorte que $\widehat{(F_d)}_{[(D_0,{{\mathcal T}}_0)]}$ est le foncteur des déformations triangulines de $(D_0,{{\mathcal T}})$ d’après le lemme \[sousfonct\]. La relation , combinée au fait que $W$ est fermé et $\Omega$ réduit, assure donc que $$T_{V_0,{{\mathcal}{F}}} \subset T_x(W)$$ pour tout raffinement ${{\mathcal}{F}}$ de $V_0$. On conclut alors par le résultat crucial suivant, démontré dans [@chU3] : “Toute déformation à l’ordre $1$ d’une représentation cristalline générique est combinaison linéaire de déformations triangulines”. [([@chU3 Thm. C]) ]{} $T_x(X) = \sum_{{{\mathcal}{F}}} T_{V_0,{{\mathcal}{F}}}$. Une généralisation ------------------ Terminons par un énoncé plus général que nous démontrons par la même méthode. Soit $d\geq 1$ un entier. Nous renvoyons à [@chdet] pour la notion de variété des caractères $p$-adiques de dimension $d$ d’un groupe profini $G$. Disons simplement ici que si l’on suppose que tout sous-groupe ouvert $U \subset G$ a la propriété que ${\mathrm{Hom}}(U,{{\mathbb F}}_p)| < \infty$, alors le foncteur ${\rm Aff} \rightarrow {\rm Ens}$ associant à $A$ l’ensemble des pseudo-caractères continus $G \rightarrow A$ de dimension $d$ est représentable par un espace analytique $p$-adique ${{\mathfrak X}}$ : c’est la variété des caractères $p$-adiques de $G$ de dimension $d$. En particulier, les points de cette variété sont en bijection avec les classes de conjugaison de représentations continues semi-simples $G \rightarrow GL_d({\overline{\mathbb{Q}}_p})$. Dans le cas particulier $G={\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$ on peut démontrer que ${{\mathfrak X}}$ est équidimensionnel de dimension $d^2+1$, et que son lieu singulier coïncide exactement avec le lieu des représentations réductibles (avec une exception quand $d=2$) : voir [@chnp]. Des arguments similaires à ceux de la preuve du théorème $A$ démontrent : Soit ${{\mathfrak X}}_d$ la variété des caractères $p$-adiques de dimension $d$ de ${\rm Gal}({\overline{\mathbb{Q}}_p}/{\mathbb{Q}_p})$. Soit $Y$ une composante irréductible de ${{\mathfrak X}}_d$ et $L$ une extension finie de ${\mathbb{Q}}_p$ telle que $Y(L)$ contiennent un point cristallin déployé absolument irréductible. Alors les points cristallins déployés de $Y(L)$ sont Zariski-dense et d’accumulation dans $Y$. Il semble raisonnable de formuler la conjecture suivante, que l’on peut voir comme un analogue local de la conjecture de modularité de Serre. \[serrelocal\] Toute composante irréductible de ${{\mathfrak X}}_d$ contient un point cristallin absolument irréductible. Si la conjecture \[serrelocal\] est vraie, alors les points cristallins sont Zariski-dense dans ${{\mathfrak X}}_d$. L’application “représentation résiduelle” réalise l’espace ${{\mathfrak X}}$ comme réunion disjointe admissible d’espaces ${{\mathfrak X}}({{\bar\rho}})$ indexés par l’ensemble des représentations continues semi-simples ${{\bar\rho}}: G_{{\mathbb{Q}}_p} \longrightarrow {\mathrm{GL}}_n({\overline{\mathbb{F}}_p})$ prises modulo isomorphisme et action du Frobenius sur l’image. Lorsque ${{\bar\rho}}$ est absolument irréductible, la composante ${{\mathfrak X}}({{\bar\rho}})$ est simplement l’espace défini dans l’introduction. Quand ${{\bar\rho}}\not \simeq {{\bar\rho}}(1)$ il est irréductible (c’est une boule!) et la conjecture ci-dessus est facile : c’est la proposition \[existcrisgen\] (i). À défaut de pouvoir proposer une argument plausible pour la conjecture ci-dessus, démontrons le résultat suivant. Il assure que pour tout ${{\bar\rho}}$, au moins une (de l’ensemble fini) des composantes irréductibles de ${{\mathfrak X}}({{\bar\rho}})$ contient un point cristallin absolument irréductible. \[liftgeneral\] Soit ${{\bar\rho}}: G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d({{\mathbb F}}_q)$ une représentation semi-simple continue. Il existe une extension finie $L/{\mathbb{Q}}_p$ et une représentation $G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d(L)$ cristalline absolument irréductible dont la représentation résiduelle est ${{\bar\rho}}$. Avant de procéder à la démonstration, il est raisonnable de commencer par la proposition plus simple suivante, qui est une application déjà frappante des résultats de cet article ‘a l’existence de représentations cristallines ayant certaines propriétés. La seconde partie est à comparer avec des observations antérieures de l’auteur et Bellaïche (voir par exemple [@bch §4] ou encore [@bchsign lemme 3.3]). Dans cet énoncé, $r$ est une représentation semi-simple quelconque et $L/{\mathbb{Q}}_p$ une extension finie de ${\mathbb{Q}}_p$. \[famcolemandeux\] Soient $x \in S_d^{\square}(r)(L)$ tel que $\delta(x) \in B_d(L)$ et $U \subset S_d^\square(r)$ un voisinage de $x$. Pour tout réel $C>0$, il existe un ouvert affinoïde $V \subset U$ et un ensemble Zariski-dense de $y \in V(L)$ tels que la représentation $V_y$ soit cristalline et tels que $\delta(y)=(\delta_i) \in {{\mathcal T}}(L)^d$ ait les propriétés suivantes : - il existe une suite strictement croissante d’entiers $k_1,\dots,k_d$ telle que $\delta_i(\gamma)=\gamma^{-k_i}$ pour tout $i$ et telle que pour toute paire de parties distinctes $I,J \subset \{1,\dots,d\}$ avec $1\leq |I|=|J| < d$, on ait $|\sum_{i \in I} k_i -\sum_{j \in J} k_j| > C$. - pour tout $i$, $v(\delta_i(p))=v(\delta(x)_i(p))$, On peut de plus supposer que les $V_y$ sont absolument irréductibles dans chacun des cas suivants: - $V_x$ est absolument irréductible, PS. - $x$ a la propriété que pour toute partie $I \subset \{1,\dots,n\}$ telle que $1\leq |I| < n$ on ait $\sum_{i \in I} v(\delta(x)_i(p)) \neq 0$. Quitte à rétrécir $U$ on peut supposer qu’il est dans l’overt affinoïde donné par la proposition \[conskliu\], en particulier $U \subset S_d^\square(r)$. Considérons un voisinage $U$ de $x$, ainsi que $\iota$ et $\Omega$ comme dans le corollaire \[corfamcoleman\]. Quitte à rétrécir $\Omega$ on peut supposer que les $\delta_i(p)$, vues comme fonctions analytiques sur $U$, sont chacune de valuation constante. L’existence de $y$ satisfaisant (a) et (b) résulte alors de la proposition \[critcris\] et de ce que l’ensemble des suites croissantes $(k_1,\dots,k_d)$ vues comme éléments de ${\mathrm{Hom}}({\mathbb{Z}}_p^\times,L)^d$ qui satisfont la condition la condition (b) (pour un $C$ fixé) est Zariski-dense dans ${\mathrm{Hom}}({\mathbb{Z}}_p^\times,\mathbb{G}_m)^d$ et s’accumule en tous les points de ${\mathbb{Z}}^d$. PS. Pour le second point, il découle dans le cas (i) de la proposition \[conskliu\] et du fait classique que si $\rho : G \rightarrow {\mathrm{GL}}_d(A)$ est une représentation d’un groupe $G$ à valeurs dans un anneau commutatif $A$ telle que pour un $x \in {\rm Spec}(A)$ la représentation $\rho_x : G \rightarrow {\mathrm{GL}}_d(k(x))$ évaluée en $x$ est absolument irréductible, alors il en va de même pour tout $\rho_y$ pour $y$ dans un voisinage ouvert de $x$ dans ${\rm Spec}(A)$ : en effet, il existe $g_1,\dots g_{d^2} \in G$ tels que $\det({\rm trace}(\rho(g_ig_j))) \in A_x^\times$ (Wedderburn) et donc dans $A_f^\times$ pour $x \in D(f)$ et $f$ bien choisie.PS. Dans le cas (ii), il suffit de voir que le $D_{\rm cris}(V_y)$ n’a pas de sous-$\varphi$-module filtré admissible non trivial. Mais si on a un tel sous-module, disons de rang $1 \leq r < d$, l’égalité des extrémités de ses polygones de Hodge et Newton implique qu’il existe deux parties $I,J \subset \{1,\dots,d\}$ avec $|I|=|J|=r$ telles que $$\sum_{i \in I}k_i = \sum_{j \in J} v(\varphi_j)$$ où $\varphi_1,\dots,\varphi_d \in L^\times$ désignent les valeurs propres du Frobenius cristallin. Mais par construction (et quitte à renuméroter les $\varphi_i$), on a $\delta_i(p)=\varphi_i p^{-k_i}$ pour tout $i$, de sorte que l’égalité ci-dessus s’écrive aussi $$\sum_{i \in I} v(\delta_i(p)) = \sum_{i \in I} k_i - \sum_{j \in J} k_j.$$ Mais par le (b), le terme de gauche est aussi $\sum_{i \in I} v(\delta(x)_i(p))$, qui est un nombre fixé disons $C'$. Ainsi, si on choisit $y$ de sorte que le (a) est vérifié pour $C > C'$, il vient que $I=J$, et on obtient la contradiction voulue. (de la proposition \[liftgeneral\]) Quitte à grossir ${{\mathbb F}}_q$ on peut supposer que ${{\bar\rho}}=\oplus_{i=1}^s {{\bar\rho}}_i$ où chaque ${{\bar\rho}}_i$ est absolument irréductible. Nous allons raisonner par récurrence sur le nombre $s$ de facteurs, le cas $s=1$ résultant de la proposition \[existcrisgen\] (i). Pour $s>1$ on peut donc trouver des représentation $L$-cristallines $V_1$ et $V_2$ de représentations résiduelles respectives ${{\bar\rho}}_1$ et $\oplus_{i=2}^r {{\bar\rho}}_i$. On pose $$a=\dim_L V_1, \,\,\,\,\,b=\dim_L V_2.$$ On note $k_1 \leq k_2 \leq \dots k_a$ (resp. $k_{a+1} \leq k_{a+2} \leq \dots \leq k_d$) les poids de Hodge-Tate de $V_1$ (resp. $V_2$) et $\varphi_1,\dots,\varphi_a$ (resp. $\varphi_{a+1},\dots,\varphi_{d}$) les valeurs propres du Frobenius cristallin de $V_1$ (resp. $V_2$). Quitte à appliquer la proposition ci-dessus (plus exactement, sa démonstration) à $V_1$ et $V_2$, munis de leur raffinements respectifs associés à $(\varphi_1,\dots,\varphi_a)$ et $(\varphi_{a+1},\dots,\varphi_d)$ on peut supposer que $(k_i)$ est strictement croissante et que si $C={\rm Sup}_i |v(\varphi_i)-k_i|$ alors pour toute paire de parties $I \neq J \subset \{1,\dots,d\}$ avec $|I|=|J| < d$ on ait $$\label{eqsomme} |\sum_{i \in I} k_i-\sum_{j \in J} k_j | > dC.$$ En particulier, les $\varphi_i$ sont distincts. Considérons une extension non triviale cristalline $$0 \longrightarrow V_1 \longrightarrow V \longrightarrow V_2 \longrightarrow 0.$$ Il y a plusieurs façons de voir qu’une telle extension existe, cela vient par exemple de ce que $H^1_f(G_{{\mathbb{Q}_p}},V_1 \otimes_L V_2^\vee)$ est de dimension au moins égale au nombre de poids de Hodge-Tate strictement négatifs de $V_1 \otimes_L V_2^\vee$ par un résultat classique de Bloch-Kato (ici ces poids sont les $k_i - k_j$ avec $i\leq a$ et $j > a$, qui sont en fait tous strictement négatifs). Considérons la permutation $\sigma \in \got{S}_d$ définie par $\sigma(i)=i+a$ si $i=1,\dots,b$ et $\sigma(i+b)=i$ si $i=1,\dots, a$, et considérons le raffinement $$(\varphi_{\sigma(1)},\varphi_{\sigma(2)},\dots,\varphi_{\sigma(d)})$$ de la représentation $V$. Soit $\tau \in \got{S}_d$ la permutation telle que $(k_{\tau(1)},k_{\tau(2)},\dots,k_{\tau(d)})$ est la suite des sauts de la filtration de Hodge de $D_{\rm cris}(V)$ définie par ${{\mathcal}{F}}$ (voir [@bch §2.4]). Il nous suffira ici de dire que si $(D,{{\mathcal T}})$ est le ${(\varphi,\Gamma)}$-module triangulin associé à $(V,{{\mathcal}{F}})$, alors ses paramètres $\delta_i$ vérifient $$\delta_i(p)=\varphi_{\sigma(i)}p^{-k_{\tau(i)}}, \, \, \delta_i(\gamma)=\gamma_i^{-k_{\tau(i)}} \,\, \,\,\,\forall \gamma \in {\mathbb{Z}}_p^\times.$$ Soit $D' \subset D$ l’unique sous-${(\varphi,\Gamma)}$-module cristallin saturé dont les valeurs propres du Frobenius cristallin sont les $\varphi_i$ avec $a+1 \leq i \leq b$. L’application naturelle $\eta : D' \rightarrow D_{\rm rig}(V_2)$ est injective (c’est un isomorphisme sur les ${D_{\rm cris}}$) et on constate que l’on a l’inégalité pour l’ordre lexico-graphique $$(\tau(1),\tau(2),\dots,\tau(b)) \leq (a+1,a+2,\dots,d).$$ Cette inégalité est une égalité si et seulement si $\eta$ est surjective, ce qui ne se produit pas car $V$ est non scindée. C’est donc une inégalité stricte, de sorte qu’il existe un $1 \leq i_0 \leq b$ tel que $\tau(i_0) \leq a$, et en particulier $$\tau(i_0) \leq a < \sigma(i_0).$$ PS. Nous allons appliquer une variante de la proposition précédente cas (i) et (ii) au point $x=(D,{{\mathcal T}},\nu)$ pour un choix quelconque de $\nu$. Cette proposition permettrait de conclure si nous savions que pour toute partie $I \subset \{1,\dots,d\}$ telle que $1\leq |I| \leq d-1$ alors $\sum_{i \in I} v(\delta_i(p)) \neq 0$. Cela n’est pas toujours satisfait dans notre cas, mais $U$ n’ayant que deux facteurs irréductibles c’est un exercice de vérifier que si aucune des représentations $V_y$ donnée par cette proposition n’est absolument iréductible (pour $y$ assez proche de $x$), alors elles ont toutes un constituent de dimension $\dim(V_1)=a$. La théorie de Sen en famille, appliquée à la famille donnée par la proposition \[conskliu\], permet d’identifier ses poids de Hodge-Tate : si ceux de $V_y$ rangés en ordre croissant sont les $k'_i$, le (ou un si $a=b$) constituent de dimension $a$ a pour poids de Hodge-Tate les $k'_{\tau^{-1}(i)}$ pour $i=1,\dots,a$. Par conséquent, l’argument de la preuve de la proposition ci-dessus montre qu’il suffit dans notre cas de vérifier que $\sum_{i \in I} v(\delta_i(p)) \neq 0$ quand $I=\tau^{-1}(\{1,\dots,a\})$. Mais $$|\sum_{i \in I} v(\delta_i(p))- (\sum_{i \in \sigma(I)} k_i-\sum_{i \in \tau(I)} k_i)| < dC.$$ Par la relation , il suffit pour conclure de voir que $\sigma(I) \neq \tau(I)$, c’est à dire que $\sigma \tau^{-1} (\{1,\dots,a\}) \neq \{1,\dots,a\}$. On conclut car $\tau(i_0) \leq a$ et $\sigma(i_0) > a$. Notons qu’il ressort de la démonstration que le corps de définition $L$ peut être précisé dans l’énoncé de la proposition \[liftgeneral\]. Par exemple, si ${{\bar\rho}}$ est la représentation triviale, alors on peut choisir $L={\mathbb{Q}}_p$. [9999]{} J. Bellaïche & G. Chenevier, [*Families of Galois representations and Selmer groups*]{}, Astérisque 324, Soc. Math. France (2009). J. Bellaïche & G. Chenevier, [*The sign of Galois representations attached to automorphic forms for unitary groups*]{}, à paraître à Compositio Math. D. Benois, [*A generalization of Greenberg’s $\mathcal{L}$-invariant*]{}, prépublication disponible sur ArXiV (2009). L. Berger, [*Représentations $p$-adiques et équations différentielles*]{}, Invent. Math. 148 No. 2 (2002), 219–284. L. Berger, [*Équations différentielles $p$-adiques et $(\phi,N)$-modules filtrés*]{}, Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi,\Gamma)$-modules. Astérisque No. 319 (2008), 13–38. L. Berger & G. Chenevier, [*Représentations potentiellement triangulines de dimension $2$*]{}, à paraître au Journal de théorie des nombres de Bordeaux. L. Berger & P. Colmez, [*Familles de représentations de de Rham et monodromie p-adique*]{}, Astérisque 319 (2008), 303–337. S. Bosch, U. Güntzer & R. Remmert, [*Non-archimedean analysis*]{}, Grundlehren der mathematischen Wissenschaften, Vol. 261, Springer-Verlag (1984). G. Chenevier, [*Variétés de Hecke des groupes unitaires et représentations galoisiennes*]{}, Cours Peccot, Collège de France (2008), disponible à l’adresse [<http://www.math.polytechnique.fr/~chenevier/courspeccot.html>]{}. G. Chenevier, [*The $p$-adic analytic space of pseudo-characters of a profinite group, and pseudo-representations over arbitrary rings*]{}, prépublication disponible sur ArXiV (2008). G. Chenevier, [*On the infinite fern of Galois representations of unitary type*]{}, prépublication disponible sur ArXiV (2009). G. Chenevier, [*Une application des variétés de Hecke des groupes unitaires*]{}, prépublication (04/2009) disponible à l’adresse [<http://www.math.polytechnique.fr/~chenevier/pub.html>]{}, à paraître dans [*“Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques”*]{} tome 2. G. Chenevier, [*Sur la variété des caractères $p$-adiques de $G_K$ où $K$ est une extension finie de ${\mathbb{Q}}_p$.*]{} Notes disponibles sur demande. R. Coleman, [*$p$-adic Banach spaces & families of modular forms*]{}, Invent. Math. 127 (1997), 417–479. R. Coleman & B. Mazur, [*The eigencurve*]{}, in Galois representations in Arithmetic Algebraic Geometry (Durham, 1996), London. Math. Soc. Lecture Notes [**254**]{}, Cambridge univ. press (1998). P. Colmez, [*Représentations triangulines de dimension 2*]{}, Astérisque 319 (2008), 213–258. P. Colmez, [*Représentations de ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ et ${(\varphi,\Gamma)}$-modules*]{}, Astérisque 330 (2010), 281–509. P. Colmez, [*Invariants $L$ et dérivées de valeurs propres de Frobenius*]{}, Astérisque 331 (2010), 13-28. B. Conrad, [*Irreducible components of rigid spaces*]{}, Ann. Inst. Fourier 49 (1999), 473–541. B. Conrad, [*Relative ampleness in rigid-analytic geometry*]{}, Ann. Inst. Fourier 56 (2006), 1049–1126. L. Herr, [*Sur la cohomologie galoisienne des corps $p$-adiques*]{}, Bull. Soc. Math. France 126 (1998). L. Herr, [*Une approche nouvelle de la dualité locale de Tate*]{}, Math. Ann. 320 (2001). K. Kedlaya, [*Slope filtrations and ${(\varphi,\Gamma)}$-modules in families*]{}, cours au trimestre Galoisien à l’Institut Henri Poincaré, Paris, disponible à l’adresse [<http://math.mit.edu/~kedlaya/papers/families.pdf>]{}. K. Kedlaya & R. Liu, [*On families of ${(\varphi,\Gamma)}$-modules*]{}, à paraître à Algebra and Number Theory (2010), disponible sur ArXiV. M. Kisin, [*Overconvergent modular forms and the Fontaine-Mazur conjecture*]{}, Invent. Math. 153 (2003), 373–454. M. Kisin, [*Potentially semi-stable deformation rings*]{}, J.A.M.S. 21 (2008), 513–546. M. Kisin, [*Deformations of $G_{{\mathbb{Q}}_p}$ and ${\mathrm{GL}}_2({\mathbb{Q}}_p)$ representations*]{}, Astérisque 330 (2010), 511–528. R. Liu, [*Cohomology and duality for $(\phi,\Gamma)$-modules over the Robba ring*]{}, Int. Math. Res. Not. no. 3 (2008). B. Mazur, [*Deforming Galois representations*]{}, Galois groups over ${\mathbb{Q}}$ (Berkeley, CA, 1987), 385–437, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989. B. Mazur, [*An ‘infinite fern’’ in the universal deformation space of Galois representations*]{}, Journées Arithmétiques (Barcelona, 1995). Collect. Math. 48 (1997), no. 1-2, 155–193. K. Nakamura, [*Déformations of trianguline $B$-pairs and Zariski-density of two-dimensional crystalline representations*]{}, prépublication disponible sur ArXiV (2010). P. Schneider, [*Nonarchimedean functional analysis*]{}, Springer Monographs in Mathematics. Springer-Verlag, Berlin (2002). P. Schneider & J. Teitelbaum, [*Algebras of p-adic distributions and admissible representations*]{}, Invent. Math. 153 (2003), no. 1, 145–196. J. Tate, [*Duality theorems in Galois cohomology over number fields*]{}, Proc. Int. Congress Math. Stokholm (1962). [^1]: L’auteur est financé par le C.N.R.S. [^2]: Cela signifie que ${{\bar\rho}}$ est irréductible et le reste après extension des scalaires à ${\overline{\mathbb{F}}_p}$. Rappelons aussi que ${{\bar\rho}}$ est nécessairement définie sur le sous-corps de ${{\mathbb{F}}}_q$ engendré par les coefficients des $\det(t-{{\bar\rho}}(g))$, $g \in G_{{\mathbb{Q}}_p}$ (l’obstruction de Schur est vide pour les corps finis), de sorte qu’il n’est pas restrictif de supposer que ce corps est exactement ${{\mathbb{F}}}_q$, ce que nous faisons désormais par commodité. [^3]: En effet, cela entraîne que $H^2(G_{{\mathbb{Q}}_p},{\rm ad}({{\bar\rho}}))=0$ et $\dim_{{{\mathbb{F}}}_q} H^1(G_{{\mathbb{Q}}_p},{\rm ad}({{\bar\rho}})) = d^2+1$ [^4]: On rappelle, suivant Colmez [@colmeztri], qu’une représentation $\rho : G_{{\mathbb{Q}}_p} \rightarrow {\mathrm{GL}}_d(L)$ est dite trianguline si son ${(\varphi,\Gamma)}$-module pris sur l’anneau de Robba est extension successive ${(\varphi,\Gamma)}$-modules de rang $1$. Nous appellerons le choix d’une telle extension une triangulation de $\rho$. [^5]: Précisément, si le Frobenius cristallin de $D_{\rm cris}(\rho_x)$ est semi-simple. [^6]: Le presque ici fait référence au fait que pour Colmez le paramètre de $(s,t)$ doit être $p$-régulier, notion qui sera introduite plus tard (cette hypothèse n’est d’ailleurs plus nécessaire grâce au théorème \[thmdegal2\]). [^7]: Ces deux familles sont les analogues locaux des deux familles de Coleman passant par les deux formes jumelles associées à une forme modulaire propre et $p$-ancienne. [^8]: Mentionnons qu’il semble difficile d’appliquer le théorème d’approximation d’Artin pour relever les germes formels de familles de ${(\varphi,\Gamma)}$-modules triangulins construits dans [@bch §2] en des vrais germes de familles analytiques. Cela vient entre autres de ce que nous ne savons pas démontrer que les foncteurs sous-jacents sont de présentation finie, question qui nous semble assez profonde. [^9]: Précisions tout de même que dans toute la suite, un ${(\varphi,\Gamma)}$-module triangulin est la donnée d’un ${(\varphi,\Gamma)}$-module [*muni*]{} d’une filtration (la filtration fait partie de la donnée). [^10]: Le fait que l’analogue “automorphe” global de ce résultat était déjà connu est aussi la raison pour laquelle nous avions étudié le cas global en premier dans [@chU3]. Ces deux contextes comportent des similarités évidentes mais aussi des différences importantes, par exemple on ne dispose malheureusement pas en global d’analogue de la proposition \[existcrisgen\] (ii). [^11]: Une conséquence de cette propriété est l’existence d’une sous-catégorie pleine, abélienne, naturelle de celle des ${\mathcal{O}}(X)$-modules que Schneider et Teitelbaum nomment “co-admissibles” : un ${\mathcal{O}}(X)$-module est dit co-admissible si il est la limite projective d’une suite de ${\mathcal{O}}(X_n)$-modules de type fini $M_n$ munis d’isomorphismes $M_{n+1} \otimes_{{\mathcal{O}}(X_{n+1})} {\mathcal{O}}(X_n) \rightarrow M_n$ pour tout $n\geq 0$. Ces modules ont par définition une topologie canonique de ${\mathcal{O}}(X)$-module de Fréchet. Les ${\mathcal{O}}(X)$-modules de présentation finie sont admissibles, ainsi que tous leurs sous-modules de type fini et plus généralement fermés. Nous renvoyons à [*loc. cit*]{} pour plus de renseignements. Nous n’aurons recours à ces résultats que dans la preuve du lemme \[topo\] (iv) (lui-même utilisé uniquement pour le théorème \[h1universel\]). [^12]: Comme $A$-module, on a donc $D^{'(n)}=D^{(n)}$... [^13]: Rappelons que d’après [@chhecke lemme 3.18], une représentation continue $\rho : G \rightarrow {\mathrm{GL}}_d(A)$ d’un groupe profini $G$ étant donnée, on peut toujours trouver un recouvrement fini de $X$ par des affinoïdes $U_i$, ainsi que des modèles ${\mathcal}{A}_i \subset {\mathcal{O}}(U_i)$, tels que pour tout $i$ la représentation $\rho \otimes_A {\mathcal{O}}(U_i)$ de $G$ sur ${\mathcal{O}}(U_i)^d$ stabilise un sous-${{\mathcal A}}_i$-module libre $L_i$ de rang $d$ tel que $L_i[1/p]={\mathcal{O}}(U_i)^d$. [^14]: Un tel générateur n’existe bien sûr que pour $p>2$. Quand $p=2$, on choisit pour $\gamma \in \Gamma$ un élément engendrant topologiquement $\Gamma/\Gamma_{\rm tors}$ où $\Gamma_{\rm tors}=\{\pm 1\}$. On définit ensuite $C(D)^\bullet$ de la même manière à ceci près que de $D$ y est partout remplacé par ses invariants $D^{\Gamma_{\rm tors}}$ sous le groupe fini $\Gamma_{\rm tors}$, ce qui n’altère aucun des arguments qui suivent. [^15]: Ils sont discrets au sens que tout ouvert affinoïde de ${{\mathcal T}}$ ne rencontre qu’un nombre fini de tels points. [^16]: Si $p=2$, il faut remplacer $A/(1-\delta(p)p^{-i})$ par $A/(\delta(-1)+1,1-\delta(p)p^{-i})$. [^17]: Pour les raisons usuelles il ne s’agit pas vraiment d’un ensemble. Pour contourner ce problème il suffit de rajouter une fois pour toutes dans la définition d’un ${(\varphi,\Gamma)}$-module triangulin régulier rigidifié de rang $d$ sur ${\mathcal{R}}_A$ que le ${\mathcal{R}}_A$-module sous-jacent est ${\mathcal{R}}_A^d$ (plutôt que simplement, isomorphe à ${\mathcal{R}}_A^d$). [^18]: Voir la note précédente! [^19]: Il est immédiat de vérifier que si $U$ et $V$ sont des affinoïdes, et si $A \subset U$ et $B \subset V$ des parties Zariski-denses, alors $A \times B$ est Zariski-dense dans $U \times V$. [^20]: C’est à dire une réunion quelconque d’ouverts affinoïdes. [^21]: Avec éventuellement ${{\bar\rho}}\simeq {{\bar\rho}}(1)$. [^22]: Il s’agit ici de poids de Hodge-Tate relativement à $L$, c’est à dire les racines du polynôme de Sen vu comme élément de $L[T]$.

--- author: - François Bolley and Ivan Gentil date: 'November, 2010 ' title: '*Phi-entropy inequalities and Fokker-Planck equations*' --- [**Keywords:**]{} Functional inequalities, logarithmic Sobolev inequality, Poincaré inequality, $\Phi$-entropies, Bakry-Emery criterion, diffusion semigroups, Fokker-Planck equation. [**AMS subject classification:** ]{} 35B40, 35K10, 60J60. Introduction ============ We consider a Markov semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ on ${\ensuremath{\mathbb{R}}}^n$, acting on functions on ${\ensuremath{\mathbb{R}}}^n$ by ${\mathbf{P}_{\!t}} f(x) = \int_{{\ensuremath{\mathbb{R}}}^n} f(y) \, p_t(x, dy)$ for $x$ in ${\ensuremath{\mathbb{R}}}^n.$ The kernels $p_t(x, dy)$ are probability measures on ${\ensuremath{\mathbb{R}}}^n$ for all $x$ and $t \geq 0$, called transition kernels. We assume that the Markov infinitesimal generator $L = \displaystyle \frac{\partial {\mathbf{P}_{\!t}}}{\partial t} \Big\vert_{t=0^+}$ is given by $$L f (x)= \sum_{i,j=1}^n D_{ij} (x) \frac{\partial^2 f}{\partial x_i \partial x_j} (x) - \sum_{i=1}^n a_i (x) \frac{\partial f}{\partial x_i} (x)$$ where $D(x) = (D_{ij}(x))_{1 \leq i,j \leq n}$ is a symmetric $n \times n$ matrix, nonnegative in the sense of quadratic forms on ${\ensuremath{\mathbb{R}}}^n$ and with smooth coefficients, and where the $a_i, 1 \leq i \leq n,$ are smooth. Such a semigroup or generator is called a [*diffusion*]{}, and we refer to Refs. [@bakrystflour], [@bakrytata], [@ledouxmarkov] for backgrounds on them. If $\mu$ is a Borel probability measure on ${\ensuremath{\mathbb{R}}}^n$ and $f$ a $\mu$-integrable map on ${\ensuremath{\mathbb{R}}}^n$ we let $\mu(f) = \int_{{\ensuremath{\mathbb{R}}}^n} f(x) \, \mu(dx).$ If, moreover, $\Phi$ is a convex map on an interval $I$ of ${\ensuremath{\mathbb{R}}}$ and $f$ an $I$-valued map with $f$ and $\Phi(f)$ $\mu$-integrable, we let $${\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(f\right)}}} = \mu(\Phi(f)) - \Phi(\mu(f))$$ be the $\Phi$-entropy of $f$ under $\mu$ (see Ref. [@chafai04] for instance). Two fundamental examples are $\Phi(x) = x^2$ on ${\ensuremath{\mathbb{R}}}$, for which ${\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(f\right)}}}$ is the variance of $f,$ and $\Phi(x) = x \ln x $ on $]0, +\infty[$, for which ${\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(f\right)}}}$ is the Boltzmann entropy of $f$. By Jensen’s inequality, ${\mathbf{Ent}_{\mu}}^\Phi (f)$ is always nonnegative and, if $\Phi$ is strictly convex, it is positive unless $f$ is a constant, equal to $\mu(f).$ The semigroup $({\ensuremath{\mathbf{P_{\!t}}}})_{t \geq 0}$ is said [*$\mu$-ergodic*]{} if ${\ensuremath{\mathbf{P_{\!t}}}}f$ tends to $\mu(f)$ as $t$ tends to infinity in $L^2(\mu)$, for all $f$. In Section \[sectone\] we shall derive bounds on ${\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(f\right)}}}$ and ${\mathbf{Ent}_{{\mathbf{P}_{\!t}}}}^\Phi{\ensuremath{{\left(f\right)}}} (x)$ which will measure the convergence of ${\ensuremath{\mathbf{P_{\!t}}}}f$ to $\mu(f)$ in the ergodic setting. This is motivated by the study of the long time behaviour of solutions to Fokker-Planck equations, which will be discussed in Section \[secttwo\]. Some results of this note with their proofs are detailled in Ref. [@bolley-gentil]. Phi-entropy inequalities {#sectone} ======================== Bounds on ${\mathbf{Ent}_{{\mathbf{P}_{\!t}}}}^\Phi{\ensuremath{{\left(f\right)}}}$ and assumptions on $L$ will be given in terms of the [*carré du champ*]{} and $\Gamma_2$ operators associated to $L$, defined by $$\Gamma(f,g)= \frac{1}{2} \Big( L(fg) - f \, Lg - g \, Lf \Big), \,\, \Gamma_2(f) = \frac{1}{2} \Big( L \Gamma(f) - 2\Gamma (f, Lf) \Big).$$ If $\rho$ is a real number, we say that the semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the [*$CD(\rho,\infty)$ curvature-dimension*]{} (or Bakry-Émery) [*criterion*]{} (see Ref. [@bakryemery]) if $$\Gamma_2(f)\geq \rho \, \Gamma(f)$$ for all functions $f$, where $\Gamma (f) = \Gamma (f,f).$ The carré du champ is explicitely given by $$\Gamma (f,g) (x) = < \nabla f (x), D(x) \, \nabla g(x) >.$$ Expressing $\Gamma_2$ is more complex in the general case but, for instance, if $D$ is constant, then $L$ satisfies the $CD(\rho,\infty)$ criterion if and only if $$\label{cdrhocst} \frac{1}{2} \big( J a (x) D + (J a (x) D )^* \big) \geq \rho \, D$$ for all $x$, as quadratic forms on ${\ensuremath{\mathbb{R}}}^n$, where $\displaystyle J a$ is the Jacobian matrix of $a$ and $M^*$ denotes the transposed matrix of a matrix $M$ (see Ref. [@arnoldcarlenju08; @amtucpde01]) . Poincaré and logarithmic Sobolev inequalities for the semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ are known to be implied by the ${CD}(\rho,\infty)$ criterion. More generally, and following Ref. [@bakrytata; @bakryemery; @chafai04], let $\rho>0$ and $\Phi$ be a $C^4$ strictly convex function on an interval $I$ of ${\ensuremath{\mathbb{R}}}$ such that $-1/\Phi''$ is convex. If $({\ensuremath{\mathbf{P_{\!t}}}})_{t \geq 0}$ is $\mu$-ergodic and satisfies the $CD(\rho, \infty)$ criterion, then $\mu$ satisfies the $\Phi$-entropy inequality $$\label{PHII} {\mathbf{Ent}_{\mu}}^\Phi (f) \leq \frac{1}{2 \rho} \, \mu (\Phi''(f) \, \Gamma (f))$$ for all $I$-valued functions $f$. The main instances of such $\Phi$’s are the maps $x \mapsto x^2$ on ${\ensuremath{\mathbb{R}}}$ and $x \mapsto x \ln x$ on $]0, +\infty[$ or more generally, for $1 \leq p \leq 2$ $$\label{eq-phip} \Phi_p(x)= \left\{ \begin{array}{cl} \frac{x^p- x}{p(p-1)}, \quad x > 0 & \text{ if }p\in ]1,2]\\ x \ln x, \quad x>0 & \text{ if }p=1. \end{array} \right.$$ For this $\Phi_p$ with $p$ in $]1,2]$ the $\Phi$-entropy inequality becomes $$\label{beckner} \frac{\mu(g^2) - \mu(g^{2/p})^p}{p-1} \leq \frac{2}{ p \rho} \, \mu (\Gamma (g))$$ for all positive functions $g$. For given $g$ the map $p \mapsto \frac{\mu(g^2) - \mu(g^{2/p})^p}{p-1}$ is nonincreasing with respect to $p >0, p\neq1$. Moreover its limit for $p \to 1$ is ${{{{\mathbf{Ent}_{\mu}}}\!\left({g^2}\right)}}$, so that the so-called [*Beckner inequalities*]{}  for $p$ in $]1,2]$ give a natural monotone interpolation between the weaker Poincaré inequality (for $p=2$), and the stronger logarithmic Sobolev inequality (for $p \to 1$). ### Long time behaviour of the semigroup {#long-time-behaviour-of-the-semigroup .unnumbered} The $\Phi$-entropy inequalities provide estimates on the long time behaviour of the associated diffusion semigroups. Indeed, let $({\ensuremath{\mathbf{P_{\!t}}}})_{t\geq 0}$ be such a semigroup, ergodic for the measure $\mu$. If $\Phi$ is a $\mathcal C^2$ function on an interval $I$, then $$\label{HI} \frac{d}{dt}{\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left({\ensuremath{\mathbf{P_{\!t}}}}f\right)}}}=-\mu{\ensuremath{{\left(\Phi''({\ensuremath{\mathbf{P_{\!t}}}}f) \, \Gamma({\ensuremath{\mathbf{P_{\!t}}}}f)\right)}}}$$ for all $t \geq 0$ and all $I$-valued functions $f.$ As a consequence, if $C$ is a positive number, then the semigroup converges in $\Phi$-entropy with exponential rate: $$\label{cv1} {\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left({\ensuremath{\mathbf{P_{\!t}}}}f\right)}}}\leq e^{-\frac{t}{C}}{\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left( f\right)}}}$$ for all $t \geq 0$ and all $I$-valued functions $f$, if and only if the measure $\mu$ satisfies the $\Phi$-entropy inequality for all $I$-valued functions $f$, $$\label{eq-phisob} {\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(f\right)}}}\leq C\mu {\ensuremath{{\left(\Phi''( f) \, \Gamma(f)\right)}}}.$$ Refined $\Phi$-entropy inequalities ----------------------------------- We now give and study improvements of  for the $\Phi_p$ maps given by : \[thm-main\] Let $\rho \in {\ensuremath{\mathbb{R}}}$ and $p \in ]1,2[$. Then the following assertions are equivalent, with ${\ensuremath{{\left(1 - e^{-2\rho t}\right)}}}/{\rho}$ and ${\ensuremath{{\left(e^{2\rho t} -1\right)}}}/{\rho}$ replaced by $2 t$ if $\rho =0$: 1. the semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the ${CD}(\rho,\infty)$ criterion; 2. $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the refined local $\Phi_p$-entropy inequality $$\frac{1}{(p-1)^2}{\ensuremath{{\left[{{\ensuremath{\mathbf{P_{\!t}}}}(f^p)}-{{\ensuremath{\mathbf{P_{\!t}}}}(f)^p}{\ensuremath{{\left(\frac{{\ensuremath{\mathbf{P_{\!t}}}}(f^p)}{{\ensuremath{\mathbf{P_{\!t}}}}(f)^p}\right)}}}^{\! \! \frac{2}{p}-1}\right]}}}\leq \frac{1-e^{-2\rho t}}{\rho}{\ensuremath{\mathbf{P_{\!t}}}}{\ensuremath{{\left(f^{p-2} \, {\Gamma(f)}\right)}}}$$ for all positive $t$ and all positive functions $f$; 3. $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the reverse refined local $\Phi_p$-entropy inequality $$\frac{1}{(p-1)^2}{\ensuremath{{\left[{{\ensuremath{\mathbf{P_{\!t}}}}(f^p)}-{{\ensuremath{\mathbf{P_{\!t}}}}(f)^p}{\ensuremath{{\left(\frac{{\ensuremath{\mathbf{P_{\!t}}}}(f^p)}{{\ensuremath{\mathbf{P_{\!t}}}}(f)^p}\right)}}}^{\! \! \frac{2}{p}\!-\!1}\right]}}} \geq \frac{e^{2\rho t}\! \!-\!1}{\rho}{\ensuremath{{\left(\frac{({\ensuremath{\mathbf{P_{\!t}}}}f)^p}{{\ensuremath{\mathbf{P_{\!t}}}}{\ensuremath{{\left(f^p\right)}}}}\right)}}}^{\! \! \frac{2}{p}\!-\!1}\!\!\!{\ensuremath{{\left({\ensuremath{\mathbf{P_{\!t}}}}f\right)}}}^{p-\!2} \, {\Gamma({\ensuremath{\mathbf{P_{\!t}}}}f)}$$ for all positive $t$ and all positive functions $f$. If, moreover, $\rho >0$ and the measure $\mu$ is ergodic for the semigroup $({\ensuremath{\mathbf{P_{\!t}}}})_{t \geq 0}$, then $\mu$ satisfies the refined $\Phi_p$-entropy inequality $$\label{eq-ad2} \frac{p^2}{(p-1)^2}{\ensuremath{{\left[{\mu(g^2)}-{\mu(g^{2/p})^p}{\ensuremath{{\left(\frac{\mu(g^2)}{\mu(g^{2/p})^p}\right)}}}^{\frac{2}{p}-1}\right]}}}\leq \frac{4}{\rho}\mu{\ensuremath{{\left(\Gamma(g)\right)}}}$$ for all positive maps $g.$ The bound  has been obtained in Ref. [@arnolddolbeault05] for the generator $L$ defined by $Lf = \textrm{div} (D \nabla f) - < \! \!D \nabla V , \nabla f \! \!>$ with $D(x)$ a scalar matrix and for the ergodic measure $\mu = e^{-V}$, and under the corresponding $CD(\rho, \infty)$ criterion. It improves on the Beckner inequality  since $$\label{comp-beckner+} \frac{\mu(g^2) - \mu(g^{2/p})^p}{p-1} \leq \frac{p}{2 (p-1)^2} \Big[ \mu(g^2) -\mu(g^{2/p})^p \, \Big(\frac{\mu(g^2)}{\mu(g^{2/p})^p}\Big)^{\frac{2}{p}-1} \Big].$$ We have noticed that for all $g$ the map $ p \mapsto \frac{\mu(g^2) - \mu(g^{2/p})^p}{p-1} $ is continuous and nonincreasing on $]0, +\infty[$, with values ${{{{\mathbf{Ent}_{\mu}}}\!\left({g^2}\right)}}$ at $p=1$ and ${{{{\mathbf{Var}_{\mu}}}\!\left({g}\right)}}$ at $p=2$. Similarly, for the larger functional introduced in , the map $$p \mapsto \frac{p}{2 (p-1)^2} \Big[ \mu(g^2) -\mu(g^{2/p})^p \, \Big(\frac{\mu(g^2)}{\mu(g^{2/p})^p}\Big)^{\frac{2}{p}-1} \Big]$$ is nonincreasing on $]1,+\infty[$ (see [@bolley-gentil Prop. 11]). Moreover its value is ${{{{\mathbf{Var}_{\mu}}}\!\left({g}\right)}}$ at $p=2$ and it tends to $ {{{{\mathbf{Ent}_{\mu}}}\!\left({g^2}\right)}}$ as $p \to 1,$ hence providing a new monotone interpolation between Poincaré and logarithmic Sobolev inequalities. The pointwise $CD(\rho, \infty)$ criterion can be replaced by the integral criterion $$\mu{\ensuremath{{\left(g^{\frac{2-p}{p-1}}\Gamma_2(g)\right)}}}\geq \rho \, \mu{\ensuremath{{\left(g^{\frac{2-p}{p-1}}\Gamma(g)\right)}}}$$ for all positive functions $g,$ and one can still get the refined $\Phi_p$-entropy inequality , even in the case of non-reversible semigroups (see [@bolley-gentil Prop. 14]). For $\rho=0$, and following Ref. [@arnolddolbeault05], the convergence of ${\ensuremath{\mathbf{P_{\!t}}}}f$ towards $\mu(f)$ can be measured on $H(t) = {\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left({\ensuremath{\mathbf{P_{\!t}}}}f\right)}}}$ as $$\vert H'(t)\vert \leq \frac{\vert H'(0) \vert}{1+ \alpha t}, \quad t \geq 0$$ where $\alpha = \frac{2-p}{p} \vert H'(0) \vert / H(0).$ This illustrates the improvement offered by  instead of , which does not give here any convergence rate. The case of the Gaussian isoperimetry function ---------------------------------------------- Let $F$ be the distribution function of the one-dimensional standard Gaussian measure. The map $\mathcal U=F' \circ F^{-1}$, which is the isoperimetry function of the Gaussian distribution, satisfies $\mathcal U''=-1/\mathcal U$ on the set $[0,1]$, so that the map $\Phi=-\mathcal U$ is convex with $-1/\Phi''$ also convex on $[0,1].$ \[theophisoperimetry\] Let $\rho$ be a real number. Then the following three assertions are equivalent, with ${\ensuremath{{\left(1 - e^{-2\rho t}\right)}}}/{ \rho}$ and ${\ensuremath{{\left(e^{2\rho t} -1\right)}}}/{ \rho}$ replaced by $2t$ if $\rho =0$: 1. the semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the ${CD}(\rho,\infty)$ criterion; 2. the semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the local $\Phi$-entropy inequality $${\mathbf{Ent}_{{\ensuremath{\mathbf{P_{\!t}}}}}}^\Phi (f) \leq \frac{1}{\Phi''({\ensuremath{\mathbf{P_{\!t}}}}f)}\log{\ensuremath{{\left(1+\frac{1 - e^{-2\rho t}}{2 \, \rho}\Phi''({\ensuremath{\mathbf{P_{\!t}}}}f) \, {\ensuremath{\mathbf{P_{\!t}}}}( \Phi''(f) \Gamma (f))\right)}}}$$ for all positive $t$ and all $[0,1]$-valued functions $f$; 3. the semigroup $({\mathbf{P}_{\!t}})_{t \geq 0}$ satisfies the reverse local $\Phi$-entropy inequality $${\mathbf{Ent}_{{\ensuremath{\mathbf{P_{\!t}}}}}}^\Phi (f) \geq \frac{1}{\Phi''({\ensuremath{\mathbf{P_{\!t}}}}f)}\log{\ensuremath{{\left(1+ \frac{e^{2\rho t} -1}{2 \, \rho}\Phi''({\ensuremath{\mathbf{P_{\!t}}}}f)^2 \Gamma ({\ensuremath{\mathbf{P_{\!t}}}}f))\right)}}}$$ for all positive $t$ and all $[0,1]$-valued functions $f$. If, moreover, $\rho>0$ and the measure $\mu$ is ergodic for the semigroup $({\ensuremath{\mathbf{P_{\!t}}}})_{t \geq 0}$, then $\mu$ satisfies the $\Phi$-entropy inequality for all $[0,1]$-valued functions $f$: $${\mathbf{Ent}_{\mu}}^\Phi (f) \leq \frac{1}{\Phi''(\mu( f))}\log{\ensuremath{{\left(1+\frac{\Phi''(\mu( f))}{2 \rho} \mu( \Phi''( f)\Gamma (f))\right)}}}.$$ The proof is based on [@bolley-gentil Lemma 4]. For $\Phi = - \mathcal U$ it improves on the general $\Phi$-entropy inequality  since $\log(1+x)\leq x$. Links with the isoperimetric bounds of Ref. [@bakryledoux] for instance will be addressed elsewhere. Long time behaviour for Fokker-Planck equations {#secttwo} =============================================== Let us consider the linear Fokker-Planck equation $$\label{eq-fp1} \frac{\partial u_t}{\partial t} = \textrm{div} {\ensuremath{{\left[ D(x)( \nabla u_t + u_t (\nabla V(x)+F(x)))\right]}}}, \quad t \geq 0, \, x \in {\ensuremath{\mathbb{R}}}^n$$ where $D(x)$ is a positive symmetric $n \times n$ matrix and $F$ satisfies $$\label{rev} \textrm{div} {\ensuremath{{\left(e^{-V} D F\right)}}}=0.$$ It is one of the purposes of Refs. [@arnoldcarlenju08] and [@amtucpde01] to rigorously study the asymptotic behaviour of solutions to -. Let us formally rephrase the argument. Assume that the Markov diffusion generator $L$ defined by $$\label{fparnold2} Lf = \textrm{div} (D \nabla f) - < D ( \nabla V-F) , \nabla f>$$ satisfies the $CD(\rho,\infty)$ criterion with $\rho>0$, that is  if $D$ is constant, etc. Then the semigroup $({\ensuremath{\mathbf{P_{\!t}}}})_{t \geq 0}$ associated to $L$ is $\mu$-ergodic with $d\mu=e^{-V}/Zdx$ where $Z$ is a normalization constant. Moreover, a $\Phi$-entropy inequality  holds with $C=1/(2\rho)$ by , so that the semigroup converges to $\mu$ according to . However, under , the solution to  for the initial datum $u_0$ is given by $u_t=e^{-V}\,{\ensuremath{\mathbf{P_{\!t}}}}(e^V u_0)$. Then we can deduce the convergence of the solution $u_t$ towards the stationary state $e^{-V}$ (up to a constant) from the convergence estimate  for the semigroup, in the form $$\label{cvedp} {\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(\frac{u_t}{e^{-V}}\right)}}}\leq e^{-2\rho{t}}{\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left( \frac{u_0}{e^{-V}}\right)}}}, \quad t \geq 0.$$ In fact such a result holds for the general Fokker-Planck equation $$\label{eq-fp2} \frac{\partial u_t}{\partial t} = \textrm{div} {\ensuremath{{\left[ D(x) (\nabla u_t + u_t a(x) )\right]}}}, \quad t \geq 0, \, x \in {\ensuremath{\mathbb{R}}}^n$$ where again $D(x)$ is a positive symmetric $n \times n$ matrix and $a(x) \in {\ensuremath{\mathbb{R}}}^n.$ Its generator is the dual (for the Lebesgue measure) of the generator $$\label{exam} Lf = \textrm{div} (D \nabla f) - < D a , \nabla f>.$$ Assume that the semigroup associated to $L$ is ergodic and that its invariant probability measure $\mu$ satisfies a $\Phi$-entropy inequality  with a constant $C\geq0$: this holds for instance if $L$ satisfies the $CD(1/(2C),\infty)$ criterion. In this setting when $a (x)$ is not a gradient, the invariant measure $\mu$ is not explicit. Moreover the relation $u_t=e^{-V}\,{\ensuremath{\mathbf{P_{\!t}}}}(e^V u_0)$ between the solution of  and the semigroup associated to $L$ does not hold, so that the asymptotic behaviour  for solutions to  can not be proved by using . However, this relation can be replaced by the following argument, for which the ergodic measure is only assumed to have a positive density $u_{\infty}$ with respect to the Lebesgue measure. Let $u$ be a solution of  with initial datum $u_0$. Then, by [@bolley-gentil Lemma 7], $$\frac{d}{dt}{\mathbf{Ent}_{\mu}}^\Phi\big( \frac{u_t}{u_\infty} \big) = \int \Phi' \big ( \frac{u_t}{u_\infty} \big)L^* u_t dx =\int L \Big[ \Phi' \big(\frac{u_t}{u_\infty} \big) \Big] \frac{u_t}{u_\infty} d\mu = - \int \Phi'' \big(\frac{u_t}{u_\infty} \big)\Gamma \big( \frac{u_t}{u_\infty} \big)d\mu.$$ Then a $\Phi$-Entropy inequality  for $\mu$ implies the exponential convergence: With the above notation, assume that a $\Phi$-entropy inequality  holds for $\mu$ and with a constant $C$. Then all solutions $u = (u_t)_{t \geq 0}$ to the Fokker-Planck equation converge to $u_{\infty}$ in $\Phi$-entropy, with $${\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left(\frac{u_t}{u_\infty}\right)}}}\leq e^{-t/C}{\mathbf{Ent}_{\mu}}^\Phi{\ensuremath{{\left( \frac{u_0}{u_\infty}\right)}}}, \quad t \geq 0.$$ [**Acknowledgment:**]{} This work was presented during the 7th ISAAC conference held in Imperial College, London in July 2009. It is a pleasure to thank the organizers for giving us this opportunity. [1]{} C. An[é]{}, S. Blach[è]{}re, D. Chafa[ï]{}, P. Foug[è]{}res, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer. *Sur les inégalités de [S]{}obolev logarithmiques*, S.M.F., Paris, 2000. A. Arnold, A. Carlen, and Q. Ju. [Comm. Stoch. Analysis]{}, **2** (1) (2008), 153–175. A. Arnold and J. Dolbeault. [J. Funct. Anal.]{}, **225** (2) (2005), 337–351. A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter. [ Comm. Partial Diff. Equations]{}, **26** (1-2) (2001), 43–100. D. Bakry. *L’hypercontractivité et son utilisation en théorie des semigroupes.* Lecture Notes in Math. **1581** Springer, Berlin, 1994. D. Bakry. *Functional inequalities for [M]{}arkov semigroups*. In [Probability measures on groups: recent directions and trends]{}. Tata Inst., Mumbai, 2006. D. Bakry and M. [É]{}mery. *Diffusions hypercontractives*. In [ Séminaire de probabilités, [XIX]{}, 1983/84]{}, Lecture Notes in Math. **1123**. Springer, Berlin, 1985. D. Bakry and M. [Ledoux]{}. [ Invent. math.]{}, **123** (1996), 259–281. F. Bolley and I. Gentil. *Phi-entropy inequalities for diffusion semigroups.* to appear in [ J. Math. Pu. Appli]{}. D. Chafa[ï]{}. [ J. Math. Kyoto Univ.]{}, **44** (2) (2004), 325–363. D. Chafa[ï]{}. [ ESAIM Proba. Stat.]{}, **10** (2006), 317–339. B. Helffer. *Semiclassical analysis, [W]{}itten [L]{}aplacians, and statistical mechanics*. World Scientific Publishing Co. Inc., River Edge, 2002. M. Ledoux. [ Ann. Fac. Sci. Toulouse Math.]{}, **9** (2) (2000) 305–366.

--- abstract: 'The probability distributions for the smeared energy densities of quantum fields, in the two and four-dimensional Minkowski vacuum are discussed. These distributions share the property that there is a lower bound at a finite negative value, but no upper bound. Thus arbitrarily large positive energy density fluctuations are possible. In two dimensions we are able to give an exact unique analytic form for the distribution. However, in four dimensions, we are not able to give closed form expressions for the probability distribution, but rather use calculations of a finite number of moments to estimate the lower bound, and the asymptotic form of the tail of the distribution. The first 65 moments are used for these purposes. All of our four-dimensional results are subject to the caveat that these distributions are not uniquely determined by the moments. One can apply the asymptotic form of the electromagnetic energy density distribution to estimate the nucleation rates of black holes and of Boltzmann brains.' author: - 'Christopher J. Fewster' - 'L. H. Ford' - 'Thomas A. Roman' title: Probability distributions for quantum stress tensors in two and four dimensions --- Introduction ============ There has been extensive work in recent decades on the definition and use of the expectation value of a quantum stress tensor operator. However, the semiclassical theory does not describe the effects of quantum fluctuations of the stress tensor around its expectation value. One way to examine these fluctuations is through the probability distribution for individual measurements of a smeared stress tensor operator. This distribution was given recently for Gaussian averaged stress tensors operators in two-dimensional flat spacetime [@FFR10:FFRPrague] using analytical methods, and more recently for averaged stress tensors in four-dimensional spacetime from calculations of a finite set of moments. (Throughout our discussion, all quadratic operators are understood to be normal-ordered with respect to the Minkowski vacuum state.) Quantum Inequalities -------------------- Quantum inequalities are lower bounds on the [*expectation values*]{} of the smeared energy density operator in arbitrary quantum states [@F78:FFRPrague], [@F91:FFRPrague], [@FR95:FFRPrague],  [@FR97:FFRPrague], [@Flanagan97:FFRPrague], [@FewsterEveson98:FFRPrague]. If we sample in time along the worldline of an inertial observer, the quantum inequality takes the form $$\int_{-\infty}^\infty f(t)\, \langle T_{\mu \nu} u^{\mu} u^{\nu} \rangle \, dt \geq -\frac{C}{\tau^d} \,, \label{eq:QI}$$ where $T_{\mu \nu} u^{\mu} u^{\nu}$ is the normal-ordered energy density operator, which is classically non-negative, $t$ is the observer’s proper time, and $f(t)$ is a sampling function with characteristic width $\tau$. Here $C$ is a numerical constant, typically small compared to unity, $d$ is the number of spacetime dimensions, and we work in units where $c=\hbar=1$. Although quantum field theory allows negative expectation values of the energy density, quantum inequalities place strong constraints on the effects of this negative energy for violating the second law of thermodynamics [@F78:FFRPrague], maintaining traversable wormholes [@FR96:FFRPrague] or warpdrive spacetimes [@PF97:FFRPrague]. The implication of Eq. (\[eq:QI\]) is that there is an inverse power relation between the magnitude and duration of negative energy density. For a massless scalar field in two-dimensional spacetime, Flanagan [@Flanagan97:FFRPrague] has found a formula for the constant $C$ for a given $f(t)$ which makes Eq. (\[eq:QI\]) an optimal inequality. This formula is $$C = \frac{1}{6 \pi} \, \int^\infty_{-\infty} du \left( \frac{d}{du} \sqrt{g(u)}\right)^2\,, \label{eq:Flanagan}$$ where $f(t)=\tau^{-1}g(u)$ and $u=t/\tau$. In four-dimensional spacetime, Fewster and Eveson [@FewsterEveson98:FFRPrague] have derived an analogous formula for $C$, but in this case the bound is not necessarily optimal. Shifted Gamma Distributions - 2D Case {#sec:SGD} ===================================== In two-dimensional Minkowski spacetime, we determined the probability distribution for individual measurements, in the vacuum state, of the Gaussian sampled energy density to be $$\rho =\frac1{\sqrt{\pi}\,\tau} \int_{-\infty}^{\infty} T_{tt}\, {\rm e}^{-t^2/\tau^2} \, dt \,.$$ This was achieved by finding a closed form expression for the generating function of the moments $\langle \rho^n\rangle$ of $\rho$, from which the probability distribution was obtained. The definition of the $n$’th moment of the distribution of a variable $x$ is given by $$a_n = \int x^n \, P(x)\, dx \,. \label{eq:moment}$$ The resulting distribution is conveniently expressed in terms of the dimensionless variable $x = \rho \, \tau^2$ and is a shifted Gamma distribution: $$P(x) =\vartheta(x+x_0) \frac{\beta^{\alpha}(x+x_0)^{\alpha-1}}{\Gamma(\alpha)} \exp(-\beta(x+x_0)) \,, \label{eq:shifted_Gamma}$$ with parameters $$x_0 = \frac{1}{12\pi},\qquad \alpha = \frac{1}{12}, \qquad \beta = \pi \,.$$ Here $x = -x_0$ is the lower bound of the distribution. {width="11cm"} \[fig:2Dprob\] The lower bound, $-x_0$, for the probability distribution for energy density fluctuations in the vacuum is exactly Flanagan’s optimum lower bound, Eq.( \[eq:Flanagan\]), on the Gaussian sampled expectation value. As was argued in Ref. [@FFR10:FFRPrague], this is a general feature, giving a deep connection between quantum inequality bounds and stress tensor probability distributions. The quantum inequality bound is the lowest eigenvalue of the sampled operator, and is hence the lowest possible expectation value and the smallest result which can be found in a measurement. That the probability distribution for vacuum fluctuations actually extends down to this value is more subtle and depends upon special properties of the vacuum state, and is implied by the Reeh-Schlieder theorem. There is no upper bound on $P(x)$, as arbitrarily large values of the energy density can arise in vacuum fluctuations. Nonetheless, for the massless scalar field, negative values are much more likely; 84% of the time, a measurement of the Gaussian averaged energy density will produce a negative value. However, the positive values found the remaining 16% of the time will typically be much larger, and the average first moment of $P(x)$ will be zero. Furthermore, the probability distribution for the two-dimensional stress tensor is uniquely determined by its moments, as a consequence of the Hamburger moment theorem  [@Simon:FFRPrague]. This condition is a sufficient, although not necessary, condition for uniqueness, and is fulfilled by the moments of the shifted Gamma distribution. The 4D Case {#sec:4D} =========== In four dimensions, the operators $\rho_S$, and $\rho_{EM}$ all have dimensions of $length^{-4}$. Their probability distributions $P(x)$ are taken to be functions of the dimensionless variable $$x = (4\pi \, \tau^2)^2 \, A\,,$$ where $A$ is the Lorentzian time average of $\rho_S$, and $\rho_{EM}$, where $\rho_S$ and $\rho_{EM}$ are the smeared energy density operators for the massless scalar field, and electromagnetic fields, respectively. The distributions were calculated numerically from 65 moments [@FFR12:FFRPrague] The situation here is less straightforward. In this case, the moments grow too rapidly to satisfy the Hamburger moment criterion. Unfortunately, this means that we cannot be guaranteed of finding a unique probability distribution $P(x)$ from these moments. These probability distributions share some of the main characteristics of their two-dimensional counterparts. They have a lower bound but no upper bound. Our our techniques allow us to give approximate lower bounds and the asymptotic forms of the tails of the distributions. Our estimates for the lower bounds are $$-x_0(\rho_{EM}) \approx -0.0472 \qquad -x_0(\rho_S) \approx -0.0236\,. \label{eq:bounds2}$$ These are also estimates of the optimal quantum inequality bounds for each field. In contrast, the non-optimal bound for $\rho_S$, given by the method of Fewster and Eveson [@FewsterEveson98:FFRPrague], is $-x_0(FE) =- 27/128 \approx -0.21$, which is an order of magnitude larger. It is of interest to note that the magnitudes of the dimensionless lower bounds, given in Eq. (\[eq:bounds2\]) are small compared to unity. The fact that the probability distribution has a long positive tail, and must have a unit zeroth moment and a vanishing first moment, implies that the total probability of a negative value to be substantial. The small magnitudes of $x_0(\rho_{S})$ and $x_0(\rho_{EM})$ imply strong constraints on the magnitude of negative energy which can arise either as an expectation value in an arbitrary state, or as a fluctuation in the vacuum. They also imply that an individual measurement of the sampled energy density in the vacuum state is very likely to yield a negative value. One can show that the asymptotic behavior of the tail of the probability distribution is determined by the moments, even if the exact probability distribution is not uniquely determined. Our fitted tail decreases asymptotically as $$P_{{\rm fit}} \sim e^{-a x^{1/3} } \,,$$ where $a$ is a constant. We are also able to show that no distribution with the same moments can have a tail which decreases at a faster rate than ours. By contrast, the tail of a Boltzmann distribution for thermal fluctuations falls off as $$P_{{\rm Boltzmann}} \sim e^{-\beta x} \,,$$ where $\beta$ is a constant. Therefore vacuum fluctuations outweigh thermal fluctuations at high energies. {width="12cm"} \[fig:Ptail-vs-Boltzmann\] Application: Black Hole Nucleation ---------------------------------- The fact that the energy density probability distribution has a long positive tail implies a finite probability for the nucleation of black holes out of the Minkowski vacuum via large, though infrequent positive fluctuations (see Ref [@FFR12:FFRPrague]). This probability cannot be too large, of course, or it will conflict with observation. Our estimate of the probability depends only on the asymptotic form of the tail. (One can use similar arguments to estimate the probability of “Boltzmann brains” [@Bbrains:FFRPrague] nucleating out of the vacuum.) Summary ======= We have found that the probability distribution for vacuum fluctuations of the Gaussian-smeared energy density for a massless scalar field in two-dimensional spacetime is [*uniquely*]{} defined by a shifted gamma distribution. The distribution has a negative lower bound but no upper bound. It has an integrable singularity (i.e., a “spike") at the lower bound. In addition, we find that there is a deep connection between the lower bound of the distribution and the quantum inequalities. In fact the lower bound of the distribution coincides [*exactly*]{} with the [*optimal*]{} quantum inequality bound for a Gaussian sampling function, derived earlier by Flanagan. The lower bound is very small in magnitude, but the probability distribution is large in the region between zero and the lower bound. As a result, rather surprisingly, the probability of obtaining a negative result in an individual measurement is $84\%$! Although the negative fluctuations are very frequent, they are small in magnitude. As a result, one would not expect to see large effects of negative energy (e.g., violations of the second law, wormholes, warpdrives, etc.) nucleating out of the vacuum. However, the distribution has a long positive tail, which guarantees that the frequent but small negative energy density fluctuations are balanced by the much rarer but larger positive energy fluctuations. Therefore, the expectation value of the energy density in the Minkowski vacuum state is zero. It is quite remarkable that the quantum inequalities which are bounds on the [*expectation value*]{} of the energy density in an [*arbitrary*]{} quantum state, should be so intimately related to the probability distribution of [*individual*]{} measurements of the energy density made in the [*vacuum*]{} state. In four dimensions, we find similarities with the two-dimensional case, in that there is a lower bound but no upper bound. We are able to give numerical estimates of the lower bounds, i.e., the optimal bounds, and the asymptotic form of the tails. The lower bounds are negative with small magnitudes. However, our methods do not allow us to determine whether there is a “spike” at the lower bound, as in two dimensions. Nonetheless, the low magnitudes of the lower bounds indicate that a significant fraction of the probability must lie in the negative region. Therefore, as in the two-dimensional case, the probability of obtaining a negative value in an individual measurement is quite high. The long positive tail drops off more slowly than that of a Boltzmann distribution, which implies that vacuum fluctuations dominate over thermal fluctuations at high energies. Unfortunately, it seems likely that it is not possible to uniquely determine the four-dimensional distributions from the moments alone, as the latter do not obey the Hamburger moment condition. Nonetheless, we are able to glean some information from the moments. For example, we can determine that no distribution with the same moments as ours can have a tail which decreases faster than ours. The asymptotic forms of the long positive tail allow us to estimate the probability of nucleation of (small) black holes and “Boltzmann brains" out of the vacuum. Clearly further work can be done on this subject. One topic would be to see what additional information can be obtained from our calculated four-dimensional probability distributions, even if they cannot be uniquely determined from their moments. For example, does the “spike” behavior persist in four dimensions as well as in two, and what is its physical significance? Another would be to determine what the optimal quantum inequality bounds actually are. It would also be useful to try various sampling functions. Can the probability distributions and optimal bounds can be obtained by other methods which do not have the limitation of the ambiguities in the moment methods? There is more to do to explore the physical content of stress-tensor fluctuations. References ========== [Z]{} De Simone, A., Guth, A.H., Linde, A., Noorbala, M., Salem, M.P. and Vilenkin, A., [*Phys. Rev. D*]{}, [**82**]{}, 063520, (2010). Fewster, C.J. and Eveson, S.P., [*Phys. Rev. D*]{}, [**58**]{}, 084010, (1998). Fewster, C.J., Ford L.H. and Roman, T.A., [*Phys. Rev. D*]{}, [**81**]{}, 121901, (2010). Fewster, C.J., Ford L.H. and Roman, T.A., [*Phys. Rev. D*]{}, [**85**]{}, 125038, (2012). Flanagan, E.E., [*Phys. Rev. D*]{}, [**56**]{}, 4922, (1997). Ford, L.H. and Roman, T.A., [*Phys. Rev. D*]{}, [**51**]{}, 4277, (1995). Ford, L.H. and Roman, T.A., [*Phys. Rev. D*]{}, [**53**]{}, 5496, (1996). Ford, L.H. and Roman, T.A., [*Phys. Rev. D*]{}, [**55**]{}, 2082, (1997). Ford, L. H., [*Proc. Roy. Soc. Lond. A*]{}, [**364**]{}, 227, (1978). Ford, L. H., [*Phys. Rev. D*]{}, [**43**]{}, 3972, (1991). Pfenning, M.J. and Ford, L.H., [*Class. Quant. Grav.*]{}, [**14**]{}, 1743, (1997). Simon, B., [*Adv. Math.*]{}, [**137**]{}, 82, (1998).

--- abstract: 'In our standard geodetic VLBI solutions, we estimate the positions of quasars assuming that their positions do not vary in time. However, in solutions estimating proper motion, a significant number of quasars show apparent proper motion greater than 50 $\mu$as/yr. For individual quasars, there are source structure effects that cause apparent proper motion. To examine how coherent the pattern of apparent proper motion is over the sky, we have estimated the vector spherical harmonic components of the observed proper motion using VLBI data from 1980 to 2002. We discuss the physical interpretation of the estimated harmonic components.' author: - 'D. S. MacMillan' title: Quasar Apparent Proper Motion Observed by Geodetic VLBI Networks --- \#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} = \#1 1.25in .125in .25in History of Radio Source Observing by Geodetic Networks ====================================================== Since 1979, geodetic VLBI networks have observed in 3555 24-hour experiment sessions. Most of these observations were made by about 40 antennas. The number of radio sources observed in geodetic sessions has grown significantly during the last two decades. Until 1986, only 65 sources had been observed and these were nearly all in the Northern Hemisphere. From 1987 to 1989, 210 more sources were observed and the distribution of sources between hemispheres was greatly improved. In our current geodetic solutions, we estimate positions for 610 radio sources. Due to the fact that most of the geodetic antennas are in the Northern hemisphere, the distribution of radio sources is still better in the Northern than the Southern hemisphere. Since 1990, there have been 2-3 24-hour geodetic sessions each week, where typically 40-50 radio sources are observed per session. Typically each session used a network of 4-7 antennas, but we also have data from the bimonthly RDV series of sessions (1997-2003) that mostly used 10 of the standard geodetic antennas along with the 10-station VLBA. In each session, the choice of radio sources to be observed depended on the geometry of the network. The result is that the sampling history of the set of all sources that have been observed is generally uneven. Observed Proper Motion Field ============================ A VLBI terrrestrial reference frame solution was performed using all VLBI observations from 1979-2003. In this type of solution, station positions, station velocities, and radio source positions and velocities are estimated from all of the data. To remove the translational and rotation degeneracies of the solution, it is necessary to apply several constraints. Here, the station positions and velocities were weakly constrained to ITRF2000 via no net translation and rotation constraints. Similarly, radio source positions were weakly constrained to a priori ICRF positions with a no net rotation constraint. Radio source velocities (proper motions) were constrained to have no net rotation velocity. The paper by Ma et al. (1990) describes the least-squares estimation program (SOLVE) used in the analysis. For each 24-hour session observing day, the estimated parameters are pole coordinates and their rates, UT1 and the UT1 rate, and nutation offsets. Several nuisance parameters are estimated as piecewise linear functions with constraints on the rate of change between segments. These parameters are station clock functions with one hour segments, wet atmospheric delay with 20 minute segments, and horizontal tropospheric delay with 6 hour segments. For the most part, the theoretical time delays follow the IERS Conventions (McCarthy 1996). Figure \[fig-1\] shows the distribution of proper motion for 580 radio sources with proper motion formal uncertainties better than 0.5 mas/yr. Since the observations of sources have not been very even, the uncertainties of the source proper motion estimates range from less than 50 $\mu$as/yr to more than 1 mas/yr. The formal uncertainties for 167 sources are better than 20 $\mu$as/yr and better than 50 $\mu$as/yr for 348 sources. There are about 50-60 sources with observed proper motion with at least 3 sigma significance. In terms of obvious systematic effects, there is not any clear declination dependence of observed proper motion in declination or in right ascension. However, the precision of determinations of proper motion for southern declination sources (below 40-50S) is poorer than for higher declination sources. The main reason for this is that most of the geodetic antennas are in the northern hemisphere. For the distribution of estimated proper motions, we find that the weighted RMS of motion in declination was 30 $\mu$as/yr and in right ascension (arc length) was 26 $\mu$as/yr. Figure \[fig-2\] shows the magnitude of observed proper motion versus redshift for sources where the proper motion uncertainty is at least as good as 50 $\mu$as/yr. The available redshifts were taken from Archinal (1997), which contains radio source characteristics compiled from a number of sources. The observed quasars (355 sources) have redshifts as large as 4.3. The observed radio galaxies (54 sources) and BL Lac objects (56 sources) generally have redshifts less than 1. There may be a trend toward lower proper motion with increasing redshift but it is not very clear. Possible Source Structure Effects ================================= Source structure variations are known to be correlated with variations in the apparent position measured by VLBI (Charlot et al. 1990). When a solution is made in which source position time series are estimated, one observes both linear and nonlinear variation. One problem is that we do not know how much of this observed linear variation is due to structure variation. Figure \[fig-3\] shows the time series for the right ascension of 4C39.25, whose apparent proper motion has been shown to be related to structure variations (Fey et al. 1997). Sovers et al. (2002) have derived structure corrections from source maps for the RDV series of VLBI sessions and found that these corrections removed about 8 ps in quadrature from solution weighted rms delay residuals (typically at a level of 30 ps). Pattern of Apparent Proper Motion ================================= We have analyzed the observed proper motion vector field to determine whether there are systematic patterns present. We have expressed the observed field as an expansion of transverse vector spherical harmonics (VSH), which conveniently comprise an orthonormal basis for vector fields on a sphere. $$\sum_{l,m} (a^{E}_{l,m} Y^{E}_{l,m} + a^{M}_{l,m} Y^{M}_{l,m})$$ The E-harmonics are the electric or poloidal harmonics and the M-harmonics are the magnetic or toroidal harmonics. Since the observed motion is real, we estimate real linear combinations of the complex amplitudes. This is similar to the approach of Gwinn et al. (1997) except that instead of estimating the expansion amplitudes (externally) from the set of proper motion estimates made for each source (solution in Section 2), the expansion amplitudes were estimated directly in the solution. We also have available another six years of data. The L=1 M-harmonics are simply rotations and are therefore indistinguishable from Earth rotation. On the other hand, the L=1 E-harmonics can arise from galactocentric acceleration or quasar/galaxy acceleration. The RMS of the proper motion for observed sources using the sum of these harmonics was found to be  8 $\mu$as/yr. For the sum of the L=2 harmonics, we obtained an RMS of 19 $\mu$as/yr. Gwinn et al. (1997) related the squared proper motion averaged over the sky to the energy density of gravitational radiation for wavelengths short compared to source distances. Figure \[fig-4\] shows the estimated L=2 vector field. A variant of the above solution was made in which for each year, the positions for all sources observed in a given year were estimated with a VSH expansion. One can clearly see the linear evolution in time of the expansion amplitudes after 1990. For years increasingly earlier than 1990, the distribution of sources poorer especially in the Southern hemisphere so that a spherical harmonic expansion becomes increasingly less reliable. Discussion ========== The position error floor of the ICRF (International Celestial Reference Frame) is 250 $\mu$as and the orientation stability is about 20 $\mu$as. The stability of the celestial reference frame depends on the level of secular change of radio source positions. The WRMS of observed proper motion from nearly 2 decades of observations by geodetic networks is at the level of about 30 $\mu$as/yr in both right ascension and declination. Given this level of observed motion and an observing duration of 1-2 decades, modeling or corrections are needed to make improvement in stability below the nominal noise floor. There does appear to be a statistically significant pattern in the observed proper motion vector field. The problem, however, is to how to determine how much the observed apparent motion is due to unmodelled source structure effects. To improve the determination of spherical expansion amplitudes, it would be desirable to make more observations of southern declination sources below about -40. Archinal, B. A., Arias, E. F., Gontier, A. -M., & Mercuri-Moreau, C. 1997, IERS Technical Note 23, III-11 Charlot, P. 1990, , 99, 1309 Fey, A., Eubanks T. M., & Kingham, K. 1997, , 114, 2284 Gwinn, C. R., Eubanks, T. M., Pyne, T., Birkinshaw, M, & Matsakis, D. N. 1997, , 485, 87 Ma, C., Sauber, J. M., Bell, L. J., Clark, T. A. ,Gordon, D., Himwich, W. E., & Ryan, J. W. 1990, J. Geophys. Res, 95, 21991 McCarthy, D. D. 1996, IERS Technical Note 21 Sovers, O. J., Charlot, P., Fey, A. L. & Gordon, D. G. 2002, IVS 2002 General Meeting Proceedings, 243

--- abstract: 'In this paper, we develop statistical inference techniques for the unknown coefficient functions and single-index parameters in single-index varying-coefficient models. We first estimate the nonparametric component via the local linear fitting, then construct an estimated empirical likelihood ratio function and hence obtain a maximum empirical likelihood estimator for the parametric component. Our estimator for parametric component is asymptotically efficient, and the estimator of nonparametric component has an optimal convergence rate. Our results provide ways to construct the confidence region for the involved unknown parameter. We also develop an adjusted empirical likelihood ratio for constructing the confidence regions of parameters of interest. A simulation study is conducted to evaluate the finite sample behaviors of the proposed methods.' address: - 'College of Applied Sciences, Beijing University of Technology, Beijing 100124, China.\' - 'Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100080, China' - 'School of Mathematics and Statistics, Yunnan University, Kunming 650091, China.\' author: - - title: 'Empirical likelihood for single-index varying-coefficient models' --- Introduction {#sec1} ============ Consider a single-index varying-coefficient model of the form $$Y={\rm g}_0^T(\beta_0^TX)Z+\varepsilon, \label{eq1.1}$$ where $(X, Z)\in R^p\times R^q$ is a vector of covariates, $Y$ is the response variable, $\beta_0$ is an $p\times1$ vector of unknown parameters, ${\rm g}_0(\cdot)$ is an $q\times1$ vector of unknown functions and $\varepsilon$ is a random error with mean 0 and finite variance $\sigma^2$. Assume that $\varepsilon$ and $(X,Z)$ are independent. For the sake of identifiability, it is often assumed that $\|\beta_0 \|=1$, and the first non-zero element is positive, where $\|\cdot\|$ denotes the Euclidean metric. Model (\[eq1.1\]) includes a class of important statistical models. For example, if $q=1$ and $Z=1$, (\[eq1.1\]) reduces to the single-index model (see, e.g., Härdle, Hall and Ichimura [@HarHalIch93], Weisberg and Welsh [@WeiWel94], Zhu and Fang [@ZhuFan96], Chiou and Müller [@ChiMul98], Hristache, Juditsky and Spokoiny [@HriJudSpo01], Xue and Zhu [@XueZhu06]). If $p=1$ and $\beta_0=1$, (\[eq1.1\]) is the varying-coefficient model (see, e.g., Chen and Tsay [@CheTsa93], Hastie and Tibshirani [@HasTib93], Wu, Chiang and Hoover [@WuChiHoo98], Fan and Zhang [@FanZha99], Cai, Fan and Li [@CaiFanLi00], Cai, Fan and Yao [@CaiFanYao00], Xue and Zhu [@XueZhu07N1]). If the last component of $\beta_0$ to be non-zero and $Z=(1,X^*T)^T$ where $X^*$ is the remaining vector of $X$ with its $p$th component deleted, (\[eq1.1\]) becomes the adaptive varying-coefficient linear model (see, e.g., Fan, Yao and Cai [@FanYaoCai03], Lu, Tjøstheim and Yao [@LuTjsYao07]). Model (\[eq1.1\]) is easily interpreted in real applications because it has the features of the single-index model and the varying-coefficient model. In addition, model (\[eq1.1\]) may include cross-product terms of some components of $X$ and $Z$. Hence it has considerable flexibility to cater for complex multivariate nonlinear structure. Xia and Li [@XiaLi99] investigated a class of single-index coefficient regression models, which include model (\[eq1.1\]) as a special example. When it is used as a nonparametric time series model, Xia and Li [@XiaLi99] obtained the estimator of ${{\rm g}}(\cdot)$ by kernel smoothing and then derived the estimator of $\beta_0$ by the least squares method and proved that the corresponding estimators are consistent and asymptotically normal. In this paper, we develop statistical inference techniques of ${{\rm g}}_0(\cdot)$ and $\beta_0$ with independent observations of $(Y, X, Z)$. We can construct an empirical likelihood ratio function for $\beta_0$ by assuming ${\rm g}_0(\cdot)$ and its derivative to be known functions. In practice, however, they are unknown, and hence the empirical likelihood ratio function cannot be used to make inference on $\beta$. This motivates us to estimate the unknown ${\rm g}_0(\cdot)$ and $\dot{{\rm g}}_0(\cdot)$ via the local linear smoother, and then obtain an estimated empirical likelihood ratio of $\beta_0$. The estimated empirical log-likelihood ratio is asymptotically distributed as a weighted sum of independent $\chi_1^2$ variables with unknown weights. This result cannot be applied directly to construct confidence region for $\beta_0$. To solve this issue, two methods may be used (see Wang and Rao [@WanRao02]). The first method is to estimate the unknown weights consistently so that the distribution of the estimated weighted sum of chi-squared variables can be estimated from the data. The second method is to adjust the estimated empirical log-likelihood ratio so that the resulting adjusted empirical log-likelihood ratio is asymptotically chi-squared. Also, we obtain a maximum empirical likelihood estimator of $\beta_0$, by maximizing the estimated empirical likelihood ratio function, and investigate its asymptotic property. In addition, we obtain the convergence rate of the estimator of $\sigma^2$ and define the consistent estimator of asymptotic variance; this allows us to construct a confidence region for $\beta_0$. Comparing with the existing methods, our estimating method has the following advantage: The asymptotic variance of our estimator for $\beta_0$ is the same as those of Härdle *et al.* [@HarHalIch93] and Xia and Li [@XiaLi99] when the model reduces to the single-index model; this shows that our estimator for $\beta_0$ is the same efficient as than those of Härdle *et al.* [@HarHalIch93] and Xia and Li [@XiaLi99]. The difference between the proposed estimating approaches and the existing estimating approaches is that we use an empirical likelihood ratio to define the estimator of $\beta_0$ while the existing work uses the least squares techniques (see, e.g., Härdle *et al.* [@HarHalIch93], Xia and Li [@XiaLi99]). Also, we develop an empirical likelihood inference for constructing a confidence region of $\beta$. The empirical likelihood method, introduced by Owen [@Owe88], has many advantages for constructing confidence regions or intervals. For example, it does not impose prior constraints on region shape, and it does not require the construction of a pivotal quantity. The empirical likelihood has been studied by many authors. The related works are Wang and Rao [@WanRao02], Wang, Linton and Härdle [@WanLinHar04], Xue and Zhu [@XueZhu06; @XueZhu07N1; @XueZhu07N2], Zhu and Xue [@ZhuXue06], Qin and Zhang [@QinZha07], Stute, Xue and Zhu [@StuXueZhu07], Xue [@Xue09N2; @Xue09N1], Wang and Xue [@WanXue11], among others. The rest of the paper is organized as follows. In Section \[sec2\], we define an estimated empirical likelihood ratio, and then obtain a maximum empirical likelihood estimator of $\beta_0$ by maximizing the empirical likelihood ratio function; the asymptotic properties of the proposed estimators are also investigated. In Section \[sec3\], we define an adjusted empirical log-likelihood and derive its asymptotic distribution. Section \[sec4\] reports a simulation study. Proofs of theorems are relegated to the . It should be pointed that some special techniques are used in the proofs. Estimated empirical likelihood {#sec2} ============================== Methodology {#sec2.1} ----------- Suppose that $ \{(Y_i,X_i, Z_i);1\leq i\leq n \}$ is an independent and identically distributed (i.i.d.) sample from (\[eq1.1\]), that is $$Y_i={\rm g}_0^T(\beta_0^TX_i)Z_i+\varepsilon_i, \qquad i=1,\ldots,n,$$ where $\varepsilon_i$s are i.i.d. random errors with mean 0 and finite variance $\sigma^2$. Assume that $ \{\varepsilon_i;1\leq i\leq n \}$ are independent of $ \{(X_i,Z_i);1\leq i\leq n \}$. To construct an empirical likelihood ratio function for $\beta_0$, we introduce an auxiliary random vector $$\eta_i(\beta)=\{Y_i-{\rm g}_0^T(\beta^TX_i)Z_i\}\dot{{\rm g}}_0^T(\beta^TX_i)Z_iX_iw(\beta^TX_i), \label{eq2.1}$$ where $\dot{{\rm g}}_0(\cdot)$ stands for the derivative of the function vector ${\rm g}_0(\cdot)$, and $w(\cdot)$ is a bounded weight function with a bounded support ${{\mathcal}U}_w$, which is introduced to control the boundary effect in the estimations of ${{\rm g}}_0(\cdot)$ and $\dot{{\rm g}}_0(\cdot)$. To convenience, we take that $w(\cdot)$ is the indicator function of the set ${{\mathcal}U}_w$. Note that $E\{\eta_i(\beta)\}=0$ if $\beta=\beta_0$. Hence, the problem of testing whether $\beta$ is the true parameter is equivalent to testing whether $E\{\eta_i(\beta)\}=0$ for $i=1,2,\ldots,n$. By Owen [@Owe88], this can be done by using the empirical likelihood. That is, we can define the profile empirical likelihood ratio function $${L}_n(\beta)=\max \Biggl\{\prod_{i=1}^n(np_i) \bigg| p_i\geq0, \sum_{i=1}^n p_i=1, \sum_{i=1}^np_i{\eta}_i(\beta)=0 \Biggr\}.$$ It can be shown that $-2\log L_n(\beta_0)$ is asymptotically chi-squared with $p$ degrees of freedom. However, $L_n(\beta)$ cannot be directly used to make statistical inference on $\beta_0$ because $L_n(\beta)$ contains the unknowns ${\rm g}_0(\cdot)$ and $\dot{{\rm g}}_0(\cdot)$. A natural way is to replace ${\rm g}_0(\cdot)$ and $\dot{{\rm g}}_0(\cdot)$ in ${L}_n(\beta)$ by their estimators and define an estimated empirical likelihood function. In this paper, we estimate the vector functions ${\rm g}_0(\cdot)$ and $\dot{{\rm g}}_0(\cdot)$ via the local linear regression technique (see, e.g., Fan and Gijbels [@FanGij96]). The local linear estimators for ${\rm g}_0(u)$ and $\dot{{\rm g}}_0(u)$ are defined as $\hat{{\rm g}}(u;\beta _0)=\hat{\rm a}$ and $\hat{\dot{{\rm g}}}(u;\beta_0)=\hat{\rm b}$ at the fixed point $\beta_0$, where $\hat{\rm a}$ and $\hat{\rm b}$ minimize the sum of weighted squares $$\sum_{i=1}^n [Y_i-\{{\rm a} + {\rm b} (\beta_0^TX_i-u)\}^TZ_{i} ]^2K_h(\beta_0^TX_i-u),$$ where $K_h(\cdot)=h^{-1}K(\cdot/h)$, $K(\cdot)$ is a kernel function, and $h=h_n$ is a bandwidth sequence that decreases to 0 as $n$ increases to $\infty$. It follows from the least squares theory that $$(\hat{{\rm g}}^T (u;\beta_0),h\hat{\dot{{\rm g}}}{}^T (u;\beta_0) )^T ={{{S}}}_n^{-1}(u;\beta_0)\xi_n(u;\beta_0),$$ where $${{{S}}}_n(u;\beta_0)=\left( \begin{array}{c@{ \quad }c} {{{S}}}_{n,0}(u;\beta_0) & {{{S}}}_{n,1}(u;\beta_0) \\ {{{S}}}_{n,1}(u;\beta_0) & {{{S}}}_{n,2}(u;\beta_0) \end{array} \right) \quad \mbox{and} \quad \xi_n(u;\beta_0)=\left( \begin{array}{c} \xi_{n,0}(u;\beta_0) \\ \xi_{n,1}(u;\beta_0) \end{array} \right)$$ with $${{{S}}}_{n,j}(u;\beta_0)=\frac{1}{n}\sum_{i=1}^nZ_iZ_i^T \biggl(\frac {\beta_0^TX_i-u}{h} \biggr)^jK_h(\beta_0^TX_i-u)$$ and $$\xi_{n,j}(u;\beta_0)=\frac{1}{n}\sum_{i=1}^nZ_iY_i \biggl(\frac {\beta_0^TX_i-u}{h} \biggr)^jK_h(\beta_0^TX_i-u).$$ Since the convergence rate of the estimator of $\dot{g}_0'(u)$ is slower than that of the estimator of ${g}_0(u)$ if the same bandwidth is used, this leads to a slower convergence rate for the estimator $\hat\beta$ of $\beta_0$ than $\sqrt n$. To increase the convergence rate of the estimator of $\dot{g}_0'(u)$, we introduce the another bandwidth $h_1$ to replace $h$ in $\hat{\dot{\rm g}}(u;\beta)$, and define as $\hat{\dot{\rm g}}_{h_1}(u;\beta)$. Let $\hat{\eta}_i(\beta)$ be $\eta_i(\beta)$, with ${\rm g}_0(\beta^TX_i)$ and $\dot{{\rm g}}_0(\beta^TX_i)$ replaced by $\hat{{\rm g}}(\beta^TX_i;\beta)$ and $\hat{\dot{{\rm g}}}_{h_1}(\beta ^TX_i;\beta)$, respectively, for $i=1,\ldots,n$. Then an estimated empirical likelihood ratio function is defined by $$\hat{L}(\beta)=\max \Biggl\{\prod_{i=1}^n(np_i) \bigg| p_i\geq0, \sum_{i=1}^n p_i=1, \sum_{i=1}^np_i\hat{\eta}_i(\beta)=0 \Biggr\}.$$ By the Lagrange multiplier method, $\log\hat{L}(\beta)$ can be represented as $$\log\hat{L}(\beta)=-\sum_{i=1}^n\log\bigl (1+\lambda^T\hat{\eta }_i(\beta) \bigr), \label{eq2.2}$$ where $\lambda$ is determined by $$\frac{1}{n}\sum_{i=1}^n\frac{\hat{\eta}_i(\beta)}{1+\lambda ^T\hat{\eta}_i(\beta)}=0. \label{eq2.3}$$ Let ${{\mathcal}B}=\{\beta\in R^p\dvt \|\beta\|=1$, and the first non-zero element is positive. Then $\beta_0$ is an inner point of the set ${{\mathcal}B}$. Therefore we need only search for $\beta_0$ over ${{\mathcal}B}$. A maximum empirical likelihood estimator for $\beta_0$ is given by $$\hat{\beta}=\arg\sup_{\beta\in{{\mathcal}B}}\hat{L}(\beta). \label{eq2.4}$$ With $\hat{\beta}$, we define the estimate of ${\rm g}(u)$ by $ \hat{{\rm g}}(u)=\hat{{\rm g}}(u, \hat{\beta}), $ and the estimate of $\sigma^2$ by $$\hat{\sigma}^2=\frac{1}{n}\sum_{i=1}^n\{Y_i-\hat{{\rm g}}{}^T(\hat {\beta}^TX_i;\hat{\beta})Z_i\}^2. \label{eq2.5}$$ It is well known that if $\beta$ is known, the optimal bandwidth $h$ for $\hat{{\rm g}}(u)$ is of order $\mathrm{O}(n^{-1/5})$. However, if $\beta$ is unknown, in order to ensure that the estimator $\hat{\beta}$ is root-$n$ consistent, the bandwidth $h$ should be smaller than $\mathrm{O}(n^{-1/5})$, if we only assume ${{\rm g}}(\cdot)$ are second-order differentiable (see Theorem \[theo2\] below). Note that once the estimator $\hat{\beta}$ is available, an optimal bandwidth of order $\mathrm{O}(n^{-1/5})$ can be used in the final estimator for ${{\rm g}}(\cdot)$. Asymptotic properties {#sec2.2} --------------------- In order to obtain the asymptotic behaviors of our estimators, we first give the following conditions: 1. The density function of $\beta^TX$, $f(u)$, is bounded away from zero for $u\in{{\mathcal}U}_w$ and $\beta$ near $\beta_0$, and satisfies the Lipschitz condition of order 1 on ${{\mathcal}U}_w$, where ${{\mathcal}U}_w$ is the support of $w(u)$. 2. The functions $g_{j}(u)$, $1\leq j\leq q$, have continuous second derivatives on ${{\mathcal}U}_w$, where $g_{j}(u)$ are the $j$th components of ${\rm g}_0(u)$. 3. $E(\|X\|^6)<\infty$, $E(\|Z\|^6)<\infty$ and $E(|\varepsilon|^6)<\infty$. 4. $nh^2/\log^2{n}\rightarrow\infty$, $nh^4\log{n}\rightarrow0$; $nhh_1^3/\log^2{n}\rightarrow\infty$, $nh_1^5=\mathrm{O}(1)$. 5. The kernel $K(\cdot)$ is a symmetric probability density function with a bounded support and satisfies the Lipschitz condition of order 1 and $\int u^2K(u)\,\mathrm{d}u\neq0$. 6. The matrix ${{{ D}}}(u)=E(ZZ^T|\beta_0^TX=u)$ is positive definite, and each entry of ${{{ D}}}(u)$ and ${{{ C}}}(u)=E(VZ^T|\beta_0^TX=u)$ satisfies the Lipschitz condition of order 1 on ${{\mathcal}U}_w$, where $V=X\dot{{\rm g}}_0^T(\beta_0^TX)Zw(\beta_0^TX)$, and ${{\mathcal}U}_w$ is defined in (C1). 7. The matrices ${B}(\beta_0)=E(VV^T)$ and ${B}_*(\beta_0)= {B}(\beta_0)-E\{C(\beta_0^TX)\dot{\rm g}_0(\beta_0^TX)E(X^T|\allowbreak \beta_0^TX)\}$ are positive definite, where $V$ is defined in (C6). Condition [(C1)]{} is used to bound the density function of $\beta^TX$ away from zero. This ensures that the denominators of $\hat{{\rm g}}(u;\beta)$ and $\hat{\dot{{\rm g}}}(u;\beta)$ are, in probability one, bounded away from 0 for $u\in{{\mathcal}U}_w$. The second derivatives in [(C2)]{} are standard smoothness conditions. [(C3)–(C5)]{} are necessary conditions for the asymptotic normality or the uniform consistency of the estimators. Conditions [(C6)]{} and [(C7)]{} ensure that the asymptotic variance for the estimator of $\beta_0$ exists. Let ${{\mathcal}B}_n=\{\beta\in{{\mathcal}B}\dvt \|\beta-\beta_0\|\leq c_0n^{-1/2}\}$ for some positive constant $c_0$. This is motivated by the fact that, since we anticipate that $\hat{\beta}$ is root-$n$ consistent, we should look for a maximum of $\hat{L}(\beta)$ which involves $\beta$ distant from $\beta_0$ by order $n^{-1/2}$. Similar restrictions were also made by Härdle, Hall and Ichimura [@HarHalIch93], Xia and Li [@XiaLi99] and Wang and Xue [@WanXue11]. The following theorem shows that $-2\log\hat{L}(\beta_0)$ is asymptotically distributed as a weighted sum of independent $\chi_1^2$ variables. \[theo1\] Suppose that conditions [(C1)–(C7)]{} hold. Then $$-2\log\hat{L}(\beta_0)\stackrel{D}{\longrightarrow}w_1\chi _{1,1}^2+\cdots +w_p\chi_{1,p}^2,$$ where $\stackrel{D}{\longrightarrow}$ represents convergence in distribution, $\chi_{1,1}^2, \ldots,\chi_{1,p}^2$ are independent $\chi_1^2$ variables and the weights $w_j$, for $1\leq j\leq p$, are the eigenvalues of ${{{ G}}}(\beta_0)={{{ B}}}^{-1}(\beta_0){{{ A}}}(\beta_0)$. Here ${{{ B}}}(\beta_0)$ is defined in condition [(C7)]{}, $${{{ A}}}(\beta_0)={{{ B}}}(\beta_0)-E\{{{{ C}}}(\beta_0^TX){{{ D}}}^{-1}(\beta _0^TX){{{ C}}}{}^T(\beta_0^TX)\}, \label{eq2.6}$$ and ${{{ C}}}(u)$ and ${{{ D}}}(u)$ are defined in condition [(C6)]{}. To apply Theorem \[theo1\] to construct a confidence region or interval for $\beta_0$, we need to consistently estimate the unknown weights $w_j$. By the “plug-in” method, ${{{ A}}}(\beta_0)$ and ${{{ B}}}(\beta_0)$ can be consistently estimated by $$\hat{{{{ A}}}}(\hat{\beta})=\frac{1}{n}\sum_{i=1}^n \{\hat {V}_i\hat{V}_i^T - \hat{{{{ C}}}}(\hat{\beta}^TX_i)\hat{{{{ D}}}}^{-1}(\hat{\beta }^TX_i)\hat{{{{ C}}}}{}^T(\hat{\beta}^TX_i) \} \label{eq2.7}$$ and $$\hat{{{{ B}}}}(\hat{\beta})=\frac{1}{n}\sum_{i=1}^n\hat{V}_i\hat{V}_i^T, \label{eq2.8}$$ respectively, where $\hat{\beta}$ is the maximum empirical likelihood estimator of $\beta_0$ defined by (\[eq2.4\]), $\hat{V}_i=X_i\hat{\dot{{\rm g}}}{}^T(\hat{\beta}^TX_i;\hat{\beta })Z_iw(\hat{\beta}^TX_i)$, $\hat{{{{ C}}}}(\cdot)=\sum_{i=1}^nW_{ni}(\cdot)\hat{V}_iZ_i^T$ and $\hat{{{{ D}}}}(\cdot)=\break\sum_{i=1}^nW_{ni}(\cdot)Z_iZ_i^T$ with $$W_{ni}(\cdot)=K_1 \biggl(\frac{\hat{\beta}^TX_i-\cdot}{b_n} \biggr) \bigg/\sum_{k=1}^nK_1 \biggl(\frac{\hat{\beta}^TX_k-\cdot }{b_n} \biggr),$$ where $K_1(\cdot)$ is a kernel function, and $b_n$ is a bandwidth with $0<b_n\rightarrow0$. This implies that the eigenvalues of $\hat{{{ G}}}(\hat{\beta})=\hat{{{ B}}}^{-1}(\hat{\beta})\hat{{{ A}}}(\hat{\beta})$, say $\hat{w}_j$, consistently estimate $w_j$ for $j=1,\ldots,p$. Let $\hat{c}_{1-\alpha}$ be the $1-\alpha$ quantile of the conditional distribution of the weighted sum $\hat{s} = \hat{w}_1\chi_{1,1}^2+\cdots +\hat{w}_p\chi_{1,p}^2$ given the data. Then an approximate $1-\alpha$ confidence region for $\beta_0$ can be defined as $${{\mathcal}R}_{\rm eel}(\alpha) = \{\beta\in{{\mathcal}B}\dvt -2\log\hat{L}(\beta)\leq\hat{c}_{1-\alpha} \}.$$ In practice, the conditional distribution of the weighted sum $\hat{s}$, given the sample $\{(Y_i,X_i, Z_i),1\leq i\leq n\}$, can be calculated using Monte Carlo simulations by repeatedly generating independent samples $\chi_{1,1}^2,\ldots,\chi_{1,p}^2$ from the $\chi_1^2$ distribution. The following theorem states an interesting result about $\hat \beta$. The asymptotic variance of $\hat\beta$ is smaller than that of Härdle *et al.* [@HarHalIch93] when our model reduces to a single-index model.=-1 \[theo2\] Suppose that conditions [(C1)–(C7)]{} hold. Then $$\sqrt{n} (\hat{\beta}-\beta_0 )\stackrel {D}{\longrightarrow}N (0,\sigma^{2}{{{ B}}_*^{-1}}(\beta_0){{{ A}}}(\beta_0){{{ B}}_*^{-1}}(\beta_0) ),$$ where ${{{ B}}_*}(\beta_0)$ and ${{{ A}}}(\beta_0)$ are defined in condition [(C7)]{} and [(\[eq2.6\])]{}, respectively. In model (\[eq1.1\]), if $q=1$ and $Z=1$, then (\[eq1.1\]) reduces to the single-index model. By Theorem \[theo2\], we derive the following result. \[cor1\] Suppose that the conditions of Theorem \[theo2\] hold. If $q=1$ and $Z=1$ in model (\[eq1.1\]), then $$\sqrt{n}(\hat{\beta}-\beta_0)\stackrel{D}{\longrightarrow}N (0,\sigma^{2}{{{ A}}}_1^-(\beta_0)),$$ where ${{{ A}}}_1(\beta_0)=E[\{X-E(X|\beta_0^TX)\}\{X-E(X|\beta_0^TX)\}^T\dot {{\rm g}}_0^2(\beta_0^TX)w(\beta_0^TX)]$ and $A_1^{-}$ represents a generalized inverse of the matrix $A_1^{-}$. Corollary \[cor1\] is the same as the results of Härdle et al. [@HarHalIch93] and Xia and Li [@XiaLi99] for the single-index model. For the estimator of the variance of error, $\hat{\sigma}^2$, we have the following result. \[theo3\] Suppose that conditions hold. Then, $$\hat{\sigma}^2-\sigma^2 = \mathrm{O}_P (n^{-1/2} ).$$ To apply Theorem \[theo2\] to construction of the confidence region of $\beta_0$, we use the estimators $\hat{\sigma}^2$ and $\hat{A}(\hat{\beta})$ defined in (\[eq2.5\]) and (\[eq2.7\]), and define the estimator of $B_*(\beta_0)$ as follows $$\hat{B}_*(\hat{\beta})=\frac{1}{n}\sum_{i=1}^n\{\hat{V}_i\hat{V}_i^T-\hat{C}(\hat{\beta}^TX_i)\hat{\dot{g}}(\hat{\beta}^TX_i;\hat\beta)\hat{\mu}^T(\hat{\beta}^TX_i) \},$$ where $\hat{\mu}(\cdot)=\sum_{i=1}^nW_{ni}(\cdot)X_i$ is the estimator of $\mu(u)=E(X|\beta_0^TX=u)$. It can be shown that $\hat{A}(\hat{\beta})\stackrel{P}{\longrightarrow}{A}(\beta_0)$ and $\hat{B}_*(\hat{\beta})\stackrel{P}{\longrightarrow}{B}_*(\beta_0)$, where $\stackrel{P}{\longrightarrow}$ denotes convergence in probability. By Theorems \[theo3\] and \[theo4\], we have $$\{\hat{\sigma}^2{\hat{{{ B}}}_*^{-1}}(\hat{\beta})\hat{{{{ A}}}}(\hat{\beta}){\hat{{{ B}}}_*^{-1}}(\hat{\beta}) \}^{-1/2} \sqrt{n}(\hat{\beta}-\beta_0) \stackrel{D}{\longrightarrow}N(0,{{{ I}}}_p).$$ Using Theorem 10.2d in Arnold [@Arn81], we obtain $$(\hat{\beta}-\beta_0)^T \{n^{-1}\hat{\sigma}^2{\hat{{{ B}}}_*^{-1}}(\hat{\beta})\hat{{{{ A}}}}(\hat{\beta}){\hat{{{ B}}}_*^{-1}}(\hat{\beta}) \}^{-} (\hat{\beta}-\beta_0) \stackrel{D}{\longrightarrow}\chi_p^2.\vadjust{\goodbreak}$$ Let $\chi_p^2(1-\alpha)$ be the $1-\alpha$ quantile of $\chi_p^2$ for $0<\alpha<1$. Then $$ \{\beta\dvt (\hat{\beta}-\beta)^T (n^{-1}\hat{\sigma}^2{\hat{{{ B}}}_*^{-1}}(\hat{\beta})\hat{{{{ A}}}}(\hat{\beta}) {\hat{{{ B}}}_*^{-1}}(\hat{\beta}) )^{-} (\hat{\beta}-\beta)\le\chi_p^2(1-\alpha) \}$$ gives an approximate $1-\alpha$ confidence region for $\beta_0$. Adjusted empirical likelihood {#sec3} ============================= In addition to the above, direct way of approximating the asymptotic distributions, we can also consider the following alternative. The alternative is motivated by the results of Rao and Scott [@RaoSco81]. By Rao and Scott [@RaoSco81] the distribution of $\rho(\beta_0)\sum_{i=1}^p w_i\chi_{1,i}^2$ can be approximated by $\chi_p^2$, where $\rho(\beta_0)=p/\operatorname{tr}\{{{{ G}}}(\beta_0)\}$. Let $\hat\rho(\hat\beta)=p/\operatorname{tr}\{\hat{{{ G}}}(\hat\beta)\}$ with $\hat{{{ G}}}(\hat{\beta})=\hat{{{ A}}}^{1/2}(\hat{\beta})\hat{{{ B}}}^{-1}(\hat{\beta})\hat{{{ A}}}^{1/2}(\hat{\beta})$, where $\hat{{{ A}}}(\hat{\beta})$ and $\hat{{{ B}}}(\hat{\beta})$ are defined in (\[eq2.7\]) and (\[eq2.8\]). Invoking Theorem \[theo1\] and the consistency of $\hat{{{ G}}}(\hat{\beta})$, the asymptotic distribution of $\hat{\rho}(\hat{\beta})\{-2\log\hat{L}(\beta)\}$ can be approximated by $\chi_p^2$. Clearly, $\hat\beta$ in $\hat\rho(\cdot)$ can be replaced by $\beta$. Therefore, an improved Rao–Scott adjusted empirical log-likelihood can be defined as $$\tilde{l}(\beta)=\hat{\rho}(\beta)\{-2\log\hat{L}(\beta)\}.$$ However, the accuracy of this approximation still depends on the values of the $w_i$s. Now, we propose another adjusted empirical log-likelihood, whose asymptotic distribution is chi-squared with $p$ degrees of freedom. The adjustment technique is developed by Wang and Rao [@WanRao02] by using an approximate result in Rao and Scott [@RaoSco81]. Note that $\hat{\rho}(\beta)$ can be written as $$\hat{\rho}(\beta)= \frac{\operatorname{tr}\{\hat{{{ A}}}^{-}(\beta)\hat{{{ A}}}(\beta)\}} {\operatorname{tr}\{\hat{{{ B}}}^{-1}(\beta)\hat{{{ A}}}(\beta)\}}.$$ By examining the asymptotic expansion of $-2\log\hat{L}(\beta)$, which is specified in the proof of Theorem \[theo4\] below, we define an adjustment factor $$\hat{r}({\beta})= \frac{\operatorname{tr}\{\hat{{{ A}}}^{-}(\beta)\hat{{{\Sigma}}}(\beta)\}} { \operatorname{tr}\{\hat{{{ B}}}^{-1}(\beta)\hat{{{\Sigma}}}(\beta)\}},$$ by replacing ${\hat A}(\beta)$ in ${\hat\rho}(\beta)$ by ${\hat {{{\Sigma}}}}(\beta) $, where $\hat{{{\Sigma}}}(\beta) = \{\sum_{i=1}^n\hat{\eta}_i(\beta) \} \{\sum _{i=1}^n\hat{\eta}_i(\beta) \}^T$. The adjusted empirical log-likelihood ratio is defined by $$\hat{l}_{\rm ael}(\beta)=\hat{r}(\beta)\{-2\log\hat{L}(\beta)\}, \label{eq3.1}$$ where $\log\hat{L}(\beta)$ is defined in (\[eq2.2\]). \[theo4\] Suppose that conditions [(C1)–(C6)]{} hold. Then $ \hat{l}_{\rm ael}(\beta_0)\stackrel{D}{\longrightarrow}\chi_p^2. $ According to Theorem \[theo4\], $\hat{l}_{\rm ael}(\beta)$ can be used to construct an approximate confidence region for $\beta_0$. Let $${{\mathcal}R}_{\rm ael}(\alpha) = \{\beta\in{{\mathcal}B}\dvt \hat{l}_{\rm ael}(\beta)\leq\chi_p^2(1-\alpha) \}.$$ Then, ${{\mathcal}R}_{\rm ael}(\alpha)$ gives a confidence region for $\beta_0$ with asymptotically correct coverage probability $1-\alpha$. Numerical results {#sec4} ================= Bandwidth selection {#sec4.1} ------------------- Various existing bandwidth selection techniques for nonparametric regression, such as the cross-validation and generalized cross-validation, can be adapted for the estimation $\hat{{\rm g}}(\cdot)$. But we, in our simulation, use the modified multi-fold cross-validation (MMCV) criterion proposed by Cai, Fan and Yao [@CaiFanYao00] to select the optimal bandwidth because the algorithm is simple and quick. Let $m$ and $Q$ be two given positive integers and $n>mQ$. The basic idea is first to use $Q$ sub-series of lengths $n-km$ $(k=1,\ldots,Q)$ to estimate the unknown coefficient functions and then to compute the one-step forecasting error of the next section of the sample of lengths $m$ based on the estimated models. More precisely, we choose $h$ which minimizes $$\operatorname{AMS}(h) = \sum_{k=1}^Q\operatorname{AMS}_k(h), \label{eq4.1}$$ where, for $k=1,\ldots,Q$, $$\operatorname{AMS}_k(h)=\frac{1}{m}\sum_{i=n-km+1}^{n-km+m}\Biggl \{Y_i -\sum_{j=1}^q\hat{g}_{j,k}(U_i)Z_{ij} \Biggr\}^2,$$ and $\{\hat{g}_{j,k}(\cdot)\}$ are computed from the sample $\{(Y_i,U_i,Z_i),1\leq i\leq n-km\}$ with bandwidth equal $h(\frac{n}{n-km})^{1/5}$. Note that for different sample size, we re-scale bandwidth according to its optimal rate, that is, $h\propto n^{-1/5}$. Since the selected bandwidth does not depend critically on the choice of $m$ and $Q$, to computation expediency, we take $m=[0.1n]$ and $Q=4$ in our simulation. Let $h_{\rm opt}$ be the bandwidth obtained by minimizing (\[eq4.1\]) with respect to $h > 0$; that is, $h_{\rm opt}=\inf_{h>0}\operatorname{AMS}(h)$. Then $h_{\rm opt}$ is the optimal bandwidth for estimating $\hat{{\rm g}}(\cdot)$. When calculating the empirical likelihood ratios and estimator of $\beta_0$, we use the approximation bandwidth $$h=h_{\rm opt}n^{-1/20}(\log n)^{-1/2}, \qquad h_1=h_{\rm opt},$$ because this insures that the required bandwidth has correct order of magnitude for the optimal asymptotic performance (see, e.g., Carroll *et al.* [@Caretal97]), and the bandwidth $\hat{h}$ satisfies condition (C4). Simulation study {#sec4.2} ---------------- We now examine the performance of the procedures described in Sections \[sec2\] and \[sec3\]. Consider the regression model $$Y_i = g_0(\beta_0^TX_i) + g_1(\beta_0^TX_i)Z_{i1} + g_2(\beta_0^TX_i)Z_{i2} + \varepsilon_i, \label{eq4.2}$$ where $\beta_0=(1/\sqrt{5},2/\sqrt{5})^T$ and the $\varepsilon_i$s are independent $N(0,0.8^2)$ random variables. The sample $\{X_i=(X_{i1},X_{i2})^T;1\leq i\leq n\}$ was generated from a bivariate uniform distribution on $[-1,1]^2$ with independent components, $\{Z_i=(Z_{i1},Z_{i2})^T;1\leq i\leq n\}$ was generated from a bivariate normal distribution $N(0,\Sigma)$ with $\operatorname{var}(Z_{i1})=\operatorname{var}(Z_{i2})=1$ and the correlation coefficient between $Z_{i1}$ and $Z_{i2}$ is $\rho=0.6$. In model (\[eq4.2\]), the coefficient functions are $g_0(u)=12\exp(-2u^2)$, $g_1(u)=10u^2$ and $g_2(u)=16\sin(\uppi u)$. For the smoother, we used a local linear smoother with a Epanechnikov kernel $K(u)=0.75(1-u^2)_+$ with a MMCV bandwidth throughout all smoothing steps. We take the weight function $w(u)=I_{[-3/\sqrt{5},3/\sqrt{5}]}(u)$. The sample size for the simulated data is $100$, and the run is 500 times in all simulations. The confidence regions of $\beta_0$ and their coverage probabilities, with nominal level $1-\alpha=0.95$, were computed from 500 runs. Four methods were used to construct the confidence regions: the estimated empirical likelihood (EEL) with a conditional approximation, the adjusted empirical likelihood (AEL), the improved Rao–Scott adjusted empirical likelihood (IRSAEL) and the normal approximation (NA). A comparison among three methods was made through coverage accuracies and coverage areas of the confidence regions. The simulated results are given in Figure \[fig1\]. ![Averages of 95% confidence regions of $(\beta_1,\beta_2)$, based on EEL (solid curve), AEL (dashed curve), IRSAEL (doted curve) and NA (dot-dashed curves) when $n=100$.[]{data-label="Fig:1"}](365f01.eps) \[fig1\] From Figure \[fig1\] we can see that EEL, AEL and IRSAEL give smaller confidence regions than NA, and the region obtained by AEL is much smaller than the others. Thus, AEL is the best of the four algorithms. The histograms of the 500 estimators of the parameter $\beta_1$ and $\beta_2$ are in Figures \[fig2\](a) and (b), respectively. The Q–Q plots of the 500 estimators of the parameter $\beta_1$ and $\beta_2$ are in Figures \[fig2\](c) and (d), respectively. [cc]{} ![(a) for $\beta_1$ and (b) for $\beta_2$: the histograms of the 500 estimators of every parameter, the estimated curve of density (solid curve) and the curve of normal density (dashed curve); (c) for $\beta_1$ and (d) for $\beta_2$: the Q–Q plot of the 500 estimators of every parameter.[]{data-label="Fig:2"}](365f02a.eps "fig:") &![(a) for $\beta_1$ and (b) for $\beta_2$: the histograms of the 500 estimators of every parameter, the estimated curve of density (solid curve) and the curve of normal density (dashed curve); (c) for $\beta_1$ and (d) for $\beta_2$: the Q–Q plot of the 500 estimators of every parameter.[]{data-label="Fig:2"}](365f02b.eps "fig:")\ (a) Histogram&(b) Histogram\ ![(a) for $\beta_1$ and (b) for $\beta_2$: the histograms of the 500 estimators of every parameter, the estimated curve of density (solid curve) and the curve of normal density (dashed curve); (c) for $\beta_1$ and (d) for $\beta_2$: the Q–Q plot of the 500 estimators of every parameter.[]{data-label="Fig:2"}](365f02c.eps "fig:") &![(a) for $\beta_1$ and (b) for $\beta_2$: the histograms of the 500 estimators of every parameter, the estimated curve of density (solid curve) and the curve of normal density (dashed curve); (c) for $\beta_1$ and (d) for $\beta_2$: the Q–Q plot of the 500 estimators of every parameter.[]{data-label="Fig:2"}](365f02d.eps "fig:")\ (c) Normal Q–Q Plot&(d) Normal Q–Q Plot \[fig2\] Figure \[fig2\] shows empirically that these estimators are asymptotically normal. The means of the estimates of the unknown parameters $\beta_1$ and $\beta_2$ are 0.44734 and 0.89502, respectively, and their biases (standard deviations) are 0.000131 (0.00302) and 0.000596 (0.00257), respectively. We also consider the average estimates of the coefficient functions ${g}_{0}(u)$, ${g}_{1}(u)$ and ${g}_{2}(u)$ over the 500 replicates. The estimators $\hat{g}_j(\cdot)$ are assessed via the root mean squared errors (RMSE); that is, $\mathrm{RMSE} = \sum_{j=0}^2{\rm RMSE}_j$, where $${\rm RMSE}_j= \Biggl[n_{\rm grid}^{-1}\sum_{k=1}^{n_{\rm grid}} \{\hat{g}_j(u_k)-g_j(u_k)\}^2 \Biggr]^{1/2},$$ and $\{u_k, k=1,\ldots,n_{\rm grid}\}$ are regular grid points. The boxplot for the 500 RMSEs is given in Figure \[fig3\]. [cc]{} ![The true cure (solid curve) and the estimated curve (dashed curve). (a) for $g_{0}(\cdot)$, (b) for $g_{1}(\cdot)$, (c) for $g_{2}(\cdot)$; (d) the boxplots of the 500 RMSE values in estimations of $g_0(\cdot)$, $g_1(\cdot)$, $g_2(\cdot)$ and the sum of the three RMSEs.[]{data-label="Fig:3"}](365f03a.eps "fig:") &![The true cure (solid curve) and the estimated curve (dashed curve). (a) for $g_{0}(\cdot)$, (b) for $g_{1}(\cdot)$, (c) for $g_{2}(\cdot)$; (d) the boxplots of the 500 RMSE values in estimations of $g_0(\cdot)$, $g_1(\cdot)$, $g_2(\cdot)$ and the sum of the three RMSEs.[]{data-label="Fig:3"}](365f03b.eps "fig:")\ (a)&(b)\ ![The true cure (solid curve) and the estimated curve (dashed curve). (a) for $g_{0}(\cdot)$, (b) for $g_{1}(\cdot)$, (c) for $g_{2}(\cdot)$; (d) the boxplots of the 500 RMSE values in estimations of $g_0(\cdot)$, $g_1(\cdot)$, $g_2(\cdot)$ and the sum of the three RMSEs.[]{data-label="Fig:3"}](365f03c.eps "fig:") &![The true cure (solid curve) and the estimated curve (dashed curve). (a) for $g_{0}(\cdot)$, (b) for $g_{1}(\cdot)$, (c) for $g_{2}(\cdot)$; (d) the boxplots of the 500 RMSE values in estimations of $g_0(\cdot)$, $g_1(\cdot)$, $g_2(\cdot)$ and the sum of the three RMSEs.[]{data-label="Fig:3"}](365f03d.eps "fig:")\ (c)&(d) \[fig3\] From Figures \[fig3\](a)–(c) we see every estimated curve agrees with the true function curve very closely. Figure \[fig3\](d) shows that all RMSEs of estimates for the unknown functions are very small. \[appm\] Appendices {#appendices .unnumbered} ========== We divide the appendices into Appendix \[appmA\] and Appendix \[appmB\]. The proofs of are presented in Appendix \[appmA\], and the proofs of Lemmas \[lemm2\] and \[lemm3\] are presented in Appendix \[appmB\]. We use $c$ to represent any positive constant which may take a different value for each appearance. Proofs of theorems {#appmA} ================== The following lemma gives uniformly convergent rates of $\hat{{\rm g}}(u;\beta)$ and $\hat{\dot{{\rm g}}}(u;\beta)$. This lemma is a straightforward extension of known results in nonparametric function estimation; for its proof, the reader may refer to Theorem \[theo2\] in Wang and Xue [@WanXue11], we hence omit the proof. \[lemm1\] Suppose that conditions [(C1)–(C3)]{}, [(C5)]{} and [(C6)]{} hold. Then $$\sup_{u\in{{\mathcal}U}_w,\beta\in{{\mathcal}B}_n}\|\hat{{\rm g}}(u;\beta)-{{\rm g}}_0(u)\| = \mathrm{O}_P \biggl(\biggl \{\frac{\log(1/h)}{nh} \biggr\}^{1/2} + h^2 \biggr)$$ and $$\sup_{u\in{{\mathcal}U}_w,\beta\in{{\mathcal}B}_n}\|\hat{\dot{{\rm g}}}(u;\beta)-\dot{{\rm g}}_0(u)\| = \mathrm{O}_P \biggl(\biggl \{\frac{\log(1/h)}{nh^3} \biggr\}^{1/2} + h \biggr).$$ Denote ${{\mathcal}G}=\{{{\rm g}}\dvt {{\mathcal}U}_w\times{{\mathcal}B}\mapsto R^q\}$, $\|{{\rm g}}\|_{{\mathcal}G}=\sup_{u\in{{\mathcal}U}_w,\beta\in{{\mathcal}B}_n}\|{{\rm g}}(u;\beta)\|$. From Lemma \[lemm1\], we have $\|\hat{{\rm g}}-{{\rm g}}_0\|_{{\mathcal}G}=\mathrm{o}_P(1)$ and $\|\hat{\dot{{\rm g}}}-\dot{{\rm g}}_0\|_{{\mathcal}G}=\mathrm{o}_P(1)$; hence we can assume that ${{\rm g}}$ lies in ${{\mathcal}G}_{\delta}$ with $\delta=\delta_n\rightarrow0$ and $\delta>0$, where $${{\mathcal}G}_{\delta}=\{{{\rm g}}\in{{\mathcal}G}\dvt \|{{\rm g}}-{{\rm g}}_0\| _{{\mathcal}G}\leq \delta,\|\dot{{\rm g}}-\dot{{\rm g}}_0\|_{{\mathcal}G}\leq\delta\}. \label{eqA.1}$$ Let ${\rm g}_0(\beta^TX;\beta)=E\{{\rm g}_0(\beta_0^TX)|\beta^TX\}$ and $\dot{\rm g}_0(\beta^TX;\beta)=E\{\dot{\rm g}_0(\beta_0^TX)|\beta^TX\}$, $$\begin{aligned} \label{eqA.2} Q({{\rm g}},\beta) & =&E[\{Y-{{\rm g}}^T(\beta^TX;\beta)Z\}\dot{{\rm g}}{}^T(\beta^TX;\beta)ZXw(\beta^TX)], \\ \label{eqA.3} Q_n({{\rm g}},\beta) & =&\frac{1}{n}\sum_{i=1}^n\{Y_i-{{\rm g}}^T(\beta^TX_i;\beta)Z_i\}\dot{{\rm g}}{}^T(\beta^TX_i;\beta )Z_iX_iw(\beta^TX_i).\end{aligned}$$ The following two lemmas are required for obtaining the proofs of the theorems; their proofs can be found in Appendix \[appmB\]. \[lemm2\] Suppose that conditions [(C1)–(C6)]{} hold. Then $$\begin{aligned} \label{eqA.4} \sup_{({{\rm g}},\beta)\in{{\mathcal}G}_{\delta}\times{{\mathcal}B}_n}\| J_1({{\rm g}},\beta)\| &=& \mathrm{o}_P(n^{-1/2}), \\ \label{eqA.5} \sup_{\beta\in{{\mathcal}B}_n} \|J_2(\hat{{\rm g}},\beta)\| &=& \mathrm{o}_P(n^{-1/2}), \\ \label{eqA.6} \sup_{({{\rm g}},\beta)\in{{\mathcal}G}_{\delta}\times{{\mathcal}B}_n}\| J_3({{\rm g}},\beta)\| &=& \mathrm{o}(n^{-1/2}), \\ \label{eqA.7} \sqrt{n}J_4(\hat{{\rm g}},\beta_0)&\stackrel{D}{\longrightarrow }&N(0,\sigma^2{{{ A}}}(\beta_0)),\end{aligned}$$ where ${{{ A}}}(\beta_0)$ is defined in (\[eq2.6\]), $$\begin{aligned} J_1({{\rm g}},\beta) & =&Q_n({{\rm g}},\beta)-Q({{\rm g}},\beta)-Q_n({{\rm g}}_0,\beta_0), \\ J_2({{\rm g}},\beta) & =& Q({{\rm g}},\beta)-Q({{\rm g}}_0,\beta) \\ &&{} -\varpi({{\rm g}}_0(\beta^TX; \beta);\beta)\{{{\rm g}}(\beta ^TX;\beta)-{{\rm g}}_0(\beta^TX; \beta)\}, \\ J_3({{\rm g}},\beta) & =&\varpi({{\rm g}}_0(\beta^TX),\beta)\{{{\rm g}}(\beta^TX;\beta )-{{\rm g}}_0(\beta^TX)\} \\ &&{} -\varpi({{\rm g}}_0(\beta_0^TX; \beta),\beta_0)\{{{\rm g}}(\beta _0^TX;\beta_0)-{{\rm g}}_0(\beta_0^TX; \beta)\}\end{aligned}$$ and $$J_4(\beta_0,{{\rm g}})=Q_n({{\rm g}}_0,\beta_0) + \varpi({{\rm g}}_0(\beta_0^TX),\beta_0)\{{{\rm g}}(\beta_0^TX;\beta _0)-{{\rm g}}_0(\beta_0^TX)\}.$$ \[lemm3\] Suppose that conditions [(C1)–(C6)]{} hold. Then $$\begin{aligned} \label{eqA.8} \sup_{\beta\in{{\mathcal}B}_n}\|Q_n(\hat{{\rm g}},\beta)\| &=& \mathrm{O}_P(n^{-1/2}), \\ \label{eqA.9} \sup_{\beta\in{{\mathcal}B}_n}\|{{{ R}}}_n(\beta)- \sigma^2{{{ B}}}(\beta _0)\|&=& \mathrm{o}_P(1), \\ \label{eqA.10} \sup_{\beta\in{{\mathcal}B}_n}\max_{1\leq i\leq n}\|\hat{\eta }_i(\beta)\| &=& \mathrm{o}_P(n^{1/2}), \\ \label{eqA.11} \sup_{\beta\in{{\mathcal}B}_n}\|\lambda(\beta)\| &=& \mathrm{o}_P(n^{-1/2}),\end{aligned}$$ where $Q_n(\hat{{\rm g}},\beta)$ is defined in [(\[eqA.3\])]{}, ${{{ R}}}_n(\beta)=n^{-1}\sum_{i=1}^n\hat{\eta}_i(\beta)\hat{\eta }_i^T(\beta)$, ${{{ B}}}(\beta_0)$ is defined in condition [(C7)]{} and $\hat{\eta}_i(\beta)$ is defined in (\[eq2.2\]). [Proof of Theorem \[theo1\]]{} Note that, when $\beta=\beta_0$, Lemma \[lemm3\] also holds. Applying the Taylor expansion to (\[eq2.2\]) and invoking Lemma \[lemm3\], we can obtain $$-2\log\hat{L}(\beta_0) = -\sum_{i=1}^n \biggl[\lambda^T\hat{\eta}_i(\beta_0)-\frac {1}{2} \{\lambda^T\hat{\eta}_i(\beta_0) \}^2 \biggr]+\mathrm{o}_P(1). \label{eqA.12}$$ By (\[eq2.3\]) and Lemma \[lemm3\], we have $$\sum_{i=1}^n \{\lambda^T\hat{\eta}_i(\beta_0)\}^2 = \sum _{i=1}^n\lambda^T\hat{\eta}_i(\beta_0)+\mathrm{o}_P(1)$$ and $$\lambda= \Biggl\{\sum_{i=1}^n\hat{\eta}_i(\beta_0)\hat{\eta }_i^T(\beta_0) \Biggr\}^{-1} \sum_{i=1}^n \hat{\eta}_i(\beta_0) + \mathrm{o}_P (n^{-1/2} ).$$ This together with (\[eqA.12\]) proves that $$-2\log\hat{L}(\beta_0) = nQ_n^T(\hat{{\rm g}},\beta_0){{{ R}}}_n^{-1}(\beta_0)Q_n(\hat{{\rm g}},\beta_0)+\mathrm{o}_P(1), \label{eqA.13}$$ where $Q_n(\hat{{\rm g}},\beta_0)$ and ${{{ R}}}_n(\beta_0)$ are defined in (\[eqA.3\]) and (\[eqA.9\]), respectively. From (\[eqA.9\]) of Lemma \[lemm3\] and (\[eqA.13\]), we obtain $$\hspace*{-10pt}-2\log\hat{L}(\beta_0)\!=\! \bigl\{(\sigma^2{{{ A}}})^{-{{1}/{2}}}\sqrt{n}Q_n(\hat{{\rm g}},\beta_0) \bigr\}^T{{{ G}}}(\beta_0) \bigl\{(\sigma^2{{{ A}}})^{-{{1}/{2}}}\sqrt{n}Q_n(\hat{{\rm g}},\beta_0) \bigr\}\!+\!\mathrm{o}_P(1), \label{eqA.14}$$ where ${{{ G}}}(\beta_0)={{{ A}}}^{1/2}(\beta_0){{{ B}}}^{-1}(\beta_0){{{ A}}}^{1/2}(\beta_0)$. Let ${{{ G}}}_0 = \operatorname{diag}(w_1,\ldots,w_p)$, where $w_i$, , are the eigenvalues of ${{{ G}}}(\beta_0)$. Then there exists an orthogonal matrix ${{{ H}}}$ such that ${{{ H}}}^T{{{ G}}}_0{{{ H}}}={{{ G}}}(\beta_0)$. Using the notations of Lemma \[lemm2\], we have $$Q_n(\hat{{\rm g}},\beta) = J_1(\hat{{\rm g}},\beta) + J_2(\hat {{\rm g}},\beta) + J_3(\hat{{\rm g}},\beta) + J_4(\hat{{\rm g}},\beta_0) + Q({{\rm g}}_0,\beta). \label{eqA.15}$$ Noting that $Q({{\rm g}}_0,\beta_0)=0$, from the above equation and Lemma \[lemm2\], we have $$Q_n(\hat{{\rm g}},\beta_0)=J_4(\hat{{\rm g}},\beta_0) + \mathrm{o}_P(n^{-1/2}).$$ Hence, by (\[eqA.7\]) of Lemma \[lemm2\], we have $${{{ H}}}\{\sigma^{-2}{{{ A}}}^{-}(\beta_0)\}^{1/2}\sqrt{n}Q_n(\hat{{\rm g}},\beta_0) \stackrel{{D}}{\longrightarrow} N(0,{{{ I}}}_{p}),$$ where ${{{ I}}}_{p}$ is the $p\times p$ identity matrix. This together with (\[eqA.14\]) proves Theorem \[theo1\]. [Proof of Theorem \[theo2\]]{} Under the conditions of Theorem \[theo2\], we can follow similar arguments to those used by Wang and Xue [@WanXue11] and show that $\hat {\beta}$ is a root-$n$ consistent estimator of $\beta_0$. Because the proof is straightforward, we do not present it here. We next demonstrate the asymptotic normality of $\hat{\beta}$. By Lemma \[lemm3\] and, similarly to the proof of (\[eqA.13\]), we can obtain $$\log\hat{L}(\beta) = -\frac{n}{2}Q_n^T(\hat{{\rm g}},\beta)\{\sigma^2{{{ B}}}(\beta)\} ^{-1}Q_n(\hat{{\rm g}},\beta)+\mathrm{o}_P(1), \label{eqA.16}$$ uniformly for $\beta\in{{\mathcal}B}_n$, where $\mathrm{o}_P(1)$ tends to 0 in probability uniformly for $\beta\in{{\mathcal}B}_n$. Since the estimator $\hat\beta$ is a maximum of $\log\hat{L}(\beta)$, and ${{{ B}}}(\beta_0)$ is a positive definite matrix, the resulting estimator $\hat{\beta}$ is equivalent to solving the estimation equation $Q_n(\hat{{\rm g}},\beta)=0$; that is, $Q_n(\hat{{\rm g}},\hat{\beta})=0$. Note that $Q({{\rm g}}_0,\beta _0)=0$, and we then have, by Taylor’s expansion, that $$Q({{\rm g}}_0,\beta) = -{{{ B}}_*}(\beta_0)(\beta-\beta_0)+\mathrm{o}(n^{-1/2}), \label{eqA.17}$$ uniformly for $\beta\in{{\mathcal}B}_n$, where ${{{ B}}_*}(\beta_0)$ is the same as that in (\[eqA.9\]). By (\[eqA.15\]), (\[eqA.17\]) and (\[eqA.4\])–(\[eqA.6\]) of Lemma \[lemm2\], we have $$Q_n(\hat{{\rm g}},\hat{\beta})=J_4(\hat{{\rm g}},\beta_0) - {{{ B}}_*}(\beta_0)(\hat{\beta}-\beta_0) + \mathrm{o}_P(n^{-1/2}).$$ Noting that $Q_n(\hat{{\rm g}},\hat{\beta})=0$, we get $$\sqrt{n} (\hat{\beta}-\beta_0 ) = \sqrt{n} {{ B}}_*^{-1}(\beta_0)J_4(\hat{{\rm g}},\beta_0) + \mathrm{o}_P(1).$$ This together with (\[eqA.7\]) of Lemma \[lemm2\] proves Theorem \[theo2\]. [Proof of Theorem \[theo3\]]{} Decomposing $\hat{\sigma}^2$ into several parts, we get $$\begin{aligned} \hat{\sigma}^2 & =& \frac{1}{n}\sum_{i=1}^n\varepsilon_i^2 + \frac{1}{n}\sum_{i=1}^n [{{\rm g}}_0(X_i^T\beta_0)-\hat{{\rm g}}(X_i^T\hat{\beta};\hat{\beta})\}^TZ_i ]^2\\ &&{} + \frac{2}{n}\sum_{i=1}^n\varepsilon_i \{{{\rm g}}_0(X_i^T\beta_0)-\hat{{\rm g}}(X_i^T\hat{\beta};\hat{\beta }) \}^T Z_i\\ & \equiv& I_1 + I_2 + I_3.\end{aligned}$$ Using the central limit theorem, we have $$\sqrt{n}(I_1-\sigma^2)=\frac{1}{\sqrt{n}}\sum_{i=1}^n(\varepsilon _i^2-\sigma^2) \stackrel{D}{\longrightarrow}N(0,\operatorname{var}(\varepsilon^2)).$$ By Lemma \[lemm1\], we can obtain $$|I_2| \leq\frac{1}{n}\sum_{i=1}^n\|Z_i\|^2 \Bigl\{\sup_{(u,\beta)\in({{\mathcal}U}_w,{{\mathcal}B}_n)}\|\hat{{\rm g}}(u;\beta)-{{\rm g}}_0(u)\| \Bigr\}^2 = \mathrm{o}_P (n^{-1/2} ).$$ For $I_3$, we have $$\begin{aligned} I_3 & =& \frac{2}{n}\sum_{i=1}^n\varepsilon_i \{{{\rm g}}_0(X_i^T\beta_0)-\hat{{\rm g}}(X_i^T\beta_0;\beta_0) \}^T Z_i\\ & &{} + \frac{2}{n}\sum_{i=1}^n\varepsilon_i \{\hat{{\rm g}}(X_i^T\beta_0;\beta_0)-\hat{{\rm g}}(X_i^T\hat{\beta};\hat {\beta}) \}^T Z_i\\ & \equiv& I_{31} + I_{32}.\end{aligned}$$ It is not hard to show that $I_{31}=\mathrm{O}_P (n^{-1/2} )$. By Theorems \[theo1\] and \[theo3\], we obtain $$|I_{32}|\leq\frac{2}{n}\sum_{i=1}^n\bigl(\|Z_i\||\varepsilon_i|\|X_i-E(X_i|\beta_0^TX_i)\|\bigr)\|\hat{\beta}-\beta_0\|\mathrm{O}_P(1) = \mathrm{O}_P(n^{-1/2}).$$ This together with above results proves Theorem \[theo3\]. [Proof of Theorem \[theo4\]]{} Note that $\hat{{{ A}}}(\beta_0)\stackrel{P}{\longrightarrow}{{{ A}}}(\beta_0)$ and $\hat{{{ B}}}(\beta_0)\stackrel{P}{\longrightarrow}{{{ B}}}(\beta_0)$. By the expansion of $\hat{l}_{\rm ael}(\beta_0)$, defined in (\[eq3.1\]) and (\[eqA.16\]), we get $$\hat{l}_{\rm ael}(\beta_0) = nQ_n^T(\hat{{\rm g}},\beta_0)\{\sigma^{-2}{{{ A}}}^{-}(\beta_0)\} Q_n(\hat{{\rm g}},\beta_0)+\mathrm{o}_P(1). \label{eqA.18}$$ This together with (\[eqA.15\]) and (\[eqA.18\]) proves Theorem \[theo4\]. Proofs of lemmas {#appmB} ================ [Proof of Lemma \[lemm2\]]{} We first prove (\[eqA.4\]). Denote $r_n({{\rm g}},\beta)=\sqrt{n}\{Q_n({{\rm g}},\beta)-Q({{\rm g}},\beta)\}$. Noting that $Q({{\rm g}}_0,\beta_0)=0$, we clearly have $$J_1({{\rm g}},\beta) = n^{-1/2}\{r_n({{\rm g}},\beta)-r_n({{\rm g}}_0,\beta_0)\}. \label{eqB.1}$$ It can be shown that the empirical process $\{r_n({{\rm g}},\beta)\dvt {{\rm g}}\in{{\mathcal}G}_1,\beta\in{{\mathcal}B}_1\} $ has the stochastic equicontinuity, where ${{\mathcal}B}_1=\{\beta\in{{\mathcal}B}\dvt \|\beta-\beta_0\|\leq1\}$ and ${{\mathcal}G}_1$ are defined in (\[eqA.1\]) with $\delta=1$, which are subsets of ${{\mathcal}B}$ and ${{\mathcal}G}$, respectively. The equicontinuity is sufficient for proof of (\[eqA.4\]) since $\delta<1$ for large enough $n$. This stochastic equicontinuity follows by checking the conditions of Theorem \[theo1\] in Doukhan, Massart and Rio [@DouMasRio95]. Therefore, we have $r_n({{\rm g}},\beta)-r_n({{\rm g}}_0,\beta_0)=\mathrm{o}_P(1)$, uniformly for $\beta\in{{\mathcal}B}_1$ and ${{\rm g}}\in{{\mathcal}G}_1$. This together with (\[eqB.1\]) proves (\[eqA.4\]). We now prove (\[eqA.5\]). Define the functional derivative $\varpi({\rm g}_0(\cdot;\beta),\beta)$ of $Q({{\rm g}},\beta)$ with respect to ${{\rm g}}(\cdot;\beta)$ at ${\rm g}_0(\cdot;\beta)$ at the direction ${{\rm g}}(\cdot;\beta)-{\rm g}_0(\cdot;\beta)$ by $$\begin{aligned} && \varpi({\rm g}_0(\cdot;\beta),\beta)\{{{\rm g}}(\cdot;\beta)-{\rm g}_0(\cdot;\beta)\} \\[-1pt] && \quad = \lim_{\tau\rightarrow 0}\bigl[Q\bigl({\rm g}_0(\cdot;\beta)+\tau\bigl({{\rm g}}(\cdot;\beta)-{\rm g}_0(\cdot;\beta)\bigr),\beta\bigr) - Q({\rm g}_0(\cdot;\beta),\beta)\bigr]\cdot\frac{1}{\tau},\end{aligned}$$ where $Q({{\rm g}},\beta)$ is defined in (\[eqA.2\]). We have $$\begin{aligned} \label{eqB.2} && \varpi({{\rm g}}_0(\beta^TX; \beta),\beta)\{{{\rm g}}(\beta^TX;\beta )-{{\rm g}}_0(\beta^TX; \beta)\} \nonumber \\[-8.5pt] \\[-8.5pt] && \quad = - E[\{{{\rm g}}(\beta^TX;\beta)-{{\rm g}}_0(\beta^TX; \beta)\} ^TZ\dot{{\rm g}}_0^T(\beta^TX; \beta)ZXw(\beta^TX)].\nonumber\end{aligned}$$ It follows from (\[eqB.2\]) that $$\begin{aligned} J_2({{\rm g}},\beta) & =& - E [\{{{\rm g}}(\beta^TX;\beta)-{{\rm g}}_0(\beta_0^TX)\} ^TZXZ^T \\[-1pt] &&\hphantom{- E [} {} \times\{{\dot{{\rm g}}}(\beta^TX;\beta)-{\dot{{\rm g}}}_0(\beta ^TX; \beta)\}w(\beta^TX) ],\end{aligned}$$ and hence we have $$\begin{aligned} \label{eqB.3} \omega^TJ_2(\hat{{\rm g}},\beta) & =& - \int\{\hat{{\rm g}}(u;\beta)-{{\rm g}}_0(u)\}^T\mu_\omega(u) \nonumber \\[-8.5pt] \\[-8.5pt] & &\hphantom{- \int}{} \times\{\hat{\dot{{\rm g}}}(u;\beta)-{\dot{{\rm g}}}_0(u)\}w(u)f(u)\,\mathrm{d}u +\mathrm{o}_P(n^{-1/2})\nonumber\end{aligned}$$ for any $p$-dimension vector $\omega$, where $\mu_\omega(u)=E\{Z\omega^TXZ^T|\beta^TX=u\}$, and $f(u)$ is the probability density of $\beta^TX$. Using the standard argument of nonparametric estimation, we can prove $$\hat{{\rm g}}(u;\beta)-{{\rm g}}_0(u) = {{{ D}}}^{-1}(u)\{f(u)\}^{-1}\xi_{n}(u;\beta) + \mathrm{O}_P(c_n), \label{eqB.4}$$ uniformly for $u\in{{\mathcal}U}_w$ and $\beta\in{{\mathcal}B}_n$, where $c_n=n^{-1/2}+h^2$ and ${{{ D}}}(u)$ is defined in condition (C6). $$\xi_{n}(u;\beta) = \frac{1}{n}\sum_{i=1}^nZ_i\{Y_i-{{\rm g}}_0^T(\beta^TX_i)Z_i\} K_h(\beta^TX_i-u).$$ This together with (\[eqB.3\]) derives that $$\begin{aligned} \omega^TJ_2(\hat{{\rm g}},\beta) & =& - \int\{{{{ D}}}^{-1}(u)\xi_{n}(u;\beta)\}^T \mu_\omega(u)\{\hat{\dot{{\rm g}}}(u;\beta)-{\dot{{\rm g}}}_0(u)\}\,\mathrm{d}u + \mathrm{O}_P(c_n) \\[-1pt] & =& -n^{-1/2}\{\gamma_n(\hat{\dot{{\rm g}}},\beta)-\gamma_n(\dot {{\rm g}}_0,\beta)\}+ \mathrm{O}_P(c_n),\end{aligned}$$ where $\gamma_n({\dot{{\rm g}}},\beta)=n^{-1/2}\sum_{i=1}^n\varepsilon _iw(\beta^TX_i)Z_i^T{{{ D}}}^{-1}(\beta^TX_i)\mu_\omega(\beta ^TX_i){\dot{{\rm g}}}(\beta^TX_i;\beta)$. Using the empirical process techniques, and similarly to the proof of (\[eqA.4\]), we can show that the stochastic equicontinuity of $\gamma_n({\dot{{\rm g}}},\beta)$, and hence $\|\gamma_n(\hat{\dot{{\rm g}}},\beta)-\gamma_n(\dot{{\rm g}}_0,\beta)\|=\mathrm{o}_P(1)$. Also, $nh^4=\mathrm{O}(1)$ implies $h^2=\mathrm{O}(n^{-1/2})$, and hence $c_n=\mathrm{O}(n^{-1/2})$. Thus, the proof of (\[eqA.5\]) is complete. We now prove (\[eqA.6\]). Denote $\psi({\dot{{\rm g}}_0},\beta)=\dot{{\rm g}}_0^T(\beta^TX;\beta)ZXw(\beta^TX)$ and $\varphi({{\rm g}},\beta)=\{{{\rm g}}(\beta^TX; \beta)-{{\rm g}}_0(\beta^TX; \beta)\}^TZ$. It follows from (\[eqB.2\]) that $$\begin{aligned} J_3({{\rm g}},\beta) & =& - E\{\varphi({{\rm g}},\beta)\psi({\dot{{\rm g}}_0},\beta)\} + E\{ \varphi({{\rm g}},\beta_0)\psi({\dot{{\rm g}}_0},\beta_0)\} \\ & =& - E[\{\varphi({{\rm g}},\beta)-\varphi({{\rm g}},\beta_0)\}\psi ({\dot{{\rm g}}_0},\beta)] \\ &&{} - E[\varphi({{\rm g}},\beta_0)\{\psi({\dot{{\rm g}}_0},\beta )-\psi({\dot{{\rm g}}_0},\beta_0)\}] \\ & \equiv& J_{31}({{\rm g}},\beta) + J_{32}({{\rm g}},\beta).\end{aligned}$$ By condition (C2), we get $$\begin{aligned} && \|\varphi({{\rm g}},\beta)-\varphi({{\rm g}},\beta_0)\| \\ && \quad = \|[\{{\rm g}(\beta^TX;\beta)-{\rm g}(\beta_0^TX;\beta_0)\}-\{{\rm g}_0(\beta^TX;\beta)-{\rm g}_0(\beta_0^TX)\}]^TZ\| \\ && \quad = \|[\{\dot{\rm g}(\beta_1^TX;\beta_1)-\dot{\rm g}_0(\beta_2^TX)\}(\beta-\beta_0)^T\{X-E(X|\beta_0^TX)\}]^TZ\| \\ && \quad \leq c\|\dot{\rm g}-\dot{\rm g}_0\|_{{\mathcal}G}\|\beta-\beta_0\|(\|X-E(X|\beta_0^TX)\|)(\|Z\|),\end{aligned}$$ where $\beta_1$ and $\beta_2$ are between $\beta$ and $\beta_0$, and $\|\psi({\dot{\rm g}_0},\beta)\|\leq c(\|Z\|)(\|X\|)$. Therefore, we have $\|J_{31}({{\rm g}},\beta)\| = \mathrm{o}(n^{-1/2})$, uniformly for ${{\rm g}}\in{{\mathcal}G}_{\delta}$ and $\beta\in{{\mathcal}B}_n$. Similarly, we can prove $\|J_{32}({{\rm g}},\beta)\|=\mathrm{o}(n^{-1/2})$, uniformly for ${{\rm g}}\in{{\mathcal}G}_{\delta}$ and $\beta\in{{\mathcal}B}_n$, and hence (\[eqA.6\]) follows. Finally, we prove (\[eqA.7\]). Let $f_0(u)$ denote the density function of $\beta_0^TX$. By (\[eqB.2\]) and (\[eqB.4\]), and using the dominated convergence theorem (Loève [@Loe00]), we can obtain $$\begin{aligned} && \varpi({{\rm g}}_0(\beta_0^TX),\beta_0)\{\hat{{\rm g}}(\beta _0^TX;\beta_0)-{{\rm g}}_0(\beta_0^TX)\} \\ && \quad = - \int{{{ C}}}(u)\{\hat{{\rm g}}(u;\beta_0)-{{\rm g}}_0(u)\}f_0(u)\,\mathrm{d}u \\ && \quad = - \frac{1}{n}\sum_{i=1}^n\varepsilon_i{{{ C}}}(\beta_0^TX_i){{{ D}}}^{-1}(\beta_0^TX_i)Z_i + \mathrm{o}_P(c_n).\end{aligned}$$ This together with (\[eqA.3\]) proves that $$J_4(\hat{{\rm g}},\beta_0) = \frac{1}{n}\sum_{i=1}^n\varepsilon_i\zeta_i + \mathrm{o}_P(c_n),$$ where $\zeta_i=V_i - {{{ C}}}(\beta_0^TX_i){{{ D}}}^{-1}(\beta_0^TX_i)Z_i$ and $V_i=X_i\dot{{\rm g}}_0^T(\beta_0^TX_i)Z_iw(\beta_0^TX_i)$. Therefore, by the central limit theorem and Slutsky’s theorem, we get $$\sqrt{n}J_4(\hat{{\rm g}},\beta_0)= \frac{1}{\sqrt{n}}\sum _{i=1}^n\varepsilon_i\zeta_i+\mathrm{o}_P(1) \stackrel{D}{\longrightarrow}N(0,\sigma^2{{{ A}}}(\beta_0)).$$ This proves (\[eqA.7\]). The proof of Lemma \[lemm2\] is complete. [Proof of Lemma \[lemm3\]]{} By (\[eqA.15\]), (\[eqA.17\]) and Lemma \[lemm2\], we can prove (\[eqA.8\]). We now prove (\[eqA.9\]). Let $$\begin{aligned} R_{ni}(\beta) & =& \varepsilon_i\dot{{\rm g}}_0^T(\beta_0^TX_i)Z_iX_i\{w(\beta ^TX_i)-w(\beta_0^TX_i)\} \\ &&{} + \varepsilon_i\{\hat{\dot{{\rm g}}}(\beta^TX_i;\beta )-\dot{{\rm g}}_0(\beta_0^TX_i)\}^TZ_iX_iw(\beta^TX_i) \\ &&{} + \{{{\rm g}}_0(\beta_0^TX_i)-\hat{{\rm g}}(\beta ^TX_i;\beta)\}^TZ_iZ_i^T\dot{{\rm g}}_0(\beta_0^TX_i)X_iw(\beta^TX_i) \\ &&{} + \{{{\rm g}}_0(\beta_0^TX_i)-\hat{{\rm g}}(\beta ^TX_i;\beta)\}^TZ_iZ_i^T \\ &&{} \times\{\hat{\dot{{\rm g}}}(\beta^TX_i;\beta)-\dot{{\rm g}}_0^T(\beta_0^TX_i)\}X_iw(\beta^TX_i). $$ Then we have $\hat{\eta}_i(\beta)=\eta_i(\beta_0) + R_{ni}(\beta)$, where $\eta_i(\cdot)$ is defined in (\[eq2.1\]), and hence $$\begin{aligned} \label{eqB.5} {{{ R}}}_n(\beta) & =& \frac{1}{n}\sum_{i=1}^n\eta_i(\beta_0)\eta_i^T(\beta_0) + \frac{1}{n}\sum_{i=1}^nR_{ni}(\beta)R_{ni}^T(\beta) \nonumber\\ &&{} + \frac{1}{n}\sum_{i=1}^n\eta_i(\beta_0)R_{ni}^T(\beta) + \frac{1}{n}\sum_{i=1}^nR_{ni}(\beta)\eta_i^T(\beta_0) \\ & \equiv& M_1(\beta_0) + M_2(\beta) + M_3(\beta) + M_4(\beta).\nonumber\end{aligned}$$ By the law of large numbers, we have $M_1(\beta_0)\stackrel{P}{\longrightarrow}\sigma^2{{{ B}}}(\beta_0)$. Therefore, to prove (\[eqA.9\]), we only need to show that $M_k(\beta)\stackrel{P}{\longrightarrow}0$ uniformly for $\beta$, $k=2,3,4$. Let $M_{2,st}(\beta)$ denote the $(s,t)$ element of $M_2(\beta)$, and $R_{ni,s}(\beta)$ denote the $s$th component of $R_{ni}(\beta)$. Then by the Cauchy–Schwarz inequality, we have $$|M_{2,st}(\beta)|\leq \Biggl(\frac{1}{n}\sum_{i=1}^nR_{ni,s}^2(\beta) \Biggr)^{1/2} \Biggl(\frac{1}{n}\sum_{i=1}^nR_{ni,t}^2(\beta) \Biggr)^{1/2}. \label{eqB.6}$$ It can be shown by a direct calculation that $$\frac{1}{n}\sum_{i=1}^nR_{ni,s}^2(\beta)\stackrel {P}{\longrightarrow}0,$$ uniformly for $\beta\in{{\mathcal}B}_n$. This together with (\[eqB.6\]) proves that $M_2(\beta)\stackrel{P}{\longrightarrow}0$, uniformly for $\beta\in{{\mathcal}B}_n$. Similarly, it can be shown that $M_3(\beta)\stackrel{P}{\longrightarrow}0$ and $M_4(\beta)\stackrel{P}{\longrightarrow}0$, uniformly for $\beta\in{{\mathcal}B}_n$. This together with (\[eqB.5\]) proves (\[eqA.9\]). Similarly to above proof, we can derive (\[eqA.10\]). (\[eqA.11\]) can be shown by using (\[eqA.8\])–(\[eqA.10\]), and employing the same arguments used in the proof of (2.14) in Owen [@Owe90]. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful for the many detailed suggestions of the editor, the associate editor and the referees, which led to significant improvements of the paper. Liugen Xue’s research was supported by the National Natural Science Foundation of China (10871013,11171012), the Beijing Natural Science Foundation (1102008), the Beijing Municipal Education Commission Foundation (KM201110005029) and the PHR(IHLB). Qihua Wang’s research was supported by the National Science Fund for Distinguished Young Scholars in China (10725106), National Natural Science Foundation of China (10671198), the National Science Fund for Creative Research Groups in China and a grant from the Key Lab of Random Complex Structure and Data Science, CAS and the Key grant from Yunnan Province (2010CC003). [34]{} (). . . : . , (). . . , (). . . , , (). . . (). . . (). . . , (). . . (). . . : . , (). . . (). . . , (). . . (). . . , (). . . (). , ed. : . , (). . . (). . . (). . . (). . . (). . . , (). . . , (). . . (). . . (). . . (). . . , (). . . (). . . (). . . (). . . (). . . (). . . (). . . (). . . (). . .

--- abstract: 'Recent studies of Tidal Disruption Events (TDEs) have revealed unexpected correlations between the TDE rate and the large-scale properties of the host galaxies. In this review, we present the host galaxy properties of all TDE candidates known to date and quantify their distributions. We consider throughout the differences between observationally-identified types of TDEs and differences from spectroscopic control samples of galaxies. We focus here on the black hole and stellar masses of TDE host galaxies, their star formation histories and stellar populations, the concentration and morphology of the optical light, the presence of AGN activity, and the extra-galactic environment of the TDE hosts. We summarize the state of several possible explanations for the links between the TDE rate and host galaxy type. We present estimates of the TDE rate for different host galaxy types and quantify the degree to which rate enhancement in some types results in rate suppression in others. We discuss the possibilities for using TDE host galaxies to assist in identifying TDEs in upcoming large transient surveys and possibilities for TDE observations to be used to study their host galaxies.' author: - 'K. Decker French' - Thomas Wevers - 'Jamie Law-Smith' - Or Graur - 'Ann I. Zabludoff' bibliography: - 'extra\_hostgal.bib' date: 'Received: date / Accepted: date' title: The Host Galaxies of Tidal Disruption Events --- Introduction {#sec:intro} ============ Tidal Disruption Events (TDEs) are observed when a star passes close enough to a supermassive black hole (SMBH) to be disrupted and torn apart by tidal forces. The rate of TDEs and the properties of the stars and SMBHs involved depend on the nuclear conditions of the host galaxies. The mass of the SMBH will affect whether TDEs are observed and which stars can be tidally disrupted outside the event horizon [e.g., @Hills1975; @Rees1988; @MacLeod2012; @Law-Smith2017a]. Because the tidal radius scales as M$_{\rm BH}^{1/3}$, while the gravitational radius scales linearly with M$_{\rm BH}$, stars will be swallowed whole if the SMBH is larger than the so-called Hills mass [@Hills1975], which is $\approx$10$^{8}$ M$_{\odot}$ for a non spinning black hole and a Solar type star. The Hills mass also depends on the SMBH spin, such that a faster spinning SMBH can disrupt less massive stars at a given M$_{\rm BH}$ [@Kesden2012]. The mass of the black hole is therefore an important parameter of a TDE. It can be estimated from observations because it is closely correlated with the mass and velocity dispersion of the galaxy’s stellar bulge [@Magorrian1998; @Ferrarese2000; @Gultekin2009; @Kormendy2013; @McConnell2013]. The mass of the stars available to be disrupted depends on the recent star formation history of the galaxy and the initial mass function. The mass of the disrupted star is much harder to infer from observations, although it has been argued that it leaves imprints on the UV/optical lightcurve [@Lodato2009; @Guillochon2015b; @Mockler2018]. Stars will be perturbed in their orbits to pass within the tidal radius depending on the distribution function of the stars in that galaxy [@Magorrian1999]. The parameter space of stars that can be tidally disrupted, called the loss cone, is thought to be re-filled mainly through two-body interactions, although other mechanisms may also play a non negligible role (see Stone et al. 2020, ISSI review). The stellar density profile in the vicinity of the SMBH and any deviations from an isotropic velocity / velocity dispersion field will hence affect the TDE rate [@Magorrian1999; @Merritt2004; @Stone2018]. As most TDEs are thought to be sourced from within the gravitational radius of influence of the black hole (@Stone2016b; this is typically 0.1-10 pc for galaxies with M$_{\rm BH} \sim 10^{6-8} M_\odot$), the galaxy properties at these scales are likely to be most important in setting the TDE rate. However, the conditions in this region will be affected by the evolution and merger history of the galaxy as a whole, and may therefore be correlated with larger scale galaxy properties. Large transient surveys such as the Palomar Transient Factory (PTF; @Law2009 [@Rau2009]), Pan-STARRS [@Chambers2016], the All Sky Automated Survey for SuperNovae (ASASSN; @Shappee2014), and the Zwicky Transient Facility (ZTF; @Bellm2019) in the optical, as well as the Roentgen Satellite (ROSAT) and the X-ray Multi-Mirror telescope (XMM; @Jansen2001) in X-rays, and *Swift* in gamma rays have enabled the detections of tens of TDEs, providing a sample large enough to study population properties. In addition to the TDE properties themselves, these new samples of TDEs also allow us to study trends in their host galaxy properties. @Arcavi2014 studied the host galaxies of seven UV/optical bright TDEs with broad H/He emission lines, and found many of the hosts showed E+A, or post-starburst, spectra. Such post-starburst spectra are characterized by a lack of strong emission lines, indicating low current star formation rates, but with strong Balmer absorption, indicating a recent burst of star formation (within the last $\sim$Gyr) that has now ended. Quiescent Balmer-strong galaxies, and the subset of post-starburst/E+A galaxies with less ambiguous star formation histories, are rare in the local universe, and yet are over-represented among TDE host galaxies [@French2016; @French2017; @Law-Smith2017; @Graur2018]. The observed correlations between the pc-scale regions of stars which can be tidally disrupted and the kpc-scale star-formation histories and stellar concentrations are a puzzle, for which many possible solutions have been proposed. Here, we review the known host galaxy properties of TDEs observed to date and with published or archival host galaxy spectra in §\[sec:knownproperties\]. We discuss possible drivers for the host galaxy preference in §\[sec:discussion\] and implications for the TDE rates in §\[sec:rates\]. We discuss possibilities for using the host galaxy information in future surveys to find more TDEs in §\[sec:surveys\] and study galaxy properties in §\[sec:galaxystudies\] and conclude in §\[sec:summary\]. TDEs Included in This Review ---------------------------- In this review, we have selected a list of TDEs to discuss from the sample compiled by @Auchettl2017 of X-ray and optical/UV - bright TDEs, as this sample has been used for recent host galaxy studies [@Law-Smith2017; @Wevers2017; @Graur2018]. Given the focus of this chapter on the host galaxy properties, we only include TDEs for which a spectrum of the host galaxy has been published or is available from archival surveys. We have added three more recent TDEs for which host galaxy spectroscopy is available from before the TDE from SDSS and BOSS: AT2018dyk, AT2018bsi, and ASASSN18zj (aka AT2018hyz), as well as a new TDE with published host galaxy information [PS18kh, @Holoien2018b]. We note the important caveat that the classification of transient events as TDEs is complicated by the heterogeneous datasets obtained for each event. For the purposes of this review we aim to balance including a large enough sample to reflect the range of published literature in this field with giving preference to the most well-justified claims of observed TDEs. We thus preserve the classifications of @Auchettl2017 for the X-ray and optical detected TDEs that rank the likelihood an event is a TDE based on the completeness of the data, and divide the data into a number of subsets. It is important to note that the host galaxy statistics may change depending on which subset of TDE candidates are used. We comment on the differences one obtains depending on the sample used throughout, though for some subclasses we are limited by small number statistics. We list in Table \[tab:tde\_info\] the TDEs considered in this review. We divide the TDEs into two classes—X-ray bright and optical/UV bright—with several sub-categories. The X-ray bright TDEs are subdivided further into X-ray TDEs [@Holoien2015; @Levan2011; @Saxton2017; @Holoien2016], likely X-ray TDEs [@Saxton2012; @Esquej2007; @Cenko2012; @Maksym2010; @Lin2015; @Lin2017] , and possible X-ray TDEs [@Komossa1999b; @Gezari2008; @Grupe1999; @Ho1995; @Greiner2000; @Maksym2014] as done by @Auchettl2017. TDEs with no known or observed X-ray emission are classed as optical/UV TDEs [@Brown2017; @Blanchard2017; @Tadhunter2017; @Gezari2009; @Chornock2014; @Komossa2009; @Wang2012; @VanVelzen2011; @Yang2013; @Holoien2015; @Arcavi2014; @Blagorodnova2019; @2018ATel11953....1A; @2018ATel12035....1G; @2018ATel12198....1D]. TDEs requiring re-classification based on new X-ray data are re-classified (ASASSN-15oi, PS18kh; K. Auchettl, private communication). TDEs that exhibited coronal lines [@Komossa2009; @Wang2012] or broad H/He lines [e.g., @Arcavi2014] are also indicated. Three X-ray bright TDEs (ASASSN-14li, ASASSN-15oi, and PS18kh) additionally had significant optical observations, including broad H/He lines in their spectra, and are categorized as noted in the text. D3-13 is classed as a possible X-ray TDE, and also had significant optical/UV flux, but did not show broad H/He lines. We note that these classes are based on observational distinctions, which may or may not reflect physically different phenomena. Some optical/UV TDEs may have produced significant X-ray flux which was missed because of a lack of simultaneous X-ray observations. Indeed, the optical to X-ray luminosity ratios show significant variation in the events so far detected in both X-ray and optical light, and the observations of ASASSN-15oi by @Gezari2017 demonstrate that X-ray emission can even be delayed well past the peak of the optical light curve, and would have been likely missed for many optical TDEs[^1]. Similarly, the coronal line detections may be a light echo from a previous TDE [@Komossa2008], and the relation between these events and the others is still unclear. These classes represent those for which samples have been aggregated in the literature, and with adequate host galaxy localization and observations to study for the purposes of this chapter. We direct the reader to the other chapters in this review, especially those by Saxton et al., Arcavi et al., Zauderer et al., Alexander et al., and Zabludoff et al., (2020, ISSI review) for further discussion of TDE classification and the question of multiple TDE classes. In particular, we note that the class of events including F01004 [@Tadhunter2017] may be a type of nuclear phenomenon other than a TDE. @Trakhtenbrot2019 argue against the TDE interpretation of this event as it has significantly narrower He lines than other broad H/He line events and the presence of Bowen fluorescence lines. However, Bowen fluorescence lines have now been found in other TDEs with broader H/He lines [@Leloudas2019], indicating the space for observed TDE features may be broader than expected. The optical features of TDEs are discussed further in Arcavi et al. (2020, ISSI review), and a comparison of observational properties of observed TDE candidates with the spectrum of possible “imposters" is discussed further in Zabludoff et al. (2020, ISSI review). We separate out 13 TDE host galaxies which are part of the SDSS main spectroscopic sample (indicated in Table \[tab:tde\_info\]) in some of the following analysis, as these host galaxies can be matched to the general galaxy population in a uniform way. [l c c c c c c]{} Name & R.A. & Dec & $z$ & Type & BL$^a$ & CL$^b$\ Known Host Galaxy Properties of all events {#sec:knownproperties} ========================================== We consider here the host galaxy properties and trends of the TDE samples discussed above. We compare the stellar mass and black hole masses to expectations given the volume-corrected mass functions and expectations from an upper cutoff in the black hole mass from event horizon suppression (§\[sec:stmass\]). We also consider the stellar populations and inferred recent star formation histories of the TDE hosts, and discuss the observed enhancement in post-starburst and quiescent Balmer-strong galaxies (§\[sec:sfh\]). We discuss the morphologies and concentrations of the stellar light and observed trends toward higher central concentration on kpc scales in the TDE hosts (§\[sec:conc\]). The presence of on-going gas accretion and AGN activity in the TDE host galaxies, as well as possible biases against identifying TDEs in such host galaxies are also discussed (§\[sec:agn\]). We summarize the extragalactic environments of the TDE host galaxies, given the efforts to identify TDEs in galaxy clusters (§\[sec:enviro\]). The redshift range of the host galaxies affects the extent to which they can be studied. Most of the TDEs discovered to date are at low redshift, such that many of the TDE host galaxies have data from the SDSS. The redshift of all of the TDEs considered in this review (see Table \[tab:tde\_info\]) ranges from 0.01 to 0.4. The median redshift is $z=0.08$, and the 50 percentile range is 0.05–0.15. This redshift range is necessarily biased by the surveys which have discovered TDEs so far. Future surveys, such as LSST, may find a larger sample of higher redshift TDEs, depending on how the intrinsic TDE rate changes with redshift. A study by @Kochanek2016 predicts that the TDE rate will drop steeply with redshift between $z=0$ and $z=1$, based on the expected evolution of the host galaxy stellar populations, black hole masses, and merger rates. However, the rising fraction of post-starburst hosts with redshift [@Yan2009; @Snyder2011; @Wild2016] may act to counter this effect. The blue continuum of TDE emission may result in a negative $k$-correction [@Cenko2016], which would result in a greater number of observed TDEs at higher redshift. Host Galaxy Stellar Mass and Black Hole Mass {#sec:stmass} -------------------------------------------- [l c c c c ]{} Name & log(M$_\star$) &M$_g$ & $\sigma$ & log(M$_{\rm BH}$)\ & (M$_{\odot}$) & (mag) & (km s$^{-1}$) & (M$_{\odot}$)\ . $\dagger$ Broad line TDEs (see Table \[tab:tde\_info\]). $^*$ Coronal line TDEs (see Table \[tab:tde\_info\]). \[tab:tde\_properties\] The black hole mass is one of the fundamental parameters for TDE studies, as it sets both the energetics (e.g. peak luminosity, accretion efficiency) and the dynamics (e.g. orbital timescales, relativistic effects) of the disruption. While theoretical predictions [@Wang2004] suggest that TDEs should preferentially occur in the lowest mass galaxies still hosting SMBHs (10$^4$–10$^6$ M$_{\odot}$), the observed distribution (using a heterogeneous set of measurements) was observed to peak around 10$^{7}$ M$_{\odot}$ (e.g. @Stone2016b [@Kochanek2016]). More recently, @Wevers2017 presented systematic measurements of black hole masses using the M-$\sigma$ relation for a sample of 12 optical TDEs, and found the peak in the TDE black hole mass distribution to be significantly lower, near 10$^6$ M$_{\odot}$, consistent with theoretical predictions (taking into account the uncertainty in the calibration of the M-$\sigma$ relation at the low mass end). In contrast to previous studies [e.g., @Stone2016b] which use scaling relations from photometric observations to infer black hole masses, @Wevers2017 use spectroscopic observations of the bulge velocity dispersions. Black hole mass measurements from TDE light curves [@Mockler2018] are consistent with measurements from galactic properties given the uncertainties in each set of measurements, but the number of TDE light curves with well-measured rises and thus more accurate black hole mass measurements is still limited. @vanvelzen2018 uses the BH masses from @Wevers2017 to infer the BH mass and luminosity functions of TDEs. Correcting for selection effects such as survey depth, cadence and area, they find that the TDE rate is constant with black hole (or galaxy stellar) mass over two orders of magnitude from $M_{\odot} = 10^{5.5} - 10^{7.5}$. Given the uncertainties, the observed black hole mass function of TDE hosts could be consistent with either the expected black hole mass function over this mass range, or with the slightly steeper trend expected given the scaling of the TDE rate with black hole mass. The dearth of BH masses $\geq$ 10$^8$ M$_{\odot}$ is consistent with the presence of BH event horizons, and the disappearance of the tidal radius for a main sequence 1$M_\odot$ star inside the event horizon. While black hole masses are difficult and time-consuming to measure, stellar masses can be more easily measured using galaxy luminosities and stellar population estimates. The stellar masses of the host galaxies are roughly correlated with the black hole masses, via the black hole mass – bulge mass relation [e.g., @McConnell2013] and the correlation between galaxy stellar mass and bulge stellar mass [e.g., @Mendel2014]. For the host galaxies in the SDSS main spectroscopic sample, we plot a histogram of their stellar masses compared to the rest of the SDSS galaxies and the volume-corrected stellar mass function (SMF) in Figure \[fig:stmass\]. The TDE host galaxies are less massive than the typical SDSS galaxies, but with a typical stellar mass near M\*. This distribution is consistent with the TDE host galaxies being drawn from a volume limited sample of galaxies with stellar mass greater than $10^9$ M$_\odot$. Are the distributions of host galaxy stellar mass or black hole mass different for different classes of TDEs? There are several predictions in the literature, and this question depends on the details of how stars are disrupted and accreted, and the origin of the observed emission. The inverse dependence of the accretion disk temperature on black hole mass suggests X-ray TDEs should have lower black hole masses than optical TDEs [@Dai2015], but if rapid circularization is required to produce X-rays, higher mass black holes may be expected to produce more X-ray emission [@Guillochon2015b]. Alternatively, if the difference between the classes is related to a viewing angle effect [@Dai2018], no difference in the host galaxy properties would be expected. @Wevers2019 have measured the host galaxy absolute magnitudes of a large sample of optical and X-ray TDEs using SDSS and PS1 photometry. They used the kcorrect software [@Blanton2017] and the Petrosian or Kron magnitudes for SDSS and PS1 to estimate the host absolute magnitude as well as the galaxy stellar mass for a sample of 35 TDEs and TDE candidates. These values are presented in Table \[tab:tde\_properties\] for all sources in the current sample. Using different subdivisions (and a smaller sample) of host galaxies than the ones used here, @Wevers2019 found that the host galaxy absolute magnitudes, stellar masses, and black hole masses for different TDE classes are consistent with being drawn from the same parent population. Sample size M$_{\star}$ M$_{\rm g}$ M$_{\rm BH}$ ---------------------- ------------- ------------- -------------- Optical 20 20 16 X-ray + likely X-ray 10 10 7 Possible X-ray 6 6 6 p-values Optical - X-ray 0.03 (0.09) 0.02 (0.02) 0.38 (0.42) X-ray - pos. X-ray 0.15 (0.05) 0.03 (0.02) 0.06 (0.10) : Summary of statistical comparison between samples for different host properties, including host galaxy stellar mass (M$_{\star}$, absolute $g$-band magnitude (M$_{\rm g}$) and the black hole mass (M$_{\rm BH}$). We give the relevant sample sizes for each parameter. We test the hypothesis that the respective samples are drawn from the same parent distribution. The p-values of an Anderson-Darling test are given, as well as the p-values for a Kolmogorov-Smirnov test (in parentheses); values below 0.05 suggest that we can reject the hypothesis at $>$95 $\%$ significance. We note that these conclusions differ from those by @Wevers2019 due to the larger sample of TDEs considered here, as well as a different class division.[]{data-label="tab:pvalues"} The larger sample considered here allows us to repeat the analysis in @Wevers2019 with more statistical power (Figure \[fig:kde\]; Table \[tab:tde\_properties\]). We group the X-ray and likely X-ray hosts, the UV/optically discovered hosts and the possible X-ray hosts and perform pairwise Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) tests for these 3 samples. For the X-ray and optical samples, we find that for both host galaxy stellar mass and absolute magnitude the hypothesis that they are drawn from the same parent distribution can be rejected. The KS and AD significance values are summarized in Table \[tab:pvalues\]. The p-values for the X-ray and possible X-ray stellar mass comparison are higher, and we cannot reject the null hypothesis that they are drawn from the same parent distribution. For the latter, this could be due to the small size of the sample. The properties of the possible X-ray sources suggest significant contamination by AGN, which favour higher mass (both stellar mass and M$_{\rm BH}$) and more luminous host galaxies. The difference with the results in @Wevers2019 can be explained by i) the larger sample considered here and ii) the different sample subdivision. In particular, the soft X-ray sample in @Wevers2019 consists of 6 likely and 6 possible X-ray TDE hosts, and the sources ASASSN–14li and ASASSN–15oi are considered as optical events. {width="60.00000%"} {width="80.00000%"} {width="80.00000%"} {width="90.00000%"} @Wevers2019 also presented velocity dispersion measurements of an additional 19 TDE candidates, yielding a sample of 29 homogeneously measured black hole masses[^2]. Figure \[fig:kde\] shows a kernel density estimate (KDE) of the black hole mass distribution, divided by type. The KDE was calculated by representing each black hole mass estimate with a Gaussian function with a full width at half maximum (FWHM) equal to the measurement uncertainty (including both the velocity dispersion uncertainty and the scatter in the M–$\sigma$ relation), and then summing over the respective samples. Using statistical tests to compare the distributions between X-ray and optical samples, we find no significant differences between the black hole mass distributions. This supports the idea that the apparent dichotomy between optical and X-ray selected TDEs could be related to (for example) viewing angle or geometry [@Watarai2005; @Coughlin2014; @Dai2015; @Metzger2016; @Roth2016; @Dai2018], and that these events intrinsically belong to the same class. This is also supported by observations of UV/optical TDEs with deep X-ray upper limits: the detection of Bowen fluorescence lines in optical spectra implies that an ionizing (X-ray) radiation field exists, although no X-rays are actually observed [@Leloudas2019]. UV emission is not sufficient to excite the Bowen fluorescence lines for the one TDE (AT2018dyb) for which measurements are available. Emission line measurements presented by @Leloudas2019 show that the Wien tail of the UV blackbody responsible for the UV/optical radiation is insufficient (by $\sim$6 orders of magnitude) to explain the observed line fluxes. This suggests that the X-ray source is completely obscured along the line of sight. While the black hole mass distribution is very similar, the host galaxy stellar mass and absolute magnitude are significantly lower for the X-ray sample, as we can reject the null hypothesis of a common parent sample at high significance. However, we caution that this effect could be due to the lesser number of TDE host galaxies with black hole mass measurements (Table \[tab:pvalues\]). A larger sample of robust X-ray TDEs is required to draw robust statistical conclusions, and test whether optical/UV TDEs might have smaller black hole masses for their stellar masses (or whether X-ray TDEs have larger black hole masses for the stellar masses). If the observed trends in stellar mass and absolute magnitude are driven by differences in the black hole mass distribution between the X-ray and optical samples, this suggests that smaller black holes have higher temperature accretion disks with higher X-ray luminosities, or a combination of this effect and a viewing angle effect are acting. Another possibility is that selection biases from the very different identification methods of TDEs in the optical vs. X-ray could lead to this effect. The light curve duration for TDEs may vary with black hole mass in different ways for X-ray compared to optical emission. Wen et al. (in prep) find lower mass black holes to have longer duration super-eddington plateaus in their predicted X-ray light curves. @Lin2018 have found one such example of a long duration X-ray light curve from an event around a small SMBH. If it were the case that smaller black holes have longer-duration light curves in the X-ray compared to the optical, coarser cadence surveys in the X-ray would be biased against detecting TDEs in more massive black holes. Differences between TDEs found in optical vs. X-ray surveys will need to be studied further in the era of eROSITA and LSST. @Wevers2019 also consider a class of hard X-ray selected TDE candidates (which are not included in the sample discussed here), finding these host galaxies to have significantly different black hole mass distributions, as well as absolute magnitude and stellar mass distributions. However, these conclusions are based on a sample of 5 hard X-ray TDE candidate host galaxies, and a larger sample is needed to confirm these findings and understand the cause of these potential differences. Star Formation Rates, Star Formation Histories, and Stellar Populations {#sec:sfh} ----------------------------------------------------------------------- [l c c c c c c c c c]{} Name & H$\alpha$ & $\sigma$(H$\alpha$) & Lick H$\delta_A$ & $\sigma$(Lick H$\delta_A$) & SFR & Type\ & \[Å\] & \[Å\] &\[Å\] &\[Å\] & M$_\odot$ yr$^{-1}$ &\ Next, we consider the current star formation rates, the past star formation history, and the stellar populations of the TDE host galaxies. While the stellar populations in the nucleus may be far removed from those of the bulk of the host galaxy, the galaxy-wide stellar populations trace the formation and evolution of the host galaxy and are closely tied to the morphologies, kinematics, interstellar medium properties, and merger histories of the host galaxies. Star formation rates (SFRs) for host galaxies are calculated using various tracers of short-lived massive O and B stars. While many SFR tracers from the UV to the radio work well for galaxies with significant star formation, these tracers can be heavily biased in galaxies with rapidly changing SFRs (as many of the TDE host galaxies are thought to have), in dusty galaxies, or in galaxies influenced by AGN activity. We consider here SFRs of TDE host galaxies derived using H$\alpha$ luminosities from the SDSS, and correct for extinction and aperture using the corrections from the MPA-JHU SDSS catalogues [@Brinchmann2004; @Tremonti2004]. The SFRs from this catalogue use the D4000 break to estimate the SFRs for galaxies with non-star-forming emission line ratios, like those of many of the TDE hosts (see §\[sec:agn\]). While the D4000-sSFR correlation will lead to accurate SFRs on average for large samples, individual galaxies will have high uncertainties. Our use of H$\alpha$ here thus is more accurate for galaxies without strong AGN, but will be in general biased towards higher values for galaxies with additional H$\alpha$ emission from non-star-forming sources. We convert the H$\alpha$ luminosities to SFRs using $\eta = 5.4\times10^{-42}$ M$_\odot$ yr$^{-1}$/(ergs s$^{-1}$) [@Kennicutt2012]. We compare the TDE hosts to galaxies from the SDSS main spectroscopic survey in SFR–stellar mass space in Figure \[fig:ms\]. While several of the TDE hosts are at the lower SFR edge of the “main sequence" of star forming galaxies, most are quiescent, with low SFRs. {width="100.00000%"} We also consider the optical colours of the TDE hosts in the context of the colour magnitude relation. The $u-r$ colours and M$_r$ absolute magnitudes are plotted in Figure \[fig:cmr\]. The TDE hosts occupy a range of red, blue, and green-valley host galaxies. Because the colours are affected by both the current SFR and recent star formation history (SFH), we explore the physical interpretation of these quantities below. {width="100.00000%"} The quiescent SFRs paired with green colours of many of the TDE hosts is suggestive of an intermediate stellar population and a recent decline in the SFH of the host. Indeed, a large number of TDEs have been observed in E+A or post-starburst galaxies. The large number of post-starburst host galaxies was first observed by @Arcavi2014 in a sample of UV/optical bright TDEs with broad H/He lines. The presence of even one post-starburst galaxy amongst the hosts would be unusual given the rarity of post-starburst galaxies. The over-representation of post-starburst galaxies in TDE hosts was then quantified by @French2016, who used tracers sensitive to recent star formation on different scales to assess the recent star formation history of the host galaxies. One such comparison is to use the H$\alpha$ emission as a tracer for the current star formation on $\sim10$ Myr timescales and the Balmer absorption as a tracer for star formation on timescales of $\sim1$ Gyr. This method of using H$\alpha$ emission vs. Lick H$\delta_A$ has been used by many [@French2016; @Law-Smith2017; @Graur2018] to study the recent star formation histories of TDE host galaxies. There are many ways to quantify the current SFR in galaxies, as described above. Many methods that are sensitive to post-starburst galaxies use H$\alpha$ emission or O\[II\] emission if the red end of the rest-frame spectra are not available. Other methods allow for residual star formation using a PCA analysis [@Wild2010] or a BPT diagram analysis [@Alatalo2016]. @French2016 require H$\alpha$ EW $<$ 3 Å in emission in the rest frame to be considered quiescent. This corresponds to a specific SFR $\lesssim 1\times10^{-11}$ yr$^{-1}$, well below the main sequence of star-forming galaxies [e.g., @Elbaz2011]. The H$\alpha$ emission is also corrected for stellar Balmer absorption, which is significant for post-starburst quiescent Balmer-strong galaxies. The moderate lifetime of A stars means that the presence of a large A star population is indicative of a burst of star formation within the last Gyr. A star spectra show strong Balmer absorption, which can be best traced using the H$\gamma$, H$\delta$, or H$\epsilon$ lines. @French2016 use the Lick H$\delta_{\rm A}$ index and its uncertainty $\sigma$(H$\delta_{\rm A})$, which is optimized for the stellar absorption from A stars [@Worthey1997], has lower emission filling than H$\beta$, and smooth nearby continuum regions. The more bursty the SFH, i.e. the greater fraction of stellar mass produced over a shorter time, the higher H$\delta$ absorption will be. A stricter cut of H$\delta_{\rm A}$ $-$ $\sigma$(H$\delta_{\rm A}$) $>$ 4Å will select galaxies with recent starbursts creating $>3$% of their current stellar mass over 25–200 Myr (referred to as post-starburst galaxies throughout), and a weaker cut of H$\delta_{\rm A} >$ 1.31Å (referred to as quiescent Balmer-strong galaxies throughout) will select galaxies with recent epochs of star formation which created $>0.1$% of their current stellar mass over 25–1000 Myr [@French2017; @French2018b]. We plot these SFH tracers in Figure \[fig:hahd\] for various subsamples of TDEs. The TDE host galaxies span a range of SFHs from star-forming galaxies, to quiescent galaxies which have been quiescent for at least the past Gyr, and galaxies which had significant star-formation within the last Gyr but are currently quiescent. This last category consists of “post-starbust" or “quiescent Balmer-strong" galaxies as defined above. Several TDE host galaxies in the classes of the coronal line TDEs and optical/UV TDEs without coronal or broad lines have low H$\delta$ absorption for their H$\alpha$ emission compared to the rest of the galaxies in the SDSS spectroscopic sample. This may not be due to a physical difference between the host galaxies, and might instead be due to filling of the H$\delta$ line by residual TDE emission, as discussed by @Graur2018. Another possibility is contamination from the nearby Bowen fluorescence line NIII $\lambda$4100 [@Blagorodnova2018; @Leloudas2019; @Trakhtenbrot2019]. {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} The over-representation of a galaxy type among the TDE host galaxies can be determined using its rate in the TDE host galaxies compared to its rate in a general galaxy sample. We describe here the analyses done by various groups, and summarize in Table \[tab:overenhancement\]. @French2016 find that 38% of a sample of eight UV/optical H/He broad line TDE hosts meet a post-starburst selection criterion with a rate of only 0.2% in the general galaxy population. Similarly, 75% of the same TDE host galaxies meet a quiescent Balmer-strong selection criterion with a rate of 2.3% in the general galaxy population. These rates imply overdensities of 33$^{+7}_{-11}\times$ in quiescent Balmer-strong galaxies and 190$^{+115}_{-100}\times$ in post-starburst galaxies. @Graur2018 considered the over-enhancement rates for several additional categories of observed TDEs using a similar but slightly different parent galaxy sample and post-starburst/ quiescent Balmer-strong definitions. For an updated sample of UV/optical bright H/He broad line TDEs, the over-enhancement rates are 34$^{+22}_{-14}\times$ in quiescent Balmer-strong galaxies and 110$^{+80}_{-50}\times$ in post-starburst galaxies, consistent with the rate enhancements found by @French2016. The over-enhancement rates in post-starburst galaxies for the X-ray bright TDEs are weaker than for the UV/optical broad line TDEs, though this comparison is limited by small number statistics. For the set of X-ray TDEs, “likely" X-ray TDEs, and “possible" X-ray TDEs identified in @Auchettl2017a, @Graur2018 find the over-enhancement rates to be 18$^{+13}_{-9}\times$ in quiescent Balmer-strong galaxies and 18$^{+22}_{-18}\times$ in post-starburst galaxies. These rates are higher once the “possible" X-ray TDEs are excluded, many of which have ambiguous light curves and may be AGN flares. Considering only the X-ray and “likely" X-ray TDEs, the over-enhancement rates are 23$^{+21}_{-13}\times$ in quiescent Balmer-strong galaxies and 29$^{+41}_{-29}\times$ in post-starburst galaxies. These rate enhancements for the post-starburst sample are driven by the one X-ray (including the “likely" and “possible" samples) TDE that meets the strictest post-starburst criterion, ASASSN-14li. Overenhancement Galaxy Sample$^a$ TDE Sample$^b$ Source --------------------------- ------------------- ---------------------------------- -------- 33$^{+7}_{-11}\times$ QBS H/He broad line \[1\] 190$^{+115}_{-100}\times$ PSB H/He broad line \[1\] 34$^{+24}_{-14}\times$ QBS H/He broad line \[2\] 110$^{+80}_{-50}\times$ PSB H/He broad line \[2\] 18$^{+13}_{-9}\times$ QBS X-ray, likely, possible \[2\] 18$^{+22}_{-18}\times$ PSB X-ray, likely, possible \[2\] 23$^{+21}_{-13}\times$ QBS X-ray, likely \[2\] 29$^{+41}_{-29}\times$ PSB X-ray, likely \[2\] 17$^{+12}_{-8}\times$ QBS Optical \[2\] 50$^{+38}_{-29}\times$ PSB Optical \[2\] 18$^{+8}_{-7}\times$ QBS X-ray, likely, possible, optical \[2\] 35$^{+21}_{-17}\times$ PSB X-ray, likely, possible, optical \[2\] 20-80$\times$ QBS/PSB X-ray, likely, possible, optical \[3\] 40-120$\times$ QBS/PSB X-ray, H/He broad line \[3\] : Summary of TDE rate overenhancement found in various samples of galaxies and TDE classifications. $^a$ We note that the definitions of Quiescent Balmer-Strong (QBS) and Post-Starburst (PSB) vary slightly between @French2016 and @Graur2018, and in this review we present the overenhancement as a function of the Balmer strength (see Fig \[fig:enhancement\]). $^b$ Similarly, the TDEs used in each classification vary. For the TDEs included in the two calculations for this review, see Table \[tab:tde\_info\]. \[1\] @French2016 \[2\] @Graur2018 \[3\] This review.[]{data-label="tab:overenhancement"} We present classifications for the TDE hosts discussed in this review in Table \[tab:tde\_sfh\], using the criteria described above. 5/41 (12%) host galaxies are post-starburst galaxies and 13/41 (32%) are either quiescent Balmer-strong or post-starburst. Of the 4 X-ray TDEs, 3 (75%) are quiescent Balmer-strong and 1 (25%) is post-starburst. Of the 15 broad H/He line TDEs, 9 (60%) are quiescent Balmer-strong and 5 (33%) are post-starburst. To account for the dependence of the TDE rate enhancement on the definition of “post-starburst" or “quiescent Balmer-strong" we plot in Figure \[fig:cumulative\_hd\] the cumulative distribution of quiescent SDSS galaxies and quiescent TDE hosts with stronger Balmer absorption than the value on the x-axis. For both the full set of TDE hosts considered here as well as the subsets of the broad H/He line TDEs and the X-ray TDEs with the strongest post-starburst enhancement, there is a significant difference in the distributions between the quiescent SDSS galaxies and TDE hosts. We also demonstrate the effect of the criteria for post-starburst or quiescent Balmer-strong on the TDE enhancement rate over normal quiescent galaxies in Figure \[fig:enhancement\]. For the full set of TDE hosts, the enhancement rate ranges between 20-80$\times$ that of normal quiescent galaxies. For the broad H/He line TDEs and the X-ray TDEs, the enhancement rate ranges from 40-120$\times$ that of normal quiescent and star-forming galaxies. We note again that these TDE classifications are tentative and subject to a number of observational biases. More observations of the host galaxies for a large sample of well-characterized TDEs are needed to overcome these uncertainties and the small-number statistics limiting our precision here. {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} @Law-Smith2017 consider the post-starburst and quiescent Balmer-strong galaxy over-enhancement rates in a somewhat different subset of the @Auchettl2017a X-ray and optical TDEs, but control the galaxy parent sample on different properties to test whether selection effects drive the observed rate enhancements. Controlling on bulge mass (and thus likely black hole mass), redshift, surface brightness, [Sérsic ]{}index, or bulge to total light ratio can affect the relative post-starburst or quiescent Balmer-strong galaxy rate by $\le 2\times$, within the error on the rates due to small number statistics. Controlling on bulge colour has the largest possible effect on the observed post-starburst or quiescent Balmer-strong rates, decreasing the observed rate enhancements by factors of $\sim4$ for the post-starburst galaxies and $\sim2.5$ for the quiescent Balmer-strong galaxies, depending on what other factors are controlled for. This may be due to selection effects against finding TDEs in dustier and thus redder galaxies, or by the unique stellar populations in post-starburst galaxies, which cause them to lie in the optical green valley [@Wong2012]. If we control on properties which are strongly correlated with post-starburst and quiescent Balmer-strong galaxies, such as green optical colors or high central concentrations, we expect that the residual enhancement of the TDE rate in such hosts would be diminished by construction. The TDE enhancement rates in such hosts are thus consistent (given the small number statistics) between the studies of @French2016, @Law-Smith2017 and @Graur2018. While the Balmer absorption is a proxy for the recent SFH, detailed stellar population fitting can be used to determine the nature of the recent SFH using the information from the full galaxy SED. @French2017 fit stellar population models to the UV/optical host galaxy photometry and optical Lick indices to determine the time elapsed since the recent starburst, the fraction of mass produced in the starburst, and the duration of the recent starburst [@French2018], for a sample of host galaxies of broad H/He line TDEs. While the lower Balmer absorption “quiescent Balmer-strong" galaxies could have had weaker H$\delta$ absorption due to longer-duration bursts, older bursts, or weaker bursts, stellar population modelling by @French2017 determined that this effect is driven by the fact that quiescent Balmer-strong TDE hosts had weaker starbursts than most post-starburst galaxies. While most post-starburst galaxies have formed $\sim3-50$% of their stellar mass in their recent starbursts, the quiescent Balmer-strong (and non-post-starburst) hosts had burst mass fractions of $0.5-2$%. The post-starburst and quiescent Balmer-strong TDE hosts have ages ranging from 60 Myr to 1 Gyr since the recent starbursts ended. This large range in age suggests the enhanced TDE rate is not limited to a specific time in their host’s post-starburst evolution. When compared to the total sample of post-starburst and quiescent Balmer-strong galaxies, @French2017 observed a statistically insignificant dearth of TDEs in older ($>600$ Myr) post-starburst galaxies. The evolution of the TDE rate enhancement after a starburst implied by this early small sample is consistent with several models for explaining the enhanced TDE rate during this phase [@Stone2018 discussed further in §\[sec:rates\]]. TDEs may also be over-represented in starbursting host galaxies. However, the extreme dust extinction present in the nucleus of starburst galaxies as well as the co-existence of AGN activity make detecting such TDEs difficult. Either a lucky dust-free sightline or transient detections in the NIR are required to find TDEs in starburst galaxies. Both such scenarios have been observed. @Tadhunter2017 observed a light-curve over 10 years and a serendipitous appearance of broad He lines similar to those observed in other TDEs in a starburst galaxy[^3]. @Mattila2018 observed a jet launched from the nucleus of the starburst galaxy Arp 299 believed to be caused by a TDE, with a transient discovered in NIR AO imaging. Arp 299 has a stellar population consistent with evolving to a typical post-starburst galaxy, with an starburst age of 70-260 Myr since the starburst began and 9-29% of the total stellar mass formed in the on-going starburst [@Pereira-Santaella2015]. @Tadhunter2017 estimate the TDE rate to be enhanced in such galaxies by 1000-10,000$\times$, to one per century or even one per decade per galaxy. From the observations thus far, it is unclear whether the TDE rate is enhanced during both the starburst and post-starburst phases, with selection effects biasing against observing TDEs in starburst galaxies, or if the TDE rate enhancement peak lags in time after the starburst. Upcoming infrared and radio surveys for TDEs, as well as concerted efforts to disambiguate TDEs from AGN will be necessary to resolve this question. Concentration and Morphology of Stellar Light; Stellar Kinematics {#sec:conc} ----------------------------------------------------------------- [l c c c c c]{} Name & [Sérsic ]{}$n^a$ & M$_{\rm BH}$ $^b$ & A$^c$ & log $(\Sigma_{M_\star}) ^d$ & $\sigma_v ^e$ (km/s)\ The TDE rate is expected to depend on various physical properties of the SMBH and the stellar population in its vicinity, including the mass of the SMBH, the density of stars within its loss cone, and their velocity dispersion. Unfortunately, it is exceedingly hard to observationally probe the parsec-scale region of influence, except for the most nearby SMBHs. However, some global host-galaxy properties, on kpc scales, are known to be correlated with local properties in galactic nuclei. The most fundamental of these is the $M$–$\sigma$ relation, which relates the mass of the SMBH, $M$, to the host galaxy stellar velocity dispersion, $\sigma$ (e.g., @Kormendy1995 [@Magorrian1998; @2000ApJ...539L..13G; @Tremaine2002; @McConnell2013]). Moreover, these central stellar velocity dispersions have been shown to be correlated over galactic scales [@Cappellari2006]. @Graur2018 compared a sample of 11 TDE host galaxies with surface stellar mass densities and velocity dispersions computed from galaxy properties measured by the Sloan Digital Sky Survey (SDSS; @York2000 [@Kauffmann2003; @Brinchmann2004]) to a volume-limited sample of SDSS galaxies with galaxy properties measured by the same pipeline. Their TDE host galaxies had surface stellar mass densities in the range $\Sigma_{M_\star}=10^{9-10}~M_\odot~{\rm kpc}^2$. The star-forming TDE hosts were significantly denser than the star-forming control sample. This effect was not significant for quiescent galaxies, which already tend to have high surface stellar mass densities. @Graur2018 also measured surface stellar mass densities for a similar sample of 9 TDE host galaxies with velocity dispersions measured by @Wevers2017, and found that they too had values in the range $10^{9-10}~M_\odot~{\rm kpc}^2$ with one exception: PS1-10jh, which had a surface stellar mass density of ${\rm log}(\Sigma_{M_\star}/M_\odot~{\rm kpc}^2)=8.7^{+0.3}_{-0.4}$. Both of these samples, in purple and gray markers, respectively, are shown in Figure \[fig:concentration\_summary\]. Because the volume-corrected quiescent galaxies have a different stellar mass distribution than the star-forming galaxy sample, we also compare the $\Sigma_{M_\star}$ and velocity dispersion for a comparison sample cut in stellar mass to be M$_\star > 10^9$ M$_\odot$ in order to match the TDE host galaxies. Even after removing the low mass galaxies, the TDE hosts still have higher stellar surface densities than the volume-corrected comparison sample. While the preference of TDE hosts for galaxies with high surface stellar mass densities was statistically significant (at least for star-forming galaxies), there was no significant dependence on the galaxy central stellar velocity dispersion. Only the quiescent host galaxies showed a hint that their velocity dispersions might be lower than those of the quiescent galaxies in the control sample. It remains to be seen whether this effect proves to be significant in a larger sample. @Law-Smith2017 also compared kpc-scale indicators of stellar mass concentration for a sample of 10 TDE hosts with data from the @Simard2011 and @Mendel2014 SDSS catalogues, comparing the [Sérsic ]{}indices of the TDE host galaxies and SDSS comparison galaxies to the black hole masses inferred from the $M-\sigma$ relation. The TDE host galaxies have [Sérsic ]{}indices in the top 10–15% of the comparison sample in bins of black hole mass, indicating the TDE host galaxies have more concentrated stellar populations. In this review, we add data for two new TDEs (ASASSN-18zj and AT2018dyk) with archival SDSS information. We also perform a volume correction for the SDSS comparison sample using the volume calculations of @Mendel2014 in order to compare this analysis to that of @Graur2018. The updated [Sérsic ]{}index–black hole mass plots are shown in Figure \[fig:concentration\_summary\]. The volume correction accounts for the larger number of galaxies with low black hole mass, but the same trend of TDE host galaxies having higher [Sérsic ]{}indices for their black hole masses is seen. We find that 50% of the TDE host galaxies have [Sérsic ]{}indices in the top 20% of the volume-corrected SDSS galaxies with M$_{\rm BH}$ $10^5-10^6$ M$_\odot$, top 10% of the volume-corrected SDSS galaxies with M$_{\rm BH}$ $10^6-10^7$ M$_\odot$, and top 30% of the volume-corrected SDSS galaxies with M$_{\rm BH}$ $10^7-10^8$ M$_\odot$. If we only compare to the volume-corrected SDSS galaxies with quiescent levels of star formation (SFR$<1$ M$_\odot$ yr$^{-1}$), we find the same result for M$_{\rm BH}$ $10^5-10^7$ M$_\odot$, but no significant enhancement of TDE hosts in higher [Sérsic ]{}index galaxies with black hole masses M$_{\rm BH}$ $10^7-10^8$ M$_\odot$. A similar trend is also seen if the bulge to total light ratio is used as a proxy for stellar concentration instead of the [Sérsic ]{}index [@Law-Smith2017]. However, the [Sérsic ]{}index measurements from @Simard2011 have lower errors and thus allow finer binning and a more detailed comparison for our analyses. The analyses by @Law-Smith2017 and @Graur2018 have established that TDE hosts are more concentrated on galaxy-wide (kpc) scales. We discuss possible mechanisms for the stellar concentration affecting the TDE rate, and the interplay between this effect and the trend with star formation history in Section \[sec:rate\_conc\], as post-starburst and quiescent Balmer-strong galaxies are also known to have high central concentrations of stellar light. {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} We also consider here the quantitative morphologies of the TDE host galaxies in asymmetry - concentration space. Spiral galaxies and elliptical galaxies separate in this space with elliptical galaxies having higher concentrations and spiral galaxies having higher asymmetries [@Abraham1996]. Mergers can be further identified, with higher asymmetries than individual galaxies [@Conselice2003]. Post-starburst galaxies often show signs of recent mergers, but at several hundred Myr past coalescence, their asymmetries as measured using HST imaging have lessened to be between those of elliptical and spiral galaxies [@Yang2008 Figure 5 reproduced in this review]. @Law-Smith2017 have compiled a sample of morphological indicators for the TDE host galaxies as well as the SDSS main spectroscopic sample, using the catalogues of @Simard2011. We compare the concentration to the residual asymmetries [@Simard2002] for the SDSS galaxies and TDE host galaxies in Figure \[fig:c\_a\], using the [Sérsic ]{}index as a proxy for concentration. We separate out star-forming, quiescent, and quiescent Balmer-strong galaxies to identify trends in this space. The star-forming galaxies have high asymmetries and low [Sérsic ]{}indices, while the quiescent galaxies have [Sérsic ]{}indices of $\sim3-5$ and low asymmetries. The quiescent Balmer-strong galaxies show high [Sérsic ]{}indices[^4], with a tail extending down to the quiescent galaxies, and low asymmetries. The TDE hosts are distributed like the quiescent galaxies, with one source (SDSSJ0952) having a high [Sérsic ]{}index of $n\sim8$. The shift towards higher asymmetry for the post-starburst or quiescent Balmer-strong galaxies is not observed in the residual asymmetries from the SDSS imaging as it was in the total light asymmetries from the HST imaging; this is likely due to the greater sensitivity of the HST data to low surface brightness tidal features. @Yang2008 found that many of the tidal features observed with HST imaging would not be observable with ground-based imaging. Thus, the lack of high asymmetries in the TDE host sample does not rule out a recent merger, even a recent major merger. Higher resolution imaging and a variety of new measures of galaxy asymmetry [e.g., @Pawlik2015] will be required to determine whether the TDE host galaxies have the trend towards intermediate asymmetries indicative of recent mergers, as seen in the HST imaging of post-starburst galaxies. {width="50.00000%"} {width="50.00000%"} [l c c c c c c l l]{} Name & H$\alpha$ & H$\beta$ & \[NII\]6584 & \[SII\]6717+6731 & \[OIII\]5007 & BPT Class$^a$ & W1-W2$^b$ & SDSS Notes$^c$\ AGN Activity {#sec:agn} ------------ We consider here the possibility that some TDEs may occur in an environment with a pre-existing accretion disk. We have compiled a BPT [@Baldwin1981] diagram in Figure \[fig:bpt\] showing the TDE host galaxies as well as galaxies from the SDSS main spectroscopic survey. We include TDE host galaxies with SDSS spectra as described above, as well as galaxies with emission line ratios measured by @French2017 and @Wevers2019. These emission line fluxes are shown in Table \[tab:agn\]. We classify galaxies into star-forming, composite, and AGN Seyfert II or LINER based on the classifications of @Kewley2001 and @Kauffmann2003b. These classifications are subject to a number of caveats, and represent the likely dominant ionisation source in the aperture probed by the spectrum. However, many galaxies in the AGN region of the BPT diagram, especially those with relatively weak emission lines, may instead have ionisation consistent with an origin from shocks or evolved stars [e.g., @Rich2015; @Yan2012]. “Composite" galaxies lie in between the star-forming and AGN regions and could be a mix of star-formation and other ionisation sources. The TDE host galaxies occupy a range of star-formation dominated, AGN-dominated, and ambiguous ionisation source galaxies. Another way of identifying AGN, especially those obscured by dust, is to look for signatures of hot dust from the WISE 3.4–4.6$\mu$m colours. @Stern2012 identify a WISE colour cut of WISE 3.4–4.6$>0.8$ Vega mag to indicate the presence of an AGN. We present these WISE colours in Table \[tab:agn\], identifying four TDE hosts which meet this criterion: the hosts of F01004, SDSS J0952, SDSS J1342, and SDSS J1350. The first host galaxy is currently experiencing a starburst [@Tadhunter2017]. The latter three galaxies all hosted TDEs with observed coronal line emission. The BPT analysis described above selects narrow-line AGN (Seyfert II galaxies), or obscured AGN, as does the infrared selection. When the broad-line regions of AGN are visible, this provides another way to identify them from their optical spectra. In Table \[tab:agn\], we also list notes from the SDSS to indicate broad-line or QSO emission. Five TDE host galaxies have such notes, with varying overlap with those galaxies classified as AGN from the BPT or WISE colour analyses. AGN selection using any of these methods is neither pure nor complete. One may note that the host galaxy of PS16dtm, a Narrow-line Seyfert I galaxy [@Blanchard2017], is not selected as an AGN using a BPT or WISE colour analysis, and requires further analysis of the optical spectrum to identify the broad components of the Balmer lines. The connection between TDEs and Narrow line Seyfert I galaxies requires further study [@Wevers2019b]. Such galaxies make up only $\sim15$% of all Seyfert I galaxies [@Williams2002] and have been observed to have optical flares similar to TDEs [@Kankare2017]. Some of these events may even belong to a different class of transient [@Frederick2019]. Caution should be taken in interpreting these results given the selection effects against identifying TDEs in AGN host galaxies. We discuss the role of AGN in either enhancing the TDE rate or as the source of selection effects against identifying TDEs in such host galaxies in §\[sec:circumnucleargas\]. TDEs will be more difficult to identify in AGN due to selection against AGN flares and higher levels of dust obscuration. These effects may also bias the types of AGN TDEs are found in. Further study will be needed to fully understand these effects. Furthermore, we note that spectra taken after the TDE may be contaminated by residual TDE emission, depending on how long the emission persists for. @Brown2016b found that narrow H$\alpha$ emission can persist for a year after the TDE, but after several years, @Wevers2019 find no residual narrow line emission. @French2017 noted a tentative offset in H$\alpha$ equivalent width and \[NII\]-6584/H$\alpha$ emission ratio between the few events with spectroscopy before vs. after the TDE. @French2017 found the host galaxies with spectroscopy from after the TDE to have higher H$\alpha$ equivalent widths and \[NII\]-6584/H$\alpha$ emission ratios than the host galaxies with spectroscopy from before the TDE, although this analysis was limited by the small number of events and the lack of events with spectra from both before and well-after the TDEs. In the sample considered in this review, we note the host galaxies with spectra taken before the TDE contain more Seyferts and the host galaxies with spectra taken after the TDE contain more star-forming and composite classifications. However, this comparison is still limited by small number statistics and selection effects between the various TDE detection methods used. A systematically collected set of follow-up spectra will be needed to better understand the presence of narrow line emission in the decade after a TDE. Further insight into the presence or absence of AGN in TDE host galaxies requires spatially resolved emission line maps from IFU data. We present one example here; the host galaxy of AT2018dyk was observed as part of the MaNGA survey [@manga]. This host galaxy is in the AGN regions of the BPT diagrams in Figure \[fig:bpt\], and in Figure \[fig:manga\] we see that there is indeed a central \[OIII\]-bright source, and that the outskirts of the galaxy have ionisation dominated by star formation. The host galaxy of the TDE ASASSN-14li additionally shows evidence of past AGN activity, with large extended ionized regions visible in \[OIII\] 5007 and \[NII\] 6584 lines, seen in MUSE observations [@Prieto2016 Figure \[fig:muse\]]. Given the light travel time to these narrow-line regions, their ionisation implies strong AGN activity $10^4-10^5$ years in the past. Such extended ionized features have been seen around other galaxies in large imaging surveys and around other post-starburst galaxies in narrow-band imaging [@Schweizer2013; @Watkins2018]. These instances of recent AGN activity in galaxies lacking strong current AGN activity further complicate our understanding of the co-existence of gas and stellar accretion by supermassive black holes in galactic nuclei, raising questions regarding the timescale for TDE rate enhancements compared to AGN duty cycles. The host galaxy of ASASSN-14li furthermore has a persistent radio source discovered in the FIRST survey which may also indicate on-going low-level AGN activity [@Holoien2016]. ![BPT [@Baldwin1981] diagrams of emission line ratios indicative of ionisation from AGN or star formation in the TDE host galaxies and SDSS main galaxy spectroscopic sample. Galaxies to the lower left of the dotted and solid lines have ionisation dominated by star formation. Galaxies to the upper right of the dotted lines have ionisation dominated by AGN, although those with relatively weak emission lines may instead have ionisation from shocks or evolved stars [e.g., @Rich2015; @Yan2012]. The dotted line is the observed star formation - AGN separation from @Kauffmann2003b and the solid lines are the theoretical maximum starburst lines from @Kewley2001. Characteristic error bars are shown in each panel. The TDE host galaxies occupy a range of star-formation dominated, AGN-dominated, and ambiguous hosts. []{data-label="fig:bpt"}](bpt_tde_NII.png "fig:"){width="50.00000%"} ![BPT [@Baldwin1981] diagrams of emission line ratios indicative of ionisation from AGN or star formation in the TDE host galaxies and SDSS main galaxy spectroscopic sample. Galaxies to the lower left of the dotted and solid lines have ionisation dominated by star formation. Galaxies to the upper right of the dotted lines have ionisation dominated by AGN, although those with relatively weak emission lines may instead have ionisation from shocks or evolved stars [e.g., @Rich2015; @Yan2012]. The dotted line is the observed star formation - AGN separation from @Kauffmann2003b and the solid lines are the theoretical maximum starburst lines from @Kewley2001. Characteristic error bars are shown in each panel. The TDE host galaxies occupy a range of star-formation dominated, AGN-dominated, and ambiguous hosts. []{data-label="fig:bpt"}](bpt_tde_SII.png "fig:"){width="50.00000%"} ![Spatially resolved BPT diagram, H$\alpha$ map, and \[OIII\]5007 map for the host galaxy of TDE AT2018dyk, made using Marvin [@marvin]. Points are colored blue, green, red, dark grey, or light grey based on their classification as star-forming, composite, Seyfert, LINER, or ambiguous. This host galaxy is in the AGN regions of the BPT diagrams in Figure \[fig:bpt\], and here we see that there is indeed a central \[OIII\]-bright source, and that the outskirts of the galaxy have ionisation dominated by star formation. Obtaining data like this for additional TDE hosts, especially those for which the ionisation source is ambiguous, or there is evidence for unusual AGN properties, will help further our understanding of the co-existence of TDEs and AGN. []{data-label="fig:manga"}](MaNGA_AT2018dyk_BPT_1.png "fig:"){width="100.00000%"} ![Spatially resolved BPT diagram, H$\alpha$ map, and \[OIII\]5007 map for the host galaxy of TDE AT2018dyk, made using Marvin [@marvin]. Points are colored blue, green, red, dark grey, or light grey based on their classification as star-forming, composite, Seyfert, LINER, or ambiguous. This host galaxy is in the AGN regions of the BPT diagrams in Figure \[fig:bpt\], and here we see that there is indeed a central \[OIII\]-bright source, and that the outskirts of the galaxy have ionisation dominated by star formation. Obtaining data like this for additional TDE hosts, especially those for which the ionisation source is ambiguous, or there is evidence for unusual AGN properties, will help further our understanding of the co-existence of TDEs and AGN. []{data-label="fig:manga"}](MaNGA_AT2018dyk_BPT_2.png "fig:"){width="32.00000%"} ![Spatially resolved BPT diagram, H$\alpha$ map, and \[OIII\]5007 map for the host galaxy of TDE AT2018dyk, made using Marvin [@marvin]. Points are colored blue, green, red, dark grey, or light grey based on their classification as star-forming, composite, Seyfert, LINER, or ambiguous. This host galaxy is in the AGN regions of the BPT diagrams in Figure \[fig:bpt\], and here we see that there is indeed a central \[OIII\]-bright source, and that the outskirts of the galaxy have ionisation dominated by star formation. Obtaining data like this for additional TDE hosts, especially those for which the ionisation source is ambiguous, or there is evidence for unusual AGN properties, will help further our understanding of the co-existence of TDEs and AGN. []{data-label="fig:manga"}](MaNGA_AT2018dyk_Ha.png "fig:"){width="32.00000%"} ![Spatially resolved BPT diagram, H$\alpha$ map, and \[OIII\]5007 map for the host galaxy of TDE AT2018dyk, made using Marvin [@marvin]. Points are colored blue, green, red, dark grey, or light grey based on their classification as star-forming, composite, Seyfert, LINER, or ambiguous. This host galaxy is in the AGN regions of the BPT diagrams in Figure \[fig:bpt\], and here we see that there is indeed a central \[OIII\]-bright source, and that the outskirts of the galaxy have ionisation dominated by star formation. Obtaining data like this for additional TDE hosts, especially those for which the ionisation source is ambiguous, or there is evidence for unusual AGN properties, will help further our understanding of the co-existence of TDEs and AGN. []{data-label="fig:manga"}](MaNGA_AT2018dyk_O.png "fig:"){width="32.00000%"} ![Three-colour RGB image constructed from a MUSE datacube of the host galaxy of ASASSN-14li. The red and blue components of this RGB image are from red and blue continuum regions and green is from the \[OIII\] 5007 line. Extended ionized features are observed via the \[OIII\] 5007 line extending beyond the continuum-dominated bulk of the galaxy, and are likely from past AGN activity $10^4-10^5$ years ago [@Prieto2016]. The star observed below the galaxy and the edge-on galaxy to the right are not associated with the TDE host galaxy. Recent AGN activity in galaxies lacking strong current AGN activity further complicates our understanding of the co-existence of gas and stellar accretion by supermassive black holes in galaxies, raising questions of the timescale for TDE rate enhancements compared to AGN duty cycles.[]{data-label="fig:muse"}](ASASSN14li_MUSE_label.png){width="50.00000%"} Environment {#sec:enviro} ----------- We consider here the extragalactic environments of the TDE host galaxies, as many mechanisms which act to change the other galaxy properties considered in this section can only act in dense cluster-scale environments. Several TDEs have been found in targeted X-ray surveys of dense galaxy clusters. J1311 [@Maksym2010] was found in a search for TDEs in Abell 1689. While it is not included in our analysis due to the lack of a host galaxy spectrum, “Wings" [@Maksym2013] was found in Abell 1795. In addition to possible selection biases towards finding TDEs in clusters, post-starburst galaxies are known to lie preferentially in group environments [@Zabludoff1996]. While groups like the Local Group are the most common galaxy environment, the groups favored by post-starburst galaxies tend to be virialized and more massive than the Local Group, with low enough velocity dispersions and high enough galaxy densities to make galaxy-galaxy mergers and tidal interactions likely [@Zabludoff1996]. We cross match the catalogue in Table \[tab:tde\_info\] with the Abell cluster catalogue [@Abell1989], to check for host galaxies within 25 arcminutes of a cluster with a similar redshift (velocities within 3000 km/s of the host galaxy redshift). We find one additional TDE host galaxy, PTF09axc, to be associated with the cluster Abell 1986. To test for TDE host galaxies in less rich clusters and groups, we cross match our TDE host galaxy catalogue with the group and cluster catalogue of @Tempel2014. Of the 41 host galaxies, 14 are matched with the @Tempel2014 catalogue. Seven are the only galaxy in their halo[^5] , six are associated with groups of 1-6 additional galaxies[^6], and one is part of a cluster of 52 galaxies (SDSS J1350). Given the set of 14 TDE host galaxies in the @Tempel2014 catalogue, and their full sample of galaxies, we find no evidence to suggest that the TDE host galaxies prefer different environments than the general galaxy sample. Using either a Kolmogorov-Smirnov or Anderson-Darling test, we cannot reject the hypothesis their environment richnesses are drawn from the same distribution. Possible Drivers for Host Galaxy Preferences {#sec:discussion} ============================================ There are several possible causes for the observed TDE rate enhancement in the host galaxy types discussed above. Many of these scenarios predict TDE enhancements in both post-starburst hosts and centrally concentrated hosts, such as a central overdensity or mechanisms related to galaxy-galaxy mergers. Increased Stellar Concentration / Central Overdensities {#sec:rate_conc} ------------------------------------------------------- The TDE rate depends on the number of stars which can be scattered into center-crossing orbits, and so a high central stellar density will result in a high TDE rate. For a Nuker surface-brightness profile, the inner stellar slope $\gamma$ is found to correlate with the TDE rate as $\dot{N}_{\rm TDE} \propto \gamma^{0.705}$ in a sample of early type galaxies from @Lauer2007 [@Stone2016b]. A high central stellar concentration may be correlated with high concentrations on larger $\sim$kpc scales. As discussed above in §\[sec:conc\], TDEs are overrepresented in galaxies with high [Sérsic ]{}indices for their black hole masses [@Law-Smith2017] and in galaxies with high stellar surface densities on scales of the half-light radius [@Graur2018]. An increased TDE rate due to a merger-induced stellar overdensity is seen in simulations by @Pfister2019, although at very early stages in the merger, before the coalescence of the two black holes. If high stellar concentrations drive the TDE rate enhancement in high [Sérsic ]{}index or stellar surface density galaxies, it may also explain the rate enhancement in post-starburst galaxies. Post-starburst galaxies have high [Sérsic ]{}indices [@Yang2004; @Quintero2004; @Yang2008] as the recent starbursts are centrally concentrated, likely due to stars formed from gas infall in the recent merger, and the young/intermediate A stars dominate the light. Once the bright young stellar population in post-starburst galaxies fades, the bulge properties are consistent with evolving to normal early type galaxies, but the stellar concentrations on scales close to the black hole radius of influence have not been measured in samples of post-starburst galaxies. However, the nearby post-starburst galaxy NGC 3156 has HST imaging with high enough resolution to measure the central slope of the Nuker profile, and @Stone2016 find the slope to be steeper than any of the early type galaxies studied previously by @Stone2016b. Given the lack of a similar TDE rate enhancement in early type galaxies, something must change in the central galaxy concentration or dynamics in the few Gyrs after the post-starburst phase. The evolution of a central overdensity with time was studied by @Stone2018, who model the stellar density profile as $\rho \propto r^{-\gamma}$, and determine how $\gamma$ changes with time. Given the post-burst ages of the post-starburst TDE hosts [@French2017], the TDE rate enhancement and its tentative evolution with time could be explained if $\gamma \ge 2.5$. The predictions made by this model can be tested with larger samples of post-starburst TDE hosts in the LSST and perhaps even ZTF eras. Based on the supposition that the TDE rate should depend on the density of stars in the SMBH loss cone, along with their velocity dispersions, @Graur2018 assumed those local properties would be correlated with their global, kpc-scale counterparts, and that the TDE rate would depend on the latter as $R_{\rm TDE}\propto \Sigma_{M_\star}^\alpha \times \sigma_v^\beta$, where $\Sigma_{M_\star}$ is the surface stellar mass density on the scale of the half-light radius and $\sigma_v$ is the kpc-scale velocity dispersion. By comparing their sample of TDE host galaxies with a volume-limited control sample drawn from the SDSS, @Graur2018 estimated the values of the power-law indices to be $\hat{\alpha}=0.9\pm0.2$ and $\hat{\beta}=-1.0\pm0.6$ using SDSS fiber measurements of the central few kpc, and assuming these global properties correlate with the properties on the smaller scales of the stars in the SMBH loss cone. @Wang2004 find that the TDE rate of an isothermal sphere ($\rho \propto r^{-2}$) depends on the SMBH mass, $M_\bullet$, and local velocity dispersion, $\sigma$, as $R_{\rm TDE}\propto M_\bullet^{-\alpha}\times \sigma^\eta$. The average surface stellar mass density of the stars orbiting the SMBH is $\Sigma = M_\bullet/\pi r_h^2=\sigma^4/\pi G^2 M_\bullet$, where $r_h=GM_\bullet /\sigma^2$ is the size of the star cluster [@Peebles1972], and $G$ is the gravitational constant. This allows us to rewrite the TDE rate as $R_{\rm TDE}\propto \Sigma^\alpha \times \sigma^{\eta-4\alpha}$. Using the @Graur2018 estimates for $\alpha$ and $\beta=\eta-4\alpha$, the values measured from the data, $\alpha=0.9\pm0.2$ and $\eta=2.6\pm1.0$ are consistent with the theoretical predictions, $\alpha=1$ and $\eta=3.5$ [@Wang2004]. This suggests that the TDE rate is indeed driven by the dynamical relaxation of stars into the loss cone of the SMBH. Could both the preference of TDE hosts to be in post-starburst or quiescent Balmer-strong host galaxies and the preference for host galaxies with high central concentrations be driven by the same effect? Similar galaxy over-representations can be found in both [Sérsic ]{}index–black hole mass and in H$\alpha$ emission–H$\delta$ absorption, where $\ge 60$% of TDEs are found in $\sim2$% of the parameter space (i.e., at high H$\delta$ absorption and low H$\delta$ emission, or at high [Sérsic ]{}index and low black hole mass)[^7]. Of the five TDE host galaxies considered by @Law-Smith2017 with high [Sérsic ]{}indices and low black hole masses, three are post-starburst or quiescent Balmer-strong, and two (PTF09ge and SDSSJ123) are not. Of the seven quiescent Balmer-strong galaxies considered by @French2016, two (ASASSN-14li and ASASSN-14ae) also meet the high [Sérsic ]{}index and low black hole mass criteria, and the remaining five do not have sufficient data to determine an accurate [Sérsic ]{}index or bulge mass. Larger numbers of observed TDEs and more detailed analyses of low concentration post-starburst hosts or high concentration non-bursty host galaxies will be an important test of which mechanisms most affect the TDE rate. Black Hole Binary ----------------- After a galaxy–galaxy merger, the supermassive black hole from each galaxy will inspiral and coalesce. The influence of supermassive black hole binary dynamics on TDEs is the subject of another chapter in this review (Coughlin et al. 2019, ISSI review). We summarize here the relevant points from Coughlin et al. (2019, ISSI review) for the present discussion of the host galaxies. The TDE rate can be very high (of order 1 per year) for a short ($\sim 1$ Myr) period during coalescence when the secondary black hole approaches the cusp of stars around the primary, at $\sim$pc scale separations. As inspiral continues, the TDE rate will then drop below that of an isolated black hole, and rise once more to a modest rate enhancement of $\sim2-10\times$ that of an isolated black hole once the binary has reached mpc separations. The lightcurves of TDEs can be altered in the case of a tightly bound binary where the debris stream interacts with the companion supermassive black hole. However, if most of the TDEs around black holes in coalescing binaries happens at pc-scale separations, the debris streams will be on significantly smaller scales, and the TDE lightcurves will show no evidence of the companion supermassive black hole. Thus, we are unlikely to see observational effects from the secondary black hole at separations of the same spatial scales which will boost the TDE rate. The main observable difference between this explanation for the TDE rate enhancement in post-starburst or centrally-concentrated galaxies and the others, is the TDE rate per galaxy. If the TDE rate is very high ($\sim 0.1-1$ per year per galaxy) for a Myr, we should observe a high instance of repeat TDEs per host galaxy, especially over the 10 year run of LSST. The other mechanisms for enhancing the TDE rate described in this section would act over 100 Myr - 1 Gyr, with more modest TDE rates per galaxy, and instances of repeat TDEs would be rare. No repeated TDEs have been observed to date, which means either there are no observed cases where the TDE rate is as high as 1 per several years, or systems hosting multiple TDEs within several years are obscured by dust or otherwise produce a different observational signature than the TDEs discussed in this review. For now, the likelihood that supermassive black hole binary effects are driving the observed host galaxy distributions can be probed statistically. @French2017 measured the star formation histories for a sample of six post-starburst TDE host galaxies to determine the time since the recent starbursts. If the starburst coincides with the coalescence of the two galaxies, this can also constrain the time since the supermassive black holes started to inspiral on kpc-scales. Most of the TDE host galaxies are less than 600 Myr since starburst. For a secondary to have in-spiraled to pc-scales in 600 Myr, that mass ratio of the two galaxies must be more equal than 12:1 given the dynamical friction timescales. This constrains the TDE rate enhancement in supermassive black hole binaries to be more strongly dependent on the mass ratio than the merger rate, since minor mergers with mass ratios less equal than 12:1 are more common than major mergers. @Stone2018 also argue against the possibility of supermassive black hole binaries causing the observed rate enhancement in post-starburst galaxies by constructing an expected delay time distribution of TDEs after a starburst. Many more TDEs would be expected at times $>1$ Gyr after a starburst, but the observed host galaxies have younger ages. Unless the timescale between the merger and the starburst is fine-tuned, compared to the dynamical friction timescale, the observed host galaxies are not compatible with rate enhancement from a black hole binary scenario. Circumnuclear Gas {#sec:circumnucleargas} ----------------- If a supermassive black hole is surrounded by a circumnuclear gas disk, this may act to enhance the TDE rate as stars interact with the disk. @Kennedy2016 predict the TDE rate could be increased by up to $10\times$ by the presence of such a disk. This effect may co-exist with the other mechanisms for affecting the TDE rate discussed here, as mergers are expected to trigger AGN activity [e.g., @Treister2012] as well as bursts of centrally-concentrated star formation. However, there are a number of selection effects which may hinder the identification of TDEs in host galaxies with pre-existing circumnuclear gas disks. As discussed by @Law-Smith2017, @Wevers2019, and others, observational searches for TDEs will select against AGN host galaxies while trying to avoid classifying AGN flares from variations in the gas accretion rate as TDEs, and many TDEs in AGN host galaxies will be heavily extincted by dust. One possibility for identifying TDEs in AGN host galaxies with heavy dust obscuration is through multi-epoch radio observations. @Mattila2018 identify a TDE in Arp 299, a Seyfert II galaxy and dust-obscured LIRG [@Pereira-Santaella2015], via a growing radio jet in one of the two nuclei in this merging system. Identifying the accretion of an individual star in an AGN with a high gas accretion rate may prove unfeasible, or only possible to assess statistically. Other Dynamical or Secular Effects ---------------------------------- There are several other effects that might increase the TDE rate in post-merger galaxies, causing the observed host galaxy preferences. A more complete description of these effects can be found in the chapter of this review by Stone et al. (2020, ISSI review); we very briefly mention two mechanisms here. First, a radial anisotropy of the orbits after a merger would increase the rate of TDEs. @Stone2018 modeled the dependence of the rate enhancement on the time since the starburst, parameterizing the radial anisotropy as $\beta \equiv 1- \frac{T_\perp}{2 T_{\|}}$, where $T_\perp$ and $T_{\|}$ are the kinetic energies of the tangential and radial motion, respectively[^8]. @Stone2018 find that the observed host galaxies could be explained if $\beta > 0.6$. A triaxial nuclear potential [@Merritt2004] could have a similar effect of enhancing the TDE rate after a nuclear starburst. Second, @Madigan2018 predict an enhanced TDE rate that declines with time after a nuclear starburst due to the formation of an eccentric stellar disk, where the TDE rate would be increased due to secular effects and could potentially reach 0.1 – 1 gal$^{-1}$ yr$^{-1}$. However, the timescale over which this effect might produce a high TDE rate is uncertain. The high spatial and spectral resolution of next-generation 30-m class telescopes will provide crucial tests of these mechanisms in nearby TDE hosts. Implications for TDE Rates {#sec:rates} ========================== The variation in TDE rates by host galaxy type has implications for the relative TDE rates in different types of galaxies. The TDE rate averaged across all galaxy types is observed to be $\sim 10^{-4}$ per galaxy per year [@vanvelzen2018]. We quantify the TDE rates in quiescent Balmer-strong and post-starburst galaxies compared to normal quiescent galaxies in Figure \[fig:rate\_change\], and how this depends on the definition of “Balmer-strong". The TDE rate in quiescent Balmer-strong and post-starburst galaxies is 1-3$\times10^{-3}$ per galaxy per year, depending on the threshold used to define such galaxies. An enhancement in the TDE rate in certain galaxies will necessarily result in a decreased TDE rate in other galaxies given a measurement of the total TDE rate. Is this lowered rate enough to cause tension with theoretical predictions for the TDE rate in normal quiescent galaxies ($\sim$ few$\times 10^{-4}$ per galaxy per year, e.g., @Stone2016b)? Depending on the definition of “Balmer-strong" vs. “normal" quiescent galaxies, the TDE rate in normal quiescent galaxies is $0.3-1\times10^{-4}$ per galaxy per year. We stress here that despite the enhanced TDE rates observed in post-starburst and quiescent Balmer-strong galaxies, the number of TDEs in normal galaxies is still high enough to avoid a crisis in their TDE rates. The large number of galaxies which do not meet the quiescent Balmer-strong or post-starburst selection compensates for the lower TDE rate in such “normal" galaxies. This rate suppression of $\lesssim3\times$ in normal quiescent galaxies is comparable to the uncertainties in predicting the theoretical TDE rate and determining the observed TDE rate in various surveys. {width="50.00000%"} {width="50.00000%"} Using the Host Galaxy Information in Transient Surveys {#sec:surveys} ====================================================== Identifying TDEs using a priori Host Galaxy Information ------------------------------------------------------- In addition to being used to understand what drives the TDE rates, the unique properties of TDE host galaxies may be useful for selecting candidate TDEs for spectroscopic follow-up observations in large transient surveys. The large volume of transient alerts faced by current and future surveys such as ZTF and LSST means that not all transients can be followed up for spectroscopic confirmation and further study with triggered space-based observations and high cadence photometry. Methods to flag likely TDEs will be essential for future TDE studies. One method to flag likely TDE candidates is to use a priori information about their likely host galaxies. Transients discovered in pre-identified likely TDE host galaxies, based on (for example) E+A spectroscopic signatures or concentration indices, can then be systematically classified with dedicated spectroscopic observations to efficiently select TDEs for further photometric and spectroscopic monitoring. For transient surveys in the Northern hemisphere where transient detections will have a significant overlap with large spectroscopic surveys (especially SDSS), the detailed properties of likely host galaxies can be used to predict which candidate detections are likely TDEs, supernovae, or other transient phenomenon, and flag interesting objects for follow-up. However, a significant portion of the LSST footprint will not be covered by large spectroscopic surveys, so photometric criteria will be useful. A technique of selecting likely TDEs for followup based on the colour (and thus star formation rate) of the host galaxy is used by iPTF/ZTF [@Hung2018] in order to reduce contamination by supernovae. Further cuts on the recent star formation history or concentration of host galaxies may further refine this selection, independent of the physical reason for the enhanced TDE rate in such galaxies. @French2018 present a method for identifying likely TDE candidates using photometrically-identified quiescent Balmer-strong galaxies. A Random Forest classifier trained on spectroscopically-identified quiescent Balmer-strong galaxies can be used to detect such galaxies in LSST using LSST photometry in addition to archival photometry from [*GALEX*]{} and [*WISE*]{}. Because these galaxies have low star formation rates and thus low supernova rates, contamination from other transients will be low. Other possible methods for identifying likely TDE hosts can use photometric information on the light concentration, given by either the [Sérsic ]{}index or bulge fraction that may predict a TDE overabundance in a related set of likely host galaxies (see discussion in §\[sec:rate\_conc\]). For instance, choosing nuclear transients in high-[Sérsic ]{}galaxies could significantly increase the success of confirming TDEs. There are several important benefits to selecting TDEs for spectroscopic follow-up. In addition to identifying a large number of events, this method can identify TDEs with properties that diverge from the known set of TDEs. The small number of optical/UV bright TDEs studied to date already show a large variety in their peak magnitude, colour evolution, and decline timescale [@Holoien2016; @Blagorodnova2017; @Wevers2017]. Furthermore, a priori selection of the likely host galaxies means candidate events can be followed-up early, even before a light curve is obtained. Early time light curve information will enable more detailed modeling, and may be able to be used to measure black hole masses [@Mockler2018]. Any method for selecting transient events for follow-up based on the host galaxy properties will bias the host galaxy properties of the resulting sample. Thus, these selection methods are complementary to other methods which are un-targeted or use information on the transient properties alone. Galaxy Studies Using TDEs {#sec:galaxystudies} ========================= Current and upcoming transient surveys, such as LSST in the optical and e-ROSITA in the X-ray, are expected to detect hundreds to thousands of TDEs each year [@VanVelzen2011; @Khabibullin2014]. In light of the theoretical predictions and observational evidence that these TDEs should occur in the lowest mass galaxies still harbouring black holes [@Wang2004], TDEs could become beacons for studying the black hole mass functions and spin distributions of quiescent supermassive black holes. While measuring velocity dispersions of large samples of increasingly faint galaxies will become prohibitively expensive, alternatives are emerging to measure the black hole mass. For example, it is expected that the shape of a TDE lightcurve depends mainly on the mass of the black hole and the mass of the disrupted star. @Mockler2018 exploit this by using lightcurve models to infer the black hole mass under some assumptions for the mass of the disrupted star. They show that this method yields similar uncertainties when compared to other ways of estimating the black hole mass (e.g. galaxy scaling relations), while further improvements (such as including realistic models for the stellar structure) could decrease potential systematic effects. This can provide the opportunity to measure black hole masses of galaxies that are too faint for velocity dispersion measurements, as well as large samples of galaxies, as long as the lightcurve is well sampled around peak and ideally in more then one filter. These mass measurements will be invaluable for TDE demographics studies, but can also potentially contribute to other areas of galaxy evolution, such as the cosmic growth of SMBHs in quiescent galaxies and the BH occupation fraction and mass function of dwarf galaxies. On the high end of the mass function, observed TDE rates will be affected by the presence of an event horizon cutoff, as a consequence of the tidal radius becoming smaller than the event horizon and making disruptions invisible to outside observers. A main sequence star disrupted by a Schwarzschild black hole of mass $\ge 10^8 M_\odot$ will not produce an observable flare. However, the spin of the SMBH will affect this cutoff mass, such that TDEs would still be observable around $10^8-10^9 M_\odot$ SMBHs at high ($a\sim0.9-0.999$) spins [@Kesden2012; @Leloudas2016] A large number of TDEs with black hole masses $\ge 10^8 M_\odot$ would imply a large fraction of supermassive black holes with high spin parameters. The spin distribution for supermassive black holes has implications for their accretion and merger history, as coherent gas accretion is expected to spin up black holes while frequent mergers will spin them down [e.g., @Volonteri2003; @Hughes2003]. The effect of the spin distribution on the observed TDE rate is discussed further by Stone et al. (2020, ISSI review). The possibilities for observing TDEs in dwarf galaxies with intermediate mass black holes are discussed by Maguire et al. (2020, ISSI review). Summary and Discussion {#sec:summary} ====================== We have summarized the observed host galaxy properties of X-ray bright and optical/UV bright TDEs and discussed the possible physical mechanisms that could drive the observed correlations between host galaxy properties and the TDE rate. While the black hole masses of TDEs have similar distributions between X-ray and optical/UV bright TDEs, compiling a large sample of known TDEs allows us to identify a significant shift in the stellar masses and absolute magnitude distributions between the two samples, with optical/UV TDEs having higher stellar masses than the X-ray TDEs. This may be due to an underlying trend in black hole mass we do not yet have the statistical power to resolve. Future work on determining the black hole masses of new TDEs and understanding the selection effects of TDEs identified in the X-ray or optical will be needed. Most TDE host galaxies are quiescent, with little current star formation, and TDEs are over-represented in galaxies with post-starburst or quiescent Balmer-strong star formation histories. We present new estimates of the TDE rate enhancement in these samples depending on the definition of these classes and the types of TDEs considered. For the TDE host galaxies considered in this review, 5/41 (12%) are post-starburst galaxies and 13/41 (32%) are either quiescent Balmer-strong or post-starburst. Of the 4 X-ray TDEs, 3 (75%) are quiescent Balmer-strong and 1 (25%) is post-starburst. Of the 15 broad H/He line TDEs, 9 (60%) are quiescent Balmer-strong and 5 (33%) are post-starburst. We note again that these TDE classifications are tentative and subject to a number of observational biases. More observations of the host galaxies for a large sample of well-characterized TDEs are needed to overcome these uncertainties and the small-number statistics limiting our precision here. While controlling for some galaxy properties, notably bulge color and central concentration, can reduce the TDE enhancement rate in post-starburst or quiescent Balmer-strong galaxies, these properties are strongly correlated with galaxy SFH during the post-starburst phase. The TDE enhancement rates in such hosts are thus consistent (given the small number statistics) between the studies of @French2016, @Law-Smith2017 and @Graur2018. While the rate enhancement in quiescent Balmer-strong and post-starburst galaxies results in higher TDE rates, the resulting suppression of TDE rates in normal quiescent galaxies is mild, at $<3\times$. TDE host galaxies have higher concentrations of stellar light than expected given their stellar mass or black hole mass. We aggregate here observations from many sources of TDE host galaxy optical light concentrations, and present new versions of two analyses from the literature. We consider the effect of volume-correcting the SDSS comparison sample on the result by @Law-Smith2017 that TDE host galaxies have high [Sérsic ]{}indices for their black hole masses, finding that despite the large number of low black hole mass galaxies inferred from the volume correction, the TDE hosts still lie at high [Sérsic ]{}indices for their black hole masses, compared with the rest of the galaxy sample. We also add measurements for two additional TDE host galaxy velocity dispersions from @Wevers2019 to the analysis of @Graur2018 that shows TDE hosts have high stellar surface densities for their velocity dispersions, and find the conclusions unchanged. The ionisation states of TDE host galaxies span a range from star-formation dominated to AGN-dominated, and we identify many TDE host galaxies with signs of on-going gas accretion by the SMBH. However, given the significant selection biases affecting this distribution, and the uncertainties in identifying especially low-luminosity AGN or LINERs, we urge caution in its interpretation. The extra-galactic environments of TDE host galaxies show no signs of being different than the general galaxy population or the post-starburst galaxy population, but this comparison is especially limited by the small numbers of TDE hosts with well-studied extra-galactic environments. We summarize the state of several possible explanations for the links between the TDE rate and host galaxy type, including the effect of stellar overdensities, black hole binaries, circumnuclear gas, and dynamical or secular effects. We present estimates of the TDE rate for different host galaxy types and quantify the degree to which rate enhancement in some groups results in rate suppression in others. We discuss the possibilities for using TDE host galaxies to assist in identifying TDEs in upcoming large transient surveys and possibilities for TDE observations to be used to study their host galaxies. We note that the TDEs considered here are a relatively small number of events, with classifications subject to observational and selection biases which may change in future studies of TDEs. We identify the following as important open questions in this field: 1. What is the primary driver of the observed correlations between galaxy-scale host properties and the TDE rate? 2. What are the distributions of stellar mass and the stellar kinematics of TDE host galaxies within the radius of gravitational influence of their supermassive black holes? 3. Do the same physical effects drive the observed enhanced TDE rates in post-starburst or quiescent Balmer-strong galaxies and galaxies with high central concentrations? 4. Are the high central concentrations at kpc scales of optical stellar light seen in TDE hosts correlated with high central concentrations of stars at the nucleus? 5. What is the unobscured TDE rate in starburst galaxies? Does the enhancement during the post-starburst phase arise from a higher absolute rate during this phase or simply a higher observed rate due to dust obscuration during the starburst phase and/or selection bias against AGN in starburst galaxies? 6. How does the presence of an existing AGN of varying accretion rates affect the rates and observability of TDEs? 7. What is the connection between TDEs and Narrow-Line Seyfert Is? 8. What is the connection between TDEs and the LINER-like emission present in most post-starburst or quiescent Balmer-strong galaxies? What can cases where a host galaxy has pre-TDE spectroscopy tell us about the evolution of narrow line emission from TDEs vs. AGN? 9. Are there differences in the host galaxies depending on the type of TDE? 10. What drives the observed difference in stellar mass between the optical and X-ray bright TDEs? Is the difference due to a difference in intrinsic black hole mass? Are the observed trends affected by selection bias in how TDEs are identified in the optical vs. X-ray? 11. How can we best use our understanding of TDE host galaxies to study supermassive black holes and their host galaxies using new observations from next-generation surveys like LSST and *eROSITA*? How can we best compare the samples of TDEs discovered using LSST vs. *eROSITA*? Planned time-domain programs in the next decade will discover hundreds to thousands of new TDEs and will enable the detailed study of TDEs and their host galaxies with significantly greater statistical power. We thank the referees for their detailed feedback and helpful suggestions, which have improved this review. The authors thank ISSI for their support and hospitality and the review organizers for their leadership in coordinating this set of reviews. K.D.F. is supported by Hubble Fellowship grant HST-HF2-51391.001-A, provided by NASA through a grant from the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. A.I.Z. acknowledges support from NASA through STScI grant HST-GO-14717.001-A. T.W. is funded in part by European Research Council grant 320360 and by European Commission grant 730980. O.G. is supported by an NSF Astronomy and Astrophysics Fellowship under award AST-1602595. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III website is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration, including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, the Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. This publication makes use of data products from the Wide-field Infrared Survey Explorer [@2010AJ....140.1868W], which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This work made use of the IPython package [@PER-GRA:2007]. This research made use of SciPy [@jones_scipy_2001]. This research made use of Astropy, a community-developed core Python package for Astronomy . This research made use of NumPy [@van2011numpy]. [^1]: See e.g. @Jonker2019 for more late-time X-ray detections of UV/optical TDEs; this article was posted to the arXiv during review of this article. [^2]: We note that the sample presented by @Wevers2019 contains a smaller but not completely overlapping sample to that presented in this review, due to differences in the TDE selection and differences in available data. [^3]: This event is included in our catalogue (F01004), although its classification as a TDE is controversial. @Trakhtenbrot2019 argue it may not be a true TDE, although the space for observed TDE features may be broader than expected [@Leloudas2019]. [^4]: See above for a more thorough discussion of the [Sérsic ]{}indices of TDE hosts, accounting for trends in stellar mass and black hole mass. [^5]: ASASSN14li, RBS1032, SDSS J1342, SDSS J0748, ASASSN14ae, ASASSN18zj [^6]: SDSS J1323, NGC 5905, SDSS J0952, PTF09ge, AT2018dyk, AT2018bsi [^7]: Although the details of this will depend on the TDE samples and comparison samples used, see previous discussions in §\[sec:sfh\] and §\[sec:conc\]. [^8]: $\beta$ is defined such that if all orbits are purely radial, $\beta=1$, and if all orbits are purely tangential, $\beta = -\infty$.

--- abstract: 'Motivated by quantum nature of gravitating black holes, higher dimensional exact solutions of conformal gravity with an abelian gauge field is obtained. It is shown that the obtained solutions can be interpreted as singular black holes. Then, we calculate the conserved and thermodynamic quantities, and also, perform thermal stability analysis of the obtained black hole solutions. In addition, we show that the critical behavior does not occur for these black holes. Finally, we consider a minimally coupled massive scalar perturbation and calculate the quasinormal modes by using the sixth order WKB approximation and the asymptotic iteration method. We also investigate the time evolution of modes through the discretization scheme.' author: - 'Seyed Hossein Hendi$^{1,2}$[^1], Mehrab Momennia$^{1}$[^2] and Fatemeh Soltani Bidgoli$^{1}$' title: 'Higher Dimensional Conformal-$U(1)$ Gauge/Gravity Black Holes: Thermodynamics and Quasinormal Modes ' --- Introduction ============ Considering the quantum effects in gravitational interaction, one may find that the higher-curvature modification of general relativity is inevitable. However, in order to have a physically ghost free theory of higher-curvature modifications, some special constraints should be applied. Fortunately, there are known higher-curvature interesting renormalizable actions with no ghosts under certain criterion. As an interesting example, we can regard the so-called Conformal Gravity (CG), which is defined by the square of the Weyl tensor [@Riegert; @Lu]. The CG is an interesting theory of modified general relativity with a remarkable property which is sensitive to angles, but not distances. In other words, it is invariant under local stretching of the metric which is called the Weyl transformation, $g_{\mu \nu }(x)\rightarrow \Omega ^{2}(x)g_{\mu \nu }(x)$. It has been shown that CG is useful for constructing supergravity theories [@Bergshoeff; @Wit] and can be considered as a possible UV completion of gravity [@Adler; @Hooft; @Mannheim]. It may be also arisen from twister-string theory with both closed strings and gauge-singlet open strings [@Berkovits]. Moreover, CG can be appeared as a counterterm in $adS_{5}/CFT_{4}$ calculations [Liu,Balasubramanian]{}. In addition to the motivations mentioned above, solving the dark matter and dark energy problems are two of the most important and interesting motivations of studying CG theory [@Mannheim]. Since CG is renormalizable [@Adler; @Stelle] and the requirement of conformal invariance at the classical level leads to a renormalizable gauge theory of gravity, it seems interesting to consider black holes in CG which permits a consistent picture of black hole evaporation [@Hasslacher]. The first attempt to obtain the spherically symmetric black hole solutions in four dimensions has been done by Bach [@Bach], and then, Buchdahl has considered a particular case of the conformal solutions in [@Buchdahl]. It is worthwhile to mention that the $4$-dimensional solution of Einstein gravity is a solution of CG as well. In addition, it has been shown that the Einstein solutions can be obtained by considering suitable boundary condition on the metric in CG [@Maldacena; @Anastasiou]. CG can also be introduced in higher dimensions ($D>4$), straightforwardly [@B]. Nevertheless, unlike the $4$-dimensional case, CG does not admit Einstein trivial solutions in higher dimensions. This nontrivial behavior is due to the fact that in contrast to the $4$-dimensional action, the Kretschmann scalar $R^{\alpha \beta \gamma \delta }R_{\alpha \beta \gamma \delta }$ contributes dynamically in the higher dimensions [@C]. Such a nontrivial behavior motivates one to investigate higher dimensional CG black hole solutions. On the other hand, when a black hole undergoes perturbations, the resulting behavior leads to some oscillations which are called quasinormal modes (QNMs). The quasinormal frequencies (QNFs) related to such QNMs are independent of initial perturbations and they are the intrinsic imprint of the black hole response to external perturbations on the background spacetime of black hole. The asymptotic behavior of the QNMs relates to the quantum gravity [@Nollert; @Hod] and the imaginary part of the frequencies in adS spacetime corresponds to the decay of perturbations of a thermal state in the conformal field theory [@Horowitz; @Lemos]. The QNM is one of the most important and exciting features of compact objects and describes the evolution of fields on the background spacetime of such objects [@Kokkotas; @Berti; @KonoplyaR]. Therefore, the QNM spectrum reflects the properties of spacetime, and consequently, we can find out about the properties of background spacetime by studying the QNMs. As a result, the QNM spectrum will be a function of black hole parameters, such as mass, charge, and angular momentum. The QNM spectrum of gravitational perturbations can be observed by gravitational wave detectors [Abbott1,Abbott2,Abbott3]{}, and after the detection of the QNFs of compact binary mergers by LIGO, investigation of the QNMs of black holes attracted attention during the past three years (for instance, see an incomplete list [Konoplyajcap,Lin,Cook,Breton,Gonzalez,Kunz,Qian,Price,Herdeiro,Zhidenko,CardosoPRL,Stuchlik,Assumpcao,Blazquez,Rincon,Panotopoulos]{} and references therein). In this paper, we consider higher dimensional charged black hole solutions in CG theory and investigate their stabilities. The outline of this paper is as follows. In the next section, we give a brief review on neutral and charged black holes of CG in $4$-dimensional spacetime. We also construct $D$-dimensional topological static black hole solutions of CG gravity in the presence of generalized Maxwell theory. Then, the thermodynamics of obtained solutions is investigated and the conserved and thermodynamic quantities are calculated. We also perform the thermal stability analysis of the solutions in the canonical ensemble, and also, by using geometrical thermodynamic approach. In addition, we investigate the possibility of the critical behavior of obtained black hole solutions. Finally, we consider a minimally coupled massive scalar perturbation in the background spacetime of the black holes and calculate the related QNMs by using the sixth order WKB approximation and the asymptotic iteration method (AIM). Then, we argue that these black holes cannot have quasi-resonance modes which are a feature of massive scalar perturbations. We finish our paper with some concluding remarks. four-dimensional exact solutions ================================ At the first step, we consider a four-dimensional conformal action as $$\begin{aligned} I_{G} &=&-\alpha \int d^{4}x\sqrt{-g}C_{\lambda \mu \nu \kappa }C^{\lambda \mu \nu \kappa } \notag \\ &\equiv &-2\alpha \int d^{4}x\sqrt{-g}\left[ R^{\mu \nu }R_{\mu \nu }-{\frac{% 1}{3}}(R_{\phantom{\alpha}\alpha }^{\alpha })^{2}\right] , \label{IG}\end{aligned}$$where the Weyl conformal tensor is $$\begin{aligned} C_{\lambda \mu \nu \kappa } &=&R_{\lambda \mu \nu \kappa }+{\frac{1}{6}}R_{% \phantom{\alpha}\alpha }^{\alpha }\left[ g_{\lambda \nu }g_{\mu \kappa }-g_{\lambda \kappa }g_{\mu \nu }\right] \notag \\ &-&{\frac{1}{2}}\left[ g_{\lambda \nu }R_{\mu \kappa }-g_{\lambda \kappa }R_{\mu \nu }-g_{\mu \nu }R_{\lambda \kappa }+g_{\mu \kappa }R_{\lambda \nu }% \right] . \label{C}\end{aligned}$$ Variation of action (\[IG\]) with respect to the metric tensor leads to the following equation of motion $$\begin{aligned} &&W^{\mu \nu }=2C_{\ \ \ \ \ ;\lambda \kappa }^{\mu \lambda \nu \kappa }-C^{\mu \lambda \nu \kappa }R_{\lambda \kappa }= \notag \\ &&\frac{1}{2}g^{\mu \nu }(R_{\phantom{\alpha}\alpha }^{\alpha })_{% \phantom{;\beta};\beta }^{;\beta }+R_{\phantom{\mu\nu;\beta};\beta }^{\mu \nu ;\beta }-R_{\phantom{\mu\beta;\nu};\beta }^{\mu \beta ;\nu }-R_{% \phantom{\nu \beta;\mu};\beta }^{\nu \beta ;\mu }-2R^{\mu \beta }R_{% \phantom{\nu}\beta }^{\nu } \notag \\ &&+\frac{1}{2}g^{\mu \nu }R_{\alpha \beta }R^{\alpha \beta }-\frac{2}{3}% g^{\mu \nu }(R_{\phantom{\alpha}\alpha }^{\alpha })_{\phantom{;\beta};\beta }^{;\beta }+\frac{2}{3}(R_{\phantom{\alpha}\alpha }^{\alpha })^{;\mu ;\nu }+% \frac{2}{3}R_{\phantom{\alpha}\alpha }^{\alpha }R^{\mu \nu }-\frac{1}{6}% g^{\mu \nu }(R_{\phantom{\alpha}\alpha }^{\alpha })^{2}=0. \label{W}\end{aligned}$$ It was shown that the static spherically symmetric solution of conformal gravity in four dimensions can be written as $$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega ^{2}, \label{metric}$$where $d\Omega ^{2}$ is the line element of a $2-$sphere, $S^{2}$, and the metric function is [@Buchdahl] $$f(r)=c_{0}+\frac{d}{r}+\frac{c_{0}^{2}-1}{3d}r-\frac{1}{3}\Lambda r^{2}. \label{f(r)}$$ It is clear that for nonvanishing $\Lambda $, Eq. (\[f(r)\]) is not a solution of Einstein gravity, while as long as $\Lambda =0$, the metric becomes identical to the Schwarzschild solution of Einstein gravity. It is worth mentioning that although $\Lambda $ plays the role of the cosmological constant, it is arisen purely as an integration constant and is not put in the action by hand. Such a constant cannot be added to the action of CG because it would introduce a length scale and hence break the conformal invariance. In order to add an action of matter, we should take care of its conformal transformation to keep the theory be conformally invariant. Fortunately, the Lagrangian of Maxwell theory is conformal invariant in four dimensions and we can add it to the gravitational sector of conformal theory as an appropriate matter field. So, we can consider the static charged adS solutions of conformal$-U(1)$ gravity in four dimensions [@Bach]. The appropriate action is$$I=\alpha \int d^{4}x\sqrt{-g}\left( \frac{1}{2}C^{\mu \nu \rho \sigma }C_{\mu \nu \rho \sigma }+\frac{1}{3}F^{\mu \nu }F_{\mu \nu }\right) , \label{I2}$$where the unusual sign in front of the Maxwell term comes from the so-called critical gravity, which is necessary to recover the Einstein gravity from conformal gravity in IR limit. The static topological solution is found in [@D] the same as Eq. (\[f(r)\]) with the following gauge potential one form $$A=-\frac{Q}{r}dt,$$which leads to the following metric function [@D] $$f(r)=c_{0}+\frac{d}{r}+c_{1}r-\frac{1}{3}\Lambda r^{2}, \label{Bach}$$ In order to have consistent solutions, three integration constants $c_{0}$, $% c_{1}$, and $d$ should obey an algebraic constraint $$3c_{1}d+\varepsilon ^{2}+Q^{2}=c_{0}^{2}, \label{Back}$$where $\varepsilon =1,-1,0$ denotes spherical, hyperbolic, and planar horizons, respectively. Therefore, the metric function takes the following compact form$$f(r)=c_{0}+\frac{d}{r}+\frac{c_{0}^{2}-\varepsilon ^{2}-Q^{2}}{3d}r-\frac{1}{% 3}\Lambda r^{2}. \label{back}$$ higher dimensional solutions \[DD\] =================================== Here, we are going to generalize the conformal action of $U(1)$-gauge/gravity coupling in higher dimensions. As we know, the Maxwell action does not enjoy the conformal invariance properties in higher dimensions, and therefore, the higher dimensional solutions in CG cannot be produced in the presence of Maxwell field (and also the other electrodynamic fields that are not conformal invariant in higher dimensions). So, we should consider a generalization of linear Maxwell action to the case that it respects the invariance of conformal transformation. To do so, we take into account the power Maxwell nonlinear theory, which its Lagrangian is a power of Maxwell invariant, $(-F_{\mu \nu }F^{\mu \nu })^{s}$. It is a matter of calculation to show that the power Maxwell action enjoys the conformal invariance, for $% s=D/4$ ($D=$dimension of spacetime) [@Mokhtar]. In other words, it is easy to show that as long as $s=D/4$, the energy-momentum tensor of power Maxwell invariant theory is traceless [@Mokhtar]. Regarding the mentioned issues, we find that the suitable action of higher dimensional conformal $U(1)$-gauge/gravity action can be written as$$I=\frac{1}{16\pi }\int d^{D}x\sqrt{-g}\left( C^{\mu \nu \rho \sigma }C_{\mu \nu \rho \sigma }+\beta (-F^{\mu \nu }F_{\mu \nu })^{\frac{D}{4}}\right) . \label{I3}$$ Hereafter, we can regard a higher dimensional static spacetime and look for exact solutions with black hole interpretation. Variation of this action with respect to the metric tensor $g_{\mu \nu }$ and the Faraday tensor $% F_{\mu \nu }$ leads to the following field equations$$\mathbf{E}_{\rho \sigma }=\left( \nabla ^{\mu }\nabla ^{\nu }+\frac{1}{2}% R^{\mu \nu }\right) C_{\rho \nu \mu \sigma }+\frac{\beta }{8}\left[ g_{\rho \sigma }(-F^{\mu \nu }F_{\mu \nu })^{\frac{D}{4}}+D(-F^{\mu \nu }F_{\mu \nu })^{\frac{D}{4}-1}F_{\sigma \delta }F_{\rho }^{\phantom{\nu}\delta }\right] =0,$$$$\partial _{\rho }\left[ \sqrt{-g}(-F^{\mu \nu }F_{\mu \nu })^{\frac{D}{4}% -1}F^{\rho \sigma }\right] =0.$$ Since we are looking for topological black hole solutions of mentioned field equations, we express the metric of a $D$-dimensional spacetime as follows$$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\Sigma _{k,D-2}^{2}, \label{Metric}$$where $k$ denotes spherical $\left( k=1\right) $, hyperbolic $\left( k=-1\right) $, and planar $\left( k=0\right) $ horizons of the $\left( D-2\right) $-dimensional manifold with the following line element$$d\Sigma _{k,D-2}^{2}=\left\{ \begin{array}{cc} d\Omega _{D-2}^{2}=d\theta _{1}^{2}+\sum\limits_{i=2}^{D-2}\prod\limits_{j=1}^{i-1}\sin ^{2}\theta _{j}d\theta _{i}^{2} & k=1 \\ d\Xi _{D-2}^{2}=d\theta _{1}^{2}+\sinh ^{2}\theta _{1}\left( d\theta _{2}^{2}+\sum\limits_{i=3}^{D-2}\prod\limits_{j=2}^{i-1}\sin ^{2}\theta _{j}d\theta _{i}^{2}\right) & k=-1 \\ dl_{D-2}^{2}=\sum\limits_{i=1}^{D-2}d\theta _{i}^{2} & k=0% \end{array}% \right. ,$$in which $d\Omega _{D-2}^{2}$ is the standard metric of a unit $\left( D-2\right) $-sphere, $d\Xi _{D-2}^{2}$ is the metric of a $\left( D-2\right) $-dimensional hyperbolic plane with unit curvature, and $dl_{D-2}^{2}$ is the flat metric of $R^{D-2}$. Using this metric and a radial gauge potential ansatz $A_{\mu }=-qr^{\left( 2s-D+1\right) /\left( 2s-1\right) }\delta _{\mu }^{0}$, one can find the nonzero components of the theory as follows$$\begin{aligned} \mathbf{E}_{tt} &=&2D_{3}r^{4}f\left( r\right) f^{\left( 4\right) }\left( r\right) +D_{3}\left[ rf^{\prime }\left( r\right) +4D_{5/2}f\left( r\right) % \right] r^{3}f^{\prime \prime \prime }\left( r\right) \notag \\ &&+\left[ \frac{D_{3}}{2}r^{2}f^{\prime \prime }\left( r\right) +\left( 2D^{2}D_{23/2}+85D_{102/85}\right) f\left( r\right) +D_{3}D_{4}\left( \frac{% 3rf^{\prime }\left( r\right) }{2}+k\right) \right] r^{2}f^{\prime \prime }\left( r\right) \notag \\ &&-\left[ 3D_{10/3}D_{3}rf^{\prime }\left( r\right) +\left( 4D^{2}D_{49/4}+187D_{228/187}\right) f\left( r\right) +5kD_{3}D_{16/5}\right] rf^{\prime }\left( r\right) \notag \\ &&+2\left( 2D^{2}D_{23/2}+84D_{99/84}\right) f^{2}\left( r\right) -4k\left( D^{2}D_{12}+45D_{54/45}\right) f\left( r\right) -2D_{3}^{2}k^{2} \notag \\ &&-\frac{D_{1}}{2}r^{4}\beta \left( 1-2s\right) \left( \frac{\sqrt{2}q\left( 2s-D_{1}\right) }{\left( 2s-1\right) r^{\left( D-2\right) /\left( 2s-1\right) }}\right) ^{2s}, \label{F1}\end{aligned}$$$$\begin{aligned} \mathbf{E}_{rr} &=&D_{3}\left[ 2f\left( r\right) -rf^{\prime }\left( r\right) \right] r^{3}f^{\prime \prime \prime }\left( r\right) -D_{3}\left[ \frac{r^{2}}{2}f^{\prime \prime }\left( r\right) -3D_{10/3}f\left( r\right) +D_{4}\left( \frac{3}{2}rf^{\prime }\left( r\right) +k\right) \right] \notag \\ &&\times r^{2}f^{\prime \prime }\left( r\right) +D_{3}\left[ 3D_{10/3}rf^{\prime }\left( r\right) -9D_{28/9}f\left( r\right) +5kD_{16/5}% \right] rf^{\prime }\left( r\right) \notag \\ &&+2D_{3}^{2}\left[ 3f^{2}\left( r\right) -4kf\left( r\right) +k^{2}\right] +% \frac{D_{1}}{2}r^{4}\beta \left( 1-2s\right) \left( \frac{\sqrt{2}q\left( 2s-D_{1}\right) }{\left( 2s-1\right) r^{\left( D-2\right) /\left( 2s-1\right) }}\right) ^{2s}, \label{F2}\end{aligned}$$$$\begin{aligned} \mathbf{E}_{\theta \theta } &=&D_{3}r^{4}f\left( r\right) f^{\left( 4\right) }\left( r\right) +D_{3}\left[ rf^{\prime }\left( r\right) +2D_{3}f\left( r\right) \right] r^{3}f^{\prime \prime \prime }\left( r\right) \notag \\ &&+\left[ \frac{D_{3}}{2}r^{2}f^{\prime \prime }\left( r\right) +\left( D^{2}D_{13}+52D_{66/52}\right) f\left( r\right) +D_{3}D_{4}\left( \frac{% 3rf^{\prime }\left( r\right) }{2}+k\right) \right] r^{2}f^{\prime \prime }\left( r\right) \notag \\ &&-\left[ 3D_{10/3}D_{3}rf^{\prime }\left( r\right) +\left( 2D^{2}D_{29/2}+121D_{156/121}\right) f\left( r\right) +5kD_{3}D_{16/5}\right] rf^{\prime }\left( r\right) \notag \\ &&+2\left( D^{2}D_{13}+51D_{63/51}\right) f^{2}\left( r\right) -2k\left( D^{2}D_{14}+57D_{72/57}\right) f\left( r\right) -2D_{3}^{2}k^{2} \notag \\ &&+\frac{D_{1}D_{2}}{4}r^{4}\beta \left( \frac{\sqrt{2}q\left( 2s-D_{1}\right) }{\left( 2s-1\right) r^{\left( D-2\right) /\left( 2s-1\right) }}\right) ^{2s}, \label{F3}\end{aligned}$$in which we used $D_{i}=D-i$ for convenience and prime refers to $d/dr$. However, we will use common $D-i$ in indices and powers for clarity of equations. Solving Eqs. (\[F1\])-(\[F3\]), and keeping in mind that $% s=D/4$, we can obtain the metric function for $D\geq 5$ as follows$$f\left( r\right) =k-\frac{C_{1}}{r^{D-3}}-\frac{\beta \mu q^{D/2}}{C_{1}}% r+C_{2}r^{2}, \label{MF}$$where $C_{1}$ and $C_{2}$ are two integration constants, and $\mu =2^{\left( D-4\right) /4}\left( D_{2}D_{3}\right) ^{-1}$ is a dimensionful constant. Interestingly, we see that although the field equations are too complicated, the solutions are quite simple. It is noticeable to mention that the solution to $D=4$ is given by (\[back\]) when we set $\beta =-2/3 $ in (\[I3\]), but it cannot be obtained by using the field equations (\[F1\])-(\[F3\]). Therefore, the four-dimensional spacetime has one more integration constant compared to the other dimensions. In order to obtain a compact form of the metric function to be valid for all dimensions, one can consider a special case $c_{0}=\varepsilon $ of the four-dimensional metric function (\[back\]). In this situation, the metric function of $D\geq 4$ is given by (\[MF\]) and the field equations ([F1]{})-(\[F3\]) can be used for $D=4$ as well. We should note that since we considered a special case, $c_{0}=\varepsilon $, the $4$-dimensional solution given in (\[MF\]) is also a solution of the field equations of (\[I2\]). $% \begin{array}{ccc} \epsfxsize=10cm \epsffile{f4.eps} & \epsfxsize=10cm \epsffile{f5.eps} & \end{array} $ Having solutions at hand, we are in a position to check that these solutions can be considered as a black hole or not. To do so, we first look for the singularities of the solutions. By calculating the Kretschmann scalar$$R^{\lambda \tau \rho \sigma }R_{\lambda \tau \rho \sigma }=2DD_{1}C_{2}^{2}-% \frac{4D_{1}D_{2}C_{2}\beta \mu q^{D/2}}{C_{1}r}+\frac{2D_{2}^{2}\beta ^{2}\mu ^{2}q^{D}}{C_{1}^{2}r^{2}}+\frac{D_{1}D_{2}^{2}D_{3}C_{1}^{2}}{% r^{2\left( D-1\right) }}, \label{K}$$one can easily find that the metric (\[Metric\]) with the metric function (\[MF\]) has an essential singularity at the origin ($\underset{% r\rightarrow 0}{\lim }\left( R^{\lambda \tau \rho \sigma }R_{\lambda \tau \rho \sigma }\right) =\infty $). In addition, Fig. \[FofR\] shows that this singularity can be covered with an event horizon, and therefore, we can interpret the solution as a singular black hole. thermodynamics \[thermo\] ========================= Thermodynamic parameters ------------------------ At this stage, we calculate temperature and entropy of the obtained solutions by using the surface gravity at the event horizon and the Wald entropy formula. Then, we will investigate thermal stability of black holes in the coming subsection. By calculating the surface gravity, $\kappa =\sqrt{-\left( \nabla _{\mu }\chi _{\nu }\right) \left( \nabla ^{\mu }\chi ^{\nu }\right) /2}$ ($\chi =\partial _{t}$ is the null Killing vector of the horizon), we can obtain the Hawking temperature of the black hole at the outermost (event) horizon, $% r_{+}$. If we redefine (just for simplicity without loss of generality) the constants of the black hole solutions (\[MF\]) as $C_{1}\equiv m$, $\beta \mu q^{D/2}/C_{1}\equiv \tilde{\mu}$, and $C_{2}\equiv -\Lambda /3$, the metric function takes the following form$$f\left( r\right) =k-\frac{m}{r^{D-3}}-\tilde{\mu}r-\frac{\Lambda }{3}r^{2}, \label{NMF}$$and accordingly, the temperature of the black hole is given as $$T=\frac{\kappa }{2\pi }=\left. \frac{f^{\prime }(r)}{4\pi }\right\vert _{r=r_{+}}=\frac{1}{12\pi r_{+}}\left( 3D_{3}k-3D_{2}\tilde{\mu}% r_{+}-D_{1}\Lambda r_{+}^{2}\right) . \label{temp}$$ In addition, the entropy of the black hole in higher derivative theories can be obtained by Wald formula [@Wald; @Iyer] which makes the dependence of entropy on gravitational action$$S=-2\pi \int_{\mathcal{M}}d^{D-2}x\sqrt{h}\frac{\delta \mathcal{L}}{\delta R_{\mu \nu \rho \sigma }}\xi _{\mu \nu }\xi _{\rho \sigma },$$where $\mathcal{L}$ is the Lagrangian density of the theory, $\xi _{\mu \nu }$ is the binormal to the (arbitrary) cross-section $\mathcal{M}$ of the horizon, and $h$ is the determinant of induced volume on $\mathcal{M}$. Therefore, the entropy of our case study black hole takes the following form$$S=-\frac{1}{8}\int_{\mathcal{M}}d^{D-2}x\sqrt{h}C^{\mu \nu \rho \sigma }\xi _{\mu \nu }\xi _{\rho \sigma }=\frac{D_{2}D_{3}m_{+}\omega _{D-2}}{4r_{+}}, \label{entropy}$$in which $\omega _{D-2}$ denotes the volume of $d\Sigma _{k,D-2}^{2}$ and $% m_{+}$ can be obtained by $f(r_{+})=0$. Here, we use the first law of thermodynamics ($\delta M=T\delta S$) to calculate the total mass of the solutions as$$\begin{aligned} M &=&\frac{\omega _{D-2}r_{+}^{D-5}}{144\pi D_{4}D_{5}}\left\{ D_{2}\left[ 3D_{3}D_{4}k\left( 1-\delta _{D,5}\right) \right] ^{2}\right. \notag \\ &&-3D_{2}D_{5}k\left[ 3D_{3}\tilde{\mu}\left( 17+2DD_{6}\right) \left( 1-\delta _{D,4}\right) +2D_{4}\Lambda \left( 5+DD_{5}\right) r_{+}\right] r_{+} \notag \\ &&+D_{3}D_{4}D_{5}\left[ \left( 3D_{2}\tilde{\mu}\right) ^{2}+3\Lambda \tilde{\mu}\left( 7+2DD_{4}\right) r_{+}+\left( D_{2}\Lambda r_{+}\right) ^{2}\right] r_{+}^{2} \notag \\ &&\left. -18D_{5}k\ln \left( \frac{r_{+}}{l}\right) \left( D_{4}\tilde{\mu}% r_{+}\delta _{D,4}-6k\delta _{D,5}\right) \right\} , \label{mass}\end{aligned}$$where $\delta _{a,b}$ is the Kronecker delta and $l$ is a constant with length dimension. Now, it is worthwhile to compare the neutral case ($\tilde{\mu}=0$) of obtained solutions (\[NMF\]) with the (a)dS Schwarzschild black hole. The metric function and temperature are totally the same as the Schwarzschild one. But they have the same entropy just for $\Lambda =3\left( D_{2}D_{3}r_{+}^{2}\right) ^{-1}\left( 6k+DD_{5}k-r_{+}^{2}\right) $. In addition, one can solve $M-\left( 48\pi \right) ^{-1}D_{1}r_{+}^{D-3}\left( 3k-\Lambda r_{+}^{2}\right) =0$ to find some conditions in order to have the same mass. However, we should note that it is not possible to find a constraint for conformal solutions to obtain the same entropy and mass of the Schwarzschild black hole, simultaneously. Thermal stability ----------------- Now, we investigate the heat capacity of constructed black hole solutions in order to find the thermally stable criteria. The heat capacity of the solutions has the following explicit form$$\begin{aligned} C &=&T\partial _{T}S= \notag \\ &&\frac{D_{2}D_{3}\omega _{D-2}r_{+}^{D-4}\left[ 3D_{4}k-\left( 3D_{3}\tilde{% \mu}+D_{2}\Lambda r_{+}\right) r_{+}\right] \left[ -3D_{3}k+\left( 3D_{2}% \tilde{\mu}+D_{1}\Lambda r_{+}\right) r_{+}\right] }{12\left( 3D_{3}k+D_{1}\Lambda r_{+}^{2}\right) }. \label{HC}\end{aligned}$$ The sign of heat capacity shows the stability condition of the solutions. The positive sign shows stable solutions whereas the negative sign indicates unstable ones. The heat capacity changes sign whenever it meets root or divergence points. The root of heat capacity indicates a bound point which separates the physical black holes (with positive temperature) from non-physical ones (with negative temperature). Moreover, divergence points may separate stable and unstable regions. Therefore, it is important to look for the roots and divergencies of Eq. (\[HC\]). This equation has one divergence point (dp) given by$$r_{+,dp}=\sqrt{-\frac{3D_{3}k}{D_{1}\Lambda }}, \label{DP}$$which is real whenever $k$ or $\Lambda $ be negative. Thus, the heat capacity has, at most, one possible divergence point under certain conditions. As a result, the obtained black hole solutions just can undergo the Hawking-Page phase transition [@HawkingPage]. The root points (rp) of the heat capacity are given by$$r_{+,rp}=\left\{ \begin{array}{c} r_{+,rp1}=-\frac{3D_{3}\tilde{\mu}+\sqrt{12D_{2}D_{4}k\Lambda +\left( 3D_{3}% \tilde{\mu}\right) ^{2}}}{2D_{2}\Lambda } \\ \\ r_{+,rp2}=-\frac{3D_{3}\tilde{\mu}-\sqrt{12D_{2}D_{4}k\Lambda +\left( 3D_{3}% \tilde{\mu}\right) ^{2}}}{2D_{2}\Lambda } \\ \\ r_{+,rp3}=-\frac{3D_{2}\tilde{\mu}+\sqrt{12D_{1}D_{3}k\Lambda +\left( 3D_{2}% \tilde{\mu}\right) ^{2}}}{2D_{1}\Lambda } \\ \\ r_{+,rp4}=-\frac{3D_{2}\tilde{\mu}-\sqrt{12D_{1}D_{3}k\Lambda +\left( 3D_{2}% \tilde{\mu}\right) ^{2}}}{2D_{1}\Lambda }% \end{array}% \right. . \label{RP}$$ It is clear that the final number of real positive roots depends on the choice of the free parameters $D$, $k$, $\Lambda $, and $\tilde{\mu}$. For example, in order to have all four roots in adS spacetime, the free parameters should obey the certain condition $-\tilde{\mu}^{2}\leq \frac{% 4D_{1}D_{3}k\Lambda }{3D_{2}^{2}}<0$. This is the strongest condition on (\[RP\]) which leads to four real positive roots for the heat capacity and there are some weaker conditions that lead to one, two or three roots. It is worthwhile to mention that due to the presence of more than one free parameter, we should fix some of them and study the conditions of appearing the roots and divergence point. However, this investigation is not enough to study the thermal stability of black holes and in order to see the positivity and negativity of the heat capacity, we should plot the related diagrams simultaneously. We do these studies for $\omega _{D-2}=1$ and $% \tilde{\mu}>0$ as follows. $% \begin{array}{cccc} \epsfxsize=5cm \epsffile{CadSk1C1.eps} & \epsfxsize=5cm % \epsffile{CadSk1C2.eps} & \epsfxsize=5cm \epsffile{CadSk1C3.eps} & \end{array} $ $% \begin{array}{ccc} \epsfxsize=7.5cm \epsffile{CdSk1C1.eps} & \epsfxsize=7.5cm % \epsffile{CdSk1C2.eps} & \end{array} $ $% \begin{array}{ccc} \epsfxsize=7.5cm \epsffile{CadSk0.eps} & \epsfxsize=7.5cm % \epsffile{CdSk0.eps} & \end{array} $ $% \begin{array}{cccc} \epsfxsize=5cm \epsffile{CadSkM1C1.eps} & \epsfxsize=5cm % \epsffile{CadSkM1C2.eps} & \epsfxsize=5cm \epsffile{CdSkM1.eps} & \end{array} $ ### Case I: spherical topology ($k=1$) Here, we fix $k=1$ and study the possibility of the presence of roots and divergence. At the same time, one may think about the asymptotic adS ($% \Lambda <0$) or dS ($\Lambda >0$) solutions. For adS solutions, we have two roots given by $r_{+,rp1}$ and $r_{+,rp2}$ under condition $% 12D_{2}D_{4}\Lambda +\left( 3D_{3}\tilde{\mu}\right) ^{2}\geq 0$ (left panel of Fig. \[CadSk1\]), and also, $r_{+,rp3}$ and $r_{+,rp4}$ for $% 12D_{1}D_{3}\Lambda +\left( 3D_{2}\tilde{\mu}\right) ^{2}\geq 0$ (middle panel of Fig. \[CadSk1\]). If we obey the stronger condition, $% 12D_{1}D_{3}\Lambda +\left( 3D_{2}\tilde{\mu}\right) ^{2}\geq 0$, all four roots will be appeared (middle panel of Fig. \[CadSk1\]), and if we violate the weaker condition, $12D_{2}D_{4}\Lambda +\left( 3D_{3}\tilde{\mu}% \right) ^{2}\geq 0$, there will be no root (right panel of Fig. \[CadSk1\]). This is while the divergence point is always present. On the other hand, for dS solutions, there is no divergence point. In the case of roots, $r_{+,rp4}$ is always present (left panel of Fig. \[CdSk1\]) and we have $r_{+,rp2}$ for $D>4$ (right panel of Fig. \[CdSk1\]). In both adS and dS cases, the behavior of the heat capacity and temperature is seen in Figs. \[CadSk1\] and \[CdSk1\]. We recall that the positive sign of the heat capacity shows stable black holes and the negative sign indicates unstable ones. In addition, the positive (negative) temperature belongs to physical (non-physical) black holes. Although we are dealing with some mathematical constraints, one should note that the obtained conditions and plotted diagrams tell us an important story about the existence possibility of obtained black holes. For example, we have physical and stable small black holes with spherical topology just in four dimensions with dS asymptote. On the other hand, the large black holes are always stable and physical in asymptotically adS spacetime. ### Case II: flat topology ($k=0$) For $k=0$, the divergence point will disappear and Eq. (\[RP\]) reduces to $r_{+,rp1}=-3D_{3}\tilde{\mu}/(D_{2}\Lambda )$ and $r_{+,rp3}=-3D_{2}\tilde{% \mu}/(D_{1}\Lambda )$. Therefore, there will be two roots for adS spacetime(left panel of Fig. \[Ck0\]) whereas there is no root for dS solutions (right panel of Fig. \[Ck0\]) which shows that dS black holes are unconditionally unstable and non-physical. ### Case III: hyperbolic topology ($k=-1$) Now, we set $k=-1$ and look for stable black holes in asymptotically adS spacetime. In this case, the divergence point is absent and there are always one root, $r_{+,rp3}$ (left panel of Fig. \[CkM1\]). For $D>4$, $% r_{+,rp1} $ will appear in addition to $r_{+,rp3}$ (middle panel of Fig. \[CkM1\]). For dS solutions, the divergence point will appear but there is no root (right panel of Fig. \[CkM1\]). Although we presented our study for positive $\tilde{\mu}$, by considering $% \tilde{\mu}=\beta \mu q^{D/2}/C_{1}$, $\tilde{\mu}$ can be negative whenever $D/2$ is an odd number and black hole has a negative net charge. This case is very interesting because the negative charge changes the spacetime geometry differently compared with a positive charge. However, since the negative charge of the black hole in some other dimensions leads to imaginary $\tilde{\mu}$ and this condition ($\tilde{\mu}<0$) is a very special case, we do not involve it in our investigation. But, even considering this matter does not change our results significantly. Possibility of critical behavior -------------------------------- Investigation of $P-V$ criticality of different types of black holes in thermodynamical extended phase space attracted attention during past decade (for instance, see an incomplete list [Kubiznak,Poshteh,Caceres,PRD,Mandal,IJMPD,Hennigar,RPT,PLB,ogun,Sun,LiuYang,Jamil]{} and references therein). The $P-V$ criticality of black holes is interesting because the black hole thermodynamics leads to very similar behavior as for the typical thermodynamic systems such as van der Waals fluid. In [@YMmassive], it was shown that in order to observe the critical behavior for black holes, a local instability between two divergence points is required. Indeed, this local instability leads to a non-analytic behavior in the isotherm diagram which results in a small-large black hole phase transition. Since we have not seen such behavior (a local instability between two divergence points) during the thermal stability investigation (see Eq. (\[DP\])), our black hole case study cannot acquire the critical behavior with obtained temperature (\[temp\]), entropy ([entropy]{}), and mass (\[mass\]). Nevertheless, one can extend the thermodynamical phase space based on the Smarr relation into$$M=\frac{2}{3}TS-\mathcal{VP}+\Xi \tilde{\mu},$$where $\Xi =\left( \partial _{\tilde{\mu}}M\right) _{S,\mathcal{P}}$, $% \mathcal{V}=\left( \partial _{\Lambda }M/\partial _{\Lambda }\mathcal{P}% \right) _{S,\tilde{\mu}}$, and $\mathcal{P}$ is given by$$\begin{aligned} \mathcal{P}_{D=4} &=&\frac{\exp \left[ -2\Lambda r_{+}^{2}/\left( k-5\tilde{% \mu}r_{+}\right) \right] }{\left[ kr_{+}\left( 9\tilde{\mu}+\Lambda r_{+}\right) -5\tilde{\mu}r_{+}^{2}\left( 3\tilde{\mu}+\Lambda r_{+}\right) -3k^{2}\right] ^{3\left( 4k^{2}+3k\tilde{\mu}r_{+}+5\tilde{\mu}% ^{2}r_{+}^{2}\right) /\left( 2k-10\tilde{\mu}r_{+}\right) ^{2}}}, \label{P4} \\ \mathcal{P}_{D=5} &=&\frac{-\exp \left[ 6x\left( 12k+143\tilde{\mu}% r_{+}\right) \arctan \left[ x\left( 9k+42\tilde{\mu}r_{+}-2\Lambda r_{+}^{2}\right) \right] \right] }{\left[ 18k^{2}\left( 3\ln (r_{+}/l)-2\right) +9kr_{+}\left( 10\tilde{\mu}-\Lambda r_{+}\right) -r_{+}^{2}\left( 45\tilde{\mu}-\Lambda r_{+}\right) \left( 3\tilde{\mu}% +\Lambda r_{+}\right) \right] ^{9}}, \label{P5} \\ x &=&3^{-1}\left[ k^{2}\left( 24\ln (r_{+}/l)-25\right) -4\tilde{\mu}% r_{+}\left( 11k+64\tilde{\mu}r_{+}\right) \right] ^{-1/2},\end{aligned}$$and for $D\geq 6$, we have$$\begin{aligned} \mathcal{P}_{D\geq 6} &=&\left\{ r_{+}^{D-3}\left[ D_{2}\left( 3kD_{3}\right) ^{2}+D_{3}D_{5}r_{+}^{2}\left( 3\tilde{\mu}+\Lambda r_{+}\right) \left( D_{4}\Lambda r_{+}-15D_{2}\tilde{\mu}\right) \right. \right. \notag \\ &&\left. \left. -6D_{5}kr_{+}\left( \Lambda r_{+}\left( 3+DD_{5}\right) -6D_{3}D_{5/2}\tilde{\mu}\right) \right] \right\} ^{-3D_{2}/D_{4}}\times \notag \\ &&\exp \left[ 3D_{2}^{-1}D_{4}^{-1}\sqrt{D_{5}}\mathcal{X}\left[ 4D_{2}k\left( 7+DD_{6}\right) +3D_{3}D_{4/3}\tilde{\mu}r_{+}\left( 13+2DD_{5}\right) \right] \times \right. \notag \\ &&\left. \arctan \left[ 3^{-1}\sqrt{D_{5}}\mathcal{X}\left( 3k\left( 3+DD_{5}\right) +D_{3}r_{+}\left( 6D_{3/2}-D_{4}\Lambda r_{+}\right) \right) % \right] \right] , \label{P6} \\ \mathcal{X} &=&\frac{1}{D_{3}\tilde{\mu}r_{+}}\left[ \frac{k^{2}\left( 219D_{57/73}+8D^{2}D_{19/2}\right) }{\left( D_{3}\tilde{\mu}r_{+}\right) ^{2}% }-D_{5}\left( \frac{2k\left( 51+6DD_{19/3}\right) }{D_{3}\tilde{\mu}r_{+}}% +9D_{7/3}^{2}\right) \right] ^{-\frac{1}{2}},\end{aligned}$$in which all the extensive and intensive parameters satisfy the first law of black hole thermodynamics as$$dM=TdS+\mathcal{V}d\mathcal{P}+\Xi d\tilde{\mu}. \label{MFL}$$ It is noticeable to mention that due to the presence of the Kronecker delta in mass parameter (\[mass\]), we could not include $D=4$ and $5$dimensions in (\[P6\]). Now, it is worthwhile to concentrate our attention on the physical interpretation of $P$ and $V$. At first glance and comparing (\[MFL\]) with modified first law of thermodynamics in the extended phase space calculated in [Kubiznak,Poshteh,Caceres,PRD,Mandal,IJMPD,Hennigar,RPT,PLB,ogun,Sun,LiuYang,Jamil]{}, one may think that the pressure, $P=-\Lambda /8\pi $, modified into ([P4]{}), (\[P5\]), and (\[P6\]) so that $P$ and $V$ be the thermodynamical pressure and volume. But these equations are dimensionless whereas the dimension of pressure is $(length)^{-2}$. Therefore, $P$ and $V$ cannot be considered as the thermodynamical pressure and volume of the system, and one can consider (\[MFL\]) just as a mathematical extension of the first law of thermodynamics at first step. However, the physical interpretation of $P$ and $V$ is arguable. This result confirms the fact that in order to have the critical behavior for black holes, a local instability between two divergencies in the heat capacity is required. Geometrical thermodynamics \[GTD\] ---------------------------------- Geometrical thermodynamics is an interesting way to investigate the thermal stability of a thermodynamical system. In this perspective, the behavior of the system is governed by the Ricci scalar of a line element so that the components of the metric tensor field are the thermodynamic variables and their derivatives. In 1975, Weinhold [@Weinhold1; @Weinhold2] introduced a line element on the space of equilibrium states which the metric components are the Hessian of internal energy. In addition, Ruppeiner and Quevedo have introduced two metrics in [@Ruppeiner1; @Ruppeiner2] and [Quevedo1,Quevedo2]{}, respectively. Ruppeiner metric is conformally equivalent to Weinhold one whereas Quevedo metric enjoys the Legendre invariant and has been introduced to solve some problems in Weinhold and Ruppeiner metrics. Moreover, these three metrics were not free of shortcoming in the context of some black hole solutions [@HPEM]. Therefore, the fourth metric was introduced [@HPEM]$$ds^{2}=\frac{S\partial _{S}M}{\prod_{i=2}\left( \partial _{\chi _{i}}^{2}M\right) ^{3}}\left( -\left( \partial _{S}^{2}M\right) dS^{2}+\sum_{i=2}\left( \partial _{\chi _{i}}^{2}M\right) d\chi _{i}^{2}\right) , \label{Met}$$where $\chi _{i}$’s are the residual extensive parameters with $\chi _{i}\neq S$. The geometrical thermodynamics of different types of black holes has been investigated in the literature [Chabab,IJMPD2,ZhangNPB,Zhang,Jafarzade,Vetsov]{} by using (\[Met\]). Due to the complex values of $\mathcal{P}$ and $\mathcal{V}$, we omit the term $% \mathcal{V}d\mathcal{P}$ from the first law (\[MFL\]) and investigate the geometrical thermodynamics of the obtained solutions. For our black hole case study, the metric (\[Met\]) reduces into$$ds^{2}=\frac{S\partial _{S}M}{\left( \partial _{\tilde{\mu}}^{2}M\right) ^{3}% }\left[ -\left( \partial _{S}^{2}M\right) dS^{2}+\left( \partial _{\tilde{\mu% }}^{2}M\right) d\tilde{\mu}^{2}\right] . \label{FM}$$ Due to the cumbersome terms of the Ricci scalar (\[FM\]), we do not show the explicit form of it for simplicity but the resulting diagrams are plotted in Fig. \[gtd\] related to the information of Figs. \[CadSk1\]-\[CkM1\]. From Fig. \[gtd\], interestingly, we can see that the singularities of the Ricci scalar totally coincide with all the points that the heat capacity changes sign, and more importantly, without introducing extra singular points. $% \begin{array}{ccc} \epsfxsize=5cm \epsffile{RadSk1C1.eps} & \epsfxsize=5cm % \epsffile{RadSk1C2.eps} & \epsfxsize=5cm \epsffile{RadSk1C3.eps} \\ \epsfxsize=5cm \epsffile{RdSk1C1.eps} & \epsfxsize=5cm \epsffile{RadSk0.eps} & \epsfxsize=5cm \epsffile{RdSk0.eps} \\ \epsfxsize=5cm \epsffile{RadSkM1C1.eps} & \epsfxsize=5cm % \epsffile{RadSkM1C2.eps} & \epsfxsize=5cm \epsffile{RdSkM1.eps}% \end{array} $ quasinormal modes \[QNM\] ========================= Setup ----- Here, we consider a massive scalar perturbation in the background of the black hole spacetime and obtain the QNFs by using two independent methods of calculations; the sixth order WKB approximation [Schutz,IyerWill,Konoplya6th]{} and the asymptotic iteration method (AIM) [@AIM]. In addition, we concentrate our attention on the asymptotically dS black holes ($\Lambda >0$) with spherical topology ($k=1$) of the obtained metric function (\[NMF\]). The asymptotic flat solutions ($k=1$, $% \Lambda =0$, and $\tilde{\mu}=0$) reduce to $D$-dimensional Schwarzschild black hole and one can use Horowitz-Hubeny method [@Horowitz] to obtain the QNMs of asymptotically adS black holes ($\Lambda <0$). A discussion on the adS black holes will appear elsewhere. The equation of motion for a minimally coupled massive scalar field is given by$$\square \Phi -\nu ^{2}\Phi =0, \label{WEQ}$$so that $\nu $ is the mass of the scalar field $\Phi $. If we consider modes$$\Phi \left( t,r,angles\right) =r^{\left( 2-D\right) /2}\Psi \left( r\right) Y\left( angles\right) e^{-i\omega t},$$where $Y\left( angles\right) $ denotes the spherical harmonics on $\left( D-2\right) $-sphere, the equation of motion (\[WEQ\]) reduces to the wavelike equation for the radial part $\Psi \left( r\right) $ in the following way$$\left[ \partial _{x}^{2}+\omega ^{2}-V_{l}\left( x\right) \right] \Psi _{l}\left( x\right) =0. \label{Weq}$$ In this equation, $x$ is the known tortoise coordinate with the definition$$dx=\frac{dr}{f(r)}, \label{tortoise}$$and the effective potential $V_{l}\left( x\right) $ is given by$$V_{l}\left( x\right) =f\left( r\right) \left[ \nu ^{2}+\frac{l\left( l+D-3\right) }{r^{2}}+\frac{\left( D-2\right) \left( D-4\right) }{4r^{2}}% f\left( r\right) +\frac{D-2}{2r}f^{\prime }\left( r\right) \right] , \label{EP}$$where $l$ is the angular quantum number and note that $r$ in the right-hand side is a function of $x$ by (\[tortoise\]). Figure \[Pot\] shows the behavior of this effective potential (\[EP\]) versus the tortoise coordinate for some fixed values of different free parameters. The spectrum of QNMs for a perturbed black hole spacetime is the solution of the wave equation (\[Weq\]). However, we have to impose some proper boundary conditions in order to obtain its solutions. The quasinormal boundary conditions imply that the wave at the event (cosmological) horizon is purely incoming (outgoing)$$\begin{array}{c} \Psi _{l}\left( r\right) \sim e^{-i\omega x}\ \ \ \ \ \ as\ \ \ \ \ \ x\rightarrow -\infty \ (r\rightarrow r_{e}), \\ \Psi _{l}\left( r\right) \sim e^{i\omega x}\ \ \ \ \ \ \ as\ \ \ \ \ \ \ \ \ x\rightarrow \infty \ (r\rightarrow r_{c}),% \end{array} \label{bc}$$where $r_{e}$ is the event horizon and $r_{c}$ is the cosmological horizon. One should consider the mentioned boundary conditions in order to obtain the QNFs. $% \begin{array}{cccc} \epsfxsize=5cm \epsffile{VD.eps} & \epsfxsize=5cm \epsffile{VD4.eps} & % \epsfxsize=5cm \epsffile{VD5.eps} & \end{array} $ WKB approximation ----------------- The method is based on the matching of WKB expansion of the wave function $% \Psi _{l}\left( x\right) $ at the event horizon and cosmological horizon with the Taylor expansion near the peak of the potential barrier through the two turning points. Therefore, this method can be used for an effective potential that forms a potential barrier and takes constant values at the event horizon ($x\rightarrow -\infty $) and cosmological horizon ($% x\rightarrow \infty $) (like Fig. \[Pot\]). The WKB approximation was first applied to the problem of scattering around black holes [@Schutz], and then extended to the third order [@IyerWill] and sixth order [Konoplya6th]{}. The WKB formula is given by$$\frac{i\left( \omega ^{2}-V_{0}\right) }{\sqrt{-2V_{0}^{\prime \prime }}}% -\sum_{j=2}^{6}\Lambda _{j}=n+\frac{1}{2};\ \ \ \ \ \ n=0,1,2,...,$$where $V_{0}$ is the value of the effective potential at its local maximum, the correction terms $\Lambda _{j}$’s correspond to the $j$th order and depend on the value of the effective potential and its derivatives at the local maximum, and $n$ is the overtone number. The explicit form of the WKB corrections is given in [@IyerWill] (for $\Lambda _{2}$ and $\Lambda _{3}$) and [@Konoplya6th] (for $\Lambda _{4}$, $\Lambda _{5}$, and $% \Lambda _{6}$). It is worthwhile to mention that the WKB approximation does not give reliable frequencies for $n\geq l$. We use this formula up to the sixth order as a semi-analytical approach to obtain the QNFs of perturbation. AIM --- The AIM has been employed to solve the eigenvalue problems and solving second-order differential equations [@Ciftci; @CiftciHall], and then it was shown that it is an accurate technique for calculating QNMs [AIM,Naylor]{}. If one wants to employ the AIM, it is convenient to use the independent variable $\xi =1/r$, and rewrite the wave equation (\[Weq\]) as$$\frac{d^{2}\Psi _{l}\left( \xi \right) }{d\xi ^{2}}+\frac{P^{\prime }}{P}% \frac{d\Psi _{l}\left( \xi \right) }{d\xi }+\left( \frac{\omega ^{2}}{P^{2}}-% \frac{V_{l}\left( \xi \right) }{P}\right) \Psi _{l}\left( \xi \right) =0, \label{Wq2}$$where $P$, $P^{\prime }$, and $V_{l}\left( \xi \right) $ are given by$$\begin{aligned} P &=&\xi ^{2}f\left( \xi \right) ;\ \ \ \ \ \ f\left( \xi \right) =\left. f\left( r\right) \right\vert _{r=1/\xi }, \\ P^{\prime } &=&\frac{dP}{d\xi }=2\xi -\left( D-1\right) m\xi ^{D-2}-\tilde{% \mu}, \\ V_{l}\left( \xi \right) &=&\left[ \frac{\nu ^{2}}{\xi ^{2}}+l\left( l+D-3\right) +\frac{\left( D-2\right) \left( D-4\right) }{4}f\left( \xi \right) +\frac{D-2}{2\xi }f^{\prime }\left( \xi \right) \right] , \\ f^{\prime }\left( \xi \right) &=&\left. \frac{df\left( r\right) }{dr}% \right\vert _{r=1/\xi }=\left( D-3\right) m\xi ^{D-2}-\tilde{\mu}-\frac{% 2\Lambda }{3\xi }.\end{aligned}$$ In order to choose the appropriate scaling behavior for quasinormal boundary conditions, one may define [@Moss; @Naylor]$$e^{i\omega x}=\prod_{j}\left( \xi -\xi _{j}\right) ^{-i\omega /\kappa _{j}}, \label{scaling}$$in which $\kappa _{j}$ is the surface gravity at $\xi _{j}$ with $f\left( \xi =\xi _{j}\right) =0$. This equation scale out the divergent behavior at some boundary $\xi _{j}$ and applies the boundary conditions (\[bc\]) to the solution. Now, by redefinition of $\Psi _{l}\left( \xi \right) $ as$$\Psi _{l}\left( \xi \right) =e^{i\omega x}\psi _{l}\left( \xi \right) , \label{redifine}$$the equation (\[Wq2\]) converts to $$P\frac{d^{2}\psi _{l}\left( \xi \right) }{d\xi ^{2}}+\left( P^{\prime }-2i\omega \right) \frac{d\psi _{l}\left( \xi \right) }{d\xi }-V_{l}\left( \xi \right) \psi _{l}\left( \xi \right) =0. \label{Weq3}$$ Based on the equations (\[scaling\]) and (\[redifine\]), the correct quasinormal condition at the event horizon, $\xi _{e}$, is$$\psi _{l}\left( \xi \right) =\left( \xi -\xi _{e}\right) ^{-i\omega /\kappa _{e}}\mathcal{U}_{l}\left( \xi \right) , \label{SC}$$where$$\kappa _{e}=\left. \frac{1}{2}\frac{df\left( r\right) }{dr}\right\vert _{r=r_{e}}=\frac{1}{2}f^{\prime }\left( \xi _{e}\right) .$$ By inserting (\[SC\]) into (\[Weq3\]), one can find the standard AIM form as follows$$\frac{d^{2}\mathcal{U}_{l}\left( \xi \right) }{d\xi ^{2}}=\lambda _{0}\left( \xi \right) \frac{d\mathcal{U}_{l}\left( \xi \right) }{d\xi }+s_{0}\left( \xi \right) \mathcal{U}_{l}\left( \xi \right) , \label{Weq4}$$so that $\lambda _{0}\left( \xi \right) $ and $s_{0}\left( \xi \right) $are$$\begin{aligned} \lambda _{0}\left( \xi \right) &=&-\frac{1}{P}\left( P^{\prime }-2i\omega -% \frac{2i\omega P}{\kappa _{e}\left( \xi -\xi _{e}\right) }\right) , \label{l0} \\ s_{0}\left( \xi \right) &=&\frac{1}{P}\left[ \frac{i\omega \left( P^{\prime }-2i\omega \right) }{\kappa _{e}\left( \xi -\xi _{e}\right) }-\frac{i\omega P% }{\kappa _{e}\left( \xi -\xi _{e}\right) ^{2}}\left( \frac{i\omega }{\kappa _{e}}+1\right) +V_{l}\left( \xi \right) \right] . \label{s0}\end{aligned}$$ We will use Eqs. (\[Weq4\])-(\[s0\]) to calculate the QNFs as a numerical method for obtained black hole solutions in the coming section (see [@AIM; @Naylor] for details of calculations). Time-domain profiles -------------------- Using the time-domain integration of the wavelike equation (\[Weq\]), one can study the contribution of all the modes for a fixed value of the angular quantum number in a single ringing profile. The time-domain profile of modes shows the behavior of the asymptotic tails after ringdown stage at late times. In order to obtain the time evolution of modes, we follow the discretization scheme given in [@Pullin] (see also [@KonoplyaR] and [@Chakraborty]). In terms of the light-cone coordinates $u=t-x$ and $% v=t+x$, the perturbation equation (\[Weq\]) takes the following form$$\left[ 4\partial _{u}\partial _{v}+V_{l}\left( u,v\right) \right] \Psi _{l}\left( u,v\right) =0.$$ By integrating the mentioned equation on the small grids on the two null surfaces $u=u_{0}$ and $v=v_{0}$ as the initial data, one can obtain the time-domain profile of modes. By applying the time evolution operator on $% \Psi _{l}\left( u,v\right) $ and expanding this operator for sufficiently small grids, one can obtain the evolution equation in the light-cone coordinates as below $$\begin{aligned} \Psi _{l}\left( u+\Delta ,v+\Delta \right) &=&\Psi _{l}\left( u+\Delta ,v\right) +\Psi _{l}\left( u,v+\Delta \right) -\Psi _{l}\left( u,v\right) \notag \\ &&-\frac{\Delta ^{2}}{8}\left[ V_{l}\left( u+\Delta ,v\right) \Psi _{l}\left( u+\Delta ,v\right) +V_{l}\left( u,v+\Delta \right) \Psi _{l}\left( u,v+\Delta \right) \right] ,\end{aligned}$$which $\Delta $ is the step size of the grids. We shall obtain the time evolution of perturbations with a Gaussian wave packet as initial data on the surfaces $u=u_{0}$ and $v=v_{0}$. Results and discussion ---------------------- The QNMs are calculated by using the sixth order WKB approximation and AIM after $15$ iterations, and results are presented in tables $I-IV$. The tables contain the fundamental QNM ($n=0$) for different values of spacetime dimension and multipole number, and fixed $m=0.5$, $\tilde{\mu}=0.3$, $% \Lambda =0.1$, and $\nu =1$. From table $I$, we find that although the WKB approximation does not give reliable frequencies for $n\geq l$, for higher dimensions, say $D\geq 6$, the WKB formula gives better results for $n=0=l$. However, for a fixed multipole number so that $l\geq 1$, as the dimension increases, the result of WKB formula gets worse. In addition, most of the results of WKB approximation are in a good agreement with numeric results (tables $II-IV$), and results get better for the higher multipole number (for example, compare the row with $D=7$ of tables $II-IV$), as we expected. On the other hand, both the real and imaginary parts of the QNFs increase with increasing in dimension which shows that there are more oscillations for higher dimensions at ringdown stage and the modes live longer for lower dimensions (see the left panel of Fig. \[TDP\]). On the other hand, as one can see from Fig. \[Pot\], the effective potential forms a potential barrier which is positive everywhere and vanishes at the event horizon and spatial infinity. This leads to the following fact$$\int_{-\infty }^{+\infty }V_{l}\left( x\right) dx>0,$$which shows that we can find dynamically stable black holes under massive scalar perturbations from the obtained solutions [@Simon]. The right panel of Fig. \[TDP\] indicates the late time behavior of modes. The power law decay of modes at late times confirms the fact that the black holes enjoy the dynamical stability. $% \begin{array}{ccc} \epsfxsize=7.5cm \epsffile{tab.eps} & \epsfxsize=7.5cm \epsffile{TDP.eps} & \end{array} $ In addition, it is worthwhile to mention that although the calculated QNMs in tables $I-IV$ are related to the dynamically stable black holes under massive scalar perturbations, these black holes are thermally stable just in four dimensions. Therefore, the obtained black hole solutions in $D>4$ are stable dynamically, but not thermally. -------------------------------------------------------------------------------------------------------------------- $D$ AIM ($\omega _{r}-i\omega _{i}$) WKB ($\omega _{r}-i\omega _{i}$) $r_{e}$ ------ -- ---------------------------------- ------------------------------------------------- --------------------- $4$ $0.41487-0.12777i$ $0.18614-1.4372i\ \left( $0.62556<r_{+,rp4}$ 55.13\%,1024.83\%\right) $ $5$ $0.52903-0.20231i$ $0.67709-0.10511i\ \left( $0.82840<r_{+,rp2}$ 27.99\%,48.05\%\right) $ $6$ $0.66789-0.32804i$ $0.67545-0.31549i\ \left( 1.13\%,3.83\%\right) $0.89131<r_{+,rp2}$ $ $7$ $0.86292-0.46763i$ $0.87357-0.46878i\ \left( 1.23\%,0.25\%\right) $0.92081<r_{+,rp2}$ $ $8$ $1.0886-0.58598i$ $1.0992-0.59847i\ \left( 0.97\%,2.13\%\right) $ $0.93779<r_{+,rp2}$ $9$ $1.3291-0.69027i$ $1.3383-0.71872i\ \left( 0.69\%,4.12\%\right) $ $0.94880<r_{+,rp2}$ $10$ $1.5801-0.78466i$ $1.5858-0.83505i\ \left( 0.36\%,6.42\%\right) $ $0.95651<r_{+,rp2}$ -------------------------------------------------------------------------------------------------------------------- Table $I$: The fundamental QNM for $l=0$. $r_{e}$ shows the value of the event horizon radius for each dimension. For $D=4$, the event horizon radius is smaller than the only root of the heat capacity $r_{+,rp4}$, and therefore, this black hole is thermally stable (see the left panel of Fig. \[CdSk1\]). However, for other dimensions, the event horizon radius is located before the smaller root of the heat capacity $r_{+,rp2}$, and therefore, these black holes are thermally unstable (see the right panel of Fig. \[CdSk1\]). ---------------------------------------------------------------------------------------------- $D$ AIM ($\omega _{r}-i\omega _{i}$) WKB ($\omega _{r}-i\omega _{i}$) ------ -- ---------------------------------- ------------------------------------------------- $4$ $0.71667-0.16408i$ $0.71680-0.16407i\ \left( 0.02\%,<0.01\%\right) $ $5$ $0.89554-0.24159i$ $0.89528-0.24209i\ \left( 0.03\%,0.21\%\right) $ $6$ $1.1454-0.34506i$ $1.1442-0.34700i\ \left( 0.10\%,0.56\%\right) $ $7$ $1.4091-0.44882i$ $1.4055-0.45391i\ \left( 0.26\%,1.13\%\right) $ $8$ $1.6807-0.54759i$ $1.6720-0.55880i\ \left( 0.52\%,2.05\%\right) $ $9$ $1.9579-0.64053i$ $1.9400-0.66216i\ \left( 0.91\%,3.38\%\right) $ $10$ $2.2397-0.72789i$ $2.2063-0.76562i\ \left( 1.49\%,5.18\%\right) $ ---------------------------------------------------------------------------------------------- Table $II$: The fundamental QNM for $l=1$. ---------------------------------------------------------------------------------------------- $D$ AIM ($\omega _{r}-i\omega _{i}$) WKB ($\omega _{r}-i\omega _{i}$) ------ -- ---------------------------------- ------------------------------------------------- $4$ $1.1161-0.17673i$ $1.1161-0.17673i\ \left( <0.01\%,<0.01\%\right) $ $5$ $1.3025-0.25160i$ $1.3025-0.25162i\ \left( <0.01\%,<0.01\%\right) $ $6$ $1.6110-0.34815i$ $1.6111-0.34813i\ \left( <0.01\%,<0.01\%\right) $ $7$ $1.9262-0.44387i$ $1.9265-0.44361i\ \left( 0.02\%,0.06\%\right) $ $8$ $2.2400-0.53549i$ $2.2411-0.53458i\ \left( 0.05\%,0.17\%\right) $ $9$ $2.5524-0.62251i$ $2.5550-0.62037i\ \left( 0.10\%,0.34\%\right) $ $10$ $2.8637-0.70509i$ $2.8690-0.70105i\ \left( 0.19\%,0.57\%\right) $ ---------------------------------------------------------------------------------------------- Table $III$: The fundamental QNM for $l=2$. ----------------------------------------------------------------------------------------------- $D$ AIM ($\omega _{r}-i\omega _{i}$) WKB ($\omega _{r}-i\omega _{i}$) ------ -- ---------------------------------- -------------------------------------------------- $4$ $1.5323-0.18053i$ $1.5323-0.18053i\ \left( <0.01\%,<0.01\%\right) $ $5$ $1.7173-0.25513i$ $1.7173-0.25513i\ \left( <0.01\%,<0.01\%\right) $ $6$ $2.0745-0.34934i$ $2.0745-0.34932i\ \left( <0.01\%,<0.01\%\right) $ $7$ $2.4341-0.44183i$ $2.4342-0.44169i\ \left( <0.01\%,0.03\%\right) $ $8$ $2.7857-0.53009i$ $2.7860-0.52960i\ \left( 0.01\%,0.09\%\right) $ $9$ $3.1299-0.61393i$ $3.1309-0.61267i\ \left( 0.03\%,0.21\%\right) $ $10$ $3.4686-0.69364i$ $3.4710-0.69094i\ \left( 0.07\%,0.39\%\right) $ ----------------------------------------------------------------------------------------------- Table $IV$: The fundamental QNM for $l=3$. $% \begin{array}{ccc} \epsfxsize=7.5cm \epsffile{Rnu.eps} & \epsfxsize=7.5cm \epsffile{Inu.eps} & \end{array} $ On the other hand, one of the motivations for considering the test massive fields comes from the fact that there are some QNMs with arbitrarily long life (purely real) modes called quasi-resonance modes [@Ohashi]. For the quasi-resonance modes, the oscillations do not decay and the situation is similar to the standing waves on a string which is fixed at its both ends. The quasi-resonance modes occur for special values of field mass and the QNMs disappear when the field mass takes higher values. However, this happens just for lower overtones whenever the effective potential is non-zero at least at one of the boundaries (the event horizon $x\rightarrow -\infty $ or cosmological horizon $x\rightarrow \infty $). Now, let us investigate the possibility of the quasi-resonance modes presence for obtained black hole solutions. The effective potential (\[EP\]) vanishes at both infinities for all possible values of different parameters, and therefore, there is no quasi-resonance modes for (\[Weq\]). In addition, if one sets the integration constant $-\Lambda /3$ equals to zero, the effective potential still vanishes at both infinities. There is only one case so that the effective potential can be non-zero at spatial infinity and that is neutral black holes ($\tilde{\mu}=0$) with zero integration constant ($-\Lambda /3=0$). In this case, the effective potential reduces to the Schwarzschild one which its quasi-resonance modes have been investigated in [@SchwMSF]. Therefore, our black hole case study has no quasi-resonant oscillations in general and the imaginary part of the frequencies never vanishes. Figure \[nuFig\] shows the behavior of QNFs with increasing in $\nu $ and confirms the above discussion. As $\nu $increases, the real part of frequencies increases too, whereas the imaginary part first decreases rapidly and then takes a constant value. Conclusions \[Conclusions\] =========================== Motivated by the importance of higher dimensional spacetime in high energy physics, we have constructed the conformal-$U(1)$ gauge/gravity black hole solutions in $D\geq 4$. Since higher-dimensional solutions in CG cannot be produced in the presence of Maxwell field (and also the other electrodynamic fields that are not conformal invariant in higher dimensions), we have used a class of nonlinear electrodynamics $(-F_{\mu \nu }F^{\mu \nu })^{s}$ (which enjoys the conformal invariance properties in higher dimensions as $% s=D/4$) to obtain black hole solutions. In addition, we have seen that the obtained solutions enjoy an essential singularity at the origin and they can be considered as black holes. We have calculated the conserved charges of the obtained black hole solutions and studied the thermal stability of these black holes in the canonical ensemble. We have calculated the root and divergence points of the heat capacity and obtained some regions where the black holes are stable and physical. It was shown that the large black holes are always physical and stable in adS spacetime whereas these black holes are unconditionally unstable and non-physical in arbitrary dimensional dS spacetime. We have also investigated the thermal stability with geometrical thermodynamics approach and we have seen that the singularities of the Ricci scalar totally coincide with all points that the heat capacity changes sign without introducing extra singular points. Then, we have shown that the obtained black hole solutions cannot undergo the van der Waals like phase transition because of the absence of local instability. Furthermore, we have considered a minimally coupled massive scalar perturbation in the background spacetime of our the black hole case study and calculated the QNFs by using the sixth order WKB approximation and the AIM after $15$ iterations. We also investigated the time evolution of modes in some diagrams. We have shown that although the WKB approximation does not give reliable frequencies for $n\geq l$, this approximation gives better results in higher dimensions for $n=0=l$. It was shown that most of the results of WKB approximation are in a good agreement with AIM and results get better for the higher multipole number. Besides, we observed that there were more oscillations for higher dimensions and the modes live longer for lower dimensions. We also showed that the four-dimensional black holes are stable both thermally and dynamically. However, the higher dimensional black holes were thermally unstable but they enjoy the dynamical stability. We should note that the previous results have been obtained for some special values of free parameters and it may be possible to find stable black holes in higher dimensions by choosing some other values for the free parameters. In addition, we argued that the black holes have no quasi-resonant oscillations even when one sets the integration constant $-\Lambda /3$equals to zero. This happens due to the presence of the linear $r$-term. Therefore, the imaginary part of the frequencies never vanishes and there are always damping modes (QNFs are always complex). Here, we finish our paper with some suggestions. One can consider (minimally or non-minimally coupled) the other kinds of perturbations such as charged scalar perturbation, electromagnetic perturbation, Proca field, and etc on the background spacetime of these black holes and calculate the QNFs and investigate the dynamical stability. Investigating the near extremal regime of these black holes in dS spacetime is an interesting work which is under examination. We wish to thank Shiraz University Research Council. MM wishes to thank A. Zhidenko and R. A. Konoplya for their helps on QNMs. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran. [99]{} R. J. Riegert, Phys. Rev. Lett. 53 (1984) 315. H. Lu and C. N. Pope, Phys. Rev. Lett. 106 (2011) 181302. E. Bergshoeff, M. de Roo and B. de Wit, Nucl. Phys. B 182 (1981) 173. B. de Wit, J. W. van Holten and A. Van Proeyen, Nucl. Phys. B 184 (1981) 77. S. L. Adler, Rev. Mod. Phys. 54 (1982) 729. G. ’t Hooft, Found. Phys. 41 (2011) 1829. P. D. Mannheim, Found. Phys. 42 (2012) 388. N. Berkovits and E. Witten, JHEP 08 (2004) 009. H. Liu and A. A. Tseytlin, Nucl. Phys. B 533 (1998) 88. V. Balasubramanian, E. G. Gimon, D. Minic and J. Rahmfeld, Phys. Rev. D 63 (2001) 104009. K. S. Stelle, Phys. Rev. D 16 (1977) 953. B. Hasslacher and E. Mottola, Phys. Lett. B 99 (1981) 221. R. Bach, Math. Z. 9 (1921) 110. A. Buchdahl, Edinburgh Math. Soc. Proc. 10 (1953) 16. J. Maldacena, \[arXiv:1105.5632\]. G. Anastasiou and R. Olea, Phys. Rev. D 94 (2016) 086008. S. Deser, H. Liu, H. Lu, C .N. Pope, T. C. Sisman and B. Tekin, Phys. Rev. D 83 (2001) 061502. C. Lanczos, Annals Math. 39 (1938) 842. H. Nollert, Phys. Rev. D 47 (1993) 5253. S. Hod, Phys. Rev. Lett. 81 (1998) 4293. G. T. Horowitz and V. E. Hubeny, Phys. Rev. D 62 (2000) 024027. V. Cardoso and J. P. S. Lemos, Phys. Rev. D 64 (2001) 084017. K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2 (1999) 2. E. Berti, V. Cardoso and A. O. Starinets, Class. Quant. Grav. 26 (2009) 163001. R. A. Konoplya and A. Zhidenko, Rev. Mod. Phys. 83 (2011) 793. B. P. Abbott et al. \[LIGO Scientific and Virgo Collaborations\], Phys. Rev. Lett. 116 (2016) 061102. B. P. Abbott et al. \[LIGO Scientific and Virgo Collaborations\], Phys. Rev. Lett. 116 (2016) 221101. B. P. Abbott et al. \[LIGO Scientific and Virgo Collaborations\], Phys. Rev. Lett. 116 (2016) 241103. R. A. Konoplya and A. Zhidenko, JCAP 12 (2016) 043. K. Lin, W. L. Qian and A. B. Pavan, Phys. Rev. D 94 (2016) 064050. G. B. Cook and M. Zalutskiy, Phys. Rev. D 94 (2016) 104074. N. Breton and L. A. Lopez, Phys. Rev. D 94 (2016) 104008. P. A. Gonzalez, R. A. Konoplya and Y. Vasquez, Phys. Rev. D 95 (2017) 124012. J. L. Blazquez-Salcedo, F. S. Khoo and J. Kunz, Phys. Rev. D 96 (2017) 064008. K. Lin, W. L. Qian, A. B. Pavan and E. Abdalla, Mod. Phys. Lett. A 32 (2017) 1750134. G. Khanna and R. H. Price, Phys. Rev. D 95 (2017) 081501. M. Wang, C. Herdeiro and J. Jing, Phys. Rev. D 96 (2017) 104035. R. A. Konoplya and A. Zhidenko, Phys. Rev. D 97 (2018) 084034. V. Cardoso, J. L. Costa, K. Destounis, P. Hintz and A. Jansen, Phys. Rev. Lett. 120 (2018) 031103. R. A. Konoplya, Z. Stuchlik and A. Zhidenko, Phys. Rev. D 98 (2018) 104033. T. Assumpcao, V. Cardoso, A. Ishibashi, M. Richartz and M. Zilhao, Phys. Rev. D 98 (2018) 064036. J. L. Blazquez-Salcedo, X. Y. Chew and J. Kunz, Phys. Rev. D 98 (2018) 044035. K. Destounis, G. Panotopoulos and A. Rincon, \[arXiv:1801.08955\]. G. Panotopoulos, \[arXiv:1807.03278\]. J. Li, H. S. Liu, H. Lu and Z. L. Wang, JHEP 02 (2013) 109. M. Hassaine and C. Martinez, Phys. Rev. D 75 (2007) 027502. R. M. Wald, Phys. Rev. D 48 (1993) 3427. V. Iyer and R. M. Wald, Phys. Rev. D 52 (1995) 4430. S. Hawking and D. Page, Commun. Math. Phys. 87 (1983) 577. D. Kubiznak and R. B. Mann, JHEP 07 (2012) 033. M. B. J. Poshteh, B. Mirza and Z. Sherkatghanad, Phys. Rev. D 88 (2013) 024005. E. Caceres, P. H. Nguyen and J. F. Pedraza, JHEP 09 (2015) 184. S. H. Hendi, S. Panahiyan, B. Eslam Panah, M. Faizal and M. Momennia, Phys. Rev. D 94 (2016) 024028. A. Mandal, S. Samanta and B. R. Majhi, Phys. Rev. D 94 (2016) 064069. S. H. Hendi, S. Panahiyan and M. Momennia, Int. J. Mod. Phys. D 25 (2016) 1650063. R. A. Hennigar, E. Tjoab and R. B. Mann, JHEP 02 (2017) 070. S. H. Hendi and M. Momennia, Eur. Phys. J. C 78 (2018) 800. S. H. Hendi and M. Momennia, Phys. Lett. B 777 (2018) 222. A. Ovgun, Adv. High Energy Phys. 2018 (2018) 8153721. Z. Sun and M. S. Ma, EPL 122 (2018) 60002. B. Liu, Z. Y. Yang and R. H. Yue, \[arXiv:1810.07885\]. M. Jamil, B. Pourhassan, A. Ovgun and I. Sakalli, \[arXiv:1811.02193\]. S. H. Hendi and M. Momennia, \[arXiv:1801.07906\]. F. Weinhold, J. Chem. Phys. 63 (1975) 2479. F. Weinhold, J. Chem. Phys. 63 (1975) 2484. G. Ruppeiner, Phys. Rev. A 20 (1979) 1608. G. Ruppeiner, Rev. Mod. Phys. 67 (1995) 605. H. Quevedo, J. Math. Phys. 48 (2007) 013506. H. Quevedo, A. Sanchez, JHEP 09 (2008) 034. S. H. Hendi, S. Panahiyan, B. Eslam Panah and M. Momennia, Eur. Phys. J. C 75 (2015) 507. M. Chabab, H. El Moumni and K. Masmar, Eur. Phys. J. C 76 (2016) 304. S. H. Hendi, S. Panahiyan, M. Momennia and B. Eslam Panah, Int. J. Mod. Phys. D 26 (2017) 1750026. M. Zhang, Nucl. Phys. B 935 (2018) 170. M. Zhang and W. B. Liu, \[arXiv:1610.03648\]. Kh. Jafarzade and J. Sadeghi, \[arXiv:1711.04522\]. T. Vetsov, \[arXiv:1806.05011\]. B. F. Schutz and C. M. Will, Astrophys. J. Lett. 291 (1985) L33. S. Iyer and C. M. Will, Phys. Rev. D 35 (1987) 3621. R. A. Konoplya, Phys. Rev. D 68 (2003) 024018. H. T. Cho, A. S. Cornell, J. Doukas, T. R. Huang and W. Naylor, Adv. Math. Phys. 2012 (2012) 281705. H. Ciftci, R. L. Hall and N. Saad, J. Phys. A 36 (2003) 11807. H. Ciftci, R. L. Hall and N. Saad, Phys. Lett. A 340 (2005) 388. H. T. Cho, A. S. Cornell, J. Doukas and W. Naylor, Class. Quant. Grav. 27 (2010) 155004. I. G. Moss and J. P. Norman, Class. Quant. Grav. 19 (2002) 2323. C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D 49 (1994) 883. S. Chakraborty, K. Chakravarti, S. Bose and S. SenGupta, Phys. Rev. D 97 (2018) 104053. B. Simon, Ann. Phys. 97 (1976) 279. A. Ohashi and M. a. Sakagami, Class. Quant. Grav. 21 (2004) 3973. R. A. Konoplya and A. Zhidenko, Phys. Lett. B 609 (2005) 377. [^1]: email address: [email protected] [^2]: email address: [email protected]

--- abstract: 'We present results obtained with the PRONAOS balloon-borne experiment on interstellar dust. In particular, the submillimeter / millimeter spectral index is found to vary between roughly 1 and 2.5 on small scales (3.5$''$ resolution). This could have implications for component separation in Cosmic Microwave Background maps.' author: - 'X. Dupac (ESA-ESTEC, Noordwijk, the Netherlands, [email protected]), J.-P. Bernard, N. Boudet, M. Giard (CESR Toulouse), J.-M. Lamarre (LERMA Paris), C. Mény (CESR), F. Pajot (IAS Orsay), I. Ristorcelli (CESR)' --- å[A&A]{} dust, extinction — infrared: ISM — submillimeter — cosmology: Cosmic Microwave Background Introduction ============ To accurately characterize dust emissivity properties represents a major challenge of nowadays astronomy. It is crucial for deriving very accurate maps of the Cosmic Microwave Background fluctuations, as well as for understanding the physics of the interstellar medium. In the submillimeter domain, large grains at thermal equilibrium (e.g. [@desert90]) dominate the dust emission. This thermal dust is characterized by a temperature and a spectral dependence of the emissivity which is usually simply modelled by a spectral index. The temperature, density and opacity of a molecular cloud are key parameters which control the structure and evolution of the clumps, and therefore, star formation. The spectral index ($\beta$) of a given dust grain population is directly linked to the internal physical mechanisms and the chemical nature of the grains. It is generally admitted from Kramers-König relations that 1 is a lower limit for the spectral index. $\beta$ = 2 is particularly invoked for isotropic crystalline grains, amorphous silicates or graphitic grains. However, it is not the case for amorphous carbon, which is thought to have a spectral index equal to 1. Spectral indices above 2 may exist, according to several laboratory measurements on grain analogs. Observations of the diffuse interstellar medium at large scales favour $\beta$ around 2 (e.g. [@boulanger96], [@dunne01]). In the case of molecular clouds, spectral indices are usually found to be between 1.5 and 2. However, low values (0.2-1.4) of the spectral index have been observed in circumstellar environments, as well as in molecular cloud cores. PRONAOS observations\[obs\] =========================== PRONAOS (PROgramme NAtional d’Observations Submillimétriques) is a French balloon-borne submillimeter experiment ([@ristorcelli98]). Its effective wavelengths are 200, 260, 360 and 580 , and the angular resolutions are 2$'$ in bands 1 and 2, 2.5$'$ in band 3 and 3.5$'$ in band 4. The data analyzed here were obtained during the second flight of PRONAOS in September 1996, at Fort Sumner, New Mexico. The data processing method, including deconvolution from chopped data, is described in Dupac (2001). This experiment has observed various phases of the interstellar medium, from diffuse clouds in Polaris ([@bernard99]) and Taurus ([@stepnik03]) to massive star-forming regions in Orion ([@ristorcelli98], [@dupac01]), Messier 17 ([@dupac02]), Cygnus B, and the dusty envelope surrounding the young massive star GH2O 092.67+03.07 in NCS. The $\rho$ Ophiuchi low-mass star-forming region has also been observed, as well as the edge-on spiral galaxy NGC 891 ([@dupac03b]). Analysis of the Galactic dust emission ====================================== We fit a modified black body law to the spectra: $\inu = \epsilon_0 \; \bnu(\lambda,T) \; (\lambda/\lambda_0)^{-\beta}$, where $\inu$ is the spectral intensity (MJy/sr), $\epsilon_0$ is the emissivity at $\lambda_0$ of the observed dust column density, $\bnu$ is the Planck function, $T$ is the temperature and $\beta$ is the spectral index. In most of the areas, we use either only PRONAOS data or PRONAOS + IRAS 100  data. We restrain the analysis to all fully independent (3.5$'$ side) pixels for which both relative errors on the temperature and the spectral index are less than 20%. This procedure allows to reduce the degeneracy effect between the temperature and the spectral index. Dupac (2001) and Dupac (2002) have shown by fitting simulated data that this artificial anticorrelation effect was small compared to the effect observed in the data. ![Spectral index versus temperature, for fully independent pixels in Orion (black asterisks), M17 (diamonds), Cygnus (triangles), $\rho$ Ophiuchi (grey asterisks), Polaris (black squares), Taurus (grey square), NCS (grey cross) and NGC 891 (black crosses). The full line is the result of the best hyperbolic fit: $\beta = {1 \over 0.4 + 0.008 T}$ []{data-label="compil"}](dupac_f1.eps) We present in Fig. \[compil\] the spectral index - temperature relation observed. The temperature in this data set ranges from 11 to 80 K, and the spectral index also exhibits large variations from 0.8 to 2.4. One can observe an anticorrelation on these plots between the temperature and the spectral index, in the sense that the cold regions have high spectral indices around 2, and warmer regions have spectral indices below 1.5. In particular, no data points with $T >$ 35 K and $\beta >$ 1.6 can be found, nor points with $T <$ 20 K and $\beta <$ 1.5. This anticorrelation effect is present for all objects in which we observe a large range of temperatures, namely Orion, M17, Cygnus and $\rho$ Ophiuchi. It is also remarkable that the few points from other regions are well compatible with this general anticorrelation trend. The temperature dependence of the emissivity spectral index is well fitted by a hyperbolic approximating function. Several interpretations are possible for this effect: one is that the grain sizes change in dense environments, another is that the chemical composition of the grains is not the same in different environments and that this correlates to the temperature, a third one is that there is an intrinsic dependence of the spectral index on the temperature, due to quantum processes such as two-level tunneling effects. Additional modeling, as well as additional laboratory measurements and astrophysical observations, are required in order to discriminate between these different interpretations. More details about this analysis and the possible interpretations can be found in Dupac et al. (2003a). Implications for high-redshift far-infrared observations ======================================================== A recent paper from Eales et al. (2003) showed that the 850 / 1200  ratio of their sample of extragalactic millimeter sources was very low, which could be explained by spectral indices of the dust around 1 in high-redshift galaxies. Though this result is still uncertain, it might confirm our measurements because high-redshift galaxies are likely to be warmer than low-redshift galaxies (because of cosmic expansion). The anticorrelation between the temperature and the emissivity spectral index can indeed have major implications for deriving the redshifts, masses, temperatures, luminosities, etc, of extragalactic objects. Implications for Cosmic Microwave Background measurements ========================================================= Even high Galactic latitude clouds can have a non-uniform spectral index (e.g. [@bernard99]). Since the Galactic dust is the major contributor to CMB foregrounds in the submillimeter and millimeter domains, even small variations of the dust spectral energy distribution can harm the CMB measurements if the component-separation methods assume a uniform dust spectral index. [1]{} Bernard, J.-P., Abergel, A., Ristorcelli, I., et al.: 1999, å, 347, 640 Boulanger, F., Abergel, A., Bernard, J.-P., et al.: 1996, å, 312, 256 Désert, F.-X., Boulanger, F., Puget, J.-L.: 1990, å, 237, 215 Dunne, L., Eales, S.A.: 2001, , 327, 697 Dupac, X., Giard, M., Bernard, J.-P., et al.: 2001, , 553, 604 Dupac, X., Giard, M., Bernard, J.-P., et al.: 2002, å, 392, 691 Dupac, X., Bernard, J.-P., Boudet, N., et al.: 2003a, , 404, L11 Dupac, X., del Burgo, C., Bernard, J.-P., et al.: 2003b, , in press, astro-ph/0305230 Eales, S., Bertoldi, F., Ivison, R., Carilli, C., Dunne, L., Owen, F.: 2003, MNRAS, in press, astro-ph/0305218 , I., [Serra]{}, G., [Lamarre]{}, J.-M., et al.: 1998, , 496, 267 Stepnik, B., Abergel, A., Bernard, J.-P., et al.: 2003, å, 398, 551

--- abstract: | We generalize to $FC^*$, the class of generalized $FC$-groups introduced in \[F. de Giovanni, A. Russo, G. Vincenzi, *Groups with restricted conjugacy classes*, Serdica Math. J. [**28**]{} (2002), 241–254\], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an $FC^*$-group $G$ coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of $G$ are subgroups and the former coincides with the hypercentre. We also give an example of an $FC^*$-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.\ [**2010 Mathematics Subject Classification:**]{} 20F45; 20F24\ [**Keywords:**]{} Engel elements, generalized $FC$-groups author: - | \ \ \ title: '**THE ENGEL ELEMENTS IN GENERALIZED $FC$-GROUPS**' --- Introduction ============ Let $n$ be a positive integer and $x,y$ be elements of a group $G$. The commutator $[x,_n y]$ is defined inductively by the rules $$[x,_1 y]=x^{-1}x^y\quad {\rm and,\, for}\; n\geq 2,\quad [x,_n y]=[[x,_{n-1} y],y].$$ An element $a\in G$ is called a [*left Engel element*]{} if for any $g\in G$ there exists $n=n(a,g)\geq 1$ such that $[g,_n a]=1$. If $n$ can be chosen independently of $g$, then $a$ is called a [*left $n$-Engel element*]{}. Moreover, $a$ is a [*bounded left Engel element*]{} if it is left $n$-Engel for some $n\geq 1$. Similarly, an element $a\in G$ is called a [*right Engel element*]{} if the variable $g$ appears on the right, i.e. for any $g\in G$ there exists $n=n(a,g)\geq 1$ such that $[a,_n g]=1$; in addition, if $n=n(a)$, then $a$ is a [*right $n$-Engel element*]{} or simply a [*bounded right Engel element*]{}. By a well-known result of Heineken [@Ro2 Theorem 7.11], the inverse of any right Engel element is a left Engel element and the inverse of any right $n$-Engel element is a left $(n+1)$-Engel element. Following [@Ro2], we denote by $L(G)$ and $\overline{L}(G)$ the sets of left Engel elements and bounded left Engel elements of $G$, respectively; and by $R(G)$ and $\overline{R}(G)$ the sets of right Engel elements and bounded right Engel elements of $G$, respectively. Thus $$\label{RL} R(G)^{-1}\subseteq L(G)\quad {\rm and}\quad \overline{R}(G)^{-1}\subseteq \overline{L}(G).$$ It is also clear that these four subsets are invariant under automorphisms of $G$, but it is still unknown whether they are subgroups. This is a very long-standing problem, even if Bludov announced recently that there exists a group $G$ for which $L(G)$ is not a subgroup [@Bl]. We mention that $L(G)$ contains the Hirsch-Plotkin radical $HP(G)$ of $G$ and $\overline{L}(G)$ contains the Baer radical $B(G)$ of $G$; whereas, $R(G)$ contains the hypercentre $\overline{Z}(G)$ of $G$ and $\overline{R}(G)$ contains $Z_{\omega}(G)$, the $\omega$-hypercentre of $G$ [@Ro2 Lemma 7.12]. Recall that $HP(G)$ is the unique maximal normal locally nilpotent subgroup containing all normal locally nilpotent subgroups of $G$ [@Ro2 Part 1, p. 58]; and $B(G)$ is the subgroup generated by all elements $x\in G$ such that $\l x\r$ is subnormal in $G$. Notice also that, by a famous example of Golod [@Go], $L(G)$ can be larger than $HP(G)$. However, if $G$ is a soluble group, then $L(G)=HP(G)$ and $\overline{L}(G)=B(G)$ [@Ro2 Theorem 7.35]. This latter result is due to Gruenberg, who also proved that in this case $R(G)$ and $\overline{R}(G)$ are always subgroups and that there exists a soluble group $G$ such that $Z_{\omega}(G)\subset \overline{R}(G), \overline{Z}(G)\subset R(G)$ and $\overline{R}(G)\subset R(G)$ [@Gr]. On the other hand, a remarkable theorem of Baer shows that groups satisfying the maximal condition have a fine Engel structure: \[Baer\] Let $G$ be a group which satisfies the maximal condition. Then $L(G)$ and $\overline{L}(G)$ coincide with the Fitting subgroup of $G$, and $R(G)$ and $\overline{R}(G)$ coincide with the hypercentre of $G$, which equals $Z_k(G)$ for some finite $k$. There are a series of wide generalizations of Theorem \[Baer\] (see [@Ro2 7.2 and 7.3] and [@Ab; @ABT] for an account). For instance, in [@Pl], Plotkin proved that $L(G)=HP(G)$ and $R(G)$ is a subgroup whenever $G$ is a group with an ascending series whose factors satisfy max locally (i.e., every finitely generated subgroup has the maximal condition). The aim of this note is to extend Theorem \[Baer\] to the class of $FC^n$-groups, which has been introduced in [@dGRV] as follows. Let $FC^0$ be the class of finite groups, and suppose by induction hypothesis that for some positive integer $n$ a group class $FC^{n-1}$ has been defined. A group $G$ is called an [*$FC^n$-group*]{} if for any element $x\in G$ the factor group $G/C_G(x^G)$ belongs to the class $FC^{n-1}$, where $x^G$ is the normal closure of $\l x\r$ in $G$. It is easy to see that the set $FC^n(G)=\{x\in G\,|\,G/C_G(x^G)$ is an $FC^{n-1}$-group$\}$ is a subgroup of $G$, the so-called [*$FC^n$-centre*]{} of $G$. Hence $G$ is an $FC^n$-group if and only if $G=FC^n(G)$. Of course, $FC^1$ is the class of $FC$-groups, namely groups with finite conjugacy classes. More generally, a group is an [*$FC^*$-group*]{} if it is an $FC^n$-group for some $n\geq 0$. The investigation of properties, that are common to finite groups and nilpotent groups, has been satisfactory for $FC^*$-groups [@dGRV; @RRV; @RomVin; @KV]. It turns out that every finite-by-nilpotent group is an $FC^*$-group and, conversely, every $FC^*$-group is locally (finite-by-nilpotent) [@dGRV Proposition 3.6]. A group $G$ is said to be [*extended residually finite*]{}, or briefly an [*$ERF$-group*]{}, if every subgroup is closed in the profinite topology, i.e. every subgroup of $G$ is an intersection of subgroups of finite index. A complete classification of $ERF$-groups in the class of $FC^*$-groups is given in [@RRV]. In Section 2 we prove that, if $G$ is an $FC^n$-group, then $L(G)$ and $\overline{L}(G)$ coincide with the Fitting subgroup of $G$; whereas $R(G)$ and $\overline{R}(G)$ are subgroups of $G$ and, in particular, $R(G)$ coincides with the hypercentre of $G$, which equals $Z_{\omega+(n-1)}(G)$. It remains an open question whether $\overline{R}(G)$ coincides with the $\omega$-hypercentre, when $G$ is an $FC^n$-group. Nevertheless, we show that $R(G)=\overline{R}(G)=Z_{\omega}(G)$ under the additional assumption that $G$ is a periodic $ERF$-group. We also give an example of a non-periodic $FC^2$-group $G$ such that $G$ is $ERF$ and $\overline{R}(G)\subset R(G)$. The results =========== Given an arbitrary group $X$, we denote by $F(X)$ the Fitting subgroup of $X$. \[lemA\] Let $G$ be an $FC^n$-group. Then the normal closure of any left Engel element of $G$ is nilpotent and, consequently, $$L(G)=\overline{L}(G)=F(G).$$ Let $a\in L(G)$. By [@dGRV Lemma 3.7] the quotient group $a^G/Z_n(a^G)$ is finite. Applying Theorem \[Baer\], we have $$L(a^G/Z_n(a^G))=F(a^G/Z_n(a^G))=F(a^G)/Z_n(a^G),$$ where $F(a^G)$ is nilpotent because so is $F(a^G/Z_n(a^G))$. From $aZ_n(a^G)\in L(a^G/Z_n(a^G))$, we get $a\in F(a^G)$. But $F(a^G)$ is characteristic in $a^G$ and hence normal in $G$. Thus $a^G=F(a^G)$ and $a^G$ is nilpotent. In particular $a\in F(G)$, that is $L(G)\subseteq F(G)$. It follows that that $L(G)=\overline{L}(G)=F(G)$, because $F(G)\subseteq B(G)\subseteq\overline{L}(G)\subseteq L(G)$ [@Ro2 Lemma 7.12]. We recall that any $FC^*$-group $G$ satisfies max locally [@dGRV Proposition 3.6] and therefore, in according to Plotkin [@Pl], the set of right Engel elements of $G$ is always a subgroup. \[gamma\] Let $G$ be an $FC^n$-group and $a\in \gamma_n(G)\cap R(G)$. Then $a\in Z_k(G)$ for some $k=k(a)$. Let $N$ be the normal closure of $a$ in $G$. Since $\gamma_n(G)$ is contained in the $FC$-centre of $G$ [@dGRV Theorem 3.2], we have that $G/C_G(N)$ is finite. Then $G=HC_G(N)$, where $H$ is a finitely generated subgroup of $G$. Now $HN$ is finitely generated and so it satisfies the maximal condition, by [@dGRV Proposition 3.6]. Hence, Theorem \[Baer\] shows that $R(HN)=Z_k(HN)$ for some $k$. But $N\leq R(G)$, so that $N\leq R(HN)=Z_k(HN)$. For any $1\leq i\leq k$, let $g_i=x_i h_i\in G$ with $x_i\in C_G(N)$ and $h_i\in H$. Thus $[a,g_1,\ldots,g_k]=[a, h_1,\ldots,h_k]=1$ and $a\in Z_{k}(G)$, as desired. Let $G$ be a group. Following [@Gr], we denote by $\rho(G)$ the set of all elements $a\in G$ such that $\l x\r$ is ascendant in $\l x, a^G\r$, for any $x\in G$; and by $\overline{\rho}(G)$ the set of all elements $a\in G$ such that $\l x\r$ is subnormal in $\l x, a^G\r$ of defect at most $k=k(a)$, for any $x\in G$. By [@Ro2 Lemma 7.31], the sets $\rho(G)$ and $\overline{\rho}(G)$ are characteristic subgroups of $G$ satisfying the following inclusions: $$\label{rho} \overline{Z}(G)\subseteq\rho(G)\subseteq R(G) \quad{\rm and}\quad Z_{\omega}(G)\subseteq\overline {\rho}(G)\subseteq \overline{R}(G).$$ The subgroups $\rho(G)$ and $R(G)$ can be different (see for instance [@Ro2 Part 2, p. 59]) and it is possible that $\overline{Z}(G)=1$ and $\rho(G)=R(G)\neq 1$ [@Gr]. In contrast with this, for $FC^n$-groups, we have: \[lemB\] Let $G$ be an $FC^n$-group. Then - $R(G)=\rho(G)=\overline{Z}(G)=Z_{\omega+(n-1)}(G)$; - $\overline{R}(G)=\overline{\rho}(G)$. In particular, if $G$ is an $FC$-group, then $$R(G)=\overline{R}(G)=Z_{\omega}(G).$$ $(i)$ Clearly $Z_{\omega+(n-1)}(G)\subseteq\overline{Z}(G)\subseteq R(G)$, by (\[rho\]). Let $a\in R(G)$. As $R(G)$ is normal in $G$, for any $x_1,\ldots,x_{n-1}\in G$, we have $[a,x_1,\ldots, x_{n-1}]\in\gamma_n(G)\cap R(G)$ which is contained in $Z_{\omega}(G)$, by Lemma \[gamma\]. Hence $a\in Z_{\omega+(n-1)}(G)$ and $R(G)\subseteq Z_{\omega+(n-1)}(G)$. $(ii)$ Let $a\in \overline{R}(G)$. By Lemma \[lemA\], jointly with (\[RL\]), we have that $a^G$ is nilpotent. It follows that $a\in \overline{\rho}(G)$, by [@Gr2 Theorem 1.6], and $\overline{R}(G)\subseteq \overline{\rho}(G)$. Thus $\overline{R}(G)=\overline{\rho}(G)$, by (\[rho\]). A group $G$ is called an [*Engel group*]{} if $R(G)=G$ or, equivalently, $L(G)=G$. Of course locally nilpotent groups are Engel, but Golod’s example [@Go] shows that Engel groups need not be locally nilpotent. As a consequence of Lemma \[lemB\] $(i)$, every Engel $FC^n$-group is hypercentral and its upper central series has length at most $\omega+(n-1)$ (compare with [@dGRV Theorem 3.9 (b)]). Moreover this bound cannot be replaced by $\omega$ when $n>1$, see Example \[ex\]. By combining Lemma \[lemA\] and Lemma \[lemB\], our main result follows. \[FC\*\] Let $G$ be an $FC^n$-group. Then $L(G)$ and $\overline{L}(G)$ coincide with the Fitting subgroup of $G$; whereas $R(G)=\rho(G)$ coincides with the hypercentre of $G$, which equals $Z_{\omega+(n-1)}(G)$, and $\overline{R}(G)=\overline{\rho}(G)$. The respective position of these subgroups is indicated in the following diagram (see also the diagram in [@Ro2 Part 2, p. 63]). $\xymatrix{ { } & F(G)=\overline{L}(G)=L(G)\ar@{-}[d] & { } \\ { } & Z_{\omega+(n-1)}(G)=\overline{Z}(G)=\rho(G)=R(G)\ar@{-}[dd]\ar@{-}[dr] & { }\\ { } & { } & \overline{\rho}(G)=\overline{R}(G)\ar@{-}[dl] \\ { } & Z_{\omega}(G) & { }\\ }$ Notice that if $G$ is a finitely generated $FC^*$-group, then $G$ is finite-by-nilpotent [@dGRV Proposition 3.6] and so, by the next result, $R(G)$ and $\overline{R}(G)$ coincide with the $\omega$-hypercentre of $G$. \[proB\] Let $G$ be a finite-by-nilpotent group. Then $$R(G)=\overline{R}(G)=Z_{\omega}(G).$$ By (\[rho\]) we have $Z_{\omega}(G)\subseteq \overline{R}(G)\subseteq R(G)$. Since $G$ is finite-by-nilpotent, there exists $i\geq 0$ such that $G/Z_i(G)$ is finite [@Ro2 Theorem 4.25]. Then, by Theorem \[Baer\], we have $$R(G)Z_i(G)/Z_i(G)\subseteq R(G/Z_i(G))=Z_j(G/Z_i(G))=Z_{i+j}(G)/Z_i(G)$$ for some $j\geq 0$. It follows that $R(G)\subseteq Z_{i+j}(G)$, so that $R(G)\subseteq Z_{\omega}(G)$. In the sequel we restrict our attention to $FC^*$-groups belonging to the class of $ERF$-groups. Let $G$ be any $FC^n$-group and denote by $T(G)$ its torsion subgroup [@dGRV Corollary 3.3]. By [@RRV Theorem 3.6], the group $G$ is $ERF$ if and only if the following conditions hold: - Sylow subgroups of $G$ are abelian-by-finite with finite exponent; - Sylow subgroups of $\gamma_{n+1}(G)$ are finite; - $G/T(G)$ is torsion-free nilpotent of finite rank and no quotient of its subnormal subgroups is of $p^{\infty}$-type for any prime $p$. \[proC\] Let $G$ be an $FC^*$-group which is $ERF$. Then every periodic right Engel element of $G$ belongs to $Z_k(G)$ for some $k=k(a)$. Hence, if $G$ is periodic, then $$R(G)=\overline{R}(G)=Z_{\omega}(G).$$ First notice that, if $N$ is a finite subgroup of $G$ of order $m$ contained in $Z_i(G)$ for some $i\geq 1$, then $N\leq Z_m(G)$. This is true for any arbitrary group and its proof is a straightforward induction on $m$. Let $a$ be any nontrivial right Engel element of $G$. We may assume that $a$ is a $p$-element, where $p$ is prime. With $x_1,\ldots,x_{n}\in G$, by Lemma \[gamma\], we have $[a,x_1,\ldots,x_n]\in Z_i(G)$ for some $i$. Suppose $[a,x_1,\ldots,x_n]\neq 1$ and denote by $N$ the normal closure of $[a,x_1,\ldots,x_{n}]$ in $G$. Then $N\leq P\cap Z_i(G)$, where $i\geq 1$ and $P$ is a Sylow $p$ subgroup of $\gamma_{n+1}(G)$. Now $P$ is finite, say of order $m$. Then the previous remark implies that $N\leq Z_m(G)$ and so $[a,x_1,\ldots,x_n]\in Z_m(G)$. But Sylow $p$-subgroups of $G$ are isomorphic [@RRV Theorem 3.9] and so $m$ is independent of $x_1,\ldots, x_n$. Hence $a\in Z_{k}(G)$ where $k=m+n$. Next we show that, in our context, the set of bounded right Engel elements can be properly contained in the set of right Engel elements. \[ex\] There exists a non-periodic metabelian $FC^2$-group $G$ such that $$\overline{R}(G)=Z_{\omega}(G)\quad{\rm and}\quad R(G)=Z_{\omega+1}(G)=G.$$ Further, $Z_{\omega}(G)$ is periodic and $G$ is an $ERF$-group. Let $p_1<p_2<\ldots$ be a sequence of odd primes and $1<n_1<n_2<\ldots$ be a sequence of integers. For any $i\geq 1$, put $$P_i=\l a_i,b_i\r$$ where $a_i$ has order $p_i^{n_i}$, $b_i$ has order $p_i^{n_i-1}$ and $a_i^{b_i}=a_i^{1+p_i}$. Then $[a_i,b_i]=a_i^{p_i}$ and therefore, for any $m\geq 1$, we have $[a_i,_m b_i]=a_i^{p_i^m}$. In particular $[a_i,_{n_i} b_i]=1$ and, consequently, the commutator $[a_i,_{n_i-1} b_i]$ is a nontrivial element of $Z(P_i)$. This leads to $[a_i,b_i]\in Z_{n_i-1}(P_i)$, so that $P_i= \l a_i, b_i\r$ is nilpotent of class exactly $n_i$. Let $b_i^ja_i^k$ be an arbitrary element of $P_i$, with $0\leq j< p_i^{n_i-1}$ and $0\leq k< p_i^{n_i}$. By [@Da Lemma 4], the map $\alpha_i$ defined by $$(b_i^j a_i^k)^{\alpha_i}=(b_i a_i^p)^j a_i^k$$ is an automorphism of $P_i$. Clearly, $a_i^{\alpha_i}=a_i$ and $b_i^{\alpha_i}=b_i a_i^{p_i}$. Now form the semidirect product $$G=\l x\r\ltimes P$$ where $\l x\r$ is infinite cyclic, $\displaystyle P=Dr_{i\geq 1} P_i$ and $$a_i^x=a_i,\quad b_{i}^x=b_i a_i^{p_i}.$$ If $A=Dr_{i\geq 1}\l a_i\r$, then $G/A$ is abelian and $A\leq C_G(x)$. It follows that $C_G(x^G)=C_G(x)$ and $G/C_G(x^G)$ is abelian, that is $x\in FC^2(G)$. On the other hand $y^G$ is finite for any $y\in P$. Then so is $G/C_G(y^G)$, which embeds in $Aut(y^G)$. Hence $P\leq FC(G)\leq FC^2(G)$ and $G=FC^2(G)$, namely $G$ is an $FC^2$-group. Of course $G$ is $ERF$ by construction. Notice also that $R(G)=Z_{\omega+1}(G)$, by Lemma \[lemB\]. But $a_i\in Z_{n_i}(G)$, so that $A\leq Z_{\omega}(G)$ and $Z_{\omega+1}(G)=G$. Finally let $g=x^{r}y$ be an arbitrary element of $\overline{R}(G)$, where $r\in \mathbb{Z}$ and $y\in P$. Since $y$ is a periodic (right Engel) element, then $y\in Z_{\omega}(G)\subseteq \overline{R}(G)$ by Proposition \[proC\]. It follows that $x^r$ is an $m$-right Engel element, for some $m\geq 1$. From $[b_i,x^r]=a_i^{rp_i}$, we get $1=[x^r,_m b_i]^{-1}=(a_i^r)^{p_i^m}$, for any $i$. This forces $r=0$ and therefore $g=y\in P\cap Z_{\omega}(G)$. We conclude that $\overline{R}(G)=Z_{\omega}(G)\leq~P$. It is well-known that, if $G$ is an $FC$-group, then $G/Z(G)$ is periodic. This fails for $FC^*$-groups, even if they are $ERF$: there exists a non-periodic $FC^2$-group with trivial centre which is $ERF$ [@RRV Example 4.4]. One more example, but with nontrivial centre, is then the group given in Example \[ex\]. [10]{} A. Abdollahi, *Engel elements in groups*, Groups St Andrews 2009 in Bath. Volume 1, 94–117, London Math. Soc. Lecture Note Ser. [**387**]{}, Cambridge, 2011. A. Abdollahi, R. Brandl and A. Tortora, *Groups generated by a finite Engel set*, J. Algebra [**347**]{} (2011), 53–59. V. Bludov, *An example of not Engel group generated by Engel elements*, abstracts of “A Conference in Honor of Adalbert Bovdi’s 70th Birthday”, November 18–23, 2005; [www.math.klte.hu/bovdiconf/abstracts.pdf](www.math.klte.hu/bovdiconf/abstracts.pdf). R.M. Davitt, *The automorphism group of a finite metacyclic $p$-group*, Proc. Amer. Math. Soc. [**25**]{} 1970, 876–879. F. de Giovanni, A. Russo and G. Vincenzi, *Groups with restricted conjugacy classes*, Serdica Math. J. [**28**]{} (2002), 241–254. E.S. Golod, [*Some problems of Burnside type*]{}, 1968 Proc. Internat. Congr. Math. (Moscow 1966), 284–289; [*Amer. Math. Soc. Transl. $(2)$*]{} [**84**]{} (1969), 83–-88. K.W. Gruenberg, *The Engel elements of a soluble group*, Illinois J. Math. [**3**]{} (1959), 151–168. K.W. Gruenberg, *The upper central series in soluble groups*, Illinois J. Math. [**5**]{} (1961), 436–466. G. Kaplan and G. Vincenzi, *On the Wielandt subgroup of generalized $FC$-groups*, Internat. J. Algebra Comput., to appear. B.I. Plotkin, [*Radicals and nil-elements in groups*]{}, Izv. Vys$\check{\rm s}$ U$\check{\rm c}$ebn. Zaved. Matematika [**1**]{} (1958), 130–135. D.J.S. Robinson, *Finiteness conditions and generalized soluble groups*, Part 1 and Part 2, Springer Verlag, Berlin, 1972. D.J.S. Robinson, A. Russo and G. Vincenzi, *On the theory of generalized $FC$-groups*, J. Algebra [**326**]{} (2011), 218–226. E. Romano, G. Vincenzi, *Pronormality in generalized FC-groups*, Bull. Aust. Math. Soc. [**83**]{} (2011), 220–230.

--- abstract: 'The evolution of the antiferromagnetic order parameter in [CeFeAsO$_{1-x}$F$_{x}$]{} as a function of the fluorine content $x$ was investigated primarily via zero-field muon-spin spectroscopy. The long-range magnetic order observed in the undoped compound gradually turns into a short-range order at $x=0.04$, seemingly accompanied/induced by a drastic reduction of the magnetic moment of the iron ions. Superconductivity appears upon a further increase in doping ($x>0.04$) when, unlike in the cuprates, the Fe magnetic moments become even weaker. The resulting phase diagram evidences the presence of a crossover region, where the superconducting and the magnetic order parameters coexist on a nanoscopic range.' author: - 'T. Shiroka' - 'G. Lamura' - 'S. Sanna' - 'G. Prando' - 'R. De Renzi' - 'M. Tropeano' - 'M. R. Cimberle' - 'A. Martinelli' - 'C. Bernini' - 'A. Palenzona' - 'R. Fittipaldi' - 'A. Vecchione' - 'P. Carretta' - 'A. S. Siri' - 'C. Ferdeghini' - 'M. Putti' title: 'Long- to short-range magnetic order in fluorine-doped CeFeAsO ' --- \[sec:intro\]Introduction ========================= The discovery of high-temperature superconductivity in the iron-based layered compound LaFeAsO$_{1-x}$F$_{x}$ [@Kamihara2008] immediately created considerable excitement among condensed matter scientists. Other superconductors belonging to the same RE-1111 family, with RE a rare-earth metal, were discovered successively and superconductivity with a transition temperature of up to 55 K was found when La is substituted by other rare earths as e.g. Sm, Ce, Nd, Pr, and Gd.[@Chen2008; @ChenNAT2008; @RenEu2008; @RenMat2008; @Cheng2008] The new compounds show strong similarities with the high-$T_{c}$ cuprates:[@Basov2011] *i*) they have layered crystal structures with alternating REO and FeAs planes, where the iron ions are arranged on a simple square lattice, *ii*) the parent compound is antiferromagnetically ordered, *iii*) superconductivity emerges upon doping the parent compound with either electrons or holes, a process which suppresses the magnetic order. However, there are also important differences between the two families, among which two are particularly significant: the semi-metallicity of the iron-based parent compound, as opposed to the Mott-insulator character of cuprates, and the moderate degree of electron correlation in the former vs. the strong correlation observed in the latter, as from comparative studies of far-infrared reflection.[@Basov2009] The REFeAsO parent compounds generally show a commensurate spin-density wave (SDW) magnetic order characterized by *strongly reduced* Fe momenta, as evidenced by standard neutron scattering studies:[@Zhao2008] Fe magnetic moments are comprised between 0.25 and 0.8 $\mu_{\mathrm{B}}$, to be compared with 4 $\mu_{\mathrm{B}}$, the spin-only value for the Fe$^{2+}$ ion. For this very reason, the determination of the temperature dependence of the magnetic order parameter via sensitive local probe techniques, such as muon-spin rotation,[@Luetkens2009; @Amato2009; @Maeter2009; @JPCarlo2009] Mössbauer spectroscopy,[@Klauss2008; @McGuire2009] electron spin resonance (ESR),[@Alfonsov2011] or nuclear magnetic resonance (NMR)[@Bobroff2010] is supposed to provide a higher accuracy than that possible using standard powder neutron scattering.[@Lumsden2010] Currently, the accepted magnetic moment value for iron at 2 K in the REFeAsO families is 0.63(1) $\mu_{\mathrm{B}}$,[@Maeter2009; @Qureshi2010] seemingly independent of the rare earth. The study of magnetism of these systems is crucial: its disappearance often signals the onset of superconductivity. Since the coexistence of the magnetic (M) and superconducting (SC) orders in interspersed nanoscopic domains has been generally associated to unconventional superconductivity, it is of fundamental interest to discover whether the antiferromagnetism persist also in the superconducting phase, or it simply disappears as soon as the SC phase is established. Also the possible role played by the ordering of the rare-earth magnetic moments is not yet clear. Among the 1111 compounds the case of [CeFeAsO$_{1-x}$F$_{x}$]{} is paradigmatic of this situation. It was first investigated via neutron diffraction[@Zhao2008; @Chi2008], whose results suggested that both the magnetic and the superconducting order parameters go to zero at the M-SC boundary, thus hinting at the possibility of a quantum critical point (QCP) separating a long-range, anti-ferromagnetically ordered phase from the superconducting phase. This, however, is in contrast with the presence of a nanoscopic coexistence of magnetism and superconductivity, as evidenced in our recent study of [CeFeAsO$_{1-x}$F$_{x}$]{} for $x=0.07$.[@Sanna2010] This apparent discrepancy can be solved by observing that standard neutron diffraction is sensitive only to long-range magnetic order. In addition, also @JPCarlo2009 have observed an anomalous $\mu$SR relaxation in a similarly doped Ce-compound. To date, though, no systematic studies exist to establish whether the long-range magnetic order becomes short-ranged, or it simply vanishes as the superconductivity appears. A study of the evolution with doping of the magnetic order in the FeAs planes can, therefore, provide us with fundamental hints on the interplay between the superconductivity and magnetism, i.e. whether these two phases are competing or reinforcing each other. To address the above issues we carried out systematic $\mu$SR investigations in the [CeFeAsO$_{1-x}$F$_{x}$]{} system in a large doping range $x$. Due to their local probe character and to the high sensitivity to internal magnetic fields, muons are perfect for unraveling the complex interplay between magnetism and superconductivity in the iron-based superconductors. In the specific case of [CeFeAsO$_{1-x}$F$_{x}$]{} we find that as the fluorine content increases the magnetically ordered phase does not disappear, even at the highest investigated $x$ concentrations, where it coexists with SC. The features of the magnetic order, on the other hand, are found to depend strongly on F doping, with the magnetic coherence range becoming shorter as $x$ increases. ![\[fig:diffrattogramma\] (Color online) X-ray diffraction pattern for the sample $x=0.04$ and the corresponding Rietveld refinement plot.](fig1){width="43.00000%"} \[sec:exp\_details\]Experimental details and results ==================================================== \[ssec:preparation\]Sample preparation and morphological and structural characterization ---------------------------------------------------------------------------------------- A series of [CeFeAsO$_{1-x}$F$_{x}$]{} samples, with a real fluorine content ranging from $x=0$ up to $x=0.07$ (see next section), was prepared via solid-state reaction methods by reacting stoichiometric amounts of CeAs, Fe$_{2}$O$_{3}$, FeF$_{2}$, and Fe. CeAs was obtained by reacting Ce chips and As pieces at 450–500$^{\circ}$C for 4 days. The raw materials were thoroughly mixed and pressed into pellets, which were then heated up to ca. $1100^{\circ}$C for 50 h. Further details concerning sample preparation and characterization have been reported elsewhere.[@Martinelli2008] A morphological and a structural characterization were carried out on all the successively investigated samples. The morphological analysis, performed via scanning electron microscopy (SEM), revealed the presence of grains whose dimensions range from 1 to 10 $\mu$m, almost independently from the doping level (see also App. \[ssec:SEM\]). This independence from doping rules out a possible influence of the grain morphology on the magnetic and superconducting properties (e.g., apparent changes in the superconducting fraction, etc.). The structural characterization was carried out using standard powder X-ray diffraction (XRD) analysis. Figure \[fig:diffrattogramma\] shows the Rietveld refinement plot for a representative $x=0.04$ sample. Figure \[fig:strutturali\] instead shows the variation of the unit cell parameters as a function of the real fluorine content. The latter was estimated by means of NMR measurements, as reported below. We note that while the length of the $a$-axis is practically insensitive to F substitution, the $c$-axis’ length decreases linearly with increasing $x$(F), in agreement with data reported in Ref. . As a result, the global effect of O$^{2-}$/F$^{-}$ substitution is that of decreasing the volume of the unit cell, a reasonable outcome reflecting the smaller ionic radius of fluorine. ![\[fig:strutturali\] (Color online) Evolution of the $a$- and $c$-axis lengths of [CeFeAsO$_{1-x}$F$_{x}$]{} as a function of the real fluorine content, $x$(F), at 290 K. Lines are best fits to the experimental data, while the estimated uncertainty $\Delta x/x$ is ca. 25%.](fig2){width="45.00000%"} \[ssec:NMR\]Determination of fluorine doping via ${}^{19}$F NMR --------------------------------------------------------------- The gradual fluorine substitution in a RE-1111 system is known to introduce free carriers onto the FeAs layers, hence inducing important modifications to the material’s electronic properties. As such, $x$(F) represents the most natural parameter for describing the phase diagram of F-doped pnictides. Since our aim is the study of the M-SC crossover region, where even tiny variations of *x*(F) could play a major role, the determination of the *real* fluorine content is of crucial importance. To this purpose quantitative fluorine-19 NMR measurements were carried out and the samples classified accordingly. To obtain a reliable estimate of the $x$(F) values, all the samples were measured using fluorine-free cabling and probehead. The resonant ${}^{19}$F NMR signal, following a conventional solid-echo sequence ($\pi/2 - \tau - \pi/2 - \tau -$ acquisition), was acquired in a fixed-field configuration, $\mu_{0}H \simeq 1.4$ T, corresponding to $\nu \simeq 56$ MHz for the ${}^{19}$F nuclei. By recording the NMR signal for different values of the delay $\tau$ we could extrapolate the exponential decay of the integrated echo intensity, $I(\tau)$, back to $\tau = 0$. The extrapolation procedure is crucial in providing an unbiased quantitative estimate of the fluorine content. In fact, samples with different F doping have, in general, different spin-spin relaxation times $T_{2}$, which would potentially impair a straightforward intensity comparison. Since quantitative NMR measurements are notoriously difficult, the *total* fluorine content for each sample was evaluated by comparing the measured $I(0)$ values with that of an SmOF reference compound.[@Sanna2010] In addition, we recall that NMR alone is unable to distinguish the fluorine signal coming from primary vs. secondary phases. Therefore, to obtain the *true* $x$(F) values, the measured NMR intensity data were finally corrected to take into account the possible presence of CeOF impurities. The relative amount of the latter was accurately quantified by means of Rietveld refinement of X-ray powder diffraction patterns. We find that, whenever present, the spurious CeOF phase never exceeds 2% vol. This analysis reveals that the real F content is systematically lower than the nominal one but, nevertheless, the use of NMR labeling is superior to the simple use of the nominal doping. Therefore, hereafter we use the former as a label for the different samples. Possible errors would uniformly affect the investigated samples, implying a rescaling of the final phase diagram, but cannot give rise to distortions or inversions. \[ssec:transport\]Transport measurements ---------------------------------------- The resistivity of the [CeFeAsO$_{1-x}$F$_{x}$]{} samples was measured using a standard four-point method. The temperature dependence of $\rho(T)$ for selected F doping values is shown in Fig. \[fig:trasporto\]. Upon cooling, the undoped sample presents the typical transport features of iron-based oxypnictides: *i*) a low-temperature resistivity in the m$\Omega\,$cm range; *ii*) a broad maximum, followed by a drop of $\rho(T)$, with an inflection point defined as the maximum of the first derivative, $\mathrm{d}\rho/\mathrm{d}T$ (arrows in Fig. \[fig:trasporto\]). The presence of a maximum in the first derivative of resistivity has been observed also in the 1111 systems containing La,[@Klauss2008; @McGuire2008] Pr[@Kimber2008] and Sm,[@Tropeano2009] and has been generally attributed to a spin-density wave (SDW) transition. As the fluorine content increases, the maxima of $\rho(T)$ and $\mathrm{d}\rho/\mathrm{d}T$ both shift towards lower temperatures, become broader and eventually disappear for $x=0.04$, in full agreement with existing experimental data on the Ce-1111 family.[@Chen2008] ![\[fig:trasporto\] (Color online) Normalized resistivity vs. temperature for a selection of [CeFeAsO$_{1-x}$F$_{x}$]{} samples. For a better visibility the curves have been shifted against each other by 1.5 units. Arrows indicate the maxima of $\mathrm{d}\rho/\mathrm{d}T$.](fig3){width="45.00000%"} \[ssec:magnetization\]Magnetization measurements ------------------------------------------------ DC magnetization measurements were performed by means of a superconducting quantum interference device (SQUID) magnetometer (Quantum Design) on all the tested samples. Both magnetization vs. temperature, from 2 up to 300 K at $\mu_0 H = 3$ T, in zero-field cooling (ZFC) and in field cooling (FC) conditions, as well as magnetization measurements vs. applied field, $m(H)$, at selected temperatures were carried out. The experimental results can be summarized as follows:\ *i*) The quantity of dilute ferromagnetic impurities, if any, is irrelevantly small. This is evinced by the linear (i.e. purely paramagnetic) behavior of $m(H)$ at both low and high temperatures (not shown).\ *ii*) Ce ions mostly retain their free-ion magnetic moment value. This result arises from numerical fits of $\chi(T)$ data using the Curie-Weiss law, $\chi(T)=C/(T-\theta)+\chi_0$, with $\chi_0$ the temperature-independent susceptibility, $C$ the Curie constant, and $\theta$ the Curie-Weiss temperature. First, we determine $\chi_0$ by considering only the high-temperature regime. Successively, we perform a linear fit of $1/(\chi(T)-\chi_0)$, as shown in Fig. \[fig:mag1\] for a typical case, $x=0$. From the resulting Curie constant one can determine the Ce magnetic moment (in the free-ion approximation), which is plotted against the F content in the inset of Fig. \[fig:mag1\]. ![\[fig:mag1\] (Color online) Linear fit of $1/(\chi - \chi_0$) for the case of the undoped sample ($x=0$). Inset: Ce free-ion magnetic moment for all the samples under test, as derived from Curie-Weiss fits of the dc magnetization data. The dotted line shows the theoretically expected value for free Ce$^{3+}$ ions (2.54 $\mu_{\mathrm{B}}$).](fig4){width="45.00000%"} This value is very close to that expected for free Ce$^{3+}$ ions (2.54 $\mu_{\mathrm{B}}$) and in perfect agreement with previous data.[@Chen2008; @Zhao2008; @CeION_1; @CeION_2]\ *iii*) An increase in F doping up to $x = 0.07$ depresses slightly the cerium antiferromagnetic (AF) ordering temperature, $T_{\mathrm{N}}$(Ce). We should here caution the reader that the Ce AF transition is somehow affected by applied fields of moderate intensity: a magnetic field of $\sim 4$ T can lower an otherwise “regular” $T_{\mathrm{N}}$(Ce) by more than 2 K, in agreement with data reported in Ref. . All $T_{\mathrm{N}}$(Ce) values, measured in similar low-field conditions, are reported in Table \[tab:table\].\ *iv*) Samples with $x \gtrsim 0.06$ display a clear transition to the superconducting state, as shown by the low-temperature ZFC magnetization data for $x=0.06$ and 0.07 reported in Fig. \[fig:magSC\]. A precise determination of the superconducting fraction from the magnetization data, though, is difficult: at lower F-doping values, the field penetration depth increases considerably and becomes comparable to the grain size (1–10 $\mu$m), thus effectively reducing the shielding volume within each grain. Nevertheless, TF-$\mu$SR data clearly demonstrate that all these samples show bulk superconductivity, as reported in detail for the $x=0.07$ case in Ref. . ![\[fig:magSC\] (Color online) ZFC magnetization data for the superconducting samples at $\mu_0 H = 0.5$ mT. Arrows denote the FeAs SC transitions and the Ce magnetic ordering temperatures, respectively.](fig5){width="45.00000%"} --------- --------- ---------- ---------- --------- -------- --------- -- 0.7mm 0.7mm 0.7mm 0.7mm 0.7mm 0.7mm \[2pt\] 0 150(2) 10(1) – 97(2) 4.0(3) 0.03(1) 106.0(9) 9(1) – 93(1) 3.5(3) 0.04(1) 37.9(9) 9(1) – 100(8) 2.78(3) 0.06(1) 28.0(7) 11(1) 26.5(5) 100(3) 2.7(3) 0.07(1) 16.5(3) 9.8(0.4) 18.3(5) 100(5) 2.8(3) --------- --------- ---------- ---------- --------- -------- --------- -- \[ssec:ZFmuSR\]ZF-$\mu$SR measurements -------------------------------------- The muon-spin relaxation measurements were carried out at the GPS instrument ($\pi$M3 beam line) of the Paul Scherrer Institut, Villigen, Switzerland. $\mu$SR experiments consist in implanting 100% spin-polarized muons in a sample and successively detecting the relevant decay positrons. Implanted muons thermalize almost instantaneously at interstitial sites, where they act as sensitive probes, precessing in the local magnetic field $B_{\mu}$ with a frequency $f_{\mu} = \gamma/(2\pi)\cdot B_{\mu}$, with $\gamma/(2\pi) = 135.53$ MHz/T the muon gyromagnetic ratio. Both zero-field (ZF) and longitudinal field (LF) $\mu$SR experiments were performed. Due to the absence of applied fields, ZF-$\mu$SR represents the best technique for investigating the spontaneous magnetism and its evolution with fluorine doping. LF-$\mu$SR measurements, instead, were used to single out the dynamic or static character of the magnetic order, as detailed in App. \[ssec:LFmuSR\]. Figure \[fig:LongShort\] shows the low-temperature ZF asymmetry (i.e. the muon-spin precession signal), $A(t)$, for the tested [CeFeAsO$_{1-x}$F$_{x}$]{} samples. As a comparison, the high-temperature asymmetry is also shown for the undoped $x=0$ case. At high temperatures, i.e. above the Néel temperature $T_{\mathrm{N}}$, the asymmetry signal is practically flat, with no oscillations and with a very small decay of the initial polarization. This behavior is typical of paramagnetic materials, where there are no significant internal magnetic fields, except for those due to the tiny randomly-oriented nuclear moments, which account for the exiguous decay of asymmetry. ![\[fig:LongShort\] (Color online) Zero-field, time-domain $\mu$SR data of [CeFeAsO$_{1-x}$F$_{x}$]{} for selected fluorine doping values $x$, recorded at $T = 5$ K. The superconductivity appears for $x \gtrsim 0.06$, whereas the antiferromagnetic order is always present. The range of the AF order, however, changes from long to short, as shown by the absence of oscillations in samples with a high F content.](fig6){width="46.00000%"} ![\[fig:volMag\] (Color online) Magnetic volume fraction vs. temperature as extracted from the longitudinal component of $\mu$SR data and fitted with an $\mathrm{erf}(T)$ function (Eq. \[eq:mag\_vol\_frac\]) for a selection of samples ($x=0$, 0.03, 0.06). The relevant fit values, including those of other samples, are reported in Table \[tab:table\].](fig7){width="45.00000%"} However, once the temperature is lowered below $T_{\mathrm{N}}$, dramatic differences develop, reflecting the emergence of a spontaneous magnetic order. The new ordered phase seems to have features which depend strongly on $x$: for a low F content we observe well-defined asymmetry oscillations which, as $x$(F) increases, change quickly to highly damped ones. The damping for $x \ge 0.04$ becomes so high that the oscillatory behavior disappears altogether, to be replaced by a fast decaying signal. We recall that asymmetry oscillations indicate the presence of a uniform magnetic field at the muon sites, while a strong decay of the asymmetry arises whenever there is a wide distribution of fields. Hence, the case $x=0$ is compatible with a *long-range* (static) magnetic order. On the other hand, a strong $\mu$SR signal decay (for $0.04\lesssim x \lesssim 0.07$) is compatible with dephasing (i.e. incoherent muon-spin precession) due to a distribution of internal fields, which can be attributed to a *short-range* magnetic order. In Fig. \[fig:LongShort\] the evolution with doping of the spin polarization of the muon ensemble is followed against the fluorine content. To achieve a better understanding from the above results, the various ZF-$\mu$SR time-domain data were fitted using the function: $$\label{eq:osc} \frac{A^{\mathrm{ZF}}(t)}{A_{\mathrm{tot}}^{\mathrm{ZF}}(0)}= \left[ a^{A}_{\perp} e^{-\frac{\sigma_A^2 t^2}{2}} f (2 \pi \gamma B_{\mu} t) \! + a^{B}_{\perp} e^{-\frac{\sigma_B^2 t^2}{2}}\right]+ a_{\parallel}e^{-\lambda t}.$$ Here $a^{A,B}_{\perp}$ are two transverse components, while $a_{\parallel}$ represents a longitudinal component, each with respective decay coefficients $\sigma_{A,B}$ and $\lambda$. The choice of two transverse decay components follows closely the calculations presented in Ref. , according to which, for the undoped CeFeAsO case (but largely valid also in presence of F doping), two distinct muon implantation sites are expected. The most populated site, named A, is located next to the FeAs planes, while a second site, named B, is close to the oxygen atoms in the CeO planes. As suggested, muons implanted in A are sensitive to a magnetic field that consists of two contributions: the molecular field generated by the Fe sublattice and the field arising from the Ce polarization, the latter being induced by the Fe magnetism (via an exchange coupling $J_{\mathrm{Fe-Ce}} \sim 43$ T/$\mu_{\mathrm{B}}$).[@Maeter2009] Muons implanted in B are sensitive mostly to this second contribution. Since the latter site is statistically the least populated one (accounting for only $\sim 15\%$ of the implanted muons, as from Eq. \[eq:osc\]), its contribution to the total asymmetry does not permit to distinguish a second precession frequency and therefore it is taken into account by a single exponential decay ($a^{B}_{\perp}$ ). The corresponding longitudinal components, nevertheless, share similar decay rates, which allows us to merge them in a single term, $a_{\parallel}$. The nature of the oscillating term $f(t)$ depends on the F content: for $x=0$ (and other low $x$ values) $A(t)$ could be fitted using $f(t)=\cos(2\pi\gamma B_{\mu} t)$; on the other hand, for $x=0.03$ the best fit was obtained with $f(t)=J_0(2\pi\gamma B_{\mu} t)$ (here $J_0$ is the zeroth order Bessel function). The cosine term is the hallmark of *commensurate* long-range magnetic order, while the presence of a Bessel function is generally attributed to *incommensurate* long-range ordered systems.[@SAVICI] Finally, $f(t)=1$ for all those samples where no coherent oscillations could be detected ($x \gtrsim 0.04$). As a result, the observed static AF order seems to be commensurate for $x < 0.03$, incommensurate for $x \sim 0.03$ and fully disordered for $0.04 \lesssim x \lesssim 0.07$.  Internal magnetic field $B_\mu$ as probed by ZF-$\mu$SR at the so-called A-sites for $x=0$ and $x=0.03$.](fig8){width="45.00000%"} Let us now determine the magnetic volume fraction from the $\mu$SR data. Since all the considered samples were available as pressed powder pellets, we can safely assume a randomly oriented internal field model: in an ideal case, where the whole sample shows static antiferromagnetic order, on average, $\nicefrac{1}{3}$ of the muons experience a field parallel to their initial spin direction, and hence do not precess (longitudinal component), while the remaining $\nicefrac{2}{3}$ will precess (transverse component). This is exactly what we find for all the measured samples. From the temperature dependence of the longitudinal component one can follow the evolution of the volume fraction, $V_M$, of the magnetically ordered phase, $V_M(T) = \frac{3}{2} \left(1-a_{\parallel}\right) \cdot 100\%$. The resulting $V_M(T)$ values for a selection of representative samples are plotted in Fig. \[fig:volMag\]. It is worth noticing that *all* samples with $x \lesssim 0.07$ become fully magnetic at low temperature ($V_M = 100\%$). To determine the average Néel temperature, the corresponding transition width and the magnetically ordered fraction, the obtained $V_M(T)$ data were fitted using the following phenomenological function: $$\label{eq:mag_vol_frac} V_M(T) = \frac{1}{2} \left[ 1 -\mathrm{erf} \left(\frac{T-T_{\mathrm{N}}}{\sqrt{2}\Delta}\right)\right].$$ The fit results, summarized in Table \[tab:table\], as well as in Fig. \[fig:volMag\] for selected cases, clearly show that as the fluorine content $x$(F) increases there is both a gradual decrease of $T_{\mathrm{N}}$ and a progressive broadening of the transition.  Dependence on fluorine doping of the average magnetic field value $\left< B_{\mu} \right>$ (for $x<0.4$) and field width $\left<\Delta B_{\mu}^2 \right>^{1/2}$ (for $x \ge 0.4$), as measured by muons stopping in sites A and B. The dashed line provides a guide to the eye, emphasizing the sharp field drop at $x \sim 0.4$, in concomitance with the onset of the short-range magnetic order.](fig9){width="48.50000%"} The internal field $B_{\mu}$, as resulting from fits of the asymmetry data with Eq. \[eq:osc\], is plotted in Fig. \[fig:Bmu\] for the case $x=0$ and 0.03. In the undoped sample $B_{\mu}$ corresponds to values of the order of $\sim 200$ mT (27 MHz), which are typical for the oxypnictides[@Amato2009] and in full agreement with those reported in Ref. . As for the temperature dependence, $B_{\mu}$ shows a characteristic low-$T$ increase (instead of a saturation), which is peculiar of CeFeAsO[@Maeter2009] and is due to the AF ordering of Ce$^{3+}$ ions. As the fluorine content increases the average Gaussian field value $\left< B_{\mu} \right>$ reduces and its width broadens. At high F doping, no more coherent oscillations are present in $A(t)$. In these cases the internal field can be described as a broadened distribution of fields whose values range from zero up to $\sigma/\gamma\! = \!(\overline{B_i}^2- \overline{B_i^2})^{1/2}$.[@Kadono2004] Figure \[fig:largRIGA\] displays the behavior of the internal field for $T_{\mathrm{N}}\mathrm{(Ce)} < T \ll T_{\mathrm{N}}$ as a function of F content for both muon sites A and B. Since the internal field is proportional (via the dipolar interaction) to the staggered magnetization due to the Fe ordered moments, plus a less relevant contribution from the Ce polarized sublattice,[@Maeter2009] Fig. \[fig:largRIGA\] shows that the Fe magnetic moment is progressively reduced as the doping content increases. This is a general feature of the iron superconductors in marked contrast with the cuprates. In fact, unlike in the former, in the cuprates the low-temperature staggered magnetization always recovers to the value of the undoped parent compound. This recovery occurs throughout the doping range corresponding to a magnetically ordered phase,[@Borsa1995] either long- or short-ranged, and irrespective of its possible coexistence with superconductivity.[@Coneri2010; @Sanna2010b] This feature is a consequence of the Mott-Hubbard character of the cuprates and the related spin freezing, both absent in the iron pnictides. \[sec:discussion\]Discussion ============================ The study of doping effects on both the magnetic and the superconducting properties, as well as the presence of a possible M-SC crossover region in the Ce-1111 family compounds, are as interesting as important to our comprehension of the new iron arsenide compounds. To this aim, we have performed a full investigation of structural, transport and magnetic properties on a series of samples with $x$(F) ranging from 0 up to 0.07. By collecting the results from all the presented investigations we obtain a coherent physical picture that provides us with important hints on the evolution with doping of the magnetic order in the FeAs planes and on the influence of the latter on the developing superconducting order. In particular, we highlight the following key results:\ *i*) The Ce$^{3+}$ AF ordering temperature decreases gradually as the doping $x$(F) increases. This behavior most likely suggests a correlation between the magnetic order of iron and that of the cerium ions.\ *ii*) In the undoped compound, a long-range commensurate AF order sets in for temperatures below $T_{\mathrm{N}}$. This long-range magnetic order is evidenced as coherent oscillations of the muon polarization at low temperatures. Once a small percentage of fluorine is substituted to oxygen, these oscillations become highly damped. In particular, for the $x=0.03$ case a zeroth-order Bessel function seems to provide a better fit to $A(t)$ than a harmonic (cosine) function, thus signaling the onset of an incommensurate AF order of iron ions.[@SAVICI; @UEMURA] At higher F doping, the shrinking dimensions of the magnetically ordered domains imply a faster dephasing and depolarization of the implanted muons. This marked depolarization is reflected in such a high asymmetry damping that it prevents the detection of any coherent oscillations.\ *iii*) Our $\mu$SR results can be summarized in the revised phase diagram shown in Fig. \[fig:phase\_diag\]. Here we report the ordering temperatures, $T_{\mathrm{N}}$, as obtained by fits of $a_{\parallel}(T)$ asymmetry using Eq. (\[eq:mag\_vol\_frac\]), and $T_c$, in the superconducting phase, as determined by dc magnetization measurements. ![\[fig:phase\_diag\] (Color online) Phase diagram of [CeFeAsO$_{1-x}$F$_{x}$]{} as determined from the current $\mu$SR measurements. M and SC coexist in the violet region, while the hatched area indicates the presence of a short-range (SR) magnetic order. The point at $x = 0.16$ was taken from Ref. . See text for details.](fig10){width="45.00000%"} Our data are in a fairly good agreement with those of Ref.  for $x<0.04$, where some long-range magnetic order still persists and, therefore, is detectable also by neutrons. However, for higher F concentrations there is a clear discrepancy, most likely due to the fact that ordinary powder diffraction methods using thermal neutrons are not sensitive to short-range order. In fact, from $x \geq 0.04$ and up to $x = 0.07$, muons are able to evidence the presence of a *short-range static magnetic order* (hatched area in Fig. \[fig:phase\_diag\]). In the narrow region, extending from $x \sim 0.05$ to 0.07, this magnetic order coexists with superconductivity on a nm-range scale, as detailed in Ref.  (thistle-colored area in Fig. \[fig:phase\_diag\]), then magnetism is expected to disappear above $x=0.07$. The narrow M-SC coexisting region clearly rules out the presence of a quantum critical point, i.e. the existence of a single critical F-doping separating the two phases. Consequently, our results are in good agreement with a recently suggested Ce-1111 phase diagram,[@Wang2011] where the long-range AF and SC phases do not overlap but are separated by an intermediate phase, which we identify with the short-range AF phase. Conclusion ========== The evolution of the magnetic order vs. F doping in [CeFeAsO$_{1-x}$F$_{x}$]{} was mapped via muon-spin spectroscopy. The experimental data and the successive analysis confirm the coexistence of two magnetic-ion subsystems, one related to Ce$^{3+}$ and the other to Fe$^{2+}$. The cerium antiferromagnetism, occurring at relatively low temperatures ($< 4$ K), seems to be correlated with the magnetic order taking place in the FeAs planes, as evidenced by the slight drop of $T_{\mathrm{N}}$(Ce) observed when the Fe magnetic order disappears. The FeAs magnetic order, a hallmark also for the Ce-1111 family, is strongly affected by an increase in F content: *i*) its Néel temperature decreases sharply; *ii*) the AF magnetic order evolves from *long-* to *short-*ranged; *iii*) superconductivity appears when the Fe$^{2+}$ magnetic moment is significantly reduced with respect to its value in the undoped case. These findings seriously question the presence of a quantum critical point in Ce-1111 and, together with previous results on Nd-[@JPCarlo2009], Sm-[@Sanna2009] and Gd-1111[@Alfonsov2011] compounds, most likely suggest that the phase diagram of the 1111 family is RE-independent and consists of a narrow M-SC crossover region, where nanoscopic coexistence takes place. Surprisingly, though, this coexisting region seems to be immeasurably small or totally absent in La-1111.[@Luetkens2009; @Khasanov2011] Future measurements using Ce-1111 samples with finely tuned fluorine doping could be useful to better determine the extension of the M-SC coexisting region. This work was performed at the Swiss Muon Source S$\mu$S, Paul Scherrer Institut (PSI), Switzerland and was partially supported by MIUR under project PRIN2008 XWLWF9. The authors are grateful to A. Amato for the instrumental support. T.S. acknowledges support from the Schweizer Nationalfonds (SNF) and the NCCR program MaNEP. \[ssec:SEM\]SEM analysis ======================== To check for possible anomalies related to the varying fluorine content, detailed SEM analysis were performed on samples having different $x$ values. The morphology of the tested specimens turned out to be almost independent of doping. As an example, Fig. \[fig:SEM\] shows the SEM image of a sample with a nominal $x=0.06$. ![\[fig:SEM\]SEM image of $x=0.06$ sample showing the grain morphology.](fig11){width="35.00000%"} \[ssec:LFmuSR\]LF-$\mu$SR measurements ====================================== The absence of oscillations in the $\mu$SR signal for fluorine dopings above $\sim 4\%$, could be due either to a wide distribution of static fields, or to strongly fluctuating (i.e. dynamic) magnetic moments. Although we could reasonably expect the static picture to reflect the physics of our system, we still carried out LF-$\mu$SR in the representative $x=0.06$ case.  Longitudinal field scan at $T=26.5$ K for the $x=0.06$ sample. The solid line represents a fit using the polarization recovery function.[@Cox1987; @Blundell2004]](fig12){width="40.00000%"} In a so-called LF-decoupling experiment[@Hayano1979] an external magnetic field $B_{\parallel}$ is applied along the initial muon-spin direction. If $B_{\parallel}$ is of the same order of or higher than the internal static fields, then it will have a large influence on the muon polarization through a “spin-locking” effect. On the other hand, in case of strongly fluctuating internal fields the effect of the external field is barely noticeable. By applying longitudinal fields in the range 0 to 500 mT, we could observe a clear polarization recovery for $B_{\parallel} \gtrsim 200$ mT (see Fig. \[fig:LFscan\]), in agreement with the hypothesis of a static distribution of the internal fields. 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--- abstract: 'We characterize isometric actions on compact Kähler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the complex projective space with a Lagrangian orbit, and we give the explicit list of these orbits.' address: - | Dipartimento di Matematica - Università di Bologna\ Piazza di Porta S. Donato 5\ 40126 Bologna\ Italy - | Dipartimento di Matematica e Appl. per l’Architettura - Università di Firenze\ Piazza Ghiberti 27\ 50100 Firenze\ Italy author: - Lucio Bedulli and Anna Gori title: Homogeneous Lagrangian submanifolds --- [^1] Introduction {#introduction .unnumbered} ============ A Lagrangian submanifold of a $2n$-dimensional symplectic manifold $(M,\omega)$ is an $n$-dimensional submanifold on which the symplectic form $\omega$ vanishes. Lagrangian submanifolds play an important role in symplectic geometry and topology.\ In the Kähler setting i.e. when $M$ admits an integrable almost complex structure $J$ such that the bilinear form $g(X,Y)=\omega(X,JY)$ defines a Riemaniann metric on $M$, the associated Riemannian properties of Lagrangian submanifolds have been studied by different authors (see [@HL], [@Ta], [@Br], [@Oh], [@Oh2]), in particular in relation to the analysis of [*minimal*]{} Lagrangian submanifolds. In [@Oh] the author asks for a group theoretical machinery producing minimal Lagrangian submanifolds in Hermitian symmetric spaces.\ In the present paper we first study the existence problem of homogeneous Lagrangian submanifolds in compact Kähler manifolds, coming to the characterization of isometric actions admitting a Lagrangian orbit, by imposing an additional hypotesis on $M$, holding for a large class of Kähler manifolds including irreducible Hermitian symmetric spaces. Namely we require the space $H^{1,1}(M)$ to be 1-dimensional. \[teoremone\] Let $K$ be a compact connected group of isometries acting in a Hamiltonian fashion on a compact Kähler manifold $M$ with $h^{1,1}(M)=1$. Then $M$ admits a $K$-homogeneous Lagrangian submanifold if and only if $K^{\mathbb{C}}$ has an open Stein orbit in $M$. In [@Akbook] it is proved that if $G:=K^{\mathbb{C}}$ acts holomorphically on a [*complex*]{} manifold with an open Stein orbit, then there exists a [*totally real*]{} $K$-orbit $\mathcal O$, i.e. at every point of $\mathcal O$ the tangent space does not contain complex lines.\ A counterexample shows that when $h^{1,1}>1$, even the presence of an open Stein $K^{\mathbb{C}}$-orbit does not guarantee the existence of a Lagrangian $K$-orbit.\ When $K$ is semisimple it turns out that the Lagrangian orbit is unique and, when $M$ is Kähler Einstein, it is also minimal. In the general case we describe the actions having infinitely many Lagrangian orbits, characterizing the minimal ones in Section 2. \[cardinalita\] A Lagrangian $K$-orbit, $K\cdot p$, is isolated (actually unique) if and only if the smallest subgroup $K'$ of $K$ such that $K\cdot p=K' \cdot p$ is semisimple. Our main tool will be the moment map, that can be defined whenever we consider an Hamiltonian group action on $M$. More precisely, let $(M,\omega,J)$ be a compact $2n$-dimensional Kähler manifold, acted on in a Hamiltonian fashion by a compact connected subgroup $K$ of its full isometry group. This means that there exists a smooth map $\mu: M\rightarrow\mathfrak{k}^*=\rm{Lie}(K)^*$, called a [*moment map*]{}, with the following properties: 1. $d\mu_p(v)(X)=\omega_p(v,\widehat {X}_p)$ for all $p\in M$, $v\in T_p M$ and $X\in \mathfrak{k}$. Here $\widehat {X}_p$ stands for the fundamental field associated to $X$, evaluated at $p$; 2. $\mu$ is $K$-equivariant with respect to the coadjoint action of $K$ on $\mathfrak{k}^*$. In general the matter of existence and uniqueness of the moment map is delicate. However, whenever the Lie group $K$ is semisimple there is a unique moment map (see e.g. [@Kir]). If $(M,\omega)$, as in our situation, is a compact Kähler manifold and $K$ is a connected compact group of holomorphic isometries then the existence problem can be easily solved: a moment map exists if and only if $K$ acts trivially on the Albanese manifold of $M$ (see e.g. [@HW]). Moreover if $\mu_1$ and $\mu_2$ are two moment maps, there exists $c$ in the dual of the Lie algebra of the center of $K$, such that $\mu_1=\mu_2+c$.\ In [@GP] the authors have studied the critical set of the squared moment map $||\mu||^2$, where $||\cdot||$ denotes the norm induced by an $Ad(K)$-invariant inner product $<,>$ on $\mathfrak{k}^*$. In particular it is proved that if a point $x\in M$ realizes the maximum of $||\mu||^2$, then the orbit $K\cdot x$ is complex; hence $K\cdot x=K^{\mathbb{C}}\cdot x$ is a closed $K^{\mathbb{C}}$-orbit; it is therefore natural to consider the “dual” problem, i.e. to investigate the $K$-orbits through points $y\in M$ that attain the minimum of $||\mu||^2$. At least when $K$ is semisimple and $K^{\mathbb{C}}$ has an open Stein orbit on $M$, Theorem \[teoremone\] is a step in this direction.\ While in Theorem \[teoremone\] we prove the existence of a Lagrangian orbit $L$, we do not exhibit an effective way to single out $L$. At least for self-dual representations, we give in Remark \[selfdual\] an explicit expression, in terms of the highest weight vector, of a point through which the orbit is Lagrangian. Using this result and several [*ad hoc*]{} arguments we finally give the complete classification of Lagrangian submanifolds of the complex projective space on which a simple group of isometries of the whole space acts transitively. \[classificazione\] Let $K$ be a simple compact Lie group acting on the complex projective space ${\mathbb{P}}(V)$, by means of a unitary representation $\rho:K\to {{\rm U}}(V)$. The group $K$ has a Lagrangian orbit in ${\mathbb{P}}(V)$ if and only if it appears in Table $1$. The paper is organized as follows: In the first section we introduce some notations and give the proof of Theorem \[teoremone\] and Theorem \[cardinalita\]. In the second section we analyse the minimality of Lagrangian submanifolds, while in the last section we give the complete classification of simple Lie groups that admit an homogeneous Lagrangian submanifold $L$ in the complex projective space.\ [*Notations and conventions*]{}. Lie groups and their Lie algebras will be indicated with capital and gothic letters respectively. Moreover, after identifying, by means of a $Ad(K)$-invariant inner product $<,>$ on $\mathfrak k ^*,$ the Lie algebra $\mathfrak k$ and its dual ${\mathfrak k}^*$, we will alternatively consider $\mu$ as a $\mathfrak{k}$-valued map.\ [**Acknowledgements.**]{} The authors would like to thank Prof. F. Podestà for his constant support and his useful advices. Existence and Uniqueness {#mainsection} ======================== Let $M$ be a compact complex manifold with a Kähler form $\omega$ and $K$ be a compact group of isometries acting on $M$ in a Hamiltonian fashion. From now on we fix a moment map $\mu$, and focus on the set of points of $M$ sent by $\mu$ to $\mathfrak{z(k)}$, we will denote this set by $\mathcal Z$. The defining properties of $\mu$ imply that the $K$-orbits through points of $\mathcal Z$ are $\omega$-isotropic, indeed, for every $X,Y$ in $\mathfrak{k}$ and $q=kp$ in $K\cdot p$ $$\omega_q(\widehat{X}_q,\widehat{Y}_q)=d\mu_q(\widehat{X}_q)(Y)={\frac{d}{dt}_|}_{t=0} \exp t X k\cdot\mu(p)(Y)=0.$$ Since the $K$-action on $M$ is holomorphic, it induces, when $M$ is compact, an action of the complexified group $G:=K^{\mathbb{C}}$ on $M.$ With these notation we state \[lemma1\] Let $p$ be in $\mathcal Z$. Then the following statements are equivalent 1. the $K$-orbit through $p$ is Lagrangian; 2. the $G$-orbit $\Omega$ through $p$ is open in $M$, i.e. $M$ is a $G$-almost homogeneous space. In this case the $G$-orbit is a Stein manifold. Denote by $\mathcal O$ the $K$-orbit through $p$.\ $(i)\Rightarrow (ii)$ The tangent space to the $G$-orbit through $p$ is given by $$T_p (G\cdot p)=T_p \mathcal O+JT_p \mathcal O$$ and the sum must be direct since $JT_p \mathcal O\cap T_p \mathcal O=\{0\}$ because $\mathcal O$ is Lagrangian. Hence $\dim T_pG\cdot p=2\dim T_p \mathcal O =\dim M$ and the $G$-orbit is open.\ $(ii)\Rightarrow (i)$ Since the $G$-orbit is open, $\dim T_p \mathcal O+ \dim JT_p \mathcal O\geq 2n.$ Now the conclusion follows recalling that $\mathcal O$ is isotropic $(\mu(p)\in \mathfrak{z(k)})$, hence $\dim \mathcal O \leq n$.\ Let $H\leq K$ be the isotropy subgroup at $p$. When we consider the complexified action, the Lie algebra of the stabilizer, $\mathfrak{g}_p$, is given by the set of vectors $W=X+iY$ such that $\widehat W_p =0$. Now, recalling that $JT_p \mathcal O={(T_p \mathcal O )}^\perp$ we get that $\widehat {X}_p=\widehat {Y}_p=0$, therefore the complex isotropy is reductive and the open orbit $\Omega=\frac{K^{\mathbb{C}}}{H^{\mathbb{C}}}$ is Stein thanks to a theorem of Matsushima [@Ma]. An immediate consequence is the following \[Cod1\] The complement of $\Omega$ in $M$ has complex codimension $1$. Theorem \[teoremone\] proves that, by imposing an additional hypothesis on the cohomology of $M$, the existence of an open Stein $G$-orbit is indeed sufficient to guarantee the presence of a Lagrangian $K$-orbit, while in Theorem \[cardinalita\] we characterize the actions having infinitely many Lagrangian orbits. Now we recall two results that will be used in proving the theorems, the first one is due to Kirwan [@Kir] \[Kir\] Let $x$ and $y$ be two points in a Kähler manifold $M$, acted on in a Hamiltonian fashion by a group of isometries $K,$ such that $\mu(x)=\mu(y)=0$. Suppose that $x$ and $y$ lie in different $K$-orbits, then there exist two $K^{\mathbb{C}}$-invariant disjoint neighborhoods $U_x$ and $U_y$ of $x$ and $y$ respectively. The following is a classical result in Kähler geometry (see e.g. [@KoMo] for a proof), it is essentially a consequence of $\partial\overline\partial$-lemma, holding for compact Kähler manifolds. \[kodaira\] Let $L\rightarrow M$ be a line bundle on a compact Kähler manifold $M$. If $\omega$ is any real, closed $(1,1)$-form such that $[\omega]=c^{{\mathbb{R}}}_1(L)\in H_{dR}^2(M)$, then there exists a Hermitian metric along the fibers of $L$ whose curvature form is $\Theta=\frac{i}{2\pi}\omega$. Now we can prove Theorem \[teoremone\].\ Note that a compact Kähler manifold $M$ with $h^{1,1}(M)=1$ is necessarily projective. Indeed, since the Kähler form $\omega$ is of type $(1,1)$, we can scale it so that we obtain an integral class $[\widetilde{\omega}]\in H^2(M;{\mathbb{Z}})$ and use the Kodaira Embedding Theorem.\ Note that the hypotheses of Theorem \[teoremone\] are naturally satisfied when $M$ is a compact irreducible Hermitian symmetric space. We need only to prove that if $G=K^{{\mathbb{C}}}$ has an open Stein orbit, then there exists a Lagrangian $K$-orbit. Denote again by $\Omega=G\cdot p$ the open Stein orbit and by $Y$ its complement in $M$. By Corollary \[Cod1\], $Y$ is a divisor of $M$ and therefore it determines a holomorphic line bundle $L$ on $M$ and a section $\sigma\in H^0(L)$ such that $Y$ is the vanishing locus of $\sigma$. We scale $\omega$ so that we obtain a positive generator of the free part of $H^2(M;{\mathbb{Z}}) \cap H^{1,1}(M)$. Since $h^{1,1}(M)=1$ the first Chern class of $L$ is a positive integer multiple of the class of the scaled Kähler form $\omega$ on $M$: $$c_1(L)=m[\omega]\in H^{1,1}(M).$$ Now, by Proposition \[kodaira\], it is possible to find a Hermitian metric $h$ on the fibers of $L$ such that its curvature form is $$\Theta=m\frac{i}{2\pi}\omega.$$ On the other hand, the curvature on $\Omega$ is exactly (see e.g. [@KoMo]) $$\partial\overline\partial \log\|\sigma\|^2,$$ where $\|\cdot\|$ is the norm induced by $h$. Thus we have found a strictly plurisubharminic real valued function $\rho$ such that $\omega=i\partial\overline\partial \rho.$ Note that by construction $\rho:\Omega\rightarrow {\mathbb{R}}$ is an exhaustion function. Observe that we can assume $\rho$ to be $K$-invariant, since this can be achieved by averaging over the [*compact*]{} group $K$.\ Starting from $\rho$ we can define a map $\phi:\Omega\rightarrow\mathfrak{k}^*$ as follows $$\phi(p)(X):=\frac{1}{2}{(J\widehat X)}_p(\rho).$$ Clearly the $K$-invariance of $\rho$ implies the $K$-equivariance of $\phi$. Moreover, for every $p\in \Omega$, $v\in T_p\Omega$ and $X\in \mathfrak{k}$, we have that $d\phi_p(v)(X)=\omega_p(v,\widehat X_p)$ (see [@HHL] for the proof). Hence $\phi$ is a moment map for the Hamiltonian action of $K$ on $\Omega$ and therefore its extension to the whole $M$ $\phi$ differs from $\mu$ by an element $z$ of $\mathfrak{z(k)}$.\ Let now $x_o\in \Omega$ be a critical point of the exhaustion function $\rho$, then $d\rho_{x_o}=0$ and $\phi(x_o)(X)=\frac{1}{2}(J\widehat X_{x_o})(\rho)=0$ for all $X\in \mathfrak{k}$. Thus $\mu(x_o)$ belongs to the Lie algebra of the center of $K$, and the $K$-orbit through $x_o$ is Lagrangian by Lemma \[lemma1\]. If the assumption of the Hodge number $h^{1,1}(M)$ in Theorem \[teoremone\] is not satisfied we cannot reach the same conclusion. Indeed consider the example of ${{\rm SU}}(3)$ acting on ${\mathbb{P}}^2 \times {{\mathbb{P}}^2}$ as follows $$A\cdot([x],[y])=([Ax],[\bar{A}y])$$ with $A \in {{\rm SU}}(3)$ and $x,y \in {\mathbb{C}}^3\setminus \{0\}$. Since $h^{1,1}({\mathbb{P}}^2\times{\mathbb{P}}^2)=2$ we can choose an ${{\rm SU}}(3)$-invariant symplectic form $\omega_\varepsilon=\omega_0\oplus (1+\varepsilon) \omega_0$ on ${\mathbb{P}}^2\times{\mathbb{P}}^2$, where $\omega_0$ is the Fubini-Study $2$-form on ${\mathbb{P}}^2$ and $\varepsilon$ is a small positive constant. In this case, there exists an open Stein $G$-orbit (see e.g. [@Ak2]), while the image of the moment map does not contain $0$ (see also [@BG] for the picture of the moment polytope in this case). If the group $K^{\mathbb{C}}$ has an open Stein orbit in $M$ with $h^{1,1}(M)=1$, the same is true for $(K\cdot Z)^{\mathbb{C}}$, where $Z$ centralizes $K$. Indeed consider $p \in M$ such that $K\cdot p$ is Lagrangian (cfr. Theorem \[teoremone\]), then $\mu(p) \in \mathfrak{z(k)}$ where $\mu$ is a moment map for the $K$-action. On the other hand $\mu$ is the composition of the moment map $\mu'$ for the action of $K':=K\cdot Z$ with the projection induced by the inclusion on the dual of the Lie algebras. Therefore $\mu'(p) \in \mathfrak{z(k')}$ and for dimensional reasons $K'\cdot p$ is Lagrangian and the claim follows from lemma \[lemma1\]. Under the same assumptions of Theorem \[teoremone\] on $M$ we prove Theorem \[cardinalita\] In the semisimple case the moment map is unique, therefore, using the same notation as in Theorem \[teoremone\] we have $\mu(x_o)=0$ at the critical point $x_o$ of $\rho$, and $\mu^{-1}(0)\cap \Omega\neq\emptyset$. Take $x$ and $y$ in $\mu^{-1}(0)\cap \Omega$, applying Lemma \[Kir\], we deduce that $x$ and $y$ belong to the same $K$-orbit, and $\mu^{-1}(0)\cap \Omega$ is therefore compact. Then, since when $M$ is compact the fibers of the moment map are connected [@Kir], we get that $\mu^{-1}(0)$ is contained in $\Omega$ and it is a single $K$-orbit.\ If the semisimple part of $K$, that will be denoted by $K_s$, has a Lagrangian orbit $L$, then $K$ has a unique Lagrangian orbit. Indeed, combining Theorem \[teoremone\] and the previous remark, we get that there exists a Lagrangian $K$-orbit, this is contained in $\mu_s^{-1}(0)=L$, where $\mu_s$ is the moment map for the $K_s$ action, and it is therefore unique.\ Now assume that $K\cdot p$ is a Lagrangian orbit and denote by $H$ the connected component of the identity of the isotropy subgroup $K_p$. At the Lie algebra level $\mathfrak{k}$ can be written as the direct sum $\mathfrak{k}_s\oplus\mathfrak{z(k)}$. Consider the projection $\pi:\mathfrak{k}\rightarrow \mathfrak{z(k)}$. Suppose that $\pi(\mathfrak{h})\neq 0$, and consider $Z' \subset Z$ a subtorus such that its Lie algebra satisfies $$\mathfrak{z(k)}=\pi(\mathfrak{h})+\mathfrak{z'},$$ and call $K'$ the group $K_s\cdot Z'$. We first prove that $K'\cdot p$ has the same dimension of $K\cdot p$ and therefore $K'\cdot p$ is Lagrangian. The set of tangent vectors to the $K'$-orbit is given by $$\widehat {\mathfrak{k}'}_{|p}=\widehat {\mathfrak{k}_s}_{|p}+\widehat {\mathfrak{z}'}_{|p}$$ while the set of vectors tangent to the $K$-orbit is given by $$\widehat {\mathfrak{k}}_{|p}=\widehat {\mathfrak{k}_s}_{|p}+\widehat{ \mathfrak{z}'}_{|p}+\widehat {\pi(\mathfrak{h})}_{|p}.$$ By construction $\widehat {\pi(\mathfrak{h})}_{|p}$ is contained in $\widehat {\mathfrak{k}_s}_{|p}$ hence the $K-$ and $K'$-orbits through $p$ coincide.\ Now denote by $H'$ the group ${K'_p}^o$; the projection $\pi(\mathfrak{h'})$ is $\{0\}$, i.e. $H'\subseteq K_s$.\ We claim that for all subtori $Z''\leq Z'$ of codimension $1$ in $Z'$, the group $K''=K_s\cdot Z''$ has no Lagrangian orbits. This can be proven observing that $H'$ coincides with ${K''_p}^o$. Indeed $K''\cdot p\subset K'\cdot p$ and $\rm{codim}_{K'\cdot p}(K''\cdot p)=1$, hence $$\dim K'_p=\dim K'-\dim K'\cdot p=\dim K''-\dim K'\cdot p-1=\dim K''-\dim K''\cdot p=\dim K''_p$$ therefore $$\dim K''\cdot p< \dim K'\cdot p$$ and the $K''$-orbit is not Lagrangian.\ Denote by $\mu''$ the moment map associated to the $K''$-action on $M$. Consider the set $M^{{K'_p}^o}$, i.e the set $$\{x\in M|\;{H'}\cdot x=x\}.$$ We first prove that $\mu''(M^{{H'}})$ is contained in $\mathfrak{z}_{\mathfrak{m''}}(\mathfrak{h'})$, where $\mathfrak{k}''=\mathfrak{h'}\oplus \mathfrak{m''}$ and $\mathfrak{z}_{\mathfrak{m''}}(\mathfrak{h'})=\{X\in \mathfrak{m''}|\;[X,\mathfrak{h'}]=0\}.$ Clearly $\mu''(M^{{H'}})$ is contained in $\mathfrak{z}_{\mathfrak k''} (\mathfrak{h'})$. Moreover let $\gamma(t)$ be a smooth curve contained in $M^{{H'}}$ joining $p$ and a point $x\in M^{{H'}}$. We get $$\frac{d}{dt}<\mu''(\gamma(t)),\mathfrak{h'}>=<d\mu''_{\gamma(t)}(\gamma'(t)),\mathfrak{h'}> =\omega_{\gamma(t)}(\gamma'(t),\widehat{\mathfrak{h'}}{_{|_{\gamma(t)}}})\equiv0$$ where the last equality holds since $\widehat{\mathfrak{h'}}_{|\gamma(t)}= 0.$ Now recall that the orbit $K'\cdot p$ is Lagrangian hence $\mu''(p)=c\in \mathfrak{z(k'')}=\mathfrak{z''}$ which is orthogonal to $\mathfrak{h'}$. Therefore $\mu''(x)$ is orthogonal to $\mathfrak{h'}$ and belongs to $\mathfrak{m''}\cap \mathfrak{z}_{\mathfrak{k''}}(\mathfrak{h'})$=$\mathfrak{z}_{\mathfrak{m''}}(\mathfrak{h'})$ as claimed.\ Now the dimension of $M^{{H'}}$ is given by $$\dim M^{H'}=2\dim {(K'\cdot p)}^{H'}=2(\dim {(K''\cdot p)}^{H'}+1)= 2(\dim\mathfrak{z}_{\mathfrak{m''}}(\mathfrak{h'})+1).$$ This will be used in proving that $Q=\mu''^{-1}(c)\cap M^{{H'}}$ is a submanifold.\ Note that $${{\rm Ker}}\; d\mu''_p=\{Y\in T_p M|\; \omega(Y,\widehat{X}_p)=0 \; \mbox{for all} \; X\in \mathfrak{k''}\}={( J\widehat {\mathfrak{k''}}_{|p})}^\perp=T_p K'\cdot p\oplus V_1$$ where $V_1$ has dimension $1$ and is contained in ${T_p K'\cdot p}^\perp$, indeed $K'\cdot p$ is Lagrangian and $T_p M=T_p K'\cdot p\oplus J\widehat {\mathfrak{k''}}_{|_p}\oplus V_1$. Moreover $V_1$ is contained in $T_p M^{{H'}}$; indeed ${H'}$ acts by isotropy on $T_p M$ and leaves $\widehat {\mathfrak{k''}}_{|_p}$ invariant hence ${( J\widehat {\mathfrak{k''}}_{|_p})}^\perp$ invariant. Therefore $V_1$ is ${H'}$-invariant, hence fixed, since it is $1$ dimensional and ${H'}$ is compact.\ We conclude that $${{\rm Ker}}\;d\mu''_p\cap T_p (M^{H'})= {(T_p K'\cdot p)}^{{H'}} \oplus V_1$$ and $$\dim ({{{\rm Ker}}\; d\mu''_p\cap T_p (M^{H'})})=\dim \mathfrak{z}_{\mathfrak{m''}}(\mathfrak{h'})+2.$$ Counting the dimension of the image, it follows that ${\mu''}_{|M^{H'}}$ is a submersion at $p$, hence $Q$ is a manifold locally around $p$ whose dimension is $\dim \mathfrak{z}_{\mathfrak{m''}}(\mathfrak{h'})+2$. Note that for dimensional reasons $Q\setminus K'\cdot p\neq \emptyset$. To complete the proof it is sufficient to observe that if we take $y\in Q\setminus K'\cdot p$, sufficiently close to $p$, then the $K'$-orbit through $y$ is Lagrangian. Indeed $K'\cdot y$ is isotropic for $\mu''(y)\in \mathfrak{z''}$, and furthermore ${H'} \subseteq K'_y$; by the Slice Theorem ${K'_y}^o$ is conjugated to a subgroup of $H'$ hence $\dim K'_y \leq \dim H'$ so that $\dim K'\cdot y=\dim K' \cdot p$. The uniqueness of the Lagrangian $K$-orbit in the semisimple case is independent of the assumption $h^{1,1}(M)$.\ Note that whenever $\mu^{-1}(0)\cap \Omega\neq\emptyset$, we can argue that $\Omega$ coincides the set of semistable points $M^{ss}:=\{x\in M| \: \overline{G\cdot x}\cap \mu^{-1}(0)\neq\emptyset\}$. Indeed $\Omega$ is always contained in $M^{ss}$, moreover (see e.g. [@Sj]) $M^{ss}$ is the smallest $G$-invariant subset of $M$ that contains $\mu^{-1}(0)$, therefore $M^{ss}=\Omega$. In the Kähler case it is easy to see that the stratum associated to the minimum critical set of $\|\mu\|^2$ (see [@Kir] for the precise definition) coincides with the set of semistable points. Lerman in [@Le] shows that this stratum retracts to the zero set; we have thus proved the following If $\mu^{-1}(0)\cap \Omega$ is not empty, then $\Omega$ has $\mu^{-1}(0)$ as a deformation retract, and thus has the same homotopy type of the Lagrangian orbit. \[retraz\] Let $K\cdot p$ be a Lagrangian $K$-orbit. One can easily show that, if $Z\in \mathfrak{z(k)}$, then $K_p=K_{\exp iZp}$ and all the orbits through $\exp iZ p$ are totally real ([*i.e.*]{} at every point the tangent space is transversal to its image via the complex structure), but not in general Lagrangian. Consider for example the action on ${\mathbb{C}}{\mathbb{P}}^N$, with $N={\frac{n^2+3n}{2}-1}$ induced by the representation $\rho$ of $T^2\times SU(n)$ on $V=S^2({\mathbb{C}}^n)\oplus {\mathbb{C}}^n$, defined by $\rho(g)(X,Y)=(\alpha AXA^t,\beta A y)$ for $g=(\alpha,\beta,A)\in T^2\times {{\rm SU}}(n)$, where we see the elements of $S^2{\mathbb{C}}^n$ as symmetric matrices. Here there are more than one $K$-Lagrangian orbit but, moving through points $\exp i tZ p$, one does not meet any (other) Lagrangian orbit. Azad, Loeb and Qureshi in [@ALQ] give necessary and sufficient conditions under which one can prove that there are infinitely many totally real orbits; more precisely this is the case whenever $N_G(G_p)/G_p$ is not finite. In the non semisimple case this condition is always satisfied. Whenever the isotropy of a Lagrangian $K$-orbit is discrete, the set of Lagrangian orbits is a manifold whose dimension equals the dimension of the center of the group [@Pa]. This situation holds whenever there exists a regular (i.e. principal or exceptional) Lagrangian $K$-orbit. Nevertheless note that if $p$ belongs to the set $$M_{\mu}\cap M_{princ}$$ where $M_{\mu}$ is the set of points $x$ in $M$ whose orbits $K\cdot \mu(x)$ has maximal dimension and $M_{princ}$ the set of principal points in $M$, and the $K$-orbit through $p$ is Lagrangian, then necessarily $K$ must be abelian. Indeed, in general when $p\in M_{\mu}\cap M_{princ}$, $K_{\mu(p)}/K_p$ is abelian (see e.g.[@HW]); since in this case $K\cdot p$ is principal and Lagrangian $K_p$ is trivial, $K_{\mu(p)}$ is abelian, but $\mu(p)\in \mathfrak{z(k)}$, hence $K_{\mu(p)}=K$ and the claim follows. As a consequence of Theorem \[teoremone\] we have incidentally proved that linear representations of complex semisimple Lie groups are [*balanced*]{}, in the sense of [@Wi], if and only if they have an open Stein orbit. Minimality of Lagrangian orbits =============================== We here give a proof of the minimality of the Lagrangian orbit in the semisimple case, however this can be proved also as a consequence of the more general fact stated in Proposition \[minimale\]. If $K$ is semisimple and $M$ is Kähler Einstein, the $K$-orbit is also minimal. If $H$ denotes the mean curvature vector of the Lagrangian orbit $\mathcal{O}$, it is known (see Dazord [@Da]) that the $1$-form $\alpha\in\Lambda^1(\mathcal O)$ which is the $\omega$-dual of $H$ restricted to $\mathcal O$ is closed. But $\alpha$ is $K$-invariant, hence for every $X,Y\in \mathfrak{k}$ we have $$0=d\alpha(\widehat X,\widehat Y)=\widehat X\alpha(\widehat Y)-\widehat Y \alpha (\widehat X)-\alpha([\widehat X,\widehat Y])=-\alpha([\widehat X,\widehat Y]),$$ so that $\alpha([\widehat{\mathfrak{k}},\widehat{\mathfrak{k}}])=\alpha(\widehat{\mathfrak{k}})=0$ and $\alpha\equiv 0$. This means $H=0$. Actually one can characterize the minimal Lagrangian orbit $L$ in the general case.\ When $(M,\omega)$ is compact we can define a [*canonical moment map*]{}, $\widetilde{\mu}$, that is characterized by the fact that $\int_M \mu \omega^n=0$. If further $M$ is Kähler-Einstein with Einstein constant $c$, then $\widetilde{\mu}$ can be explicitly written (see e.g. [@Fu], [@Po]): $$\widetilde{\mu}(p)(Y):=\frac{1}{2c}\mbox{div} (J\widehat{Y}_p)$$ for every $Y\in \mathfrak{k}.$ \[minimale\] Let $\widetilde\mu$ be the canonical moment map of a Kähler-Einstein manifold, then a Lagrangian orbit $\mathcal O$ is minimal if and only if $\widetilde\mu(\mathcal O)=0.$ The previous result is stated and proved in [@Pa] assuming that the Lagrangian orbit is [*principal*]{}. Actually Proposition \[minimale\] holds without any assumption on the type of Lagrangian orbits. Indeed, since $L$ is Lagrangian, in order to prove that $L$ is minimal, it is sufficient to show that the mean curvature vector $H$ at some point $p$ of $L$ is orthogonal to $J\widehat{\mathfrak{k}}_p$ as done in [@Pa] in Proposition $5$. Once an orthonormal frame $\{e_i\}$ at $p$ is fixed, we have $$\begin{aligned} \langle H, J\widehat{Y} \rangle & = & \langle \nabla_{e_i}e_i,J\widehat{Y}\rangle\\ & = & e_i \sum\langle e_i , J\widehat{Y} \rangle - \sum\langle e_i,\nabla_{e_i}J\widehat{Y} \rangle\\ & = & -\sum \langle e_i,\nabla_{e_i}J\widehat{Y} \rangle\\ & = & -\frac{1}{2} \mbox{div}J\widehat{Y}\\ & = & c\widetilde{\mu}_p(Y) = 0.\end{aligned}$$ Combining the previous proposition and the fact that the zero level set of the moment map is a single orbit when it meets the open Stein $K^{\mathbb{C}}$-orbit (see proof of Theorem \[cardinalita\]), we get \[unica\] Let $K$ be a compact connected group of isometries acting in a Hamiltonian fashion on a compact Kähler-Einstein manifold $M$. Then $M$ admits at most one minimal Lagrangian $K$-orbit. Under the same hypotheses of Theorem \[teoremone\], assuming further that $K^{\mathbb{C}}$ is simply connected, $M$ is Kähler-Einstein and the isotropy subgroup at a point of the Stein orbit has finite connected components we get that $M$ admits a unique $K$-orbit wich turns out to be minimal. From Theorem \[teoremone\] we get that there is a Lagrangian $K$-orbit $L$; moreover any other Lagrangian $K$-orbit has the same homotopy type of $L$ by Proposition \[retraz\] and therefore has finite fundamental group. But, according to Chen (see Theorem 5.1 in [@Chen] and the reference therein), in a Kähler-Einstein manifold, the mean curvature of every compact Lagrangian submanifolds with $b_1=0$ must vanish somewhere. The homogeneity implies that all the Lagrangian orbits are minimal. The conclusion follows from Corollary \[unica\]. Obviously the same result holds if $K^{\mathbb{C}}$ is only supposed to have finite fundamental group. The classification of Simple Lie groups with a Lagrangian orbit in the complex projective space =============================================================================================== In this section we give the complete classification of simple compact Lie groups $K$ with a Lagrangian orbit in the complex projective space. We give also an explicit description of Lagrangian orbits, except in case $K=E_7$. This part can be treated combining the results of section \[mainsection\] with the work of Sato and Kimura [@SK] and Kimura [@Ki].\ Consider a finite-dimensional unitary representation of a compact Lie group $K$ on a Hermitian vector space $(V,\langle,\rangle)$. Endow ${\mathbb{P}}(V)$ with the Fubini-Study Kähler form and consider the induced $K$-action. Note that this action is automatically Hamiltonian since ${\mathbb{P}}(V)$ is simply connected. The map $\mu: {\mathbb{P}}(V) \to \mathfrak{k}^*$ defined for every $v \in V$ and $X \in \mathfrak{k}$ by $$\label{mmproj} \mu([v])(X)=\frac{1}{i}\frac{\langle X\cdot v,v\rangle}{\langle v, v \rangle}$$ is a moment map for the $K$-action on ${\mathbb{P}}(V)$.\ Here we recall notations and results from [@SK]. Given a connected complex linear algebraic group $G$, and a rational representation $\rho$ of $G$ on a finite dimensional complex vector space $V$, a triplet $(G,\rho,V)$ is [*prehomogeneous*]{} if $V$ has a Zariski dense $G$-orbit.\ We give here an easy-to-prove lemma that allows to find relations between almost homogeneous actions on the projective space and prehomogeneus triplets. \[sato\] Let $G$ be any complex, connected Lie group. $G$ acts with an open dense orbit on ${\mathbb{C}}{\mathbb{P}}^{n-1}$ if and only if $G \times GL(1)$ acts with an open dense orbit on ${\mathbb{C}}^n$ i.e. $(G\times {{\rm GL}}(1),\rho,\mathbb{C}^n)$ is a prehomogeneus triplet. Hence, thanks to Theorem \[teoremone\], in order to classify the action of compact [*simple*]{} Lie groups on the projective space admitting a Lagrangian orbit, it is sufficient to go through the list of prehomogeneous triplets in [@SK], and consider those that have reductive generic isotropy, i.e. those that have an open Stein $G$-orbit. They are exactly [*regular*]{} PV spaces of [@SK] (p. 59). These spaces are characterized by the existence of a [*relative invariant*]{}, i.e. a rational function $f$ such that there exists a rational character $\chi$ of $G$ satisfying $f(\rho(g)x)=\chi(g)f(x)$ for any $g\in G$ and $x\in V$. We here enclose a lemma that will be useful in the sequel; the proof can be found in [@SK] p.64 . If $\rho$ is an irreducible representation, then the polynomial $f$ that defines the hypersurface $Y$ is irreducible. In [@SK] prehomogeneous vector spaces are classified up to an equivalence relation which we are going to describe. Two triplets $(G,\rho,V)$ and $(G',\rho',V')$ are called [*equivalent*]{} if there exist a rational isomorphism $\sigma:\rho(G)\to\rho'(G')$ and an isomorphism $\tau:V\to V'$, both defined over ${\mathbb{C}}$ such that the diagram $$\xymatrix{ V\ar[r]^-\tau \ar[d]_-{\rho(g)} & V' \ar[d]^-{\sigma(\rho(g))} \\ V\ar[r]^-\tau & V'}$$ is commutative for all $g\in G$. This equivalence relation will be denoted by $(G,\rho,V)\cong (G',\rho',V')$. We say that two triplets $(G,\rho,V)$ and $(G',\rho',V')$ are [*castling transforms*]{} of each other when there exist a triplet $(\tilde{G},\tilde{\rho},V(m))$ and a positive number $n$ with $m>n\geq 1$ such that $$(G,\rho,V)\cong (\tilde{G}\times {{\rm SL}}(n) ,\tilde{\rho}\otimes \Lambda_1,V(m)\otimes V(n))$$ and $$(G',\rho',V')\cong (\tilde{G}\times {{\rm SL}}(m-n) ,{\tilde{\rho}}^*\otimes \Lambda_1,V(m)^*\otimes V(m-n)),$$ where ${\tilde{\rho}}^*$ is the dual representation of ${\tilde{\rho}}$ on the dual vector space $V(m)^*$ of $V(m)$. We recall that $V(n)$ is a complex vector space of dimension $n.$ A triplet $(G,\rho,V)$ is called [*reduced*]{} if there is no castling transform $(G',\rho',V')$ with $\dim V'<\dim V.$\ Note that in fact in each class there is only one representative of the form $G \times {{\rm GL}}(1)$ where $G$ is simple and it is necessarily reduced. This can be seen [*a posteriori*]{} as follows. Suppose that $(G',\rho',V')$ is a reduced and castling equivalent to $(G \times {{\rm GL}}(1),\rho,V)$, then there should exist a representation $\widetilde{\rho}: \widetilde{G} \to {{\rm GL}}(V(m))$ such that $(G\times{{\rm GL}}(1),\rho,V)\cong (\tilde{G}\times {{\rm SL}}(n) ,\tilde{\rho}\otimes \Lambda_1,V(m)\otimes V(n))$. But now we would have (at least locally) $G\simeq{{\rm SL}}(n)$ and ${{\rm GL}}(1)=\widetilde{G}$ since $G$ is simple, hence $G'={{\rm GL}}(1)\times{{\rm SL}}(m-n)$, but the correspondent triple does not appear in the list of [@SK] (p. 144–146). Stabilizer and fundamental group -------------------------------- We here collect some results and remarks that will be used in order to single out Lagrangian homogeneous submanifolds in the complex projective space.\ Assume that a complex Lie group $G=K^{\mathbb{C}}$ acts with an open Stein orbit $\Omega={\mathbb{P}}^n\setminus Y$ on ${\mathbb{P}}^n$. Denote by $L$ the Lagrangian $K$-orbit. Thanks to Proposition \[retraz\], we get that $\Omega$ has the same homotopy type of $L$. We give here a well known result on the topology of the complement of an algebraic hypersurface $Y$ in ${\mathbb{P}}^n$ (see e.g.[@Li]): Let $Y$ be an algebraic hypersurface of ${\mathbb{P}}^n$. If its irreducicible components $Y_1,\ldots,Y_r$ have degree $d_1,\ldots,d_r$ respectively, then $H_1({\mathbb{P}}^n\setminus Y;{\mathbb{Z}})={\mathbb{Z}}^r/(d_1,\ldots,d_r)$. From the previous proposition it follows that, if $Y$ is irreducible of degree $d>1$, then $H_1({\mathbb{P}}^n\setminus Y;{\mathbb{Z}})$ is cyclic of order $d$. The open Stein orbit ${\mathbb{P}}^n\setminus Y$ contains the Lagrangian orbit $L=K/K_p$ and retracts onto it. From the homotopy sequence, whenever $K$ is simply connected $\pi_1(K/K_p)\simeq K_p/K_p^o.$ Hence we get a method in order to determine the number of connected components of the stabilizer $K_p$. If $N_K(K_p^o)/K_p^o$ is abelian then $K_p/K_p^o={\mathbb{Z}}_d$. Indeed $K_p/K_p^o\subset N_K(K_p^o)/K_p^o$ is abelian, hence $$K_p/K_p^o=\pi_1(K/K_p)=H_1(K/K_p)={\mathbb{Z}}_d.$$ ([*Self-dual representations*]{})\[selfdual\] Let $V$ be a $(N+1)$-dimensional complex self-dual representation of a compact Lie group $K$ and $\mu$ be the corresponding moment map. Assume that $G=K^{\mathbb{C}}$ has an open Stein orbit $\Omega=G/H$ in ${\mathbb{P}}^N$. Assume also that the highest weight $\lambda$ of the representation satisfies $2\lambda \notin R^+$. Denote by ${\mathcal{P}}=-{\mathcal{P}}$ the set of weights. If $v_{\pm 1}\in V_{\pm \lambda}$ are two non zero vectors with the same norm, then $[v]:=[v_1+v_{-1}] \in {\mathbb{P}}^N$ is a point in $\mu^{-1}(0)$ (see [@DK]). If moreover $2\lambda \notin R^++R^+$, then $$(\mathfrak{k}_{[v]})^{\mathbb{C}}=\ker \lambda \oplus_{\pm\alpha\in A_\lambda}\mathfrak{k}_\alpha,$$ where $A_\lambda=\{\alpha \in R^+: -\lambda +\alpha \notin {\mathcal{P}}\}=\{\alpha \in R^+: \langle\lambda,\alpha\rangle=0\}$. Indeed $$X=H+\sum_{\alpha \in {\mathbb{R}}^+}c_\alpha E_\alpha+\sum_{\alpha \in {\mathbb{R}}^-}d_\alpha E_\alpha$$ belongs to $(\mathfrak{k}_{[v]})^{\mathbb{C}}$ if and only if $$X\cdot v =\lambda(H)(v_1-v_{-1})+\sum_{\alpha\in R^+}c_\alpha E_\alpha v_{-1}+\sum_{\alpha\in R^-}d_\alpha E_\alpha v_1=c\cdot v$$ and the conclusion follows from the fact that the weight spaces $V_{-\lambda+\alpha}$ and $V_{\lambda-\beta}$ are distinct for $\alpha, \beta \in R^+$. If $\Omega=K^{\mathbb{C}}/H^{\mathbb{C}}$ is the open Stein orbit, then there exists $p\in \Omega$ such that $K_p=H$. Now, by the $K$-equivariance of $\mu$, $H=K_p\subseteq K_{\mu(p)}$ which is the centralizer of a torus $T$ in $K$.\ In some situation the only centralizer of a torus which contains $H$ is the whole group $K$. In this case we have $K_{\mu(p)}=K$ and we can conclude that $\mu(p)=0$ if $K$ is semisimple. The case-by-case classification ------------------------------- In what follows a compact Lie group $K$ acts on the complex finite-dimensional vector space $V$ by a linear representation $\rho$. Moreover we will identify the fundamental highest weights $\Lambda_l$ with the corresponding irreducible representations. 1. \[2L1\] $K={{\rm SU}}(n),\rho=2\Lambda_1$. Identify the representation space $V$ with the set of symmetric $n$ by $n$ complex matrices. Now the Hermitian product on $V$ preserved by $K$ is explicitly given by $\langle A, B\rangle={{\rm tr}}(A\overline{B})$ and we get immediately $\mu(I_n)=0$. Moreover if $Q$ is the $n$ by $n$ matrix $\mbox{diag}(-1,1,\dots,1)$, the stabilizer at $I_n$ is $$\{\alpha\cdot {{\rm SO}}(n): \alpha^n=1\} \cup \{\alpha Q\cdot {{\rm SO}}(n):\alpha^n=-1\}$$ Therefore the $K$-orbit through $I_n$ is Lagrangian in ${\mathbb{P}}(V)$ and $K_{I_n}/K_{I_n}^o\simeq Z_{n}$. Indeed it is generated by $e^{i\frac{\pi}{n}}$ if $n$ is even, and by $e^{i\frac{2\pi}{n}}$ if $n$ is odd. 2. [*$K={{\rm SU}}(n)$, $\rho=\Lambda_1\oplus \Lambda_1^*$*]{}. Identify $V$ with ${\mathbb{C}}^n\oplus {{\mathbb{C}}^n}^*$. Take $p=(e_1,e_1^*)$. A direct calculation shows that $\mu(p)=0$. The real isotropy is ${{\rm SU}}(n-1)\cdot {\mathbb{Z}}_2$. 3. [*$K={{\rm SU}}(n)$, $\rho=\Lambda_1\oplus\cdots\oplus\Lambda_1$ $n$ times*]{}. Identify $V$ with ${\mathbb{C}}^n\oplus\cdots\oplus{\mathbb{C}}^n$. Take $p=(e_1,e_2,\ldots,e_n)$. A slightly more complicated calculation shows that $\mu(p)=0$. The complex isotropy of $p$ is discrete while the real one is ${\mathbb{Z}}_n$. 4. $K={{\rm SU}}(2n)$, $\rho=\Lambda_2$. Identify the representation space $V$ with the set of anti-symmetric $2n$ by $2n$ complex matrices. The argument of case \[2L1\] applies to $p=J_n=\left[ \begin{array}{cc} 0 & -I_n \\ I_n & 0 \end{array} \right]$. The real stabilizer is $$\{\omega\cdot {{\rm Sp}}(n): \omega^{4n}=1\}.$$ Since $-I_{2n}\in {{\rm Sp}}(n)$ we have $K_{J_n}={{\rm Sp}}(n)\cdot{\mathbb{Z}}_{2n}$ and the $K$-orbit through $J_n$ is Lagrangian. 5. [*$K={{\rm SU}}(2n+1)$, $\rho=\Lambda_2\oplus\Lambda_1$*]{}. Identify the $\Lambda_2$ part of $V$ with anti-symmetric complex matrices and take $p=(\widetilde{J}_n,e_1)$ where $\widetilde{J}_n=\left[ \begin{array}{cc} 1 & 0 \\ 0 & J_n \end{array} \right]$. Again, if $\mu$ is the moment map associated to the hermitian metric $h((X,v),(Y,w))={\mbox{Tr}}(^t\!X \overline{Y})+2 ^t\!v\overline{w}$, a straightforward computation proves that $\mu(p)=0$, and the real isotropy at $p$ is ${{\rm Sp}}(n){\mathbb{Z}}_{n+1}$. 6. $K={{\rm SU}}(2)$, $\rho=3\Lambda_1$. This case has also been treated in [@Ch]. The representation is self-dual, hence we apply remark 7. Here $\lambda=3\epsilon_1$ and the set of simple roots $R=\{\pm\alpha\}$ with $\alpha=\epsilon_1-\epsilon_2.$ Hence $\mathcal P$ is the set $\{\lambda,\lambda-\alpha,\lambda-2\alpha,\lambda-3\alpha\}$ and $\mathfrak{k}_{[v]}=\{0\}$. Explicitly, identifying the representation space with the space of complex homogeneous polynomial of degree 3, we can take $[v]=z_1^3+z_2^3$ and $K_{[v]}$ is a non-abelian group of order 12 whose abelianization is isomorphic to ${\mathbb{Z}}_4$. More precisely $K_{[v]}$ is isomorphic to the unique non-trivial semidirect product ${\mathbb{Z}}_3\rtimes {\mathbb{Z}}_4$ in which ${\mathbb{Z}}_3$ is normal. 7. $K={{\rm SU}}(6)$ $\rho=\Lambda_3$. The representation is again self-dual, here $\lambda=\epsilon_1+\epsilon_2+\epsilon_3$ and $\mathcal P=\{\epsilon_i+\epsilon_j+\epsilon_k; i<j<k\}$ and $A_\lambda=\{\epsilon_i-\epsilon_j; i<j<3\}\cup\{\epsilon_i-\epsilon_j;4\leq i<j\}$ hence $\mathfrak{k}_{[v]}=\mathfrak{su}(3)\oplus\mathfrak{su}(3)$. Explicitly $[v]=[e_1\wedge e_2\wedge e_3+e_4\wedge e_5\wedge e_6]$ and $K_{[v]}$ has four connected components given by $$\left \{ \left[ \begin{array}{cc} A & 0 \\ 0 & D \end{array} \right]: A,D\in {{\rm SU}}(3) \right \} \cup \left \{ \left[ \begin{array}{cc} A & 0 \\ 0 & D \end{array} \right]: A,D\in {{\rm U}}(3), \det A=\det D=-1\right\}$$ $$\cup \left \{ \left[ \begin{array}{cc} 0 & B \\ C & 0 \end{array} \right]: \det B=\det C=i \right \}\cup\left \{ \left[ \begin{array}{cc} 0 & B \\ C & 0 \end{array} \right]: \det B=\det C=-i \right \}$$ Hence the fundamental group of the Lagrangian orbit has order $4$. But,since $H_1(L,{\mathbb{Z}})$ is equal to ${\mathbb{Z}}_4$ (indeed the invariant has degree $4$ [@SK] (p.144)), $\pi_1(L)={\mathbb{Z}}_4$. 8. \[SU(7)\] $K={{\rm SU}}(7)$, $\rho=\Lambda_3$. Take $p$ such that $K\cdot p$ is the Lagrangian $K$-orbit in ${\mathbb{P}}(V)$. By [@SK] (p. 144) we know that $K_p^o={{\rm G}}_2$. Let $g\in N_K({{\rm G}}_2)$, then $g$ induces an automorphism of the Lie algebra $\mathfrak{g}_2$ which is necessarily inner, since $\mathfrak{g}_2$ has only inner automorphisms. Therefore there exists $h\in {{\rm G}}_2$ such that $gh$ induces the identity on $\mathfrak{g}_2$, i.e. centralizes ${{\rm G}}_2$. Now, recalling that ${{\rm G}}_2$ acts irreducibly on ${\mathbb{C}}^7$, we get that $gh$ is a scalar multiple of the identity and $N_K({{\rm G}}_2)\subset {{\rm G}}_2\cdot {\mathbb{Z}}_7,$ where ${\mathbb{Z}}_7$ is the center of ${{\rm SU}}(7)$, and $K_p={{\rm G}}_2\cdot {\mathbb{Z}}_7$. 9. $K={{\rm SU}}(8)$, $\rho=\Lambda_3$. In this case, if $p$ is such that $K\cdot p$ is the Lagrangian $K$-orbit in ${\mathbb{P}}(V)$, following the explicit calculations in [@SK] (p.87–90), we know that $K_p^o$ is the image in ${{\rm SU}}(8)$ of ${{\rm SU}}(3)$ via the map $Ad^{{\mathbb{C}}}:{{\rm SU}}(3)\rightarrow {{\rm Aut}}(\mathfrak{sl}(3,{\mathbb{C}}))$, hence $K_p^o\simeq {{\rm SU}}(3)/{\mathbb{Z}}_3$. We claim that the cardinality of $N_K(K_p^o)/K_p^o$ is not greater than $16$, therefore the cardinality of $H_1(K/K_p,{\mathbb{Z}})$ cannot be greater than $16$, while from [@SK] we know that its cardinality is exactly $16$. Recall that every automorphism of $\mathfrak{su}(3)$ is given by the composition of an inner and an outer (the conjugation $\sigma$) automorphism; let $g$ be in $N_K(K_p^o)$ and $\phi_g$ the induced automorphism on $K_p^o$. Then two possibilities arise. In the first case there exists $h\in K_p^o$ such that $\phi_g=\phi_h$, in other words $gh^{-1}$ commutes with $K_p^o$, which acts irreducibly on ${\mathbb{C}}^8$, hence, by the Schur Lemma, it is a scalar multiple of the identity, i.e. an element of the center ${\mathbb{Z}}_8$ of ${{\rm SU}}(8)$. Otherwise there exists $h\in K_p^o$ such that $\phi_g=\phi_h\circ \sigma$; in this case put $g_o=h^{-1}\circ g$. Therefore $N_K(K_p^o)=K_p^o\cdot({\mathbb{Z}}_8\cup g_o{\mathbb{Z}}_8)$, and has at most order $16$. Now, since $K_p^o$ has no center, we conclude that $K_p=K_p^0\cdot{\mathbb{Z}}_{16}$. 10. $K={{\rm Sp}}(n)$, $\rho=\Lambda_1\oplus\Lambda_1$. Identify $V$ with ${\mathbb{C}}^n\oplus {\mathbb{C}}^n$. Take $p=(e_1,e_2)$, $\mu(p)=0$. The complex isotropy at $p$ is locally isomorphic to ${{\rm Sp}}(n-1,{\mathbb{C}})$ while the real isotropy is ${{\rm Sp}}(n-1)\cdot {\mathbb{Z}}_2$. 11. $K={{\rm Sp}}(3)$, $\rho=\Lambda_3$. The action is the restriction of the ${{\rm SU}}(6)$ action on the same space. Therefore the stabilizer is given by the intersection of ${{\rm Sp}}(3)$ with the stabilizer obtained in $(7)$. Hence $K_{[v]}$ is $$\left \{ \left[ \begin{array}{cc} A & 0 \\ 0 & \overline{A} \end{array} \right]: A\in {{\rm SU}}(3) \det A=1\right \} \cup \left \{ \left[ \begin{array}{cc} A & 0 \\ 0 & \overline{A} \end{array} \right]: A\in {{\rm U}}(3), \det A=-1\right\} \cup$$ $$\left \{ \left[ \begin{array}{cc} 0 & B \\ -\overline {B} & 0 \end{array} \right]: B\in {{\rm U}}(3); \det B=i \right \}\cup\left \{ \left[ \begin{array}{cc} 0 & B \\ -\overline{B} & 0 \end{array} \right]: B\in {{\rm U}}(3); \det B=-i \right \}$$ And we conclude as in $(7)$. 12. [*$K={{\rm SO}}(n)$, $\rho=\Lambda_1$*]{}. The representation $\rho$ is self-dual, nevertheless it is easier to see that $\mu(p)=0$, where $p=[1:0:\dots:0]$. and $K_p={{\rm SO}}(n-1)\cdot{\mathbb{Z}}_2$. 13. [*$K={{\rm Spin}}(7)$, $\rho=\mbox{spin rep.}$*]{} \[Spin(7)\] The orbits of ${{\rm Spin}}(7)$ are the same of ${{\rm SO}}(8)$ (see the previous case), therefore the Lagrangian orbit is $$\frac{{{\rm Spin}}(7)}{{{\rm G}}_2\cdot{\mathbb{Z}}_2}=\frac{{{\rm SO}}(8)}{{{\rm SO}}(7)\cdot{\mathbb{Z}}_2}={\mathbb{R}}{\mathbb{P}}^7.$$ 14. [*$K={{\rm Spin}}(9)$, $\rho=\mbox{spin rep.}$*]{} The case is completely analogous to the previous one considering the inclusion ${{\rm Spin}}(9) \subset {{\rm SO}}(16)$. Thus the Lagrangian orbit is $$\frac{{{\rm Spin}}(9)}{{{\rm Spin}}(7)\cdot{\mathbb{Z}}_2}=\frac{{{\rm SO}}(16)}{{{\rm SO}}(15)\cdot{\mathbb{Z}}_2}={\mathbb{R}}{\mathbb{P}}^{15}.$$ 15. $K={{\rm Spin}}(10)$, $\rho=\Lambda_e\oplus\Lambda_e$ where $\Lambda_e$ is the even half-spin representation. The complex isotropy through the point $p=(1+e_{1234},e_{15}+e_{2345})$ is locally isomorphic to $G_2^{\mathbb{C}}$ (see [@SK] also for notations and conventions on the spin representation space). Moreover a direct computation using formula \[mmproj\] shows that $\mu(p)=0$, thus $p$ belongs to a Lagrangian orbit. 16. [*$K={{\rm Spin}}(11)$, $\rho=\mbox{spin rep.}$*]{} This case and the next one (to which we refer) can be treated simultaneously since ${{\rm Spin}}(11)$ and ${{\rm Spin}}(12)$ have the same orbits on ${\mathbb{P}}^{31}$. This can be easily seen noting that ${{\rm Spin}}(11) \subset {{\rm Spin}}(12)$ and computing the cohomogeneity of these actions. In the case of ${{\rm Spin}}(11)$ the isotropy of the Lagrangian orbit is locally isomorphic to ${{\rm SU}}(5)$. 17. [*$K={{\rm Spin}}(12)$, $\rho=\Lambda_e$*]{} The computation of the fundamental group of the Lagrangian orbit is done by several steps.\ [*Step 1*]{}. The representation $\rho$ is of quaternionic type, so it preserves a quaternionic structure $J \in \mbox{End}({\mathbb{C}}^{32})$, such that $J^2=-\mbox{id}$. Denote by $\lambda$ the maximal weight of $\rho$ and by $T$ a fixed maximal torus of $K$. Note first that the Weyl group $W_{{{\rm Spin}}(12)}$ contains $-1$. Let $w\in N_{{{\rm Spin}}(12)}(T)$ induce $-1$ on $\mathfrak{t}$. Since $w(\lambda)=-\lambda$, $w$ preserves also $\lambda^\perp$, therefore $w\in N_K({{\rm SU}}(6))\subset N_K({{\rm U}}(6))$. On the other hand $w$ cannot lie in ${{\rm U}}(6)$ because otherwise $w$ should belong to $W_{{{\rm SU}}(6)}$ but ${-1}\notin W_{{{\rm SU}}(6)}$. Hence $w$ generates $N({{\rm U}}(6)/{{\rm U}}(6)$, and by [@BR] we know that $N_{{{\rm SO}}(12)}({{\rm U}}(6)/{{\rm U}}(6)\simeq {\mathbb{Z}}_2$.\ [*Step 2*]{}. Take $p\in {\mathbb{P}}^{31}$ with $K\cdot p$ Lagrangian and $K_p^o$ locally isomorphic to ${{\rm SU}}(6)$. Since $\rho$ is self dual, Remark \[selfdual\] implies that $p=u_1+u_{-1}$ with $u_{\pm 1} \in V_{\pm \lambda}$ and $\|u_1\|=\|u_{-1}\|$. Now $K_p\subset N_{{{\rm SO}}(12)}({{\rm SU}}(6))\subset N_{{{\rm SO}}(12)}({{\rm U}}(6))$, hence if $k\in K_p$ then $k\in w^i{{\rm U}}(6)=w^i T^1\cdot {{\rm SU}}(6)$ for $i=0,1$, where $T'$ is the center of ${{\rm U}}(6)$.\ [*Step 3*]{}. Let $v_1\in V_{\lambda}$ be fixed and take $v_2=Jv_1\in V_{-\lambda}$. It is possible to choose $x\in{{\rm U}}(1)$ such that $p=v_1+xv_2$ and $w\cdot p\in {\mathbb{C}}\cdot p$, and the $K$-orbit through $p$ is Lagrangian.\ [*Step 4*]{}. Let $T^1\in{{\rm U}}(6)$ be the center of ${{\rm U}}(6)$. We consider the homomorphism $c:T^1\to {{\rm U}}(1)$ such that, for every $t\in T^1$, $t\cdot v_1=c(t)\cdot v_1$, with $c(t)\neq 1$. By easy computations we get that $t\cdot v_2=\overline{c(t)} v_2$. Let $k\in K_p\subset w^i \cdot T^1\cdot {{\rm SU}}(6)=T^1\cdot w^i {{\rm SU}}(6)$, since both $w$ and ${{\rm SU}}(6)$ fix $[p]$, then $k\in H\cdot w^i \cdot {{\rm SU}}(6)$ where $H:=\{t\in T^1, t[p]=[p]\}$. Now $t\cdot p\in {\mathbb{C}}\cdot p$ if and only if $c(t)=\pm 1$ i.e. $H={{\rm Ker}}(c^2)\subset T^1$ is cyclic.\ [*Step 5*]{}. Recall that $w^2\in T$, hence it commutes with $H$. In ${{\rm SO}}(12)$, $w$ can be taken as ${{\rm diag}}(B,B,\ldots,B)$ where $B= \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]$; then $w\in {{\rm Spin}}(12)$, taken in $\pi^{-1}(w_{{{\rm SO}}(12))}$, is such that $w^2\in \pi^{-1}(e)\simeq {\mathbb{Z}}_2$ i.e. $w^4=id$.\ [*Step 6*]{}. Now we determine $H$. If $uJ$, with $u\in i{\mathbb{R}}$, is a generic element of $\mathfrak{t}^1$, and $J:={{\rm diag}}(A,A,\ldots,A)$ where $A= \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right]$ we have $H=\{uJ| \exp (uJ)v_1=\pm v_1\}$, and recalling that $\lambda=\frac{1}{2}(\omega_1+\omega_2+\cdots+\omega_6)$, where $\omega_i$ are the fundamental weights, we get $$H=\{uJ|u=\frac{\pi}{3}i\cdot k, k\in{\mathbb{Z}}\}.$$ Obviously $H\cap {{\rm SU}}(6)={\mathbb{Z}}_6$ therefore $[k]\in K_p/K_p^o$ is generated by $[w]$ and by $\alpha$, where $\alpha$ is a non trivial element of $H/H\cap {{\rm SU}}(6)\simeq {\mathbb{Z}}_2$. Now $w^2\in H$, thus $[w]^2$ equals $\alpha$ or $1$ and $|K_p/K_p^o|\leq 4$. The claim follows from the fact that the invariant has degree $d=4$. 18. [*$K={{\rm Spin}}(14)$, $\rho=\Lambda_e$*]{} Let $p$ be such that $K\cdot p$ is the Lagrangian $K$-orbit in ${\mathbb{P}}(V)$. From [@SK] we know that $G_p^o$ is ${{\rm G}}_2^{\mathbb{C}}\times{{\rm G}}_2^{\mathbb{C}}$ and from the inclusion $${{\rm G}}_2\times{{\rm G}}_2 \subset {{\rm SO}}(7)\times{{\rm SO}}(7) \subset SO(14)$$ which lifts to ${{\rm Spin}}(14)$ (${{\rm G}}_2\times{{\rm G}}_2$ is simply connected), we get $K_p^o={{\rm G}}_2\times{{\rm G}}_2$. Now we claim that $\pi_1(K_p)=K_p/K_p^o$ is exactly ${\mathbb{Z}}_8$. Since in this case the degree of the invariant is 8 (see [@SK]) to prove this fact it is sufficient to show that $|N_K(K_p^o)/K_p^o|$ is at most 8.\ First we compute $N_{{{\rm SO}}(14)}(K_p^o)/K_p^o$. As an automorphism of ${{\rm G}}_2\times{{\rm G}}_2$ an element $g$ of $N_{{{\rm SO}}(14)}(K_p^o)$ can either preserve or interchange the ${{\rm G}}_2$ factors. Since $\mathfrak{g}_2$ has no outer automorphism and the centralizer of ${{\rm G}}_2$ in ${{\rm O}}(7)$ is $\{\pm \mbox{Id}\}$ we have that $N_{{{\rm SO}}(14)}(K_p^o)$ is given by the following four connected components: $$\left \{ \left[ \begin{array}{cc} A & 0 \\ 0 & B \end{array} \right]: A,B\in {{\rm G}}_2 \right\} \cup \left \{ \left[ \begin{array}{cc} -A & 0 \\ 0 & -B \end{array} \right]: A,B\in{{\rm G}}_2 \right\} \cup$$ $$\left \{ \left[ \begin{array}{cc} 0 & A \\ -B & 0 \end{array} \right]: A,B\in {{\rm G}}_2 \right \}\cup \left \{ \left[ \begin{array}{cc} 0 & -A \\ B & 0 \end{array} \right]: A,B\in{{\rm G}}_2 \right \} .$$ Thus $N_{{{\rm SO}}(14)}(K_p^o)/K_p^o \cong {\mathbb{Z}}_4$. Now note that the Lie group covering map ${{\rm Spin}}(14)\to {{\rm SO}}(14)$ induces an epimorphism $$\frac{N_{{{\rm Spin}}(14)}(K_p^o)}{K_p^o} \to \frac{N_{{{\rm SO}}(14)}(K_p^o)}{K_p^o}$$ whose kernel is ${\mathbb{Z}}_2$ and the claim follows. 19. [*$K={{\rm E}}_6$, $\rho=\Lambda_1$*]{}. As before let $p$ be such that $K\cdot p$ is the Lagrangian $K$-orbit in ${\mathbb{P}}(V)$. From [@SK] we know that $G_p^o$ is ${{\rm F}}_4^{\mathbb{C}}$ but ${{\rm F}}_4 \subset {{\rm E}}_6$, hence $K_p^o={{\rm F}}_4$. Following the same argument as in (\[SU(7)\]), since ${{\rm F}}_4$ has only inner automorphisms, we get that $N_K ({{\rm F}}_4)$ is contained in ${{\rm F}}_4\cdot C_K({{\rm F}}_4)$. Now ${{\rm F}}_4$ acts on ${\mathbb{C}}^{27}={\mathbb{C}}\oplus{\mathbb{C}}^{26}$ irreducibly on the second summand, hence $C_K({{\rm F}}_4)$ acts on each summand as scalar multiplication. Therefore $C_K({{\rm F}}_4)$ is contained in a 2-dimensional torus and $N_K({{\rm F}}_4)/{{\rm F}}_4$ is abelian. Since in this case the invariant has degree 3, we have $K_p={{\rm F}}_4\cdot{\mathbb{Z}}_3$. Note that ${\mathbb{Z}}_3$ is the center of ${{\rm E}}_6$ which acts trivially on $\ P^{26}$. 20. [*$K={{\rm E}}_7$, $\rho=\Lambda_1$*]{}. As before let $p$ be such that $K\cdot p$ is the Lagrangian $K$-orbit in ${\mathbb{P}}(V)$. From [@SK] we know that $G_p^o$ is ${{\rm E}}_6^{\mathbb{C}}$ but ${{\rm E}}_6 \subset {{\rm E}}_7$, hence $K_p^o={{\rm E}}_6$. This representation is self-dual. 21. [*$K={{\rm G}}_2$, $\rho=\Lambda_2$*]{}. The orbits of ${{\rm G}}_2$ are the same of ${{\rm SO}}(7)$ (see case (\[Spin(7)\])), therefore the Lagrangian orbit is $$\frac{{{\rm G}}_2}{{{\rm SU}}(3)\cdot{\mathbb{Z}}_2}=\frac{{{\rm SO}}(7)}{{{\rm SO}}(6)\cdot{\mathbb{Z}}_2}={\mathbb{R}}{\mathbb{P}}^6.$$ We have thus proved Theorem \[classificazione\] [ [**Table 1:**]{} Lagrangian orbits of simple Lie Groups in Projective spaces ]{} $$\begin{array}{|l|l|l|l|l|l|l|l|l|l} \hline & \quad K &\quad \rho &\dim_{\mathbb{C}}{\mathbb{P}}(V)&\text{cond.}& K_p^0& K_p/K_p^0& d \\\hline 1&{{\rm SU}}(n) & 2 \Lambda_1 & \ \frac{n(n+1)}{2}-1 & & {{\rm SO}}(n)& {\mathbb{Z}}_{n} & n\\ 2&{{\rm SU}}(n) & \Lambda_1\oplus \Lambda_1^* & \ 2n-1 & & {{\rm SU}}(n-1) & {\mathbb{Z}}_2 & 2 \\ 3&{{\rm SU}}(n) & \underbrace{\Lambda_1\oplus\cdots\oplus \Lambda_1}_n &\ n^2-1 & &\{1\} & {\mathbb{Z}}_n & n \\ 4&{{\rm SU}}(2n) & \Lambda_2 & \ n(2n-1)-1 & n \geq 3 & {{\rm Sp}}(n)&{\mathbb{Z}}_{2n} & 2n \\ 5&{{\rm SU}}(2n+1)& \Lambda_2\oplus \Lambda_1 &\ 2n^2+3n+1 & n\geq 2 & {{\rm Sp}}(n) & {\mathbb{Z}}_{n+1} & n+1 \\ 6&{{\rm SU}}(2) & 3 \Lambda_1 & \ 3 & & \{1\}&{\mathbb{Z}}_3\rtimes {\mathbb{Z}}_4& 4\\ 7&{{\rm SU}}(6) & \Lambda_3 & \ 19 & & ({{\rm SU}}(3)\times{{\rm SU}}(3)) & {\mathbb{Z}}_4 & 4\\ 8&{{\rm SU}}(7) & \Lambda_3 & \ 34 & & {{\rm G}}_2 & {\mathbb{Z}}_7 & 7\\ 9&{{\rm SU}}(8) & \Lambda_3 & \ 55 & & {\rm Ad}({{\rm SU}}(3)) & {\mathbb{Z}}_{16} & 16 \\ 10&{{\rm Sp}}(n) & \Lambda_1\oplus \Lambda_1 & \ 4n-1 & &{{\rm Sp}}(n-1) & {\mathbb{Z}}_2 & 2 \\ 11&{{\rm Sp}}(3) & \Lambda_3 & \ 13 & &{{\rm SU}}(3) & {\mathbb{Z}}_4 & 4 \\ 12&{{\rm SO}}(n) & \Lambda_1 & \ n-1 &n \geq 3 & {{\rm SO}}(n-1) & {\mathbb{Z}}_2 & 2\\ 13&{{\rm Spin}}(7) & {\rm spin \ rep.} & \ 7 & & {{\rm G}}_2 & {\mathbb{Z}}_2 & 2\\ 14&{{\rm Spin}}(9) & {\rm spin \ rep.} & \ 15 & &{{\rm Spin}}(7) & {\mathbb{Z}}_2 & 2\\ 15&{{\rm Spin}}(10) & \Lambda_e\oplus \Lambda_e & \ 31 & & {{\rm G}}_2 & - & 4 \\ 16&{{\rm Spin}}(11) & {\rm spin \ rep.} & \ 31 & & {{\rm SU}}(5) & {\mathbb{Z}}_4 & 4\\ 17&{{\rm Spin}}(12) & \Lambda_e & \ 31 & & {{\rm SU}}(6) & {\mathbb{Z}}_4 & 4\\ 18&{{\rm Spin}}(14) & \Lambda_e & \ 63 & & ({{\rm G}}_2\times{{\rm G}}_2) & {\mathbb{Z}}_8 & 8 \\ 19&{{\rm E}}_6 & \Lambda_1 & \ 26 & & {{\rm F}}_4 & {\mathbb{Z}}_3 & 3 \\ 20&{{\rm E}}_7 & \Lambda_1 & \ 55 & & {{\rm E}}_6 & - & 4 \\ 21&{{\rm G}}_2 & \Lambda_2 & \ 6 & & {{\rm SU}}(3) & {\mathbb{Z}}_2 & 2 \\ \hline \end{array}$$ In the Table the connected components of the isotropy subgroups $K_p$ of points $p$ through which the $K$-orbit is Lagrangian are listed in the fifth column. [Xyz]{} : [*Lie group actions in complex analysis*]{}, Aspects of Mathematics E27, (1995) : [*Equivariant completions of homogeneous algebraic varieties by homogeneous divisors*]{}, Ann. Glob. Anal. Geom. [**2**]{}, (1983) 49–78 : , Nagoya Math. J.[**139**]{}, (1995), 87–92 : : [*Minimal Lagrangian submanifolds of Kähler-Einstein manifolds*]{}, in “Differential geometry and differential equations (Shanghai, 1985)”, 1–12, Lecture Notes in Math., [**1255**]{}, Springer, 1987. : [*Twistor theory for Riemannian Symetric Spaces*]{}, Lecture Notes in Math., [**1424**]{}, Springer, 1990. : Taiwanese J. Math. [**5**]{}, (2001) 681–723 : Int. Math. Res. Not. [**45**]{}, (2004) 2437–2441 : [*Polar representations*]{} J. Algebra [**92**]{} (1985), 504–524 : [*Sur la géométrie des sous-fibrés et des feuilletages lagrangiens*]{} Ann. Sci. École Norm. Sup. [**13**]{} (1981) 465–480 : [*Kähler-Einstein metrics and integral invariants*]{} Lecture Notes in Math., [**1314**]{}, Springer, Berlin, 1988. : [*A note on the moment map on compact Kähler manifolds*]{}. Ann. Global Anal. Geom. [**26**]{} (2004), 315–318. : [*Calibrated geometries*]{}. Acta Math. [**148**]{} (1982), 47–157. : , J. Reine Angew. Math. [**455** ]{}, (1994) 123–140 New York-London: Academic Press-inc (1978) Math. Annalen [**286**]{}, (1990) 261–280 J. Algebra [**83**]{}, (1983) 72–100 : [*Cohomology of quotiens in symplectic and algebraic Geometry*]{}, Math. Notes [**31**]{}, Princeton (1984) : [*Foundations of differential geometry. Vol. II.*]{} John Wiley & Sons, Inc. New York (1996) Holt, Rinehart and Winston, Inc., New York-Montreal-London, (1971) : [*Gradient flow of the norm squared of a moment map*]{} Enseign. Math. [**51**]{} (2005), 117–127 : [*Alexander polynomial of plane algebraic curves and cyclic multiple planes*]{} Duke Math. J. [**49**]{} (1982), 833–851 Nagoya Math. J. [**16**]{}, (1960) 205–218 Math. Z. [**216**]{}, (1994) 471–482 Invent. Math. [**101**]{} (1990), 501–519 : [*Mean curvature flow, orbits, moment maps*]{} Trans. Amer. Math. Soc. [**355**]{} (2003), 3343–3357 : [*A note on moment maps and Kähler-Einstein manifolds*]{} [preprint]{} (2005), Nagoya Math. J. [**65**]{}, (1977) 1–155 : Ann. of Math. [**141**]{}, (1995) 87–129 Tohoku Math. J. (2) [**36**]{} (1984), 293–314 : (1992) 257–268 [^1]: [*Mathematics Subject Classification.*]{} 32J27, 53D20, 53D12.

--- abstract: 'We evaluate the orbital evolution and several plausible origins scenarios for the mutually inclined orbits of c and d. These two planets have orbital elements that oscillate with large amplitudes and lie close to the stability boundary. This configuration, and in particular the observed mutual inclination, demands an explanation. The planetary system may be influenced by a nearby low-mass star, $\upsilon$ And B, which could perturb the planetary orbits, but we find it cannot modify two coplanar orbits into the observed mutual inclination of $30^\circ$. However, it could incite ejections or collisions between planetary companions that subsequently raise the mutual inclination to $>30^\circ$. Our simulated systems with large mutual inclinations tend to be further from the stability boundary than $\upsilon$ And, but we are able to produce similar systems. We conclude that scattering is a plausible mechanism to explain the observed orbits of $\upsilon$ And c and d, but we cannot determine whether the scattering was caused by instabilities among the planets themselves or by perturbations from $\upsilon$ And B. We also develop a procedure to quantitatively compare numerous properties of the observed system to our numerical models. Although we only implement this procedure to $\upsilon$ And, it may be applied to any exoplanetary system.' author: - 'Rory Barnes, Richard Greenberg, Thomas R. Quinn, Barbara E. McArthur, and G. Fritz Benedict' title: 'Origin and Dynamics of the Mutually Inclined Orbits of $\upsilon$ Andromedae c and d' --- 2[W m$^{-2}$]{} Introduction ============ The $\upsilon$ Andromedae ($\upsilon$ And) planetary system is the first multiple planetary system discovered beyond our own Solar System around a solar-like star (Butler 1999). Not surprisingly, it has received considerable attention from theoreticians, and in many ways has been a paradigm for gravitational interactions in multiplanet extrasolar planetary systems. At first, research focused on its stability (Laughlin & Adams 1999; Rivera & Lissauer 2000; Lissauer & Rivera 2001; Laskar 2000; Barnes & Quinn 2001; Goździewski 2001). These investigations showed the system appeared to lie near the edge of instability, although an additional body could survive in between planets b and c (Rivera & Lissauer 2001). Later, attention turned to the apsidal motion (Stepinski 2000; Chiang & Murray 2002; Malhotra 2002; Ford 2005; Barnes & Greenberg 2006a,c, 2007a), with most investigators considering how the apsidal behavior provides clues to formation mechanisms such as planet-planet scattering or migration via disk torques. The recent direct measurement of the actual masses of planets c and d, and especially their 30$^\circ$ mutual inclination, via astrometry (McArthur 2010) maintains this system’s prominence among known exoplanetary systems. In this investigation, we evaluate planets c and d’s gravitational interactions (ignoring b as its orbit is still only constrained by radial velocity (RV) observations), as presented in McArthur (2010), which place strict constraints on the system’s origin. Such large mutual inclinations among planets are unknown in our Solar System and hence indicate different processes occurred during or after the planet formation process. We assume c and d formed in coplanar, low eccentricity orbits in a standard planetary formation models (see Hubickyj 2010; Mayer 2010), but then additional phenomena, occurring late or after the formation process, altered the system’s architecture. We consider two plausible mechanisms: a) If the distant stellar companion B (Lowrance 2002) is on a significantly inclined (relative to the initial orbital plane of the planets) and/or eccentric orbit, it may pump up eccentricities and inclinations through “Kozai” interactions (Kozai 1962; Takeda 2007); or b) If the planets form close together they may interact and scatter into mutually inclined orbits, as shown in previous studies (Weidenschilling & Marzari 2002; Chatterjee 2008; Raymond 2010). The mutual inclinations are obviously of the most interest, but other features of the system are also important. As represents the first exoplanetary system with full three-dimensional orbits and true masses directly measured (aside from the pulsar system PSR 1257+12 \[Wolszczan 1994\]), we may exploit this information when evaluating formation models. To that end, we develop a simple metric that quantifies the success of a model at reproducing numerous observed aspects of the $\upsilon$ And planetary system. Although we apply this approach to $\upsilon$ And, it is generalizable to any planetary system. In this investigation we limit the analysis to just c and d, and to the stable fit presented in McArthur (2010). As described in that paper, the inclination and longitude of ascending node of b are not detectable with the [*Hubble Space Telescope*]{}, so rather than explore the range of architecture permitted by RV data, we focus on the known properties of c and d. Furthermore, we do not consider the range of uncertainties in c and d as the stable fit in McArthur et al. (2010) is surrounded by unstable fits. As we see below, even limiting our scope in this way, we still must perform a large number of simulations over a wide range of parameter space. We first ($\S$ 2) analyze the current orbital oscillations of c and d with an N-body simulation. We then use the dynamical properties to constrain our two inclination-raising scenarios, which we explore through $>$50,000 N-body simulations. Then in $\S$ 3 we show that B by itself cannot raise the mutual inclinations to $30^\circ$. In $\S$ 4 we show that planet-planet scattering is a likely inclination-raising mechanism. However, we find that the most stringent constraint on scattering is the combination of its $30^\circ$ mutual inclination *and* extreme proximity to the stability boundary. In $\S$ 5 we discuss the results and place them in context with previous dynamical and stability studies of exoplanets. Table 1: Current Configuration of c and d\ Planet $m$ (M$_J$) a (AU) e $i$ ($^\circ$) $\varpi$ ($^\circ$) $\Omega$ ($^\circ$) $n$ ($^\circ$) -------- ------------- -------- ------- ---------------- --------------------- --------------------- ---------------- c 14.57 0.861 0.24 16.7 290 295 270.5 d 10.19 2.70 0.274 13.5 240.8 115 266.1 Orbital Evolution of c and d ============================ In this section we determine the orbital evolution of c and d. As the mass and orbit of planet b are unknown and the four-body interactions between A, b, c and d are extremely complex and depend sensitively on a secular resonance, general relativity and the stellar oblateness (McArthur 2010), we have chosen to leave them out of this analysis, but will address them in a future study. We examine the secular behavior of the system through an N-body simulation using the Mercury code (Chambers 1999). Here and below we used the “hybrid” integrator in Mercury. Energy was conserved to 1 part in $10^8$. For this integration we change the coordinate system from that in the discovery paper, which is based on the viewing geometry, and instead reference our coordinates to the invariable (or fundamental) plane. This plane is perpendicular to the total angular momentum vector of the system (although we ignore planetary spins), we rotated the coordinate system. The planets’ orbital elements in this coordinate system are listed in Table 1 at epoch JD 2452274.0. The system shown in Table 1 was found to be stable in McArthur , but it was also noted that the system is close to the stability boundary. Therefore we cannot exclude the possibility that other solutions may result in significantly different dynamical behavior. Here and below we assume that such a situation is not the case. The orbits of planets c and d undergo mutual perturbations which cause periodic variations in orbital elements over thousands of orbits. The long-term changes can be conveniently divided into two parts: the apsidal evolution (changes in eccentricity $e$ and longitude of periastron $\varpi$) and the nodal evolution (changes in inclination $i$, longitude of ascending node $\Omega$, and hence the mutual inclination $\Psi$). The variations, starting with the conditions in Table 1, are shown in Fig. \[fig:secular\]. In this figure, we chose a Jacobi coordinate system in order to minimize frequencies due to the reflex motion of the star. In Fig. \[fig:secular\] the left panels show the apsidal behavior, and the right show the nodal behavior. The lines of apse oscillate about $\Delta\varpi = \pi$, anti-aligned major axes. This revision once again changes the expected apsidal evolution. Initially Chiang & Murray (2002) and Malhotra (2002) found the major axes oscillated about alignment. Then Ford (2005; see also Barnes & Greenberg 2006a,c) found the system was better described as “near-separatrix,” meaning the apsides lie close to the boundary between libration and circulation. Now we find that the system found by McArthur (2010) librates in an anti-aligned sense! Substantial research has examined the secular behavior of exoplanetary systems, yet the story of $\upsilon$ And shows that predicting the dynamical evolution of a planetary system based on minimum masses and poorly constrained eccentricities is uncertain at best and foolhardy at worst. Even now, without full three-dimensional information about planets b and e (a trend seen in McArthur ) our analysis should only be considered preliminary. Table 2: Dynamical Properties of c and d\ Property Value ---------------------- -------------- $e^{min}_c$ 0.069 $e^{min}_d$ 0.074 $e^{max}_c$ 0.39 $e^{max}_d$ 0.365 $\epsilon$ 0.17 $i^{min}_c$ $16.0^\circ$ $i^{min}_d$ $11.6^\circ$ $i^{max}_c$ $20.4^\circ$ $i^{max}_d$ $16.7^\circ$ $\Psi^{min}$ $27.6^\circ$ $\Psi^{max}$ $37.1^\circ$ $\beta/\beta_{crit}$ 1.075 \ The eccentricities and inclinations undergo large oscillations, as does the mutual inclination. The slow $\sim$15,000 year oscillations are expected from analytical secular theory. The high-frequency oscillation is probably a combination of coupling between eccentricity and inclination and velocity changes that occur during conjunction (note that the impulses at each conjunction can change $e$ by more than 0.01). Note also that the ratio of the orbital period, 5.32, is not very close to the low-order 5:1 and 11:2 resonances, so this high frequency evolution is not due to a mean motion resonance. The numerical integration also allows a calculation of the proximity to the apsidal separatrix $\epsilon$ (Barnes & Greenberg 2006c). When $\epsilon$ is small ($\lsim 0.01$), two planets are near the boundary between librating and circulating major axes. Although $\upsilon$ And was the first system to be identified as near-separatrix (Ford 2005), we find $\epsilon = 0.17$ indicating that the system is actually [*not*]{} close to the separatrix, according to the McArthur (2010) model. Also of interest is the system’s proximity to dynamical instability, the ejection of a planet. Previous studies found planets c and d are close to this limit (Rivera & Lissauer 2000; Barnes & Greenberg 2001, 2004; Goździewski 2001). Here we calculate c and d’s proximity to the Hill stability boundary with the quantity $\beta/\beta_{crit}$ (Barnes & Greenberg 2006b,2007b; see also Marchal & Bozis 1982; Gladman 1993; Veras & Armitage 2004). If $\beta/\beta_{crit} > 1$ then the pair is stable, if $<1$, it is unstable. We find $\beta/\beta_{crit} = 1.075$ and therefore these two planets are very close to the dynamical stability boundary. We caution that constraints based on $\beta/\beta_{crit}$ may be misleading, as Hill stability is strictly only applicable to three-body systems. In Table 2 we list some statistics of our $10^6$ year integration based on 36.5 day output intervals in astrocentric coordinates. Superscripts $min$ and $max$ refer to the minimum and maximum values achieved, respectively. Clearly the actual dynamics of this system depend on the presence and properties of planet b, and, in principle, any additional unconfirmed planets, but without better data on these objects, Table 2 is the best available characterization of the orbital evolution of these two planets. They may also be used to evaluate the origins scenarios described in the following two sections. Perturbations from B ==================== B is a distant M4.5, 0.2 $\msun$ companion star to A (Lowrance 2002; McArthur 2010). Its orbit can not be estimated yet, hence we do not know if it is gravitationally bound (Patience ; Raghavan ). Estimates for their separation range from 702 AU (Raghavan 2006) to as much as 30,000 AU (McArthur 2010). If planet c and d formed on circular, coplanar orbits, then could $\upsilon$ And B have pumped up the mutual inclinations of c and d? Given the uncertainty in B’s orbit, we consider a broad parameter space sweep: 13,200 simulations that cover the range $500 \le a_B \le 2000$ AU, $0.5 \le e_B \le 0.85$ and $30^\circ \le i_B \le 80^\circ$. Here $i_B$ is referenced to the initial orbital plane of the planets, not the invariable plane of the four-body system. The angular elements were varied uniformly from 0 to 2$\pi$. Note that this range was chosen to increase the perturbative effects of B and does not reflect any expectation of its actual orbit. Each simulation was run for $5 \times 10^6$ years, which corresponds to $\sim 250$ orbits of B. While not a long time, we find that B may destabilize the planetary system, which could lead to large mutual inclinations of planet c and d (see $\S$ 4). For the vast majority of these cases, the orbits of c and d remain coplanar, with $\Psi < 1^\circ$. However, 3 simulations ejected planet d; 192 led to planets with $\Psi^{max} > 1^\circ$; 26 with $\Psi^{max} > 10^\circ$; and 1 case out of the 13,200 reached $\Psi^{max} = 34^\circ$. The 192 non-planar cases were spread throughout parameter space, with no significant clustering. To explore effects on longer timescales, we integrated 20 cases to 1 Gyr. Four of the previous simulations that had led to significant mutual inclinations (including that which led to $\Psi^{max} = 34^\circ$) were tested, in order to examine stability. The other 16 were chosen from among those in which $\Psi^{max}$ stayed $<1^\circ$ over 250 orbits of B, in order to determine if mutual inclinations could develop over longer timescales. We find that most of these simulations, in fact, ejected a planet. Therefore the orbit of B appears able to destabilize a circular, coplanar system. Next we relax the requirement that the planets began on coplanar orbits and ran simulations with initial mutual inclinations $\Psi_0 = 3^\circ$, $10^\circ$, and $30^\circ$, and with $a_B = 700$, $e_B = 0$, $i_B = 30^\circ$. The two planets began with their current best fit semi-major axes and masses, but on circular orbits, and one inclination was set to $3^\circ$, $10^\circ$ or $30^\circ$ with the masses held constant. These systems were integrated for 1 Gyr. In each of these cases, shown in Fig. \[fig:kozainonzero\], the initial value of $\Psi$ is maintained for the duration of the simulation. The widths of the libration increase with $\Psi_0$ because the interactions among the planets are driving a secular oscillation. It therefore seems that even if the planets began with a nonzero relative inclination, B is unable to pump it to the range shown in Fig. \[fig:secular\]. These simulations also demonstrate that plausible orbits of B will not destabilize the observed system. These simulations indicate that it is unlikely that B could have twisted the orbits into the mutually inclined system we see today. However, it could have destabilized the system, which we will see in the next section is a process which can lift coplanar orbits to $\Psi \gsim 30^\circ$. Planet-Planet Scattering ======================== Our second hypothesis considers the possibility that the planetary system formed in an unstable configuration independent of B, and that encounters between planets ultimately ejected an original companion leaving a system with high mutual inclinations. Such impulsive interactions could drive inclinations to large values, perhaps as large as $60^\circ$ (Marzari & Weidenschilling 2002; Chatterjee 2008; Raymond 2010). We considered 41,000 different initial configuration of the system, one or two additional planets with masses in the range 1 – 15 M$_{\textrm{Jup}}$. At the end of this section we summarize the results of all these simulations, but initially we focus on one subset. We completed 5,000 simulations that began with three 10 – 15 M$_{\textrm{Jup}}$ mass objects (uniformly distributed in mass) separated by 4–5 mutual Hill radii (Chambers 1996), with $e < 0.05$, $i < 1^\circ$ and $0.75 < a < 4$ AU. We integrated these cases for $10^6$ years with Mercury’s hybrid integrator, conserving energy to 1 part in $10^4$. About 1% of cases failed to conserve energy at this level and were thrown out. These ranges are somewhat arbitrary but follow the recent study by Raymond (2010), which considered smaller mass planets at larger distances. They found that a system consisting of three 3 M$_{\textrm{Jup}}$ planets could, after removal of one planet and settling into a stable configuration, end up with $\Psi > 30^\circ$ about 15% of the time (down from 30% for three Neptune-mass planets). However, they also found that only 5% of systems of three 3 M$_{\textrm{Jup}}$ planets settled into a configuration with $\beta/\beta_{crit} < 1.1$. Therefore we expect that these two parameters will be the hardest to reproduce via scattering. As we see below, this expectation is borne out by our modeling. In our study, a successful model conserved energy adequately (1 part in $10^4$), removed the extra planet, and the remaining planets all had orbits with $a < 10$ AU. 2072 trials met these requirements (1416 collisions and 656 ejections). We ran each of these final two-planet configurations for an additional $10^5$ years (again validating the simulation via energy conservation) to assess secular behavior for comparison with the system presented in Table 2. In Fig. \[fig:example\] we show the outcome of one such trial in which a hypothetical planet was ejected. The format of this figure is the same as Fig. \[fig:secular\]. The behavior is qualitatively similar as in Fig. \[fig:secular\], including anti-aligned libration of the apses, the magnitudes of the eccentricities and the inclinations, and the short period oscillation superposed on the longer oscillation. The mutual inclination for this case is even larger than the observed system. This system’s $\beta/\beta_{crit}$ is 1.06, slightly lower than the observed system. This simulation, which is one of the closest matches to the observed system, shows that the ejection of a single additional planet could have produced the system. Figure \[fig:example\] is but one outcome. We next explore the statistics of this suite of simulations and consider the other orbital elements and dynamical properties. We divide the outcomes into two cases: Ejections and Collisions. These two phenomena could produce significantly different outcomes. For example, collisions tend to occur near periastron of one planet and apoastron of the other, and we might expect the merged body to have a lower eccentricity than either of the progenitors. We show the cumulative distributions of the properties listed in Table 2 in Fig. \[fig:single\]. Comparing the values of orbital elements at a given time is not ideal, but as it has been done many times (see Ford 2001; Ford & Rasio 2008; Juric & Tremaine 2008; Chatterjee 2008; Raymond 2010), we do so here as well. In Figs. \[fig:single\]a–c, we show the values of $e$, $i$, and $\Psi$ at the end of the initial $10^5$ year integration. In panels d–h we show the ranges of $i^{min}$, $i^{max}$, $\Psi^{min}$ and $\Psi^{max}$. We find that 8.9% of successful models produced a system with $\Psi^{max} > 30^\circ$, consistent with Raymond (2010). Note that ejections produce $\Psi^{max} > 30^\circ$ about 20% of the time. In Fig. \[fig:single\]i we show the $\epsilon$ distribution, which is bimodal with one peak near 0.1 and another near $10^{-3}$. The observed value of 0.17 is not an unusual value, and we find that systems with this $\epsilon$ value can have appropriate values of $e^{min}$ and $e^{max}$. We note that the significant fraction of systems near the apsidal separatrix contradicts the results of Barnes & Greenberg (2007a), which found that scattering only produced $\epsilon < 0.01$ a few percent of the time. The most likely explanation for this difference is that Barnes & Greenberg considered coplanar orbits and forbade collisions, whereas here we explore non-planar motion. Our results also indicate that near-separatrix motion is likely a result of collisions, rather than ejections, and $\epsilon < 10^{-4}$ (which is unlikely to be measured any time soon) only result from collisions. Figure \[fig:single\]j shows the distribution of $\beta/\beta_{crit}$ after scattering. Here the difference between collisions and ejections is starkest: Ejections have a much broader distribution than collisions. Barnes (2008) noted that systems are “packed” (no additional planets can lie in between two planets) when $\beta/\beta_{crit} \lsim 2$, which is near the peak of the ejection distribution. Our results consistent with Raymond (2009). In this model the mutual inclinations and proximity to instability are the strongest constraints on the system’s origins, but we may quantify scattering’s ability to reproduce all the observed features of the system. Most previous studies of scattering have focused on reproducing eccentricity distributions, but all available information should be used. With this goal in mind, we lay out here a simple method to quantify any model’s ability to reproduce observations. For each simulated system, we calculate its “parameter space distance” $\rho$ from the best fit. We define this quantity as $$\label{eq:rho} \rho = \sqrt{\sum{\Big(\frac{\eta - \eta_j}{\eta}\Big)^2}},$$ where $\eta$ represents $e^{min}_j...\beta/\beta_{crit}$ and $j$ = c,d (see Table 2). This statistic has several limitations: It ignores correlations between parameters; is dependent on the coordinate system; ignores uncertainties in the observations (as discussed above), and possibly overweights some parameters by including combinations of variables that are not independent. Although crude, $\rho$ does provide a quantitative estimate of how close a modeled system is to the observed system. Smaller values of $\rho$ signal a system that is a closer match to the actual system. In Fig. \[fig:single\]k we plot the distributions of $\rho$. These distributions resemble the $\beta/\beta_{crit}$ distributions (panel j), suggesting it is the most important constraint on the system. In Table 3 we show some statistics of our runs, where “min” is the minimum value of the set, “avg” is the mean, $\sigma$ is the standard deviation, and “max” is the maximum value. For most of the parameters we consider, ejections and collisions do a reasonable job of producing the observed values. However, this representation does not show any cross-correlation, does a system with high $\Psi$ also have low $\beta/\beta_{crit}$? We explore this relationship in Fig. \[fig:psibbc\]. We see that post-collision systems (blue points) cluster heavily at low $\beta/\beta_{crit}$ and $\Psi^{max}$, but post-ejection systems (red points) have a much broader range. Nonetheless, the two outcomes seem equally likely to reproduce the system, represented by the “+” (recall that there are three times as many blue points as red). However, from our models the actual probability that instabilities can reproduce the $\upsilon$ And system is less than 1%. From our analysis of these 2072 systems, we see that it is possible for planetary ejections and collisions to reproduce the observed $\upsilon$ And system, albeit with low probability. This suite of simulations is obviously limited in scope, so we performed 36,000 more simulations relaxing constraints on planetary mass (allowing uniform values between 1 and 15 M$_{\textrm{Jup}}$), separation (uniform distribution between 2 and 5 mutual Hill radii), and number of planets (3 or 4). These other simulations began with two planets with approximately the same semi-major axes as observed, and placed planets interior and/or exterior to these two planets. These simulations show that the equal-mass case we considered here is the best method to achieve large $\Psi$, as expected from Raymond (2010). For additional planets with masses less than 5 M$_{\textrm{Jup}}$, $\Psi$ values greater than $30^\circ$ are very unlikely, but $\beta/\beta_{crit} \sim 1$ is more likely. The removal of two planets does not make much difference in the resulting system. We conclude that the removal of 1 or more planets with mass(es) larger the 5 M$_{\textrm{Jup}}$ is a viable process to produce the observed configuration of $\upsilon$ And c and d. Table 3: Distribution of Properties After Collision/Ejection Property ------------------------- ---------------------- ------ ---------- ------ ---------------------- ------- ---------- ------ min avg $\sigma$ max min avg $\sigma$ max $e^{min}_c$ $3.5 \times 10^{-5}$ 0.29 0.17 0.74 $6 \times 10^{-6}$ 0.1 0.1 0.64 $e^{min}_d$ $7.6 \times 10^{-5}$ 0.28 0.19 0.75 $8 \times 10^{-6}$ 0.03 0.07 0.51 $e^{max}_c$ 0.097 0.53 0.16 0.83 0.024 0.26 0.15 0.79 $e^{max}_d$ 0.064 0.28 0.16 0.88 0.05 0.25 0.12 0.77 $\epsilon$ $10^{-4}$ 0.26 0.22 1.36 $3.7 \times 10^{-5}$ 0.066 0.12 1.1 $i^{min}_c$ ($^\circ$) 0.02 7.2 7.50 34.4 0.013 1.2 2.7 30.4 $i^{min}_d$ ($^\circ$) 0.002 4.1 5.9 43.7 0.005 0.77 2.1 30.4 $i^{max}_c$ ($^\circ$) 0.08 11.9 10.8 50.8 0.014 1.7 3.8 43.0 $i^{max}_d$ ($^\circ$) 0.03 8.3 8.7 48.7 0.007 1.26 3.4 49.7 $\Psi^{min}$ ($^\circ$) 0.06 11.7 9.7 46.8 0.036 2.0 4.5 39.8 $\Psi^{max}$ ($^\circ$) 0.1 19.8 15.5 63.9 0.032 2.9 6.7 62.1 $\beta/\beta_{crit}$ 0.8 1.95 0.49 4.4 0.98 1.11 0.08 2.14 $\rho$ 1.17 5.9 1.9 11.9 1.08 3.3 0.75 8.6 Conclusions =========== We have shown that the orbital behavior of the model for Ups And c and d proposed by McArthur et al. (2010) is quite different from that of orbital models identified by previous studies that had no knowledge of the inclination or actual mass of the planets. The major axes librate in an anti-aligned configuration, and their mutual inclination is substantial and oscillates with an amplitude of about $10^\circ$. We find that the companion star B by itself cannot pump the mutual inclination up to large values, even if the planets began with a significant relative inclination. However, it may have sculpted the planetary system by inciting an instability that ultimately led to ejections of formerly bound planets. The timescale to develop these instabilities is long. The configurations of B that could have such effects are sparsely distributed over parameter space, and the orbits of previously bound planets cannot be specified. These factors make the role of B complicated, but suggest an in-depth analysis of its role merits further research. Even without B, planet-planet scattering may have driven the system to the observed state. That process can easily reproduce the apsidal motion, but pumping the mutual inclination up to the observed values is difficult and probably requires the removal of a planet with mass $> 5$ M$_\textrm{Jup}$. Removing two planets does not increase this probability significantly. The other important constraint on the scattering hypothesis is the system’s close proximity to the stability boundary, $\beta/\beta_{crit}$. Collisions may leave a system near that boundary, whereas ejections tend to spread out the planets. Furthermore, we find that collisions tend to produce systems with low $\beta/\beta_{crit}$ and low $\Psi$, while ejections produce a broad range of $\Psi$, but large values of $\beta/\beta_{crit}$. Nonetheless, Fig. \[fig:example\] demonstrates that scattering can produce systems similar to $\upsilon$ And. Although scattering is a reasonable process to produce the observed architecture, we cannot determine the triggering mechanism. Did scattering occur because B destabilized the planetary system? Or did the planet formation process itself, independent of B, ultimately lead to instabilities? The presence of B makes distinguishing these possibilities very difficult. A larger census of mutual inclinations and stellar companions can resolve this open issue. Alternatively, our decisions about the system at the onset of scattering could be mistaken. We assumed the planets formed inside the original protoplanetary disk with inclinations $<1^\circ$. It may be that larger initial inclinations are possible prior to scattering, in which case the planets could be pumped to larger mutual inclinations (Chatterjee 2008). However, it remains to be seen if such configurations are possible prior to scattering. Future studies should explore the inclinations of giant planets during formation. We have also ignored the effects of planet b, stellar companion B, and a possible fourth planetary companion (McArthur 2010) in our analysis. These bodies could significantly change the secular behavior, and/or the observed fundamental plane. Furthermore, planet b is tidally interacting with its host star, which could alter the long-term secular behavior (Wu & Goldreich 2002). Hence future revisions to this system, and the inclusion of tidal effects, could significantly alter the interpretations described above, possibly making scattering more likely to produce the observed system. We are currently exploring the wide range of $i$’s and $\Omega$’s of b and the subsequent orbital evolution of the entire system. Figure \[fig:single\]i shows that scattering tends to produce two types of apsidal behavior: near-separatrix ($\epsilon < 10^{-3}$) and motion far from the separatrix ($\epsilon > 0.01$) with a desert in between. Adding a second scatterer to the mix does not erase this bimodality. This result contrasts with the case with no inclinations (Barnes & Greenberg 2007a), in which near-separatrix motion ($\epsilon < 0.01$) is not a common outcome of scattering. Fig. \[fig:single\]i shows that, in fact, both outcomes are likely, at least in systems similar to $\upsilon$ And. Although there are hints of this structure in the observed exoplanet population (Barnes & Greenberg 2006c), those results are based on radial velocity data. We now know that minimum masses are not necessarily a good indicator of apsidal motion. For $\upsilon$ And, the large mutual inclination and proximity to instability are strong constraints on the origin of its planetary system. However, for other systems, this may not be the case. We outlined in $\S$ 4 a method in which all aspects of a planetary system can be combined to quantify the validity of a formation model. When the mass and three dimensional orbits of a planetary system are known, the properties presented in Table 1 can be combined into a single parameter $\rho$ which provides a statistic for quantitatively comparing models. In our analysis we ignored the observational errors, which is regrettable, but necessary due to the system’s extreme proximity to dynamical instability (McArthur 2010). We encourage future studies that strive to reproduce the system of McArthur to find $\rho$ values less that those listed in Table 3, as lower values imply a closer match to the system, assuming that other stable solutions show similar behavior to the one we describe here. Furthermore, investigations into exoplanet formation could compare distributions of observed and simulated properties as a quantitative method for model validation. The revisions of McArthur (2010) reveal the importance of the mass-inclination degeneracy in dynamical studies of exoplanets. Clearly in some cases masses can be much larger than the minimum value measured by radial velocity, which in turn changes secular frequencies and eccentricity amplitudes. However, large changes in mass due to the mass-inclination degeneracy should be rare, hence, trends using minimum masses may still be valid. Nonetheless, we urge caution when exploring trends among dynamical properties (Zhou & Sun 2003; Barnes & Greenberg 2006c), as they may be misleading. Even if $30^\circ$ mutual inclinations turn out to be rare, systems with $\Psi \sim 10^\circ$ probably are not (Fig. \[fig:single\]; Chatterjee 2008; Raymond 2010). If these systems host planets with habitable climates, they may be very different worlds than Earth. Planetary inclinations can drive obliquity variations in terrestrial planets (Atobe 2004, Armstrong 2004; Atobe & Ida 2007), unless they have a large moon (Laskar 1997). Therefore future analyses of potentially habitable worlds should pay particular attention to the mutual inclinations, and climate modeling should explore the range of possibilities permitted by large mutual inclinations. Terrestrial planets will likely be discovered in their star’s habitable zone in the coming years. The orbital configuration and evolution of $\upsilon$ And warns us that habitability assessment hinge on the orbital architecture of the entire planetary system. RB acknowledges support from the NASA Astrobiology Institute’s Virtual Planetary Laboratory lead team, supported by cooperative agreement No. NNH05ZDA001C. RG acknowledges support from NASA’s Planetary Geology and Geophysics program, grant No. NNG05GH65G. BEM and GFB acknowledge support from NASA through grants GO-09971, GO-10103, and GO-11210 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under NASA contract NAS5-26555. We also thank Sean Raymond for helpful discussions. Armstrong, J.C., Leovy, C.B., & Quinn, T.R. 2004, Icarus, 171, 255\ Atobe, K., Ida, S. & Ito, T. 2004, Icarus, 168, 223\ Atobe, K., Ida, S. 2007, Icarus, 188, 1\ Barnes, R., Goździewski, K., & Raymond, S.N. 2008, ApJ, 680, L57\ Barnes, R. & Greenberg, R. 2006a, ApJ, 652, L53\ ————. 2006b, ApJ, 638, 478\ ————. 2006c, ApJ, 647, L163\ ————. 2007a, ApJ, 665, L67\ ————. 2007b, ApJ, 659, L53\ Barnes, R. & Quinn, T.R. 2001, ApJ, 554, 884\ ————. 2004, ApJ, 611, 494\ Barnes, R. & Raymond, S.N. 2004 ApJ, 617, 569\ Butler, R.P. 1999, ApJ, 526, 916\ Chambers, J., 1999, MNRAS, 304, 793\ Chatterjee, S. 2008, ApJ, 686, 580\ Chiang, E. I., Tabachnik, S., & Tremaine, S. 2001, AJ, 122, 1607\ Ford, E.B., Lystad, V., & Rasio, F.A. 2005, Nature, 434, 873\ Ford, E.B. & Rasio, F.A. 2008, ApJ, 686, 621\ Gladman, B. 1993, Icarus, 106, 247\ Goździewski, K. 2001, A&A, 378, 569\ Hubickyj, O. 2010, in *Formation and Evolution of Exoplanets*, Rory Barnes (ed), Wiley-VCH, Berlin.\ Jurić, M. & Tremaine, S. 2008, ApJ, 686, 603\ Kozai, Y. 1962, AJ, 67, 591\ Laskar, J. 2000, PhRvL, 84, 3240\ Laskar, J., Joutel, F., & Robutel, P. 1993, Nature, 615\ Laughlin, G. & Adams, F.C. 1999, ApJ, 526, 881\ Lissauer, J.J. & Rivera, E.J. 2001, ApJ, 554, 1141\ Lowrance, P.J., Kirkpatrick, J.D., & Beichman, C.A. 2002, ApJ, 572, L79\ Malhotra, R. 2002, ApJ, 575, L33\ Marchal, C. & Bozis, G. 1982, CeMDA, 26, 311\ Marzari, F. & Weidenschilling, S. 2002, Icarus, 156, 570\ Mayer, L. 2010 in *Formation and Evolution of Exoplanets*, Rory Barnes (ed), Wiley-VCH, Berlin.\ McArthur, B. 2010, ApJ, 715, 1203\ Murray, C.D. & Dermott, S.F. 1999, *Solar System Dynamics*, Cambridge UP, Cambridge\ Patience, J. 2002, ApJ, 581, 654\ Raymond, S.N. 2009, ApJ, 696, L98\ ————. 2010, ApJ, 711, 772\ Rivera, E.J. & Lissauer, J.J. 2000, ApJ, 530, 454\ Stepinski, T.F., Malhotra, R. & Black, D.C. 2000, 545, 1044\ Takeda, G. & Rasio, F.A. 2005, ApJ, 627, 1001\ Veras, D., & Armitage, P. 2004, Icarus, 172, 349\ Wolf, S. & Klahr, H, 2002, ApJ, 578, L79\ Wolszczan, A. 1994, Science, 264, 538\ Wu, Y., & Goldreich, P. 2002, ApJ, 564, 1024\ Zhou, J.-L., & Sun, Y.-S. 2003, ApJ, 598, 1290\

--- abstract: 'Weighted histograms are used for the estimation of probability density functions. Computer simulation is the main domain of application of this type of histogram. A review of chi-square goodness of fit tests for weighted histograms is presented in this paper. Improvements are proposed to these tests that have size more close to its nominal value. Numerical examples are presented in this paper for evaluation of tests and to demonstrate various applications of tests.' address: 'University of Akureyri, Borgir, v/Nordurslód, IS-600 Akureyri, Iceland' author: - 'N.D. Gagunashvili' title: 'Chi-square goodness of fit tests for weighted histograms. Review and improvements.' --- probability density function ,histogram ,goodness of fit test ,multinomial distribution ,Poisson histogram 02.30.Zz ,07.05.Kf ,07.05.Fb Introduction ============ A histogram with $m$ bins for a given probability density function (PDF) $p(x)$ is used to estimate the probabilities $$p_i=\int_{S_i}p(x)dx, \; i=1,\ldots ,m \label{p1}$$ that a random event belongs to bin $i$. Integration in (\[p1\]) is done over the bin $S_i$. A histogram can be obtained as a result of a random experiment with PDF $p(x)$. Let us denote the number of random events belonging to the $i$th bin of the histogram as $n_{i}$. The total number of events $n$ in the histogram is equal to $$n=\sum_{i=1}^{m}{n_i}.$$ The quantity $$\hat{p}_i= n_{i}/n$$ is an estimator of probability $p_i$ with expectation value $$\textrm E\,[\hat{p_i}]=p_i.$$ The distribution of the number of events for bins of the histogram is the multinomial distribution [@kendall] and the probability of the random vector $(n_1,\ldots ,n_m)$ is $$P(n_1,\ldots ,n_m)=\frac{n!}{n_1!n_2! \ldots n_m!} \; p_1^{n_1} \ldots p_m^{n_m},\text{ } \sum_{i=1}^{m} p_i=1.$$ A weighted histogram or a histogram of weighted events is used again for estimating the probabilities $p_i$ (\[p1\]), see Ref. [@gagunash]. It is obtained as a result of a random experiment with probability density function $g(x)$ that generally does not coincide with PDF $p(x)$. The sum of weights of events for bin $i$ is defined as: $$W_i= \sum_{k=1}^{n_i}w_i(k), \label{ffffweight}$$ where $n_i$ is the number of events at bin $i$ and $w_i(k)$ is the weight of the $k$th event in the $i$th bin. The statistic $$\hat{p_i}=W_i/n \label{west}$$ is used to estimate $p_i$, where $n=\sum_{i=1}^{m}{n_i}$ is the total number of events for the histogram with $m$ bins. Weights of events are chosen in such a way that the estimate (\[west\]) is unbiased, $$\textrm E [\hat{p_i}]=p_i.$$ The usual histogram is a weighted histogram with weights of events equal to 1. The two examples of weighted histograms are considered below:\ Example 1 --------- To define a weighted histogram let us write the probability $p_i$ (\[p1\]) for a given PDF $p(x)$ in the form $$p_i= \int_{S_i}p(x)dx = \int_{S_i}w(x)g(x)dx, \label{weightg}$$ where $$w(x)=p(x)/g(x) \label{fweightg}$$ is the weight function and $g(x)$ is some other probability density function. The function $g(x)$ must be $>0$ for points $x$, where $p(x)\neq 0$. The weight $w(x)=0$ if $p(x)=0$, see Ref. [@sobol]. The weighted histogram is obtained from a random experiment with a probability density function $g(x)$, and the weights of the events are calculated according to (\[fweightg\]).\ Example 2 --------- The probability density function $p_{rec}(x)$ of a reconstructed characteristic $x$ of an event obtained from a detector with finite resolution and limited acceptance can be represented as $$p_{rec}(x) \propto \int_{\Omega'} p_{tr}(x')A(x')R(x|x') \,dx', \label{p1_main}$$ where $p_{tr}(x')$ is the true PDF, $A(x')$ is the acceptance of the setup, i.e. the probability of recording an event with a characteristic $x'$, and $R(x|x')$ is the experimental resolution, i.e. the probability of obtaining $x$ instead of $x'$ after the reconstruction of the event. The integration in (\[p1\_main\]) is carried out over the domain $\Omega'$ of the variable $x'$. Total probability that an event will not be registered is equal to $$\overline{p}= \int_{\Omega'} p_{tr}(x')(1-A(x')) \,dx'. \label{p1_main8}$$ The sum of probabilities $$\int_{\Omega} \int_{\Omega'}p_{tr}(x')A(x')R(x|x') \,dx'dx+\int_{\Omega'} p_{tr}(x')(1-A(x')) \,dx'=1 \label{p1_main3}$$ because $$\int_{\Omega}\int_{\Omega'} p_{tr}(x')A(x')R(x|x') \,dx'dx=\int_{\Omega'} p_{tr}(x')A(x'), \,dx',$$ where $\Omega$ domain of the variable $x$. A histogram of the PDF $p_{rec}(x)$ can be obtained as a result of a random experiment (simulation) that has three steps [@sobol]: 1. A random value $x'$ is chosen according to a PDF $p_{tr}(x')$. 2. We go back to step 1 again with probability $1-A(x')$, and to step 3 with probability $A(x')$. 3. A random value $x$ is chosen according to the PDF $R(x|x')$. The quantity $\hat {p_i}= n_{i}/n$, where $n_{i}$ is the number of events belonging to the $i$th bin for a histogram with total number of events $n$ in random experiment (at step 1), is an estimator of $p_{i}$, $$p_{i}= \int_{S_i} \int_{\Omega'}p_{tr}(x')A(x')R(x|x') \, dx' \, dx,\; i=1,\ldots ,m, \label{p2i}$$ with the expectation value of the estimator $$\textrm E \,[\hat{p_{i}}]=p_{i}.$$ The quantity $\hat{\overline{p}}= \overline{n}/n$, where $\overline{n}$ is the number of events that were lost, is an estimator of $\overline{p}$ (\[p1\_main8\]) with the expectation value of the estimator $$\textrm E \,[\hat{\overline{p}}]= \overline{p}.$$ Notice that $$\sum_{i=1}^{m} p_i+\overline{p}=1 \,\, \text{and} \,\, \sum_{i=1}^{m} n_i+\overline{n}=n.$$ In experimental particle and nuclear physics, step 3 is the most time-consuming step of the Monte Carlo simulation. This step is related to the simulation of the process of transport of particles through a medium and the rather complex registration apparatus. To use the results of the simulation with some PDF $g_{tr}(x')$ for calculating a weighted histogram of events with a true PDF $p_{tr}(x')$, we write the equation for $p_{i}$ in the form $$p_{i}= \int_{S_i} \int_{\Omega'} w(x')g_{tr}(x')A(x')R(x|x') \, dx' \, dx, \label{p23}$$ where $$w(x')=p_{tr}(x')/g_{tr}(x') \label{fweight}$$ is the weight function. The weighted histogram for the PDF $p_{rec}(x)$ can be obtained using events with reconstructed characteristic $x$ and weights calculated according to (\[fweight\]). In this way, we avoid step 3 of the simulation procedure, which is important in cases where one needs to calculate Monte Carlo reconstructed histograms for many different true PDFs. The probability that an event will not be registered can be represented as $$\overline{p}= \int_{\Omega'} w(x')g_{tr}(x')(1-A(x')) \,dx', \label{p1_main2}$$ and is estimated the same way using events with weights calculated according formula (\[fweight\]). Goodness of fit tests ===================== The problem of goodness of fit is to test the hypothesis $$H_0: p_1=p_{10},\ldots, p_{m-1}=p_{m-1,0} \text{ vs. } H_a: p_i \neq p_{i0} \text{ for some } i,$$ where $p_{i0}$ are specified probabilities, and $\sum_{i=1}^{m} p_{i0}=1$. The test is used in a data analysis for comparing theoretical frequencies $np_{i0}$ with observed frequencies $n_i$. This classical problem remains of current practical interest. The test statistic for a histogram with unweighted entries $$X^2=\sum_{i=1}^{m} \frac{(n_i-np_{i0})^2}{np_{i0}} \label{basic11}$$ was suggested by Pearson [@pearson]. Pearson showed that the statistic (\[basic11\]) has approximately a $\chi^2_{m-1}$ distribution if the hypothesis $H_0$ is true. The contemporary proof of Pearson’s result ------------------------------------------ The expectation values of the observed frequency $n_i$, if hypothesis $H_0$ is valid, equal to: $$\textrm E [n_i]=np_{i0}, \; i=1,\ldots,m \label{ew}$$ and its covariance matrix $\mathbf{\Gamma}$ has elements: $$\! \! \! \! \! \! \! \!\! \! \! \! \! \! \! \!\! \! \! \! \! \! \!\gamma_{ij}= \begin{cases} np_{i0}(1-p_{i0}) \text { for } i=j\\ -np_{i0}p_{j0} \text{\hspace *{0.7cm} for } i \neq j \end{cases}$$ Notice that the covariance matrix $\mathbf{\Gamma}$ is singular [@kendall2]. Let us now introduce the multivariate statistic $$(\textbf{n}-n\textbf{p}_0)^t \mathbf{\Gamma_k^{-1}}(\textbf{n}-n\textbf{p}_0),\label{hot}$$ where\ $\textbf{n}=(n_1,\ldots, n_{k-1},n_{k+1},\ldots,n_{m})^t$, $\textbf{p}_0=(p_{10},\ldots,p_{k-1,0},p_{k+1,0},\ldots,p_{m0})^t$ and $\mathbf{\Gamma_k}=(\gamma_{ij})_{(m-1)\times(m-1)}$ is the covariance matrix for a histogram without bin $k$. The matrix $\mathbf{\Gamma_k}$ has the form $$\mathbf{\Gamma_k}=n\,\textrm{diag}\,(p_{10}, \ldots ,p_{k-1,0},p_{k+1,0}, \ldots , p_{m 0})-n\textbf{p}_0\textbf{p}_0^t.$$ The special form of this matrix permits one to find analytically $\mathbf{\Gamma_k^{-1}}$ [@woodbury]: $$\mathbf{\Gamma_k^{-1}}=\frac{1}{n}\textrm{diag}\,(\frac{1}{p_{10}}, \ldots ,\frac{1}{p_{k-1,0}},\frac{1}{p_{k+1,0}}, \ldots ,\frac{1}{p_{m 0}})+\frac{1}{np_{k,0}}\mathbf{\Theta},$$ where $\mathbf{\Theta}$ is $(m-1) \times (m-1)$ matrix with all elements unity. Finally the result of the calculation of expression (\[hot\]) gives us the $X^2$ test statistic (\[basic11\]). Notice that the result will be the same for any choice of bin number $k$. Asymptotically the vector $\textbf{n}$ has a normal distribution $\mathcal{N}(n \textbf{p}_0,\mathbf{\Gamma_k^{1/2}})$, see Ref. [@kendall2], and therefore the test statistic (\[basic11\]) has $\chi^2_{m-1}$ distribution if hypothesis ${H_0}$ is true $$X^2 \sim \chi^2_{m-1}.$$ Generalization of the Pearson’s chi-square test for weighted histograms ----------------------------------------------------------------------- The total sum of weights of events in $i$th bin $W_{i}$, $i=1,\ldots,m$, as proposed in Ref. [@gagunash], can be considered as a sum of random variables $$W_i= \sum_{k=1}^{n_i}w_i(k), \label{ffffweight}$$ where also the number of events $n_i$ is a random value and the weights $w_i(k),k=1,...,n_i$ are independent random variables with the same probability distribution function. The distribution of the number of events for bins of the histogram is the multinomial distribution and the probability of the random vector $(n_1,\ldots ,n_m)$ is $$P(n_1,\ldots ,n_m)=\frac{n!}{n_1!n_2! \ldots n_m!} \; g_1^{n_1} \ldots g_m^{n_m},\text{ } \sum_{i=1}^{m} g_i=1,$$ where $g_i$ is the probability that a random event belongs to the bin $i$. Let us denote the expectation values of the weights of events from the $i$th bin as $$\textrm E[w_{i}]= \mu_i$$ and the variances as $$\textrm {Var}[w_{i}]= \sigma_i^2.$$ The expectation value of the total sum of weights $W_{i}, i=1,\ldots,m$, see Ref. [@gnedenko], is: $$\textrm E[W_i]= \textrm E[\sum_{k=1}^{n_i}w_{i}(k)]= \textrm E[ w_{i}] \textrm E[n_i] = n\mu_ig_i.\label{eew}$$ The diagonal elements $\gamma_{ii}$ of the covariance matrix of the vector $(W_1,\ldots ,W_m)$, see Ref. [@gnedenko], are equal to $$\gamma_{ii}=\sigma_i^2 g_in+\mu_i^2 g_i(1-g_i)n= n\alpha_{2i}g_i-n\mu_i^2g_i^2, \label{giii}$$ where $$\alpha_{2i}= \textrm E[w_{i}^2].$$ The non-diagonal elements $\gamma_{ij},\,i\neq j$ are equal to: $$\begin{split} \gamma_{ij}=\sum_{k=0}^n\sum_{l=0}^{n}\textrm E \,[\sum_{u=1}^k\sum_{v=1}^l w_i(u)w_j(v)]h(k,l) -\textrm E[W_i]\textrm E[W_j]\\ =\sum_{k=0}^n\sum_{l=0}^{n}\textrm E [w_{i}w_{j}]h(k,l)kl-\mu_ing_i\mu_jng_j\quad\quad\quad\quad\,\\ =\mu_i\mu_j(-g_ig_jn+g_ig_jn^2 )- \mu_ing_i\mu_jng_j\quad\quad\quad\quad\;\:\,\,\\=-n\mu_i\mu_j g_ig_j,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, \label{gij} \end{split}$$ where $h(k,l)$ is the probability that $k$ events belong to bin $i$ and $l$ events to bin $j$. For weighted histograms again the problem of goodness of fit is to test the hypothesis $$H_0: p_1=p_{10},\ldots ,p_{m-1}=p_{m-1,0} \text{ vs. } H_a: p_i \neq p_{i0} \text{ for some } i,$$ where $p_{i0}$ are specified probabilities, and $\sum_{i=1}^{m} p_{i0}=1$. If hypothesis $H_0$ is true then $$\textrm E[W_i]=n\mu_ig_i=np_{i0}, \; i=1,\ldots,m \label{ew}$$ and $$g_i=p_{i0}/\mu_i, \; i=1,\ldots,m. \label{pi}$$ We can substitute $g_i$ to Eqs. (\[giii\]) and (\[gij\]) which gives the covariance matrix $\mathbf{\Gamma}$ with elements: $$\! \! \! \! \! \! \! \!\! \! \! \! \! \! \! \!\! \! \! \! \! \! \!\gamma_{ij}= \begin{cases} np_{i0}( r_i^{-1} -p_{i0})\,\,\, \text { for } i=j\\ -np_{i0}p_{j0}\text{\hspace *{1.3cm} for } i \neq j \end{cases}$$ where $$r_i=\mu_i/ \alpha_{2i}$$ is the ratio of the first moment of the distribution of weights of events $\mu_i$ to the the second moment $\alpha_{2i}$ for a particular bin $i$. Notice that for usual histograms the ratio of moments $r_i$ is equal to 1 and the covariance matrix coincides with the covariance matrix of the multinomial distribution. The multivariate statistic is represented as $$(\textbf{W}-n\textbf{p}_0)^t\mathbf{\Gamma_k^{-1}}(\textbf{W}-n\textbf{p}_0),$$ where\ $\textbf{W}=(W_1,\ldots,W_{k-1},W_{k+1},\ldots,W_{m})^t$, $\textbf{p}_0=(p_{10},\ldots,p_{k-1,0},p_{k+1,0},\ldots,p_{m0})^t$ and $\mathbf{\Gamma_k}=(\gamma_{ij})_{(m-1)\times(m-1)}$ is the covariance matrix for a histogram without bin $k$. The matrix $\mathbf{\Gamma_k}$ has the form $$\mathbf{\Gamma_k}=n\,\textrm{diag}\,(\frac{p_{10}}{r_1}, \ldots ,\frac{p_{k-1,0}}{r_{k-1}},\frac{p_{k+1,0}}{r_{k+1}}, \ldots , \frac{p_{m 0}}{r_m})-n \textbf{p}_0\textbf{p}_0^t.$$ The special form of this matrix permits one to find analytically the inverse matrix $$\mathbf{\Gamma_k^{-1}}=\frac{1}{n}\textrm{diag}\,(\frac{r_1}{p_{10}}, \ldots ,\frac{r_{k-1}}{p_{k-1,0}},\frac{r_{k+1}}{p_{k+1,0}}, \ldots ,\frac{r_m}{p_{m 0}})+\frac{1}{n(1-\sum_{i \neq k}r_i p_{i0})}\mathbf{r}\mathbf{r}^t,$$ where $\textbf{r}=(r_1,\ldots,r_{k-1},r_{k+1},\ldots,r_{m})^t$. After that, the multivariate statistic can be written as $$X^2_k= \sum_{i \neq k} r_i \frac{(W_i-np_{i0})^2}{np_{i0}}+\frac{(\sum_{i \neq k} r_i(W_i-np_{i0}))^2} {n(1-\sum_{i \neq k}r_ip_{i0})},\label{stdd1}$$ and can also be transformed to form $$X^2_k=\frac{1}{n} \sum_{i \neq k} \frac{r_iW_i^2}{p_{i0}}+\frac{1}{n} \frac{(n-\sum_{i \neq k}r_iW_i)^2}{1-\sum_{i \neq k}r_i p_{i0}}-n\label{stdd2}$$ which is convenient for numerical calculations. Asymptotically the vector $\textbf{W}$ has a normal distribution $\mathcal{N}(n \textbf{p}_0,\mathbf{\Gamma_k^{1/2}})$ [@robins] and therefore the test statistic (\[stdd1\]) has $\chi^2_{m-1}$ distribution if hypothesis ${H_0}$ is true $$X_k^2 \sim \chi^2_{m-1}.$$ For usual histograms when $r_i=1$, $i=1,\ldots, m$ the statistic (\[stdd1\]) is Pearson’s chi-square statistic (\[basic11\]). The expectation value of statistic (\[stdd1\]), as shown in Ref. [@gagunash], is equal to $$\textrm E[X^2_k]=m-1,$$ as for Pearson’s test [@kendall]. The ratio of moments $r_i=\mu_i/ \alpha_{2i}$, that is used for the test statistic calculation, is not known in majority of cases. An estimation of $r_i$ can be used: $$\hat{r_i}=W_i/W_{2i},$$ where $W_{2i}=\sum_{k=1}^{n_i}w_i^2(k)$. Let us now replace $r_i$ with the estimate $\hat r_i$ and denote the estimator of matrix $\mathbf{\Gamma_k}$ as ${\mathbf{\hat \Gamma_k}}$. Then for positive definite matrices $ {\mathbf{ \hat\Gamma_k}}$, $k=1,\ldots,m$ the test statistic is given as $$\hat{X^2_k}= \sum_{i \neq k} \hat r_i \frac{(W_i-np_{i0})^2}{np_{i0}}+\frac{(\sum_{i \neq k} \hat r_i(W_i-np_{i0}))^2} {n(1-\sum_{i \neq k} \hat r_i p_{i0})}.\label{stdd333}$$ Formula (\[stdd333\]) for usual histograms does not depend on the choice of the excluded bin, but for weighted histograms there can be a dependence. A test statistic that is invariant to the choice of the excluded bin and at the same time is a Pearson’s chi square statistic (\[basic11\]) for the unweighted histograms can be represented as the median value for the set of statistics $\hat X^2_k$ (\[stdd333\]) with positive definite matrixes ${\mathbf{\hat\Gamma_k}}$ $$\hat X_{Med}^2= \textrm {Med }\, \{\hat X_1^2, \hat X_2^2, \ldots , \hat X_m^2\}.\label{stdav}$$ Statistic $\hat X_{Med}^2$ first time was proposed in Ref. [@gagunash] and approximately has $\chi^2_{m-1}$ distribution if hypothesis ${H_0}$ is true $$\hat X_{Med}^2 \sim \chi^2_{m-1}.$$ The usage of $\hat X_{Med}^2$ to test the hypothesis $H_0$ with a given significance level is equivalent to making a decision by voting. It was noticed that size of test can be slightly greater than nominal value of size of test even for large value of total number of events $n$. New generalizations of Pearson’s chi-square test for weighted histograms ------------------------------------------------------------------------ Set of statistics $\{\hat X_1^2, \hat X_2^2, \ldots , \hat X_m^2 \}$, with positive definite matrixes ${\mathbf{\hat \Gamma_k}}$ only, is used for calculating the median statistic $\hat X_{Med}^2$ (\[stdav\]). It can be used for any weighted histograms, including histograms with unweighted entries. One bin is excluded because the full covariance matrix of an unweighted histogram is singular and hence can not be inverted. Let us consider estimation of a full covariance matrix $\mathbf{\hat \Gamma}$ for the weighted histogram with more detail. The symmetric matrix is positive definite if the minimal eigenvalue of the matrix larger then 0. We denote minimal eigenvalue of the matrix $n^{-1}\mathbf{\hat \Gamma}$ by $\lambda_{min}$ then follow to Ref. [@wilkin] it can be shown that $$\min_i\{\frac{p_{i0}}{\hat r_{i}}\} - \sum_{i=1}^{m}p_{i0}^2 \leq \lambda_{min} \leq \min_i\{ \frac{p_{i0}}{\hat r_{i}}\}.$$ and the eigenvalue $\lambda_{min}$ is the root of secular equation $$1-\sum_{i=1}^m \frac{p_{i0}^2}{p_{i0}/\hat r_i-\lambda}=0. \label{sekular}$$ In case of a histogram with unweighted entries, all $\hat r_i=1$ and $\lambda=0$ is zero of equation (\[sekular\]). Matrix $\mathbf{\hat \Gamma}$ for this case is not positive definite and is singular, but matrix $\mathbf{\hat \Gamma_k}$ is positive definite and therefore invertible. Number of events $n_i$ in bins of usual histogram satisfy to equation $n_1+n_2,...,+n_m=n$ that is why the covariance matrix of multinomial distribution is not positive definite and is singular. Matrix $\mathbf{\hat \Gamma}$ for a histogram with weighted entries can be also non-positive definite. There are two reasons why this can be. First of all, the total sums of weights $W_i$ in bins of a weighted histogram are related with each other, because satisfy the equation $\textrm E[\sum_{i=1}^m{W_i}]=n$ and second, due fluctuations of matrix elements. The test statistic obtained with full matrix $\mathbf{\hat \Gamma}$ is unstable and can have large variance especially for the case of low number $n$ of events in a histogram. The fact the matrix is not positive definite is equivalent to the fact that the minimal eigenvalue $\lambda_{min}$ of the matrix $\mathbf{\hat \Gamma}$ is $\leq 0$. A case when the minimal eigenvalue is positive but rather small is also not desirable, especially for computer calculations.\ Due to the above mentioned reasons it is wise to use the test statistic for a weighted histograms $$\hat X^2 =\hat X^2_k = \mathlarger{\sum}\limits_{i \neq k} \hat r_i \dfrac{(W_i-np_{i0})^2}{np_{i0}}+\dfrac{(\sum_{i \neq k} \hat r_i(W_i-np_{i0}))^2} {n(1-\sum_{i \neq k} \hat r_i p_{i0})}\label{newx}$$ for $k$ where $$\frac{p_{k0}}{\hat r_k}=\min_i\{\frac{p_{i0}}{\hat r_{i}}\}.$$ A secular equation for the new minimal eigenvalue can be solved numerically, by bisection method, to check whether a matrix ${\mathbf{\hat \Gamma_k}}$ is ​​positive definite or not. Numerical experiments show that it is very rare that the matrix ${\mathbf{\hat \Gamma_k}}$ is not positive definite and it happens only for histograms with a small number $n$ of events in a histogram. If hypothesis $H_0$ is valid, statistic $\hat X^2$ asymptotically has distribution $$\hat X^2 \sim \chi_{m-1}^2.$$ It is plausible that power of the new test is not lower than power of tests with statistic $\hat X_{Med}^2$ and with other statistics $\{\hat X^2_i, i \neq k\} $. The distribution of the statistic $\hat X^2$ is closer to $\chi_{m-1}^2$ then distribution of median statistic $\hat X_{Med}^2$. Also the statistic $\hat X^2$ is easier to calculate than the statistic $\hat X_{Med}^2$. Goodness of fit tests for weighted histograms with deviations from main model ============================================================================= Here, different deviations from the main model of weighted histograms will be considered as well as goodness of fit tests for those cases. Goodness of fit test for weighted histogram with unknown normalization ---------------------------------------------------------------------- In practice one is often faced with the case that all weights of events are defined up to an unknown normalization constant $C$ see Ref. [@gagunash]. It happens because in some cases of computer simulation is rather difficult give analytical formula for the PDF, but the PDF up to multiplicative constant is possible, that is enough for the generation of events according to the PDF, for example, by very popular Neumann’s method [@neiman]. For the goodness of fit test it means that if hypothesis $H_0$ is valid $$\textrm E [W_i]\cdot C=np_{i0}, \; i=1,\ldots,m. \label{ew}$$ with unknown constant $C$. Then the test statistic (\[stdd2\]) can be written as $$_c\hat{X^2_k}= \sum_{i \neq k} \hat r_i \frac{(W_i-np_{i0}/C)^2}{np_{i0}/C}+\frac{(\sum_{i \neq k} \hat r_i(W_i-np_{i0}/C))^2} {n(1-C^{-1}\sum_{i \neq k} \hat r_i p_{i0})}.\label{stdd33}$$ An estimator for the constant $C$ can be found by minimizing Eq. (\[stdd33\]). $$\hat C_k=\sum_{i \neq k}\hat{r}_ip_{i0}+ \sqrt{\frac{\sum_{i \neq k}\hat{r}_ip_{i0}}{\sum_{i \neq k}\hat{r}_i W_i^2/p_{i0}}}(n-\sum_{i \neq k}\hat{r}_i W_i), \label{consti}$$ where $\hat C_k$ is an estimator of $C$. Substituting (\[consti\]) to (\[stdd33\]), we get the test statistic $$_c\hat{X^2_k}= \sum_{i \neq k} \hat r_i \frac{(W_i-np_{i0}/\hat C_k)^2}{np_{i0}/\hat C_k}+\frac{(\sum_{i \neq k} \hat r_i(W_i-np_{i0}/\hat C_k))^2} {n(1-\hat C_k^{-1}\sum_{i \neq k} \hat r_i p_{i0})}.\label{stdd}$$ The statistic (\[stdd\]) has a $\chi^2_{m-2}$ distribution if hypothesis ${H_0}$ is valid. Formula (\[stdd\]) can be also transformed to $$_c\hat{X}^2_k =\frac{s_k^2}{n}+2s_k, \label{sss}$$ where $$s_k=\sqrt{\sum_{i \neq k}\hat{r}_i p_{i0} \sum_{i \neq k} \hat{r}_i{W}_i^2/p_{i0}} - \sum_{i \neq k}\hat{r}_i {W}_i$$ which is convenient for calculations, see [@gagunash]. Median statistics can be used for the same reason as in section 2.2 $$_c\hat{X}_{Med}^2= \textrm {Med }\, \{_c\hat X_1^2,\, _c\hat X_2^2, \ldots , \,_c\hat X_m^2\}. \label{stdav2}$$ and has approximately $\chi^2_{m-2}$ distribution if hypothesis ${H_0}$ valid, see Ref. [@gagunash] $$_c\hat{X}_{Med}^2 \sim \chi_{m-2}^2.$$ New goodness of fit test for weighted histogram with unknown normalization -------------------------------------------------------------------------- The new estimator of constant $C$ is $$\hat C=\sum_{i \neq k}\hat{r}_ip_{i0}+ \sqrt{\frac{\sum_{i \neq k}\hat{r}_ip_{i0}}{\sum_{i \neq k}\hat{r}_i W_i^2/p_{i0}}}(n-\sum_{i \neq k}\hat{r}_i W_i),$$ for $k$ where $$\frac{p_{k0}}{\hat r_k}=\min_i\{\frac{p_{i0}}{\hat r_{i}}\}.$$ And the test statistic can be written as $$_c\hat{X^2} = {\mathlarger{\sum}\limits_{i \neq k}} \hat r_i \dfrac{(W_i-np_{i0}/\hat C)^2}{np_{i0}/\hat C}+\dfrac{(\sum_1^m \hat r_i(W_i-np_{i0}/\hat C))^2} {n(1- \hat C^{-1}\sum_1^m \hat r_i. p_{i0})}\label{unknown}$$ Statistic $_c\hat X^2$ asymptotically has $\chi^2_{m-2}$ distribution if hypothesis $H_0$ is valid $$_c\hat{X}^2 \sim \chi^2_{m-2}.$$ Goodness of fit test for weighted Poisson histograms ---------------------------------------------------- Poisson histogram [@cousine] can be defined as histogram with multi-Poisson distributions of a number of events for bins $$P(n_1,\ldots ,n_m)=\prod_{i=1}^m e^{-n_0p_i}(n_0p_i)^{n_i}/n_i!,$$ where $n_0$ is a free parameter. The discrete probability distribution function (probability mass function) of a Poisson histogram can be represented as a product of two probability functions: a Poisson probability mass function for a number of events $n$ with parameter $n_0$ and a multinomial probability mass function of the number of events for bins of the histogram, with total number of events equal to $n$, see Ref. [@kendall] $$P(n_1,\ldots ,n_m)=e^{-n_0}(n_0)^{n}/n!\times\frac{n!}{n_1!n_2! \ldots n_m!} \; p_1^{n_1}\ldots p_m^{n_m}.$$ A Poisson histogram can be obtained as a result of two random experiments, namely, where the first experiment with Poisson probability mass function gives us the total number of events in histogram $n$, and then a histogram is obtained as a result of a random experiment with PDF $p(x)$ and the total number of events is equal to $n$. As in the case of multinomial histograms, also for Poisson histograms there is the problem of goodness of fit test with the hypothesis: $$H_0: p_1=p_{10},\ldots, p_{m-1}=p_{m-1,0} \text{ vs. } H_a: p_i \neq p_{i0} \text{ for some } i,$$ where $p_{i0}$ are specified probabilities, and $\sum_{i=1}^{m}p_{i0}=1$. If $n_0$ is known, then the statistic, see Ref. [@zech]: $$X_{pois}^2=\sum_{i=1}^{m} \frac{(n_i-n_0p_{i0})^2}{n_0p_{i0}}, \label{basic}$$ can be used and has asymptotically a $\chi^2_{m}$ distribution if the hypothesis $H_0$ is valid $$X_{pois}^2 \sim \chi^2_{m}.$$ The hypothesis $H_0$ becomes complex if parameter $n_0$ is unknown for the Poisson histogram. This is an opposite situation to the case of a multinomial histogram, where the hypothesis is simple. In [@zech] there are proposed statistics for goodness of fit test for a weighted Poisson histogram with known parameter $n_0$ $$X^2_{corr0}= \sum_{i=1}^m \frac{(W_i-n_0p_{i0})^2}{W_{2i}n_0p_{i0}/W_i}, \label{stx15}$$ and for the case the $n_0$ is not known: $$X^2_{corr}= \sum_{i=1}^m \frac{(W_i-\hat n_0p_{i0})^2}{W_{2i}\hat n_0p_{i0}/W_i}, \label{stx12}$$ with estimation of $n_0$ obtained by minimization of equation (\[stx15\]) $$\hat n_0= \left[\frac{\sum_{i=1}^m W_i^3/(W_{2i}p_{0i})}{\sum_{i=1}^m W_ip_{0i}/W_{2i}}\right]^{1/2}. \label{sx}$$ The distribution of statistic $X^2_{corr0}$ in case hypothesis $H_0$ is valid $$X_{corr0}^2 \sim \chi^2_{m}$$ and for the statistic $X^2_{corr}$ is $$X_{corr}^2 \sim \chi^2_{m-1}$$ according Ref. [@zech]. Generally, the power of the tests for Poisson histograms will be slightly lower than for multinomial histograms with the number of events $n=n_0$ which is explained by the fact that the total number of events for Poisson histograms fluctuates. The choice of the type of the histogram depends on what type of a physical experiment is produced. If the number of events $n$ is constant, then it is a multinomial histogram; if the number of events $n$ is a random value that has Poisson distribution, then it is a Poisson histogram. A weighted histogram very often is the result of modeling and the number of simulated events is known exactly, and therefore the choice of a multinomial histogram is reasonable. It is also reasonable to use tests developed for the multinomial histograms in the case, if the number of events $n$ is random value but with unknown distribution [@edie]. Restriction for goodness of fit tests applications ================================================== For the histograms with unweighted entries, the use of Pearson’s chi-square test (\[basic11\]) is inappropriate if any expected frequency $np_{i0}$ is below 1 or if the expected frequency is less than 5 in more than 20% of bins [@moore]. Restrictions for weighted histograms, due to fluctuation of the estimation of ratio of moments $\hat r_i$, can be made stronger. Namely, the use of new chi-square tests (\[newx\]) and (\[unknown\]) is inappropriate if any expected frequency $\textrm E[n_i]$ is less than 5. Following Ref. [@cochran] a disturbance is regarded as unimportant when the nominal size of the test is 5% and the size of the test lies between 4% and 6% for a goodness of fit tests. Numerical evaluation of the tests’ power and sizes ================================================== The main parameters which characterizes the effectiveness of a test are size and power. The nominal significance level was taken to be equal to 5% for calculating of size of tests in presented numerical examples. Hypothesis $H_0$ is rejected if test statistic $\hat X^2$ is larger than some threshold. Threshold $k_{0.05 }$ for a given nominal size of test 5% can be defined from the equation $$0.05 = P\,(\chi^2_l>k_{0.05})=\int_{k_{0.05}}^{+\infty} \frac{x^{l/2-1} e^{-x/2}}{2^{l/2}\Gamma(l/2)}dx, \label {kalfa}$$ where $l=m-1$. Let us define the test size $\alpha$ for a given nominal test size 5% as the probability $$\alpha = P\,(\hat X^2>k_{0.05}|H_0).\label {alfas}$$ This is the probability that hypothesis $H_0$ will be rejected if the distribution of weights $W_i$ for bins of the histogram satisfies hypothesis $H_0$. Deviation of the test size from the nominal test size is an important test characteristic. A second important test characteristic is the power. Let us define the test power as $$P\,(\hat X^2 >k_{0.05}|H_a). \label {beta}$$ This is the probability that hypothesis $H_0$ will be rejected if the distribution of weights $W_i$ for bins of the histogram does not satisfy hypothesis $H_0$. Notice that the power calculated by formula (\[beta\]) can give misleading result in case of comparing of different tests. To overcome this problem here we define the power of test $\pi$ as $$\pi = P\,(\hat X^2 >\mathcal{K}_{0.05}|H_a) \label {beta2}$$ with the threshold $\mathcal{K}_{0.05}$ calculated by Monte-Carlo method from equation $$0.05 = P\,(\hat X^2>\mathcal{K}_{0.05}|H_0).\label {alfass}$$ All definitions proposed above for statistics $\hat X^2$ can be used for other test statistics with appropriate number of degree of freedom $l$ in the formula (\[kalfa\]). The size and power of tests depend on the number of events and the binning that was discussed for usual histograms in Ref. [@kendall]. The power for weighted histograms also depends on the choice of PDF $g(x)$ (subsection 1.1) or $g_{tr}$ (subsection 1.2) and can be even higher than for histograms with unweighted entries as well as lower. Below we demonstrate two examples of an application of the previously discussed tests. The size and power of the tests are calculated for a different total number of events in the histograms. In numerical examples were demonstrated applications of: - Pearson’s goodness of fit test [@pearson], see subsection 2.1 and first paragraph of section 2. The test statistic is $X_2$ (\[basic11\]). - goodness of fit test for weighted histograms with normalized weights [@gagunash], see subsection 2.2. The test statistic is $\hat X_{Med}^2$ (\[stdav\]). - goodness of fit test for weighted histograms with unnormalized weights [@gagunash], see subsection 3.1. The test statistic is $_{c}\hat X_{Med}^2$ (\[stdav2\]). - new goodness of fit test for weighted histograms with normalized weights, see subsection 2.3. The test statistic is $\hat X^2$(\[newx\]). - new goodness of fit test for weighted histograms with unnormalized weights, see subsection 3.2. The test statistic is $_{c}\hat X^2$(\[unknown\]). - goodness of fit test for Poisson histograms with unweighted entries and known parameter $n_0$ [@zech], see subsection 3.3. The test statistic is $X_{pois}^2$(\[basic\]). - goodness of fit test for weighted Poisson histograms with known parameter $n_0$ [@zech], see subsection 3.3. The test statistic is $X_{corr0}^2$(\[stx15\]). - goodness of fit test for weighted Poisson histograms with unknown parameter $n_0$ [@zech], see subsection 3.3. The test statistic is $X_{corr}^2$(\[stx12\]). The published program, see Ref. ([@gagunash3]), was used for the calculation of the test statistics with minor modification needed for the new tests. Numerical example 1 ------------------- A simulation study was done for the example from Ref. [@gagunash]. Weighted histograms described in subsection 1.1, are used here. The PDF for hypothesis $H_0$ is: $$p_0(x) \propto \frac{2}{(x-10)^2+1}+\frac{1.15}{(x-14)^2+1}\label{basicpar}$$ against alternative $H_a$: $$p(x) \propto \frac{2}{(x-10)^2+1}+\frac{1}{(x-14)^2+1}$$ represented by the weighted histogram. Both PDF’s are defined on the interval $[4,16]$. A calculation was done for three cases of a PDF, used for the event generation, see Fig. \[fig:picture1\] $$g_1(x)=p(x) \label{case1}$$ $$g_2(x)=1/12 \label{case2}$$ $$g_3(x) \propto \frac{2}{(x-9)^2+1}+\frac{2}{(x-15)^2+1} \label{case3}.$$ Distribution (\[case1\]) gives an unweighted histogram. Distribution (\[case2\]) is a uniform distribution on the interval $[4, 16]$. Distribution (\[case3\]) has the same type of parameterizations as Eq. (\[basicpar\]), but with different values of the parameters. Histograms with 20 bins and equidistant binning were used. At Fig. \[fig:hyalbw\] presented probabilities $p_i, i=1,...,20$ for the PDF $p(x)$ and $p_{0i}, i=1,...,20$ for the PDF $p_{0}(x)$. Size and power of tests with statistics $\hat X^2$ (\[newx\]), $_{c}\hat X^2$ (\[unknown\]), $\hat X_{Med}^2$ (\[stdav\]) and $_{c}\hat X_{Med}^2$ (\[stdav2\]) were calculated for weighted histograms with weights of events equal to $p(x)/g_2(x)$ and $p(x)/g_3(x)$. Statistics $\hat X^2$ (\[newx\]) and $\hat X_{Med}^2$ (\[stdav\]) coincide with Pearson’s statistic $X^2$ (\[basic11\]) and were used for histograms with unweighted entries. The results of calculations for 100000 runs are presented in Table \[tab:result31\].\ $n$ 200 400 600 800 1000 3000 5000 7000 9000 w(x) -- -- ---------- -------- -------- -------- ------ ------ ------ ------ ------- ------- ------ $\alpha$ *5.7* *5.4* 5.3 5.2 5.2 5.1 5.0 5.0 5.1 $\pi$ *6.0* *7.1* 8.2 9.8 11.2 29.9 52.7 71.6 84.9 $\alpha$ 5.5 5.3 5.2 5.1 5.0 5.1 5.1 5.1 4.9 $\pi$ 6.1 7.0 8.2 9.2 10.5 26.2 45.8 64.0 78.7 $\alpha$ 5.0 5.1 5.0 5.0 4.9 5.0 5.0 5.2 4.9 $\pi$ 6.0 7.0 8.1 9.1 10.4 26.0 45.6 63.0 78.1 $\alpha$ 5.4 5.4 5.3 5.2 5.1 5.3 5.2 5.3 5.0 $\pi$ 6.0 6.9 8.0 9.1 10.3 25.7 45.3 63.1 78.2 $\alpha$ 5.6 5.8 5.7 5.7 5.5 5.7 5.7 5.8 5.5 $\pi$ 5.9 6.9 8.0 9.1 10.2 25.4 44.9 62.5 77.5 $\alpha$ *7.3* *6.6* *6.1* 5.8 5.6 5.2 5.2 4.9 5.0 $\pi$ *16.2* *29.7* *40.1* 48.5 56.1 95.7 99.8 100.0 100.0 $\alpha$ *4.7* *4.9* *5.0* 5.0 5.1 5.1 5.0 4.9 5.0 $\pi$ *6.9* *8.3* *9.9* 11.6 13.4 36.5 61.6 80.5 91.2 $\alpha$ *5.5* *5.3* *5.4* 5.3 5.5 5.4 5.2 5.3 5.2 $\pi$ *7.9* *11.8* *15.8* 20.2 25.0 75.3 96.6 99.8 100.0 $\alpha$ *5.4* *5.5* *5.7* 5.6 5.8 5.7 5.6 5.7 5.5 $\pi$ *6.8* *8.4* *9.7* 11.4 13.1 36.0 60.7 79.2 90.7 : Numerical example 1. Size ($\alpha$) and power ($\pi$) of different test statistics $X^2$(\[basic11\]), $\hat X^2$(\[newx\]), $_{c}\hat X^2$(\[unknown\]), $\hat X_{Med}^2$(\[stdav\]), $_{c}\hat X_{Med}^2$(\[stdav2\]) obtained for different weighted functions $w(x)$. Italic type marks a size of test with inappropriate number of events in the bins of histograms.[]{data-label="tab:result31"} Conclusion and interpretation of results presented in Table \[tab:result31\]. - The size of new tests $\hat X^2$ (\[newx\])(rows 2, 6) and $ _{c}\hat X^2$(\[unknown\])(row 3, 7) are generally closer to nominal value 5% then median tests $\hat X_{Med}^2$ (\[stdav\])(rows 4, 8) and $_{c}\hat X_{Med}^2$ (\[stdav2\])(rows 5, 9) when the application of the test satisfies restrictions formulated in section 4. - The power of new tests $\hat X^2$ (\[newx\]) (rows 2, 6) are greater than for analogous median tests $\hat X_{Med}^2$ (\[stdav\])(rows 4, 8). The power of tests $ _{c}\hat X^2 $(\[unknown\])(rows 3, 7) are greater than for analogous median tests $_{c}\hat X_{Med}^2$ (\[stdav2\])(rows 5, 9). - The power of all tests calculated for histograms with weights of events equal to $ p(x)/g_2(x)$ (rows 2-5) are lower then for histogram with unweighted entries (row 1), but the power of all tests calculated for histograms with weights of events equal to $p(x)/g_3(x)$ (rows 6-9) are greater. The explanation is that in latter case we increase the statistics of events for domains with high deviation of the distribution presented by the histogram from the tested distribution. Properties of tests in applications to Poisson histograms with the same weighted functions and distributions of events were investigated. In this case, the total number of events $n$ is random and was simulated according Poisson distribution for a given parameter $n_0$. Size and power of tests $X_{pois}^2$ (\[basic\]), $X_{corr0}^2$ (\[stx15\]) with exactly known parameter $n_0$ and $X_{corr}^2$ (\[stx12\]) developed ad hoc for the Poisson histogram in [@zech] also was calculated. Results of the calculations are presented in Table \[tab:result25\] . $n_0$ 200 400 600 800 1000 3000 5000 7000 9000 w(x) -- -- ---------- -------- -------- -------- ------ ------ ------ ------ ------- ------- ------ $\alpha$ 6.0 5.6 5.2 5.2 5.2 5.1 5.1 5.1 5.1 $\pi$ 5.9 7.0 8.3 9.6 11.1 29.2 50.9 70.0 83.8 $\alpha$ *5.6* *5.5* 5.1 5.2 5.1 5.1 5.1 5.1 5.0 $\pi$ *6.0* *7.0* 8.4 9.8 11.1 30.0 52.2 71.2 85.0 $\alpha$ 5.4 5.3 5.2 5.1 5.1 5.0 5.0 5.1 5.0 $\pi$ 6.0 6.7 7.8 8.8 10.0 25.0 43.9 61.5 76.2 $\alpha$ 3.9 4.4 4.6 4.7 4.7 5.0 5.0 5.0 4.9 $\pi$ 6.0 7.0 8.0 9.0 10.3 25.5 45.0 62.9 77.4 $\alpha$ 5.5 5.2 5.2 5.1 5.0 5.1 5.0 5.0 5.0 $\pi$ 6.1 7.1 8.1 9.2 10.6 26.3 46.0 64.1 78.5 $\alpha$ 5.1 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 $\pi$ 6.0 7.0 8.1 9.2 10.5 26.0 45.5 63.4 77.6 $\alpha$ 5.1 5.0 5.2 5.0 5.2 5.1 5.1 5.1 4.9 $\pi$ 6.3 7.5 8.8 10.7 12.3 35.3 60.5 79.6 91.3 $\alpha$ 3.5 4.1 4.5 4.7 4.8 5.0 5.0 5.0 4.9 $\pi$ 7.0 8.4 9.7 11.6 13.4 36.0 60.4 79.2 90.8 $\alpha$ *7.2* *6.5* *6.0* 5.6 5.6 5.3 5.1 5.1 4.9 $\pi$ *16.4* *30.1* *40.1* 48.8 56.1 95.7 99.8 100.0 100.0 $\alpha$ *4.6* *4.9* *4.9* 5.0 5.1 5.0 5.0 5.0 5.0 $\pi$ *7.0* *8.5* *10.0* 11.7 13.5 37.0 61.7 80.0 91.2 : Numerical example 1. Size ($\alpha$) and power ($\pi$) of different test statistics $X_{pois}^2$(\[basic\]), $X^2$(\[basic11\]) , $X_{corr0}^2$(\[stx15\]), $X_{corr}^2$(\[stx12\]) , $\hat X^2$(\[newx\]) , $_{c}\hat X^2 $(\[unknown\]) in application for Poisson histograms. Italic type marks a size of test with inappropriate number of events in the bins of histograms.[]{data-label="tab:result25"} \ Conclusion and interpretation of results presented in Table \[tab:result25\]. - The size of all tests are close to nominal value 5%. - The power of new tests $\hat X^2$(\[newx\])(rows 5, 9) and $ _{c}\hat X^2$(\[unknown\])(rows 6, 10) used for Poisson histograms are greater than the power of tests developed ad hoc for the Poisson histograms $X_{corr0}^2$(\[stx15\])(rows 3, 7) with the exactly known parameter $n_0$ and $X_{corr}^2$(\[stx12\])(rows 4, 8) with the unknown parameter $n_0$ in Ref. [@zech]. - The power of Pearson’s test $X^2$(\[basic11\])(row 2) used for Poisson histograms is greater than test $X_{pois}^2$(\[basic\])(row 1) with the exactly known parameter $n_0$ proposed in Ref. [@zech]. Numerical example 2 ------------------- A simulation study was done for the example described in Ref. [@zechebook] and also in Ref. [@gagpar]. Weighted histograms described in subsection 1.2 are used here. The PDF $p_0(x)$ for the hypothesis $H_0$ is taken according to formula (\[p1\_main\]) with: $$p_{0 tr}(x')=0.4(x'-0.5)+1; \,\, x'\in [0,1]$$ $$A(x')=1-(x'-0.5)^2$$ $$R(x|x')=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[ -\frac{(x-x')^2}{2\sigma^2}\right], \text{with}\,\,\, \sigma=0.3.$$ For the alternative $H_a$, $p(x)$ is taken with the same acceptance and resolution function according to formula (\[p1\_main\]) with: $$p_{tr}(x')=0.6666(x'-0.5)+1; \,\, x'\in [0,1]$$ that is presented by the weighted histogram. A calculation was done for two cases of PDFs used for event generation, see Fig. \[fig:trhyal1\]. $$h_1(x')=0.6666(x'-0.5)+1; \,\,\, x'\in [0,1]$$ and\ $$h_2(x')=-0.6666(x'-0.5)+1; \,\,\, x'\in [0,1].$$ In the first case, a weighted histogram is the histogram with weights of events equal to 1 ( histogram with unweighted entries) and, in the second case, weights of events equal to $h_1(x')/h_2(x')$. The results of this calculation for 100000 runs are presented in tables 3. We use a histogram with 20 bins on interval $[-0.3,1.3]$. Fig. \[fig:hyal\] presented probabilities $p_i, i=1,...,20$ for the PDF $p(x)$ and $p_{0i}, i=1,...,20$ for the PDF $p_{0}(x)$. Here, we add two bins for events with $x \leqslant -0.3$ and $x > 1.3$ as well as one bin for events that were not registered due to limited acceptance. Total number of bins $m$ is used in test equal to 23. The results of calculations of the sizes and power of tests for 100000 runs are presented in Table \[tab:result82\].\ $n$ 200 400 600 800 1000 3000 5000 7000 9000 w(x) -- -- ---------- -------- -------- ------ ------ ------ ------ ------ ------ ------ ------ $\alpha$ *5.1* 5.1 5.1 5.0 5.2 5.1 5.1 5.0 5.0 $\pi$ *5.6* 6.6 7.5 8.8 9.8 25.9 45.7 64.9 79.4 $\alpha$ *7.0* *6.2* 5.8 5.6 5.5 5.1 5.0 4.9 4.9 $\pi$ *8.4* *9.4* 10.9 12.8 14.6 40.9 67.1 85.3 94.5 $\alpha$ *5.6* *5.6* 5.5 5.4 5.3 5.1 5.0 5.0 4.9 $\pi$ *6.4* *7.4* 8.4 9.9 11.0 28.0 47.9 66.4 80.5 $\alpha$ *10.9* *7.4* 6.6 6.1 6.1 5.7 5.6 5.6 5.6 $\pi$ *9.1* *10.1* 11.5 13.9 15.8 43.7 70.9 87.8 95.8 $\alpha$ *7.8* *6.6* 6.3 5.9 5.9 5.7 5.7 5.7 5.6 $\pi$ *6.1* *7.2* 8.4 9.7 10.9 27.4 46.9 65.0 79.2 : Numerical example 2. Sizes ($\alpha$) and powers ($\pi$) of different test statistics $X^2$(\[basic11\]), $\hat X^2$(\[newx\]), $_{c}\hat X^2$(\[unknown\]), $\hat X_{Med}^2$(\[stdav\]), $_{c}\hat X_{Med}^2$(\[stdav2\]) obtained for different weighted functions $w(x)$. Italic type marks a size of test with inappropriate number of events in the bins of histograms.[]{data-label="tab:result82"} Conclusion and interpretation of results presented in Table \[tab:result82\]. - The size of new tests $\hat X^2$(\[newx\]) and $ _{c}\hat X^2$(\[unknown\]) (row 2, 3 ) is more close to the nominal value 5% then the size of median tests $\hat X_{Med}^2$(\[stdav\]) and $_{c}\hat X_{Med}^2$(\[stdav2\]) (rows 4, 5). - The power of new tests $\hat X^2$(\[newx\]) and $_{c}\hat X^2$(\[unknown\]) (rows 2,3) is roughly the same compared with analogous median tests $\hat X_{Med}^2$(\[stdav\]) and $_{c}\hat X_{Med}^2$(\[stdav2\]) (rows 4, 5). - All tests demonstrate greater power then Pearson’s test $X^2$(\[basic11\]) (row 1) used for the histogram with unweighted entries. The property of tests in application for Poisson histograms is investigated with the same weighted functions and distributions of events. In this case, the number of events $n$ in a histogram was simulated according Poisson distribution with given parameter $n_0$. The size and power of tests developed for the Poisson histogram in [@zech] was also calculated. Results of calculations are presented in Table \[tab:result41\]. $n_0$ 200 400 600 800 1000 3000 5000 7000 9000 w(x) -- -- ---------- ------- ------- ------ ------ ------ ------ ------ ------ ------ ------ $\alpha$ 5.5 5.3 5.3 5.0 5.2 5.1 5.1 5.0 5.2 $\pi$ 5.5 6.4 7.4 8.7 9.7 25.6 45.0 64.0 77.9 $\alpha$ *5.1* 5.1 5.1 4.9 5.2 5.0 5.1 5.0 5.2 $\pi$ *5.5* 6.5 7.5 8.9 9.8 26.3 45.9 65.0 78.8 $\alpha$ 5.8 5.8 5.5 5.3 5.3 5.0 5.1 5.0 5.0 $\pi$ 5.8 6.6 7.7 9.0 10.2 27.2 47.0 66.5 80.7 $\alpha$ 4.2 4.9 4.9 4.8 4.9 4.9 5.0 4.9 5.0 $\pi$ 6.3 7.2 8.4 9.7 11.1 27.5 46.9 65.5 79.5 $\alpha$ *6.8* *6.1* 5.8 5.6 5.4 5.1 5.0 5.0 4.9 $\pi$ *8.4* *9.4* 11.0 12.7 14.8 41.1 67.2 85.1 94.4 $\alpha$ *5.4* *5.6* 5.5 5.3 5.3 5.0 4.9 5.0 4.9 $\pi$ *6.5* *7.4* 8.5 9.8 11.0 28.0 48.2 66.4 80.4 : Numerical example 2. Size ($\alpha$) and power ($\pi$) of different test statistics $X_{pois}^2$(\[basic\]), $X^2$, $X_{corr0}^2$(\[stx15\]), $X_{corr}^2$(\[stx12\]), $\hat X^2$(\[newx\]), $_{c}\hat X^2 $(\[unknown\]) in application for Poisson histograms. Italic type marks a size of test with inappropriate number of events in the bins of histograms.[]{data-label="tab:result41"} Conclusion and interpretation of results presented by Table \[tab:result41\]. - The size of all tests are close to nominal value 5%. - Basically, the power of new tests $\hat X^2$(\[newx\]) and $ _{c}\hat X^2$(\[unknown\])(rows 5, 6) in applying for Poisson histograms are greater than the power of tests developed ad hoc for the Poisson histograms $X_{corr0}^2$(\[stx15\]) with the exactly known parameter $n_0$ and $X_{corr}^2$(\[stx12\]) (rows 3, 4) with the unknown parameter $n_0$ in Ref. [@zech]. - The power of Pearson’s test $X^2$ (row 2) used for Poisson histograms is greater than power of test $X_{pois}^2$(\[basic\])(row 1) with the exactly known parameter $n_0$. Generally the numerical example 1 and example 2 demonstrate the superiority of new goodness of fit tests under existing tests for weighted histograms, see Ref. [@gagunash] and for weighted Poisson histograms, see Ref. [@zech]. Conclusion ========== A review of goodness of fit tests for weighted histograms was presented. The bin content of a weighted histogram was considered as a random sum of random variables that permits to generalize the classical Pearson’s goodness of fit test for histograms with weighted entries. Improvements of the chi-square tests with better statistical properties were proposed. Evaluation of the size and power of tests was done numerically for different types of weighted histograms with different numbers of events and different weight functions. Generally the size of new tests is closer to nominal value and power is not lower than have existing tests. Except direct application of tests in data analysis, see for example Ref. [@physlet], the proposed tests are necessary bases for generalization of test in the case when some parameters must be estimated from the data, see Ref. [@cramer], as well as for the generalisation of test for comparing weighted and unweighted histograms or two weighted ones (homogeneity test), see Refs. [@cramer; @gagcomp; @gagcpc]. Parametric fit of data obtained from detectors with finite resolution and limited acceptance is one of important application of methods developed for weighted histograms that can be used for experimental data interpretation, see Refs. [@gagpar].\ [**[Acknowledgements]{}**]{}\ The author is grateful to Johan Blouw for useful discussions and careful reading of the manuscript and thanks the University of Akureyri and the MPI for Nuclear Physics for support in carrying out the research. [9]{} M. G. Kendall, A. S. Stuart, The Advanced Theory of Statistics, Vol. 2, ch. 30, sec. 30.4, Griffin Publishing Company, London, 1973. N. D. Gagunashvili, Nucl. Instr. Meth. A 596(2008)439. I. M. Sobol, Numerical Monte Carlo methods, ch.5, Nauka, Moscow, 1973. K. Pearson, Phil. Mag. 50(1900)157. M. G. Kendall, A. S. Stuart, The Advanced Theory of Statistics, Vol. 1, ch.15, sec. 15.10, Griffin Publishing Company, London, 1973. B.V. Gnedenko, V.Yu. Korolev, Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton, Florida, 1996. H. V. Henderson, S. R. Searle, SIAM Rev. 23(1981)53. H. Robbins, Bull. Amer. Math. Soc. 54(1948)1151. N. D. Gagunashvili, Nucl. Instr. Meth. A614(2010)287. G. H. Golub, SIAM Rev. 15(1973)318. J. Neumann, Various techniques used in connection with random digits, NBS Appl. Math. series, 12(1951)36-38. S. Baker, R. D. Cousins, Nucl. Instr. Meth. 221(1984)437. G. Zech, Nucl. Instr. Meth. A691(2012)178. W. T. Eadie, D. Dryard, F. E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics, North-Holland Publishing Company, ch. 4, sec.4.1.2, Amsterdam,London, 1971. D. S. Moore, G. P. McCabe, Introduction to the Practice of Statistics, W. H. Freeman Publishing Company, New York, 2007. W. G. Cochran, Ann. of Math. Stat. 23(1952)315. N. D. Gagunashvili, Comput. Phys. Commun. 183(2012)418. G. Bohm, G. Zech, Introduction to Statistics and Data Analysis for Physicists, Verlag Deutsches Elektronen-Synchrotron, 2010. N. D. Gagunashvili, Nucl. Instr. Meth. A635(2011)86. D0 Collaboration, V. M. Abazov et al., Phys. Lett. B 693(2010)515. H. Cramer, Mathematical methods of statistics, ch.  30, Princeton University Press, Princeton, 1999. N. D. Gagunashvili, Nucl. Instr. Meth. A614(2010)287. N. D. Gagunashvili, Comput. Phys. Commun. 183(2012)193.

--- abstract: | Many investment models in discrete or continuous-time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change-of-variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank-dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton’s portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well-posedness, attainability and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law-invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari’s dual model, Lopes’ SP/A model, and optimal stopping models under CPT or RDUT.\ [<span style="font-variant:small-caps;">Key Words</span>:]{} Portfolio choice/selection, behavioral finance, law-invariant, quantile formulation, probability weighting/distortion function, change of variable, relaxation method, calculus of variations, CPT, RDUT, time consistency, atomic, atomless/non-atomic, functional optimization problem. author: - | <span style="font-variant:small-caps;">Zuo Quan Xu</span>[^1]\ *The Hong Kong Polytechnic University* date: 07 April 2014 title: '**A NOTE ON THE QUANTILE FORMULATION**' --- INTRODUCTION ============ Classical expected utility theory (EUT) as a model of choice under uncertainty fails to explain a number of paradoxes. Among the alternative models proposed, Kahneman and Tversky’s (1979, 1992) cumulative prospect theory (CPT) provides one of the best explanations of these paradoxes. This theory consists of three components: an $S$-shaped utility function[^2], a reference point, and probability weighting/distortion functions. The last two are missing in EUT. In light of these theoretical developments, it is natural to consider investment problems that involve probability weighting functions. However, the probability weighting functions make these problems time-inconsistent so that these problems cannot be studied using only classical dynamic programming or probabilistic approaches. Jin and Zhou (2008) initiated the study of portfolio choice problems under CPT with probability weighting functions in continuous-time settings. They solved the problem by assuming the monotonicity of a function related to the pricing kernel and probability weighting function. However, this assumption is so restrictive that it excludes most probability weighting functions that are typically used, including that proposed by Tversky and Kahneman (1992), in the Black-Scholes market setting. Jin, Zhang, and Zhou (2011) considered the same portfolio choice problem under the scenario of a loss constraint with the same assumption. He and Zhou (2011) investigated general models with law-invariant preference measures, including the classical Merton’s portfolio choice model under EUT, the mean-variance model, the goal reaching model, the Yaari’s dual model, the Lopes’ SP/A model, the behavioral model under CPT, and those explicitly involving VaR and CVaR in their objectives and/or constraints. Their work took a step forward and reduced the monotonicity assumption in Jin and Zhou (2008) to a piece-wise monotonicity assumption. The results cover the probability weighting functions proposed by Tversky and Kahneman (1992), Tversky and Fox (1995), and Prelec (1998). Xu and Zhou (2013) initiated the study of continuous-time optimal stopping problem under CPT and solved the problem under the same assumption of piece-wise monotonicity as He and Zhou (2011). By adopting the calculus of variations, Xia and Zhou (2012) achieved a breakthrough. They proposed and solved a portfolio choice problem under rank-dependent utility theory (RDUT) with no monotonicity assumptions. Their method also works for general models with law-invariant preference measures. However, they use techniques from the calculus of variations and have extensive recourse to convex analysis, so their arguments are lengthy, technical, and difficult to follow. In this paper, without making any monotonicity assumptions, we propose a new and easy-to-follow method to study the portfolio choice problem under RDUT. A complete and compact argument replaces the lengthy calculus of variations argument in Xia and Zhou (2012). The main idea is as follows. After transforming the portfolio choice problem into its quantile formulation, we make a change of variable to remove the probability weighting function from the objective and reveal the essence of the problem. In the literature, the optimal solution is commonly obtained by point-wise maximizing the Lagrangian in the objective. However, such a solution may not be a quantile function. Our idea is to replace a part of the Lagrangian to relax the problem so that the new problem can be solved by point-wise maximizing the new Lagrangian, and then to show that there is no gap between the old and new Lagrangians in this point-wise solution. Through this approach, we show that solving a portfolio choice problem under RDUT reduces to solving a classical Merton’s portfolio choice problem under EUT with the same utility function but a different pricing kernel, which is determined by the given pricing kernel and probability weighting function. Moreover, the quantile optimization problem is avoided in the latter. As with Xia and Zhou (2012), the method is applicable to general models with law-invariant preference measures. In the literature, there is no study on feasibility, well-posedness, attainability and uniqueness issues for the portfolio choice problem under RDUT[^3]. We investigate these issues by linking the portfolio choice problem under RDUT to a classical Merton’s portfolio choice problem under EUT for which the issues have been completely solved in Jin, Xu and Zhou (2008). The remainder of this paper is organized as follows. In Section 2, we formulate a portfolio choice problem under RDUT and define its quantile formulation. In Section 3, we introduce a key step — making a change of variable — to formulate an equivalent quantile optimization problem, in which the probability weighting function is removed from the objective. The problem is then completely solved by a new relaxation method in Section 4. In Section 5, we demonstrate how to transform the portfolio choice problem under RDUT into an equivalent classical Merton’s portfolio choice problem under EUT. The feasibility, well-posedness, attainability and uniqueness issues for the portfolio choice problem under RDUT are also investigated in this section. We conclude the paper in Section 6. <span style="font-variant:small-caps;">PROBLEM FORMULATION</span> ================================================================= Using martingale representation theory (see, e.g., Pliska (1986), Karatzas, Lehoczky, and Shreve (1987), Cox and Huang (1989, 1991)), the dynamic portfolio choice problem under RDUT in a complete market setting[^4] reduces to finding a random outcome $X$ to $$\begin{aligned} \label{objective} \sup_X \quad & \int_0^{\infty} u(x)\operatorname{\mathrm{d\!}}\:\big(1-w(1-F_X(x))\big),\\ \nonumber\textrm{subject to}\quad & \operatorname{\mathbf{E}}[ {\rho} X]=x_0,\quad X\geqslant 0,\end{aligned}$$ where $F_X(\cdot)$ is the probability distribution function of $X$; $w(\cdot)$ is the probability weighting function which is differentiable and strictly increasing on $[0,1]$ with $w(0)=0$ and $w(1)=1$; $u(\cdot)$ is the utility function which is strictly increasing and second order differentiable on $\operatorname{\mathbb{R}}^+$ with $u''(\cdot)<0$; and $\rho>0$ is the pricing kernel, also called the stochastic discount factor or state pricing density. We always have that $\operatorname{\mathbf{E}}[\rho]<+\infty$. If $w(\cdot)$ is the identity function, i.e., $w(x)=x$ for all $x\in[0,1]$, then $$\begin{aligned} \int_0^{\infty} u(x)\operatorname{\mathrm{d\!}}\:\big(1-w(1-F_X(x))\big)=\int_0^{\infty} u(x)\operatorname{\mathrm{d\!}}F_X(x)=\operatorname{\mathbf{E}}[u(X)],\end{aligned}$$ for any $X\geqslant 0$, and consequently, problem reduces to a classical Merton’s portfolio choice problem under EUT: $$\begin{aligned} \sup_X \quad & \operatorname{\mathbf{E}}[u(X)],\\ \textrm{subject to}\quad & \operatorname{\mathbf{E}}[ {\rho} X]=x_0,\quad X\geqslant 0.\end{aligned}$$ To tackle problem , in the literature (see, e.g., Jin and Zhou (2008), Jin, Zhang, and Zhou (2011), He and Zhou (2011, 2012), Xia and Zhou (2012)), it is always assumed that \[assmp:atomless\] The pricing kernel is atomless[^5]. Under this assumption, solving problem then reduces to solving a quantile[^6] optimization problem $$\begin{aligned} \label{objective0} \sup\limits_{G(\cdot)\in\operatorname{\mathcal{G}}_{x_0}} \int_{0}^{1}u(G(x))w'(1-x) \operatorname{\mathrm{d\!}}x,\end{aligned}$$ where the set $\operatorname{\mathcal{G}}_{x_0}$ is given by $$\begin{aligned} \operatorname{\mathcal{G}}_{x_0}&:=\left\{G(\cdot)\in\operatorname{\mathcal{G}}:\int_0^1 G(x)F^{-1}_{\rho}(1-x)\operatorname{\mathrm{d\!}}x=x_0\right\},\end{aligned}$$ the set $\operatorname{\mathcal{G}}$ denotes the set of all quantile functions: $$\begin{aligned} \operatorname{\mathcal{G}}&:=\left\{ G(\cdot): (0,1)\mapsto\operatorname{\mathbb{R}}^+, \text{ increasing and right-continuous with left limits (RCLL)} \right\},\end{aligned}$$ and $F^{-1}_{\rho}(\cdot) \in\operatorname{\mathcal{G}}$ denotes the quantile function of the pricing kernel $\rho$. By Assumption \[assmp:atomless\], $\rho$ is atomless, so $F^{-1}_{\rho}(\cdot)$ is strictly increasing. Problem and problem are linked as follows. The optimal solution $X^*$ to problem and the optimal solution $G^*(\cdot)$ to problem satisfy $$\begin{aligned} \label{xstar0} X^*=G^*(1-F_{\rho}(\rho)).\end{aligned}$$ For this reason, problem is called the quantile formulation of problem . Before Xia and Zhou (2012), problem was partially solved under certain monotonicity assumptions in the literature. Xia and Zhou (2012) used the calculus of variations to tackle it without making those monotonicity assumptions, but their arguments are lengthy and complex. Moreover, they did not study the feasibility, well-posedness, attainability or uniqueness issues for problem . In this paper, we propose a simple change-of-variable and relaxation method to tackle problem without making any assumptions. We also solve the feasibility, well-posedness, attainability and uniqueness issues for problem by linking the problem to a classical Merton’s portfolio choice problem under EUT. In the literature, $\operatorname{\mathcal{G}}_{x_0}$ is often replaced by $$\begin{aligned} \overline{\operatorname{\mathcal{G}}}_{x_0}&:=\left\{G(\cdot)\in\operatorname{\mathcal{G}}:\int_0^1 G(x)F^{-1}_{\rho}(1-x)\operatorname{\mathrm{d\!}}x\leqslant x_0\right\}.\end{aligned}$$ However, there is no difference between considering problem for $\operatorname{\mathcal{G}}_{x_0}$ or $\overline{\operatorname{\mathcal{G}}}_{x_0}$ because the optimal solution to problem in $\overline{\operatorname{\mathcal{G}}}_{x_0}$, if it exists, must belong to $\operatorname{\mathcal{G}}_{x_0}$. Here we assume that the pricing kernel is atomless as according to convention. However, if one studies economic equilibrium models with law-invariant preference measures (see, e.g., Xia and Zhou (2012)), the pricing kernel will be a part of the solution, so one cannot make a priori any assumption on it. The quantile formulation problem with an atomic pricing kernel is solved in Xu (2014). <span style="font-variant:small-caps;">CHANGE OF VARIABLE</span> ================================================================ To tackle problem , our first main idea in this paper is to make a change of variable to remove the probability weighting function from the objective. Let $\inversew: [0,1]\mapsto [0,1]$ be the inverse mapping of $x\mapsto 1-w(1-x)$, that is $$\inversew(x):=1-w^{-1}(1-x), \quad x\in[0,1].$$ Then $\inversew(\cdot)$ is also a probability weighting function that is differentiable and strictly increasing on $[0,1]$. It follows that $$\begin{gathered} \int_{0}^{1}u(G(x))w'(1-x) \operatorname{\mathrm{d\!}}x=\int_{0}^{1}u(G(x))\operatorname{\mathrm{d\!}}\; (1-w(1-x))\\ =\int_{0}^{1}u(G(x))\operatorname{\mathrm{d\!}}\; (\inversew^{-1}(x))=\int_{0}^{1}u(G(\inversew(x)))\operatorname{\mathrm{d\!}}x=\int_{0}^{1}u(Q(x))\operatorname{\mathrm{d\!}}x,\end{gathered}$$ where $$Q(x)=G(\inversew(x)),\quad x\in(0,1).$$ Note that $$\begin{gathered} \operatorname{\mathcal{G}}_{x_0}=\left\{G(\cdot)\in\operatorname{\mathcal{G}}:\int_0^1 G(x)F^{-1}_{\rho}(1-x)\operatorname{\mathrm{d\!}}x=x_0 \right\}\\ =\left\{G(\cdot)\in\operatorname{\mathcal{G}}:\int_0^1 G(\inversew(x))F^{-1}_{\rho}(1-\inversew(x))\inversew'(x)\operatorname{\mathrm{d\!}}x=x_0 \right\}.\end{gathered}$$ Therefore, we conclude that $G(\cdot)\in\operatorname{\mathcal{G}}_{x_0}$ if and only if $Q(\cdot)\in\operatorname{\mathcal{Q}}$, where $$\begin{gathered} \operatorname{\mathcal{Q}}:=\left\{Q(\cdot): (0,1)\mapsto\operatorname{\mathbb{R}}^+,\text{ increasing and RCLL with }\int_0^1 Q(x)\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0\right\}\\ =\left\{Q(\cdot)\in \operatorname{\mathcal{G}}:\int_0^1 Q(x)\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0\right\},\end{gathered}$$ and $$\begin{gathered} \label{defi:variphi} \varphi(x):=-\int_x^1 F^{-1}_{\rho}(1-\inversew(y))\inversew'(y)\operatorname{\mathrm{d\!}}y=-\int_{\inversew(x)}^1F^{-1}_{\rho}(1-y)\operatorname{\mathrm{d\!}}y \\ =-\int_0^{1-\inversew(x)}F^{-1}_{\rho}(y)\operatorname{\mathrm{d\!}}y=-\int_{0}^{w^{-1}(1-x)}F^{-1}_{\rho}(y)\operatorname{\mathrm{d\!}}y , \quad x\in[0,1].\end{gathered}$$ Note that $\varphi(\cdot)$ is a differentiable and strictly increasing function on $[0,1]$ with $\varphi(0)=-\operatorname{\mathbf{E}}[\rho]$ and $\varphi(1)=0$. By making this change of variable, problem has now been transformed into an equivalent problem: $$\begin{aligned} \label{objective0'} \sup\limits_{Q(\cdot)\in\operatorname{\mathcal{Q}}} \int_{0}^{1}u(Q(x)) \operatorname{\mathrm{d\!}}x,\end{aligned}$$ in which the probability weighting function does not appear in the objective. From now on, we focus on this problem. We point out here that although the objective of problem does not involve the probability weighting function, the constraint set $\operatorname{\mathcal{Q}}$ does. So problem is different from the special scenario of problem , in which $w(\cdot)$ is replaced by the identity function. We will study their relationship in Section 5. This change in the formulation of problem is mathematically simple, but reveals the essence of the problem. In problem , the function $\varphi(\cdot)$, rather than the probability weighting function and the quantile function of the pricing kernel, plays a key role; whereas, in problem , the probability weighting function and the quantile function of the pricing kernel play separate roles in the objective and the constraint. Because the probability weighting function does not appear in the objective of problem , we can solve it by a new relaxation approach. Moreover, this also suggests that it may be possible to link problem to a problem under EUT. This will be investigated after solving it. We also point out here that the new formulation explains why the function $\varphi'(\cdot)$ plays such an important role in many existing models, such as those introduced by Jin and Zhou (2008), He and Zhou (2011), and Xia and Zhou (2012). In those works, the mysterious function $\varphi'(\cdot)$ is derived after lengthy analysis, and an explanation of why it should appear and play the key role is never provided. In tackling problem , some studies assume $\varphi(\cdot)$ to satisfy various properties which are not generally true in practice, and under these assumptions, the problem is partially solved. Here are some examples. In Jin and Zhou (2008), the function $ \frac{F^{-1}_{\rho}(\cdot)}{w'(\cdot)}$ is assumed to be increasing in Assumption 4.1. This is equivalent to $\varphi'(\cdot)$ being decreasing, i.e., $\varphi(\cdot)$ is a concave function. In fact, we have $$1-w(1-\inversew(x))=x,\quad x\in[0,1],$$ so $$\inversew'(x)=\frac{1}{w'(1-\inversew(x))}, \quad x\in[0,1].$$ And consequently, by , $$\begin{aligned} \label{phi-rho-w} \varphi'(x)=F^{-1}_{\rho}(1-\inversew(x))\inversew'(x)=\frac{F^{-1}_{\rho}(1-\inversew(x))}{w'(1-\inversew(x))},\quad x\in[0,1].\end{aligned}$$ The equivalence follows immediately as $\inversew(\cdot)$ is increasing. In He and Zhou (2011), the function $\frac{w'(1-\cdot )}{F^{-1}_{\rho}(1-\cdot)}$ is assumed to be first strictly increasing and then strictly decreasing in Assumption 3.5 and many of the following results. By , this is equivalent to $\varphi'(\cdot)$ being first strictly decreasing and then strictly increasing, i.e., $\varphi(\cdot)$ is a strictly reverse $S$-shaped function. In He and Zhou (2012), the function $\frac{w'(1-\cdot )}{F^{-1}_{\rho}(1-\cdot)}$ is assumed to be nondecreasing in Theorem 2, which is equivalent to $\varphi'(\cdot)$ being decreasing, i.e., $\varphi(\cdot)$ is a concave function. In Proposition 4-7, Theorem 4-6, and Corollary 1, the same function $\frac{w'(1-\cdot )}{F^{-1}_{\rho}(1-\cdot)}$ is assumed to be first strictly decreasing and then strictly increasing. This is equivalent to $\varphi'(\cdot)$ being first strictly increasing and then strictly decreasing, i.e., $\varphi(\cdot)$ is a strictly $S$-shaped function. <span style="font-variant:small-caps;">A NEW RELAXATION APPROACH</span> ======================================================================= Our second main idea in this paper is to introduce a simple relaxation method to tackle problem . The objective of problem is concave with respect to the decision quantiles, so we can apply the Lagrange multiplier method. Problem is equivalent to problem $$\begin{aligned} \label{objective1} \sup\limits_{Q(\cdot) \in\operatorname{\mathcal{G}}}J(Q (\cdot))&=\int_{0}^{1}\Big(u(Q(x))-\lambda Q(x)\varphi'(x)\Big)\operatorname{\mathrm{d\!}}x,\end{aligned}$$ for some Lagrange multiplier $\lambda> 0$ in the sense that they admit the same optimal solution. A naive approach to tackling the foregoing problem is to point-wise maximize its Lagrangian (the integrand in ) to get a point-wise solution $$Q_0(x):=\arg\max\Big\{y:u(y)-\lambda y\varphi'(x)\Big\}=(u')^{-1}(\lambda\varphi'(x)),\quad x\in(0,1).$$ However, this point-wise solution may not be a quantile function in $\operatorname{\mathcal{G}}$. In fact, $Q_0(\cdot)$ is a quantile function if and only if it is increasing, that is equivalent to $\varphi(\cdot)$ being concave. This is exactly what has been assumed in Jin and Zhou (2008) so as to solve the problem. The novel idea in this paper is to replace $\varphi(\cdot)$ by some function $\delta(\cdot)$ in the Lagrangian of problem so that: (i) The new cost function gives an upper bound to that in ; (ii) The new problem can be solved by point-wise maximizing the new Lagrangian; and (iii) There is no gap between the new and old cost functions in the point-wise solution. This approach allows us to solve the problem completely without making any assumptions on the function $\varphi(\cdot)$. We first need to find a relaxed cost function. To this end, let $\delta(\cdot)$ be an absolutely continuous function such that $$\begin{aligned} \label{deltarequirement} \int_{0}^{1}\Big(u(Q(x))-\lambda Q(x)\varphi'(x)\Big)\operatorname{\mathrm{d\!}}x\leqslant \int_{0}^{1}\Big(u(Q(x))-\lambda Q(x)\delta'(x)\Big)\operatorname{\mathrm{d\!}}x,\end{aligned}$$ for every $Q(\cdot) \in\operatorname{\mathcal{G}}$. Setting $\delta(0)=\varphi(0)$ and $\delta(1)=\varphi(1)$ and applying Fubini’s theorem, the inequality is equivalent to $$\begin{aligned} \label{determinedelta1} \int_0^1\Big(\varphi(x)-\delta(x)\Big)\operatorname{\mathrm{d\!}}Q(x)\leqslant 0,\end{aligned}$$ for every $Q(\cdot) \in\operatorname{\mathcal{G}}$, which is clearly equivalent to $\delta(\cdot)$ dominating $\varphi(\cdot)$ on $[0,1]$. In this case, we have $$\begin{gathered} \label{keyineq} \int_{0}^{1}\Big(u(Q(x))-\lambda Q(x)\varphi'(x)\Big)\operatorname{\mathrm{d\!}}x\leqslant \int_{0}^{1}\Big(u(Q(x))-\lambda Q(x) \delta' (x)\Big)\operatorname{\mathrm{d\!}}x\\ \leqslant \int_{0}^{1}\Big(u(\overline{Q}(x))-\lambda \overline{Q}(x) \delta' (x)\Big)\operatorname{\mathrm{d\!}}x,\end{gathered}$$ where the last inequality is obtained by point-wise maximizing the new Lagrangian: $$\begin{aligned} \label{overlineQ} \overline{Q}(x):=\arg\max\Big\{y:u(y)-\lambda y\delta'(x)\Big\}=(u')^{-1}(\lambda \delta' (x)),\quad x\in[0,1].\end{aligned}$$ To make $ \overline{Q}(\cdot)$ a quantile function, we require $\delta(\cdot)$ to be concave. To make $\overline{Q}(\cdot)$ an optimal solution to problem , it is sufficient, by , to have $$\begin{aligned} \label{optimal:barQ2} \int_{0}^{1}\Big(u(\overline{Q}(x))-\lambda \overline{Q}(x)\varphi'(x)\Big)\operatorname{\mathrm{d\!}}x=\int_{0}^{1}\Big(u(\overline{Q}(x))-\lambda \overline{Q}(x) \delta' (x)\Big)\operatorname{\mathrm{d\!}}x,\end{aligned}$$ or equivalently, $$\begin{aligned} \int_{0}^{1} (u')^{-1}(\lambda \delta' (x))\Big(\varphi'(x)-\delta' (x)\Big)\operatorname{\mathrm{d\!}}x=0.\end{aligned}$$ Applying Fubini’s theorem and using $\delta(0)=\varphi(0)$ and $\delta(1)=\varphi(1)$, the above identity is equivalent to $$\begin{gathered} \label{optimal:barQ} \int_{0}^{1} (u')^{-1}(\lambda \delta' (x)) \Big( \varphi'(x)-\delta' (x)\Big)\operatorname{\mathrm{d\!}}x=\int_{0}^{1} \Big(\delta(x)-\varphi(x)\Big)\operatorname{\mathrm{d\!}}\; \Big( (u')^{-1}(\lambda \delta' (x))\Big)\\ =\lambda\int_{0}^{1} \Big(\delta(x)-\varphi(x)\Big)\frac{1}{u''\Big((u')^{-1}(\lambda \delta' (x))\Big)}\operatorname{\mathrm{d\!}}\delta' (x)=0.\end{gathered}$$ Since $\delta(\cdot)$ dominates $\varphi(\cdot)$ on $[0,1]$, $u''(\cdot)<0$, and $\delta(\cdot)$ is concave, by the last identity, $\delta'(\cdot)$ must be constant on any sub interval of $\{x\in[0,1]: \delta(x)>\varphi(x)\}$. Putting all of the requirements on $\delta(\cdot)$ obtained thus far together, we see that $\delta(\cdot)$ should (i) dominate $\varphi(\cdot)$ on $[0,1]$ with $\delta(0)=\varphi(0)$ and $\delta(1)=\varphi(1)$; (ii) be concave on $[0,1]$; and (iii) be affine on $\{x\in[0,1]: \delta(x)>\varphi(x)\}$. Therefore, we conclude that $\delta(\cdot)$ must be the concave envelope of $\varphi(\cdot)$ on $[0,1]$: $$\begin{aligned} \label{envelope} \delta(x)=\sup\limits_{0\leqslant a\leqslant x\leqslant b\leqslant 1}\frac{(b-x)\varphi(a)+(x-a)\varphi(b)}{b-a},\quad x\in[0,1].\end{aligned}$$ On the other hand, if $\delta(\cdot)$ is the concave envelope of $\varphi(\cdot)$ on $[0,1]$, then and hold true. This further implies, by and , that $\overline{Q}(\cdot)$ defined in is an optimal solution to problem . Putting all of the results obtained thus far together and noting that $u(\cdot)$ is strictly concave, we conclude that \[maintheorem\] Problem admits a unique optimal solution $$\begin{aligned} (u')^{-1}(\lambda \delta' (x)),\quad x\in(0,1), \end{aligned}$$ where $\delta(\cdot)$ defined in is the concave envelope of $\varphi(\cdot)$ on $[0,1]$. Problem admits an optimal solution if and only if $$\begin{aligned} \int_{0}^{1} (u')^{-1}(\lambda \delta' (x))\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0\end{aligned}$$ admits a solution $\lambda>0$, in which case $$\begin{aligned} (u')^{-1}(\lambda \delta' (x)), \quad x\in(0,1), \end{aligned}$$ is the unique optimal solution to problem . The foregoing argument shows that $$(u')^{-1}(\lambda \delta' (x)),\quad x\in(0,1),$$ is an optimal solution to problem . Since $u(\cdot)$ is strictly concave, the optimal solution is unique. Suppose problem admits an optimal solution. Then the solution must be an optimal solution to problem for some $\lambda>0$, so it must be of the form $$(u')^{-1}(\lambda \delta' (x)),\quad x\in(0,1).$$ This should be a feasible solution to problem , so $$\int_{0}^{1} (u')^{-1}(\lambda \delta' (x))\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0.$$ On the other hand, suppose that $$\int_{0}^{1} (u')^{-1}(\lambda \delta' (x))\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0$$ holds true for some $\lambda>0$. Note that $$\int_{0}^{1} Q(x)\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0$$ for all $Q(\cdot) \in\operatorname{\mathcal{Q}}$, so $$\begin{gathered} \sup\limits_{Q(\cdot) \in\operatorname{\mathcal{Q}}} \int_{0}^{1}u(Q(x)) \operatorname{\mathrm{d\!}}x= \sup\limits_{Q(\cdot) \in\operatorname{\mathcal{Q}}} \int_{0}^{1}\Big(u(Q(x))-\lambda Q(x)\varphi'(x)\Big)\operatorname{\mathrm{d\!}}x+\lambda x_0\\ \leqslant \sup\limits_{Q(\cdot) \in\operatorname{\mathcal{G}}}\int_{0}^{1}\Big(u(Q(x))-\lambda Q(x)\varphi'(x)\Big)\operatorname{\mathrm{d\!}}x+\lambda x_0,\end{gathered}$$ where the last inequality is due to $\operatorname{\mathcal{Q}}\subseteq \operatorname{\mathcal{G}}$. The optimization problem on the right-hand side is nothing but problem , so the unique solution is$$(u')^{-1}(\lambda \delta' (x)),\quad x\in(0,1).$$ This solution belongs to $\operatorname{\mathcal{Q}}$ as $ \int_{0}^{1} (u')^{-1}(\lambda \delta' (x))\varphi'(x)\operatorname{\mathrm{d\!}}x=x_0$, so it is a feasible solution to the problem on the left-hand side, and consequently, it is an optimal solution to problem . Since $u(\cdot)$ is strictly concave, the optimal solution to problem is unique. The proof is complete. By Theorem \[maintheorem\], the optimal solution to problem is given by $$G^*(x)=(u')^{-1}(\lambda \delta' (\inversew^{-1}(x)))=(u')^{-1}(\lambda \delta' ( 1-w(1-x))),\quad x\in(0,1),$$ which is the same as the last identity on page 14 in Xia and Zhou (2012). That is, our approach yields the same result as in Xia and Zhou (2012). It is clear that our change-of-variable and relaxation approach is much simpler and neater than the calculus of variations approach in Xia and Zhou (2012), which has extensive recourse to convex analysis. If $\varphi(\cdot)$ is assumed to take special shape, such as reverse $S$-shaped function in He and Zhou (2011), $S$-shaped function in He and Zhou (2012), then we can get explicit expression for $\delta(\cdot)$, and consequently, $G^*(\cdot)$ reduces to the results obtained in those works. The feasibility, well-posedness, attainability and uniqueness issues for problem are very important and hard to answer. To avoid these issues, various assumptions are used in the literature to ensure the existence and uniqueness of solutions (see, e.g., Jin and Zhou (2008), Jin, Zhang, and Zhou (2011), He and Zhou (2011, 2012)). In the following section, with Theorem \[maintheorem\], we will link problem to a classical Merton’s portfolio choice problem under EUT, for which the feasibility, well-posedness, attainability and uniqueness issues are studied in Jin, Xu, and Zhou (2008). This connection also develops a new way to solve problem , which avoids dealing with the quantile formulation problem . <span style="font-variant:small-caps;">A LINK BETWEEN MODELS UNDER RDUT AND EUT</span> ====================================================================================== By Theorem \[maintheorem\], it is clear that a quantile function is an optimal solution to problem if and only if it is an optimal solution to problem $$\begin{aligned} \label{objective2} \sup\limits_{Q(\cdot)\in \operatorname{\widetilde{\operatorname{\mathcal{Q}}}}} \int_{0}^{1}u(Q(x)) \operatorname{\mathrm{d\!}}x,\end{aligned}$$ where $$\begin{aligned} \operatorname{\widetilde{\operatorname{\mathcal{Q}}}}:=\left\{Q(\cdot)\in \operatorname{\mathcal{G}}:\int_0^1 Q(x)\delta'(x)\operatorname{\mathrm{d\!}}x=x_0\right\}.\end{aligned}$$ Since $\delta'(\cdot)$ is decreasing, function $$F_{\widetilde{\rho}}^{-1}(x):=\delta'(1-x),\quad x\in(0,1),$$ belongs to $\operatorname{\mathcal{G}}$ and can be regarded as the quantile function of some positive random variable $\widetilde{\rho}$. It is possible to choose $\widetilde{\rho}$ to be comonotonic[^7] with $\rho$, which is henceforth assumed.[^8] Then $$\begin{aligned} \operatorname{\widetilde{\operatorname{\mathcal{Q}}}}=&\left\{Q(\cdot)\in \operatorname{\mathcal{G}}:\int_0^1 Q(x)\delta'(x)\operatorname{\mathrm{d\!}}x=x_0\right\}\\ =&\left\{Q(\cdot)\in \operatorname{\mathcal{G}}:\int_0^1 Q(x)F_{\widetilde{\rho}}^{-1}(1-x) \operatorname{\mathrm{d\!}}x=x_0\right\}.\end{aligned}$$ Now, we see that problem can be regarded as a special case of problem , in which the probability weighting function $w(\cdot)$ is replaced by the identity function and the pricing kernel $\rho$ is replaced by $\widetilde{\rho}$. We point out here that the new pricing kernel $\widetilde{\rho}$ may be atomic, which does not satisfy Assumption \[assmp:atomless\]. In fact, $ \widetilde{\rho}$ is atomless if and only if its quantile function $F_{\widetilde{\rho}}^{-1}(\cdot)$ is strictly increasing. This is equivalent to $\delta(\cdot)$ being strictly concave as $F_{\widetilde{\rho}}^{-1}(\cdot)=\delta'(1-\cdot)$, and also equivalent to $\varphi(\cdot)$ being strictly concave as $\delta(\cdot)$ is the concave envelope of $\varphi(\cdot)$. Recalling the relationship between problem and problem , it is natural to link problem to a portfolio choice problem $$\begin{aligned} \sup_X \quad & \int_0^{\infty} u(x) \operatorname{\mathrm{d\!}}F_X(x),\\ \nonumber\textrm{subject to}\quad & \operatorname{\mathbf{E}}[ \widetilde{\rho} X]=x_0,\quad X\geqslant 0.\end{aligned}$$ Note that $$\int_0^{\infty} u(x) \operatorname{\mathrm{d\!}}F_X(x)=\operatorname{\mathbf{E}}[u(X)],$$ for any $X\geqslant 0$, so the above problem is the same as problem $$\begin{aligned} \label{equivalentEUT} \sup_X \quad & \operatorname{\mathbf{E}}[u(X)],\\ \nonumber\textrm{subject to}\quad & \operatorname{\mathbf{E}}[ \widetilde{\rho} X]=x_0,\quad X\geqslant 0.\end{aligned}$$ This is a classical Merton’s portfolio choice problem under EUT. Under the assumption that $\rho$ is atomless, we have linked problem to problem . However, we cannot directly link problem to problem as before, because the new pricing kernel $ \widetilde{\rho}$ in problem may not be atomless. The following result from Xu (2014), where no assumption on $\widetilde{\rho}$ is required, links problem to problem . \[noatem\] If $\widetilde{X}^*$ is an optimal solution to problem , then its quantile function is an optimal solution to problem . On the other hand, if $\widetilde{Q}^*(\cdot)$ is an optimal solution to problem , then $$\widetilde{X}^*:=\widetilde{Q}^*(1-U)$$ is an optimal solution to problem , where $U$ is any random variable uniformly distributed on the unit interval $(0,1)$ and comonotonic with $\widetilde{\rho}$. With this result, we can link problem to problem . Let $\widetilde{X}^*$ be an optimal solution to problem and $\widetilde{Q}^*(\cdot)$ be its quantile function. Then $$\begin{aligned} X^*:=\widetilde{Q}^*(1-w(F_{\rho}(\rho)))\end{aligned}$$ is an optimal solution to problem . On the other hand, if $X^*$ is an optimal solution to problem , then there exists a unique quantile function $\widetilde{Q}^*(\cdot)$ such that $$X^*=\widetilde{Q}^*(1-w(F_{\rho}(\rho))).$$ Moreover, $\widetilde{Q}^*(1-U)$ is an optimal solution to problem , where $U$ is any random variable uniformly distributed on the unit interval $(0,1)$ and comonotonic with $\widetilde{\rho}$. Suppose that $\widetilde{X}^*$ is an optimal solution to problem and $\widetilde{Q}^*(\cdot)$ is its quantile function. By Theorem \[noatem\], $\widetilde{Q}^*(\cdot)$ is an optimal solution to problem and problem . Consequently, $$G^*(x):=\widetilde{Q}^*(\inversew^{-1}(x)),\quad x\in(0,1),$$ is an optimal solution to problem . Hence, by , $$\begin{aligned} X^*=G^*(1-F_{\rho}(\rho))=\widetilde{Q}^*(\inversew^{-1}(1-F_{\rho}(\rho)))=\widetilde{Q}^*(1-w(F_{\rho}(\rho)))\end{aligned}$$ is an optimal solution to problem . On the other hand, if $X^*$ is an optimal solution to problem . Then by , $$X^*=G^*(1-F_{\rho}(\rho)),$$ where $G^*(\cdot)$ is an optimal solution to problem . Consequently, $$\widetilde{Q}^*( x):=G^*(\inversew(x)),\quad x\in(0,1),$$ is an optimal solution to problem and problem . By Theorem \[noatem\], $\widetilde{Q}^*(1-U)$ is an optimal solution to problem . The proof is complete. The above result shows that solving the portfolio choice problem under RDUT is equivalent to solving problem under EUT, which is much easier than the former. Moreover, the latter does not require solving a quantile optimization problem. This provides us a new way to solve the portfolio choice problem . In the literature, various conditions are assumed so as to avoid studying the feasibility, well-posedness, attainability or uniqueness issues for problem (see, e.g., Jin and Zhou (2008), Jin, Zhang, and Zhou (2011), He and Zhou (2011, 2012)). By the above result, these issues for problem reduce to that for problem . However, these issues for problem are solved in Jin, Xu, and Zhou (2008), so are for problem . Similarly, these issues for problems , and are solved as well. The optimal solution to problem can be obtained by the Lagrange multiplier method directly. Consequently, its quantile function can be obtained without solving problem . Such approach to solving an investment problem under RDUT without using quantile optimization technique has never appeared in the literature to the best of our knowledge. On the other hand, this result also tells us that a functional optimization problem can be solved via solving a probabilistic optimization problem . It is an important and challenging question whether we can apply this idea to other functional optimization problems. The new pricing kernel $\widetilde{\rho}$ does not depend on the utility function $u(\cdot)$. Problem is time-inconsistent, whereas problem is time-consistent. It would be interesting to study their relationships as time changes. <span style="font-variant:small-caps;">CONCLUDING REMARKS</span> ================================================================ In this paper, we consider a portfolio choice problem under RDUT. We propose a short, neat, and easy-to-follow method to solve the problem. The method consists of two key ideas. The first is making a change of variable to reveal the key function that we need to consider in the quantile formulation problem. The second is relaxing the Lagrangian so as to find an achievable upper bound. Our approach can also be adopted to deal with portfolio choice and optimal stopping problems under CPT/RDUT as well as many other models with law-invariant preference measures. The second contribution of this paper is showing that solving a portfolio choice problem under RDUT is equivalent to solving a classical Merton’s portfolio choice problem under EUT. The latter avoids studying the quantile optimization problem and can be solved by the classical dynamic programming and probabilistic approaches. Theorem \[noatem\] obtained by Xu (2014) plays a key role in connecting these two problems as the new pricing kernel cannot be assumed to be atomless in general. The third contribution of this paper is solving the feasibility, well-posedness, attainability and uniqueness issues for the portfolio choice problem under RDUT. Last but not least, we show that solving functional optimization problems may reduce to solving probabilistic optimization problems. This idea may be applicable to other functional optimization problems. <span style="font-variant:small-caps;">Acknowledgments.</span> The author is grateful to the editors and anonymous referees for carefully reading the manuscript and making useful suggestions that have led to a much improved version of the paper. [99]{} <span style="font-variant:small-caps;">Cox, J. C., and C. F. Huang (1989):</span> Optimum Consumption and Portfolio Policies When Asset Prices Follow a Diffusion Process, *Journal of EconomicTheory*, Vol. 49, pp. 33-83 <span style="font-variant:small-caps;">Cox, J. C., and C. F. Huang (1991):</span> A Variational Problem Occurring in Fnancial Economics, *Journal of Mathematical Economics*, Vol. 20, pp. 465-487 <span style="font-variant:small-caps;">He, X. D. , and X. Y. Zhou (2011):</span> Portfolio Choice via Quantiles, *Mathematical Finance*, Vol. 21, pp. 203-231 <span style="font-variant:small-caps;">He, X. D. , and X. Y. Zhou (2012):</span> Hope, Fear and Aspirations, to appear in *Mathematical Finance* <span style="font-variant:small-caps;">Jin, H., Z. Q. Xu, and X. Y. Zhou (2008):</span> A Convex Stochastic Optimization Problem arising from Portfolio Selection, *Mathematical Finance*, Vol. 18, pp. 171-183 <span style="font-variant:small-caps;">Jin, H., S. Zhang, and X. Y. Zhou (2011):</span> Behavioral Portfolio Selection with Loss Control, *Acta Mathematica Sinica*, Vol. 27, pp. 255-274 <span style="font-variant:small-caps;">Jin, H., and X. Y. Zhou (2008):</span> Behavioral Portfolio Selection in Continuous Time, *Mathematical Finance*, Vol. 18, pp. 385-426 <span style="font-variant:small-caps;">Kahneman, D., and A. Tversky (1979):</span> Prospect Theory: An Analysis of Decision Under Risk, *Econometrica*, Vol. 46, pp. 171-185 <span style="font-variant:small-caps;">Karatzas, I., J. P. Lehoczky, and S. E. Shreve (1987):</span> Optimal Portfolio and Consumption Decisions for a Small Investor on a Finite Time-Horizon, *SIAM Journal on Control and Optimization*, Vol. 25, pp. 1557-1586 <span style="font-variant:small-caps;">Pliska, S. R. (1986):</span> A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios, *Mathematics of Operations Research*, Vol. 11, pp. 371-382 <span style="font-variant:small-caps;">Prelec, D. (1998):</span> The Probability Weighting Function, *Econometrica*, Vol. 66, pp. 497-527 <span style="font-variant:small-caps;">Tversky, A., and C. R. Fox (1995):</span> Weighing Risk and Uncertainty, *Psychological Review*, Vol.102, pp. 269-283 <span style="font-variant:small-caps;">Tversky, A., and D. Kahneman (1992):</span> Advances in Prospect Theory: Cumulative Representation of Uncertainty, *J. Risk Uncertainty*, Vol. 5, pp. 297-323 <span style="font-variant:small-caps;">Xia, J. M., and X. Y. Zhou (2012):</span> Arrow-Debrew Equilibria for Rank-Dependent Utilities, *preprint*, <http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2146353> <span style="font-variant:small-caps;">Xu, Z. Q. (2014):</span> A New Characterization of Comonotonicity and its Application in Behavioral Finance, *J. Math. Anal. Appl.*, in press, <http://dx.doi.org/10.1016/j.jmaa.2014.03.053> <span style="font-variant:small-caps;">Xu, Z. Q., and X. Y. Zhou (2013):</span> Optimal Stopping under Probability Distortion, *Annals of Applied Probability*, Vol. 23, pp. 251-282 [^1]: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. Email: [[email protected]]([email protected]). The author acknowledges financial supports from Hong Kong General Research Fund (No. 529711), Hong Kong Early Career Scheme (No. 533112), and The Hong Kong Polytechnic University. [^2]: A function is called $S$-shaped if it is convex on the left and concave on the right; and reverse $S$-shaped if concave on the left and convex on the right. [^3]: see, e.g., Jin, Xu and Zhou (2008) for the definitions of feasibility, well-posedness, attainability and uniqueness issues for a portfolio choice problem [^4]: See, e.g., Xia and Zhou (2012). [^5]: A random variable is called atomless or non-atomic if its cumulative distribution function is continuous, and called atomic otherwise. [^6]: The quantile function $Q(\cdot)$ of a real-valued random variable is defined as the right-continuous inverse function of its cumulative distribution function $F(\cdot)$, that is $Q(x)=\sup\{t\in\operatorname{\mathbb{R}}: F(t)\leqslant x\}$, for all $x\in (0,1)$, with convention $\sup\emptyset=-\infty$. A real-valued random variable is atomless if and only if its quantile function is strictly increasing. [^7]: Two random variables $X$ and $Y$ are said to be comonotonic if $(X(\omega')-X(\omega)) (Y(\omega')-Y(\omega))\geqslant 0$ almost surely under $\operatorname{\mathbf{P}}\otimes\operatorname{\mathbf{P}}$. [^8]: In fact, $\widetilde{\rho}=\delta'(1-F_{\rho}(\rho))$ in the current setting. Xu (2014) proved that $\widetilde{\rho}$ can be chosen to be comonotonic with $\rho$ even if $\rho$ is not atomless.

[**Statistical approximation by $(p,q)$-analogue of Bernstein-Stancu Operators**]{} **Asif Khan** and **Vinita Sharma** Department of Mathematics, Aligarh Muslim University, Aligarh–202002, India\ [email protected]; [email protected]\ **Abstract** [In this paper, some approximation properties of $(p,q)$-analogue of Bernstein-Stancu Operators has been studied. Rate of statistical convergence by means of modulus of continuity and Lipschitz type maximal functions has been investigated. Monotonicity of $(p,q)$-Bernstein-Stancu Operators and a global approximation theorem by means of Ditzian-Totik modulus of smoothness is established. A quantitative Voronovskaja type theorem is developed for these operators. Furthermore, we show comparisons and some illustrative graphics for the convergence of operators to a function]{}.\ [*Keywords and phrases*: $(p,q)$-integers; $(p,q)$-Bernstein-Stancu operators; Positive linear operators; Korovkin type approximation; Statistical convergence; Monotonicity for convex functions; Ditzian-Totik modulus of smoothness; Voronovskaja type theorem.]{}\ [*AMS Subject Classifications (2010)*: [41A10, 41A25, 41A36, 40A30.]{}]{} Introduction and preliminaries ============================== Mursaleen et al. [@mka1] first applied the concept of $(p,q)$-calculus in approximation theory and introduced the $(p,q)$-analogue of Bernstein operators. Later on, based on $(p,q)$-integers, some approximation results for Bernstein-Stancu operators, Bernstein-Kantorovich operators, Bleimann-Butzer and Hahn operators, $(p,q)$-Lorentz operators, Bernstein-Shurer operators, $(p,q)$-analogue of divided difference and Bernstein operators etc. have also been introduced by them in [@mur8; @mka3; @mka5; @m4; @zmn; @mnfa].\ For similar works in approximation theory [@pp] based on $q$ and $(p,q)$-integers, one can refer [@acar1; @acar2; @acar3; @acar4; @aral; @cai; @ali; @lupas; @ma1; @sofia; @kang; @kadak2; @kac; @mahmudov1; @wafi]. Motivated by the work of Mursaleen et al [@mka1], the idea of $(p,q)$-calculus and its importance.\ Very recently, Khalid et al. [@khalid1; @khalid2; @khalid3; @kblossom] has given a nice application in computer-aided geometric design and applied these Bernstein basis for construction of $(p,q)$-B$\acute{e}$zier curves and surfaces based on $(p,q)$-integers which is further generalization of $q$-B$\acute{e}$zier curves and surfaces [@bezier; @hp; @pl; @phillips; @pp; @lp]. For similar works, one can refer [@bezier; @hp]. Another advantage of using the parameter $p$ has been discussed in [@m4].\ Let us recall certain notations of $(p,q)$-calculus .\ For any $p>0$ and $q>0,$ the $(p,q)$ integers $[n]_{p,q}$ are defined by $$[n]_{p,q}=p^{n-1}+p^{n-2}q+p^{n-3}q^2+...+pq^{n-2}+q^{n-1}\\ =\left\{ \begin{array}{lll} \frac{p^{n}-q^{n}}{p-q},~~~~~~~~~~~~~~~~\mbox{when $~~p\neq q \neq 1$ } & \\ & \\ n~p^{n-1},~~~~~~~~~~~~~~\mbox{ when $p=q\neq1$ } & \\ & \\ [n]_q ,~~~~~~~~~~~~~~~~~~~\mbox{when $p=1$ }& \\ n ,~~~~~~~~~~~~~~~~~~~~~\mbox{ when $p=q=1$ } \end{array}\right.$$  where $[n]_q $ denotes the $q$-integers and $n=0,1,2,\cdots$.\ Obviously, it may be seen that $[n]_{p,q}= p^{n-1}[n]_{\frac{q}{p}}.\\$ The $(p,q)$-factorial is defined by $$[0]_{p, q}!:=1~~\text{and}~~[n]!_{p, q}=[1]_{p, q}[2]_{p, q}\cdots [n]_{p, q}~~\text{if}~~n\ge 1.$$ Also the $(p,q)$-binomial coefficient is defined by $${n \brack k}_{p, q}=\frac{[n]_{p, q}!}{[k]_{p, q}!~[n-k]_{p, q}!}~~\text{for all}~~n, k\in \mathbb N~~\text{with}~~n\ge k.$$ The formula for $(p,q)$-binomial expansion is as follows: $$(ax+by)_{p,q}^{n}:=\sum\limits_{k=0}^{n}p^{\frac{(n-k)(n-k-1)}{2}}q^{\frac{k(k-1)}{2}} \left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}a^{n-k}b^{k}x^{n-k}y^{k},$$ $$(x+y)_{p,q}^{n}=(x+y)(px+qy)(p^2x+q^2y)\cdots (p^{n-1}x+q^{n-1}y),$$ $$(1-x)_{p,q}^{n}=(1-x)(p-qx)(p^2-q^2x)\cdots (p^{n-1}-q^{n-1}x),$$\ Details on $(p,q)$-calculus can be found in [@mah; @jag; @mka1; @khalid1; @khalid2; @kblossom].\ The $(p,q)$-Bernstein Operators introduced by Mursaleen et al. for $0<q<p\leq 1$ in [@mka1] are as follow: $$\label{ee1} B_{n,p,q}(f;x)=\frac1{p^{\frac{n(n-1)}2}}\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~~f\left( \frac{ [k]_{p,q}}{p^{k-n}[n]_{p,q}}\right) ,~~x\in \lbrack 0,1].$$ Note when $p=1,$ $(p,q)$-Bernstein Operators given by turns out to be $q$-Bernstein Operators.\ Also, we have $$\begin{aligned} (1-x)^{n}_{p,q}&=\prod\limits_{s=0}^{n-1}(p^s-q^{s}x) =(1-x)(p-qx)(p^{2}-q^{2}x)...(p^{n-1}-q^{n-1}x)\\ &=\sum\limits_{k=0}^{n} {(-1)}^{k}p^{\frac{(n-k)(n-k-1)}{2}} q^{\frac{k(k-1)}{2}}\left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}x^{k}\end{aligned}$$ Motivated by the above mentioned work on $(p,q)$-approximation and its application, this paper is organized as follows: In Section 2, some basic results for $(p,q)$-analogue of Bernstein-Stancu Operators as given in [@mka3] has been recalled and based on it, second order moment is computed. In section 3, Korovkin’s type statistical approximation properties has been studied for these operators. In section 4, rate of statistical convergence by means of modulus of continuity and Lipschitz type maximal functions has been investigated. Section 5 is based on monotonicity of $(p,q)$-Bernstein-Stancu Operators. In section 6, a global approximation theorem by means of Ditzian-Totik modulus of smoothness and a quantitative Voronovskaja type theorem is established. The effects of the parameters $p$ and $q$ for the convergence of operators to a function is shown in section 7 .\ $(p,q)$- Bernstein Stancu operators =================================== Mursaleen et. al in [@mka3] introduced $(p,q)$-analogue of Bernstein-Stancu operators as follow: $$\label{ee2} S_{n,p,q}(f;x)=\frac1{p^{\frac{n(n-1)}2}}\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~~f\left( \frac{ {p^{n-k}[k]_{p,q}+\alpha}}{[n]_{p,q}+\beta}\right) ,~~x\in \lbrack 0,1].$$ where $\alpha$ and $\beta$ are real numbers which satisfy $0\leq\alpha \leq \beta$.\ Note that for $\alpha=\beta=0,$ $(p,q)$-Bernstein-Stancu operators given by reduces into $(p,q)$-Bernstein operators as given in [@mka1].\ Also for $p=1$, $(p,q)$-Bernstein-Stancu operators given by turn out to be $q$-Bernstein-Stancu operators.\ For $p=q=1,$ it reduces to classical Bernstein-Stancu operators.\ We have the following auxiliary lemmas: **Lemma 2.1.** For $x\in \lbrack [0,1],~0<q<p\leq 1$, and $\alpha,\beta \in\mathbb{R}$ with $0\leq\alpha \leq \beta$, we have\ (i)  $S_{n,p,q}(1;x)=~1$,(ii) $S_{n,p,q}(t;x)=~\frac{[n]_{p,q}}{[n]_{p,q}+\beta}x+\frac{\alpha}{[n]_{p,q}+\beta}$,(iii) $S_{n,p,q}(t^{2};x)=\frac{q[n]_{p,q}[n-1]_{p,q}}{([n]_{p,q}+\beta)^2}x^2+\frac{[n]_{p,q}(2\alpha+p^{n-1})}{([n]_{p,q}+\beta)^2}x+\frac{\alpha^2}{([n]_{p,q}+\beta)^2}$. **Proof:** Proof is given in [@mka3] using the identity $$\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)={p^{\frac{n(n-1)}2}}.$$ We give complete proof of Lemma 1 (iii) \(iii) $$\begin{aligned} S_{n,p,q}(t^2;x) &=&\frac1{p^{\frac{n(n-1)}2}}\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~~{\bigg(\frac{p^{n-k}[k]_{p,q}+\alpha}{[n]_{p,q}+\beta}\bigg)}^2\\ &=&\frac{1}{([n]_{p,q}+\beta)^2}~\frac1{p^{\frac{n(n-1)}2}}\Bigg[p^{2n}\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~\frac{[k]_{p,q}^2}{p^{2k}}\\ &&+2\alpha ~p^n\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~\frac{[k]_{p,q}}{p^{k}}\\ &&+\alpha^2~\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)\Bigg].\\\end{aligned}$$ $$S_{n,p,q}(t^2;x)=\frac{1}{([n]_p,q+\beta)^2} [(A)+(B)+(C)]$$ $$\begin{aligned} (A)&=&\frac {1}{p^{\frac{n(n-1)}2}} p^{2n}\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q}p^{\frac{k(k-1)}2}x^{k}\prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x)~\frac{[k]_{p,q}^2}{p^{2k}}\\ &=&\frac{p^{2n}}{p^{\frac{n(n-1)}2}}\sum\limits_{k=0}^{n} \frac{[n]}{[k]}\left[ \begin{array}{c} n-1 \\ k-1 \end{array} \right] x^{k}(1-x)^{n-k}~\frac{[k]^2}{p^{2k}}\end{aligned}$$ On shifting the limits and using $[k+1]_{p,q}=p^k+q[k]_{p,q}$, we get our desired result. $$\begin{aligned} (A)&=&\frac{p^{2n}}{p^{\frac{n(n-1)}2}}\sum\limits_{k=0}^{n-1} \left[ \begin{array}{c} n-1 \\ k \end{array} \right] x^{k}(1-x)^{n-k-1}~\frac{p^k+q[k]}{p^{2k+2}}\\ &&= \frac{p^{2n-2}[n]x}{p^{\frac{n(n-1)}{2}}}\Bigg[ p^{\frac{(n-1)(n-2)}{2}}+ \frac{q[n-1]x}{p}\sum\limits_{k=0}^{n-2} \left[ \begin{array}{c} n-2 \\ k \end{array} \right] x^{k}(1-x)^{n-k-2}\Bigg]\\ &&=p^n[n]x+q[n][n-1]x^2\end{aligned}$$ n\ Similarly $$(B)=\frac{2\alpha ~p^n}{p^{\frac{n(n-1)}{2}}}\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}x^{k}(1-x)^{n-k}~\frac{[k]_{p,q}}{p^{k}}=2\alpha [n]x$$ and $$(C)=\frac{\alpha^2}{p^{\frac{n(n-1)}{2}}}~\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q}x^{k}(1-x)^{n-k}={\alpha}^2$$ **Lemma 2.2**. For $x\in [0,1],~0<q<p\leq 1$ and $\alpha,\beta \in\mathbb{R}$ with $0\leq\alpha \leq \beta$,\ Let n be any given natural number, then $$\begin{aligned} S_{n,p,q}\bigl{(}(t-x)^2;x\bigl{)}&= \big\{ \frac{q[n]_{p,q}[n-1]_{p,q}-[n]_{p,q}^2+{\beta}^2}{([n]_{p,q}+\beta)^2}\big\}x^2+\big\{ \frac{p^{n-1}[n]_{p,q}-2 \alpha \beta}{([n]_{p,q}+\beta)^2} \big\}x+\frac{\alpha^2}{([n]_{p,q}+\beta)^2}\\ &\leq\frac{[n]_{p,q}p^{n-1} - 2\alpha\beta}{2([n]_{p,q}+\beta)^2}\phi^2(x)\leq \frac{[n]_{p,q}}{[n]_{p,q}+\beta}\phi^2(x)\\\end{aligned}$$ Main Results ============= Korovkin type approximation theorem ------------------------------------ We know that $C[a,b]$ is a Banach space with norm $$\Vert f\Vert _{C[a,b]}:=\sup\limits_{x\in \lbrack a,b]}|f(x)|,~f\in C[a,b].$$ For typographical convenience, we will write $\Vert .\Vert $ in place of $\Vert .\Vert _{C[a,b]}$ if no confusion arises. [*Let $C[a,b]$ be the linear space of all real valued continuous functions $f$ on $[a,b]$ and let $T$ be a linear operator which maps $C[a,b]$ into itself. We say that $T$ is $positive$ if for every non-negative $f\in $ $C[a,b],$ we have $T(f,x)\geq 0$ for all $x\in $ $[a,b]$ .*]{} The classical Korovkin type approximation theorem can be stated as follows [@brn; @korovkin];\ Let $T_n: C[a, b] \to C[a, b]$ be a sequence of positive linear operators. Then $\lim_{n\to\infty}\|T_{n}(f; x)-f(x)\|_\infty=0,\,\,\textrm{for all}~f\in C[a, b]$ if and only if $\lim_{n\to\infty}\|T_{n}(f_{i}; x)-f_i(x)\|_\infty=0,\,\,\textrm{for each}\,\,i=0,1,2,$ where the test function $f_i(x)=x^i$. In next section, we study a statistical approximation properties of the operator $S_{n,p, q}$. Statistical approximation ------------------------- The statistical version of Korovkin theorem for sequence of positive linear operators has been given by Gadjiev and Orhan [@go39]. Let $K$ be a subset of the set $\mathbb{N}$ of natural numbers. Then, the asymptotic density $\delta(K)$ of $K$ is defined as $\delta(K)=\lim_{n}\frac{1}{n}\big|\{k\leq n~:~k \in K\}\big|$ and $|.|$ represents the cardinality of the enclosed set. A sequence $x=(x_k)$ said to be statistically convergent to the number $L$ if for each $\varepsilon >0$, the set $K(\varepsilon)=\{k\leq n:|x_k-L|>\varepsilon\}$ has asymptotic density zero (see [@erd37; @fast]), i.e., $$\begin{aligned} \label{117} \lim_{n}\frac{1}{n}\big|\{k\leq n:|x_k-L|\geq \varepsilon\}\big|=0.\end{aligned}$$ In this case, we write $st-\lim x =L$. Let us recall the following theorem:\ \[ta\][@go39] Let $A_n$ be the sequence of linear positive operators from $C[0,1]$ to $C [0,1]$ satisfies the conditions $st-\lim\limits_{n}\|S_{n,p,q}(( t^\nu;x))- (x)^\nu \|_C[0,1] = 0 $ for $\nu = 0,~ 1,~ 2.$ then for any function $f\in C[0,1],$ $st-\lim\limits_{n} \|S_{ n,p,q}(f) - f \|_C[0,1] = 0.$ Korovkin Type statistical approximation properties -------------------------------------------------- The main aim of this paper is to obtain the korovkin type statistical approximation properties of operators defined in (\[ee2\]) with the help of Theorem (\[ta\]).\ \[r5.1\] For $q\in(0,1)$ and $p\in(q,1]$, it is obvious that $\lim\limits_{n\to\infty}[n]_{p,q}=0 $ or $\frac1{p-q}$. In order to reach to convergence results of the operator $L^{n}_{p,q}(f;x),$ we take a sequence $q_n\in(0,1)$ and $p_n\in(q_n,1]$ such that $\lim\limits_{n\to\infty}p_n=1,$ $\lim\limits_{n\to\infty}q_n=1$ and $\lim\limits_{n\to\infty}p_n^n=1,$ $\lim\limits_{n\to\infty}q_n^n=1$. So we get $\lim\limits_{n\to\infty}[n]_{p_n,q_n}=\infty$. Let $S_{n,p,q}$ be the sequence of operators and the sequence $ p=p_n$ and $q=q_n$ satisfying Remark $(\ref{r5.1})$ then for any function $f\in C[0,1]$\ $$st-\lim\limits_{n} \|~S_{n,p_n,q_n}{(f,.)}-f\|=0$$ **Proof:** Clearly for $\nu=0,$ $$S_{n,p,q}{(1,x)}=1,$$ which implies $$st-\lim\limits_{n}\|S_{n,p,q}(1;x)~-1~\|~~=~~0.$$\ For $\nu~=~1$\ $$\begin{aligned} \|S_{n,p,q}~(t;x)~-~x~~\|&\leq~\bigg|\frac{[n]_{p,q}}{[n]_{p,q}+\beta}x~~+~~\frac{\alpha}{[n]_{p,q}+\beta}~~-~~x\bigg|\\ &= \bigg|\bigg(\frac{[n]_{p,q}}{[n]_{p,q}+\beta}~~-~~1\bigg)x~+~\frac{\alpha}{[n]_{p,q}+\beta}\bigg|\\ &\leq \bigg|\frac{[n]_{p,q}}{[n]_{p,q}+\beta}~~-~~1\bigg|~~+~~\bigg|\frac{\alpha}{[n]_{p,q}+\beta}\bigg|.\end{aligned}$$ For a given $ \epsilon >0$, let us define the following sets.\ $$U = \{n : \|S_{n,p,q}(t;x) -x\|\geq\epsilon\}$$\ $$U^{\prime} = \{n: 1 - \frac{[n]_{p,q}}{[n]_{p,q}+\beta}\} \geq \epsilon$$\ $$U^{\prime\prime} = \{n: \frac{\alpha}{[n]_{p,q} +\beta}\geq\epsilon\}$$ So using $\delta \{k\leq n:1-\frac{[n]_{p,q}}{[n]_{p,q}+\beta}~~\geq \epsilon\}, $ then we get $$st-\lim\limits_{n}\|S_{n,p,q}(t;x) - x\|=0.$$ Lastly for $\nu=2,$ we have\ $$\begin{aligned} \|S_{n,p,q}(t^2:x)- x^2\|&\leq \big|\frac{q[n]_{p,q}[n-1]_{p,q}}{{([n]_{p,q}+\beta)}^{2}}~~-1\big|\\ &+\big|\frac{[n]_{p,q}(2\alpha+p^{n-1})}{[n]_{p,q}+\beta}^{2}x\big|+\big|\frac{\alpha^2}{{([n]_{p,q}+\beta)}^{2}}\big|.\end{aligned}$$ If we choose $$\alpha_n=\frac{q[n]_{p,q}[n-1]_{p,q}}{{([n]_{p,q}+\beta)}^{2}}~~-1$$\ $$\beta_n=\frac{[n]_{p,q}(2\alpha+p^{n-1})}{[n]_{p,q}+\beta}^{2}$$\ $$\gamma_n=\frac{\alpha^2}{{([n]_{p,q}+\beta)}^2}$$\ $st-\lim\limits_{n}\alpha_n~~=~~st-\lim\limits_{n}\beta_n~~=~~st-\lim\limits_{n}\gamma_n~~=~~0$\ Now given $\epsilon >0$, we define the following four sets:\ $$U~~=~~\|S_{n,p,q}(t^2:x)- x^2\|\geq \epsilon$$\ $$U_{1} =\{n:\alpha_{n} \geq \frac{\epsilon}{3}\}$$\ $$U_{2}=\{n:\beta _{n}\geq \frac{\epsilon }{3}\}$$\ $$U_{3}=\{n:\gamma _{n}\geq \frac{\epsilon }{3}\}.$$\ It is obvious that$ U \subseteq U_1\bigcup U_2\bigcup U_3. $ Thus we obtain\ $\delta\{K\leq n:\|S_{n,p,q}(t^2:x)- x^2\|\geq\epsilon\}$\ $\leq\delta\{K\leq n:\alpha_{n} \geq \frac{\epsilon}{3} \}~+~\delta\{K\leq n:\beta _{n}\geq \frac{\epsilon }{3}\}+\delta\{K\leq n:\gamma _{n}\geq \frac{\epsilon }{3}\}$\ So the right hand side of the inequalities is zero by $( \ref{117}).$\ Then\ $$st-\lim\limits_{n}\|S_{n,p,q}(t;x) -x\|=0$$ holds and thus the proof is completed. If we choose $$\alpha_n=\frac{q[n]_{p,q}[n-1]_{p,q}}{{([n]_{p,q}+\beta)}^{2}}~~-1$$\ $$\beta_n=\frac{[n]_{p,q}(2\alpha+p^{n-1})}{[n]_{p,q}+\beta}^{2}$$\ $$\gamma_n=\frac{\alpha^2}{{([n]_{p,q}+\beta)}^2}$$\ $st-\lim\limits_{n}\alpha_n~~=~~st-\lim\limits_{n}\beta_n~~=~~st-\lim\limits_{n}\gamma_n~~=~~0$\ Now given $\epsilon >0$, we define the following four sets:\ $$U~~=~~\|S_{n,p,q}(t^2:x)- x^2\|\geq \epsilon$$\ $$U_{1} =\{n:\alpha_{n} \geq \frac{\epsilon}{3}\}$$\ $$U_{2}=\{n:\beta _{n}\geq \frac{\epsilon }{3}\}$$\ $$U_{3}=\{n:\gamma _{n}\geq \frac{\epsilon }{3}\}.$$\ It is obvious that$ U \subseteq U_1\bigcup U_2\bigcup U_3. $ Thus we obtain\ $\delta\{K\leq n:\|S_{n,p,q}(t^2:x)- x^2\|\geq\epsilon\}$\ $\leq\delta\{K\leq n:\alpha_{n} \geq \frac{\epsilon}{3} \}~+~\delta\{K\leq n:\beta _{n}\geq \frac{\epsilon }{3}\}+\delta\{K\leq n:\gamma _{n}\geq \frac{\epsilon }{3}\}$\ So the right hand side of the inequalities is zero by $( \ref{117}).$\ Then\ $$st-\lim\limits_{n}\|S_{n,p,q}(t;x) -x\|=0$$ holds and thus the proof is completed. Rate of Statstical Convergence ============================== In this part, rates of statistical convergence of the operators $(\ref{ee2} )$ by means of modulus of continuity and LIPSCHITZ TYPE maximal functions are introduced.\ The modulus of continuity for the space of function $ f\in C[0,1]$ is defined by\ $$w(f;\delta)=\sup\limits_{x,t\in C[0,1],~~ |t-x|<\delta} |f(t)-f(x)|$$\ where ${w}(f;\delta)$ satisfies the following conditions:  for all $f\in C[0,1],$\ $$\label{e118} \lim\limits_{\delta\rightarrow0}~w (f;\delta) = 0.$$ and $$\label{e119} |f(t)-f(x)|\leq w(f;\delta)\bigg(\frac{|t-x|}{\delta}+ 1\bigg)$$ Let the sequence $ p=p_n$ and $q=q_n$ satisfy for $0<q_n<p_n\leq1$, so we have\ $$|S_{n,p,q}(t;x) -f(x)|\leq w(f;\sqrt{\delta_n(x)})(1+q_n)$$\ where $$\label{e10} \delta_n(x)=\frac{1}{([n]_{p,q}+\beta)^{2}}[(q[n]_{p,q}[n-1]_{p,q} -{[n]}^2+\beta^{2})x^2~~+~~([n]_{p,q}p^{(n-1)}-2\alpha\beta)x~+\alpha^2].$$ Proof: $|S_{n,p,q}(t;x) -f(x)|\leq S_{n,p,q}(|f(t)-f(x)|:x)$\ by using $(\ref{e119}),$ we get\ $$|S_{n,p,q}(t;x) -f(x)|\leq w(f;\delta)\{S_{n,p,q}(1;x)+\frac{1}{\delta}S_{n,p,q}(|t-x|:x)\}.$$\ By using Cauchy Schwarz inequality, we have\ $$\begin{aligned} |S_{n,p,q}(t;x)-f(x)|&\leq w(f;\delta_n)\bigg(1+\frac{1}{\delta_n}[(S_{n,p,q}(t-x)^2;x)]^{\frac {1}{2}}~~[S_{n,p,q}(1;x)]^{\frac {1}{2}}\bigg)\\ &\leq w(f;\delta_n)\bigg(1+\frac{1}{\delta_n}\bigg\{\frac{1}{([n]_{p,q}+\beta)^{2}}[(q[n]_{p,q}[n-1]_{p,q} -{[n]}^2\\~~ &+~~\beta^2)x^2~~+~~([n]_{p,q}p^{(n-1)}~-~2\alpha\beta)x~~ +\alpha^2]\bigg\}\bigg)\end{aligned}$$ so it is obvious by choosing $\delta_n$ as in $(\ref{e10})$ the theorem is proved.\ Notice that by the condition in (\[e118\]) $st-\lim\limits_{n} \delta_n =0,$ by $(\ref{e118})$ we have\ $$st-\lim\limits_{n} w(f;\delta)~~=~~0.$$ This gives us the pointwise rate of statistical convergence of the operators $S_{n,p,q}(f;x)~ \text{to}~ f(x).$\ Monotonicity for convex functions ================================= Oruç and Phillips proved that when the function $f$ is convex on $[0,1]$, its $q$-Bernstein operators are monotonic decreasing. In this section we will study the monotonicity of $(p,q)$-Bernstein Stancu operators.\ If f is convex function on $[0,1],$ then $S_{n,p,q}(f;x)\geq f(x), $ $0\leq x \leq1$\ for all $n\geq 1$ and $0 < q < p \leq 1$ **Proof:** We consider the knots $x_k = \frac{p^{n-k}[k]_{p,q}}{[n]_{p,q}},$ $$\lambda_k = \left[ \begin{array}{c} n \\ k\end{array}\right] _{p,q} p^{\frac{k(k-1)-n(n-1)}{2}} x^k \prod\limits_{s=0}^{n-k-1}(p^{s}-q^{s}x),~~~~ 0 \leq k \leq n.$$ Using Lemma 2.1, it follows that\ $$\lambda_0+\lambda_1+\lambda_2+................\lambda_n = 1$$\ $$x_0\lambda_0+x_1\lambda_1+x_2\lambda_2+................x_n\lambda_n = x.$$\ From the convexity of the function $f,$ we get\ $S_{n,p,q}(f;x) =\sum\limits_{k=0}^{n}\lambda_k f(x_k)\geq f\bigg(\sum\limits_{k=0}^{n}\lambda_k x_k\bigg ) =f(x).$\ Let f be convex on $[0,1]$. Then$S_{{n-1},p,q}(f;x)\geq S_{n,p,q}(f;x)$ for $0 < q < p\leq 1, $ $0 \leq x \leq 1,$ and $ n \geq 2 $. If $f \in C[0,1]$ the inequality holds strictly for $0 < x < 1 $ unless f is linear in each of the intervals between consecutive knots $\frac{p^{n-k-1}[k]_{p,q}}{[n]_{p,q}}$, $0 \leq k\leq n-1 $, in which case we have the equality.\ **Proof:** For $0<q<p\leq1,$ we begin by writing\ $$\prod\limits_{s=0}^{n-1}(p^s -q^sx)^{-1} [S_{n-1,p,q}(f;x) - S_{n,p,q}(f;x)]$$\ $$\begin{aligned} &=&\prod\limits_{s=0}^{n-1}(p^s -q^sx)^{-1}\bigg [\sum\limits_{k=0}^{n-1} \left[ \begin{array}{c} n-1 \\ k \end{array} \right] _{p,q} p^{\frac{k(k-1)-(n-2)(n-1)}{2}} x^k \prod \limits_{s=0}^{n-k-2}(p^s -q^sx)f \bigg (\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\\ &&-\sum\limits_{k=0}^{n} \left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q} x^k p^{\frac{k(k-1)-n(n-1)}{2}} \prod\limits_{s=0}^{n-k-1}(p^s -q^sx)f\bigg(\frac {p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\bigg]\\ &=&\sum\limits_{k=0}^{n-1} \left[ \begin{array}{c} n-1 \\ k \end{array} \right] _{p,q} p^{\frac{k(k-1)-(n-2)(n-1)}{2}} x^k \prod\limits_{s=n-k-2}^{n-1}(p^s -q^sx)^{-1}f\bigg (\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\\ &&-\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q} x^k p^{\frac{k(k-1)-n(n-1)}{2}}\prod\limits_{s=n-k-1}^{n-1}(p^s -q^sx)^{-1} f\bigg (\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg).\end{aligned}$$ Denote\ $$\label{e14} \psi_{k}(x) = p^{\frac{k(k-1)}{2}} x^k \prod\limits_{s=n-k-1}^{n-1}(p^s -q^sx)^{-1}$$ and using the following relation:\ $$\begin{aligned} p^{n-1} p^{\frac{k(k-1)}{2}} x^k \prod\limits_{s=n-k-1}^{n-1}(p^s -q^sx)^{-1} = p^k \psi_{k}(x)+q^{n-k-1}\psi_{k+1}(x).\\\end{aligned}$$ We find\ $$\prod\limits_{s=0}^{n-1}(p^s -q^sx)^{-1}[S_{n-1,p,q}(f;x) - S_{n,p,q}(f;x)]\\$$ $$\begin{aligned} &=&\sum\limits_{k=0}^{n-1}\left[ \begin{array}{c} n-1\\ k \end{array} \right] _{p,q} p^{\frac{-(n-2)(n-1)}{2}} p^{-(n-1)}(p^k \psi_{k}(x)+q^{n-k-1}\psi_{k+1}(x)) f\bigg (\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\\ &&-\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right] _{p,q} p^{\frac{-n(n-1)}{2}}\psi_{k}(x)f\bigg (\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\\ &=&p^{\frac{-n(n-1)}{2}}\bigg[\sum\limits_{k=0}^{n-1}\left[ \begin{array}{c} n-1\\ k \end{array} \right]_{p,q} p^k \psi_{k}(x)f\bigg(\frac{p^{n-k-1}[k]_{p,q}+\alpha}{[n]_{p,q}+ \beta}\bigg)\\ &&+ \sum\limits_{k=1}^{n}\left[ \begin{array}{c} n-1 \\ k-1 \end{array} \right]_{p,q} q^{n-k} \psi_{k}(x) f \bigg(\frac {p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)-\sum\limits_{k=0}^{n}\left[ \begin{array}{c} n \\ k \end{array} \right]_{p,q} \psi_{k}(x) f \bigg(\frac {p^{n-k}[k]_{p,q} + \alpha}{[n]_{p,q} + \beta} \bigg)\bigg]\\ &=&p^{\frac{-n(n-1)}{2}} \sum \limits_{k=1}^{n-1}\Bigg\{\left[ \begin{array}{c} n-1\\ k \end{array} \right]_{p,q} p^k f \bigg(\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\\ &&+\left[ \begin{array}{c} n-1\\ k-1 \end{array} \right] _{p,q} q^{n-k} f\bigg(\frac{p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg) -\left[ \begin{array}{c} n \\ k \end{array} \right]_{p,q} f \bigg (\frac {p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\Bigg\} \psi_{k}(x)\\ &=&p^{\frac{-n(n-1)}{2}} \sum\limits_{k=1}^{n-1}\left[ \begin{array}{c} n\\ k \end{array} \right]_{p,q}\Bigg\{\frac {[n-k]_{p,q}}{[n]_{p,q}}p^k f\bigg(\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\\ &&+\frac{[k]_{p,q}}{[n]_{p,q}} q^{n-k} f\bigg(\frac{p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)-f\bigg(\frac {p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)\Bigg\}\psi_{k}(x)\\ &=&p^{\frac{-n(n-1)}{2}} \sum \limits_{k=1}^{n-1}\left[ \begin{array}{c} n\\ k \end{array} \right]_{p,q} a_k \psi_{k}(x)\end{aligned}$$ where\ $a_k =\frac{[n-k]_{p,q}}{[n]_{p,q}} p^k f \bigg(\frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)+\frac{[k]_{p,q}}{[n]_{p,q}} q^{n-k} f\bigg(\frac{p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg)- f \bigg (\frac {p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}\bigg).$\ From it is clear that each $ \psi_k(x)$ is non-negative on $[0,1]$ for $0< q < p \leq 1 $ and, thus, it suffices to show that each $a_k$ is non-negative.\ Since $f$ is convex on $[0,1]$ then for any $t_0 ,t_1$ and $\lambda \in [0,1]$ it follows that\ $$f(\lambda t_0 + (1-\lambda)t_1) \leq \lambda f(t_0) + (1-\lambda)f(t_1).$$\ If we choose $t_0 = \frac {p^{n-k}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta}$, $ t_1 = \frac {p^{n-k-1}[k]_{p,q}+ \alpha}{[n]_{p,q}+ \beta},$ and\ $\lambda = \frac {[k]_{p,q}}{[n]_{p,q}}q^{n-k},$ then $t_0 ,t_1$ $\in $ $[0,1]$ and $\lambda \in (0,1)$ for $1 \leq k \leq n-1,$ and we deduce that\ $$a_k = \lambda f( t_0) + (1-\lambda)f(t_1)- f(\lambda t_0 + (1-\lambda)t_1)\geq0$$\ Thus $ S_{n-1,p,q}(f;x)\geq S_{n,p,q}(f;x).$\ We have equality for $x=0$ and $x=1,$ since the Bernstein polynomials interpolate $f$ on these end points.The inequality will be strict for $ 0 < x < 1 $ unless when $f$ is linear in each of the intervals between consecutive knots $$\frac{p^{n-k-1}[k]_{p,q} + \alpha}{[n]_{p,q} + \beta},~~ 0 \leq k \leq n-1,$$ then we have\ $$S_{n-1,p,q}(f;x) = S_{n,p,q}(f;x)$$ for $ 0 \leq x \leq 1.$\ A Global Approximation theorem ============================== In this section, we establish a global approximation theorem by means of Ditzian-Totik modulus of smoothness and Voronovskaja type approximation result.\ In order to prove our next result, we recall the definitions of the Ditzian-Totik first order modulus of smoothness and the K-functional. Let $\phi(x) = \surd{x(1-x)}$ and $f \in C[0,1]$. The first order modulus of smoothness is given by\ $$\label{e15} \omega_\phi(f;t) = \sup \limits _ {0< h \leq t} \Big\{\big|f(x + \frac {h\phi(x)}{2}) - f(x - \frac {h\phi(x)}{2})\bigg| , x \pm \frac {h\phi(x)}{2} \in [0,1]\Big\}\\$$ The corresponding k-functional to is defined by\ $k_\phi(f;t) = \inf\limits_{g \in W_\phi [0,1]}\Big\{\|f - g\| + t \|\phi g^{\prime}\|\Big\} $ $(t > 0),$\ where $W_\phi[0,1] = \{g: g \in AC_{loc}[0,1] , \|\phi g^{\prime}\|< \infty\} $ and $g \in AC_{loc}[0,1]$ means that $g$ is absolutely continuous on every interval $[a,b]\subset[0,1]$. It is well known [@dzk1] that there exists a constant $C >0$ such that\ $$\label{ab} k_\phi(f;t) \leq Cw_\phi(f;t).\\$$ Let $f \in C[0,1]$ and $\phi(x) = \surd{x(1-x)},$ then for every $x \in [0,1]$ we have\ $\bigg| S_{n,p,q}(f;x)- f(x)\bigg| \leq C \omega_\phi\bigg(f;\frac{[n]_{p,q}}{\surd([n]_{p,q}+\beta)}\bigg)$ where C is a constant independent of n and x.\ **Proof:** Using the representation\ $$g(t) = g(x)+\int_{x}^{t} g^{\prime}(u) du ,$$\ we get\ $$\label{e15} \bigg| S_{n,p,q}(g;x)- g(x)\bigg| = \bigg| S_{n,p,q}\bigg(\int_{x}^{t} g^{\prime}(u) du ;x\bigg)\bigg|.\\$$ For any $x \in (0,1)$ and $ t \in [0,1],$ we find that\ $$\label{e16} \bigg |\int_{x}^{t} g^{\prime}(u) du\bigg |\leq \| \phi g^{\prime}\| \bigg|\int_{x}^{t} \frac {1}{\phi(u)}du \bigg|$$ Further,\ $$\begin{aligned} \label{e17} \bigg| \int_{x}^{t} \frac {1}{\phi(u)}du\bigg| &= \bigg|\int_{x}^{t} \frac {1}{\surd u(1-u)} du\bigg| \notag\\ &\leq \bigg| \int_{x}^{t} \bigg( \frac{1}{\surd u} + \frac{1}{\surd 1-u}\bigg)du\bigg |\notag\\ &\leq 2 ( |\surd{t} -\surd {x}|+ | \surd {1-t} - \surd {1-x}|)\notag\\ & = 2|t - x |\bigg (\frac {1}{\surd t + \surd x} + \frac{1}{\surd {1-t} + \surd {1-x}}\bigg)\notag\\ & < 2 |t - x | \bigg (\frac {1}{\surd x} +\frac {1}{\surd {1-x}}\bigg) \leq \frac { 2\surd 2 |t - x |}{\phi(x)}\end{aligned}$$ From (\[e15\]) - (\[e17\]) and using the Cauchy - Schwarz inequality, we obtain\ $$\begin{aligned} | S_{n,p,q}(g;x)- g(x)|&< 2\surd2 \|\phi g^{\prime}\| \phi^{-1}(x)S_{n,p,q}(|t - x|;x)\\ &\leq 2\surd2 \|\phi g^{\prime}\| \phi^{-1}(x)(S_{n,p,q}((t - x)^{2};x))^{\frac {1}{2}}.\end{aligned}$$ Using lemma (2.2), we get $$\begin{aligned} | S_{n,p,q}(g;x)- g(x)| \leq \frac {2\surd2[n]_{p,q}}{\surd([n]_{p,q}+\beta} \|\phi g^{\prime}\|.\end{aligned}$$ Now using the above inequality we can write\ $$\begin{aligned} | S_{n,p,q}(f;x)- f(x)|&\leq | S_{n,p,q}(f-g ;x)| + |f(x) - g(x)| + | S_{n,p,q}(g;x)- g(x)|\\ & \leq 2\surd 2 \bigg(\|f - g\|+ \frac {[n]_{p,q}}{\surd([n]_{p,q}+\beta)} \|\phi g^{\prime}\|\bigg).\end{aligned}$$ Taking the infimum on the right hand side of the above inequality over all $g \in W_\phi[0,1],$ we get\ $$\label{e18} | S_{n,p,q}(f;x)- f(x)| \leq CK_\phi \bigg(f;\frac{[n]_{p,q}}{\surd([n]_{p,q}+\beta)} \bigg).$$ Using equation $(\ref{ab})$ this theorem is proven. where $g\in W_\phi [0,1]$.On the other hand, for any $ m = 1,2,.......$ and $ 0< q < p \leqslant 1$, there exists a constant $C_m > 0 $ such that\ $$\label{e20} \vert S_{n,p,q}((t-x)_{p,q}^{m};x)\vert \leqslant C_m \frac{\phi^2(x)[n]_{p,q}}{([n]_{p,q}+ \beta)^{\lfloor \frac{m+1}{2}\rfloor}},$$ where $x\in [0,1]$ and $\lfloor a \rfloor $ is the integral part of $a \geq 0.$\ Throughout this proof, C denotes a constant not necessarily the same at each occurrence.\ Now combining (\[e18\]) -(\[e20\]) and applying lemma (2.2), the cauchy-schwarz inequality,\ We get\ $\bigg| S_{n,p,q}(f;x)- f(x)\frac {p^{n-1}[n]_{p,q} - 2\alpha \beta}{2([n]_{p,q}+ \beta)^2}f^{\prime\prime}(x)\bigg|$\ $$\begin{aligned} &\leq 2 \|f^{\prime\prime} - g \|S_{n,p,q}((t- x )^{2};x) + 2 \|\phi g^{\prime} \|\phi^{-1}(x)S_{n,p,q}(|t - x|^3;x)\\ &\leq2\|f^{\prime\prime} - g\|\frac{\phi^2(x)[n]_{p,q}}{([n]_{p,q}+ \beta)} + 2\|\phi g ^{\prime} \|\phi^{-1}(x)\{S_{n,p,q}((t- x )^{2};x)\}^{\frac{1}{2}}\{S_{n,p,q}((t- x )^{4};x)\}^{\frac{1}{2}}\\ &\leq2\|f^{\prime\prime} - g \|\frac{\phi^2(x)[n]_{p,q}}{([n]_{p,q}+ \beta)} + 2\frac{C}{([n]_{p,q}+ \beta)}\|\phi g ^{\prime} \|\frac{ \phi(x)[n]_{p,q}}{([n]_{p,q}+ \beta)^\frac{1}{2}}\\ &\leq \frac{C[n]_{p,q}}{([n]_{p,q}+ \beta)}\{\phi^2(x)\|f^{\prime\prime} - g \| + ([n]_{p,q}+ \beta)^\frac{-1}{2}\phi(x) \| \phi g ^{\prime}\|\}.\\\end{aligned}$$ since $\phi^2(x)\leq \phi(x)\leq 1,x\in[0,1],$ We obtain\ $$\begin{aligned} \bigg|([n]_{p,q}+ \beta)^2[S_{n,p,q}(f;x)- f(x)]-\frac {p^{n-1}[n]_{p,q}-2\alpha\beta}{2}\phi^2(x)f^{\prime\prime}(x)\bigg| & \leq C\{\|f^{\prime\prime}-g\| \\ &+([n]_{p,q}+ \beta)^\frac{-1}{2}\phi(x)\|\phi g^{\prime}\|\}.\\\end{aligned}$$ Also, the following inequality can be obtained:\ Voronovskaja type theorem ------------------------- Using the first order Ditzian-Totik modulus of smoothnes, we prove a quantitative Voronovskaja type theorem for the $(p,q)$-Bernstein operators. For any $f$ $\in C^2[0,1],$ the following inequalities holds:\ $$\vert([n]_{p,q}+ \beta)[ S_{n,p,q}(f;x)- f(x)] - \frac{p^{n-1}- 2 \alpha\beta}{2} \phi^2(x)f^{\prime\prime}(x)\vert \leqslant C\omega_\phi(f^{\prime\prime}\phi(x)n^{\frac{-1}{2}}),$$ $$\vert([n]_{p,q}+\beta)[ S_{n,p,q}(f;x)- f(x)] - \frac{p^{n-1}- 2 \alpha\beta}{2}\phi^2(x)f^{\prime\prime}(x)\vert \leqslant C\phi(x)\omega_\phi(f^{\prime\prime},n^{\frac{-1}{2}}),$$ where C is a positive constant.\ **Proof:** Let $f \in C^2[0,1]$ be given and $t,x \in[0,1]$ using Taylor’s expansion, we have\ $$f(t)-f(x) = (t-x)f^{\prime}(x)+ \int_{x}^{t}(t-u)f^{\prime\prime}(u)du$$ Therefore $$\begin{aligned} f(t) -f(x) - (t-x)f^{\prime}(x)- \frac{1}{2}(t-x)^2f^{\prime\prime}(x) &= \int_{x}^{t}(t-u)f^{\prime\prime}(u)du - \int_{x}^{t}(t-u)f^{\prime\prime}(x)dx \\ &= \int_{x}^{t}(t-u)[f^{\prime\prime}(u) - f^{\prime\prime}(x)]du\end{aligned}$$ in view of lemma (2.2), we get\ $$\bigg| S_{n,p,q}(f;x)- f(x) - \frac {p^{n-1}[n]_{p,q} - 2\alpha \beta}{2([n]_{p,q}+ \beta)^2}\phi^2(x)f^{\prime\prime}(x)\bigg| \leq S_{n,p,q} \bigg(\bigg| \int_{x}^{t}|(t-u)||f^{\prime\prime}(u)- f^{\prime \prime}(x)|du \bigg|;x \bigg).$$ The quantity $ |\int_{x}^{t} |f^{\prime\prime}(u)- f^{\prime\prime}(x)||(t-u)|du|$ was estimated in \[ \],p- , as follows:\ $$\bigg|\int_{x}^{t}f^{\prime\prime}(u)- f^{\prime \prime}(x)||t-u|du \bigg| \leq 2 \|f^{\prime\prime} - g \|(t - x)^2 + 2\| \phi g^{\prime}\| \phi^{-1}(x)|t - x|^3,$$ where $g\in W_\phi [0,1]$ . On the other hand, for any $ m = 1,2,.......$ and $ 0< q < p \leqslant 1$, there exists a constant $C_m > 0 $ such that\ $$\vert S_{n,p,q}((t-x)_{p,q}^{m};x)\vert \leqslant C_m \frac{\phi^2(x)[n]_{p,q}}{([n]_{p,q}+ \beta)^{\lfloor \frac{m+1}{2}\rfloor}}$$ where $x\in [0,1]$ and $\lfloor a \rfloor $ is the integral part of $a \geq 0.$\ Throughout this proof, C denotes a constant not necessarily the same at each occurrence.\ Now combining (8.4) -(8.5) and applying lemma (2.2), the cauchy-schwarz inequality,\ We get\ $\bigg| S_{n,p,q}(f;x)- f(x)\frac {p^{n-1}[n]_{p,q} - 2\alpha \beta}{2([n]_{p,q}+ \beta)^2}f^{\prime\prime}(x)\bigg|$\ $$\begin{aligned} &\leq 2 \|f^{\prime\prime} - g \|S_{n,p,q}((t- x )^{2};x) + 2 \|\phi g^{\prime} \|\phi^{-1}(x)S_{n,p,q}(|t - x|^3;x)\\ &\leq2\|f^{\prime\prime} - g\|\frac{\phi^2(x)[n]_{p,q}}{([n]_{p,q}+ \beta)} + 2\|\phi g ^{\prime} \|\phi^{-1}(x)\{S_{n,p,q}((t- x )^{2};x)\}^{\frac{1}{2}}\{S_{n,p,q}((t- x )^{4};x)\}^{\frac{1}{2}}\\ &\leq2\|f^{\prime\prime} - g \|\frac{\phi^2(x)[n]_{p,q}}{([n]_{p,q}+ \beta)} + 2\frac{C}{([n]_{p,q}+ \beta)}\|\phi g ^{\prime} \|\frac{ \phi(x)[n]_{p,q}}{([n]_{p,q}+ \beta)^\frac{1}{2}}\\ &\leq \frac{C[n]_{p,q}}{([n]_{p,q}+ \beta)}\{\phi^2(x)\|f^{\prime\prime} - g \| + ([n]_{p,q}+ \beta)^\frac{-1}{2}\phi(x) \| \phi g ^{\prime}\|\}\\\end{aligned}$$ since $\phi^2(x)\leq \phi(x)\leq 1,x\in[0,1],$ We obtain\ $$\bigg|([n]_{p,q}+ \beta)^2[S_{n,p,q}(f;x)- f(x)] - \frac {p^{n-1}[n]_{p,q} - 2\alpha \beta}{2}\phi^2(x)f^{\prime\prime}(x)\bigg| \leq C\{\|f^{\prime\prime} - g\| + ([n]_{p,q}+ \beta)^\frac{-1}{2}\phi(x) \| \phi g ^{\prime}\|\}\\$$ Also, the following inequality can be obtained:\ $$\bigg|([n]_{p,q}+ \beta)^2[S_{n,p,q}(f;x)- f(x)] - \frac {p^{n-1}[n]_{p,q} - 2\alpha \beta}{2}\phi^2(x)f^{\prime\prime}(x)\bigg| \leq C\phi(x)\{\|f^{\prime\prime} - g\| + ([n]_{p,q}+ \beta)^\frac{-1}{2}\| \phi g ^{\prime}\|\}$$ Taking the infimum on the right - hand side of the above relations over $g\in W_\phi[0,1],$ we get\ $$\bigg|([n]_{p,q}+ \beta)^2[S_{n,p,q}(f;x)- f(x)] - \frac {p^{n-1}[n]_{p,q} - 2\alpha \beta}{2}\phi^2(x)f^{\prime\prime}(x)\bigg| \leq {C\phi(x)K_\phi( f^{\prime\prime};([n]_{p,q}+ \beta)^\frac{-1}{2})}{CK_\phi(f^{\prime\prime};\phi(x)([n]_{p,q}+ \beta)^\frac{-1}{2})},$$ Using (8.9) and(7.2) the theorem is proved.\ Graphical Analysis ================== With the help of Matlab, we show comparisons and some illustrative graphics for the convergence of operators $(\ref{ee2})$ to the function $f(x)=1+x^3~ sin(14x)$ under different parameters.\ From figure \[f1\](a), it can be observed that as the value the $q ~\text{and}~ p $ approaches towards $1$ provided $0<q<p\leq1$, $(p, q)$-Bernstein Stancu operators given by $(\ref{ee2})$ converges towards the function.\ From figure \[f1\](a) and (b), it can be observed that for $\alpha=\beta=0,$ as the value the $n$ increases, $(p, q)$-Bernstein Stancu operators given by \[ee2\] converges towards the function $f(x)=1+x^3~ sin(14x)$.\ Similarly from figure \[f2\](a), it can be observed that for $\alpha=\beta=3,$ as the value the $q ~\text{and}~ p $ approaches towards $1$ provided $0<q<p\leq1$, $(p, q)$-Bernstein Stancu operators given by \[ee2\] converges towards the function.\ From figure \[f2\](a) and (b), it can be observed that as the value the $n$ increases, $(p, q)$-Bernstein Stancu operators given by $f(x)=1+x^3~ sin(14x)$ converges towards the function. \[f1\] \[f2\] [99]{} T Acar, A. Aral, S. A Mohiuddine, Approximation By Bivariate $(p, q)$−Bernstein-Kantorovich operators, Iranian Journal of Science and Technology, Transactions A: Science, DOI:10.1007/s40995-016-0045-4. T. Acar, S.A. Mohiudine, M. Mursaleen, Approximation by $(p, q)$−Baskakov-Durrmeyer–Stancu operators, Communicated. T. Acar, $(p,q)$-Generalization of Szász-Mirakyan operators, *Mathematical Methods in the Applied Sciences*, DOI: 10.1002/mma.3721. T. Acar, A. Aral, S.A. Mohiuddine, On Kantorovich modification of $(p,q)$-Baskakov operators, Journal of Inequalities and Applications, 98 (2016). A. Aral, V.Gupta and Ravi P. Agarwal, Applications of $q$-calculus in operator theory. Berlin: *Springer*, 2013. S. N. Bernstein, Constructive proof of Weierstrass approximation theorem, *Comm. Kharkov Math. Soc*. (1912) P.E. B$\acute{e}$zier, Numerical Control-Mathematics and applications, *John Wiley and Sons*, London, 1972. Qing-Bo Cai, Guorong Zhoub, On $(p, q)$-analogue of Kantorovich type Bernstein-Stancu-Schurer operators, *Applied Mathematics and Computation*, Volume 276, 5 March 2016, Pages 12–20. G. Gasper, M. Rahman, Basic hypergometric series, *Cambridge University Press*, Cambridge, 1990. Mahouton Norbert Hounkonnou, Joseph Désiré Bukweli Kyemba, $\mathcal{R}(p,q)$-calculus: differentiation and integration, *SUT Journal of Mathematics,* Vol. 49, No. 2 (2013), 145-167. R. Jagannathan, K. Srinivasa Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, *Proceedings of the International Conference on Number Theory and Mathematical Physics*, 20-21 December 2005. A. Karaisa, On the approximation properties of bivariate $(p,q)$−Bernstein operators arXiv:1601.05250 P. P. Korovkin, Linear operators and approximation theory, *Hindustan Publishing Corporation*, Delhi, 1960. A. Lupaş, A $q$-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, *University of Cluj-Napoca*, 9(1987) 85-92. S. Ersan and O. Doğru, Statistical approximation properties of $q$-Bleimann, Butzer and Hahn operators, Mathematical and Computer Modelling, 49 (2009) 1595–1606. H. Fast, Sur la convergence statistique, Colloq. Math. 2(1951) 241-244. A. D. Gadjiv and C. Orhan, Some approximzation theorems via statistical convergence, Rocky Mount. J. Math., 32 (2002) 129–138. M. Mursaleen, K. J. Ansari, Asif Khan, On $(p,q)$-analogue of Bernstein Operators, *Applied Mathematics and Computation,* 266 (2015) 874-882, (Erratum to ‘On $(p,q)$-analogue of Bernstein Operators’ Appl. Math. Comput. 266 (2015) 874-882.) M. Mursaleen, K. J. Ansari and Asif Khan, Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators, *Applied Mathematics and Computation* 264,(2015), 392-402. (Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators(Revised)arXiv:1602.06288v1 \[math.CA\].) M. Mursaleen, K. J. Ansari and Asif Khan, Some approximation results for Bernstein-Kantorovich operators based on $(p, q)$-calculus, *U.P.B. Sci. Bull., Series A,* Vol. 78, Iss. 4, 2016. M. Mursaleen, Md. Nasiruzzaman, Asif Khan and K. J. Ansari, Some approximation results on Bleimann-Butzer-Hahn operators defined by $(p,q)$-integers, Filomat 30:3 (2016), 639-648, DOI 10.2298/FIL1603639M. M. Mursaleen, Asif Khan, Generalized $q$-Bernstein-Schurer Operators and Some Approximation Theorems, *Journal of Function Spaces and Applications* Volume 2013, Article ID 719834, 7 pages http://dx.doi.org/10.1155/2013/719834 M. Mursaleen, F. Khan and Asif Khan, Approximation by $(p,q)$-Lorentz polynomials on a compact disk, Complex Anal. Oper. Theory, DOI: 10.1007/s11785-016-0553-4. Mursaleen, Md. Nasiruzzaman, F. Khan, A. Khan, On $(p,q)$-analogue of divided differences and Bernstein operators, Journal of Nonlinear Functional Analysis, Vol. 2017 (2017), Article ID 25, pp. 1-13. M. Mursaleen and Md. Nasiruzzaman and Ashirbayev Nurgali, Some approximation results on Bernstein-Schurer operators defined by $(p,q)$-integers, *Journal of Inequalities and Applications* (2015), 249, DOI 10.1186/s13660-015-0767-4. N. Mahmudov and P. Sabancigil, A $q$-analogue of the Meyer-K ōnig and Zeller Operators. Bull. Malays. Math. Sci. Soc. (2), 35(1) (2012) 39-51 Halil Oruç, George M. Phillips, $q$-Bernstein polynomials and B$\acute{e}$zier curves, *Journal of Computational and Applied Mathematics* 151 (2003) 1-12. Sofiya Ostrovska, On the Lupaş¸ $q$-analogue of the bernstein operator, *Rocky mountain journal of mathematics* Volume 36, Number 5, 2006. G.M. Phillips, Bernstein polynomials based on the $q$-integers, The heritage of P.L.Chebyshev, *Ann. Numer. Math.*, 4 (1997) 511–518. G.M. Phillips, A De Casteljau Algorithm For Generalized Bernstein Polynomials, *BIT* 36 (1) (1996), 232-236. Ugur Kadak, On weighted statistical convergence based on $(p,q)$-integers and related approximation theorems for functions of two variables, Journal of Mathematical Analysis and Applications, May 2016, DOI: 10.1016/j.jmaa.2016.05.062. Shin Min Kang, Arif Rafiq, Ana-Maria Acu, Faisal Ali and Young Chel Kwun, Some approximation properties of (p,q)-Bernstein operators, *Journal of Inequalities and Applications* 2016:169, DOI: 10.1186/s13660-016-1111-3. Khalid Khan, D.K. Lobiyal, Adem Kilicman, A de Casteljau Algorithm for Bernstein type Polynomials based on $(p,q)$-integers, *arXiv* 1507.04110v4. Khalid Khan, D.K. Lobiyal, B$\acute{e}$zier curves based on Lupas $(p,q)$-analogue of Bernstein functions in CAGD, *Journal of Computational and Applied Mathematics* Volume 317, June 2017, Pages 458-477. Khalid Khan, D.K. Lobiyal, B$\acute{e}$zier curves and surfaces based on modified Bernstein polynomials, *arXiv*:1511.06594v1. Khalid Khan, D.K. Lobiyal, Algorithms and identities for $(p,q)$-B$\acute{e}$zier curves via $(p,q)$-Blossom. *arXiv*: P. P. Korovkin, Linear operators and approximation theory, *Hindustan Publishing Corporation*, Delhi, 1960. V. Kac, P. Cheung, Quantum Calculus, in: Universitext Series, vol. IX, Springer-Verlag, 2002. A. Lupaş, A $q$-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, *University of Cluj-Napoca*, 9(1987) 85-92. N. I. Mahmudov and P. Sabancıgil, Some approximation properties of Lupaş $q$-analogue of Bernstein operators, *arXiv*:1012.4245v1 \[math.FA\] 20 Dec 2010. A. Wafi, Nadeem Rao, Approximation properties of $(p,q)$-variant of Stancu-Schurer and Kantorovich-Stancu-Schurer operators, arXiv:1508.01852v2.

--- abstract: 'Gaussian processes (GPs) are widely used as surrogate models for emulating computer code, which simulate complex physical phenomena. In many problems, additional boundary information (i.e., the behavior of the phenomena along input boundaries) is known beforehand, either from governing physics or scientific knowledge. While there has been recent work on incorporating boundary information within GPs, such models do not provide theoretical insights on improved convergence rates. To this end, we propose a new GP model, called BdryGP, for incorporating boundary information. We show that BdryGP not only has improved convergence rates over existing GP models (which do not incorporate boundaries), but is also more resistant to the “curse-of-dimensionality” in nonparametric regression. Our proofs make use of a novel connection between GP interpolation and finite-element modeling.' author: - '$^*$' - '$^\dagger$' - '$^\ddagger$' bibliography: - 'imsart.bib' title: 'BdryGP: a new Gaussian process model for incorporating boundary information' --- , and \ *[Hong Kong University of Science and Technology$^*$, Duke University$^\dagger$ and Georgia Institute of Technology$^\ddagger$]{.nodecor}* Introduction ============ With advances in mathematical modeling and computation, complex phenomena can now be simulated via computer code. This code numerically solves a system of governing equations which represents the underlying science of the problem. Due to the time-intensive nature of these numerical simulations [@yeh2018common], Gaussian processes (GPs; [@sacks1989]) are often used as surrogate models to emulate the expensive computer code. Let ${\bold x}\in \mathcal{X} = [0,1]^d$ be a vector of $d$ code inputs, and let $f({\bold x})$ be its corresponding code output. The idea is to adopt a GP prior for $f(\cdot)$, then use the posterior process given data to infer code output at an unobserved input. GP emulators are now widely used to study a broad range of scientific and engineering problems, such as rocket engines [@Mak18], universe expansions [@kaufman2011efficient] and high energy physics [@goh2013prediction]. In many applications, there is additional knowledge on the phenomenon than simply computer code output, and incorporating such knowledge can improve GP predictive performance. This “physics-integrated” GP modeling has garnered much attention in recent years [@Wheeler14; @Golchi15; @WangBerger16]. We consider here a specific type of information called *Dirichlet boundaries* [@bazilevs2007weak], which specifies the values of $f$ along certain input boundaries. Dirichlet boundaries are often available from governing physics or from simple physical considerations [@Matt18]. One example is the simulation of viscous flows [@white2006viscous], widely used in climatology and high energy physics. Such flows are dictated by the complex Navier-Stokes equations [@temam2001navier], and can be very time-consuming to simulate. At the limits of certain variables (e.g., zero viscosity or fluid incompressibility), the Navier-Stokes equations can be greatly simplified for efficient, even closed-form, solutions [@kiehn2001some; @humphrey2016introduction]. Incorporating this boundary information within the GP can allow for improved predictive performance. Despite its promise, the integration of GPs with boundary information is largely unexplored in the literature, with the only reference being a recent paper by [@Matt18]. In this paper, a flexible *Boundary Modified Gaussian Process* (BMGP) is proposed, which can integrate a broad range of boundaries by modifying the mean and variance structure of a stationary GP. Due in part to its modeling flexibility, the BMGP model is quite complicated and difficult to analyze theoretically. This raises an important open question: to what extent does incorporating boundary information improve convergence rates for GPs? To address this, we propose a new GP model, called BdryGP, which has provably improved error rates when incorporating boundary information. The key novelty is a new Boundary Constrained Mat[é]{}rn (BdryMatérn) covariance function, which incorporates boundary information of the form: $$\mathcal{F}_j^{[0]} := \{f({\bold x}) : x_j = 0\}, \quad \text{or} \quad \mathcal{F}_j^{[1]} := \{f({\bold x}) : x_j = 1\}. \label{eq:bound}$$ The BdryMatérn covariance inherits the same smoothness properties as the tensor Matérn kernel, while constraining GP sample paths to satisfy almost surely. Assuming boundaries of the form is known for each variable $j = 1, \cdots, d$, we prove two main results for BdryGP. The first is a *deterministic* $L^p$ convergence rate for a *fixed* function $f \in {\mathcal H}^{1,c}_{mix}(\mathcal{X})$: $$\|f - \hat{f}_n^{\rm BM}\|_{L^p} = { \mathchoice {{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(n^{-1}), \quad 1 \leq p < \infty. \label{eq:lp}$$ Here, $n$ is the sample size, $\hat{f}_n^{\rm BM}$ is the BdryGP predictor, and ${\mathcal H}^{1,c}_{mix}(\mathcal{X})$ is the Sobolev space with mixed first derivatives satisfying . The second is a *probabilistic* uniform bound for a *random* function $Z(\cdot)$ following a GP with sample paths in ${\mathcal H}^{1,c}_{mix}(\mathcal{X})$: $$\sup_{{\bold x}\in[0,1]^d}|Z({\bold x})-\mathcal{I}_n^{\rm BM}Z({\bold x})|={\mathcal O}_{\mathbb{P}}(n^{-1}[\log n]^{2d-\frac{3}{2}}), \label{eq:unif}$$ where $\mathcal{I}_n^{\rm BM}$ is the BdryGP interpolation operator satisfying $\mathcal{I}_n^{\rm BM}f = \hat{f}_n^{\rm BM}$. Both rates require a sparse grid design [@Bungartz04]. Compared to existing GP rates (which do not incorporate boundary information), our BdryGP rates decay much faster in sample size $n$. (A full comparison is given in Section \[sec:comp\].) Furthermore, by incorporating boundaries, our rates are also more resistant to the well-known “curse-of-dimensionality” in nonparametric regression [@geenens2011curse]. Our proof makes use of a novel connection between GP interpolation and finite-element modeling (FEM). This paper is organized as follows. In Section 2, we present the new BdryGP model and derive the BdryMatérn kernel. In Section 3, we establish a novel connection between the BdryGP predictor and the FEM interpolator. In Section 4, we connect the function space for FEM with the native space for the BdryMatérn kernel. Using these results, we then derive in Section 5 the main convergence rates for BdryGP, and verify these rates in Section 6 via numerical simulations. Section 7 concludes the paper. The BdryGP model ================ We first give a brief review of GP modeling, then present a model specification for BdryGP. Gaussian process modeling ------------------------- Let ${\bold x}\in \mathcal{X}$ be an input vector on domain $\mathcal{X} = [0,1]^d$, with $f({\bold x})$ denoting its corresponding computer code output. In Gaussian process emulation [@sacks1989; @santner2003], $f(\cdot)$ is assumed to be a realization of a Gaussian process with mean function $\mu: \mathcal{X} \to {\mathbb R}$, and covariance function $k: \mathcal{X} \times \mathcal{X} \to {\mathbb R}$. Further details on GP modeling can be found in [@Adler81]. Suppose the code is evaluated at $n$ input points ${\bold X}=\{{\bold x}_1,\cdots,{\bold x}_n\}\subset \mathcal{X}$, yielding observations $f({\bold X})=[f({\bold x}_1),\cdots,f({\bold x}_n)]^\intercal$. Given data $f({\bold X})$, one can show that the conditional process $f(\cdot)|f({\bold X})$ is still a GP, with mean function: $$\label{eq:Kriging_posteriorMean} \hat{f}_n({\bold x})=\mu({\bold x})+k({\bold x},{\bold X})k^{-1}({\bold X},{\bold X})\left[f({\bold X})-\mu({\bold X})\right],$$ and covariance function: $$\label{eq:Kriging_posteriorCov} k_n({\bold x},{\bold y})=k({\bold x},{\bold y})-k({\bold x},{\bold X})k^{-1}({\bold X},{\bold X})k({\bold X},{\bold y}).$$ Here, $k({\bold X},{\bold x}) = [k({\bold x},{\bold x}_1),\cdots,k({\bold x},{\bold x}_n)]^\intercal$ denotes the covariance vector between design ${\bold X}$ and a new point ${\bold x}$, $k({\bold x},{\bold X}) = k({\bold X},{\bold x})^\intercal$, and $k({\bold X},{\bold X})$ is the covariance matrix ${{[k({\bold x}_i},{\bold x}_j)]_{i=1}^n}_{j=1}^n$ over design points. The posterior mean $\hat{f}_n(\cdot)$ is typically used as a predictor (or emulator) for unknown code output $f(\cdot)$, since it is optimal under quadratic and absolute error loss [@santner2003]. The posterior variance $k_n({\bold x},{\bold x})$ then quantifies the uncertainty of the predictor $\hat{f}_n({\bold x})$ at a new input setting ${\bold x}$. The kernel $k$ is also associated with an important function space ${\mathcal H}_k(\mathcal{X})$, called the *reproducing kernel Hilbert space* (RKHS) or *native space* of $k$ [@Wendland10]. For a symmetric, positive definite kernel $k$, the RKHS ${\mathcal H}_k(\mathcal{X})$ of $k$ is defined as the closure of the linear function space: $$\left\{\sum_{i=1}^n c_ik(\cdot,{\bold x}_{i}):c_i\in{\mathbb R},{\bold x}_{i}\in {\mathcal X}, n \in \mathbb{N} \right\}. \label{eq:rkhs}$$ This RKHS is also endowed with an inner product $\langle \cdot,\cdot\rangle _k$ which satisfies the so-called *reproducing property*: $$f({\bold x})=\langle f,k(\cdot,{\bold x})\rangle_k, \quad {\bold x}\in \mathcal{X},$$ for any function $f\in{\mathcal H}_k({\mathcal X})$. Both the RKHS ${\mathcal H}_k(\mathcal{X})$ and its reproducing property will play a key role in the derivation and analysis of the BdryGP. Boundary information -------------------- In many problems, boundary information on $f$ is available from governing physical principles or scientific knowledge. We consider here a common type of boundary called Dirichlet boundaries [@bazilevs2007weak], which specify the values of $f$ along certain boundaries of the input domain $\mathcal{X} = [0,1]^d$. In particular, we consider boundaries of the form $\mathcal{F}_j^{[0]}$ or $\mathcal{F}_j^{[1]}$ in , which quantify the values of $f$ along the left hyperplane $\mathcal{S}_j^{[0]} := \{{\bold x}:x_j = 0\}$ or right hyperplane $\mathcal{S}_j^{[1]} := \{{\bold x}:x_j = 1\}$ of a variable $x_j$, respectively. We will call $\mathcal{F}_j^{[0]}$ and $\mathcal{F}_j^{[1]}$ the *left* and *right* boundary condition of variable $x_j$. Boundaries of this form arise naturally in many limiting simplifications of physical systems, and provide closed-form expressions for the BdryMatérn kernel. ![Proposed mean function $\mu(\cdot)$ in for the three boundary cases in Figure \[fig:boundary\].[]{data-label="fig:mean_func_boundary"}](figures/boundary/full_boundary.pdf "fig:"){width="32.50000%"} ![Proposed mean function $\mu(\cdot)$ in for the three boundary cases in Figure \[fig:boundary\].[]{data-label="fig:mean_func_boundary"}](figures/boundary/partial_boundary.pdf "fig:"){width="32.50000%"} ![Proposed mean function $\mu(\cdot)$ in for the three boundary cases in Figure \[fig:boundary\].[]{data-label="fig:mean_func_boundary"}](figures/boundary/partial_boundary_2.pdf "fig:"){width="32.50000%"} ![Proposed mean function $\mu(\cdot)$ in for the three boundary cases in Figure \[fig:boundary\].[]{data-label="fig:mean_func_boundary"}](figures/boundary/mean_full_boundary.jpg "fig:"){width="32.50000%"} ![Proposed mean function $\mu(\cdot)$ in for the three boundary cases in Figure \[fig:boundary\].[]{data-label="fig:mean_func_boundary"}](figures/boundary/mean_partial_boundary.jpg "fig:"){width="32.50000%"} ![Proposed mean function $\mu(\cdot)$ in for the three boundary cases in Figure \[fig:boundary\].[]{data-label="fig:mean_func_boundary"}](figures/boundary/mean_partial_boundary_2.jpg "fig:"){width="32.50000%"} To distinguish which boundaries are known beforehand, let $I^{[0]} \subseteq [d] = \{1, \cdots, d\}$ denote the variables with known *left* boundary condition, and let $I^{[1]} \subseteq [d]$ denote the variables with known *right* boundary condition. With $I^{[0]} = \emptyset$ and $I^{[1]} = \emptyset$, this reduces to the standard setting with no boundary information; with $I^{[0]} = [d]$ and $I^{[1]} = [d]$, this implies knowledge of $f$ along the full boundary of $\mathcal{X}$. Figure \[fig:boundary\] illustrates this in two dimensions, with known boundaries of $f$ in blue. The left plot shows the case of $I^{[0]} = I^{[1]} = \{1, 2\}$, with the value of $f$ known on all boundaries of $[0,1]^2$. The middle plot shows $I^{[0]} = \{1,2\}, I^{[1]} = \emptyset$, with $f$ known only on the left boundaries of each variable. The right plot shows $I^{[0]} = \{1,2\}$, $I^{[1]} = \{1\}$, with $f$ known on both boundaries of $x_1$ and on the left boundary of $x_2$. To integrate such boundary information, we need to specify two ingredients for BdryGP. First, a mean function $\mu(\cdot)$ is needed which satisfies known boundary conditions on $f$. Second, a covariance function $k(\cdot,\cdot)$ is needed which satisfies $k({\bold x},{\bold x})=0$ for any ${\bold x}\in \mathcal{S}_j^{[0]}$, $j \in I^{[0]}$ and any ${\bold x}\in \mathcal{S}_j^{[1]}$, $j \in I^{[1]}$. This ensures the BdryGP model satisfies the desired boundary information on $f$ almost surely. Mean function specification {#sec:mean_func} --------------------------- Consider first the specification of the BdryGP mean function $\mu(\cdot)$. We adopt a simple strategy for constructing $\mu(\cdot)$ via an interpolator on known boundary conditions. For a point ${\bold x}\in \mathcal{X}$, let $\mathcal{P}_j^{[0]}{\bold x}$ and $\mathcal{P}_j^{[1]}{\bold x}$ denote the projection of ${\bold x}$ onto the subspaces $\mathcal{S}_j^{[0]}$ and $\mathcal{S}_j^{[1]}$, respectively. These projected points can be written explicitly as: $$\begin{aligned} \begin{split} &[\mathcal{P}_j^{[0]}{\bold x}]_k=\begin{cases} x_k, &\text{if} \ j\neq k\\ 0,&\text{if} \ j = k \end{cases}\ \ \ \ \text{for}\ j \in I^{[0]},\\ &[\mathcal{P}_j^{[1]}{\bold x}]_k=\begin{cases} x_k, &\text{if} \ j \neq k\\ 1,&\text{if} \ j = k \end{cases}\ \ \ \ \text{for}\ j \in I^{[1]}. \end{split}\end{aligned}$$ Furthermore, let $\mathbf{P}({\bold x})=\{ \mathcal{P}_j^{[0]}{\bold x}: j \in I^{[0]} \} \cup \{ \mathcal{P}_j^{[1]}{\bold x}: j \in I^{[1]} \}$ be the set of all such projected points of ${\bold x}$ on known boundaries. With this, the mean function $\mu(\cdot)$ can then be constructed as: $$\mu({\bold x})=\phi({\bold x},\mathbf{P}({\bold x})) [\phi(\mathbf{P}({\bold x}),\mathbf{P}({\bold x}))]^{-1}f(\mathbf{P}({\bold x})). \label{eq:ourmu}$$ where $\phi(\cdot,\cdot)$ is a compactly supported, positive definite, radial basis kernel [@Wendland10]. Here, $\mu({\bold x})$ can be interpreted as a GP interpolant at ${\bold x}$, using boundary information at projected points $\mathbf{P}({\bold x})$ as data. By the interpolation property of GPs, the proposed mean function must therefore satisfy the desired boundary conditions: $$\left\{\mu({\bold x}): {\bold x}\in \mathcal{S}_j^{[0]}\right\} = \mathcal{F}_j^{[0]}, \; \forall j \in I^{[0]}, \quad \left\{\mu({\bold x}): {\bold x}\in \mathcal{S}_j^{[1]}\right\} = \mathcal{F}_j^{[1]}, \; \forall j \in I^{[1]}. \label{eq:meanbound}$$ We find the radial basis kernel $\phi({\bold x}_1,{\bold x}_2)=\max\{(1-||{\bold x}_1-{\bold x}_2||)^\nu,0\}$ [@Wendland10] to work well in practice. Figure \[fig:mean\_func\_boundary\] illustrates the proposed mean function $\mu(\cdot)$ in using the earlier two-dimensional example (with known boundaries marked in blue). From the left plot, which shows the mean function $\mu(\cdot)$ for the full boundary case of $I^{[0]} = I^{[1]} = \{1,2\}$, we see that $\mu(\cdot)$ satisfies the desired boundary conditions from Figure \[fig:boundary\]. The same is true for the middle and right plots, which shows the proposed $\mu(\cdot)$ given partial boundary information. Covariance function specification --------------------------------- Consider next the specification of the covariance function $k(\cdot,\cdot)$ for BdryGP. We present below a new BdryMatérn covariance function which incorporates boundary information of the form . We first discuss the properties of the BdryMatérn kernel for modeling, then provide an explicit derivation of this kernel. ### The BdryMatérn kernel For variable $x_j$, the one-dimensional (1-d) BdryMatérn kernel is defined as: $$\small k_{\omega_j}^{\rm BM}(x,y)=\begin{cases} \dfrac{\sinh{[\omega_j (x\wedge y)]}\sinh{[\omega_j(1- x\vee y)]}} {\sinh(\omega_j)},& \ j \in I^{[0]} \cap I^{[1]} \quad \text{(full)}\\ \sinh{[\omega_j (x\wedge y)]}\exp{[\omega_j(- x\vee y)]},& \ j \in I^{[0]} \cap \overline{I^{[1]}} \quad \text{(left)}\\ \exp[\omega_j ( x\wedge y)]\sinh[\omega_j(1- x\vee y)],& \ j \in \overline{I^{[0]}} \cap I^{[1]} \quad \text{(right)}\\ \exp(-\omega_j{|x-y|}) ,&\ j \in \overline{I^{[0]}} \cap \overline{I^{[1]}} \quad \text{(none)}.\\ \end{cases} \label{eq:BdryMatern1d}$$ Here, $x \wedge y = \min(x,y)$ and $x\vee y = \max(x,y)$, and $\sinh(\cdot)$ and $\cosh(\cdot)$ are the hyperbolic sine and cosine functions. The first case corresponds to known boundaries for both the left and right endpoints of $x_j$ (i.e., *full* boundary information). The second and third cases correspond to known boundaries for only the left and only the right endpoints of $x_j$, respectively (i.e., *partial* boundary information). The last case is for *no* boundary information on $x_j$; this reduces to the Matérn-1/2 correlation function. Using , we adopt the following product form for the BdryMatérn covariance over all $d$ variables: $$k^{\rm BM}_\omega({\bold x},{\bold y})= \sigma^2 \prod_{j=1}^d k^{\rm BM}_{\omega_j}(x_j,y_j), \label{eq:BdryMatern}$$ where $\sigma^2$ is a variance parameter. This product form of the BdryMatérn kernel $k_\omega^{\rm BM}({\bold x},{\bold y})$ yields a very useful native space, which can be connected to FEM for proving improved GP convergence rates. To see why the BdryMatérn kernel can incorporate boundary information, consider a simple 1-d setting with $\sigma^2 = 1$ and $\omega=1$. Figure \[fig:variance\_plot\] visualizes the process variance $k_\omega^{\rm BM}(x,x)$ as a function of ${\bold x}$ over $[0,1]$, for the first three cases in . The left plot shows $k_\omega^{\rm BM}(x,x)$ for the full boundary case, where *both* left and right boundaries are known. Here, the process variance equals zero at the endpoints $x=0$ and $x=1$, meaning the BdryGP constrains sample paths to satisfy the left and right boundaries almost surely. The middle plot show $k^{\rm BM}_\omega(x,x)$ when *only* the left boundary is known. Here, the process variance equals zero only when $x=0$, meaning all BdryGP sample paths satisfy the left boundary almost surely. A similar interpretation holds for the right plot, where *only* the right boundary known. The wavelength parameter $w_j$ in the BdryMatérn kernel plays a similar role as the scale parameter in the Matérn kernel: it controls the smoothness of sample paths from the BdryGP. To visualize this, Figure \[fig:wavelength\_plot\] plots the process covariance $k_\omega^{\rm BM}(0.5,x)$ between a point $x \in [0,1]$ and a fixed point at 0.5, for difference choices of wavelength $\omega$. The left plot shows, as $\omega \rightarrow \infty$, this covariance converges to zero everywhere *except* at $x=0.5$, which suggests that for larger wavelengths $\omega$, the sample paths from BdryGP become more rugged. The right plot shows, as $\omega \rightarrow 0^+$, this covariance converges to zero everywhere, *including* at $x=0.5$. This suggests that the process variance $k_\omega^{\rm BM}(x,x)$ becomes smaller as $\omega \rightarrow 0^+$, which results in smoother sample paths. The 1-d BdryMatérn kernel also has an inherent connection to the covariance functions for the Brownian bridge and the Brownian motion. Suppose either the left or right boundary is known for variable $x_j$. Taking wavelength $\omega_j \rightarrow 0^+$ for the normalized BdryMatérn kernel, we get: $$\label{eq:brownian_kernel1d} k_j^{\rm BR}(x,y)=\lim_{\omega_j \to 0^+}\frac{k_{\omega_j}(x,y)}{\omega_j}=\begin{cases} (x\wedge y)(1-x\vee y),& \; j \in I^{[0]} \cap I^{[1]} \quad \text{(full)}\\ x\wedge y,& \; j \in I^{[0]} \cap \overline{I^{[1]}} \quad \text{(left)}\\ 1-x\vee y,& \; j \in \overline{I^{[0]}} \cap I^{[1]} \quad \text{(right)}. \end{cases}$$ The first case is the covariance function of a Brownian bridge, and the second and third cases are variants of the covariance function for a Brownian motion. We will call $k_j^{\rm BR}(x,y)$ the 1-d *Brownian kernel*, and its product form: $$k^{\rm BR}({\bold x},{\bold y}) = \prod_{j=1}^d k_j^{\rm BR}(x_j,y_j) \label{eq:brownian_kernel}$$ the *Brownian kernel*. The link between the BdryMatérn kernel (used in BdryGP) and the Brownian kernel will serve as the basis for proving improved convergence rates via finite-element modeling. We note that the Brownian kernel is *not* used for modeling purposes, but rather as a theoretical tool for bridging the BdryGP model with FEM. ![Visualizing the BdryGP variance $k^{\rm BM}_\omega(x,x)$ over $x \in [0,1]$ for full boundary (left), left boundary (middle), and right boundary (right) information.[]{data-label="fig:variance_plot"}](figures/wavelength/BM_full.pdf "fig:"){width="30.00000%"} ![Visualizing the BdryGP variance $k^{\rm BM}_\omega(x,x)$ over $x \in [0,1]$ for full boundary (left), left boundary (middle), and right boundary (right) information.[]{data-label="fig:variance_plot"}](figures/wavelength/BM_left.pdf "fig:"){width="30.00000%"} ![Visualizing the BdryGP variance $k^{\rm BM}_\omega(x,x)$ over $x \in [0,1]$ for full boundary (left), left boundary (middle), and right boundary (right) information.[]{data-label="fig:variance_plot"}](figures/wavelength/BM_right.pdf "fig:"){width="30.00000%"} ![Visualizing the BdryGP covariance $k_\omega^{\rm BM}(0.5,x)$ for full boundary information, with wavelength $\omega \rightarrow \infty$ (left) and $\omega \rightarrow 0^+$ (right).[]{data-label="fig:wavelength_plot"}](figures/wavelength/wavelength_large.pdf "fig:"){width="40.00000%"} ![Visualizing the BdryGP covariance $k_\omega^{\rm BM}(0.5,x)$ for full boundary information, with wavelength $\omega \rightarrow \infty$ (left) and $\omega \rightarrow 0^+$ (right).[]{data-label="fig:wavelength_plot"}](figures/wavelength/wavelength_small.pdf "fig:"){width="40.00000%"} ### Derivation under boundary conditions We now provide a derivation of the 1-d BdryMatérn kernel . Consider the 1-d Matérn kernel: $$k_{\nu,\omega}(x,y) = \frac{2^{1-\nu}}{\Gamma(\nu)}(\sqrt{2\nu}\omega|x-y|)^\nu K_\nu(\sqrt{2\nu}\omega|x-y|), \label{eq:matern1d}$$ where $\nu$ is the smoothness parameter, $\omega$ is the scale parameter, and $K_\nu$ is the modified Bessel function [@abramowitz1965handbook]. Let $S_{\nu,\omega}(s)$ denote the spectral density of $k_{\nu,\omega}$ [@cressie1991]. With $\simeq$ denoting equality up to an independent constant, the inner product of the RKHS ${\mathcal H}_{k_{\nu,\omega}}$ can be written as: $$\langle f,g \rangle_{k_{\nu,\omega}}\simeq\int_{\mathbb R}\frac{\hat{f}(s)\hat{g}(s)}{S_{\nu,\omega}(s)}ds, \quad f, g \in {\mathcal H}_{k_{\nu,\omega}}, \label{eq:materninprod}$$ where $\hat{f}$ is the Fourier transform of $f$. Let $m=\nu+\frac{1}{2}$, and suppose $m\in {\mathbb N}$. Equation then simplifies to: $$ \langle f,g \rangle_{k_{\nu,\omega}} \simeq \langle f,\mathcal{L}_{\nu,\omega}g \rangle_{L^2}, \label{eq:materninprod2}$$ where $D^l= d/dx^l$, $\mathcal{L}_{\nu,\omega}$ is the self-adjoint differential operator: $$\mathcal{L}_{\nu,\omega}:=\sum_{l=0}^m(-1)^l{m \choose l}C^{m-l}_{\nu,\omega}{C'}^{l}_{\nu,\omega}D^{2l}, \; C_{\nu,\omega} = 2\nu\omega^2C_{\nu,\omega}'', \; C'_{\nu,\omega} = 4\pi^2 C_{\nu,\omega}'', \label{eq:diffop}$$ and $C_{\nu,\omega}'' =\{{2\pi^{{1}/{2}}\Gamma(\nu+{1}/{2})(2\nu)^{\nu}\omega^{2\nu}}/{\Gamma(\nu)}\}^{-1/(\nu+{1}/{2})}$. With this, the reproducing property of the RKHS gives: $$f(x)=\langle k_{\nu,\omega}(x,\cdot),f\rangle_{k_{\nu,\omega}}=\int_{\mathbb R}k_{\nu,\omega}(x,y)\mathcal{L}_{\nu,\omega}f(y) dy, \quad \forall f \in {\mathcal H}_{k_{\nu,\omega}}. \label{eq:reprod}$$ This suggests that the Matérn kernel $k_{\nu,\omega}(x,y)$ is the Green’s function of the differential operator $\mathcal{L}_{\nu,\omega}$, and can therefore be *uniquely* obtained by solving the following differential equation for $k$ [@ODEHandbook]: $$\mathcal{L}_{\nu,\omega}k(x,y)=\delta(x-y), \label{eq:diff}$$ where $\delta(x-y)$ is the Dirac delta function. This link serves as the basis for deriving the 1-d BdryMatérn kernel. Consider next the case of full boundary information on $f$. This information can be incorporated into the Matérn RKHS ${\mathcal H}_{k_{\nu,\omega}}$ by restricting all functions $f \in {\mathcal H}_{k_{\nu,\omega}}$ to satisfy $f(0) = f(1) = 0$. The corresponding kernel $k$ for this constrained function space must satisfy the reproducing property: $$f(x) = \int_{\mathbb R}k(x,y)\mathcal{L}_{\nu,\omega}f(y) dy, \quad \forall f \in {\mathcal H}_{k_{\nu,\omega}}, \quad f(0) = f(1) = 0, \label{eq:reprod2}$$ or equivalently, the following constrained differential equation: $$\mathcal{L}_{\nu,\omega}k(x,y)=\delta(x-y), \quad k(0,y) = k(1,y) = 0. \label{eq:diff2}$$ For the cases of only left and only right boundary information, a similar reasoning gives the differential equations: $$\begin{aligned} \mathcal{L}_{\nu,\omega}k(x,y)=\delta(x-y), &\quad k(0,y) = 0, \quad \text{and}\label{eq:diff3}\\ \mathcal{L}_{\nu,\omega}k(x,y)=\delta(x-y), &\quad k(1,y) = 0, \label{eq:diff4}\end{aligned}$$ respectively. The following proposition shows that the 1-d BdryMatérn kernel cases in satisfy the above differential equations with $\nu = 1/2$, with their corresponding RKHS related to the weighted first-order Sobolev space: $$\label{eq:sobolev1} \mathcal{H}^1_{\omega} = \left\{f : \omega \|f\|_{L^2}^2 + \frac{1}{\omega} \|Df\|_{L^2}^2 < \infty\right\}.$$ Suppose $\nu = 1/2$. The unique kernel $k$ solving , and are the first three cases of the 1-d BdryMatérn kernel , with corresponding RKHS equal to $\mathcal{H}^1_{\omega}$ with the additional constraint of $\{f(0)=0,f(1)=0\}$, $\{f(0)=0\}$ and $\{f(1)=0\}$, respectively. \[prop:BdryMatern1d\] This can be proven by simply showing each of the first three cases of the 1-d BdryMatérn kernel satisfies its corresponding differential equation , or . Since a solution to these differential equations is unique, the uniqueness of the kernel then follows. The RKHS claim follows from the equivalence between the differential equations and its corresponding reproducing property. This shows that the proposed BdryMatérn kernel indeed inherits the same smoothness properties as the Matérn-1/2 kernel, while also satisfying the desired boundary conditions. Unfortunately, the same kernel derivation does not appear to extend for more general smoothness parameters $\nu > 1/2$, since more constraints are needed on kernel $k$ in order to solve the corresponding differential equation. For example, when $\nu = 3/2$, a unique solution to requires boundary conditions on both $k$ and its first derivative, which implies further boundary information on $f$ than the Dirchlet boundaries assumed in the paper. Interpolation: BdryGP and FEM {#sec:fem} ============================= With the BdryGP in hand, we now reveal a useful connection between FEM and the BdryGP predictor. This connection allows us to extend results from FEM to prove improved convergence rates for BdryGP. Finite-Element Modeling ----------------------- We begin with a brief review of FEM. Consider first the following partial differential equation (PDE) system: $$\begin{cases} \mathcal{L} f({\bold x}) = g({\bold x}), \quad &{\bold x}\in \mathcal{X},\\ f({\bold x}) = 0, \quad &{\bold x}\in \partial \mathcal{X}. \end{cases} \label{eq:pde}$$ Here, $f$ is a solution on a Hilbert space $\mathcal{V}$, $\partial \mathcal{X}$ is the boundary of $\mathcal{X}$, and $\mathcal{L}$ is a differential operator on $\mathcal{V}$. Under regularity conditions, the Lax-Milgram Theorem [@Evans15] ensures the existence of a unique weak solution satisfying . The idea behind FEM is to approximate on a discretization of $\mathcal{X}$. This requires two ingredients: a discretization mesh on $\mathcal{X}$, and a finite-dimensional function space constructed from this mesh. Given a multi-index ${\boldsymbol{\alpha}}= (\alpha_1, \cdots, \alpha_d) \in \mathbb{N}^d$, let $\mathcal{X}$ be discretized on the full grid mesh: $${\bold X}_{{\boldsymbol{\alpha}}} = \left\{{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}=[\beta_j2^{-\alpha_j}]_{j=1}^d : \beta_j \in \mathcal{B}_{{\boldsymbol{\alpha}}} \right\}, \label{eq:fullgrid}$$ where $\mathcal{B}_{{\boldsymbol{\alpha}}}$ is the index set: $$\mathcal{B}_{{\boldsymbol{\alpha}}} = \left\{ \mathbf{1}_{\{j \in I^{[0]}\}}, \cdots, 2^{\alpha_j} - \mathbf{1}_{\{j \in I^{[1]}\}} \right\}. \label{eq:indexset}$$ The mesh size of ${\bold X}_{{\boldsymbol{\alpha}}}$ then becomes $h_{{\boldsymbol{\alpha}}}=(h_{\alpha_1},\cdots,h_{\alpha_d})=(2^{-\alpha_1},\cdots,2^{-\alpha_d})$. Next, given the mesh ${\bold X}_{{\boldsymbol{\alpha}}}$, let $\mathcal{V}_{{\boldsymbol{\alpha}}}$ be the finite-dimensional function space spanned by first-order polynomials within each hypercube formed by ${\bold X}_{{\boldsymbol{\alpha}}}$ (we discuss $\mathcal{V}_{{\boldsymbol{\alpha}}}$ in greater detail in Section \[sec:funcspace\]). The FEM solution is then defined as the projection of the weak solution $f$ on the finite-dimensional space $\mathcal{V}_{{\boldsymbol{\alpha}}}$. Using a connection to Lagrange polynomial interpolation (see Chapter 15.2 of [@Wendland10]), this FEM solution can be equivalently represented as: $$\label{eq:LagrangePoly} \mathcal{I}_{{\boldsymbol{\alpha}}} f = \sum_{{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\in{\bold X}_{{\boldsymbol{\alpha}}}} f({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})\phi_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}({\bold x}),$$ where: $$\phi_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}({\bold x})=\prod_{j=1}^d\max\left\{1-\frac{|x-x_{\alpha_j,\beta_j}|}{2^{-\alpha_j}},0\right\}$$ are piecewise-linear basis functions over each cube. FEM and the Brownian kernel --------------------------- We now reveal a novel connection between the FEM interpolator and the GP predictor under the Brownian kernel $k^{\rm BR}$, first for full grid designs then for sparse grid designs. ### Full Grids We first make this connection for full grid designs: \[thm:LagrangeKrigingEquivalent\] Suppose $I^{[0]} \cup I^{[1]} = [d]$, and assume the full grid design ${\bold X}_{{\boldsymbol{\alpha}}}$ with $n=|{\bold X}_{{\boldsymbol{\alpha}}}|$ points. For any $f\in {\mathcal H}^{1,c}_{mix}$, the posterior predictor $\hat{f}_n^{\rm BR}$ of a GP with mean function $\mu(\cdot)$ in and Brownian kernel $k^{\rm BR}$ is equivalent to the FEM solution $\mathcal{I}_{{\boldsymbol{\alpha}}} f$, i.e.: $$\label{eq:LagrangeKrigingEquivalent} \hat{f}_n^{\rm BR}(\cdot) = \mu(\cdot)+k^{\rm BR}(\cdot,\bold{X_{{\boldsymbol{\alpha}}}}) [k^{\rm BR}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})]^{-1} \left[f(\bold{X}_{{\boldsymbol{\alpha}}})-\mu(\bold{X}_{{\boldsymbol{\alpha}}})\right]=\mathcal{I}_{{\boldsymbol{\alpha}}} f(\cdot).$$ In other words, assuming $I^{[0]} \cup I^{[1]} = [d]$ (i.e., there exists left or right boundary information for each of the $d$ variables), the predictor $\hat{f}_n^{\rm BR}$ for a GP with Brownian kernel $k^{\rm BR}$ is equivalent to the FEM solution $\mathcal{I}_{{\boldsymbol{\alpha}}} f$, under the full grid design (or mesh) ${\bold X}_{{\boldsymbol{\alpha}}}$. The key idea in proving Theorem \[thm:LagrangeKrigingEquivalent\] is to show that, under the Brownian kernel $k^{\rm BR}$, the matrix inverse $[k^{\rm BR}(\bold{X},\bold{X})]^{-1}$ has an explicit closed-form expression. Under this expression, the desired equivalence can be shown via an inductive argument on dimension $d$. This result can be viewed as an extension of Proposition 2 in [@DingZhang18]. Without loss of generality (WLOG), we assume the setting of only known left boundaries, i.e., $I^{[0]} = [d]$, $I^{[1]} = \emptyset$, since the setting of $I^{[0]} \cup I^{[1]} = [d]$ follows immediately. Furthermore, since the mean function $\mu(\cdot)$ in satisfies the desired boundary conditions, we can simply show that the claim holds for a zero-mean GP with Brownian kernel $k^{\rm BR}$, under a boundary condition of zero. Let $k({\bold x},{\bold y}):=k^{\text{BR}}({\bold x},{\bold y})$ and $k_j(x_j,y_j):=k^{\text{BR}}_j(x_j,y_j)$, $j = 1, \cdots, p$. We first prove the theorem for the base cases of $d=1$ and $d=2$, then show the claim holds for $d>2$ via induction. Consider first the base case of $d=1$. Under the assumption of known left boundaries, $\bold{X}_{{\boldsymbol{\alpha}}}=\{x_i=i 2^{-{\boldsymbol{\alpha}}}:i=1,\cdots , 2^{{\boldsymbol{\alpha}}}\}$ with $x_{0}=0$ and $n:=|{\bold X}_{{\boldsymbol{\alpha}}}|$. By Theorem 2 of [@DingZhang18], $k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})$ is a symmetric tridiagonal matrix with entries $$\begin{aligned} [k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})]_{i,i}&= \begin{cases} \frac{x_{i+1}-x_{i-1}}{(x_{i}-x_{i-1})(x_{i+1}-x_{i})}, & \text{if} \ i< n\\ \frac{1}{x_n-x_{n-1}}, & \text{if} \ i=n \end{cases},\\ [k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})]_{i-1,i}&=[k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})]_{i,i-1}=-\frac{1}{x_{i}-x_{i-1}}\end{aligned}$$ for $i=1,2,\cdots,n$. Given any point $x \in [0,1]$, assume $x_{i}<x<x_{i+1}$. By straightforward calculations, we have $$\hat{f}^{\text{BR}}_n(x)=\frac{x_{i+1}-x}{x_{i+1}-x_{i}}f(x_{i})+\frac{x-x_{i}}{x_{i+1}-x_{i}}f(x_{i+1})=\sum_{x_{{{\boldsymbol{\alpha}}},\beta}\in {\bold X}_{{\boldsymbol{\alpha}}}}f(x_{{{\boldsymbol{\alpha}}},\beta})\phi_{{{\boldsymbol{\alpha}}},\beta}(x).$$ This proves the base case of $d=1$. For clarity in the later inductive step, we will also show the case of $d=2$. Here, $\bold{X}_{{\boldsymbol{\alpha}}}=\bigtimes_{j=1}^2\bold{X}_{\alpha_j} = \bigtimes_{j=1}^2 \{x_{\alpha_j,1}, x_{\alpha_j,2}, \cdots\}$, and $k(\bold{x},\bold{y})=k_1(x_1,y_1)k_2(x_2,y_2)$. Let ${\bold x}$ be a point in $\mathcal{X}$, and let $K_{{\bold x}}$ be the hypercube in ${\bold X}_{{\boldsymbol{\alpha}}}$ containing ${\bold x}$ with vertices $\{{\bold x}_{{{\boldsymbol{\alpha}}},(i_1,i_2)}:i_j=\beta_j,\beta_j+1,j=1,2\}$. The vector $k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})k(\bold{X}_{{\boldsymbol{\alpha}}},\bold{x})$ can be decomposed as: $$\begin{aligned} &\ \ \ \ k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})k(\bold{X}_{{\boldsymbol{\alpha}}},\bold{x})\\ &=\big\{k_1^{-1}(\bold{X}_{\alpha_1},\bold{X}_{\alpha_1})\bigotimes k_2^{-1}(\bold{X}_{\alpha_2},\bold{X}_{\alpha_2})\big\}\text{vec}\big([k_1(x_{1},x_{\alpha_1,\beta_1})k_2(x_{2},x_{\alpha_2,\beta_2})]_{\beta_1,\beta_2}\big)\\ &=\text{vec}\big(k_2^{-1}(\bold{X}_{\alpha_2},\bold{X}_{\alpha_2})[k_1(x_{1},x_{\alpha_1,\beta_1})k_2(x_{2},x_{\alpha_2,\beta_2})]_{\beta_1,\beta_2}k_1^{-1}(\bold{X}_{\alpha_1},\bold{X}_{\alpha_1})\big)\end{aligned}$$ where vec$\big(\bold{M}\big)$ denotes the vectorization of the matrix $\bold{M}$. Define $$\bold{\Phi}^{[2]}({\bold x}):=k_2^{-1}(\bold{X}_{\alpha_2},\bold{X}_{\alpha_2})[k_1(x_{1},x_{\alpha_1,\beta_1})k_2(x_{2},x_{\alpha_2,\beta_2})]_{\beta_1,\beta_2}k_1^{-1}(\bold{X}_{\alpha_1},\bold{X}_{\alpha_1}).$$ By straightforward calculations similar to the 1-d case, it follows that $\bold{\Phi}^{[2]}({\bold x})$ has only the four non-zero entries: $$\begin{aligned} &\bold{\Phi}^{[2]}_{\beta_1,\beta_2}({\bold x})=\phi_{\alpha_1,\beta_1}(x_1)\phi_{\alpha_2,\beta_2}(x_2),\ \ \bold{\Phi}^{[2]}_{\beta_1+1,\beta_2}({\bold x})=\phi_{\alpha_1,\beta_1+1}(x_1)\phi_{\alpha_2,\beta_2}(x_2),\\ &\bold{\Phi}^{[2]}_{\beta_1,\beta_2+1}({\bold x})=\phi_{\alpha_1,\beta_1}(x_1)\phi_{\alpha_2,\beta_2+1}(x_2), \ \ \bold{\Phi}^{[2]}_{\beta_1+1,\beta_2+1}({\bold x})=\phi_{\alpha_1,\beta_1+1}(x_1)\phi_{\alpha_2,\beta_2+1}(x_2),\end{aligned}$$ where $\phi_{{{\boldsymbol{\alpha}}},\beta}$ is the hat function defined previously. Thus, the predictor $\hat{f}_n^{\rm BR}$ can be rewritten as: $$\begin{aligned} \hat{f}^{\text{BR}}_n({\bold x})&=\text{vec}\big[\Phi^{[2]}({\bold x})\big]^\intercal f({\bold X}_{{\boldsymbol{\alpha}}})\\ &=\sum_{i_1=\beta_1}^{\beta_1+1}\sum_{i_2=\beta_2}^{\beta_2+1}\bold{\Phi}^{[2]}_{i_1,i_2}(x)f(x_{\alpha_1,i_1},x_{\alpha_2, i_2})\\ &=\sum_{{\bold x}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}\in{\bold X}_{{\boldsymbol{\alpha}}}}f({\bold x}_{{{\boldsymbol{\alpha}}},\beta})\phi_{{{\boldsymbol{\alpha}}},\beta}({\bold x}),\end{aligned}$$ which proves the theorem for $d=2$. Consider next the inductive step on $d$. Here, the full grid becomes $\bold{X}_{{\boldsymbol{\alpha}}}=\bigtimes_{j=1}^d\bold{X}_{\alpha_j}= \bigtimes_{j=1}^d \{x_{\alpha_j,1},x_{\alpha_j,2},\ldots\}$. Let ${\bold x}\in\mathcal{X}$ and let $K_i$ be the hyper-cube in ${\bold X}_{{\boldsymbol{\alpha}}}$ containing ${\bold x}$ with vertices $\{{\bold x}_{{{\boldsymbol{\alpha}}},(i_1,\cdots,i_{d})}:i_j=\beta_j,\beta_j+1,j=1,\cdots,d\}$ as before. Suppose the inductive hypothesis: $$\begin{aligned} \hat{f}^{\text{BR}}_n({\bold x})&=k({\bold x},{\bold X}_{{\boldsymbol{\alpha}}})\left[k({\bold X}_{{\boldsymbol{\alpha}}},{\bold X}_{{\boldsymbol{\alpha}}})\right]^{-1}f({\bold X}_{{\boldsymbol{\alpha}}})\\ &=\text{vec}\big[\Phi^{[d]}({\bold x})\big]^\intercal f({\bold X}_{{\boldsymbol{\alpha}}})\\ &=\sum_{i_1=\beta_1}^{\beta_1+1}\sum_{i_2=\beta_2}^{\beta_2+1}\cdots\sum_{i_d=\beta_d}^{\beta_d+1}\bold{\Phi}^{[d]}_{i_1,\cdots,i_d}({\bold x})f(x_{\alpha_1,i_1},x_{\alpha_2,i_2},\cdots,x_{\alpha_d,i_d}),\end{aligned}$$ where: $$\bold{\Phi}^{[d]}_{i_1,\cdots,i_d}({\bold x})=\prod_{j=1}^d\phi_{\alpha_j,i_j}(x_j).$$ From this hypothesis, we can see that there are at most $2^{d}$ non-zeros entries on $\bold{\Phi}^{[d]}({\bold x})$, namely, the entries $\bold{\Phi}^{[d]}_{i_1,\cdots,i_{d}}({\bold x})$ with $i_j=\beta_j$ or $\beta_j+1$. Since $\hat{f}^{\text{BR}}_n$ is the Lagrange polynomial interpolation of $f$ and is continuous, this assumption is equivalent to $\hat{f}^{\text{BR}}_n=\mathcal{I}_{{\boldsymbol{\alpha}}} f$. Under this inductive hypothesis, consider the case for dimension $d+1$. Here, the full grid design becomes $\bold{X}_{{\boldsymbol{\alpha}}}={\bold X}_{{{\boldsymbol{\alpha}}}_{1:d}}\bigtimes{\bold X}_{\alpha_{d+1}}$. Now let ${\bold x}\in \mathcal{X}$ and let $K_i$ be a hyper-cube in ${\bold X}_{{\boldsymbol{\alpha}}}$ containing ${\bold x}$ with vertices $\{{\bold x}_{{{\boldsymbol{\alpha}}},(i_1,\cdots,i_{d})}:i_j=\beta_j,\beta_j+1,j=1,\cdots,d+1\}$. Let $k({\bold x},\bold{y})=k({\bold x}_{1:d},\bold{y}_{1:d})k_{d+1}(x_{d+1},y_{d+1})$. Then the vector $k^{-1}({\bold X}_{{\boldsymbol{\alpha}}},{\bold X}_{{\boldsymbol{\alpha}}})k({\bold X}_{{\boldsymbol{\alpha}}},{\bold x})$ becomes: $$\begin{aligned} &\ \ \ \ \ k^{-1}(\bold{X}_{{\boldsymbol{\alpha}}},\bold{X}_{{\boldsymbol{\alpha}}})k(\bold{X}_{{\boldsymbol{\alpha}}},\bold{x})\\ &=\big\{k^{-1}(\bold{X}_{{{\boldsymbol{\alpha}}}_{1:d}},\bold{X}_{{{\boldsymbol{\alpha}}}_{1:d}})\bigotimes k_{d+1}^{-1}(\bold{X}_{\alpha_{d+1}},\bold{X}_{\alpha_{d+1}})\big\}\\ & \quad \quad \quad \quad \text{vec}\big([k({\bold x}_{1:d},{\bold x}_{{{\boldsymbol{\alpha}}}_{1:d},{\boldsymbol{\beta}}_{1:d}})k_{d+1}(x_{d+1},x_{\alpha_{d+1},\beta_{d+1}})]_{{\boldsymbol{\beta}}_{1:d},\beta_{d+1}}\big)\\ &=\text{vec}\Big\{k_{d+1}^{-1}(\bold{X}_{\alpha_{d+1}},\bold{X}_{\alpha_{d+1}})\\ & \quad \quad \quad [k({\bold x}_{1:d},{\bold x}_{{{\boldsymbol{\alpha}}}_{1:d},{\boldsymbol{\beta}}_{1:d}})k_{d+1}(x_{d+1},x_{\alpha_{d+1},\beta_{d+1}})]_{{\boldsymbol{\beta}}_{1:d},\beta_{d+1}}k^{-1}(\bold{X}_{{{\boldsymbol{\alpha}}}_{1:d}},\bold{X}_{{{\boldsymbol{\alpha}}}_{1:d}})\Big\}.\end{aligned}$$ Similarly, define: $$\begin{aligned} \bold \Phi({\bold x})^{[d+1]}& = k_{d+1}^{-1}(\bold{X}_{\alpha_{d+1}},\bold{X}_{\alpha_{d+1}})\\ & \quad \quad [k({\bold x}_{1:d},{\bold x}_{{{\boldsymbol{\alpha}}}_{1:d},{\boldsymbol{\beta}}_{1:d}})k_{d+1}(x_{d+1},x_{\alpha_{d+1},\beta_{d+1}})]_{{\boldsymbol{\beta}}_{1:d},\beta_{d+1}}k^{-1}(\bold{X}_{{{\boldsymbol{\alpha}}}_{1:d}},\bold{X}_{{{\boldsymbol{\alpha}}}_{1:d}}).\end{aligned}$$ From the inductive hypothesis, we know that $$k^{-1}({\bold X}_{{{\boldsymbol{\alpha}}}_{1:d}},{\bold X}_{{{\boldsymbol{\alpha}}}_{1:d}})k({\bold X}_{{{\boldsymbol{\alpha}}}_{1:d}},{\bold x}_{1:d})=\text{vec}\left(\bold{\Phi}^{[d]}({\bold x}_{1:d})\right)$$ which is the vectorization of the sparse matrix $\bold{\Phi}^{[d]}({\bold x}_{1:d})$, which has at most $2^d$ non-zero entries. Hence, $\Phi^{[d+1]}({\bold x})$ can be decomposed as: $$\begin{aligned} & \bold{\Phi}^{[d+1]}({\bold x})=k_{d+1}^{-1}({\bold X}_{\alpha_{d+1}},{\bold X}_{\alpha_{d+1}})\bigg[\text{vec}\big(\bold{\Phi}^{[d]}({\bold x}_{1:d})\big)k_{d+1}(x_{d+1},x_{\alpha_{d+1},\beta_{d+1}}) \bigg]_{\beta_{d+1}}\end{aligned}$$ So there are at most $2^{d+1}$ non-zeros entries on $\bold{\Phi}^{[d+1]}({\bold x})$, namely, the entries $\bold{\Phi}_{i_1,\cdots,i_{d+1}}({\bold x})$ where $i_j=\beta_j$ or $\beta_j+1$. Incorporating this, we then have: $$\begin{aligned} \hat{f}^{\text{BR}}_n(x)&=\sum_{i_1=\beta_1}^{\beta_1+1}\sum_{i_2=\beta_2}^{\beta_2+1}\cdots\sum_{i_{d+1}=\beta_{d+1}}^{\beta_{d+1}+1}\bold{\Phi}_{i_1,\cdots,i_d}({\bold x})f(x_{\alpha_1,i_1},x_{\alpha_2,i_2},\cdots,x_{\alpha_{d+1},i_{d+1}})\\ &=\sum_{{\bold x}_{{{\boldsymbol{\alpha}}},\beta}\in{\bold X}}\phi_{{{\boldsymbol{\alpha}}},{\boldsymbol{\beta}}}({\bold x})f({\bold x}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}})=\mathcal{I}_{{\boldsymbol{\alpha}}} f({\bold x}),\end{aligned}$$ which completes the inductive step. ### Sparse Grids One disadvantage of full grid designs is the so-called *curse-of-dimensionality*: both the design size and its corresponding prediction error grow exponentially in dimension $d$. To this end, we extend next the earlier equivalence between FEM and the Brownian kernel for a broader class of designs called *sparse grids* [@Bungartz04], which “sparsify” a full grid by retaining only certain subgrids of interest. These designs are used later to prove the improved convergence rates for BdryGP. We first provide a brief review of sparse grid designs. A sparse grid of level $k$, denoted as ${\bold X}^{\rm SP}_k$, is defined as follows: $${\bold X}_k^{\rm SP}=\bigcup_{k\leq|{{\boldsymbol{\alpha}}}|\leq k+d-1}{\bold X}_{{\boldsymbol{\alpha}}}, \quad |{\boldsymbol{\alpha}}| := \sum_{j=1}^d \alpha_j. \label{eq:sparsegrid}$$ In words, the sparse grid ${\bold X}_k^{\rm SP}$ is the union of full grids ${\bold X}_{{\boldsymbol{\alpha}}}$ whose multi-indices ${\boldsymbol{\alpha}}$ sums between $k$ and $k+d-1$. Figure \[fig:sgg\] shows sparse grids of levels 1 to 4 in two dimensions; we see that sparse grids provide a sizable reduction in design size compared to full grids. This reduction plays a key role in providing relief from dimensionality in many numerical approximation problems [@Wendland10; @Dick13]. ![Sparse grid designs of levels 1 to 4 in two dimensions.[]{data-label="fig:sgg"}](figures/grid/order1.pdf "fig:"){width="24.00000%"} ![Sparse grid designs of levels 1 to 4 in two dimensions.[]{data-label="fig:sgg"}](figures/grid/order2.pdf "fig:"){width="24.00000%"} ![Sparse grid designs of levels 1 to 4 in two dimensions.[]{data-label="fig:sgg"}](figures/grid/order3.pdf "fig:"){width="24.00000%"} ![Sparse grid designs of levels 1 to 4 in two dimensions.[]{data-label="fig:sgg"}](figures/grid/order4.pdf "fig:"){width="24.00000%"} The FEM solution $\mathcal{I}_{{\boldsymbol{\alpha}}}f$ in , previously defined for the full grid ${\bold X}_{{\boldsymbol{\alpha}}}$, can be extended analogously for sparse grids. Similar to before, let $\mathcal{V}_k^{\rm SP}$ be the sum of the finite-dimensional function spaces for each of the component full grids in the sparse grid . The FEM solution on sparse grid ${\bold X}_k^{\rm SP}$, defined as the projection of the weak solution $f$ on $\mathcal{V}_k^{\rm SP}$, can be shown (Equation (28) of [@Garcke12]) to have the form: $$\label{eq:combinationtechnique} \begin{aligned} \mathcal{I}_k^{\rm SP} f &=\sum_{j=0}^{d-1}(-1)^j {d-1 \choose j}\sum_{|{{\boldsymbol{\alpha}}}|=k+d-1-j}\mathcal{I}_{{\boldsymbol{\alpha}}}f. \end{aligned}$$ With this in hand, we show that under sparse grid designs, the FEM solution $\mathcal{I}_k^{\rm SP} f$ is also equivalent to the GP posterior mean $\hat{f}_n^{\rm BR}$ with the Brownian kernel $k^{\rm BR}$: \[thm:sggFEMkriging\] Suppose $I^{[0]} \cup I^{[1]} = [d]$, and assume the sparse grid design ${\bold X}_k^{\rm SP}$ with $n=|{\bold X}_{{\boldsymbol{\alpha}}}|$ points. For any $f\in {\mathcal H}^{1,c}_{mix}$, the posterior predictor $\hat{f}_n^{\rm BR}$ of a GP with mean function and Brownian kernel $k^{\rm BR}$ is equivalent to the FEM solution $\mathcal{I}_k^{\rm SP} f$ in . In Theorem \[thm:LagrangeKrigingEquivalent\], we have shown the equivalence between the FEM solution $\mathcal{I}_{{\boldsymbol{\alpha}}}f$ and the GP predictor $\hat{f}_n^{\rm BR}$ on the full grid design ${\bold X}_{{\boldsymbol{\alpha}}}$. Hence, $\mathcal{I}_{{\boldsymbol{\alpha}}}f$ can be replaced by $\hat{f}_n^{\rm BR}$ in Equation (\[eq:combinationtechnique\]). The result then follows by Algorithm 1 of [@Plumlee14]. FEM and the BdryMatérn kernel ----------------------------- Having proved the connection between FEM and the Brownian kernel $k^{\rm BR}$, we then show how this relates to the BdryMatérn kernel $k^{\rm BM}_\omega$ used in BdryGP. Of course, the GP predictors $\hat{f}_n^{\rm BR}$ and $\hat{f}_n^{\rm BM}$ under the Brownian and BdryMatérn kernels are not equivalent. However, we show below that the BdryGP approximation error $|f - \hat{f}_n^{\rm BM}|$ can be upper bounded by the approximation error $|f - \hat{f}_n^{\rm BR}|$ from the Brownian kernel: \[thm:diffBdryGP\] Suppose $I^{[0]} \cup I^{[1]} = [d]$, and assume the sparse grid design ${\bold X}_k^{\rm SP}$ with $n=|{\bold X}_k^{\rm SP}|$ points. Let $\hat{f}_n^{\rm BM}$ be the BdryGP predictor with mean function and BdryMatérn kernel $k^{\rm BM}_{\omega}$. For any $f\in {\mathcal H}^{1,c}_{mix}$ and any $\omega > 0$: $$\begin{aligned} &||f-\hat{f}_n^{\rm BR}||_{L^\infty} \leq C||f-\hat{f}_n^{\rm BM}||_{L^{\infty}}. \end{aligned} \label{eq:diffBdryGP}$$ for some constant $C$ independent of $f$, $\omega$ and $n$. The proof of this theorem can be found in Appendix \[apdix:pf4diffBdryGP\]. Function spaces: BdryGP and FEM {#sec:funcspace} =============================== Next, we prove the equivalence between the RKHS of the Brownian kernel $k^{\rm BR}$, the constrained Sobolev space with mixed first derivatives, and the weak solution space for FEM. Brownian Kernel RKHS and the Constrained Mixed Sobolev Space ------------------------------------------------------------ We first establish the equivalence between the Brownian kernel RKHS and the mixed Sobolev space under boundary constraints. Let ${\mathcal H}^1_{mix}$ be the Sobolev space of functions with mixed first derivative: $${\mathcal H}^1_{mix} := \left\{f: D^{{\boldsymbol{\alpha}}} f\in L^2({\mathbb R}^d), |{\boldsymbol{\alpha}}|_\infty\leq 1\right\}. \label{eq:mixsob}$$ where $|{\boldsymbol{\alpha}}|_\infty=\max_{j\in[d]}|\alpha_j|$. Further let ${\mathcal H}^{1,c}_{mix}$ be the space of functions in ${\mathcal H}^1_{mix}$ with boundary value zero on known boundaries: $${\mathcal H}^{1,c}_{mix}:=\left\{f\in{\mathcal H}^1_{mix}:f({\bold x})=0 \text{ if } x_i \leq 0, i \in I^{[0]} \text{ or if } x_i \geq 1, i\in I^{[1]} \right\}. \label{eq:mixsobbound}$$ We will call ${\mathcal H}^{1,c}_{mix}$ the *constrained* mixed Sobolev space. The following proposition shows that the Brownian kernel RKHS $\mathcal{H}_{k^{\rm BR}}$ and the constrained Sobolev space ${\mathcal H}^{1,c}_{mix}$ are equivalent function spaces: \[prop:NativeSpaceBB\] The function spaces $\mathcal{H}_{k^{\rm BR}}$ and $\mathcal{H}^{1,c}_{mix}$ are equivalent. By a straight-forward extension of Proposition \[prop:BdryMatern1d\], the Brownian kernel $k^{\rm BR}$ satisfies the following equation for any $f\in \mathcal{H}^{1,c}_{mix}$: $$\label{eq:Brownian_RKHS} f({\bold x})=\int_{{\mathbb R}^d}D^{\bold{1}}f(\bold{s})D^{\bold{1}}k^{\text{BR}}({\bold x},\bold{s})d\bold{s}$$ From equation (\[eq:Brownian\_RKHS\]), the inner product of $\mathcal{H}_{k^{\rm BR}}$ is: $$\langle f,g \rangle_{\mathcal{H}_{k^{\rm BR}}}=\int_{{\mathbb R}^d}D^{{\boldsymbol{\alpha}}}fD^{{\boldsymbol{\alpha}}}gd{\bold x}, \quad f, g \in \mathcal{H}^{1,c}_{mix}.$$ Thus, we only need to show the norm equivalence identity: $$C_1||f||^2_{{\mathcal H}_{k^{\text{BR}}}}\leq ||f||^2_{\mathcal{H}^{1,c}_{mix}}\leq C_2||f||^2_{{\mathcal H}_{k^{\text{BR}}}}.$$ Obviously, $C_1=1$. By the 1-d Poincaré inequality for locally absolutely continuous functions, there exists some constant $C$ such that: $$\int_{{\mathbb R}^d}[D^{{\boldsymbol{\alpha}}} f]^2\leq C \int_{{\mathbb R}^d}[D^{\bold{1}}f]^2, \quad \text{for any } |{\boldsymbol{\alpha}}|_\infty\leq 1 \text{ and any } f\in \mathcal{H}^{1,c}_{mix}.$$ Iteratively applying the Poincaré inequality again, we get: $$||f||^2_{\mathcal{H}^{1,c}_{mix}}\leq 2^dC||f||^2_{{\mathcal H}_{k^{\text{BR}}}}$$ which proves the norm equivalence identity. Hierarchical Difference Spaces ------------------------------ Next, we introduce the idea of a hierarchical difference space, which is widely used in FEM analysis. These spaces will allow for a multi-level decomposition of the finite-dimensional function spaces for FEM, and thereby the FEM solution as well. Let us define the finite-dimensional function space $\mathcal{V}_{{{\boldsymbol{\alpha}}}}$ for the FEM solution on full grid ${\bold X}_{{\boldsymbol{\alpha}}}$: $$\label{eq:va} \mathcal{V}_{{{\boldsymbol{\alpha}}}}:=\text{span}\{\phi_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}: {\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\in {\bold X}_{{\boldsymbol{\alpha}}} \}$$ where $\phi_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}$ is the earlier hat function with $\phi_{0,1}(x)=x$ and $\phi_{0,2}(x)=1-x$. It is clear that $\mathcal{V}_{{\boldsymbol{\alpha}}}$ is the tensor product of these 1-d spaces, i.e., $\mathcal{V}_{{{\boldsymbol{\alpha}}}}=\bigotimes_{j=1}^d\mathcal{V}_{\alpha_j}$. Furthermore, $\mathcal{V}_{{\boldsymbol{\alpha}}}$ can be represented as the following multi-level subspace decomposition: $$\label{eq:subspacedecomp} \mathcal{V}_{{\boldsymbol{\alpha}}}=\bigoplus_{ \bm{0} \leq {\boldsymbol{\alpha}}' \leq {\boldsymbol{\alpha}}} W_{{\boldsymbol{\alpha}}'},$$ where $W_{{{\boldsymbol{\alpha}}}}=\text{span}\{\phi_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}:{{\boldsymbol{\beta}}}\in B_{{{\boldsymbol{\alpha}}}}\}$ is called a *hierarchical difference space*. Further details on these spaces can be found in [@Yse86] and [@Bungartz04]. We note that, in order to incorporate partial boundaries, the hierarchical difference space used here is slightly modified from that in the literature. In the case of full boundaries (i.e., $I^{[0]} = I^{[1]} = [d]$), the two spaces are equivalent. The subspace decomposition allows for the following useful multi-level decomposition of the FEM solution. Consider first the FEM solution $\mathcal{I}_{{\boldsymbol{\alpha}}}f$ on the full grid ${\bold X}_{{\boldsymbol{\alpha}}}$. From equation , $\mathcal{I}_{{\boldsymbol{\alpha}}}f$ can be decomposed as: $$\label{eq:hieraDiffSpa_proj} \mathcal{I}_{{\boldsymbol{\alpha}}}f = \sum_{ \bm{0} \leq {\boldsymbol{\alpha}}' \leq {\boldsymbol{\alpha}}}f_{{\boldsymbol{\alpha}}'}({\bold x}).$$ Here, $f_{{{\boldsymbol{\alpha}}}}$ is the projection of $f$ on $W_{{\boldsymbol{\alpha}}}$, given by: $$\label{eq:Proj_W_a} f_{{{\boldsymbol{\alpha}}}}({\bold x})=\sum_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}}c_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}\phi_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}({\bold x}),$$ and the constant $c_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}$ is known as the *hierarchical surplus*, defined as: $$\label{eq:HierarchicalSurplus} c_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}=\bigg(\prod_{j=1}^dA_{\alpha_j,\beta_j}\bigg)f(\bold{X}_{{\boldsymbol{\alpha}}}), \quad A_{\alpha_j,\beta_j}=\begin{cases} [-\frac{1}{2}\ \ 1 \ \ -\frac{1}{2}]& \text{if}\ \alpha_j\geq 1\\ [ -1\ \ 1] & \text{if}\ \alpha_j=0 \end{cases}.$$ Here, $\prod_{j=1}^dA_{\alpha_j,\beta_j}$ denotes the Kronecker product of vectors $A_{\alpha_j,\beta_j}$; this is the standard stencil notation used in numerical analysis. Similarly, the sparse grid FEM solution $\mathcal{I}_k^{\rm SP}$ can be decomposed as: $$\label{eq:hieraDiffSpa_proj2} \mathcal{I}_k^{\rm SP} f = \sum_{0 \leq |{\boldsymbol{\alpha}}| \leq k+d-1} f_{{\boldsymbol{\alpha}}}({\bold x}).$$ This decomposition, along with the equivalences in Section \[sec:fem\], provides the basis for proving improved convergence rates for BdryGP. Brownian Kernel RKHS and Hierarchical Difference Spaces ------------------------------------------------------- Consider now the limiting function space $\mathcal{V}$: $$\label{eq:hierarchicalInterpolation2Kriging} \mathcal{V}=\bigotimes_{j=1}^d\overline{\lim_{\alpha_j\to\infty}\mathcal{V}_{\alpha_j}}.$$ In other words, $\mathcal{V}$ is the tensor product of the limiting 1-d finite-dimensional spaces in equation . The space $\mathcal{V}$ can be viewed as the weak solution space on which FEM aims to solve the PDE system in the limit. The following proposition shows the equivalence of $\mathcal{V}$ to the native space of $k^{\text{BR}}$: \[thm:hierarchicalInterpolation2Kriging\] The function spaces $\mathcal{V}$ and $\mathcal{H}_{k^{\rm BR}}$ are equivalent. \[thm:hierdiff\] The proof of this proposition requires the following lemma, which shows that the finite-dimensional RKHS of $k^{\rm BR}$ on grid ${\bold X}_{{\boldsymbol{\alpha}}}$ is equivalent to $\mathcal{V}_{{\boldsymbol{\alpha}}}$. The finite-dimensional spaces $\mathcal{V}_{{\boldsymbol{\alpha}}}$ and $\{k^{\rm BR}({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}},\cdot):{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\in{\bold X}_{{\boldsymbol{\alpha}}}\}$ are equivalent for any ${{\boldsymbol{\alpha}}}\in\mathbb{N}^d$. \[lem:finitespace\] The proof of Lemma \[lem:finitespace\] is given in Appendix \[apdix:pf4hierdiff\]. From Lemma \[lem:finitespace\], we know that the projectors to $\mathcal{V}_{{\boldsymbol{\alpha}}}$ and $\{k^{\text{BR}}({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}},\cdot):{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\in{\bold X}_{{\boldsymbol{\alpha}}}\}$ are equal for any ${{\boldsymbol{\alpha}}}\in\mathbb{N}^d$. Since $\mathcal{H}_{k^{\rm BR}}$ is the completion of the space $\lim_{\{\alpha_j\to\infty\}_{j=1}^d}\{k^{\text{BR}}({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}},\cdot):{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\in{\bold X}_{{\boldsymbol{\alpha}}}\}$, it is therefore the function space defined in equation . Combining Propositions \[prop:NativeSpaceBB\] and \[thm:hierdiff\], we can then prove the desired equivalence between the two RKHSs ${\mathcal H}_{k^{\text{BR}}}$ and ${\mathcal H}_{k^{\text{BM}}_\omega}$, the constrained mixed Sobolev space ${\mathcal H}^{1,c}_{mix}$, and the weak solution space $\mathcal{V}$: ${\mathcal H}_{k^{\text{BR}}}$, ${\mathcal H}_{k^{\text{BM}}_\omega}$, ${\mathcal H}^{1,c}_{mix}$ and $\mathcal{V}$ are equivalent function spaces. \[thm:4spacesequal\] Proposition \[prop:NativeSpaceBB\] shows the equivalence between the RKHS ${\mathcal H}_{k^{\text{BR}}}$ and the constrained mixed Sobolev space ${\mathcal H}^{1,c}_{mix}$. Following the same reasoning (i.e., via the norm equivalence identity), the equivalence between the two RKHSs ${\mathcal H}_{k^{\text{BR}}}$ and ${\mathcal H}_{k^{\text{BM}}_\omega}$ can also be shown for any $\omega\in(0,\infty)$. The equivalence between $\mathcal{H}_{k^{\rm BR}}$ and $\mathcal{V}$ (Proposition \[thm:hierdiff\]) then completes the proof. This function space equivalence allows for the decomposition of the RKHS ${\mathcal H}_{k^{\text{BR}}}$ (and its corresponding interpolator) into hierarchical difference subspaces (and its corresponding projections) of different levels. This decomposition plays a key role in proving the following convergence rates. Convergence rates for BdryGP {#sec:convrates} ============================ With these equivalences in hand, we now prove the desired rates for BdryGP under sparse grids. All of these rates assume that $I^{[0]} \cup I^{[1]} = [d]$, i.e., at least one boundary is known for each of the $d$ variables. Of course, the same rates also hold in the full boundary setting of $I^{[0]} = I^{[1]} = [d]$, where *all* boundaries of $f$ are known. Let $n$ denote the number of observations ; let $\hat{f}$, $\hat{f}_\omega$ and $\hat{f}_0$ denote the BLUE of general BdryGP, BdryGP with BdryMatérn kernel and BdryGP with Brownian kernel conditioned on a sparse grid design respectively ; let $Y$ denote the BdryGP; let $f$ denote the true underlying function; let $\hat{Y}({\bold x})$ denote the expectation of $Y({\bold x})$ conditioned on spare grid. We prove four results herein. Firstly, the difference $||\hat{f}_\omega-\hat{f}_0||_\infty={\mathcal O}(n^{-2}[\log n]^{d-1})$ for any $\omega>0$; secondly the $L^p$ convergence rate of $\hat{f}$ to $f$ is ${\mathcal O}(n^{-1})$ given $f\in{\mathcal H}^1_{I^{[1]},mix}$; thirdly, $\sup_{{\bold x}\in T^d}\big|Y({\bold x})-\hat{Y}({\bold x})\big|={\mathcal O}_{p}(n^{-1}[\log n]^{2d-\frac{3}{2}})$; finally, the similar $L^p$ convergence rate without boundary condition is ${\mathcal O}\big(n^{-\frac{1}{2}}[\log n]^{\frac{5}{2}(d-1)}\big)$. Sparse grids ------------ We will use the following *sparse grids* [@DONGBIN10] as the design of choice ${\bold X}$ for proving convergence results. Given dimension $d$, a sparse grid of order $k = 1, \cdots, d$ (denoted as ${\bold X}_k^d$ ) is defined as: $${\bold X}_k^{\rm SP}=\bigcup_{k\leq|{\boldsymbol{\alpha}}|\leq k+d-1}{\bold X}_{{\boldsymbol{\alpha}}}, \label{eq:sparsegrid}$$ where $\bm{X}_{{\boldsymbol{\alpha}}}$ is the full grid design defined in . In words, the sparse grid ${\bold X}_k^d$ is the union of full grids ${\bold X}_{{\boldsymbol{\alpha}}}$ with $|{\boldsymbol{\alpha}}| = \sum_{j=1}^p \alpha_j$ between $k$ and $k+d-1$. Sparse grid designs play a crucial role in lifting the “curse-of-dimensionality” for polynomial approximation [@DONGBIN10], and we show that such designs also play a similar role in the convergence rates for BdryGP. In general, the design size $n$ of the sparse grid $|{\bold X}^{\rm SP}_k|$ cannot be written in closed form. The following lemma, given by [@Bungartz04], provides an upper bound on the design size $n=|{\bold X}^{\rm SP}_k|$: \[lem:num4SPG\] $$|{\bold X}^{\rm SP}_k|=\mathcal{O}(2^k[\log 2^k]^{d-1}).$$ This upper bound will prove useful in deriving the later convergence rates. BdryGP predictors with the BdryMat’ern and Brownian kernels ----------------------------------------------------------- We first directly state the theorem which can simplify the proofs of following theorems. Assume a sparse grid design . Let $\hat{f}_n^{\rm BM}$ and $\hat{f}_n^{\rm BR}$ be the posterior mean of the BdryGP with the BdryMatérn kernel and the Brownian kernel , respectively. For any wavelength $\omega$, we have: $$||\hat{f}_n^{\rm BM}-\hat{f}_n^{\rm BR}||_\infty ={\mathcal O}(n^{-2}[\log n]^{d-1}). \text{{{\color{red}{(Simon: should stick with either $|| \cdot ||_{\infty}$ or $||\cdot||_{L^{\infty}}$.)}}}} \label{eq:bmbr}$$ \[thm:diffBMandBR\] We can see that the difference between two equivalent kernels is in a order smaller than the convergence rates in later subsections. As a result, we only need to prove the convergence rates for Brownian kernel then the one with BdryMatérn kernel will have the same convergence rates. The proof of the above theorem is left in appendix. $L^p$ and $L^\infty$ Convergence Rates {#sec:lprate} -------------------------------------- Suppose $f$ is a *deterministic* function from the constrained mixed Sobolev space $\mathcal{H}^{1,c}_{mix}$. Under boundary information, the following theorem proves the $L^p$ and $L^\infty$ convergence rates for the proposed BdryGP (with BdryMatérn kernel $k^{\rm BM}_\omega$): \[thm:L1error\] Suppose $I^{[0]} \cup I^{[1]} = [d]$, and assume the sparse grid design ${\bold X}_k^{\rm SP}$ with $n = |{\bold X}_k^{\rm SP}|$ points. For any $f \in \mathcal{H}^{1,c}_{mix}$ and any wavelength $\omega$, the BdryGP has an $L^p$ convergence rate of: $$\label{eq:L1error} ||f-\hat{f}_n^{\rm BM}||_{L^p}={ \mathchoice {{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(n^{-1}), \quad 1 \leq p < \infty$$ and an $L^\infty$ convergence rate of: $$\label{eq:Linferror} ||f-\hat{f}_n^{\rm BM}||_{L^\infty}=\mathcal{O}(n^{-1}[\log n]^{2(d-1)}).$$ The proof of Theorem \[thm:L1error\] requires the following three lemmas. The first lemma (from [@Bungartz04]) provides a big-O approximation of the number of points in the sparse grid ${\bold X}_k^{\rm SP}$: \[lem:num4SPG\]\[Lemma 3.6 in [@Bungartz04]\] Let $n = |{\bold X}_k^{\rm SP}|$ be the number of points in a $d$-dimensional sparse grid of level $k$. Then: $$n=\mathcal{O}(2^k[\log 2^k]^{d-1}).$$ The second lemma upper bounds the hierarchical surplus in $c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ : \[prop:BoundofSurplus\] Let $f\in\mathcal{H}^{1,c}_{mix}$. Then there exists constants $C>0$ and $\gamma\in(0,1]$ independent of $f$, ${\boldsymbol{\alpha}}$ and ${\boldsymbol{\beta}}$, such that: $$|c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}|\leq C2^{-(\gamma|\alpha|_{\infty}+|\alpha|)} \label{eqn:lemi}$$ for almost all $({\boldsymbol{\alpha}},{\boldsymbol{\beta}})$, where ${\boldsymbol{\alpha}}\in {\mathbb N}^d$ and ${\boldsymbol{\beta}}\in B_{{\boldsymbol{\alpha}}}$. Moreover, for any ${\boldsymbol{\alpha}}\in {\mathbb N}^d$: $$\sup_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}}|c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}|\leq C2^{-|{\boldsymbol{\alpha}}|}. \label{eqn:lemii}$$ The last lemma provides a useful identity: For any $x \in (0,1)$, $$\sum_{i=0}^\infty x^i{i+k+d-1 \choose d-1} = \sum_{j=0}^{d-1}{k+d-1 \choose j}\left(\frac{x}{1-x}\right)^{d-1-j}\frac{1}{1-x}.$$ \[lem:id\] The proofs of Lemma \[prop:BoundofSurplus\] and \[lem:id\] are found in the Appendix. Consider first the prediction error $f - \hat{f}_n^{\rm BR}$ for some $f \in \mathcal{H}_{mix}^{1,c}$, where $\hat{f}_n^{\rm BR}$ is the GP predictor using the Brownian kernel $k^{\rm BR}$. Using (i) the function space equivalence $\mathcal{H}_{mix}^{1,c} = \mathcal{V}$ (Theorem \[thm:4spacesequal\]) and (ii) the equivalence between $\hat{f}_n^{\rm BR}$ and the sparse grid FEM solution $\hat{f}_n^{\rm BR}$ (Theorem \[thm:sggFEMkriging\]), this prediction error can be decomposed via : $$\label{eq:error4SPG} f-\hat{f}_n^{\rm BR} = f-\mathcal{I}_k^{\rm SP} f =\sum_{{{\boldsymbol{\alpha}}}\in\mathbb{Z}_{\geq 0}^d}f_{{\boldsymbol{\alpha}}}-\sum_{0\leq |{{\boldsymbol{\alpha}}}|\leq k+d-1}f_{{\boldsymbol{\alpha}}}=\sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}f_{{\boldsymbol{\alpha}}}.$$ Therefore, the error can be bounded by the infinite series: $$\|f-\hat{f}_n^{\rm BR}\|\leq \sum_{|\alpha|\geq k+d}||f_\alpha|| \label{eq:infser}$$ for any norm $||\cdot||$. Let us first take the $L^p$ norm for $\| \cdot \|$ in . Note that $$\begin{aligned} \|f-\hat{f}_n^{\rm BR}\|_{L^p}&\leq \sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}||f_{{\boldsymbol{\alpha}}}||_{L^p}\\ &=\sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}\left\|\sum_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}}c_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}\phi_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}({\bold x})}\right\|_{L^p}\stepcounter{equation}\tag{\theequation}\label{eq:deriv1}\\ &=\sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}\left[\sum_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}}c_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}^p\int^{{\bold x}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}+\bold{h}_{{\boldsymbol{\alpha}}}}_{{\bold x}_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}-\bold{h}_{{\boldsymbol{\alpha}}}}\phi_{{{\boldsymbol{\alpha}}},{{\boldsymbol{\beta}}}}^p({\bold x}) \; d{\bold x}\right]^{\frac{1}{p}}\\ &=\sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}\left[\frac{2^{d-1}}{{(p+1)}^d |B_{{\boldsymbol{\alpha}}}|} \sum_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}} c^p_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}} \right]^{\frac{1}{p}}\\ &\leq C \sum_{|{{\boldsymbol{\alpha}}}|\geq k+d} 2^{-(\gamma|{{\boldsymbol{\alpha}}}|_{\infty}+|{{\boldsymbol{\alpha}}}|)}\\ &\leq C \sum_{|{{\boldsymbol{\alpha}}}|\geq k+d} 2^{-(1+\varepsilon)|{{\boldsymbol{\alpha}}}|},\end{aligned}$$ where $C$ and $\epsilon$ are positive constive constant independent of ${\boldsymbol{\alpha}}$ (note that the constant $C$ is used to show big-O convergence, and may change in value throughout the proof). Here, the third line follows from the fact that $\{\phi_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\}_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}}$ is pairwise disjoint, the fourth line follows from the fact that $\int^{{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}+\bold{h}_{{\boldsymbol{\alpha}}}}_{{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}-\bold{h}_{{\boldsymbol{\alpha}}}} \phi_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}^p({\bold x}) d{\bold x}=[2/(p+1)]^d2^{-|{{\boldsymbol{\alpha}}}|}$, and the fifth line follows from Lemma \[prop:BoundofSurplus\] (Equation \[eqn:lemi\]). We can further upper bound the last equation as follows: $$\begin{aligned} \begin{split} C \sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}2^{-(1+\varepsilon)|{{\boldsymbol{\alpha}}}|}&=C \sum_{i=k+d}^\infty2^{-(1+\varepsilon)i}\sum_{|{{\boldsymbol{\alpha}}}|=i}1\\ &=C \sum_{i=k+d}^\infty2^{-(1+\varepsilon)i}{i-1 \choose d-1}\\ &\leq C 2^{-(1+\varepsilon)k} \cdot 2^{-(1+\varepsilon)d}\sum_{i=0}^\infty2^{-i}{i+k+d-1 \choose d-1}, \end{split} \label{eq:deriv2}\end{aligned}$$ where the second line follows since there are ${i-1 \choose d-1}$ ways to represent $i$ as a sum of $d$ natural numbers. With $x=2^{-1}$, Lemma \[lem:id\] gives: $$\begin{aligned} \begin{split} \sum_{i=0}^\infty2^{-i}{i+k+d-1 \choose d-1} = 2\sum_{j=0}^{d-1}{k+d-1 \choose j}=2\frac{k^{d-1}}{(d-1)!}+\mathcal{O}(k^{d-2}). \end{split} \label{eq:deriv3}\end{aligned}$$ Plugging into , we get: $$\begin{aligned} \begin{split} \|f-\hat{f}_n^{\rm BR}\|_{L^p}&\leq C\sum_{|{{\boldsymbol{\alpha}}}|\geq k+d}2^{-(1+\varepsilon)|{{\boldsymbol{\alpha}}}|}\leq C2^{-(1+\varepsilon)(k+d)}\frac{k^{d-1}}{(d-1)!}\\ &=C2^{-(1+\varepsilon)k}\left[2^{-(1+\varepsilon)(d-1)}\frac{k^{d-1}}{(d-1)!}\right]. \end{split} \label{eq:deriv4}\end{aligned}$$ Using the upper bound on grid points for sparse grids (Lemma \[lem:num4SPG\]), the above prediction error can be stated in terms of sample size $n$: $$\begin{aligned} \begin{split} \|f-\hat{f}_n^{\rm BR}\|_{L^p} &\leq C2^{-\varepsilon k}2^{-k}[\log 2^k]^{d-1}=2^{-\varepsilon k}n^{-1}[k\log 2]^{2d-2}\\ &=\mathcal{O}(n^{-(1+\delta)}) = o(n^{-1}) \label{eq:deriv5} \end{split}\end{aligned}$$ for some $\delta>0$. For $L^{\infty}$ convergence, we can take the $L^\infty$ norm for $\| \cdot \|$ in and mimic the same proof technique for $L^{p}$ convergence, with the key distinction being the use of Lemma \[prop:BoundofSurplus\] (ii) in to upper bound $\|\sum_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}} c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\phi_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\|_{L^{\infty}} = \sup_{{{\boldsymbol{\beta}}}\in B_{{\boldsymbol{\alpha}}}}|c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}|={\mathcal O}(2^{-|{{\boldsymbol{\alpha}}}|})$. This yields the following $L^{\infty}$ rate in $n$: $$\|f-\hat{f}_n^{\rm BR}\|_{L^\infty} = {\mathcal O}(n^{-1}[\log n]^{2(d-1)}). \label{eq:deriv6}$$ Finally, using Theorem \[thm:diffBdryGP\], the $L^p$ and $L^\infty$ convergence rates for $\|f-\hat{f}_n^{\rm BR}\|$ also hold for the BdryGP error $\|f-\hat{f}_n^{\rm BR}\|$ as well, which completes the proof. *Remark 1*: In Theorem \[thm:L1error\], the intuition behind the slower $L^{\infty}$ rate (compared to the $L^{p}$ rate, $1 \leq p < \infty$), is that $D^{\bold{1}}f({\bold x})$ can be ill-behaved on a measure-zero set on ${\mathcal X}$. Because of this, the pointwise convergence rate on this set can be be much slower. The effect from this measure-zero set can be ignored under integration for $\mathcal{L}^p$ with $p < \infty$. *Remark 2*: We can further improve the convergence rate in Theorem \[thm:L1error\] if we restrict $f$ to the smaller function space ${\mathcal H}^{2,c}_{mix}$, the constrained Sobolev space with mixed *second* derivatives. Using the same proof strategy, but plugging in Lemma 3.5 in [@Bungartz04], we can then show that $||\hat{f}^{\text{BM}}_n-f||_{L^2}={\mathcal O}(n^{-2}[\log n]^{d-1})$ for $f \in {\mathcal H}^{2,c}_{mix}$. The function space equivalence (Theorem \[thm:4spacesequal\]), however, does not hold under this extension, since ${\mathcal H}^{2,c}_{mix}$ is smaller than the RKHS of the BdryMatérn kernel ${\mathcal H}^{1,c}_{mix}$. Probabilistic Uniform Rate {#sec:probrate} -------------------------- Next, we prove a probabilistic convergence rate for BdryGP, where $f$ is assumed to be *random*, following a GP with sample paths in the constrained mixed Sobolev space $\mathcal{H}^{1,c}_{mix}$. This is motivated by the probabilistic convergence rates in [@Wang18] for GPs without boundary constraints. Define first the following kernel space: $${\mathcal H}^{1,c}_{mix}( {\mathcal X}\times {\mathcal X}) := \{k({\bold x},{\bold y}): k({\bold x},\cdot),k(\cdot,{\bold y})\in {\mathcal H}^{1,c}_{mix}({\mathcal X}), \; \forall {\bold x},{\bold y}\in {\mathcal X}\}. \label{eq:kerspace}$$ Such a space ensures that a GP with kernel $k \in {\mathcal H}^{1,c}_{mix}( {\mathcal X}\times {\mathcal X})$ has sample paths in $\mathcal{H}_{mix}^{1,c}$. The following theorem gives a probabilistic uniform rate for BdryGP when $f$ follows a GP with kernel $k \in {\mathcal H}^{1,c}_{mix}( {\mathcal X}\times {\mathcal X})$: \[thm:proberror\] Suppose $I^{[0]} \cup I^{[1]} = [d]$, and assume the sparse grid design ${\bold X}_k^{\rm SP}$ with $n = |{\bold X}_k^{\rm SP}|$. Let $Z(\cdot)$ be a GP with kernel $k \in {\mathcal H}^{1,c}_{mix}({\mathcal X}\times{\mathcal X})$, and $\mathcal{I}_n^{\rm BM}$ be the BdryGP interpolation operator satisfying $\mathcal{I}_n^{\rm BM} f = \hat{f}_n^{\rm BM}$. Then: $$\mathbb E\left[\sup_{{\bold x}\in{\mathcal X}}|Z({\bold x})-\mathcal{I}_n^{\rm BM} Z({\bold x})|^p\right]^{\frac{1}{p}}={\mathcal O}(n^{-1}[\log n]^{2d-\frac{3}{2}}), \quad 1 \leq p < \infty, \label{eq:lpprob}$$ and: $$\sup_{{\bold x}\in {\mathcal X}}|Z({\bold x})-\mathcal{I}_n^{\rm BM}Z({\bold x})|={\mathcal O}_{\mathbb{P}}(n^{-1}[\log n]^{2d-\frac{3}{2}}). \label{eq:linfprob}$$ Let $Z(\cdot)$ be a GP with kernel $k \in {\mathcal H}^{1,c}_{mix}({\mathcal X}\times{\mathcal X})$, and let $\mathcal{I}^{\rm BM}_n|_{{\bold x}}$ and $\mathcal{I}^{\rm BM}_n|_{{\bold y}}$ be the projection operator $\mathcal{I}^{\rm BM}_n$ in arguments ${\bold x}$ and ${\bold y}$. Consider the following hierarchical expansion of the so-called “natural distance” $\boldsymbol{\sigma}$: $$\begin{aligned} \boldsymbol{\sigma}^2({\bold x},{\bold y})&:=\mathbb{E}\big[\big(Z({\bold x})-\mathcal{I}^{\rm BM}_nZ({\bold x})\big)\big(Z({\bold y})-\mathcal{I}^{\rm BM}_nZ({\bold y})\big)\big]\\ &=k({\bold x},{\bold y})-\mathcal{I}^{\rm BM}_n|_{{\bold y}}k({\bold x},{\bold y})-\mathcal{I}^{\rm BM}_n|_{{\bold x}}k({\bold x},{\bold y})-\mathcal{I}^{\rm BM}_n|_{{\bold x}} \mathcal{I}^{\rm BM}_n|_{{\bold y}}k({\bold x},{\bold y})\\ &=\{\text{I}-\mathcal{I}^{\rm BM}_n|_{{\bold x}}\}\{\text{I}-\mathcal{I}^{\rm BM}_n|_{{\bold y}}\}k({\bold x},{\bold y}).\end{aligned}$$By Theorem \[thm:L1error\], we have: $$\boldsymbol{\sigma}({\bold x},{\bold y})={\mathcal O}(n^{-1}[\log n)]^{2(d-1)})$$ for any ${\bold x},{\bold y}\in{\mathcal X}$. This can then be plugged into the proof of Theorem 1 of in [@Wang18] to prove the result. Comparison with Existing Results {#sec:comp} -------------------------------- ----------------- ---------------- ------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------- *Work* *Design* *Type* *Rate* \[1ex\] Current Sparse grid Deterministic, $L^p$ ${ \mathchoice {{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(n^{-1})$ Current Sparse grid Deterministic, uniform ${\mathcal O}(n^{-1}[\log n]^{2(d-1)})$ [@Geer00] Optimal design Deterministic, $L^2$ ${\mathcal O}(n^{-\frac{1}{2+d}})$ [@Wu1993] Optimal design Deterministic, uniform ${\mathcal O}(n^{-\frac{1}{2d}})$ Current Sparse grid Probabilistic, uniform ${\mathcal O}_\mathbb{P}(n^{-1}[\log n]^{2d-\frac{3}{2}})$ [@Wang18] Optimal Design Probabilistic, uniform ${\mathcal O}_{\mathbb{P}}(n^{-\frac{1}{2d}}[\log n^{\frac{1}{2d}}]^{\frac{1}{2}})$ [@Stein1990] Full grid Mean square, pointwise ${\mathcal O}(n^{-\frac{1}{2d}})$ [@Ritter00] Optimal design Mean square, $L^2$ ${\mathcal O}(n^{-\frac{1}{2d}})$ ----------------- ---------------- ------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Convergence rates for BdryGP and existing rates in the literature. “Optimal design” refers to optimally-chosen points under a statistical criterion or error bound.[]{data-label="tab:convergence"} We now compare these BdryGP rates to existing GP rates which do not incorporate boundary information. Table \[tab:convergence\] summarizes several key results for the latter. Consider first the *deterministic* rates, where $f$ is a deterministic function within a function space. For $f \in {\mathcal H}^1(\mathcal{X})$ (the first-order Sobolev space), [@Wu1993] proved a $L^\infty$ minimax rate of $\mathcal{O}(n^{1/(2d)})$ for radial basis interpolators. Under the same assumptions, [@Geer00] and [@Gu02] also proved a $L^2$ minimax rate of ${\mathcal O}(n^{-{1}/(2+d)})$ for kernel ridge regression. Without additional information on $f$, these rates are in general not improvable [@Stone82]. To contrast, by incorporating boundary information, the proposed BdryGP enjoys quicker convergence rates in sample size $n$, with an $L^p$ rate of ${ \mathchoice {{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(n^{-1})$ and an $L^\infty$ rate of $\mathcal{O}(n^{-1}[\log n]^{2(d-1)})$. Furthermore, the BdryGP rates are more resistant to the “curse-of-dimensionality”. As dimension $d$ grows large, the existing error rate $\mathcal{O}(n^{1/(2d)})$ grows exponentially in sample size $n$, whereas the BdryGP rates grow exponentially in a lower-order term $\log n$ (for $L^\infty$) or in constants (for $L^p$). This shows that, by incorporating boundary information, the BdryGP not only yields lower prediction errors for fixed dimension $d$, but maintains relatively good performance as dimension $d$ grows large. Consider next the *probabilistic* uniform rates, where $f$ follows a GP with kernel $k \in {\mathcal H}^{1,c}_{mix}( {\mathcal X}\times {\mathcal X})$, which ensures sample paths are contained in the constrained mixed Sobolev space $\mathcal{H}_{mix}^{1,c}$. These probabilistic uniform GP rates were first studied in [@Tuo17] for the Matérn kernel without boundary information. There, the authors proved an $L^p$ rate over the stochastic process (uniform in $x$) of $\mathcal{O}(n^{-1/(2d)} \sqrt{[\log n^{1/(2d)}]})$, and a probabilistic rate (uniform in $x$) of ${\mathcal O}_{\mathbb{P}}(n^{-1/(2d)} \sqrt{[\log n^{1/(2d)}]})$. To contrast, by incorporating boundary information, the same uniform rates are improved to $\mathcal{O}(n^{-1} [\log n]^{2d - 3/2})$ and ${\mathcal O}_{\mathbb{P}}(n^{-1} [\log n]^{2d - 3/2})$ in Theorem \[thm:proberror\], respectively. This again shows that, by incorporating boundary information, the BdryGP can yield lower prediction errors. It is worth mentioning that the constrained *mixed* Sobolev space used here imposes greater smoothness than the Sobolev spaces used in existing rates, which may also contribute to our rate improvements. To parse out the effect from different function spaces, we can directly extend results from [@rieger17] and [@Tuo17] to show that, under *unconstrained* function spaces of comparable smoothness to Theorems \[thm:L1error\] and \[thm:proberror\], we achieve only $L^p$ rates of ${\mathcal O}(n^{-1/2}[\log n]^{(5/2)(d-1)})$ and ${\mathcal O}_{\mathbb{P}}(n^{-1/2}[\log n]^{(5/2)d-2})$ (a full proof is provided in the Appendix). These rates are of an order slower than the BdryGP rates in Theorems \[thm:L1error\] and \[thm:proberror\], which confirms that boundary information indeed improves predictive performance. Comparing theorem \[thm:L1error\] with theorem \[thm:lp\_noBoundary\], we can see that the boundary information does play an important role in improving the prediction performance. To summarize, We make some remarks on the relationship between our results and some existing results. In table \[tab:convergence\], we list some related results in the literature concerning the prediction of some underlying function, which is either a realization of a stochastic Gaussian process or a deterministic function in a reproducing kernel Hilbert spaces. ----------------- ------------------------------------------ ------------------ ------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------- Paper Model Design Convergence Type Convergence Rate \[1ex\] Current GP with boundaries Sparse Grid Probabilistic ${\mathcal O}_{\mathbb{P}}(n^{-1}[\log n]^{2d-\frac{3}{2}})$ Current GP with boundaries Sparse Grid $L^p$ ${ \mathchoice {{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(n^{-1})$ [@Wang18] GP with misspecification Scattered Points Probabilistic ${\mathcal O}_{\mathbb{P}}(n^{-\frac{\nu}{d}}[\log n^{\frac{\nu}{d}}]^{\frac{1}{2}})$ [@Stein1990] Stochastic process with misspecification Regular Grid Mean Square $n^{-\frac{\nu}{d}}$ [@Ritter00] GP Optimal Design Mean Square $n^{-\frac{\nu}{d}}$ [@Wu1993] Deterministic Kriging Scattered Points Uniform $n^{-\frac{\nu}{d}}$ ----------------- ------------------------------------------ ------------------ ------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------- : A comparison of convergence rates for the proposed BdryGP model and existing results in the literature.[]{data-label="tab:convergence"} We can see BdryGP achieves the best convergence rate among the references. The price of fast convergence is that the RKHS is smaller compared to current GP predictor. Nevertheless, $\mathcal{H}^1_{0,mix}$, the RKHS of BdryGP, is still large enough for many engineering applications especially for building emulators. In fact, $\mathcal{H}^1_{0,mix}$ is widely used in the field of numerical analysis for building the underlying function with finite elements. Extension to General Functions ============================== In general, we can relax the zero-boundary condition assumption. In this section, we prove that the convergence rate remains for a broader class of functions, which have non-zero boundary condition on a set contained by $T^d$. The first and common approach is to make the following assumption of the GP: $$Y(\cdot)\sim\mathcal{N}\bigg(\mu(\cdot),k(\cdot,\cdot)\bigg)$$ where $\mu$ is some mean function defined on $T^d$. We can easily derive that $Y\big|_{\partial T^d}=\mu\big|_{\partial T^d}$ almost surely because $k(\cdot,\cdot)=0$ on the boundary of $T^d$. As a result, the probabilistic convergence rate remains as long as the boundary value is given. Similarly, given any underlying function $f$, we can build our GP model of mean function $\mu$ to satisfy the boundary condition. As long as the difference of two functions $\mu-f$ is in the Sobolev space ${\mathcal H}_{0,mix}^1$, all the convergence results remain even if $f$ is not in ${\mathcal H}_{0,mix}^1$. Another approach is the extend the underlying function to a function with compact domain. In most cases, the assumption that functions are with support on $T^d$ is too strict in practice. So we modify the Sobolev extension theorem to show that we can put regularity constrain on the boundary to extend the underlying function. The following theorem is a modification of the Sobolev extension theorem. \[thm:sobolevExtension\] Assume a compact set $U\subset T^d$ with boundary $\partial U$. If for each point ${\bold x}^0\in\partial U$, there exist $r>0$ and a function $\gamma:{\mathbb R}^{d-1}\to{\mathbb R}$ such that, upon relabeling and reorienting the coordinates axes if needed, we have $$\begin{aligned} &U\cap B_r({\bold x}^0)=\{{\bold x}\in B_r({\bold x}^0)\big|x_d>\gamma(x_1,\cdots,x_{d-1})\}\\ &\frac{\partial^{d-1}\gamma}{\partial x_1\cdots\partial x_{d-1}} \in \mathcal{C}\end{aligned}$$ then there exists a bounded linear operator $$\bold{E}:{H}^1_{mix}(U) \to {H}^1_{0,mix}(T^d)$$ such that for any $f\in {H}^1_{mix}(U)$: $$\begin{aligned} &\bold{E}f=f \ \ \text{a.e. \ \ in} \ U,\\ &\bold{E}f =0 \ \ \text{on} \ \partial T^d\\ &||\bold{E}f||_{\mathcal{H}^1_{0,mix}(T^d)}\leq C ||f||_{\mathcal{H}^1_{mix}(U)},\end{aligned}$$ where the the constant $C$ is independent of $f$. We call $\bold{E}f$ an extension of $f$. Now, given a function $f\in{\mathcal H}^1_{mix}$ defined on $U$ with non-zero boundary condition, as long as the boundary $\partial U$ satisfies the regularity condotion in the previous theorem, we can extend it to a function $\bold{E}f$ with zero boundary condition on $T^d$. As a result, any regression method w.r.t. $\bold{E}f$ gives a version of $f$ on $U$ and the convergence rate remains if our method is applied to $\bold{E}f$ on $T^d $. In general, given a sparse grid ${\bold X}^{\rm SP}_k$, the predictor with observations $f({\bold X}^{\rm SP}_k)$ and the predictor with observations $\bold{E}f({\bold X}^{\rm SP}_k)$ are equal when $k$ is not too large because the space needed for the extension can be very small actually according to the proof of theorem \[thm:sobolevExtension\]. As a result, when the observable boundary condition of a underlying system satisfies the regularity condition in $\ref{thm:sobolevExtension}$ and the order of the sparse grid design is not too large, we can directly run our algorithm on the original underlying function $f$ without any modification of it. Numerical Experiment ==================== We now provide a small simulation study verifying the improved error convergence rate of the proposed BdryGP model over standard GP models (which do not incorporate boundary information). The set-up is as follows. We use three $d=10$-dimensional test functions from the emulation literature, taken from [@Surjano16]: $$\begin{aligned} \textit{Corner peak:} \quad f({\bold x}) &= \left(1+\frac{\sum_{j=1}^dx_j}{d}\right)^{-d-1},\\ \textit{Product peak:} \quad f({\bold x}) &= \prod_{j=1}^d\left(1+10(x_j-0.25)^2\right)^{-1},\\ \textit{Rosenbrock:} \quad f({\bold x}) &= 4\sum_{j=1}^{d-1}(x_j-1)^2+400\sum_{j=1}^{d-1}\left((x_j-0.5)-2(x_j-0.5)^2\right)^2.\end{aligned}$$ We will compare two variants of the BdryGP model: (i) the BdryGP with *full* boundary information (i.e., $I^{[0]} = I^{[1]} = [d]$), and (ii) the BdryGP with only *partial* information on left boundaries (i.e., $I^{[0]} = [d]$, $I^{[1]} = \emptyset$), with a standard GP model with the product Matérn-1/2 kernel. All models use a wavelength parameter of $\omega=1.0$, and are compared on the prediction error $\|f - \hat{f}\|_{L^1}$, which is approximated using 1000 uniformly sampled points in $\mathcal{X}$. ![(Top): Log-error $\log \|\hat{f}_n^{\rm BM}-f\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). (Bottom): The $L^1$ error ratio of the full boundary GP over the partial boundary GP, as a function of sample size $n$ (and sparse grid level $k$).[]{data-label="fig:logError"}](figures/logerror/Corn_Peak.pdf "fig:"){width="32.00000%"} ![(Top): Log-error $\log \|\hat{f}_n^{\rm BM}-f\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). (Bottom): The $L^1$ error ratio of the full boundary GP over the partial boundary GP, as a function of sample size $n$ (and sparse grid level $k$).[]{data-label="fig:logError"}](figures/logerror/prod_peak.pdf "fig:"){width="32.00000%"} ![(Top): Log-error $\log \|\hat{f}_n^{\rm BM}-f\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). (Bottom): The $L^1$ error ratio of the full boundary GP over the partial boundary GP, as a function of sample size $n$ (and sparse grid level $k$).[]{data-label="fig:logError"}](figures/logerror/Rosenbrock.pdf "fig:"){width="32.00000%"} ![(Top): Log-error $\log \|\hat{f}_n^{\rm BM}-f\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). (Bottom): The $L^1$ error ratio of the full boundary GP over the partial boundary GP, as a function of sample size $n$ (and sparse grid level $k$).[]{data-label="fig:logError"}](figures/logerror/Corn_Peak_rat.pdf "fig:"){width="30.50000%"} ![(Top): Log-error $\log \|\hat{f}_n^{\rm BM}-f\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). (Bottom): The $L^1$ error ratio of the full boundary GP over the partial boundary GP, as a function of sample size $n$ (and sparse grid level $k$).[]{data-label="fig:logError"}](figures/logerror/prod_peak_rat.pdf "fig:"){width="30.50000%"} ![(Top): Log-error $\log \|\hat{f}_n^{\rm BM}-f\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). (Bottom): The $L^1$ error ratio of the full boundary GP over the partial boundary GP, as a function of sample size $n$ (and sparse grid level $k$).[]{data-label="fig:logError"}](figures/logerror/Rosenbrock_rat.pdf "fig:"){width="30.50000%"} Figure \[fig:logError\] (top) plots the log-error $\|f - \hat{f}\|_{L^1}$ as a function of sample size $n$ (and sparse grid level $k$). For all three functions, these log-errors appear to be linearly decreasing in sparse grid level $k$. Furthermore, the two BdryGP models (both of which incorporate some form of boundary information) yield much lower errors than the standard GP without boundary information, with the error decay slopes for BdryGP roughly double that for the standard GP model. This is in line with the convergence rates proven in Section \[sec:convrates\], which show that the $L^1$ error rates for BdryGP are on the order of ${\mathcal O}(2^{-k})$, but increase to ${\mathcal O}(2^{-{k}/{2}})$ without boundary information. To highlight the error gap between full and partial boundary information, Figure \[fig:logError\] (bottom) plots the $L^1$ error ratio of the full boundary BdryGP over the partial boundary BdryGP. All ratios are above 1.0, which shows that full boundaries indeed yield more information on $f$ compared to partial boundaries. However, this improvement seems to diminish as sample size $n$ grows large; this suggests that the information on $f$ from design points can outweigh the additional information from full boundaries (over partial boundaries) for large sample sizes. ![log of MSE versus the level of associated sparse grid design on $[0,1]^5$\[fig:logError\] ](figures/logerror/logMSE.pdf){width="50.00000%"} We can notice that the log MSE curve is approximately linear in level $k$ when $k$ becomes large enough. According to theorem \[thm: HighProb\_con\], the slope of log MSE is close to $-1$ for large $k$ which coincides with our numerical experiments. Conclusion ========== This paper presents a new Gaussian process model, called BdryGP, for incorporating one type of boundary information with provably improved convergence rates. The key novelty in BdryGP is a new BdryMatérn covariance function, which inherits the same smoothness properties of a tensor Matérn kernel, while constraining sample paths to satisfy boundary information almost surely. Using a new connection between finite-element modeling and GP interpolation, we then show that under sparse grid designs, BdryGP enjoys improved convergence rates over standard GP models, which do not account for boundary information. By incorporating boundaries, our BdryGP rates are also more resistant to the well-known “curse-of-dimensionality” in nonparametric regression. Numerical simulations confirm these improved convergence rates, and demonstrate the improved performance of BdryGP over standard GP models. While this paper provides an appealing theoretical framework for the BdryGP model, there are further developments which would be useful for practical implementation. For computational efficiency, one can leverage the equivalence between FEM and BdryGP (Section \[sec:fem\]) to eliminate matrix computation steps for prediction and likelihood evaluations, which improves the scalability of BdryGP for big datasets. It would also be useful to investigate the behavior of BdryGP (e.g., consistency and convergence rates) under maximum likelihood estimation of model parameters. Proof for Lemma \[lem:finitespace\] {#apdix:pf4hierdiff} ----------------------------------- WLOG, we assume $I^{[0]}=[d]$ and $I^{[1]}=\emptyset$. Let $$\bigotimes_{j=1 }^d\bigtriangleup_if({\bold x}_{{\boldsymbol{\alpha}},{{\boldsymbol{\beta}}}}):=\bigg(\prod_{j=}^d A_{\alpha_i,\beta_i}\bigg)f(\bold{X})$$ denote the Hierarchical Surplus $c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ where the operator $\bigtriangleup_i$ is defined as: $$\bigtriangleup_if({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}):= \begin{cases} -\frac{1}{2}f({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}+2^{-\alpha_i}e_i)+f({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})-\frac{1}{2}f({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}-2^{-\alpha_i}e_i) &\text{if} \ \alpha_i\geq 1,\\ f(1) - f(0) &\text{if} \ \alpha_i=0. \end{cases}$$ Obviously, $\bigtriangleup_i$ is a linear operator. Let $\pi_{\mathcal{V}_{{\boldsymbol{\alpha}}}}[\cdot]$ be the projector to the space $\mathcal{V}_{{\boldsymbol{\alpha}}}$ as the one in equation (3.19) in [@Bungartz04]. The projection operator $\pi_{\mathcal{V}_\bold{n}}[\cdot]$ can be written as: $$\begin{aligned} &\ \ \ \ \ \pi_{\mathcal{V}_\bold{n}}[\cdot]\\ &=\sum_{|{\boldsymbol{\alpha}}|_\infty\leq \bold{n}}\sum_{{\boldsymbol{\beta}}\in B_{{\boldsymbol{\alpha}}}}\phi_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}\bigotimes_{i=1}^d\bigtriangleup_i\\ &=\sum_{|{{\boldsymbol{\alpha}}}|_\infty\leq \bold{n}}\sum_{\beta_d\in B_{{\boldsymbol{\alpha}}}}\phi_{\alpha_d,\beta_d}\bigtriangleup_d\sum_{\beta_{d-1}\in B_{{\boldsymbol{\alpha}}}}\phi_{\alpha_{d-1},\beta_{d-1}}\bigtriangleup_{d-1}\cdots\sum_{\beta_1\in B_{{\boldsymbol{\alpha}}}}\phi_{\alpha_1,\beta_1}\bigtriangleup_1\\ &=\sum_{\alpha_d\leq n_d}\sum_{\beta_d\in B_{{\boldsymbol{\alpha}}}}\phi_{\alpha_d,\beta_d}\bigtriangleup_d\sum_{\alpha_{d-1}\leq n_{d-1}}\sum_{\beta_{d-1}\in B_{{\boldsymbol{\alpha}}}}\phi_{\alpha_{d-1},\beta_{d-1}}\bigtriangleup_{d-1}\cdots\sum_{\alpha_1\leq n_1}\sum_{\beta_1\in B_{{\boldsymbol{\alpha}}}}\phi_{\alpha_1,\beta_1}\bigtriangleup_1.\end{aligned}$$ As a result the posterior mean of BdryGP with Brownian kernel $\hat{f}_\bold{n}^{\text{BR}}$ and $\pi_{\mathcal{V}_{\bold{n}}}$ lie in tensor product of spaces and we only need to show the equation holds for 1-d functions. We prove the equation by induction. when $n=1$, then according to Theorem \[thm:LagrangeKrigingEquivalent\]: $$\begin{aligned} \hat{f}_1^{\text{BR}}(x)&= \begin{cases} & 2f(\frac{1}{2})x \ \ \text{if} \ x\leq \frac{1}{2}\\ & 2f(\frac{1}{2})(1-x)+f(1)(2x-1)\ \ \text{if} \ x>\frac{1}{2} \end{cases}\\ &=[-\frac{1}{2}f(0)+f(\frac{1}{2})-\frac{1}{2}f(1)]\phi_{1,1}(x)+[-f(0)+f(1)]\phi_{0,1}(x)\\ &=\pi_{\mathcal{V}_1}[f](x).\end{aligned}$$ Suppose the equation holds for $n=k$, and WLOG, suppose $x\in(x_{k,\beta_k},x_{k,\beta_k+1})$ for some $x_{k,\beta_k},x_{k,\beta_k+1}\in{\bold X}_\bold{n}$ and $\beta_k$ is odd. So we have: $$\begin{aligned} \hat{f}_k^{\text{BR}}(x)&=f(x_{k,\beta_k})\phi_{k,\beta_k}(x)+f(x_{k,\beta_k+1})\phi_{k,\beta_k+1}(x)\\ &=\sum_{n\leq k}\sum_{\beta\in B_n}c_{n,\beta}\phi_{n,\beta}(x)=\pi_{\mathcal{V}_k}[f]\end{aligned}$$ with $x\in \text{supp}[\phi_{n,\beta_n}]$ and $\beta_n\in B_n$. When $n=k+1$, we have: $$\begin{aligned} \pi_{\mathcal{V}_{k+1}}[f]&=\sum_{n\leq k+1}\sum_{\beta\in B_n}c_{n,\beta}\phi_{n,\beta}(x)\\ &=f(x_{k,\beta_k})\phi_{k,\beta_k}(x)+f(x_{k,\beta_k+1})\phi_{k,\beta_k+1}(x)+c_{k+1,\beta_{k+1}}\phi_{k+1.\beta_{k+1}}(x).\end{aligned}$$ According to the following identities: $$\begin{aligned} &x_{k,\beta_k}=x_{k+1,\beta_{k+1}-1}\\ &x_{k,\beta_k}=x_{k+1,\beta_{k+1}+1}\end{aligned}$$ and, WLOG, conditioned on the assumption $x\in(x_{k,\beta_k},x_{k+1,\beta_{k+1}})$, we have $$\begin{aligned} &\phi_{k,\beta_k}(x)=\frac{x_{k,\beta_k+1}-x}{2^{-k}}\\ &\phi_{k,\beta_k+1}(x)=\frac{x-x_{k,\beta_k}}{2^{-k}}\\ &\phi_{k+1,\beta_{k+1}}(x)=\frac{x-x_{k,\beta_k}}{2^{-k-1}}.\end{aligned}$$ Now we plug in equation (\[eq:HierarchicalSurplus\]) and the above identities, we can have the result: $$\begin{aligned} \pi_{\mathcal{V}_{k+1}}[f]&=f(x_{k+1,\beta_{k+1}-1})\phi_{k+1,\beta_{k+1}-1}(x)+f(x_{k+1,\beta_{k+1}})\phi_{k+1,\beta_{k+1}}(x)\\ &=\hat{f}^{\text{BR}}_{k+1}(x).\end{aligned}$$ Proof of Theorem \[thm:diffBdryGP\] {#apdix:pf4diffBdryGP} ----------------------------------- WLOG, we assume that the mean function $\mu=0$. Let $f\in{\mathcal H}^{1,c}_{mix}$, then the difference between the two interpolator $\hat{f}^{\rm BR}_{n}$ and $\hat{f}^{\rm BM}_{n}$ conditioned on ${\bold X}^{\rm SP}_k$ can be written as $$\begin{aligned} \delta({\bold x})&:= |\hat{f}^{\rm BR}_{n}({\bold x})-\hat{f}^{\rm BM}_{n}({\bold x})|\\ &=|[f({\bold x})-\hat{f}^{\rm BR}_{n}({\bold x})]-[f({\bold x})-\hat{f}^{\rm BM}_{n}({\bold x})]|.\end{aligned}$$ We first define the following hierarchical difference functions: $$\begin{aligned} \Delta^{\text{BR}}_{\alpha_j}[f](x_j) &:=\{k^{\text{BR}}(x_j,{\bold X}_{\alpha_j})[k^{\text{BR}}({\bold X}_{\alpha_j},{\bold X}_{\alpha_j})]^{-1}-\\ &\ \ \ \ k^{\text{BR}}(x_j,{\bold X}_{\alpha_j-1})[k^{\text{BR}}({\bold X}_{\alpha_j-1},{\bold X}_{\alpha_j-1})]^{-1}\}f({\bold X}_{\alpha_j})\\ \Delta^{\text{BM}}_{\alpha_j}[f](x_j) &:=\{k^{\text{BM}}_{\omega_j}(x_j,{\bold X}_{\alpha_j})[k^{\text{BM}}_{\omega_j}({\bold X}_{\alpha_j},{\bold X}_{\alpha_j})]^{-1}-\\ &\ \ \ \ k^{\text{BM}}_{\omega_j}(x_j,{\bold X}_{\alpha_j-1})[k^{\text{BM}}_{\omega_j}({\bold X}_{\alpha_j-1},{\bold X}_{\alpha_j-1})]^{-1}\}f({\bold X}_{\alpha_j}).\end{aligned}$$ According to equation (2) and (3) in [@Barthelmann00], we have the following expansion of the error terms: $$\begin{aligned} &f({\bold x})-\hat{f}^{\rm BR}_{n}({\bold x})=\sum_{|{\boldsymbol{\alpha}}|\geq k+d}\bigotimes_{j=1}^d\Delta^{\text{BR}}_{\alpha_j}[f](x_j)\\ &f({\bold x})-\hat{f}^{\rm BM}_{n}({\bold x})=\sum_{|{\boldsymbol{\alpha}}|\geq k+d}\bigotimes_{j=1}^d\Delta^{\text{BM}}_{\alpha_j}[f](x_j)\end{aligned}$$ where $$\sum_{|{\boldsymbol{\alpha}}|\geq k+d}\bigotimes_{j=1}^d\Delta^{\text{BR}}_{\alpha_j}[f](x_j)= \sum_{|{\boldsymbol{\alpha}}|\geq k+d}f_{{\boldsymbol{\alpha}}}({\bold x})$$ We now want to write the expansion of $f({\bold x})-\hat{f}^{\rm BM}_{n}({\bold x})$ in terms of $\{\Delta^{\text{BR}}_{\alpha_j}[f](x_j)\}$. According to Theorem 2 in [@DingZhang18], for any 1-d function $f\in\mathcal{H}^{1,c}_{mix}$, we can write the BLUE of kernel $k^{\text{BM}}_{\omega_j}$ explicitly: $$\begin{aligned} & \ \ \ \ k^{\text{BM}}_{\omega_j}(x_j,{\bold X}_{\alpha_j})[k^{\text{BM}}_{\omega_j}({\bold X}_{\alpha_j},{\bold X}_{\alpha_j})]^{-1}f({\bold X}_{\alpha_j})\\ &=\frac{\sinh[\omega_j(x_{\alpha_j,\beta_j+1}-x_j)]}{\sinh[\omega_j2^{-\alpha_j}]}f(x_{\alpha_j,\beta_j})+\frac{\sinh[\omega_j(x_j-x_{\alpha_j,\beta_j})]}{\sinh[\omega_j2^{-\alpha_j}]}f(x_{\alpha_j,\beta_j+1})\\ &=\frac{x_{\alpha_j,\beta_j+1}-x_j}{2^{-\alpha_j}}f(x_{\alpha_j,\beta_j})+\frac{x_j-x_{\alpha_j,\beta_j}}{2^{-\alpha_j}}f(x_{\alpha_j,\beta_j+1})+{\mathcal O}(2^{-2\alpha_j})\\ &=k^{\text{BR}}(x_j,{\bold X}_{\alpha_j})[k^{\text{BR}}({\bold X}_{\alpha_j},{\bold X}_{\alpha_j})]^{-1}f({\bold X}_{\alpha_j})+{\mathcal O}(2^{-2\alpha_j})\end{aligned}$$ where $x_{\alpha_j,\beta_j}$ and $x_{\alpha_j,\beta_j+1}$ are the points that satisfy $x_j\in [x_{\alpha_j,\beta_j},x_{\alpha_j,\beta_j+1}]$, the second equality of the above equation is from Taylor expansion, and the last equality is from the proof of Theorem \[thm:LagrangeKrigingEquivalent\]. So the following equality holds: $$\begin{aligned} \bigotimes_{j=1}^d\Delta^{\text{BM}}_{\alpha_j}(x_j)&=\bigotimes_{j=1}^d\{\Delta^{\text{BR}}_{\alpha_j}[f](x_j)+{\mathcal O}(2^{-2\alpha_j})\}\\ &=\bigotimes_{j=1}^d\Delta^{\text{BR}}_{\alpha_j}[f](x_j)+\sum_{j=1}^d{\mathcal O}\big(2^{-\alpha_j}\bigotimes_{j=1}^d\Delta^{\text{BR}}_{\alpha_j}[f](x_j)\big)\end{aligned}$$ where the second equality is from the fact that $\Delta^{\text{BR}}_{\alpha_j}[f](x_j)$ is in an order no smaller than ${\mathcal O}(2^{-2\alpha_j})$. Therefore, we can have the final result: $$\begin{aligned} f({\bold x})-\hat{f}^{\rm BM}_{n}({\bold x})&=\sum_{|{\boldsymbol{\alpha}}|\geq k+d}\bigotimes_{j=1}^d\Delta^{\text{BM}}_{\alpha_j}[f](x_j)\\ &=\sum_{|{\boldsymbol{\alpha}}|\geq k+d}\left[1+\sum_{j=1}^d{\mathcal O}(2^{-\alpha_j})\right]f_{{\boldsymbol{\alpha}}}({\bold x})\\ &={\mathcal O}\left(\sum_{|{\boldsymbol{\alpha}}|\geq k+d}f_{{\boldsymbol{\alpha}}}\right).\end{aligned}$$ Proof of Lemma \[prop:BoundofSurplus\] -------------------------------------- Let $\bold{i}$ and $\bold{h}$ denote $(i_1,i_2,\cdots,i_d)$ and $(2^{-\alpha_1},\cdots,2^{-\alpha_d})$ respectively, and let $\bold{i}\bold{h}$ denote $(i_12^{-\alpha_1},\cdots,i_d2^{-\alpha_d})$. Let $f({\bold x}_{I};{\bold x})$ denote $f$ with fixed $x_i, i\not\in I$. According to equation (\[eq:HierarchicalSurplus\]), we write $c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ as $$\begin{aligned} c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}&=\bigg(\prod_{i=1}^dA_{\alpha_i,\beta_i}\bigg)f(\bold{X})\\ &=\sum_{i_d=-1}^{1}\cdots\sum_{i_1=-1}^1\bigg(\frac{-1}{2}\bigg)^{\sum_{j=1}^d|i_j|}f({\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}+\bold{ih})\\ &=\sum_{i_d=-1}^{1}\cdots\sum_{i_{2}=-1}^1\bigg(\frac{-1}{2}\bigg)^{\sum_{j=2}^{d}|i_j|}\left(-\frac{1}{2}\right)\cdot\\ &\quad \quad \quad \quad \quad \int_{{\bold x}_{\alpha_1,\beta_1}}^{{\bold x}_{\alpha_1,\beta_1}+h_1}\partial_{x_1}[f(s_1;{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})-f(s_1-h_1;{\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}})]ds_1\\ &=\left(-\frac{1}{2}\right)^d\int_{{\bold x}_{\alpha_d,\beta_d}}^{{\bold x}_{\alpha_d,\beta_d}+h_d}\cdots\int_{{\bold x}_{\alpha_1,\beta_1}}^{{\bold x}_{\alpha_1,\beta_1}+h_1}D^{\bold{1}}\sum_{i_1,\cdots,i_d=0}^1(-1)^{\sum_{j=1}^d|i_j|}f(\bold{s}-\bold{ih})d\bold{s}.\end{aligned}$$ When $d=1$, any function in ${\mathcal H}^{1,c}_{mix}\subset{\mathcal H}^1({\mathcal X})$ can be extended to trace-zero function, which is the limit of a sequence of smooth functions under the ${\mathcal H}^{1,c}_{mix}$ norm (Theorem 5.5.2 of [@Evans15]). When $d>1$, any $f\in{\mathcal H}^{1,c}_{mix}$ is also the limit of a sequence of smooth functions $\{g^n\}$ under the ${\mathcal H}^{1,c}_{mix}$ norm because ${\mathcal H}^{1,c}_{mix}$ is the tensor product of 1-d function spaces. Therefore, according to Lebesgue differentiation theorem, for almost all ${\bold x}\in\mathcal{X}$: $$\begin{aligned} &\ \ \ \ \int_{{\bold x}}^{{\bold x}+h}|D^{\bold{1}}f(\bold{s})-D^{\bold{1}}f(\bold{s}-h)|d\bold{s}\\ &\leq \int_{{\bold x}-h}^{{\bold x}+h}|D^{\bold{1}}f(\bold{s})-D^{\bold{1}}g^n(\bold{s})|d\bold{s}+\int_{{\bold x}}^{{\bold x}+h}|D^{\bold{1}}g^n(\bold{s})-D^{\bold{1}}g^n(\bold{s}-h)|d\bold{s}\\ &\leq Ch^{1+\gamma}\end{aligned}$$ where the last line is because the first term of the second line can be arbitrarily small by letting $n$ large from the trace-zero theorem and the second term is the difference of two smooth functions and hence we can use Hölder’s condition to have an upper bound. Now, WLOG, we assume $\alpha_1=|{{\boldsymbol{\alpha}}}|_\infty$ and then, as long as there is no singular point ${\bold x}$ that does not satisfy the Hölder condition near ${\bold x}_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$, we can have: $$\begin{aligned} |c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}|&=\left(\frac{1}{2}\right)^d|\int_{{\bold x}_{\alpha_d,\beta_d}}^{{\bold x}_{\alpha_d,\beta_d}+h_d}\cdots\int_{{\bold x}_{\alpha_1,\beta_1}}^{{\bold x}_{\alpha_1,\beta_1}+h_1}D^{\bold{1}}\sum_{i_1,\cdots,i_d=0}^1(-1)^{\sum_{j=1}^d|i_j|}f(\bold{s}-ih)d\bold{s}|\\ &\leq \left(\frac{1}{2}\right)^d\int_{{\bold x}_{\alpha_d,\beta_d}}^{{\bold x}_{\alpha_d,\beta_d}+h_d}\cdots\int_{{\bold x}_{\alpha_1,\beta_1}}^{{\bold x}_{\alpha_1,\beta_1}+h_1}\\ & \quad \quad \quad \quad \sum_{i_2,\cdots,i_d=0}^1|D^{\bold{1}}f(s_1;\bold{s}-\bold{i}\bold{h})-D^{\bold{1}}f(s_1-h_1;\bold{s}-\bold{i}\bold{h})|d\bold{s}\\ &\leq Ch_1^\gamma\prod_{i=1}^dh_i\\ &=C2^{-\{\gamma|{{\boldsymbol{\alpha}}}|_{\infty}+|{{\boldsymbol{\alpha}}}|\}},\end{aligned}$$ where the third line is from the inequality from Lebesgue differentiation theorem we proved previously. If the Hölder condition fails at a specific point, then we can begin with the second line of the above equation to derive that $|c_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}|={\mathcal O}(2^{-|{{\boldsymbol{\alpha}}}|})$ via the inequality: $$\begin{aligned} & \left(\frac{1}{2}\right)^d \int_{{\bold x}_{\alpha_d,\beta_d}}^{{\bold x}_{\alpha_d,\beta_d}+h_d}\cdots\int_{{\bold x}_{\alpha_1,\beta_1}}^{{\bold x}_{\alpha_1,\beta_1}+h_1}\\ & \ \ \ \ \sum_{i_2,\cdots,i_d=0}^1|D^{\bold{1}}f(s_1;\bold{s}-ih)-D^{\bold{1}}f(s_1-h_1;\bold{s}-ih)|d\bold{s}\\ &\leq \frac{1}{2^d}2^{-|{{\boldsymbol{\alpha}}}|}\sum_{i_2,\cdots,i_d=0}^1 ||D^{\bold{1}}f(s_1;\bold{s}-ih)-D^{\bold{1}}f(s_1-h_1;\bold{s}-ih)||_{L^2(\mathcal{X})}.\end{aligned}$$ This proves the claim. Proof of Lemma \[lem:id\] ------------------------- The result follows from direct calculations: $$\begin{aligned} & \ \ \ \ \sum_{i=0}^\infty x^i{i+k+d-1 \choose d-1}\\ &=\frac{x^{-k}}{(d-1)!}(\sum_{i\geq 0}x^{i+k+d-1})^{(d-1)}\\ &=\frac{x^{-k}}{(d-1)!}\left(x^{k+d-1}\frac{1}{1-x}\right)^{(d-1)}\\ &=\frac{x^{-k}}{(d-1)!}\sum_{j=0}^{d-1}{d-1 \choose j}(x^{k+d-1})^{(j)}\left(\frac{1}{1-x}\right)^{(d-1-j)}\\ &=\sum_{j=0}^{d-1}{d-1 \choose j}\frac{(k+d-1)!}{(k+d-1-j)!}x^{d-1-j}\frac{(d-1-j)!}{(d-1)!}\left(\frac{1}{1-x}\right)^{d-1-j+1}\\ &=\sum_{j=0}^{d-1}{k+d-1 \choose j}\left(\frac{x}{1-x}\right)^{d-1-j}\frac{1}{1-x}.\end{aligned}$$ $L^P$ Convergence Rate without Boundary Information {#apdix:pflperror_noboundary} --------------------------------------------------- \[thm:lp\_noBoundary\] Let $f\in\mathcal{H}^1_{mix}$ and $\Phi$ be the kernel whose native space is equivalent to $\mathcal{H}^1_{mix}$. Let $\hat{f}^{\rm SP}_k$ be the posterior mean of the GP with kernel $\Phi$ conditioned on a sparse grid design ${\bold X}^{\rm SP}_k$ with $n$ design points. Then: $$||f-f^s_k||_{L^\infty}={\mathcal O}(n^{-\frac{1}{2}}[\log n]^{\frac{5}{2}(d-1)}).$$ We replace $|f(x+\delta)-f(x)|$ with $[\int_0^1|f(x+\delta)-f(x)|^pdx]^{\frac{1}{p}}$ in equation (20) in [@rieger17] and use Hölder’s inequality to get: $$\int_0^1|f(x+\delta)-f(x)|^pdx=\int_0^1\left|\int_x^{x+\delta}f'(s)ds\right|^pdx\leq \delta^{\frac{1}{2}}||f'||_{L^2}.$$ As a result, we can have the following inequality: $$\mathcal{E}(f;\pi_m(I))_{L^p}\leq cm^{-1+\frac{1}{2}}||f||_{\mathcal{H}^1}$$ where $\mathcal{E}(f;V)_{L^p}$ is the best approximation error for a given $f$ from $V$ measured in $L_p$ norm and $\pi_m$ is the set of polynomials of degree less than $m$. On the other hand, Theorem 8 in [@Barthelmann00] also holds true for $L^p$ norm, therefore, by following the proof for Theorem 9 in [@rieger17], we can have the following inequality: $$||f||_{L^p(T^d)}\leq C{q-1 \choose d-1 }n^{-\frac{1}{2}}[\log n]^{\frac{3}{2}(d-1)}||f||_{\mathcal{H}^1_{mix}}+{q-d \choose d-1 }\max |f({\bold X}^s_q)|.$$ We then replace $f$ with $f-f^s_k$ . Because $f^s_k$ is exact on ${\bold X}^{\rm SP}_q$ the second term on the right hand side vanishes. We have shown in Theorem \[thm:L1error\] that ${q-1 \choose d-1 }={\mathcal O}([\log n]^{d-1})$ which leads to the final result. Probabilistic Convergence Rate without Boundary Information {#apdix:pfproberror_noboundary} ----------------------------------------------------------- \[thm:proberror\_noboundary\] Suppose $I^{[0]} \cup I^{[1]} = \emptyset$, and assume the sparse grid design ${\bold X}_k^{\rm SP}$ with $n = |{\bold X}_k^{\rm SP}|$. Let $Z(\cdot)$ be a GP with kernel $k \in {\mathcal H}^{1}_{mix}({\mathcal X}\times{\mathcal X})$ with no boundary information, and $\mathcal{I}_n^{\rm s}$ be the GP interpolation operator satisfying $\mathcal{I}_s^{\rm BM} f = \hat{f}_n^{\rm s}$, where $\hat{f}_n^{\rm s}$ is the posterior mean. Then: $$\mathbb E\left[\sup_{{\bold x}\in{\mathcal X}}|Z({\bold x})-\mathcal{I}^s_kZ({\bold x})|^p\right]^{\frac{1}{p}}={\mathcal O}(n^{-\frac{1}{2}}[\log n]^{\frac{5}{2}d-2}), \quad 1 \leq p < \infty,$$ and $$\sup_{{\bold x}\in {\mathcal X}}|Z({\bold x})-\mathcal{I}^s_kZ({\bold x})|={\mathcal O}_{\mathbb{P}}(n^{-\frac{1}{2}}[\log n]^{\frac{5}{2}d-2}).$$ We can follow the proof for Theorem \[thm:proberror\], with the only difference being that for any kernel $k({\bold x},{\bold y})\in{\mathcal H}^1_{mix}({\mathbb R}^d\times{\mathbb R}^d)$ without boundary information, the uniform bound of the induced natural distance becomes: $$\boldsymbol{\sigma}({\bold x},{\bold y})={\mathcal O}(n^{-\frac{1}{2}}[\log n]^{\frac{5}{2}(d-1)}).$$ The claim can then be shown by performing the same substitution as in the proof in Theorem \[thm:proberror\]. Proof of Theorem \[thm:sobolevExtension\] ----------------------------------------- We first do coordinate transform near ${\bold x}^0$ to “flatten out” the boundary. Define $$\begin{cases} y_i=x_i, \ \ \ i=1,\cdots,d-1\\ y_{d}=x_d-\gamma(x_1,\cdots,x_{d-1}). \end{cases}$$ Let ${\bold x}'$ denote $(x_1,\cdots,x_{d-1})$, ${\bold y}'$ denote $(y_1,\cdots,y_{d-1})$ and $u'({\bold y})=u({\bold y}',y_d+\gamma({\bold y}'))=u({\bold x}',x_d).$ We first consider the simple case $D^{\bold{1}}u\in\mathcal{C}(\overline{U})$ and $\partial U$ is flat near ${\bold x}^0$, lying in the plane $\gamma=0$ so ${\bold x}={\bold y}$ and $u({\bold x})=u'({\bold y})$. Then we nay assume there exists an open ball $B_r({\bold x}^0)$ such that $$\begin{cases} B^+:=B_r({\bold x}^0)\cap\{x_d\geq 0\}\subset U \\ B^-:=B_r({\bold x}^0)\cap\{x_d\leq 0\}\subset {\mathbb R}^d-U \end{cases}$$ We define a higher-order reflection from $B^+$ to $B^-$: $$\overline{u}({\bold x})= \begin{cases} u({\bold x}) \ \ \ \text{if} \ {\bold x}\in B^+\\ -3u({\bold x}',-x_d)+4u({\bold x}',-\frac{x_d}{2}) \ \ \ \text{if} \ {\bold x}\in B^- \end{cases}$$ Write $u^+:=\overline{u}\big|_{B^+}$ and $u^-:=\overline{u}\big|_{B^-}$. We then can easily check that $D^{\bold{1}}u^+=D^{\bold{1}}u^+$ on $\{x_d=0\}$. Also, we can readily check as well $$||\overline{u}||_{{\mathcal H}^1_{mix}(B_r({\bold x}^0))}\leq C ||\overline{u}||_{{\mathcal H}^1_{mix}(B^+)}$$ where the constant $C$ is independent of $u$. $$\begin{aligned} \partial_{y_k}u'=u_{x_k}+u_{x_d}\frac{\partial(y_d+ \gamma)}{\partial y_k}\end{aligned}$$ so according to the assumption of $\gamma$, $u$ and compactnes of $B_r({\bold y}^0)$, for any $\{i_1,\cdots,i_k\}\subseteq\{1,\cdots,d\}$, we must have $$\begin{aligned} &\frac{\partial^ku'}{\partial y_{i_1}\cdots\partial y_{i_k}}\in L^2(B_r({\bold y}^0))\\ &||\frac{\partial^ku'}{\partial y_{i_1}\cdots\partial y_{i_k}}||_{L^2(B_r({\bold y}^0))}\leq C||\frac{\partial^ku'}{\partial y_{i_1}\cdots\partial y_{i_k}}||_{L^2(B^+)}.\end{aligned}$$ Therefore, we must have $$||\overline{u'}||_{{\mathcal H}^1_{mix}(B_r({\bold y}^0))}\leq C ||\overline{u'}||_{{\mathcal H}^1_{mix}(B^+)}$$ and hence, by following the construction of $\bold{E}$ in [@Evans15], we can find a $V$ such that $U\subset V\subset T^d$, $\bold{E}u=0$ on $\partial V$ and $\bold{E}u=u$ on $U$.

--- abstract: 'Recent years have seen many examples of how the strong present in iridates can stabilize new emergent states that are difficult or impossible to realize in more conventional materials. In this review we outline a representative set of studies detailing how heterostructures based on and perovskite iridates can be used to access yet more novel physics. Beginning with a short synopsis of iridate thin film growth, the effects of the heterostructure morphology on the iridates including and SrIrO$_3$ are discussed. Example studies explore the effects of epitaxial strain, laser-excitation to access transient states, topological semimetallicity in SrIrO$_3$, 2D magnetism in artificial iridates, and interfacial magnetic coupling between iridate and neighboring layers. Taken together, these works show the fantastic potential for controlled engineering of novel quantum phenomena in iridate heterostructures.' address: - 'Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA' - 'Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA' author: - Lin Hao - 'D. Meyers' - 'M. P. M. Dean' - Jian Liu bibliography: - 'refs.bib' title: 'Novel spin-orbit coupling driven emergent states in iridate-based heterostructures' --- Iridates,Heterostructures,Spin-Orbit Coupling Introduction {#intro} ============ Within the last several decades, transition metal perovskites have dominated a significant portion of condensed matter research efforts due to their myriad of fascinating properties derived from their strong electron-electron correlations, with leading the charge [@lee2006doping; @Scalapino2012; @keimer2015quantum; @Armitage_EdopedCuprateReview2010; @Tsuei_PairCupratesReview2000; @anderson_1997cupratetheory]. Traditionally, perovskites hosting $3d$ orbitals, where the electron-electron repulsion, crystal field splitting, and bandwidth are the dominant energy scales, have garnered the most attention due to their abundance and relative ease of synthesis [@Cao2017; @galasso2013structure]. In contrast, perovskites with $4d$, $5d$ transition metal ions have only recently, within the last two decades, started to gain traction within the field [@Crawford1994_214mag; @Cao2002_327structure; @maeno1994superconductivity; @Witczak_correlatedSOC2014]. Particularly, parallels relating these materials to the $3d$ materials and their importance for the field of topologically protected phases stoked interest in these materials [@Cao2017; @Witczak_correlatedSOC2014; @jackeli2009mott; @kim2009phase; @kim2008novel; @Kim2012_327RIXS; @gao_214SC_2015; @yan2015electron; @kim2016observation; @Gretarsson2016_dop214RIXS; @Mitchell2015_214; @wang2011twisted; @meng2014odd]. Part of the justification for the lag of these materials relative to their $3d$ counterparts can be attributed to the expected trends as one moves down the periodic table. Electron-electron correlation, which is chiefly responsible for the myriad of exciting states found in $3d$ materials, is strongly reduced as one goes from $3d$ to $4d$ to $5d$ orbitals, owing to the larger spatial extension which effectively reduces how strongly two electrons interact within an orbital, while, concurrently, the bandwidth increases through stronger covalency with surrounding ions. Taken together, these trends were expected to push these materials towards weakly correlated metallic phases, with increased having only minor effects [@kim2008novel; @Ryden_IrO2RuO21970; @Meyers_competition2014]. ![\[Jeff\] (a) Cooperative effects of both crystal field splitting and on Ir$^{4+}$ (5$d^5$) ion in $O_h$ symmetry oxygen octahedra. (b) The final, singly occupied state is then split by electron-electron interactions to form an insulating state [@kim2008novel]. (c) orbital density profile for isospin up state [@jackeli2009mott]. ](Jeff.pdf){width="1.0\columnwidth"} Despite this, pioneering work exposed the error of these assumptions, showing, for instance, antiferromagnetic insulating states in Sr$_2$IrO$_4$ and many others [@Crawford1994_214mag; @Nakatsuji_CSRuO42000; @Lee_OptRu2001; @Mandrus_OsMIT2001; @Cava_RuMIT1994; @Shimura_Ir1195; @cao2013frontiers]. These deviations were eventually put on a strong conceptual basis as the stabilization of a new ground state [@kim2009phase; @kim2008novel]. To understand this new phenomenology, one first must consider the orbital structure of the basic structural motif of the octahedral-coordinated iridates, Fig. \[Jeff\] – an Ir $5d^5$ atom contained within an O octahedron. Assuming this octahedron has $O_h$ symmetry, the usual LS or Russell-Saunder’s scheme relevant to $3d$ materials predicts a triply degenerate $t_{2g}$ orbital with a single hole in a low-spin configuration. Since the bandwidth of this $t_{2g}$ state is much larger than the small Hubbard $U$, one expects the material to be a simple band metal. However, this scheme has assumed negligible . In iridates, both and crystal field splitting, Fig. \[Jeff\](a), have appreciable magnitude and their effects must be considered cooperatively. Introducing the effects to the $t_{2g}$ orbital scheme via jj-coupling, taken as $l_{\text{eff}} = 1$ states, yields a single orbital and a doubly degenerate $J_{\text{eff}} = 3/2$ orbital. Interestingly, the orbital is higher in energy, which can be understood as a consequence of the $L_{\text{eff}} = 1$ manifold being more than half full, or the $J=5/2$ shell being less then half-full, in accordance with Hund’s third rule. Thus, we are left with a full $J_{\text{eff}} = 3/2$ quartet and half-filled doublet. The states derived from the orbital have a strongly reduced bandwidth compared with the original $t_{2g}$ states, allowing the relatively weak coulomb repulsion, $\sim~1-2~\text{eV}$, to generate upper and lower Hubbard bands with a finite gap \[Fig. \[Jeff\] (b)\]. This then stabilizes a novel local and isotropic valence configuration, Fig. \[Jeff\](c), composed of an equal mixture of $xy$, $yz$, and $zx$ orbitals dubbed the -stabilized Mott state. The so-called isospin states are formed from a linear combination of the $t_{2g}$ orbitals with mixed up and down spins as $ | J_{\text{eff}} = 1/2,m_{J_{\text{eff}}} = \pm 1/2 \rangle= (|yz,m_s = \pm1/2\rangle\mp i|zx,m_s = \pm1/2\rangle \mp |xy,m_s = \mp1/2\rangle)/\sqrt{3}$ [@kim2008novel]. The equal mixing of the $t_{2g}$ orbitals requires perfect uniformity of the octahedra, which is, however, not exact in a real material; despite this, the state often forms a good starting point for more in-depth consideration. The total magnetic moment of this -stabilized Mott state is disparate from both the spin only $S = 1/2$ state seen with $3d$ orbitals and the atomic-like $J = 1/2$ state seen in rare-earths. In the former, the orbital moment is quenched and the total moment comes only from $\langle S_z\rangle$. In the latter, the total moment is $\langle L_z+2S_z\rangle = \pm1/3~(\mu_B)$, where the spin and orbital moments are anti-parallel with $|\langle S_z\rangle| = 1/6$ and $|\langle L_z\rangle| = 2/3$. In contrast, the state is branched from the $J = 5/2$ manifold and therefore has $L_{\text{eff},~z} = -L_z$, giving $\langle L_z+2S_z\rangle = \pm1$, while in practice the observed local moment is smaller owing to hybridization effects [@kim2008novel]. ![\[Mag\_struc\] (a) Crystal and (b) $ab$-plane canted antiferromagnetic structure of , with ferromagnetic moment along $b$-axis [@boseggia2013_214]. (c) Crystal and (d) $c$-axis collinear antiferromagnetic structure found in [@Boseggia2012_327REXS; @Kim2012_327REXS]. ](Mag_struc.pdf){width="1.0\columnwidth"} These moments typically order antiferromagnetically, as superexchange dominates the magnetic interactions in corner-sharing octahedra iridate systems. Owing to the strong , two distinct insulating magnetic ground states have been observed in the Ruddlesden-Popper (RP) iridates. For isolated IrO$_2$ layers, as found in Sr$_2$IrO$_4$ and Ba$_2$IrO$_4$, the moment lies within the basal plane, Fig. \[Mag\_struc\](a-b) [@boseggia2013_214; @cao1998_214; @Boseggia_B2IO4Mag2013]. While in Ba$_2$IrO$_4$ the in-plane Ir-O-Ir bond angle is $180^{\circ}$, in a large $c$-axis octahedral rotation is present changing the bond angle significantly. This distortion is especially important in $5d$ materials, due to the anisotropic magnetic interaction being of first-order in the . This coupling links the octahedral structural rotation with the moment, leading to significant canting of the basal plane magnetic moments giving a small total ferromagnetic moment of $\sim0.1\mu_B$ observed in Sr$_2$IrO$_4$ but not Ba$_2$IrO$_4$ [@Crawford1994_214mag; @kim2009phase; @Moriya_Anis1960; @dzyaloshinsky_thermodynamic1958]. In contrast to these two materials, for the bilayer iridate, , interlayer Ir-O-Ir bonds are straight, forbidding the term by inversion symmetry, and collinear moments along the $c$-axis are observed as shown in Fig. \[Mag\_struc\](c-d) [@Cao2002_327structure; @Boseggia2012_327REXS; @Kim2012_327REXS]. This dimensionality-driven change in the observed magnetic ground state is quite intriguing in light of the similar octahedral environments of the two systems. Employing , it was shown that the $c$-axis Néel state is stabilized by the interlayer pseudodipolar anisotropic coupling that is of second order in the [@Kim2012_327RIXS]. As the dimensionality increases, a correlated paramagnetic metallic state is found in perovskite SrIrO$_3$ [@Longo_structure1971; @Moon_PRL_2008; @Liu2016_SIO]. Originally, this dimensional crossover was attributed to the three-dimensional (3D) structure which often increases the bandwidth and overcomes the Hubbard $U$. Subsequent work, however, has found this conventional picture is insufficient due to the strong SOC, as discussed in Sec. \[113\_subseciton\] [@Nie_ARPES_PRL]. For the state in the single-layer iridates, many phenomenological parallels to the $S = 1/2$ state of cuprates can be drawn. For instance, both host a lone electron or hole in the valence orbital, ordering with moments oriented in the CuO$_2$ (IrO$_2$) plane with finite canting, and very similar crystal structures. However, the different $t_{2g}$ vs $e_g$ character of the valence orbitals is an important caveat to this comparison. More detailed experimental and theoretical works have drawn more comprehensive comparisons, showing, for instance, similarly strong magnetic exchange interactions [@Kim2012_327RIXS; @Coldea2001], persistent magnetic excitations in the doped state [@Cao2017; @Gretarsson2016_dop214RIXS; @dean2013persistence; @Meyers_LSCOmagnons2017], similar electronic structure [@wang2011twisted], and etc. Alluringly, evidence for close proximity to a possible state has also been found in several works, making this system an excellent candidate for applying perturbations towards realizing this state [@gao_214SC_2015; @yan2015electron; @kim2016observation; @Mitchell2015_214]. Beyond the possible applicability to , iridates hosting edge sharing octahedra, among other structural motiffs beyond the scope of this review, are expected to host many exotic quantum phenomena, including spin liquid, topological Mott, Weyl semi-metal, etc. [@Witczak_correlatedSOC2014]. In light of the desire to tractably alter the properties of the iridates towards these phases, a great deal of recent effort has focused on stabilizing thin films of the Ruddlesden-Popper iridates [@Liu2016_SIO; @Lu_14_APL_113_strain; @Nichols_13_APL_214_substrate; @Lupascu2014strain214; @Rayan_Serrao_PRB_2013; @Domingo_15_Nano]. Application of epitaxial strain has been shown to allow controlled modulations to the material’s properties [@Liu2016_SIO; @Rayan_Serrao_PRB_2013; @Oswaldo_theoryStrain2005; @Dawber_FerroEleStrain2005; @Chakhalian_ReviewSL2014; @Hwang_ReviewSL2012]. Furthermore, layering of SrIrO$_3$ with band-insulating SrTiO$_3$ allows the creation of artificial analogues of the RP series iridates, with the advantage of enhanced strain control and the ability to synthesize structures not obtainable in bulk form [@Matsuno2015_SIOSTO; @Hao_PRL_2017]. Finally, replacing the band-insulating layer with active 3d magnetic oxides creates interfacial magnetic interactions which can capitalize on the strong SOC of the 5$d$ state and introduce novel magnetic behavior [@Hirai_APLM_2015; @Yi_arxiv_2017; @moon_IrMn2017; @Nichols_emerging2016; @okamoto_charge2017]. Exploring the results of progress along these routes for altering the physical properties of the iridates forms the basis of this work. In this review, we begin by detailing the synthesis techniques that allow the stabilization of the series iridates in thin film form. The application of this method to is then described, taking results of representative studies on strain manipulation and ultrafast laser excitation experiments as examples. We then move on to archetypal studies on films containing SrIrO$_3$, beginning with the effects of strain on the topological semimetallic state and then detailing the effects of interspacing these layers with band insulating and magnetically active layers. We end with an outlook on future possible directions of experimental and theoretical work in this rapidly expanding field. Further, beyond the corner-sharing octahedral iridates discussed here, great work has also been undertaken for several systems towards the realization of other quantum phenomena, e.g., Kitaev model spin liquids, topological states, etc [@Witczak_correlatedSOC2014; @jackeli2009mott; @Zhang_PyroIr2017; @Kimchi_IrKitaev2014; @Zhou_QSLreview2017]. Synthesis techniques ==================== Layered iridates of the phases can be thought of as a simple alternating stacking sequence of one SrO rocksalt layer and $n$ perovskite SrIrO$_3$ layers along the $c$-axis. The early members ($n = 1$ and $n = 2$) and the infinite end member were reported to be synthesized using conventional routes [@Crawford1994_214mag; @kim2009phase; @cao1998_214; @subraman_MRB_1994; @Boseggia2012_327REXS]. For example, perovskite SrIrO$_3$ has been stabilized under high pressure (albeit only in polycrystalline form [@Longo_structure1971; @Zhao2008_SIOstructure]), though bulk SrIrO$_3$ forms the 6H hexagonal structure at ambient conditions instead of the perovskite phase. This thermodynamic instability has hindered investigations on perovskite iridates, but can be solved by epitaxial stabilization [@Jang_JPCM_2010; @Nishio_16_Ir_growthdiagram; @Jang_JKPS_2010]. Meanwhile, synthesis of bulk samples with $n$ in between is extremely challenging since these materials will naturally separate into mixtures of lower $n$ members and the $n = \infty$ phase, a general trend in materials [@Ruddlesden_AC_1957; @Elcombe_ACSB_1991]. ![\[growth\_phase\] Growth phase diagram of partial oxygen pressure $P_{\text{O}_2}$ vs substrate temperature $T_{\text{sub}}$ for perovskite iridate films. Solid and dotted lines are calculated phase boundaries from chemical equilibrium [@Nishio_16_Ir_growthdiagram]. ](growth_phase.pdf){width="8cm"} To date, most of the known thin film deposition techniques have been applied to iridate phase growth, such as magnetron sputtering [@Fruchter_16_sputtering], [@Kislin_113substrates; @Horak_113; @Zhang_15_PRB_LSAT; @113_liu_arxiv; @Nishio_16_Ir_growthdiagram] and [@Nie_ARPES_PRL; @Liu_ARPES_SR]. In general, during a deposition process, the material of a target source first converts into vapor phase and then transports onto the surface of a substrate. The process of sputtering is realized through momentum transfer between accelerated positive ions and the target. Taking advantage of the plentiful amount of plasma, sputtering is well known for large-scale production. The deposition process is triggered through thermal evaporation in . Featuring better control of vapor flow combined with in situ reflection high energy electron diffraction (RHEED), enables thickness control at the atomic level. During a growth, a target is ablated by a pulsed laser with controllable fluence and frequency to reach a tunable deposition rate. This is crucial, especially for the growth of iridate films, due to the volatile nature of iridium at high temperature [@Groenendijk_16_APL]. Additionally, when equipped with a RHEED system, can also realize thickness monitoring down to the atomic level. A detailed description of the other techniques can be found elsewhere [@Martin_review]. Epitaxial SrIrO$_3$ films have been successfully realized on a series of substrates by means of and growth. The growth has been recently reviewed in Ref. [@113_review_Biswas]. Contrary to SrIrO$_3$ films, growth of Sr$_2$IrO$_4$ films is usually more challenging due to its complex layer structure as mentioned before, which is common to many oxides. Nevertheless, successful growth by and has been reported in Refs. [@Liu2016_SIO; @Lu_14_APL_113_strain; @Nichols_13_APL_214_substrate; @Lupascu2014strain214; @Rayan_Serrao_PRB_2013; @Domingo_15_Nano; @Lee_12_PRB_doping214]. Recently, the stabilization of Sr$_3$Ir$_2$O$_7$ film has also been achieved on SrTiO$_3$ substrates, by precise control of substrate temperature and oxygen partial pressure [@Nishio_16_Ir_growthdiagram], as can be seen from the growth phase diagram of all three phases in Fig. \[growth\_phase\]. Sr$_2$IrO$_4$ based systems =========================== The stabilization of the RP iridates through thin-film growth techniques enables the study of numerous phenomena associated with the effect of the heterointerface and also facilitates experiments which are unfeasible in bulk single crystals. In this section, we discuss the findings for the influence of epitxial strain on and review one example of a ground-breaking experiment enabled by the availability of thin films. Strain-control over Ruddlesden-Popper iridate films --------------------------------------------------- As discussed in \[intro\], the state is highly sensitive to deviations from isotropic octahedra, with some deviation nearly always being present in real materials. This sensitivity can be exploited towards either enhancing or reducing the character, which is expected to have dramatic consequences for the physical properties. Specifically, the electronic gap, being set by the -stabilized Mott state, is expected to collapse upon straightening of the $Ir-O-Ir$ bond angle due to the increased bandwidth [@Moon_214Tdep2009]. Variation of transport behavior was indeed observed in films on (LaAlO$_3$)$_{0.3}$(SrAl$_{0.5}$Ta$_{0.5}$O$_3$)$_{0.7}$ (LSAT) substrates [@Lee_12_PRB_doping214] after chemical doping, which changes the bond angle and at the same time. Further, the magnetic ground state is also highly sensitive to the degree of tetragonal distortion, which is strongly dependent upon the local octahedral environment [@jackeli2009mott]. ![\[Serrao\_XLD\] Top panel: spectrum with horizontal and vertical polarization, as depicted in the inset, for each sample. Bottom panel: for each film showing recovery of leading to more uniform occupation of $t_{2g}$ orbitals. Applied in-plane strains are 0.17%, 0.23%, and 0.31% for 60, 10, and 5 nm films respectively [@Rayan_Serrao_PRB_2013]. ](Serrao_XLD.pdf){width="1.0\columnwidth"} Along these lines, a well-established method for modulating the local octahedral environment in perovskites is the application of epitaxial strain, which was successfully utilized for films of iridates [@Liu2016_SIO; @Lu_14_APL_113_strain; @Nichols_13_APL_214_substrate; @Lupascu2014strain214; @Rayan_Serrao_PRB_2013; @Domingo_15_Nano; @Nichols_13_APL_214_orientation; @miao_epitaxial2014; @lu_214crossover2014]. In the work of Serrao *et al.*, was deposited upon SrTiO$_3$ (001) substrates with varying thicknesses of 60, 10, and 5 nm [@Rayan_Serrao_PRB_2013]. By this methodology, various strains of 0.17%, 0.23%, and 0.31% (-0.31%, -0.59%, and -1.40%), with positive (negative) indicating tensile (compressive) strain, respectively, were surmised through diffraction to be applied in the $ab$-plane ($c$-axis). ![\[Serrao\_trans\] Thermal activation energy, $\delta$, as function of temperature, $T$, for each film and a bulk reference [@Ge_PRB_2011]. At room temperature the energy is strongly reduced as tensile strain is increased in the thinner films [@Rayan_Serrao_PRB_2013]. ](Serrao_trans.pdf){width="1.0\columnwidth"} Dramatic changes in the local octahedral environment were then evidenced by significant changes in measurements at the O $K$-edge with in- or out-of-plane x-ray polarization, Fig. \[Serrao\_XLD\]. Here, the pre-peak feature from 527-529 eV comes from hybridization of the Ir $5d$ $t_{2g}$ and apical oxygen and planar oxygen $2p$ orbitals. With minimum applied strain, the local electronic anisotropy, evidenced by the spectra of the two polarizations and their difference spectrum shown in Fig. \[Serrao\_XLD\], closely resembles previous experiments on the bulk [@kim2008novel] where the internal structure imposes octahedral compression in the $ab$-plane. As the applied tensile strain increases and suppresses the internal compression, the observed electronic anisotropy in Fig. \[Serrao\_XLD\], bottom panel, significantly decreases, indicating enhanced strain-induced distortion of the octahedra, as expected, corroborating the modification of the state. Modulation of the electronic properties was then deduced from changes in the electronic transport behavior. Interestingly, room temperature resistivity was nearly identical for all applied strains, and transport measurements to low-$T$ showed all films displayed insulating behavior, inset of Fig. \[Serrao\_trans\](a). However, despite these similarities there exists a clear difference in the slope of these curves, which evidences significant changes in the thermal excitation gap. Utilizing the thermally activated transport model, $\rho~\sim~e^{\delta/2T}$, a clear difference in the gap size at higher temperatures is observed, Fig. \[Serrao\_trans\](a). The thick, nearly relaxed film has a 200 meV gap at room temperature, while the highest strain film has only 50 meV, a surprisingly large reduction. The gap size was previously reasoned to be mostly a function of the bonding angle, with a critical value of $\sim~170^{\circ}$, wherein the gap disappears, based upon calculations [@Moon_214Tdep2009]. However, for the strained films XRD refinement showed the upper bound on the bound angle of $\sim~160^{\circ}$, with theoretical results showing a bond angle change of $<~1^{\circ}$, hardly justifying the large change in thermal activation energy, Fig. \[Serrao\_trans\]. Instead, the collapse of the $c$-axis (up to -1.4%) leads to the favoring of the $xy$ orbital, which enhances the in-plane transport. Finally, the associated lessening of the Ir-O$_A$ bond also increases the hybridization further contributing to the lower gap, and corroborating the reduction of the local electronic anisotropy discussed previously. The strain effect on the electronic structure was also investigated in the work by Nichols *et al.*[@Nichols_13_APL_214_substrate], which grew Sr$_2$IrO$_4$ on four different substrates and found that the optical transition energy characteristic of the Hubbard $U$ increases from compressive to tensile strain. The width of the optical transition was, however, found to increase at the same time. As a result, the overall optical gap is rather unchanged down to the cutoff of the spectrum at $\sim$ 0.25 eV. The rather large alteration of the thermal excitation gap with relatively small applied strain evidences the power of epitaxial strain to control the properties in -stabilized Mott systems, which are particularly sensitive to structural modulation [@Liu2016_SIO; @Lu_14_APL_113_strain; @Nichols_13_APL_214_substrate; @Lupascu2014strain214; @Rayan_Serrao_PRB_2013; @Domingo_15_Nano; @Nichols_13_APL_214_orientation; @miao_epitaxial2014; @lu_214crossover2014]. By way of comparison, nearly 30 GPa of external pressure was required to achieve similar changes within this system [@Haskel_214XMCD]. Having established that the effects of applied strain on the electronic structure are quite dramatic, the strong implies strong perturbation of the magnetic behavior should also be expected. Indeed, theoretical calculations predict in the extreme case the magnetic ground state can be changed between in-plane canted and out-of-plane collinear orders with sufficient epitaxial strain [@jackeli2009mott; @Kim2017_STSIOtheory; @Kim2017_327theory]. While this phenomena is yet to be observed, the effect of strain on the magnetic behavior was still found to be significant [@Lupascu2014strain214]. Over a range of 1% applied strain, spanning both compressive and tensile applied strain, a change in $T_{\text{N\'eel}}$ of 60K was observed. Further, utilizing , a modulation of the magnetic exchange coupling was inferred from changes to the zone boundary magnetic excitation [@Lupascu2014strain214]. These findings display the sensitivity of the state to applied strain, with critical consequences for both electronic and magnetic behavior. Ultra-fast control of magnetic states with laser excitation ----------------------------------------------------------- Recent years have shown that ultra-fast laser excitation provides a compelling new tuning parameter to modify the behavior of correlated oxide based thin films and heterostructures [@Zhang2014dynamics; @Aoki2014; @Dean2016; @wall2016recent; @Giannetti2016ultrafast; @Gandolfi2017]. The need to match the pumped and probed sample volume, where the corresponding photons differ greatly in energy, is a particular challenge that films can help mitigate. Light pulses with different energy and polarization interact with materials in several different ways and can therefore be used to drive numerous different changes in materials. These include using optical energies to excite carriers from below to above the Fermi level [@Giannetti2016ultrafast] or using infra-red/terahetertz pulses to distort the lattice [@Zhang2014dynamics]. Within this field, iridate films occupy a special place, firstly because the Ir $L_3$ resonance opens new opportunities for probing magnetism in transient states and secondly due to the new channels of excitation afforded by . {width="1.3\columnwidth"} Figure \[STO\_TR\_cartoon\_Dean\] shows the setup used in a recent breakthrough experiment designed to ascertain the nature of magnetism in transient states. Sr$_2$IrO$_4$ was photo-doped and the resultant state was probed by performing using a free electron laser [@Dean2016]. Laser-induced photo-doping has potential to create transient versions of the various exotic states that are accessible via chemical doping, with the advantage that the resulting states are tunable and reversible, while provides a highly incisive probe of the magnetic quasiparticle spectrum [@dean2015insights]. This spectrum is a fundamental expression of the nature of the correlated electron state as it is the spatial and temporal Fourier transform of the spin-spin correlation function, and it encodes the interactions present in the magnetic Hamiltonian. Iridate films are very well suited to such experiments; as Ir sits in the 6th row of the periodic table its $L$-edge x-ray resonance occurs at a much higher energy than elements from the 4th row such as Cu [@Lupascu2014strain214]. Ir x-ray resonant scattering can therefore easily access large momentum transfers in order to measure magnetic Bragg peaks. X-rays of this energy also propagate long distances in air, thus avoiding the requirements for ultra-high vacuum required for studies of cuprates [@dean2015insights; @sala2013high]. The magnetic dynamics of photo-doped Sr$_2$IrO$_4$ are summarized in Fig. \[SIO\_TR\_timescales\_Dean\]. It was found that the out-of-equilibrium state, 2 ps after the photo-excitation, exhibits an almost complete suppression of long-range magnetic order provided the excitation fluence exceeds 6 mJ/cm$^2$ [@Dean2016; @krupin2016ultrafast]. The recovery of these correlations was found to be highly anisotropic. Two-dimensional (2D) in-plane Néel correlations recover within a few ps; whereas the 3D long-range magnetic order restores on a fluence-dependent timescale of a several hundred ps. This was linked to the large difference in the in and out-of-plane magnetic exchange constants. The in-plane exchange is about 60 meV and the out-of-plane has been estimated to be on the order of 1 $\mathrm{\mu}$eV [@kim2012magnetic; @kim2014excitonic; @Vale_PRB_2015; @Fujiyama_PRL_2014]. The marked difference in these two timescales implies that the dimensionality of magnetic correlations is vital for our understanding of ultrafast magnetic dynamics. Going forward, there are many exciting opportunities to realize new physics in photoexcited iridates [@nembrini2016tracking; @hsieh2012observation]. Of particular note is the fact that strong breaks the usual dipole selection rules expected for atomic transitions in iridates. Such an effect has already been exploited in O $K$-edge studies of Sr$_2$IrO$_4$ [@liu_214singlemagnon2015]. These experiments probe magnons, excitations that are forbidden at the $K$-edges of light elements, but become allowed due to the exchange of orbital and spin angular momentum facilitated by in iridates [@liu_214singlemagnon2015; @kim2015resonant]. ![\[SIO\_TR\_timescales\_Dean\] A summary of the timescales governing magnetism (red squares) and charge (blue circles) in photo-doped Sr$_2$IrO$_4$. $\tau_\text{decay}$ is the timescale for the decay of magnetic correlations, $\tau_\text{2D}$ ($\tau_\text{3D}$) are the timescales for the recovery of two (three)-dimensional magnetism. $T_\text{fast}$/ $T_\text{slow}$ are the fast (slow) timescales for the recovery of the equilibrium charge distribution [@Dean2016]. ](SIO_TR_timescales_Dean.pdf){width="0.8\columnwidth"} SrIrO$_3$ based systems ======================= As compared to Sr$_2$IrO$_4$ with an alluring 2D magnetic structure, the other end member of the phases, perovskite SrIrO$_3$ is paramagnetic and actually famous for its exotic electronic structure [@Nie_ARPES_PRL; @Zeb_PRB_2012; @Carter_PRB_2012]. Nevertheless, the intrinsic strong in SrIrO$_3$ may manifest in introducing a novel magnetism when combined with other materials. Moreover, SrIrO$_3$ adopts a perovskite lattice structure when grown epitaxially, which can match with most of the well-known transition metal oxides, such as manganites. In this section, we will first introduce the strain engineering on the electronic structures of SrIrO$_3$ epitaxial films, and then list recent exciting findings on heterostructures composed of SrIrO$_3$ and other important materials. Topological semimetallicity in iridate films {#113_subseciton} -------------------------------------------- Earlier studies on bulk SrIrO$_3$ suggested the typical transport behavior of a paramagnetic metal at high temperatures [@Longo_structure1971; @Zhao2008_SIOstructure], and a low-temperature upturn attributed to a metal-to-insulator transition [@Zhao2008_SIOstructure]. Transport measurements on epitaxial thin film samples, however, indicated a semimetallic nature of the electronic ground state due to the small carrier density [@Jang_JPCM_2010; @113_liu_arxiv]. The semimetallicity complicates the temperature-dependence of the resistivity, which may increase or decrease with decreasing temperature without invoking a true metal-to-insulator transition because both carrier density and mobility may significantly change upon thermal fluctuations. Moreover, semimetals usually have both electron- and hole-like Fermi surfaces, imposing additional complications to transport analysis. Nevertheless, the terms of “metallic” and “insulating” are often used in literature to describe the resistivity phenomenology [@Jang_JPCM_2010; @Kislin_113substrates; @Zhang_15_PRB_LSAT; @biswas_JAP_2014; @gruenewald_JMR_2014; @Hirai_APL_2015; @Wu_JPCM_2013], which was found to sensitively depend on epitaxial strain and thickneses in film samples. The detailed mechanism of this behavior transition is still unclear. Theoretical study by Carter and Kee suggested not only a semimetallic state but also a nontrivial Dirac nodal ring around the $U$-point near the Fermi level [@Carter_PRB_2012]. Such a semimetallic band crossing prevents the Hubbard interaction from opening a charge gap, despite that most of the density of states are gaped away from the Fermi level [@Zeb_PRB_2012]. The semimetallic nature of the electronic ground state was demonstrated in two studies. Namely, the bandwidth was found to be small and even narrower than Sr$_2$IrO$_4$, but without any charge gap [@Nie_ARPES_PRL]. Instead, a coexistence of hole-like and electron-like pockets were found [@Nie_ARPES_PRL; @Liu_ARPES_SR]. The former is caused by a hole-like flat band near the zone center, whereas the latter is associated with a linearly dispersing electron-band at the zone boundary. While the electron-pocket can be assigned to the upper Dirac cone, the crossing to the lower Dirac cone was not observed, i.e., a small gap is found between the $\delta$ and $\gamma$ bands shown in Fig. \[ARPES\_113\] [@Liu_ARPES_SR].  Experimental (symbols) and theoretical (line) band dispersions of SrIrO$_3$ film grown on SrTiO$_3$. (b) The second derivative images with respect to the energy along $Z-U$ (1) and $U-R$ (2) high-symmetry directions, confirmed a direct gap at $U$ points [@Liu_ARPES_SR].](ARPES_113.pdf){width="8cm"} The absence of the band crossing was rather surprising because the Dirac nodal ring was originally thought to be associated with the mirror-symmetry of the $Pbnm$ space group [@Carter_PRB_2012], which should be preserved even in a thin film structure. This result pointed to the need for resolving the exact crystal structure and space group in epitaxial films. In general, there are three main symmetry operations in addition to the inversion and time reversal symmetry in the $Pbnm$ space group, *i.e.* $b$-glide, $n$-glide and mirror. Under a $b$-glide symmetry constraint, lattice structure is the same after reflecting in a plane perpendicular to the $a$-axis and translating along the $b$-vector. Similarly, $n$-glide symmetry allows crystal lattice to reflect in a plane perpendicular to the $b$-axis and translate along with a diagonal direction within the $ac$-plane. On the contrary, there is no translation operation in the mirror symmetry which is perpendicular to the $c$-axis. Therefore, without breaking translational symmetry or unit cell expansion, the mirror and glide symmetries can only be removed by breaking the reflection operation, *i.e.* the angular lattice parameters. To fully resolve the crystal structure in an epitaxial film, Liu *et al.* have deposited SrIrO$_3$ films on orthorhombic (110)-oriented GdScO$_3$ substrates [@Liu2016_SIO]. The refined lattice structure based on more than 70 structural Bragg peaks recorded using synchrotron x-ray diffraction found a deviation of $\gamma$ away from 90$^{\circ}$. Combining with dynamical diffraction theory calculations which fit the subtrate peaks, film peaks and their interference simultaneously, the new space group for the strained film was revealed as monoclinic $P112_{1}$/$m$, where the two glide symmetries are broken and the mirror is preserved. ![\[n-glide\_liu\_PRB\] (a) The calculated electronic structure based on the experimental unit cell parameters and atomic positions. (b) and (c) show the electronic structures of a $P12$$_1$$/n1$ and $P2$$_1$$/b11$ structures, respectively. The right panels show the only preserved symmetries under the corresponding space group. The $D$, $B$, and $E_0$ points now equal to $U$, $X$, and $R$ points of the orthorhombic Brillouin zone [@Liu2016_SIO]. ](n-glide_liu_PRB.pdf){width="8cm"} Resolving the lattice symmetry and atomic positions makes it possible to perform accurate band structure calculations using density functional theory. Interestingly, as shown in Fig. \[n-glide\_liu\_PRB\](a), a clear gap opening around the $U$ point can be seen between the upper and lower Dirac cones, indicative of a lifted Dirac degeneracy. This was unexpected because the mirror symmetry is the only symmetry operation of the three that survived in $P112_{1}$/$m$, suggesting that the two glide operations might play important roles. Band structure calculations were further performed on another two artificial structures with only the $n$-glide ($P12$$_1$$/n1$) and $b$-glide ($P2$$_1$$/b11$) symmetry preserved, respectively. In Fig. \[n-glide\_liu\_PRB\](b), the Dirac nodal ring is fully persevered in the $P12$$_1$$/n1$ structure, revealing that the $n$-glide symmetry protects the Dirac degeneracy in addition to inversion and time-reversal symmetry. The Dirac nodal ring shrinks into a pair of Dirac points along the $D-E_0$ line ($U-R$ line in an orthorhombic zone) under $P2$$_1$$/b11$ \[Fig. \[n-glide\_liu\_PRB\](c)\]. This observation suggests that, in the absence of the $n$-glide, the $b$-glide symmetry will take over to protect the Dirac degeneracy on the high-symmetry line of the $BZ$ boundary. Since both operations are glide planes, this symmetry protection renders SrIrO$_3$ a three-dimensional nonsymmorphic semimetal [@Fang_PRB_2015]. This conclusion and the original study [@Carter_PRB_2012] by Carter and Kee can actually be well reconciled. It was proposed that the Dirac nodal ring will be turned into a pair of Dirac points in an artificial structure, Sr$_2$IrRhO$_6$, where the mirror symmetry is broken by alternating layers of Ir and Rh [@Carter_PRB_2012]. While the $b$-glide will be preserved, such a layer stacking would also remove the $n$-glide. The $n$-glide symmetry thus can be broken in multiple ways in epitaxial thin films and superlattices, highlighting the vital role of the symmetry identification in studies of SrIrO$_3$-based thin-film samples. This work paves another route to engineer the electronic structure for obtaining a novel topological phase [@Fang_NatureP_2016], in addition to the geometric frustration as seen in iridates with edge-sharing IrO$_6$ octahedra [@Zhang_PyroIr2017; @Kimchi_IrKitaev2014]. Toward 2D magnetism in artificial Ruddlesden-Popper iridates ------------------------------------------------------------ ![\[SISTO\_Matsuno\_PRL\] Temperature dependence of resistivity (a) and $-d(\ln\rho)/d(T)$ (b). The arrows indicate the temperatures of the observed anomalies. (c) In-plane magnetization of (SrIrO$_3$)$_m$/(SrTiO$_3$)$_1$ SLs as function of temperature. Arrows in panel (c) indicate the onset temperatures of magnetic ordering [@Matsuno2015_SIOSTO]. ](SISTO_Matsuno_PRL.pdf){width="8cm"} Dimensionality is another important effect that can significantly impact the electronic structure. An intuitive consideration is that dimensionality varies the number of the nearest-neighboring sites, which generally changes the bandwidth [@Moon_PRL_2008]. Experimentally, a dimensionality-controlled metal to insulator transition was indeed observed in the iridates. However, due to the nonsymmorphic semimetallic state in SrIrO$_3$, the dimensional crossover in layered iridates is, strictly speaking, a semimetal-to-insulator transition, where the dimensionality-driven confinement of the IrO$_2$ layers removes the semimetallicity and introduces a full gap. Additionally, considering the structural flexibility in an artificially designed lattice, epitaxial layering provides new avenues to further study and tailor such correlations [@kim2014electronic]. For example, the layered structure was recently mimicked in an artificial superlattice (SL) [@Matsuno2015_SIOSTO]. In this pioneering work, the electronically and magnetically inert SrO monolayer in the unit cell of Sr$_{n+1}$Ir$_n$O$_{3n+1}$ was replaced with a monolayer of nonmagnetic dielectric SrTiO$_3$ in epitaxial superlattices of (SrIrO$_3$)$_m$/(SrTiO$_3$)$_1$. Using this strategy, the authors prepared not only the artificial counterparts of Sr$_2$IrO$_4$ and Sr$_3$Ir$_2$O$_7$, but also those of Sr$_4$Ir$_3$O$_{10}$ and Sr$_5$Ir$_4$O$_{13}$, which have never been reported in bulk synthesis. Transport measurements as shown in Fig. \[SISTO\_Matsuno\_PRL\](a) revealed a semimetal to insulator transition upon decreasing the thickness of the SrIrO$_3$ layers, the same trend as that in bulk phases [@Moon_PRL_2008]. A monotonic increase of magnetization was also observed simultaneously in Fig. \[SISTO\_Matsuno\_PRL\](c), indicating a strong coupling between electronic transport and magnetic moments. This coupling was further manifested in a resistivity anomaly originating from magnetic ordering in the $m = 1$ and $m = 2$ SLs \[Fig. \[SISTO\_Matsuno\_PRL\](b)\]. Further evidence of the similarity between the superlattices and phases comes from their magnetic structures. Sr$_2$IrO$_4$ hosts an ground state and the directions of the magnetic moments are tied to the octahedral rotation due to the strong [@jackeli2009mott]. The octahedra rotation induces canting of the magnetic moments, which gives rise to a weak net moment in the $ab$-plane [@kim2008novel]. Similarly, a weak net magnetization was also observed in the $m = 1$ SL as seen from Fig. \[SISTO\_Matsuno\_PRL\](c). Meanwhile, magnetic scattering investigation found magnetic Bragg peaks at (0.5 0.5 $L$) for the SL, demonstrated an configuration in the basal plane and coupling along the stacking direction. As compared to Sr$_2$IrO$_4$, the key difference is that, while the net moments of adjacent layers are spontaneously aligned with each other and lead to a macroscopic magnetization of the SL, they are anti-aligned in Sr$_2$IrO$_4$ and cancel each other [@kim2008novel]. This distinct interlayer coupling is likely due to the different stacking structure; while the adjacent layers have a (0.5, 0.5) lateral shift in Sr$_2$IrO$_4$, they are perfectly matched in the superlattices. Therefore, this work presents a new strategy to explore the physics of the dimensionality controllable metal-to-insulator transition. ![\[SISTOop\_KIM\_PRB\] Schematic diagrams (a) and calculated optical conductivity, $\sigma_1(\omega)$, (b) of Sr$_2$IrO$_4$ and $m = 1$ SL. (c) and (d) displays the band structures of Sr$_2$IrO$_4$ and $m = 1$ SL, respectively [@Kim2016_OptDFT]. ](SISTOop_KIM_PRB.pdf){width="8cm"} A subsequent optical study further confirms the similarity between the two series [@Kim2016_OptDFT]. However, red shifts were observed for the peaks characteristic of the transition from the lower to upper Hubbard bands in the SLs \[Fig. \[SISTOop\_KIM\_PRB\](b)\]. Meanwhile, the low-energy spectral weight was also enhanced, indicating a reduction in the effective electron correlations in the SLs. Density functional theory calculations verified the difference in the electronic structure. As shown in Fig. \[SISTOop\_KIM\_PRB\](d), the $J_\text{eff} = 1/2$ bands of the $m = 1$ SL are more extended in comparison to that in Sr$_2$IrO$_4$ \[Fig. \[SISTOop\_KIM\_PRB\](c)\], suggesting a larger bandwidth in the former. This result was ascribed to the additional Ir-Ir hopping channels through the SrTiO$_3$ spacer layers in SLs, as can be seen from the different atomic arrangements in the blocking layer between the quasi 2D layered structures, for example, Sr$_2$IrO$_4$ v.s. $m = 1$ SL \[Fig. \[SISTOop\_KIM\_PRB\](a)\]. This result implies that details of the blocking layers provide extra flexibility in controlling physical properties of the confined 2D SrIrO$_3$ layers. To get a deeper insight, experimental realization of SLs with different SrTiO$_3$ blocking layer thicknesses, (SrIrO$_3$)$_m$/(SrTiO$_3$)$_n$ ((SrIrO$_3$)$_n$/(SrTiO$_3$)$_m$ was originally used), was recently achieved [@Hao_PRL_2017]. We use $m/n$-SL to denote these SLs. Inserting more SrTiO$_3$ blocking layer into the super unit cell is expected to separate neighboring SrIrO$_3$ layers and suppress the additional interlayer hopping channels in the superlattice structure, reinforcing the effective electron-electron correlations. Indeed, the insulating behavior manifested by the temperature-dependence of the $ab$-plane resistivity of 1/2- and 1/3-SLs is strengthened compared to 1/1-SL, as shown in Fig. \[SISTO\_R\_Hao\](a). More interestingly, the resistivity anomalies observed in 2/1- and 1/1-SL are invisible in 1/2- and 1/3-SLs with more than one SrTiO$_3$ layer in their super unit cell, as seen from the $\rho$-T plots \[Fig. \[SISTO\_R\_Hao\](a)\] and also the $d(\ln\rho)/d(1/T)$ data \[Fig. \[SISTO\_R\_Hao\](b)\]. Note that the resistivity anomaly coincides with the magnetic long-range order in $m$/1-SLs. Despite the absence of such an anomaly, the 1/2- and 1/3-SLs also show a clear magnetic phase transition at low temperatures \[Fig. \[SISTO\_R\_Hao\](c)\], which indicates the coupling between transport and magnetism is weakened by the reduction of interlayer coupling. ![\[SISTO\_R\_Hao\] Temperature dependence of in-plane resistivity (a) and $d(\ln\rho)/d(1/T)$ (b). The arrows denote the temperatures of the resistivity anomalies. (c) In-plane remnant magnetization as function of temperature. Before measurement in zero-field, samples were cooled down under 5 kOe in-plane magnetic field [@Hao_PRL_2017]. ](SISTO_R_Hao.pdf){width="8cm"} The importance of interlayer coupling can also be seen from the much smaller onset temperature of the magnetic ordering in 1/2-SL (40 K) compared to that of 1/1-SL (150 K). The rapid decay of the ordering temperature when approaching the 2D limit indicates an exponential decrease of interlayer coupling with increasing blocking layer thickness. Further decrease of interlayer coupling does not significantly change the magnetic ordering stability, evidenced from the almost identical onset temperatures of 1/2- and 1/3-SLs, suggesting that the long-range magnetic order therein is maintained by spin anisotropy. Note that the magnetic interaction in the square lattice $J_\text{eff} = 1/2$ materials is dominated by the isotropic 2D Heisenberg term in the Hamiltonian [@jackeli2009mott; @kim2012magnetic; @Takayama_PRB_2016], while long-range magnetic order in a 2D Heisenberg magnet is unstable at any non-zero temperatures according to the Mermin-Wagner theorem [@Mermin_PRL_1966] (La$_2$CuO$_4$ for example [@dean2012spin]). Therefore, the stabilization of the order in the 2D limit manifests the leading anisotropy term [@Cuccoli_PRB_2003; @Cuccoli_PRL_2003], which is led by the easy-plane anisotropy driven by the anisotropic exchange coupling [@jackeli2009mott]. In the absence of other anisotropy, the transition would be a Beresinskii-Kosterlitz-Thouless transition [@Berezi_JETP_1971; @koster_JPCSSP_1973]; however, in this case, the small but finite corrections from higher-order terms, such as the compass-like term [@jackeli2009mott] and the residual interlayer coupling (see below) changes the transition into a second order phase transition of long-range order. ![\[SISTO\_M\_Hao\] (a) Magnetic structures and rotation patterns of 1/1- and 1/2-SLs. Black arrows indicate $J_\text{eff}$ moments, while orange arrows denote net moments in each SrIrO$_3$ layer. (b) $L$-dependence of peak intensity of (0.5 0.5 $L$) magnetic reflections. The measurement was performed at 10 K [@Hao_PRL_2017].](SISTO_M_Hao.pdf){width="8cm"} Another intriguing observation is the macroscopic net magnetization in SLs with varied spacer thickness. For the 1/1-SL, a $c^-$ rotation occurs involving in-phase octahedral rotation of the adjacent SrIrO$_3$ layers around the $c$-axis [@Matsuno2015_SIOSTO]. Combined with a interlayer coupling, it aligns canted moments in each SrIrO$_3$ layer along the same direction \[Fig. \[SISTO\_M\_Hao\](a)\]. However, under the same rotation pattern, the adjacent SrIrO$_3$ layers become out-of-phase in 1/2-SL. In this situation, a interlayer coupling would cancel out all the canted moments, contrary to the observed net magnetization. This can be reconciled if the interplayer coupling is . Such an anti-parallel alignment of the local moments between the adjacent layers, combined with the anti-phase rotation of the two layers, will lead to a parallel alignment of canted moment in each SrIrO$_3$ layer along the same direction, as can be seen from the Fig. \[SISTO\_M\_Hao\](b). This hypothesis was confirmed through magnetic scattering. As shown in Fig. \[SISTO\_M\_Hao\](c), magnetic peaks only appear at ${L + 1/2}$ in the 1/2-SL, where $L$ is an integer, rather than at ${L}$ as in the 1/1-SL. This result unambiguously shows that the interlayer coupling is in 1/2-SL opposite to the coupling in 1/1-SL. The comparison between their magnetic structures indicates that the interlayer coupling, although attenuated with thicker blocking layers, still plays a role in aligning the canted moments due to its variable sign commensurate with the switching phase relation of the octahedral rotation of the adjacent SrIrO$_3$ layers. Exploiting interfacial coupling between 5$d$ and 3$d$ states for novel magnetic states -------------------------------------------------------------------------------------- ![\[Natco\_Fina\_2014\] Temperature dependent resistance of (a) La$_{2/3}$Sr$_{1/3}$MnO$_3$/Sr$_2$IrO$_4$ heterostructure. Inset shows the schematic diagram of current-perpendicular-to-plane and current-in-plane geometry. (b) Anisotropic magnetoresistance in /Sr$_2$IrO$_4$ and /LaNiO$_3$/Sr$_2$IrO$_4$ heterostructures measured at 4.2 K. (c) In-plane anisotropy of the density of states [@Fina_natcom_2014]. ](Natco_Fina_2014.pdf){width="7.5cm"} ![\[PNAS\_Yi\_2016\] (a) Dependence of on SrIrO$_3$ layer number ($m$) in ()[$_3$]{}/(SrIrO$_3$)$_m$ (SL3$m$) supperlattices. Insets show the schematic diagrams of the two magnetic configurations. (b) Normalized spectra of a series of SLs. During measurement at 10 K, a 1 T magnetic field was applied along the $<100>$ direction. (c) Emergent ferromagnetism and anisotropy as functions of SrIrO$_3$ layer number based on theoretical calculations. [@Yi_PNAS_2016]. ](PNAS_Yi_2016.pdf){width="7.5cm"} In $5d$ iridates, large entangles the charge and spin degrees of freedom, opening up the possibility of controlling the electronic properties through the magnetic state [@jackeli2009mott]. The $J_\text{eff}$ wavefunctions are encoded with spin components, crystal field orbital components, and the complex phases between different components. Change to any of these feature would affect the others and lead to different entangled effects. For instance, one may potentially rotate the spin-orbit moment relative to crystal axis and create significant changes to the density of states around the Fermi level. Such a rotation in an state requires rotating the axis, which is, however, difficult. One way to circumvent this issue is by dragging the moments through interfacial coupling, generally known as the exchange spring effect [@Morales_PRL_2015]. For example, Fina *et al.* observed in a heterostructure comprising of Sr$_2$IrO$_4$ and FM La$_{2/3}$Sr$_{1/3}$MnO$_3$ [@Fina_natcom_2014]. The of Sr$_2$IrO$_4$ was found to have a four-fold symmetry as the magnetic field rotates the magnetization of the , while there is no such by separating Sr$_2$IrO$_4$ through inserting a paramagnetic LaNiO$_3$ slab \[Fig. \[Natco\_Fina\_2014\](b)\]. The origin of the observed was attributed to changes in the density of states near the band edges, above and below the charge gap, as the quantization axis of the spin-orbit wavefunction rotates within the 2D plane and senses the symmetry of the local environment. This effect can be measured by magnetoresistance because of the small band gap of Sr$_2$IrO$_4$, i.e., a narrow gap semiconductor. This study demonstrated the use of Sr$_2$IrO$_4$ as an semiconductor for novel spintronic applications. It is worthwhile to note that Fina *et al.* used current perpendicular-to-plane geometry \[Fig. \[Natco\_Fina\_2014\](a)\] in this study, taking advantage of the semiconducting nature of Sr$_2$IrO$_4$ [@cao1998_214]. Considering the in-plane configuration of semiconductor Sr$_2$IrO$_4$ [@kim2009phase; @kim2008novel; @Vale_PRB_2015; @Takayama_PRB_2016; @Fujiyama_PRL_2014], there is no contribution due to varying the angle between the spin-axis and current. The observed is solely due to spin-axis rotation relative to the crystal axis [@Rushforth_PRL_2007], originating from changes of the electronic structure induced by rotating the $J_\text{eff}$ moment. Similar behavior was also observed in a bulk crystal [@Wang_PRX_2014]. Further theoretical study verified this argument, *i.e.* the density of states displays a strong sensitivity to the direction of the effective ${J}_{\rm eff}$ moment, as shown in Fig. \[Natco\_Fina\_2014\](c). Note that though effect has been reported in devices composed of an metal electrode or a semiconductor [@Marti_NM_2014; @Shick_PRB_2010], from the application of view, the former is unsuitable for most of data processing electronics while synthesis of the latter with a high operating temperature is still technically challenging [@Dietl_book_2008]. This proof-of-concept study, thus, opens the possibility to integrate semiconducting and spintronic functionalities by utilizing AFMs. ![\[Natco\_Nichols\_2016\] (a) spectra near the $L_3$ and $L_2$ edges of both Mn and Ir in superlattices composed of SrMnO$_3$ and SrIrO$_3$. Shifts of peaks can be seen in both elements indicating a charge transfer in between. Inset shows the extracted oxidation states of the two elements. (b) spectra of Mn and Ir $L$ edges. Finite intensity peaks revealed emergence of net magnetic moment in both elements. [@Nichols_natcom_2016]. ](Natco_Nichols_2016.pdf){width="8cm"} While the La$_{2/3}$Sr$_{1/3}$MnO$_3$ layer was only used as a spin gate to drag the Ir spin-orbit moment through the interface, the interfacial coupling between 3$d$ and 5$d$ states is not a one-way street. The strongly spin-orbit-entangled state of 5$d$ electrons, while being highly susceptible to the coupling to the 3$d$ states across the interface, may also bring significant impact to the properties of the 3$d$ magnetic oxides [@Yin2013]. The first attempt toward this end was achieved in heterostructures comprising of and paramagnetic SrIrO$_3$, taking the merits of the large spin moments in the former and the pronounced spin-orbit coupling in the latter [@Yi_PNAS_2016]. Though the overall magnetic behaviors of the heterostructures were governed by the , its shows a systematic change with the number of SrIrO$_3$ monolayers. As shown in Fig. \[PNAS\_Yi\_2016\](a), while the easy axis of prefers to lie along the pseudocubic $<110>$ direction, inserting only one monolayer of SrIrO$_3$ per three monolayers of switched the easy axis to $<100>$. As the SrIrO$_3$ layer thickness increases, the magnetic easy axis of rotates systematically back to $<110>$, reaching a remarkable tunability [@Yi_PNAS_2016]. To investigate the original of this phenomenon, structural analysis was carried out and demonstrated that such a change of the is not due to shape anisotropy or symmetry changes. The possibility of charge transfer at interfaces was also excluded through measurements. Instead, measurements revealed the emergence of a weak moment in the nominal paramagnetic SrIrO$_3$ layer upon decreasing its thickness. Furthermore, there is a strong at the Ir $L_2$ edge \[Fig. \[PNAS\_Yi\_2016\](b)\], indicative of a breakdown of the usual picture in the superlattices [@kim2008novel; @Haskel_214XMCD] . Indeed, sum rules analysis (also theoretical calculation) revealed a dramatic increase of the ratio between orbital moment and spin moment in the heterostructures compared to an ideal ${J}_{\text{eff}} = 1/2$ quantum state [@kim2009phase]. The enhanced orbital component features a mixture of ${J}_{\text{eff}} = 1/2$ and ${J}_{\text{eff}} = 3/2$ states [@jackeli2009mott]. Unlike the ${J}_{\text{eff}}$ = 1/2 state which has an isotropic orbital character [@kim2008novel], the ${J_{\text{eff}} = 3/2}$ states are rather anisotropic. Their shapes resemble $t_{2g}$ crystal field orbitals but with strong spin-entanglement. In other words, the spin quantization axis strongly prefers to be aligned with crystal field quantization axis, *i.e.* the Ir-O bond directions. This character locks the ${J}_{\text{eff}} = 3/2$ component to $<100>$, which is transfered via interfacial coupling to the spin moment in the layers [@Yi_PNAS_2016]. ![\[PMA\_Yi\_2017\] (a) magnetic hysteresis loops of SrMnO$_3$/SrIrO$_3$ with magnetic field applied in-plane and out-of-plane [@Nichols_natcom_2016]. (b) Dependence of and on doping ratio $x$ of La$_{1-x}$Sr$_{x}$MnO$_{3}$/SrIrO$_3$ superlattices [@Yi_arxiv_2017]. ](PMA_Yi_2017.pdf){width="8cm"} While the easy-axis of the manganite remains in the plane of the film in the above study, rotation to the out-of-plane direction is possible when combined with charge transfer at interfaces [@Nichols_natcom_2016]. Nichols *et al.* found various magnetic ground states in heterostructures composed of SrMnO$_3$, as well as SrIrO$_3$ slabs. Interestingly, although both Mn and Ir are in a 4+ valence state in each individual compound, electrons are transferred from Ir to Mn across interface when they are combined, turning on order in the nominally SrMnO$_3$. The interfacial charge transfer is revealed by as a blue shift of Mn $L_3$ edge and red-shift of Ir $L_3$ edge with increasing SrMnO$_3$, as well as SrIrO$_3$, slab thickness. These peak shifts indicate deviation of Mn and Ir valence states from their nominal values, as shown in the inset of Fig. \[Natco\_Nichols\_2016\](a). Appreciable ferromagnetism was observed in heterostructures with only one SrMnO$_3$ and SrIrO$_3$ unit cells, where the effective charge modulation is strongest. This emerging ferromagnetism was shown to decay rapidly with increasing slab thickness. More interestingly, a is realized in the SrMnO$_3$/SrIrO$_3$ heterostructure, deduced from the smaller coercive field in the out-of-plane magnetic hysteresis, as shown in Fig. \[PMA\_Yi\_2017\](a) [@Nichols_natcom_2016]. Compared to the in-plane anisotropy in the /SrIrO$_3$ heterostructure [@Yi_PNAS_2016], this result suggests the ability to change the $A$-site cation in the manganite layer in order to rotate the magnetic easy axis. Controlling is an important step toward realizing magnetic storage devices. The also leads to an anomalous Hall effect [@Nichols_natcom_2016]. To shed light on the underlying mechanism, Yi *et al.* recently performed a systematic study of the interfacial anisotropy by preparing a series of La$_{1-x}$Sr$_{x}$MnO$_{3}$/SrIrO$_3$ superlattices with doping ratio $x$ ranging from 0 to 1 [@Yi_arxiv_2017]. The was shown to emerge at $x = 0.5$ and then increases rapidly with $x$. The variation of was demonstrated can not be explained in modification of , which displays a linear dependence on doping ratio based on results. On the other hand, x-ray diffraction investigation revealed a similar dependence of on doping ratio \[Fig. \[PMA\_Yi\_2017\](b)\], indicating a close correlation between the and . This result thus reveals the importance of the connectivity of oxygen octahedra in addition to the spin-orbit entanglement for tailoring interfacial functionalities. Outlook ======= Recent research has uncovered a wealth of interesting new physics in the iridates derived from the interplay between electronic correlation and strong . The fact that even small structural modifications can impart large changes in electronic and magnetic behavior makes heterostructuring these materials an excellent route towards realizing new quantum phenomena. For example, tuning the tetragonal distortion and bond angle distortion under epitaxial strain has potential to realize quantum critical phenomena at the crossover between different antiferromagnetic states [@jackeli2009mott; @Meyers_RIXS2017] as well as metal-to-insulator transition [@Kim2017_STSIOtheory]. The strong tie between the lattice degree of freedom and the electron and magnetic states highlights the importance of the thorough structural analysis in future investigations. Another important area for future development is the realization of nontrivial topological states. Although the 3D topological semimetallic state protected by nonsymmorphic symmetries in ortho-perovskite iridates has been predicted in theoretical studies, direct experimental observation of the Dirac nodal ring has yet to be achieved. If it is experimentally verified, it would open the door to realizing a variety of topological electronic states that have been theoretically proposed [@Carter_PRB_2012; @Fang_NatureP_2016; @Chen_NatCom_2015; @Chen_PRB_2014]. One may also realize 2D topological states in heterostructures by utilizing the strong . For instance, several theoretical proposals suggest that nontrivial band topology may emerge in bilayer SrIrO$_3$ grown along the cubic \[111\] direction [@Xiao_NatCom_2011; @Wang_PRB_2011; @Okamoto_PRL_2013; @Rau_ARCMP_2016; @Lado_PRB_2013]. The relatively strong electron-electron interaction in oxides also provide unique opportunities to investigate topological phases under electronic correlation [@Pesin_NatPhy_2010]. Towards this goal, the growth of perovskite iridates along the \[111\] direction was indeed recently realized [@Hirai_APLM_2015; @Anderson_APL_2016]. Advances in combining atomic layering with photoemission spectroscopy and scanning tunnelling microscopy will boost the development in this area. Finally, the exploration of the proximity effect and interfacial coupling of $5d$ states and $3d$ states has just begun. By engineering atomic stacking patterns, it should be possible to integrate the merits of these spin-orbit coupled oxides with other functional oxides. This includes realizing controllable in, for example, iridate-manganite heterostructures. An important future direction will be to utilize magnetic controls of this kind to achieve new switching mechanism in magnetic devices. Another fertile area involves studies of the topological Hall effect driven by artificially introducing at the interface of iridate and ruthenates [@Matsunoe_ScienceA_2016; @Pang_ACSinterface_2017]. A major target of such work is the realization of an appreciably sized magnetic skyrmion. The wide variety of opportunities offered by interfacial charge transfer to modulate the electron-filling and obtain a desired quantum state without explicit dopants are only just being explored [@okamoto_charge2017; @Nichols_natcom_2016]. Developments in theoretical calculations are also likely to play a key role in targeting the ideal heterostructures for the experimental realization of functional spin-orbit coupled interfaces. Acknowledgements {#acknowledgements .unnumbered} ================ The authors acknowledge useful discussion with Neil J. Robinson and Yue Cao. J.L. acknowledges support from the Science Alliance Joint Directed Research & Development Program and the Transdisciplinary Academy Program at the University of Tennessee. J.L. also acknowledges support by the DOD-DARPA under Grant No. HR0011-16-1-0005. M.P.M.D. and D. M. are supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Early Career Award Program under Award No. 1047478. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC00112704.

--- abstract: | In a previous paper we demonstrated that non-radial hydrodynamic oscillations of a thermally-supported (Bonnor-Ebert) sphere embedded in a low-density, high-temperature medium persist for many periods. The predicted column density variations and molecular spectral line profiles are similar to those observed in the Bok globule B68 suggesting that the motions in some starless cores may be oscillating perturbations on a thermally supported equilibrium structure. Such oscillations can produce molecular line maps which mimic rotation, collapse or expansion, and thus could make determining the dynamical state from such observations alone difficult. However, while B68 is embedded in a very hot, low-density medium, many starless cores are not, having interior/exterior density contrasts closer to unity. In this paper we investigate the oscillation damping rate as a function of the exterior density. For concreteness we use the same interior model employed in Broderick et al. (2007), with varying models for the exterior gas. We also develop a simple analytical formalism, based upon the linear perturbation analysis of the oscillations, which predicts the contribution to the damping rates due to the excitation of sound waves in the external medium. We find that the damping rate of oscillations on globules in dense molecular environments is always many periods, corresponding to hundreds of thousands of years, and persisting over the inferred lifetimes of the globules. author: - 'Avery E. Broderick, Ramesh Narayan, Eric Keto & Charles J. Lada' title: The Damping Rates of Embedded Oscillating Starless Cores --- Introduction ============ The small dark molecular clouds known as starless cores are are significant in the interstellar medium as the potential birthplaces of stars [review by @BerginTafalla2007]. As their name implies, the starless cores do not yet contain stars, but their properties are nearly the same as similar small clouds that do [@MyersLinkeBenson1983; @MyersBenson1983; @BensonMyers1989]. Furthermore, there is a compelling similarity in the mass function of the starless cores [@Lada2008] and the initial mass function (IMF) of stars. Observations of Bok globules (Bok 1948) and starless cores [@Ward-Thompson1994; @Tafalla1998; @LeeMyers1999; @LeeMyersTafalla2001; @AlvesLadaLada2001; @Ward-ThompsonMotteAndre1999; @Bacmann2000; @Shirley2000; @Evans2001; @ShirleyEvansRawlings2002; @Young2003; @Tafalla2004; @Keto2004] suggest that many of these small (M $< 10$ M$_\odot$) clouds are well described as quasi-equilibrium structures supported mostly by thermal pressure, approximately Bonnor-Ebert (BE) spheres [@Bonnor1956]. Observed molecular spectral lines [@Zhou1994; @Wang1995; @Gregersen1997; @Launhardt1998; @GregersenEvans2000; @LeeMyersTafalla1999; @Williams1999; @Lada2003; @LeeMyersPlume2004; @LeeBerginEvans2004; @Keto2004; @Crapsi2004; @Sohn2004; @Aguti2007] show complex profiles that further suggest velocity and density perturbations within these cores. We have previously shown that in at least one case, B68, these profiles can be produced by the non-radial oscillations of an isothermal sphere [@Keto2006; @Broderick2007]. To match the spectral line profiles, we have previously found that the oscillations have to be large enough (25 %) that the amplitudes are non-linear [@Keto2006]. If such large amplitude oscillations are to be a viable explanation for the observed complex velocity patterns then their decay rate must be no faster than the sound-crossing or free-fall times of the globules, $10^{5-6}$ yrs. Numerical simulations in which the external medium is substantially less dense than the gas inside the core have found that mode damping is sufficiently slow that modes will persist for many periods [@Broderick2007]. However, it is not clear that this description is appropriate for many starless cores. B68 is one of only a handful of cores surrounded by hot, rarefied gas in the Pipe Nebula, where the vast majority of cores, like those in the Taurus & Perseus molecular clouds, appear to be embedded in cold, dense molecular gas [@Lada2008; @Goldsmith2008]. Embedded oscillating clouds will generally act as sonic transducers, exciting sound waves in the external medium and thereby loosing energy. A simple one-dimensional analysis, discussed in detail in the Appendices, suggests that when the density contrast between the core interior and its bounding medium is close to unity this process can be very efficient, potentially limiting the lifetime of oscillations. In this paper we investigate the damping rate of oscillations of isothermal spheres as a function of the density contrast with the external bounding medium. We do this using numerical simulations of a large amplitude oscillation superimposed upon an isothermal sphere. To evaluate our results we also derive analytical and semi-analytical estimates of the damping rate associated with the excitation of sound waves in the exterior medium. A brief discussion of the numerical simulation is discussed in section \[sec:NHM\], the presentation and discussion of the numerical results are in section \[sec:RaD\], and conclusions can be found in section \[sec:C\]. The details of the analytical and semi-analytical computation of the damping rates of small amplitude oscillations are relegated to the appendices. We find that a reduced density contrast between the core and the exterior bounding medium does result in increased dissipation of oscillations as momentum and energy are transferred out of the core through the boundary. However, if the oscillations are non-radial the dissipation through the boundary is always less than the dissipation rate due to mode-mode coupling. Numerical Hydrodynamic Model {#sec:NHM} ============================ ![The initial, unperturbed density profile of the embedded isothermal sphere. The density contrast for each is roughly 50 (black), 25 (blue), 10 (green) and 4 (red). This color scheme will be consistent throughout the other figures.[]{data-label="fig:ds"}](f1.eps){width="\columnwidth"} [cccc]{} black & $10^6$ & 0.017 & 8.5\ blue & $300$ & 0.037 & 6.5\ green & $10$ & 0.087 & 4.8\ red & $3$ & 0.22 & 3.44\ Because the thermal gas heating and cooling time, via collisional coupling to dust at high densities [@Burke1983] and molecular line radiation at low densities [@Goldsmith2001], is short in comparison to the typical oscillation period (on the order of $10^5\,{{\rm yr}}$), the oscillations of dark-starless cores are well modeled with an isothermal equation of state [@KetoField2005]. In our simulations, we use a barotropic equation of state with an adiabatic index of unity (as opposed to $5/3$, for example), in which case the isothermal evolution is also adiabatic. This allows us to replace the energy equation with a considerably simpler adiabatic constraint. Since the unbounded isothermal gas sphere is unstable, we truncate the solution at a given center-to-edge density contrast ($\rho(R)/\rho_c\simeq12$, not to be confused with the interior/exterior density contrast), producing a stable Bonnor-Ebert sphere solution with radius ($R$) and mass determined by the central density ($\rho_c$) and temperature ($T$). This is done by inducing a phase change in the equation of state, [[i.e.]{}]{}, $$P = \left\{ \begin{array}{rl} \rho \frac{kT}{\mu m_p} & \mbox{if } \rho\ge\rho(R)\,,\\ \left[ \frac{\rho}{\rho(R)} \right]^\epsilon \rho(R) \frac{k T}{\mu m_p} & \mbox{if } \rho(R) > \rho \ge \beta\rho(R)\frac{T}{T_{\rm ext}}\,,\\ \rho \frac{kT_{\rm ext}}{\mu m_p} & \mbox{otherwise}\,, \end{array} \right.$$ where the intermediate state, with $\beta=0.8$ and $\epsilon = \log(\beta)/\log(\beta T/T_{\rm ext}) \ll 1$ chosen to make $P(\rho)$ continuous, is introduced to avoid numerical artifacts at the surface, and $T_{\rm ext}\gg T$. The specific values of $T_{\rm ext}$ that we chose, their corresponding density contrasts and decay timescales, are presented in Table \[tab:T\]. The associated radial density profiles are shown in Figure \[fig:ds\]. Note that there is a transition region which distributes the sudden drop in density over many grid zones, though the pressure is continuous (necessarily) and nearly constant outside the surface of the Bonnor-Ebert sphere. We employ the same three-dimensional, self-gravitating hydrodynamics code as in @Broderick2007 to follow the long term mode evolution. Details regarding the numerical algorithm and specific validation for oscillating gas spheres (in that case a white dwarf) can be found in @BroderickRathore2006, and thus will only be briefly summarized here. The code is a second-order accurate (in space and time) Eulerian finite-difference code, and has been demonstrated to have a low diffusivity. Because the equation of state is barotropic, we use the gradient of the enthalpy instead of the pressure in the Euler equation since this provides better stability in the unperturbed configuration [@BroderickRathore2006]. The Poisson equation is solved via spectral methods, with boundary conditions set by a multipole expansion of the matter on the computational domain. As described in @Broderick2007, the code was validated for this particular problem finding convergence by a resolution of $128^3$ grid zones. The initial perturbed states are computed in the linear approximation using the standard formalism of adiabatic [which in this case is also isothermal, @Keto2006] stellar oscillations [@Cox1980]. The initial conditions of the perturbed gas sphere are then given by the sum of the equilibrium state and the linear perturbation. We employ the same definition for the dimensionless mode amplitude as given in the Appendix of @Keto2006. Throughout the evolution, the amplitude of a given oscillation mode, denoted by its radial ($n$) and angular ($l$ & $m$) quantum numbers, may be estimated by the explicit integral: $$A_{nlm} = \omega_{nlm}^{-1}\int {{\rm d}}^3\!x \,\rho\, {\mbox{\boldmath$\rm v$}}\cdot{\mbox{\boldmath$\rm \xi$}}_{nlm}^\dagger\,, \label{eq:amp}$$ where ${\mbox{\boldmath$\rm v$}}$ is the gas velocity, ${\mbox{\boldmath$\rm \xi$}}_{nlm}$ is the mode displacement eigenfunction, $\omega_{nlm}$ is the mode frequency and the dagger denotes Hermitian conjugation.[^1] Results & Discussion {#sec:RaD} ==================== In principle, a number of hydrodynamic mechanisms exist by which the oscillations can be damped. In the absence of a coupling to an external medium, these are dominated by nonlinear mode–mode coupling, in which large scale motions excite smaller scale perturbations [@Broderick2007]. However, when embedded in a dense external medium it is possible for the mode to damp by exciting motions in the exterior gas. That is, it is possible for the oscillating sphere to act as a transducer, generating sound waves in the external gas which then propagate outward resulting in a net radial energy flux and damping the oscillations. Generally, this will be a strong function of the density contrast between the interior of the core and the exterior medium. If the external density has a low density, and thus little inertia, outwardly moving sound waves will contain little energy density despite their large amplitude and thus inefficiently damp the mode (as is the case for B68). Similarly, if the exterior medium has a very high density the excited outgoing waves will have small amplitudes, which despite the large gas inertia will again carry away only a small amount of energy. Conversely, when the density contrast is nearly unity, there will be an efficient coupling between the interior and exterior waves, resulting in a rapid flow of energy out of the cloud. This may be made explicit via the standard three-wave analysis, in which propagating wave solutions for the incident, reflected and transmitted waves on either side of the density continuity are inserted into the continuity and Euler equations, which can subsequently be solved for the reflected and transmitted energy flux. In this idealization the ratio of the reflected to transmitted flux is $4\zeta/(1+\zeta)^2$, where $$\zeta\equiv\rho_e c_{s,e} / \rho_i c_{s,i} = \sqrt{\rho_e/\rho_i}\,,$$ is the ratio of the sonic impedances on either side of the discontinuity [@LandauLifshitz1987]. As anticipated, when the density of the external gas is comparable to the surface density of the isothermal sphere we may expect efficient transmission of sound waves from the interior to the exterior ([[i.e.]{}]{}, conversion of the oscillation into traveling waves in the exterior), and therefore efficient damping of the pulsations. This conclusion is also supported by a 1D analysis of the decay of a standing wave confined to a high density region (section \[sec:1DT\]), for which the damping rate, $\gamma$ is given by $$\gamma = \frac{c_{s,i}}{2 R}\ln\left(\frac{|1-\zeta|}{1+\zeta}\right) \simeq \frac{c_{s,i}}{R} \zeta \,,$$ where $c_{s,i}$ is the sound speed in the interior and $R$ is the radius of the high-density region. Thus, as expected lower external densities (corresponding to smaller $\zeta$) have lower damping rates and larger damping timescales. Note that when $\zeta$ is of order unity the wave decays on the sound crossing time of the dense region, as would be expected if, e.g., we were to decompose the standing wave into traveling waves. More directly applicable to the damping of oscillating gas spheres is the case of the decay of a standing multipolar sound wave in a uniform density sphere, discussed in section \[sec:SDT\]. This is distinct from the 1D case in an important way: the oscillation wave vectors are no longer only normal to the boundary. As a consequence, the damping rate is a strong function of not only $\zeta$ but also the multipole structure of the underlying oscillation. In particular, the damping rate for the $l^{\rm th}$ multipole mode is proportional to $\zeta^{2l+3}$ for small $\zeta$ (see, eqs. \[eq:uniform\_sphere\_damping\] & \[eq:full\_linear\_damping\]), which is a very strong function of $\zeta$ for even low $l$! This limiting behavior is also found in semi-analytic calculations (section \[sec:FLP\]) in which the linear oscillation of an isothermal sphere is properly matched to outgoing sound waves in the exterior. The damping timescales (inverse of the damping rates) from both the uniform-sphere approximation and the full linear mode analysis are shown in Figure \[fig:tA\] for the first few multipoles. All of this implies that the damping rates of oscillations of cores embedded in cold molecular regions, where the density contrast between the core and the cloud is near unity, will be much more rapid than those in cores isolated in hot, lower-density regions. ![The damping timescales in units of the dynamical time of the cloud, $(GM/R^3)^{-1/2}$, as a function of the ratio of the exterior and surface densities. For reference the corresponding damping times in ${{\rm M}{{\rm yr}}}$ is shown on the right-hand axis for a cloud similar to Barnard 68. The solid lines show the damping timescale predicted by a linear mode analysis of the isothermal sphere for monopole (011), dipole (111) and quadrupole (122,thick) modes (see the appendices). The asymptotic behavior of the linear mode analysis is shown by the short dashed lines for each, which may be obtained by treating the oscillations of the isothermal sphere as sound waves in a uniform density sphere (see the appendices). The solid circles show the damping timescales measured via numerical simulations, and are colored to match the curves in Figures \[fig:ds\] & \[fig:sims\]. Finally, we show a power-law fit to the measured damping timescales, which have a power-law index of $-0.37$, and is considerably flatter than we might have expected (though these represent lower limits upon the decay timescale). For comparison the density contrasts observed in some well known examples are also shown, including those for Barnard 68 ($10^{-5}$), L1544 ($10^{-2}$) and the cores in the Pipe Nebula ($0.1$–$0.7$).[]{data-label="fig:tA"}](f2.eps){width="\columnwidth"} ![The mode evolution for various external gas models. The density contrast for each is roughly 50 (black), 25 (blue), 10 (green) and 4 (red), and the lines are colored to match those in Figs. \[fig:ds\] & \[fig:tA\]. The decay time for the red curve is 0.43 Myr, compared to 1.05 Myr for the black curve. Quadratic fits for each are shown by the dashed lines. Matching these up to specific models of the mode decay is complicated by (i) the non-linear mode dynamics, and (ii) understanding the energy losses due to traveling waves in the external medium (see the appendices).[]{data-label="fig:sims"}](f3.eps){width="\columnwidth"} However, this is not borne out by numerical simulations of high amplitude oscillations. This can readily be seen by the damping timescales measured via the numerical simulations. Figure \[fig:sims\] shows the evolution of the $n=1$, $l=2$, $m=2$ mode discussed in @Broderick2007. In all cases the initial amplitude was 0.25, and the evolution of the mode well fit by a decaying exponential. Oscillations on isothermal cores embedded in higher density regions did indeed damp more rapidly, as is apparent in the figure. Explicit values of the decay timescale are presented in Table \[tab:T\], though as in @Broderick2007 these should be seen as lower limits only due to coupling in the artificial atmosphere. These are plotted as a function of the surface density contrast in Figure \[fig:tA\], shown by the colored filled circles. For reference, we show the approximate surface density contrasts of Barnard 68 [@AlvesLadaLada2001], L1544 [@Ward-ThompsonMotteAndre1999] and typical for cores in the Pipe Nebula [@Lada2008]. While there is a clear power-law dependence of the damping timescale upon $\zeta$, with $\tau_A \propto (\rho_e/\rho_i)^{-0.37}$ this dependence is considerably weaker than that predicted by the linear analysis ($\tau_A\propto(\rho_e/\rho_i)^{-3.5}$). This disparity may be understood in terms of the relative importance of the excitation of sound waves in the exterior medium to the damping of the oscillations. In particular, it is notable that the numerically measured damping timescales are always less, and for most surface density contrasts considerably so, than those implied by the linearized analysis of external sound wave excitation. This is true even with low amplitude oscillations ([[e.g.]{}]{}, $A=10^{-4}$, for which the damping timescale is roughly a factor of 3 larger than for $A=0.25$ ) and thus is likely due to mode coupling in the transition region immediately outside the cloud (seen in Fig. \[fig:ds\] immediately outside $r/R=1$ as the region of rapidly decreasing density). Within this region the densities are all within a single order of magnitude despite their very different asymptotic densities at infinity (which differ by many orders of magnitude). As a consequence, the damping rates inferred from the isolated isothermal spheres are only a weak function of the properties of the external medium. Thus, even for surface density contrasts on the order of $1/4$, the decay timescale for the quadrupolar oscillation is still larger than $4\times10^5\,{{\rm yr}}$, corresponding to many oscillations and comparable to the inferred lifetimes of globules. Conclusions {#sec:C} =========== Despite the expectations of the linear mode analysis of the embedded isothermal sphere, the damping timescale of large-amplitude oscillations of embedded Bonnor-Ebert spheres is only a weak function of the density of the external medium. This is a result of the dominance of nonlinear mode–mode coupling in the damping of large oscillations. Even in cold molecular environments, the quadrupolar oscillation discussed in @Keto2006 and @Broderick2007 has a lifetime of at least $0.4\,{{\rm M}{{\rm yr}}}$, and is thus comparable to the inferred lifetimes of globules [@Lada2008]. This suggests that globules supporting large-amplitude oscillations may be common even in these environments. The presence of large-amplitude oscillations on starless cores has a number of consequences for the physical interpretation of observations of starless cores [@Keto2006; @Broderick2007]. Necessarily they would imply that starless cores are stable objects, existing for many sound-crossing times. In addition, the signatures of collapse, expansion and rotation in observations of self-absorbed molecular lines ([[e.g.]{}]{}, CS) are degenerate with molecular line profiles produced by oscillating globules [see, [[e.g.]{}]{}, figure 6 of  @Broderick2007]. Consequently, it may be difficult to determine the true dynamical nature of motions in a given globule or dense core from self-absorbed, molecular-line profiles alone. This suggests that studies of such profiles in dense cores may be of only statistical value in determining the general status of motions in dense core populations. We would like to thank Mark Birkinshaw for bringing the problem of damping starless core pulsations in dense media to our attention. Damping of an Oscillating Cavity ================================ We will first review some basic facts about sound waves in uniform media. Beginning in subsection \[sec:1DT\] we will discuss the evolution of a one-dimensional standing sound wave as a result of the excitation of waves in the external medium. In subsection \[sec:SDT\] we discuss the application to oscillating uniform density spheres, and treat the full linearized mode analysis of isothermal spheres in section \[sec:FLP\]. Equations of Motion ------------------- The governing equations are the linearized continuity and the Euler equations: $$\begin{aligned} \dot{\rho'} + {\mbox{\boldmath$\rm \nabla$}}\cdot \rho_0 {\mbox{\boldmath$\rm \delta v$}} &= 0\,,\nonumber\\ \dot{{\mbox{\boldmath$\rm \delta v$}}} + \frac{c_s^2}{\rho_0} {\mbox{\boldmath$\rm \nabla$}} \rho' &= 0\,,\end{aligned}$$ where $\rho_0$ and $c_s$ are the unperturbed density and sound speed, respectively, $\rho'$ is the Eulerian perturbation in the density and ${\mbox{\boldmath$\rm \delta v$}}$ is the Lagrangian perturbation in the velocity (which is identical to the Eulerian perturbation since the initial velocity field is assumed to vanish). These may be combined in the normal way to produce the wave equation $$\ddot{\rho'} - c_s^2 \nabla^2 \rho' = 0\,, \label{eq:rho_wave_eq}$$ where we have assumed that the background density is uniform. Solutions will generally obey the dispersion relation $k^2 = c_s^2 \omega^2$, and may then be inserted into the Euler equation to obtain the associated velocity perturbation. Decay of a One-Dimensional Sound Wave {#sec:1DT} ------------------------------------- We will consider the decay of a sound wave traveling inside of a uniform high density region (called the [*interior*]{} and denoted by sub-script $i$’s) surrounded by a uniform low density region (called the [*exterior*]{} and denoted by sub-script $e$’s). The interior wave oscillation is given by $$\rho'_i = A_i \rho_i \sin(k_i x) {{\rm e}}^{-i\omega t} \quad{\rm and}\quad \delta v^x_i = - i A_i c_{s,i} \cos(k_i x) {{\rm e}}^{-i\omega t}\,.$$ The exterior waves are outgoing, and we will focus upon the boundary condition at the $+R$ boundary. At this point the exterior sound wave is given by $$\rho'_e = A_e \rho_e {{\rm e}}^{i k_e x - i\omega t} \quad{\rm and}\quad \delta v^x_e = A_e c_{s,e} {{\rm e}}^{i k_e x - i\omega t}\,.$$ At the interface we require (i) pressure equilibrium and (ii) continuity of the displacement and hence velocity. Since we have assumed the $\rho_i$ and $\rho_e$ are constant, the first condition is simply $\rho'_i c_{s,i}^2 = \rho'_e c_{s,e}^2$. The second is trivially $\delta v^x_i = \delta v^x_e$. Thus $$A_i \rho_i c_{s,i}^2 \sin(k_i R) {{\rm e}}^{-i\omega t} = A_e \rho_e {{\rm e}}^{i k_e R - i\omega t} \quad{\rm and}\quad - i A_i c_{s,i} \cos(k_i R) {{\rm e}}^{-i\omega t} = A_e c_{s,e} {{\rm e}}^{i k_e R - i\omega t}\,,$$ give two complex equations from which we may determine $A_e$ and $\omega$ as functions of $A_i$ and the background fluid quantities (note that $k_i = \omega/c_{s,i}$ and $k_e = \omega/c_{s,e}$ by the dispersion relations in each region). In particular, note that by taking the ratio of the two equations we find $$\tan(k_i R) = - i \zeta\,, \label{eq:1Dmatching}$$ where $\zeta\equiv \rho_e c_{s,e}/\rho_i c_{s,i} = \sqrt{\rho_e/\rho_i}$ is the ratio of the exterior and interior sonic impedances. Generally this may be solved to find that $k_i R = n\pi + (i/2) \ln\left[|1-\zeta|/(1+\zeta)\right]$, however we will solve this explicitly in the $\zeta \ll 1$ limit to illustrate how we will do this for spherical geometries later. Let us begin by assuming that the damping rate is small. Specifically, let us assume that $\omega_0\equiv\Re(\omega)$ is much larger than $\gamma\equiv-\Im(\omega)$. Then we may Taylor expand the left-hand side of eq. (\[eq:1Dmatching\]) around $\gamma=0$: $$\tan(k_i R) = \tan\left(\frac{\omega_0 R}{c_{s,i}}\right) - \left.\frac{\partial\tan}{\partial \omega}\right|_{\omega=\omega_0} i \gamma + \dots = 0 - i\zeta\,,$$ and thus, equating real and imaginary parts $$\tan\left(\frac{\omega_0 R}{c_{s,i}}\right) = 0 \quad\Rightarrow\quad\omega_0 = n \pi \frac{c_{s,i}}{R}$$ and $$\left[ 1 + \tan^2\left(\frac{\omega_0 R}{c_{s,i}}\right) \right] \frac{\gamma R}{c_{s,i}} = \zeta \quad\Rightarrow\quad \gamma = \frac{c_{s,i}}{R} \zeta\,.$$ We may in principle then insert this into eq. (\[eq:1Dmatching\]) to obtain the relationship between $A_i$ and $A_e$. However, we will be concerned with only the damping rate here. Damping Timescale of an Oscillating Sphere {#sec:SDT} ------------------------------------------ In the previous section we discussed the simple problem of the damping of a one-dimensional wave due to the excitation of outgoing exterior sound waves. In this section we will address the more relevant problem of the damping timescale of multipolar oscillations on a sphere. While generally we would use the linearized analysis of the oscillating isothermal sphere, here we will limit ourselves to the spherical analog of the previous section: a sound wave in a uniform density sphere surrounded by a uniform density exterior. Before we can discuss the excitation of sound waves outside of an oscillating sphere we must first determine the explicit form of these waves. We do this by separating the radial and angular dependencies using spherical harmonics (primarily because these were used in determining the mode spectrum of the isothermal sphere). That is, we let $\rho' = {{\rm e}}^{-i\omega t} \mathcal{R}(r) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})$, and insert this into equation (\[eq:rho\_wave\_eq\]), producing $$\frac{1}{r} \frac{\partial^2}{\partial r^2} r \mathcal{R} - \frac{l(l+1)}{r^2} \mathcal{R} + k^2 \mathcal{R} = 0\,,$$ where $k\equiv\omega/c_s$. The general solutions of this equation are simply the spherical Bessel functions: $$\mathcal{R}(r) = a j_l\left(kr\right) + b n_l\left(kr\right)\,.$$ However, we must still separate the inward and outward traveling waves. This naturally occurs if we consider spherical Bessel functions of the 3rd kind, which are necessarily simply linear combinations of spherical Bessel functions of the 1st and 2nd kind ($j_l$ and $n_l$, respectively): $$h_l^{(1,2)}(z) \equiv j_l(z) \pm i n_l(z) \propto \frac{{{\rm e}}^{\pm iz}}{z^{l+1}}\,.$$ Outwardly directed waves are given by $h_l^{(1)}(z)$ and inwardly directed waves are given by $h_l^{(2)}(z)$. In terms of this the density perturbation of the exterior sound waves associated with the $l,m$ multipole is $$\rho'_e = A_e \rho_e {{\rm e}}^{-i\omega t} h_l^{(1)}(k_e r) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}}) \,,$$ where $A_e$ is a dimensionless wave amplitude. For $l=0$ this results in the standard spherical wave solutions, $\propto {{\rm e}}^{\pm ikr}/r$. The velocity perturbation may be determined via the linearized Euler equation: $$\delta v^r_e = - i \frac{c_s^2}{\omega \rho_e} \frac{\partial\rho'_e}{\partial r} = - i A_e c_{s,e} {{\rm e}}^{-i\omega t} \frac{\partial h_l^{(1)}}{\partial z_e}(z_e) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})\,, \label{eq:sph_dv}$$ where here and henceforth we have defined $z_e\equiv k_e r$ for convenience. In contrast, the interior wave must be regular at the origin, and is given by $$\rho'_i = A_i \rho_i {{\rm e}}^{-i\omega t} j_l(k_i r) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}}) \quad{\rm and}\quad \delta v^r_i = - i A_i c_{s,i} {{\rm e}}^{-i\omega t} \frac{\partial j_l}{\partial z_i}(z_i) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})\,,$$ where $z_i \equiv k_i r = z_e/\zeta$. At the interface we again require pressure equilibrium and continuity in the radial velocity: $$A_i \rho_i c_{s,i}^2 {{\rm e}}^{-i\omega t} j_l(k_i r) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}}) = A_e \rho_e c_{s,e}^2 {{\rm e}}^{-i\omega t} h_l^{(1)}(k_e r) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})$$ and $$- i A_i c_{s,i} {{\rm e}}^{-i\omega t} \frac{\partial j_l}{\partial z_i}(z_i) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})\,. = - i c_{s,e} {{\rm e}}^{-i\omega t} \frac{\partial h_l^{(1)}}{\partial z_e}(z_e) Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})$$ As before, together with $\omega=k_i/c_{s,i}=k_e/c_{s,e}$ and $z_e=\zeta z_i$, these are sufficient to determine $A_e$ and $\omega$. In particular, these imply $$\frac{\partial \ln j_l}{\partial z_i} (z_i) = \frac{1}{\zeta} \frac{\partial \ln h^{(1)}_l}{\partial z_e} (z_e)$$ The right-hand side may be simplified in the $z_e \ll 1$ limit. Since $z_i$ is typically of order unity, this implies that $\zeta\ll 1$. In this regime, to leading order in small $z_e$, $$\frac{1}{\zeta} \frac{\partial \ln h^{(1)}_l}{\partial z_e} (z_e) \simeq -\frac{l+1}{z} + i \frac{z_e^{2l}}{\left[(2l-1)!!\right]^2}\,,$$ where $(2l-1)!! = (2l-1)\cdot(2l-3)\cdots 5 \cdot 3 \cdot 1$. Again let us begin with the ansatz that the damping rate is small for small $\zeta$. In which case $$\left.\frac{\partial \ln j_l}{\partial z_i}\right|_{\omega_0} - \left.\frac{\partial^2 \ln j_l}{\partial z_i^2}\right|_{\omega_0} i\frac{\gamma R}{c_{s,i}} + \dots = -\frac{l+1}{\zeta^2 z_i} + i \frac{\zeta^{2l-1} z_i^{2l}}{\left[(2l-1)!!\right]^2}\,,$$ and thus $$\begin{aligned} &&\displaystyle \left.\frac{\partial \ln j_l}{\partial z_i}\right|_{\omega_0} = \left[ \frac{j_{l-1}(z_i)}{j_{l}(z_i)} - \frac{l+1}{z_i} \right]_{\omega_0} = -\frac{l+1}{\zeta^2 z_i}\nonumber\\ &&\displaystyle \qquad\qquad\qquad\qquad\qquad\quad\Rightarrow\quad \frac{j_{l-1}(z_i)}{j_{l}(z_i)} \simeq -\frac{l+1}{\zeta^2 z_i} \quad{\rm and}\quad \omega_0 \simeq \frac{c_{s,i}}{R} \mathcal{Z}_{nl}\end{aligned}$$ where $\mathcal{Z}_{nl}$ is the $n^{\rm th}$ root of $j_l(z)$. With $$\left.\frac{\partial^2 \ln j_l}{\partial z_i^2}\right|_{\omega_0} \simeq -\left[\frac{j_{l-1}(z_i)}{j_{l}(z_i)}\right]^2 \simeq - \frac{(l+1)^2}{\zeta^4 \mathcal{Z}_{nl}^2}\,,$$ (where we used the fact that $\zeta\ll 1$) we find $$\gamma = \frac{c_{s,i}}{R} \frac{\zeta^{2l+3} \mathcal{Z}_{nl}^{2l+2}}{(l+1)^2 \left[(2l-1)!!\right]^2}\,. \label{eq:uniform_sphere_damping}$$ We note that this is quite different than the one-dimensional case. In particular, the decay rate decreases rapidly with decreasing $\zeta$, and is a strong function of $l$, with higher multipoles decaying considerably more rapidly than lower multipoles. This is a consequence of our assumption that not just $\zeta \ll 1$, but $\zeta z_i \ll 1$ (justified in our case), which implies that $\lambda_e \gg R$. That is, the exterior propagating sound wave necessarily knows that the geometry is converging, with higher multipoles converging more rapidly. We should also emphasize that the scaling of the damping rate, $\gamma$, with $\zeta$ is dependent primarily upon the structure of the traveling sound waves in the exterior. Rather, the properties of the interior perturbation are encoded in the definition of the $\mathcal{Z}_{nl}$, and thus the normalization. Therefore, the damping rates associated with the linearized perturbations of the pressure-supported isothermal sphere (Bonnor-Ebert sphere) should be quite similar, differing only in the particular values of $\mathcal{Z}_{nl} \simeq 2\pi R/\lambda_i$. Indeed, in the next section we find this to be the case. Damping of Oscillating Isothermal Spheres in the Linear Regime {#sec:FLP} ============================================================== The rather surprisingly strong scaling of the damping timescale with the density contrast motivates a more careful analysis of the damping rates of a self-gravitating, pressure supported isothermal sphere. The mode analysis of an isothermal sphere has been treated in considerable detail elsewhere [[[e.g.]{}]{}, @Cox1980; @Keto2006] and thus we summarize it only briefly here. The perturbed quantities are most conveniently described by the Dziembowski variables [@Dziembowski1971] $${\mbox{\boldmath$\rm \eta$}}({\mbox{\boldmath$\rm r$}}) = \left(\frac{\delta r}{r} , \frac{P'+\rho_0 \psi'}{\rho_0 g r} , \frac{\psi'}{g r} , \frac{1}{g}\frac{\partial \psi'}{\partial r} \right)\,,$$ which may be separated into radial and angular parts, $\eta_i ({\mbox{\boldmath$\rm r$}}) = \eta_i (r) {{\rm e}}^{-i\omega t} Y_{lm}(\hat{{\mbox{\boldmath$\rm r$}}})$. In terms of these, the linearized hydrodynamic equations are given by $$\begin{aligned} &&r \frac{\partial \eta_1}{\partial r} = (V-3) \eta_1 + \left[\frac{l(l+1)}{\sigma^2 C} - V\right] \eta_2 + V \eta_3 \label{eq:dz_a}\\ &&r \frac{\partial \eta_2}{\partial r} = \sigma^2 C \eta_1 + (1-U)\eta_2\\ &&r \frac{\partial \eta_3}{\partial r} = (1-U) \eta_3 + \eta_4\\ &&r \frac{\partial \eta_4}{\partial r} = U V \eta_2 + \left[ l(l+1) - UV \right] \eta_3 - U \eta_4\,, \label{eq:dz_eqs}\end{aligned}$$ where $$U \equiv \frac{\partial \ln r^2 g}{\partial \ln r} \,,\quad V \equiv - \frac{\partial \ln P_0}{\partial \ln r} \,,\quad C \equiv -\frac{G M}{r^2 g} \left(\frac{r}{R}\right)^3 \,,\quad \sigma^2 \equiv \frac{R^3\omega^2}{G M}\,,$$ are functions of the background equilibrium configuration. These are solved subject to a normalization condition $\eta_1(R)=1$ and the inner boundary conditions: $$\sigma^2 C \eta_1 \big|_{r=0} = l \eta_2 \big|_{r=0} \qquad{\rm and}\qquad \eta_4\big|_{r=0} = l \eta_3\big|_{r=0}\,,$$ arising from regularity at the origin. A third boundary condition is a result from the continuity of the gravitational potential at the sphere’s surface, $$\eta_4\big|_{r=R} = -(l+1) \eta_3\big|_{r=R}\,.$$ If the exterior density vanishes, as was assumed in @Keto2006, the final boundary condition is given by requiring that the Lagrangian pressure perturbation, $$\delta P = P' + {\mbox{\boldmath$\rm \delta r$}}\cdot{\mbox{\boldmath$\rm \nabla$}} P_0 = \rho_0 g r \left( \eta_2 - \eta_3 - \eta_1 \right) = 0\,.$$ However, we now properly match this onto the outgoing multipolar wave solutions described in section \[sec:SDT\]. Namely, $$\delta P = \rho'_e c_{s,e}^2 = A_e \rho_e h^{(1)}_l (z_e)\,,$$ subject to $$\delta r = \eta_1 R = \frac{c_{s,e}^2}{\omega^2 \rho_e} \frac{\partial\rho'_e}{\partial r} = A_e \frac{c_{s,e}}{\omega} \frac{\partial h^{(1)}_l}{\partial z_e}\,.$$ Therefore, $$\left. \eta_2 - \eta_3 - \eta_1\right|_{r=R} = \left.\eta_1 \frac{h^{(1)}}{\partial h^{(1)}/\partial z_e} \frac{\rho_e c_{s,e} \omega}{\rho_0 g}\right|_{r=R}\,.$$ As before we assume that the damping rate is small, set $\omega=\omega_0 - i\gamma$ and employ the boundary condition $$\left. \eta_2 - \eta_3 - \eta_1 \right|_{r=R} = \Re\left[ \eta_1 \frac{h^{(1)}}{\partial h^{(1)}/\partial z_e} \frac{\rho_e c_{s,e} \omega}{\rho_0 g} \right]_{r=R, \omega=\omega_0}\,.$$ The damping rate is then estimated by $$\gamma = -\Im\left[ \eta_1 \frac{h^{(1)}}{\partial h^{(1)}/\partial z_e} \frac{\rho_e c_{s,e} \omega}{\rho_0 g} \right]_{r=R, \omega=\omega_0} \bigg/ \left. \frac{\partial(\eta_2-\eta_3)}{\partial\omega} \right|_{r=R}\,,$$ where we used the fact that $\partial\eta_1/\partial\omega = 0$ by the normalization condition. In practice $\partial(\eta_2-\eta_3)/\partial\omega$ is evaluated by solving the eigenvalue problem for the $\eta_i$ and $\sigma^2$ using $$\left. \eta_2 - \eta_3 - \eta_1 \right|_{r=R} = \alpha \Re\left[ \eta_1 \frac{h^{(1)}}{\partial h^{(1)}/\partial z_e} \frac{\rho_e c_{s,e} \omega}{\rho_0 g} \right]_{r=R, \omega=\omega_0} \bigg/ \left. \frac{\partial(\eta_2-\eta_3)}{\partial\omega} \right|_{r=R}\,,$$ and setting $$\frac{\partial(\eta_2-\eta_3)}{\partial \omega} = \frac{\partial(\eta_2-\eta_3)/\partial\alpha}{\partial\omega/\partial\alpha}\,.$$ This amounts to finding $\partial(\eta_2-\eta_3)/\partial\omega$ holding the other boundary conditions constant. The results of this procedure are explicitly shown in Figure \[fig:tA\]. Of particular note is that the intuition obtained from the analysis of sound waves in a uniform density sphere is borne out in the low $\rho_e/\rho_0(R)$ limit, for which $$\gamma \propto \left(\frac{\rho_e}{\rho_0(R)}\right)^{(2l+3)/2} = \zeta^{2l+3} \,. \label{eq:full_linear_damping}$$ Aguti, E.D., Lada, C.J., Bergin, E.A., Alves, J.F., Birkinshaw, M., 2007, , 665, 457 Alves, J., Lada, C., & Lada, E. 2001, , 409, 159 Bacmann, A., Andre, A.P., Puget, J.-L., Abergel, A., Bontemps, S., & Ward-Thompson, D. 2000, å, 361, 555 Benson, P., & Myers, P. 1989, , 307, 337 Bergin, E. A., & Tafalla, M., 2007, , 45, 339 Bonnor, W., 1956, , 116, 351 Broderick, A. E., Keto E., Lada, C. J. & Narayan, R., 2007, , 000, 000 Broderick, A. E. and [Rathore]{}, Y., 2006, , 372, 923 Burke, J. R. & Hollenbach, D. J., 1983, , 265, 223 Cox, J.P., 1980, Theory of Stellar Pulsations, Princeton University Press: Princeton Crapsi, A., Caselli, P., Walmsley, C.M., Tafalla, M., Lee, C.W., Bourke, T.L., Myers, P.C., 2004, å, 420, 957 Dziembowski, W., 1971 Acta Astron., 21, 289 Evans, N., Rawlings J., Shirley, Y., & Mundy, L. 2001, , 557, 193 Goldsmith, P., 2001, , 557, 736 Goldsmith, P. et al., 2008, [*arXiv:0802.2206v1 \[astro-ph\]*]{} Gregersen, E., Evans, N., Zhou, S., Choi, M., 1997, , 484, 256 Gregersen, E., Evans, N., 2000, , 538, 260 Keto, E., Broderick, A.E., Lada, C.J. & Narayan, R., 2006, , 652, 1366 Keto, E., Rybicki, G., Bergin, E., Plume, R., 2004, , 613, 355 Keto, E. & Field, G., 2005, , 635, 1151 Lada, C., Bergin, E., Alves, J., Huard, T., 2003, , 586, 286 Lada, C. et al., 2008, Landau, L.D. & Lifshitz, E.M., 1987, Course of Theoretical Physics, Vol. 6, 2nd Ed., Butterworth-Heinemann: Oxford Launhardt, R., Evans, N., Wang, Y., Clemens, D., Henning, T., Yun, J., 1998, , 119, 59 Lee, J., Bergin, E., Evans, N., 2004, , 617, 360 Lee, C.W. & Myers, P.C. 1999, , 123, 233 Lee, C.W., Myers, P., Plume, R., 2004, , 153, 253 Lee, C., Myers, P., & Tafalla, M. 1999, , 576, 788 Lee, C., Myers, P., & Tafalla, M. 2001, , 136, 703 Myers, P., & Benson, P. 1983, , 266, 309 Myers, P., Linke, R., & Benson, P. 1983, , 264, 517 Shirley, Y., Evans, N., Rawlings, J., Gregersen, E., 2000, , 131, 249 Shirely, Y. L., Evans, N. J., Rawlings, J. M. C., 2002, , 575, 337 Sohn, J., Lee, C., Lee, H., Park, Y-S., Myers, P., Lee, Y., Tafalla, M., 2005, JKAS, 37, 261 Tafalla, M., Mardones, D., Myers, P. C., Caselli, P., Bachiller, R. & Benson, P. J., 1998, , 504, 900 Tafalla, M., Myers, P., Caselli, P., Walmsley, C. & Comito, C., 2002, , 569, 815 Tafalla, M., Myers, P. C., Caselli, P., Walmsley, C. M., 2004, å, 416, 191 Wang, Y., Evans, N., Shudong, Z., Clemens, D., 1995, , 454, 742 Ward-Thompson, D., Motte, F. & Andre, P. 1999, , 305, 143 Ward-Thompson, D., Scott, P., Hills, R., & Andre, P. 1994, , 268, 274 Williams, J., Myers, P., Wilner, D., Di Francesco, J., 1999, , 513, L61 Young, C., Shirley, Y., Evans, N., Rawlings, J., 2003, , 145, 111 Zhou, S., Evans, N., Wang, Y., Peng, R., Lo, K., 1994, , 433, 131 [^1]: For stellar oscillations the mode amplitudes may be determined from the density perturbation alone using the velocity potential. However, when there is a non-vanishing surface density, as is the case for the Bonnor-Ebert sphere, this is not possible and the velocity field must be used.

--- abstract: 'Atrial fibrillation (AF) increases the risk of stroke by a factor of four to five and is the most common abnormal heart rhythm. The progression of AF with age, from short self-terminating episodes to persistence, varies between individuals and is poorly understood. An inability to understand and predict variation in AF progression has resulted in less patient-specific therapy. Likewise, it has been a challenge to relate the microstructural features of heart muscle tissue (myocardial architecture) with the emergent temporal clinical patterns of AF. We use a simple model of activation wavefront propagation on an anisotropic structure, mimicking heart muscle tissue, to show how variation in AF behaviour arises naturally from microstructural differences between individuals. We show that the stochastic nature of progressive transversal uncoupling of muscle strands (e.g., due to fibrosis or gap junctional remodelling), as occurs with age, results in variability in AF episode onset time, frequency, duration, burden and progression between individuals. This is consistent with clinical observations. The uncoupling of muscle strands can cause critical architectural patterns in the myocardium. These critical patterns anchor micro-re-entrant wavefronts and thereby trigger AF. It is the number of local critical patterns of uncoupling as opposed to global uncoupling that determines AF progression. This insight may eventually lead to patient specific therapy when it becomes possible to observe the cellular structure of a patient’s heart.' author: - 'Kishan A. Manani$^{1,2,3}$, Kim Christensen$^{1,3}$, Nicholas S. Peters$^{2}$' bibliography: - 'References.bib' title: Myocardial Architecture and Patient Variability in Clinical Patterns of Atrial Fibrillation --- A key challenge in the mathematical modelling of diseases is to link microscopic variation in individuals (e.g., genetic, metabolic or tissue structure) to variation in disease outcomes (e.g., the occurrence, recurrence or persistence of AF). In this paper we show how variation in microstructure affects the behaviour of AF, suggesting a single mechanism for the origin of clinically observed variability of AF behaviour. Atrial fibrillation is characterised by the apparently random propagation of multiple activation wavefronts in atrial muscle (myocardium). This gives rise to AF episodes of variable duration. Typically, short self-terminating episodes become longer with time until they do not terminate spontaneously. Current clinical guidelines (ACA/AGA/ESC) define AF by the episode duration as paroxysmal ($< 7$ days), persistent ($> 7$ days), long-standing persistent ($> 1$ year) and permanent (clinical decision to not treat) [@camm2010guidelines]. However, AF episodes will in fact lie on a continuum of durations. The natural history of AF is usually discussed using this classification scheme which by its technical definition allows for progression of paroxysmal to persistent but not the reverse. However, this classification scheme becomes problematic in cases where episodes lasting longer than 7 days terminate and are followed by episodes shorter than 7 days which is observed to occur frequently in patients [@Veasey2015]. Indeed, Sugihara *et al* [@Sugihara2015]. could not consistently apply these guidelines to their continuously monitored patients because AF episode durations were observed to remit from more than 7 days to less than 7 days. Hence, they defined a different classification scheme based on AF burden (fraction of time in AF) to describe their observations rather an arbitrary 7 day cut-off to distinguish between patients. It has been suggested that AF induces atrial electrophysiological changes (e.g., action potential duration shortening) and microstructural changes (e.g., fibrosis or gap junctional uncoupling), which promote further AF. This self-perpetuation has been termed “AF begets AF” [@Wijffels1995]. Whilst fibrosis promotes AF, the quantitative relationship and the mechanism by which fibrosis promotes AF is not fully understood [@DeJong2011]. Sugihara *et al.* [@Sugihara2015] monitored AF patients continuously in a long term study (1031 cumulative patient-years, mean 3.2 years per patient) using dual chamber permanent pacemakers. It was observed that progression to persistent AF was not inevitable, that is, some patients remained paroxysmal for the duration of the long term follow-up and some patients’ AF burden (fraction of time in AF) could remit from 100 % to less than 100 % and relapse to 100% again. Indeed it has been observed that some patients do not progress from paroxysmal to persistent AF, using current clinical guidelines, after as many as 22 years [@Kottkamp2013]. Veasey *et al.* [@Veasey2015] also used continuous monitoring data to show that after a mean 7 year follow up, 35 % of patients that were initially classified as persistent AF using the current clinical guidelines were re-classified as paroxysmal AF. Other research has shown that the time course of AF is seen to vary between patients with similar fibrosis burden: some patients progress rapidly from paroxysmal AF to persistent AF (on the order of months) whilst other patients do not progress at all (measured over decades) [@Kottkamp2013]. Furthermore, patients with a high fibrosis burden can remain paroxysmal and those with low fibrosis burden can be in persistent AF [@Kottkamp2013; @Oakes2009; @Teh2012; @Boldt2004]. Thus we have the following recent clinical observations: (1) AF burden does not inexorably increase and can even spontaneously decrease therefore not all patients appear to progress to persistent AF, (2) persistent AF can remit to paroxysmal AF, (3) the common conception that fibrosis correlates with AF progression needs to be reconciled with the observed variability in AF burden for patients with similar levels of fibrosis. Individually and collectively these studies challenge the contemporary view of how AF evolves. Although it has been suggested that different pathological processes (mitral valve disease, diabetes etc.) occurring in different patients may contribute to variability in AF progression [@Schotten2011], it is fair to say that there is no understanding of what causes the clinical observations summarised above. The clinical patterns of AF are studied in the domain of populations on long time-scales (months to years), whereas the microstructure of myocardium is often studied in “wet labs” within the domain of cellular electrophysiology on short time-scales (form seconds to hours). These two vastly different time scales cannot be related experimentally. Similarly, many models of AF are computationally intensive due to their complexity. As a result only short time periods (seconds or minutes) have been investigated. Thus, these models cannot address questions pertaining to the long time scales of disease progression. Previous work by Chang *et al.* [@Chang2015] has explored the two time scales by modelling AF as a simple binary process which flips between normal sinus rhythm (SR) and arrhythmia at “patient” specific rates. But, this study does not address the question of the microscopic origin of variability in clinical observations. We use a very simple computational model to link the two time domains. The model is specifically designed to address the hypothesis that the stochastic nature of transversal uncoupling is an important factor in the temporal patterns of AF. We model a “patient” by simulating “patient” specific tissue using a simple stochastic process, and then assess the resulting temporal AF patterns. The incidence of AF increases with age and is strongly associated with the accumulation of fibrosis [@DeJong2011]. In this letter we propose that the clinically observed diversity in AF progression can be caused by a single process, the progressive stochastic accumulation of transversal cellular uncoupling. Using a simple computer model we show that different time-courses of AF can occur between patients despite a similar degree of transversal cellular uncoupling. Thus the model provides an explanation to the aforementioned clinical observations, namely that (1) the time-course of AF progression can vary significantly between patients, (2) persistent AF can remit to paroxysmal AF, and (3) macrostructurally similar myocardium can show very different AF behaviour as a result of these microstructural differences. In addition to this, the model identifies specific critical architectural patterns of uncoupling between myocytes as the primary cause of AF induction. When access to the microstructure becomes available in the future this insight has the potential to result in patient-specific therapy. We have previously developed a model in which the activation wavefronts propagate on an anisotropic structure mimicking the branching network of heart muscle cells [@Manani2015] (see supplementary information for a complete description of the model). The tissue is represented by a $L \times L$ square grid of discrete cells where each cell is always coupled to its longitudinal neighbors but with probability $\nu$ to its transversal neighbors. This generates a lattice with anisotropic coupling, mimicking the uncoupling of transversal cell-to-cell connectivity through the parameter $\nu$. We use the simplest model of cell kinetics to mimic the action potential so that a cell may be in one of three states: resting (repolarized), excited (depolarizing), or refractory. An excited cell causes neighboring coupled resting cells to become excited. Thus the wavefront is a coherent propagation of this excitation through the simulated tissue. For each “patient” the initial conditions are created by assigning the same number of vertical connections (identical initial $\nu$) but at different random positions. Next the accumulation of transversal cellular uncoupling is implemented by reducing $\nu$ (e.g., to mimic the progression of fibrosis or gap junctional uncoupling). To do this, we run simulations for a period of $T = 4.3 \times 10^7$ time steps in the computer model and vertical connections are removed at a rate of one connection every nine thousand time steps. We note that the actual rate at which transversal uncoupling accumulates in humans is unknown and may differ between patients. Hence, the rates used in the model are set to be identical between simulated “patients” with the aim of capturing the generic phenomenon of the accumulation of uncoupling over time thought to occur frequently in humans [@Kottkamp2013]. We observe the dynamics of activation wavefronts as the transversal uncoupling accumulates in the tissue. All other model parameters are set to physiological values as described in [@Manani2015] (see supplementary information). We ran 32 lattice simulations, representing 32 patients, with the same initial fraction of vertical connections distributed randomly in each simulated tissue. We start at $\nu = 0.25$ where all the simulated heart muscle tissue are in sinus rhythm (SR) but when lowering $ \nu $, fibrillation may emerge. The number of excited cells can be used to determine when the system is in fibrillation, see Fig. 1 for example and associated electrogram, and hence determine the associated AF burden which we define as the amount of time in AF divided by the total observed time. To define paroxysmal and persistent AF in the model we use a scheme similar to that of Sugihara *et al.* based on AF burden. That is, we call periods of AF burden being 1-99% paroxysmal AF and burden of 100% persistent AF. Permanent AF traditionally refers to the clinical decision not to treat and thus is not informative of the dynamics of AF. ![Numerical simulation for a 200 by 200 system where pacemaker cells activate periodically. **a**, during sinus rhythm (SR, blue curve) the number of excited cells varies with the same period as the pacemaker cells. When the number of excited cells exceeds a threshold (220, see dashed gray line) it implies that the system is in fibrillation (AF, black curve). The system is defined to return to SR when the system is below threshold for more than one normal sinus rhythm beat. **b**, a rectangular electrode of size $1 {mm}^2$ (10 x 10 cells) placed at the center of the tissue is used to simulate the electrogram. During fibrillation (black curve) the rate of the electrogram increases by a factor of 2-5. **c**, the associated binary signal of the time series in **a** into periods of SR (blue filled area) and AF (black filled area). Over long-time simulations, the AF burden can be computed from this as the fraction of time in AF.[]{data-label=""}](Figure1.pdf){width="48.00000%"} Figure 2 shows the time-course of four particular simulated “patients”. “Patient” A undergoes what would be considered the standard progression from paroxysmal AF, with low AF burden, to persistent AF, with maximal AF burden. “Patient” B, however, shows isolated short-lived episodes of paroxysmal AF with a sudden cross-over to persistent AF. Hence, “Patient” B lacks a gradual progression from paroxysmal AF to persistent AF. “Patient” C also had a few isolated episodes of paroxysmal AF before entering a much more disordered relapsing-remitting phase between paroxysmal and persistent AF with different AF burdens compared to “Patient” B. “Patient” D underwent a sudden transition from sinus rhythm to persistent AF, but has phases of sinus rhythm interrupting persistent AF during the time-course of the disease. ![AF burden varies with time, calculated from the time in AF in a sliding window of $5 \times 10^6$ time steps for four different“patients”. Each simulation begins with an initial fraction of vertical connections of $\nu = 0.25$ and is depleted to $\nu=0.13$ mimicking progression of stochastic uncoupling over a simulated time period of $ 4.3 \times 10^7$ time steps. **a**, “Patient” A develops paroxysmal AF for $\nu \lesssim 0.188$ which eventually develops into persistent AF at $\nu \lesssim 0.138$. **b**, “Patient” B develops short lived episodes of AF for $0.206 \lesssim \nu \lesssim 0.238$ and a sudden transitions into persistent AF $\nu \approx 0.206$. This relapses into paroxysmal AF three times before remitting back into persistent AF at $\nu \approx 0.166$. **c**, “Patient” C shows a phase of relapsing-remitting paroxysmal to persistent AF for a considerable period of time until AF becomes persistent at $\nu \approx 0.159$. **d**, “Patient” D shows isolated short lived episodes of AF and a sudden transition into persistent AF $\nu \approx 0.164$. Remission back into sinus rhythm occurs twice ($\nu \approx 0.156$ and $\nu \approx 0.145$) before AF becomes persistent. The vertical dashed lines are values of fractions of vertical connections at which simulations are re-run without progressive uncoupling, that is, with a fixed fraction of vertical connections (see Fig. 3).[]{data-label="Fig:ModelDynamics"}](Figure2.pdf){width="48.00000%"} These four “patients” are archetypes of the time-course of AF that we observe in our simulations. In addition to the variability in the progression to persistent AF we note that the onset of AF occurs at significantly different amounts of uncoupling, that is, fractions of vertical connections. These findings are consistent with the clinical observations that macroscopically similar myocardium can show large variability in AF burden [@Sugihara2015; @Kottkamp2013]. Furthermore, we observe that AF activity tends to change suddenly. That is, the frequency and duration of AF episodes change rapidly rather than gradually with progressive uncoupling (see Fig. 3). This is consistent with the clinical observation that macrostructurally similar myocardium (e.g., quantified by the global average fibrosis burden) show very different AF characteristics [@Kottkamp2013]. The frequency and duration of AF events are different in each simulated patients. However, in addition to clinical studies we can identify the microscopic origins of the observed behaviour. ![Seven snapshots at $\nu=0.220,$ $0.206,$ $0.178,$ $0.172,$ $0.160,$ $0.156$ and $0.131$ of variability in event duration (expressed as a fraction of the simulation time $1.2 \times 10^7$ time steps), number of AF events and AF burden for repeated simulations as a function of fraction of vertical connections. We re-run the simulations shown in Fig. 2 for $1.2 \times 10^7$ time steps starting from particular values of the fraction vertical connections (see vertical dashed lines in Fig. 2) in which at least one of the four simulations displayed non sinus rhythm behavior. Note that we see variability both within a “patient” as the time-course of AF progresses as well as between “patients”. These three observables of event duration, number of AF events and AF burden are seen to vary significantly between real patients as well [@Sugihara2015]. []{data-label="Fig:StoryBoard"}](Figure3_2.pdf){width="45.00000%"} The differences in the behaviour of these simulations are explained to a large degree by the number of localised critical regions with specific architectural patterns of coupling observed in the simulated tissues. In the model we detect these critical regions by detecting complete loops of wavefront activation, that is, micro-re-entrant circuits. However, these critical patterns of uncoupling might also be determined structurally as these local patches of tissue have connections and dysfunctional cells arranged in a configuration which allows the formation of pinned micro-re-entrant circuits [@Manani2015]. These regions are characterised by large contiguous regions of uncoupled cell akin to the obstructive fibrosis found to promote AF in goats [@Angel2015]. We observe that the first occurrence of AF coincide with the (chance) emergence of the first critical structure at $\nu = 0.188,0.238,0.228$ and $0.177$ for “Patient” A, B, C and D, respectively, see Fig. 4. Furthermore, the number of these critical regions vary between “patients” despite having the same fraction of vertical connections (macroscopic measure). It is the variation in the number of these critical regions that causes the variability in the observed AF behaviour shown in Figures 2 and 3, see Fig. 4 **e** (see also full set of 32 “patients” in supplementary materials). ![**a-d**, Accumulation of initiating critical regions (black curve) versus the fraction of vertical connections for “patients” A-D. The onset of AF in each “patient” coincide with the first appearance of a critical region. As uncoupling progresses (the fraction of vertical connections is reduced), the number of initiating critical regions increases. **e**, The average AF burden for each “patient” as a function of the number of critical regions. The differences in the accumulation of critical regions between “patients” better predicts the variability in the AF behavior observed.[]{data-label="Fig:TimeInAF"}](Figure4_2.pdf){width="48.00000%"} The progression of AF and its variability is poorly understood. Fibrosis, among other factors, is known to be important. However, the reason why so much heterogeneity in AF behaviour occurs in patients with similar fibrosis burden is not known. In addition to this, insights from studies in cardiac tissue slices (short time-scales) have yet to be reconciled with the clinical patterns of AF development (long time scales). We bridge this gap using computational modelling and identify how structural characteristics of myocardium and uncoupling alone can give rise to patient variability. We note that the variability in the simulated patients is strictly due to specific architectural patterns of vertical uncoupling between cells. Fibrosis is one mechanism of cellular uncoupling. An additional mechanism of uncoupling is gap junctional remodelling, whereby the passive high resistance pathways between cells are redistributed to enhance anisotropic conduction and may result in the failure of action potential propagation [@Spach2001; @Spach2000; @Hubbard2007]. Indeed, gap junctional remodelling is known to be arrhythmogenic. We show that a simple model of heart muscle tissue can display the clinically observed variability in AF progression. This variability originates from the chance occurrence of critical regions characterised by poor vertical connectivity. The model reproduces clinical observations: variability in AF episode onset time, frequency, duration and progression (Fig. 3) along with (1) significant variability in the time-course of AF progression between patients (2) persistent AF remitting to paroxysmal AF and (3) macrostructurally similar myocardium can show very different AF behaviour (Figs. 2 and 4) Thus we show that a single pathological mechanism, namely uncoupling, can result in patterns of AF observed clinically. Specific architectural patterns of uncoupling rather than the global uncoupling (e.g., total fibrosis burden) were observed to drive AF. Thus, our work suggests that the tissue microstructure is essential in determining the time-course of AF in a given patient. This is a first step in relating structural features of myocardium, greatly studied in a basic science context, to patterns of AF in patients, studied in a clinical context. When experimental access to in vivo tissue microstructure becomes available in the future insight from this work might potentially lead to patient-specific therapy. *Acknowledgements*: We are very grateful to Nick Linton for his pertinent comments regarding the manuscript. This work was supported by the British Heart Foundation (RG/10/11/28457), the ElectroCardioMaths Programme of Imperial BHF Centre of Research Excellence, and the National Institute for Health Research Biomedical Research Centre. K.A.M and K.C conceived the experiment and its interpretation. K.A.M performed the simulations and created the figures. N.S.P provided essential comments in relating the model to a clinical context. K.C. and N.S.P supervised the project. All authors discussed the results, commented on and contributed to the writing of the manuscript.

--- abstract: 'Using an exact diagonalization technique on small clusters, we study spin and density excitations of the triangular-lattice $t$-$J$ model with multiple-spin exchange interactions, whereby we consider anomalous properties observed in the doped Mott region of the two-dimensional liquid $^3$He adsorbed on a graphite surface. We find that the double-peak structure consistent with experiment appears in the calculated temperature dependence of the specific heat; the low-temperature sharp peak comes from the spin excitations reflecting the frustrated nature of the spin degrees of freedom and high-temperature broad peak comes from the density excitations extending over the entire band width. The clear separation in their energy scales is evident in the calculated spin and density excitation spectra. The calculated single-particle excitation spectra suggest the presence of fermionic quasiparticles dressed by the spin excitations, with an enhanced effective mass consistent with experiment.' author: - 'K. Seki,$^1$ T. Shirakawa,$^{1,2}$ and Y. Ohta$^1$' title: 'Spin and density excitations in the triangular-lattice $t$-$J$ model with multiple-spin exchange interactions: $^3$He on graphite' --- Introduction ============ $^3$He atoms adsorbed on a graphite surface is known to be an ideal two-dimensional correlated spin-1/2 fermion system. A solidified commensurate phase of $^3$He atoms is stabilized at a $4/7$ density of the underlying layer of $^4$He atoms due to the substrate potential corrugation and thus a triangular lattice of $^3$He atoms is formed, which is a realization of a gapless quantum spin liquid (QSL).[@Fukuyama] Theoretically, this $4/7$ phase of spin-1/2 $^3$He atoms has been studied by using the triangular-lattice Heisenberg model with the multiple-spin exchange interactions.[@Roger1; @Roger2; @Misguich; @Momoi1; @Momoi2; @Bauerle] Importance of the density fluctuations has recently been pointed out as well.[@Watanabe] A finite amount of vacancies of $^3$He atoms can be introduced into this $4/7$ phase in a stable manner, where the vacancies can hop from site to site of the triangular lattice via quantum-mechanical tunneling motions even at absolute zero temperature. The presence of such vacancies, called the zero-point vacancies (ZPVs), was predicted a few decays ago.[@Andreev1; @Matsuda] Quite recently, the experimental evidence for the ZPVs has been reported in the monolayer of $^3$He adsorbed on a surface of graphite preplated by a solid monolayer of $^4$He:[@Matsumoto1; @Matsumoto2] i.e., heat capacity measurements of the system show an anomalous coexistence of a magnetic round-peak near 1 mK and a broad peak at several tens mK that are associated with the ZPVs doped into the commensurate Mott-localized solid. The ZPVs are maintained up to the doping of almost 20% of the lattice sites, which we call the doped Mott region of monolayer $^3$He. Theoretically, Fuseya and Ogata [@Fuseya] have proposed the triangular-lattice $t$-$J$ model with four-spin ring-exchange interactions as an effective model for the doped Mott region of the system and obtained its ground-state phase diagram. The low-energy excitations of the model were also discussed. They have thereby argued that there is a new-type anomalous quantum-liquid phase, characteristic of the “spin-charge separation”, which may be relevant with the anomalous features observed in the doped Mott region of the monolayer $^3$He adsorbed on a graphite surface. Motivated by such developments in the field, we study in this paper the triangular-lattice $t$-$J$ model with the multiple-spin exchange interactions further. In particular, we directly calculate the spin and density excitation spectra and single-particle spectra as well as the temperature dependence of the specific heat and uniform magnetic susceptibility by using an exact-diagonalization technique on small clusters. We thereby consider the anomalous properties observed in the doped Mott region of the two-dimensional liquid $^3$He adsorbed on a graphite surface. We will thus demonstrate that the double-peak structure actually appears in the temperature dependence of the specific heat, which is quantitatively consistent with experiment; the low-temperature sharp peak comes from the spin excitations and high-temperature broad peak comes from the density excitations. The spectral weight for the calculated spin excitation spectra is concentrated on a very low-energy region that scales with the exchange interactions, while that of the density excitations extends over an entire band width that scales with the hopping parameter of the vacancy. The clear separation between spin and density excitations in their energy scales is thus found. The accumulation of the low-energy spectral weight of the spin excitations comes from the frustrated nature of the spin degrees of freedom of the system; i.e., the ferromagnetic two-spin interactions compete with the antiferromagnetic four-spin interactions on the geometrically frustrated triangular lattice. The single-particle excitation spectra suggest that the vacancies behave like fermionic quasiparticles dressed by the spin excitations, with the enhanced effective mass consistent with experiment. Preliminary results of our work have been presented in Ref. [@Seki]. This paper is organized as follows. In Sec. II, we present our model and method of calculation. In Sec. III, we present our results of calculations for the specific heat, magnetic susceptibility, spin and density excitation spectra, and single-particle excitation spectra. We compare our results with experiment in Sec. IV. We summarize our work in Sec. V. \ Model and method ================ The triangular-lattice $t$-$J$ model with the multiple-spin exchange interactions is defined by the Hamiltonian $$\begin{aligned} {\cal H}&=&-t\sum_{\langle ij\rangle,\sigma} \big({\tilde c}_{i\sigma}^{\dagger}{\tilde c}_{j\sigma}+{\rm H.c.}\big) +J\sum_{\langle ij\rangle} \Big({\bf S}_i\cdot{\bf S}_j-\frac{n_in_j}{4}\Big)\cr &&+K\sum_{\langle ijkl\rangle}(P_4+P_4^{-1}) +R\sum_{\langle ijklmn\rangle}(P_6+P_6^{-1})\end{aligned}$$ where ${\tilde c}_{i\sigma}=c_{i\sigma}(1-n_{i,-\sigma})$ is the projected annihilation operator of a fermion ($^3$He atom) at site $i$ and spin $\sigma$ $(=\uparrow,\downarrow))$ allowing no doubly occupied sites, ${\bf S}_i$ is the spin-1/2 operator, and $n_i$ $(=n_{i\uparrow}+n_{i\downarrow})$ is the number operator. The summation in the $t$-$J$ part of the model is taken over all the nearest-neighbor pairs $\langle ij\rangle$ on the triangular lattice. $P_4$ and $P_6$ are the four-spin and six-spin exchange operators defined as $P_4=P_{il}P_{ik}P_{ij}$ and $P_6=P_{in}P_{im}P_{il}P_{ik}P_{ij}$, respectively, where $P_{ij}=(1+{\bm\sigma}_i\cdot{\bm\sigma}_j)/2$ with the Pauli spin matrix ${\bm\sigma}_i$. The summation is taken over all the possible combinations of four nearest-neighbor sites $\langle ijkl\rangle$ for $P_4$ and over all the equilateral hexagons $\langle ijklmn\rangle$ for $P_6$. In this paper, we study the dynamical properties of the model under the introduction of vacancies in the 4/7 commensurate solid phase of $^3$He, i.e., removal of particles ($^3$He atoms) or addition of ZPVs. We thus define the filling $n$ of particles as $n=N/L$ where $N$ is the total number of particles and $L$ is the total number of lattice sites in the system; in particular, $n=1$ is referred to as “half filling”, which corresponds to the 4/7 solid phase. The noninteracting band structure and Fermi surface at half filling of the tight-binding model with the nearest-neighbor hopping parameter $t$ are shown in Fig. 1; we find no nesting features in the Fermi surface and no singularities in the density of states for $n<1$. The nearest-neighbor hopping parameter $t$ and two-spin and four-spin exchange interaction parameters $J$ and $K$ have been estimated as follows:[@Fuseya] $t\simeq 50-100$ mK, $-J\simeq 1-10$ mK, and $K/|J|\sim 0.2$. We note that, in this parameter region, the two-spin exchange term favors the ferromagnetic spin polarization ($J<0$) but the four-spin exchange term gives the antiferromagnetic spin correlations between neighboring spins, and thus we have the situation where the strong frustration in the spin degrees of freedom of the system appears. Geometrical frustration also appears on the triangular lattice when the interaction between spins is antiferromagnetic. We have examined the effects of the six-spin exchange interaction term on the ground state and excitation spectra and found that the effects are very small, in particular when doped with vacancies. This is because the $P_6$ exchange interaction is easily cut by the presence of vacancies. We will therefore present the results at $R=0$ in this paper, the model of which we refer to as the $t$-$J$-$K$ model. Throughout the paper, we use $t=1$ as the unit of energy unless otherwise stated and we set $\hbar=k_{\rm B}=1$. \ We use the Lanczos exact-diagonalization technique on small clusters to calculate the ground state and excitation spectra of the model. In particular, we calculate the dynamical spin and density correlation functions defined, respectively, as $$\begin{aligned} S({\bf q},\omega)=-\frac{1}{\pi}\Im \langle\Psi_0| {S_{\bf q}^z}^\dagger \frac{1}{\omega+i\eta-({\cal H}-E_0)} S_{\bf q}^z |\Psi_0\rangle\end{aligned}$$ and $$\begin{aligned} N({\bf q},\omega)=-\frac{1}{\pi}\Im \langle\Psi_0| n_{\bf q}^\dagger \frac{1}{\omega+i\eta-({\cal H}-E_0)} n_{\bf q} |\Psi_0\rangle ,\end{aligned}$$ where $\Psi_0$ and $E_0$ are the ground-state wave function and energy, respectively. $S_{\bf q}^z$ and $n_{\bf q}$ are the Fourier transforms of the spin and particle-number operators defined, respectively, as $$\begin{aligned} S_{\bf q}^z&=&\frac{1}{\sqrt{L}}\sum_i e^{i{\bf q}\cdot{\bf r}_i}S_i^z\\ n_{\bf q}&=&\frac{1}{\sqrt{L}}\sum_i e^{i{\bf q}\cdot{\bf r}_i}n_i ,\end{aligned}$$ where ${\bf r}_i$ is the position of the lattice site $i$. We also calculate the single-particle excitation spectrum defined as $$\begin{aligned} A({\bf q},\omega)=A^-({\bf q},-\omega)+A^+({\bf q},\omega)\end{aligned}$$ with the particle removal spectrum $$\begin{aligned} A^-({\bf q},\omega)=-\frac{1}{\pi}\Im \langle\Psi_0| {\tilde c}_{{\bf q}\sigma}^\dagger \frac{1}{\omega+i\eta-({\cal H}-E_0)} {\tilde c}_{{\bf q}\sigma} |\Psi_0\rangle\end{aligned}$$ and particle addition spectrum $$\begin{aligned} A^+({\bf q},\omega)=-\frac{1}{\pi}\Im \langle\Psi_0| {\tilde c}_{{\bf q}\sigma} \frac{1}{\omega+i\eta-({\cal H}-E_0)} {\tilde c}_{{\bf q}\sigma}^\dagger |\Psi_0\rangle ,\end{aligned}$$ where $\eta\rightarrow +0$, which is replaced by a small positive number in the actual calculations to give an artificial broadening of the spectra. We use a cluster of 20 sites with periodic boundary condition for these calculations (see Fig. 2), where the independent available momenta in the Brillouin zone, ${\bf q}_0$, $\cdots$, ${\bf q}_{10}$, are also shown. They are at ${\bf q}_i=m{\bf b}_1+n{\bf b}_2$ with ${\bf b}_1=2\pi/5(1,1/\sqrt{3})$ and ${\bf b}_2=(0,\pi/\sqrt{3})$, where $(m,n)$ are $(0,0)$ for ${\bf q}_0$, $(1,0)$ for ${\bf q}_1$, $(3,-1)$ for ${\bf q}_2$, $(2,-1)$ for ${\bf q}_3$, $(1,1)$ for ${\bf q}_4$, $(0,2)$ for ${\bf q}_5$, $(0,1)$ for ${\bf q}_6$, $(2,0)$ for ${\bf q}_7$, $(1,-1)$ for ${\bf q}_8$, $(2,-2)$ for ${\bf q}_9$, and $(1,-2)$ for ${\bf q}_{10}$. In the following, we will in particular examine the cluster with two vacancies, i.e., $n=0.9$. To calculate the temperature $T$ dependence of the specific heat $C(T)$, magnetization $M(T)$ under uniform magnetic field $h$, and uniform magnetic susceptibility $\chi(T)=\lim_{h\rightarrow 0}(\partial M\big/\partial h)_T$, the Hamiltonian for a smaller-size cluster of 12 sites (see Fig. 2) is fully diagonalized to calculate the partition function.[@Shannon; @Elstner] We add the Zeeman term $-h\sum_iS_i^z$ to the Hamiltonian Eq. (1) when we calculate the magnetic response of the system. In the following, we will in particular examine this cluster with one vacancy ($n=0.92$) and two vacancies ($n=0.83$). Results of calculation ====================== Ground-state phase diagram -------------------------- The ground-state phase diagram in the parameter space of the present model at $R=0$ has been obtained by Fuseya and Ogata,[@Fuseya] which we have also reproduced successfully. Here, we briefly review their results. The phases obtained are as follows: the region of phase separation (phase-I), the region of Fermi liquid with strong spin fluctuations (phase-II), the region of new-type anomalous quantum liquid (phase-III), and the region of ferromagnetism (phase-IV), where we follow their notations of the phases. They have put special emphasis on the phase-III, which has been argued to be the region of “spin-charge separation” and may be relevant with the anomalous features of the doped Mott region of the $^3$He monolayer. In the following, we in particular examine the region of this new-type anomalous quantum liquid (phase-III) using the parameter values $J=-0.3$ and $K=0.06$, which we compare with the results of other regions when necessary, i.e., the region of ferromagnetism (phase-IV) using $J=-0.3$ and $K=0$ and the region of Fermi liquid (phase-II) using $J=-0.3$ and $K=0.15-0.2$. \ \ Specific heat and entropy ------------------------- The calculated results for the temperature dependence of the specific heat $C(T)$ at $n=0.83$ and $0.92$ are shown in Fig. 3. We find that there appears a double-peak structure in $C(T)$ at $K=0.06$ (corresponding to the new-type anomalous quantum-liquid phase); i.e., a sharp peak at low temperatures and a very broad peak extending over high temperatures. The double-peak structure is not clearly seen at $K=0$ and $0.15-0.2$. We should note here that the specific heat coefficient $\gamma$, where $C(T)=\gamma T$ at low temperatures, cannot be deduced from the present calculations since $C(T)$ decays exponentially at low temperatures due to the discreteness of the energies of finite-size systems. We will show in Sec. III D that the low-temperature sharp peak comes from the excitation of the spin degrees of freedom of the system and the broad high-temperature peak comes from the excitations of the density degrees of freedom of the system. In other words, the width of the sharp low-energy peak scales with the exchange interactions between spins (a combination of $J$ and $K$) and the width of the broad high-energy peak scales with the hopping parameter $t$ of the vacancy. It is interesting to note that the double-peak structure in the specific heat $C(T)$ due to the separation in their energy scales between spin and density degrees of freedom has previously been discussed in the context of the low-energy excitations in the Hubbard ladder systems with charge ordering instability although the latter is for the insulating systems with a charge gap.[@Ohta] In Sec. IV, we will compare the obtained double-peak structure with experiment[@Matsumoto1; @Matsumoto2]. We also calculate the temperature dependence of the entropy $S(T)$ (not shown here), which will be compared with experiment[@Matsumoto2] also in Sec. IV. \ Uniform magnetic susceptibility ------------------------------- The calculated results for the temperature dependence of the uniform magnetic susceptibility $\chi(T)$ are shown in Fig. 4. We find that the temperature variation is strongly dependent on the value of $K$: (i) When $0\le K\lesssim 0.04$, $\chi(T)$ is strongly enhanced in comparison with the Curie susceptibility $\chi(T)=C/T$, resulting in the ferromagnetic spin polarization at low temperatures. (ii) When $K=0.06$ at which the frustration in the spin degrees of freedom is the largest, $\chi(T)$ is slightly suppressed in comparison with the Curie law. Here, the ground state is highly degenerate due to the frustration of the spin degrees of freedom; in our cluster, the degeneracy is 15 fold. (iii) When $K\gtrsim 0.07$, $\chi(T)$ is rapidly suppressed with decreasing temperatures. Here, the ground state of the system is spin singlet without degeneracy. It should be noted that these results come basically from the finite-size effects of small clusters. However, we may infer the intrinsic nature of the infinite-size system and its $K$ dependence from the low-energy behavior under the magnetic field. The results for $\chi(T)$ thus obtained are compared with experiment in Sec. IV. \ Spin and density excitation spectra ----------------------------------- The calculated results for the spin and density excitation spectra at $K=0.06$ are shown in Fig. 5. The excitation spectra for the corresponding noninteracting infinite-size system, $N_0({\bf q},\omega)=2S_0({\bf q},\omega)$, are also shown for comparison. We find the following. The spectral weight for the spin excitations is concentrated on a very low-energy region of around $\omega\lesssim 1$. This reflects the presence of a large number of nearly degenerate low-energy states coming from the frustrated nature of the spin degrees of freedom. The low-energy spectral weight is extended over the entire Brillouin zone, rather than special momenta, reflecting the spatially localized nature of the spin fluctuations. The spectral weight for the density excitations, on the other hand, extends over a wide energy range of about $0<\omega\lesssim 9$ (entire band width), which is more or less resembles the spectrum of the noninteracting system. From these results, we may say that the spin and density excitations are clearly separated in their energy scales: i.e., the spin excitations concentrate on the low energy regions, the width of which scales with the exchange interactions (a combination of $J$ and $K$), and the charge excitations extend over the entire band width, which scales with the hopping parameter $t$ of the vacancy. With increasing $K$, we find the upward shift of the low-energy spectral weight of the spin excitations; e.g., at $K=0.15-0.2$, we find the peaks at a higher-energy region of around $0<\omega\lesssim 5$, where the momentum dependence of the positions of the peaks becomes significant as well. Thus, the separation between the energy scales of the spin and density excitations becomes weaker. We should note that the energy range where the spectral weight of the spin excitations accumulates, i.e., $0<\omega\lesssim 0.1$, corresponds well to the temperature range where the low-temperature sharp peak in the calculated specific heat $C(T)$ appears. We should also note that the broad spectra extending over the entire band width correspond well to the very broad high-temperature peak in the calculated result for $C(T)$. We may therefore conclude that the separation between the spin and density excitations in their energy scales is responsible for the double-peak structure of the temperature dependence of the specific heat. \ Single-particle excitation spectra ---------------------------------- The calculated results for the single-particle excitation spectra $A({\bf q},\omega)$ are shown in Fig. 6. From the results, we can deduce the possible quasiparticle band structure and hence the Fermi-surface topology. We find the following. There are broad and incoherent spectral features over a wide energy range corresponding to the total band width of the noninteracting dispersion, i.e., $-5\lesssim\omega\lesssim 5$, but there emerge the sharp quasiparticle-like peaks with a characteristic dispersion in the vicinity of the Fermi energy. This result is similar to the case of the square-lattice $t$-$J$ model near half filling.[@Eder1; @Eder2] Let us assume this to be the consequence of the presence of fermionic quasiparticles. Then, we find the quasiparticle band structure to be fitted well by the noninteracting band dispersion with a reduced hopping parameter $t_{\rm eff}$ or with an enhanced effective mass $m^*$ of the quasiparticle (see Fig. 7). From the fitting, we find the value $$t_{\rm eff}\,\,=\,\,\frac{m}{m^*}\,t\,\,\simeq\,\,(1/6)t$$ or $m^*/m\simeq 6$ at $n=0.9$ and $K=0.06$. The present results also suggests that the Fermi-surface topology of the quasiparticles is equivalent to that of the noninteracting system since the quasiparticle band can be obtained only by assuming that the band width is reduced (or the band mass is enhanced). Thus, the Fermi surface is large (i.e., its area $\propto n$) rather than small (i.e., its area $\propto(1-n)$). The doping dependence of $m^*$ should be interesting, in particular whether $m^*$ diverges or not at $n\rightarrow 1$. However, we cannot answer this question in our small-cluster study; the behavior of $m^*$ even in the square-lattice $t$-$J$ model still remains to be a puzzle. \ We may also assume that the quasiparticle band width shown in Fig. 7 may scale well with the energy of the spin excitations, i.e., a combination of the exchange parameters $J$ and $K$, while the entire band width ($\sim 9t$) of the broad spectral features shown in Fig. 6 scales with $t$, as in the case of the square-lattice $t$-$J$ model near half filling.[@Eder1; @Eder2] Thus, we again find the separation between the spin and density degrees of freedom in their energy scales. The quasiholes (or quasiparticles as their conjugate) are thus the vacancies dressed by the spin excitations. We should note therefore that the spin and density degrees of freedom are not exactly separated in this sense, unlike in the Tomonaga-Luttinger liquid in the one-dimensional interacting fermion systems.[@TLL] Only the energy scales are different. Further experimental and theoretical studies will be required to clarify the true low-energy physics of the system. We may also point out that the enhanced effective mass of the quasiparticle band structure may partly be responsible for the enhancement of the effective mass $m^*/m$ determined from the specific heat coefficient $\gamma$ although the latter cannot be obtained from our finite-size calculations. In Sec. IV, we compare the effective mass obtained from the quasiparticle band dispersion with experiment. \ Comparison with experiment ========================== The calculated results for the heat capacity $C(T)$ at $J=-0.3$ and $K=0.06$ (see Sec. III B) at the fillings of $n=0.92$ and $0.83$ are compared with experiment in Fig. 8(a). We here assume the value of $t$ determined so as to reproduce the higher-temperature peak observed in $C(T)$ at $n=0.92$, i.e., $t=43.8$ mK, so that we have $|J|=13.1$ mK and $K=2.63$ mK with keeping the ratio $t$ : $|J|$ : $K$ = $1 : 0.3 : 0.06$. We also assume that the total area of the sample used in experiment (556 m$^2$) is uniformly active and contributes to the heat capacity. We should note that the results for the temperature region $T\lesssim 1$ mK are not reliable because of the finite-size effects where the discreteness of the energies in the system gives the exponential decay of the heat capacity at low temperatures. We then find the fair agreement with experiment; in particular, the double-peak structure in $C(T)$, i.e., the lower-temperature peak that comes from the spin excitations and higher-energy peak that comes from the density excitations of the system, are reasonably well reproduced. More precisely, we find that our calculated results reproduce the experimental tendency that, near half filling, the lower-temperature peak is high but with increasing the vacancy concentration, the higher-temperature peak becomes larger and simultaneously the peaks shift to higher temperatures. Note that the specific heat coefficient $\gamma$ (or the effective mass) cannot be estimated from the present calculations of $C(T)$ due to finite-size effects. However, we find that the value of the enhanced effective mass $m^*/m\simeq 6$ estimated from the calculated quasiparticle band structure (see Sec. III E) is consistent with the experimental value $\sim 7.5-10$ estimated from the observed temperature dependence of $C(T)$ at $n=0.89$.[@Matsumoto2] Here, we should note that the definition of $m^*$ in the experimental specific heat coefficient is two-fold: one is the value deduced from the lower-temperature peak and the other is the value deduced from the higher-temperature peak. The two values are, however, not very different at least for $n\lesssim 0.9$, so that we can make comparison with our theoretical value. Then, it seems reasonable to assume that the renormalization of the band structure due to the spin excitations is mainly responsible for the observed enhancement of the effective mass.[@Matsumoto2] Thus, stated differently, the specific heat coefficient $\gamma$ should be determined predominantly by the spin excitations of the system. The calculated results for the entropy $S(T)$ are also compared with experimentally determined[@Matsumoto2] entropy in Fig. 8(b). We again find the fair agreement in their general tendencies. More precisely, we find that the calculated curves of $C(T)$ cross the line of the entropy of $N$ free spins $Nk_{\rm B}\ln 2$ at $\sim$10 mK, at which the lower-temperature peak in $C(T)$ terminates. This result also supports that the lower-temperature peak in $C(T)$ comes from the excitations of the spin degrees of freedom of the system. The higher-energy peak should therefore come from the density degrees of freedom or motions of vacancies in the system. Note that the experimentally determined[@Matsumoto2] entropy is significantly smaller than $Nk_{\rm B}\ln 2$ even in the vicinity of half filling, $n=0.997$, and even at temperatures of $10-20$ mK where the lower-temperature peak in $C(T)$ terminates. The missing entropy may reside in the region of much lower temperatures that the present experiment does not approach.[@Fukuyama2] \ The calculated results for the magnetization under the uniform magnetic field are compared with experiment in Fig. 9. We should note that the experimentally applied magnetic field, which is $h=0.006$ in our calculations, is very small in comparison with the energy scales of the $^3$He system, so that the behavior of the magnetization under the uniform magnetic field is the same as that of the uniform magnetic susceptibility defined at $h\rightarrow 0$. We find that, although the finite-size effect is strong at low temperatures, the calculated magnetization at $K=0.06$ is consistent with experiment in the sense that the value is somewhat smaller than the value expected from the Curie law $M(T)=\chi(T)h=Ch/T$. For more quantitative comparison, however, one would need the techniques appropriate for treating infinite-size systems, so that experimentally observed plateau-like behavior[@Murakawa] in the temperature dependence of the magnetization can be explained. Summary ======= We have used an exact-diagonalization technique on small clusters to study the low-energy physics of the triangular-lattice $t$-$J$ model with the multiple-spin exchange interactions, whereby we have considered the anomalous properties observed in the doped Mott region of the two-dimensional liquid $^3$He adsorbed on a graphite surface. We have calculated the temperature dependence of the specific heat, entropy, and uniform magnetic susceptibility, as well as the spin and density excitation spectra and single-particle spectra for the model, and have considered their implications. We have shown the following: \(1) The double-peak structure appears in the temperature dependence of the specific heat. The result is quantitatively consistent with experiment. The low-temperature sharp peak comes from the spin excitations and high-temperature broad peak comes from the density excitations. \(2) The spectral weight for the spin excitations is concentrated on a very low-energy region, the width of which scales with the exchange interactions, while that of the density excitations extends over an entire band width, which scales with the hopping parameter of the vacancy. The clear separation between spin and density excitations in their energy scales is thus found. \(3) The accumulation of the spectral weight of the spin excitations comes from the frustrated nature of the spin degrees of freedom of the system; i.e., the ferromagnetic two-spin interactions $J$ compete with the antiferromagnetic four-spin interactions $K$ on the geometrically frustrated triangular lattice. \(4) The single-particle excitation spectra suggest that the vacancies behave like the fermionic quasiparticles dressed by the spin excitations, of which the effective band mass is estimated to be $m^*/m\simeq 6$ at $n=0.9$, in consistent with the effective mass measured from the specific heat coefficient. \(5) The temperature dependence of the spin susceptibility shows a suppressed Curie-like behavior, reflecting the situation where the ground state is highly degenerate due to the frustrated nature of the spin degrees of freedom. We hope that the present study will shed more light on the physics of the two-dimensional $^3$He systems and stimulate further experimental and theoretical studies of the systems in greater details. We have focused on the doped Mott region of the monolayer $^3$He in this paper. However, it has recently been reported[@Neumann] that the bilayer $^3$He systems also contain rich physics concerning heavy fermions with quantum criticality, which we want to leave for future study. Enlightening discussions with Professors Hiroshi Fukuyama and John Saunders are gratefully acknowledged. This work was supported in part by Grants-in-Aid for Scientific Research (Nos. 18028008, 18043006, 18540338, and 19014004) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. TS acknowledges financial support from JSPS Research Fellowship for Young Scientists. A part of computations was carried out at the Research Center for Computational Science, Okazaki Research Facilities, and the Institute for Solid State Physics, University of Tokyo. [99]{} For a recent review, see H. Fukuyama, J. Phys. Soc. Jpn. **77**, at press in No. 11 (2008); also see references therein. M. Roger, Phys. Rev. Lett. **64**, 297 (1990). M. Roger, C. Bäuerle, Y. M. Bunkov, A. S. Chen, and H. Godfrin, Phys. Rev. Lett. **80**, 1308 (1998). G. Misguich, B. Bernu, C. Lhuillier, and C. Waldtmann, Phys. Rev. Lett. **81**, 1098 (1998). T. Momoi, H. Sakamoto, and K. Kubo, Phys. Rev. B [**59**]{}, 9491 (1999). T. Momoi, P. Sindzingre, and N. Shannon, Phys. Rev. Lett. **97**, 257204 (2006). C. Bäuerle, Y. M. Bunkov, A. S. Chen, D. J. Cousins, H. Godfrin, M. Roger, and S. Triqueneaux, Physica B **280**, 95 (2000). S. Watanabe and M. Imada, J. Phys. Soc. Jpn. **76**, 113603 (2007). A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP **29**, 1107 (1969). H. Matsuda and T. Tsuneto, Prog. Theor. Phys. Suppl. **46**, 411 (1970). Y. Matsumoto, D. Tsuji, S. Murakawa, H. Akisato, H. Kambara, and H. Fukuyama, J. Low Temp. Phys. **138**, 271 (2005). Y. Matsumoto, D. Tsuji, S. Murakawa, C. Bäuerle, H. Kambara, and H. Fukuyama, unpublished (2007). Y. Fuseya and M. Ogata, preprint cond-mat/0804.4329. K. Seki, T. Shirakawa, and Y. Ohta, J. Phys.: Conf. Ser., in press (2008). N. Shannon, B. Schmidt, K. Penc, and P. Thalmeier, Eur. Phys. J. B [**38**]{}, 599 (2004). N. Elstner and A. P. Young, Phys. Rev. B [**50**]{}, 6871 (1994). Y. Ohta, T. Nakaegawa, and S. Ejima, Phys. Rev. B **73**, 045101 (2006). R. Eder and Y. Ohta, Phys. Rev. B **50**, 10043 (1994). R. Eder, Y. Ohta, and S. Maekawa, Phys. Rev. Lett. **74**, 5124 (1995). See, e.g., T. Giamarchi, [*Quantum Physics in One Dimension*]{} (Oxford University Press, New York, 2004). H. Fukuyama, private communication. S. Murakawa [*et al.*]{}, AIP Conf. Proc. **850**, 311 (2006). M. Neumann, J. Nyéki, B. Cowan, and J. Saunders, Science **317**, 1356 (2007).

--- abstract: 'A new type of ’two-in-one’ wire scanner is proposed. Recent advances in linear motors’ technology make it possible to combine translational and rotational movements. This will allow to scan the beam in two perpendicular directions using a single driving motor and a special fork attached to it. Vertical or horizontal mounting will help to escape problems associated with the 45 deg scanners. Test results of the translational part with linear motors is presented.' author: - 'V. Gharibyan' - 'A. Delfs' - 'I. Krouptchenkov' - 'D. Noelle' - 'H. Tiessen' - 'M. Werner' - 'K. Wittenburg' title: Twisting wire scanner --- INTRODUCTION ============ Wire scanners serve as an essential part of accelerator diagnostic systems and are used mostly for beam transverse profile measurements (for a review see [@Wittenburg:2006zz]). Depending on scanning wire trajectory the profilers could be classified as rotational [@Fischer:1988ft] or linear [@Loos:2010zzc]. When its necessary to measure vertical and horizontal beam profiles at the same longitudinal position one has to use two independent scanners. Alternatively two profiles could be sampled by using a single driver mounted at 45deg with two wires stretched horizontally and vertically over a fork attached to this linear driver. However, wire vibration in the scanning direction is a known problem for the 45deg scanners [@Frisch:2008zz; @Iida:1998ua]. ![Linear-rotary motor from LinMot company. []{data-label="lrmot"}](fig1){width="85mm"} Different types of driver motors have been employed in order to move and control scanning wires which are normally mounted on cards or forks connected to the motors. Stepper or servo rotating motors are among the most popular drivers and linear motors are at developing stage. Here we explore commercially available translational-rotational motor units to propose a wire scanner solution which will perform beam scans in mutually perpendicular directions using a single linear-rotary motor and a simple wire hosting construction attached to it. The construction is a key-like wire holder which makes twisting (helical) motion during a 2-D scan. Next will follow a more detailed description of the translational part with linear motors. In conclusion we will estimate technical feasibility of the proposed twisting scanner. LINEAR-ROTARY MOTORS ==================== A linear-rotary motor produced by company LinMot [@linmot] is shown in Fig. \[lrmot\]. The motor consists of a linear and a rotary part merged together. Translational and rotational motions are decoupled and organized independently. However, linear and rotary motion synchronization is foreseen by motor controller logic. The motors are provided in different configurations with variable sizes and strengths reaching up to $1~kN$ linear force and $7.5~Nm$ rotating torque. [ll]{} **Parameter** & **Value**\ \ Extended Stroke ES mm (in) & 100 (3.94)\ Standard Stroke SS mm (in) &100 (3.94)\ Peak Force E12x0 - UC N (lbf)& 255 (57.3)\ Cont. Force N (lbf) &51 (11.5)\ Cont. Force Fan cooling N (lbf)& 92 (20.7)\ Force Constant N/A (lbf/A) &17 (3.8)\ Max. Current @ 72VDC A& 15\ Max. Velocity m/s (in/s) &3.9 (154)\ Position Repeatability mm (in) & $\pm0.05$ ($\pm0.0020$)\ Linearity % & $\pm0.10$\ \ Peak Torque Nm (lbfin)& 2 (17.7)\ Constant Torque (Halt) Nm (lbfin) &0.5 (4.4)\ Max. Number of revolutions Rpm& 1500\ TorqueConstantNm/Arms(lbfin/Arms)& 0.46 (4.07)\ Max. Current @ 72VDC Arms& 6.2\ Repeatability $\deg$ &$\pm0.05$\ \[linmot\] Motor controllers use advanced and flexible software/firmware which should help to perform slow or fast scans with minimal programming efforts. An operational voltage of 72VDC and maximal current of 15A complies to general Electro-Magnetic Interference (EMI) requirements in accelerator environments. Described features make the linear-rotary motor as an attractive tool for driving the proposed twisting wire scanner. A closer look to specifications of a linear-rotary motor LinMot is presented in Table \[linmot\] as an example. Listed values for the Repeatability are quoted for built-in, internal position and angle sensors. One can improve these parameters considerably by using external, finer sensors which is foreseen by controller software. In following we demonstrate that for linear motors. KEY-BIT SCANNER =============== In order to apply 2-D helical motion of a linear-rotary motor for scanning a beam, one needs to invent a suitable construction with stretched wires and a holding frame which stays out of (does not cross the) beam during the scan. For that we propose a key-bit like assembly which fulfills above requirements. The construction is schematically presented in Fig. \[fork\]. ![A key-bit holder scheme with horizontal and vertical scanning wires. A small ellipse on the right depicts a beam running normal to the page.[]{data-label="fork"}](fig2){width="85mm"} As it’s indicated by arrows, for this arrangement translational motion will first scan the beam in horizontal direction and next, when the beam will be inside the key-bit, a proper rotation will perform vertical scan. It is necessary to limit the rotation angle in order to escape crossing of the wire holder with the beam. For that there is sufficient space between the holder frame and the beam, remaining after the rotational scan is over. Denoting vertical key-bit and wire size by $L$ and $l_w$ respectively, the beam to holder distance could be expressed as $$L\left( \arccos{\frac{x}{L}} - \arccos{\frac{x}{l_w}} \right)$$ where $x$ is distance between the beam and rotational axis. Applying this formula for fast ($>1~m/sec$) rotational scans with some realistic accelerator parameters we obtain sufficient space to accelerate the wire while for deceleration the space is limited and one needs to use mechanical dumps to stop the scanner. 3-D KEY-BIT HOLDER FOR 2-D FAST SCANS ===================================== An improved, slightly more complicated design, for the fast scans could be achieved ![Scanning scheme of a three dimensional key-bit wire holder. Vertical green line depicts the beam to be scanned.[]{data-label="fork3d"}](fig3){width="75mm"} by tilting the second quadrant of the key-bit wire holder out of the construction’s plane by some angle. A 90 deg tilted key-bit holder is sketched in Fig. \[fork3d\]. Scanning sequence is exactly the same as for the flat key-bit scanner with an advantage of more room after the second scan is over. This should give sufficient time to decelerate and stop the frame by the motor alone, without mechanical dumpers. In addition the tilted key-bit’s moment of inertia is considerably smaller than in flat case. This will allow easier and improved handling of rotations with more ergonomic acceleration and deceleration of the key-bit structure. LINEAR MOTOR PERFORMANCE AS A SCANNER DRIVER ============================================ We are developing wire scanners with linear motor drivers for European XFEL accelerator. ![Designed horizontal and vertical wire scanners mounted on vacuum chamber. []{data-label="xfel3d"}](fig4){width="95mm"} Here we present some of the results obtained during recent laboratory experiment with test scanners. Planned test setup is displayed in Fig. \[xfel3d\] while experimental realization is presented by Fig. \[test1\]. For horizontal and vertical scans two identical and independent profilers are mounted to a special vacuum chamber dedicated to beam transverse diagnostics. ![XFEL wire scanners’ test setup []{data-label="test1"}](fig5){width="70mm"} Position feedback for the linear servo-motor is provided by an external Heidenhain optical system which is accurate to $1\mu m$. With the setup we have tested triggered fast scans and mechanical as well linear motors performance during/after tens of thousand scanning strokes. Important specifications of the XFEL wire scanners are shown in Table \[xfel-tab\]. **Parameter** **Value** ------------------------------ ------------------------------- Stroke 53mm Measurement duration 5 sec / 4 scanners Scanning modes Fast (1m/s), Slow Motor to beam sync $< 1 \mu sec$ (RMS) Position accuracy in a cycle 2 $\mu m$ (RMS) Width accuracy per cycle 2 % (RMS) Wire positioning error 1 $\mu m$ Number of wires per fork 3 + 2 ( 3x$90^o$, $\pm 60^o$) Wire material Tungsten Fork gap 15mm Wire-wire distance 5mm ( $90^o$ ) : European XFEL Wire Scanner Specifications \[xfel-tab\] Tests have marked most of the listed specifications as achieved. During the test mechanical design and construction precision has been justified while linear motors have demonstrated reliable performance. ![Linear motor parameters recorded during a stroke: position(mm, red), velocity(m/s, blue), current (A, green), demand and actual velocity difference (m/s, brown) []{data-label="motpar"}](fig6){width="\columnwidth"} To verify motor’s dynamic behavior we have recorded essential parameters during nominal strokes. An example is shown on Fig. \[motpar\] where together with position and velocity also the motor’s current and velocity deviation are displayed for a fast ($1m/s$) scanning stroke. An important issue for the XFEL wire profilers and fast scans in general is mechanical jitter magnitude for triggered scans. We have investigated this by recording time intervals between the trigger and fine position system reference mark traversing time. ![Motor triggering mechanical jitter distribution for the forward (upper plot) and backward (lower plot) strokes. Superimposed are shown fitting gaussian functions with displayed RMS (one sigma) values. []{data-label="trigg"}](fig7){width="\columnwidth"} Measurement results for many forward and backward strokes are summarized in Fig. \[trigg\]. Distributions show time jitter below $1 \mu sec$ which, in our case of $1 m/sec$ velocity, is equivalent to a sub-micrometer mechanical jitter. This could also be quoted as a repeatability of the tested linear motor with fine position feedback and triggering systems. DISCUSSION ========== In the last section we have demonstrated an outstanding performance of contemporary linear motors as wire scanner drivers. We have proposed to use linear-rotary motor with attached key-bit wire card as 2-D twisting wire scanner. Estimated planes of possible oscillations of the key-bit wires differ from critical planes in 45 deg forks which should cure associated vibrational problems reported at LCLS and other centers. This will become possible mainly due to different alignment of the scanning wires relative to driver unit. In addition the vibrations are normally dumped along the motion direction. An apparent difficulty for twisting wire scanner development is the linear-rotary motion transfer into the vacuum chamber where the key-bit card should operate. For that one should combine linear bellows with either wobble [@Bosser:1984us] or torsional [@tbellows] bellows. [00]{} K. Wittenburg, “Overview of recent halo diagnosis and non-destructive beam profile monitoring,” Conf. Proc.39th ICFA Advanced Beam Dynamics Workshop on High Intensity High Brightness Hadron Beams 2006 (HB2006) C. Fischer, G. Burtin, R. Colchester, B. Halvarsson, R. Jung and J. M. Vouillot, “Studies Of Fast Wire Scanners For Lep,” Conf. Proc. C [**880607**]{}, 1081 (1988). H. Loos, R. Akre, A. Brachmann, R. Coffee, F. -J. Decker, Y. Ding, D. Dowell and S. Edstrom [*et al.*]{}, “Operational Performance of LCLS Beam Instrumentation,” SLAC-PUB-14121. J. Frisch, R. Akre, F. J. Decker, Y. Ding, D. Dowell, P. Emma, S. Gilevich and G. Hays [*et al.*]{}, SLAC-REPRINT-2012-018. N. Iida, T. Suwada, Y. Funakoshi, T. Kawamoto and M. Kikuchi, “A method for measuring vibrations in wire scanner beam profile monitors,” Conf. Proc. C [**9803233**]{}, 546 (1998). http://www.LinMot.com J. Bosser, J. Camas, L. Evans, G. Ferioli, R. Hopkins, J. Mann and O. Olsen, “Transverse Emittance Measurement With A Rapid Wire Scanner At The Cern Sps,” Nucl. Instrum. Meth. A [**235**]{}, 475 (1985). ,\ http://www.youtube.com/watch?v=C3WTtMCU3lE

--- abstract: 'Let $(N,g)$ be a complete Riemannian manifold of dimension $n+1$ whose Riemannian metric $g$ is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues $\overline{\sigma}_k(\Omega)$ of a bounded domain $\Omega$ in $N$ are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain $\Omega$ in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above : $\overline{\sigma}_k(\Omega)\leq C(n) k^{2/(n+1)},$ where $C(n)$ is a constant depending only on the dimension. On a compact surface $\Sigma$ with boundary, the normalized Steklov eigenvalues are uniformly bounded above in terms of genus : $\overline{\sigma}_k(\Sigma)\leq C\left(1+\mbox{genus}(\Sigma)\right)k.$ We also obtain a relationship between the Steklov eigenvalues of a domain $\Omega$ and the eigenvalues of the Laplace-Beltrami operator on the hypersurface bounding $\Omega$.' address: - 'Université de Neuchâtel, Institut de Mathématiques, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel Switzerland' - 'Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083, Université François Rabelais de Tours, Parc de Grandmont, 37200 Tours, France' - 'Université de Neuchâtel, Institut de Mathématiques, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel Switzerland ' author: - Bruno Colbois - Ahmad El Soufi - Alexandre Girouard bibliography: - 'biblioCEG.bib' title: Isoperimetric control of the Steklov spectrum --- Introduction ============ The goal of this paper is to obtain geometric upper bounds for the spectrum of the Dirichlet-to-Neumann map. Let $N$ be a complete Riemannian manifold. Let $\Omega$ be a relatively compact domain in $N$ with smooth boundary $\Sigma$. The Dirichlet-to-Neumann map $\Lambda: C^{\infty}(\Sigma)\rightarrow C^{\infty}(\Sigma)$ is defined by $$\Lambda f=\partial_n(Hf)$$ where $Hf$ is the harmonic extension of $f$ to the interior of $\Omega$ and $\partial_n$ is the outward normal derivative. The Dirichlet-to-Neumann map is a first order elliptic pseudodifferential operator [@taylorPDEII]. Because $\Sigma$ is compact, the spectrum of $\Lambda$ is positive, discrete and unbounded [@band p. 95]: $$\begin{gathered} 0 = \sigma_1\le \sigma_2(\Omega) \leq \sigma_3(\Omega) \leq \cdots \nearrow\infty.\end{gathered}$$ The spectrum of this operator is also called the Steklov spectrum of the domain $\Omega$. Physical interpretation ----------------------- Prototypical in inverse problems, the Dirichlet-to-Neumann map is closely related to the Calderón problem [@cldr] of determining the anisotropic conductivity of a body from current and voltage measurements at its boundary. This point of view makes it useful as a model for Electrical Impedance Tomography. A particularly striking related result [@LaTaUh] is that if the manifold $M$ is real analytic of dimension at least 3, then the knowledge of $\Lambda$ determines $M$ up to isometry. The study of the spectrum of $\Lambda$ was initiated by Steklov in 1902 [@stek]. Eigenvalues and eigenfunctions of this operator are used in fluid mechanics, heat transmission and vibration problems [@foxkut; @kokrein]. Optimization ------------ The general question we are interested in is to give upper bounds for the eigenvalues in terms of natural geometric quantities. Because the eigenvalues are not invariant under scaling of the Riemannian metric, we consider normalized eigenvalues $$\bar\sigma_k(\Omega):=\sigma_k(\Omega)|\Sigma|^{\frac{1}{n}} \quad\mbox{ with } \quad|\Sigma|=\int_{\Sigma}dv_{\Sigma}$$ where $n$ is the dimension of the boundary $\Sigma$ and $dv_{\Sigma}$ is the measure induced by the Riemannian metric of $N$ restricted to $\Sigma$. Given a complete Riemannian manifold $N$, is $\overline{\sigma}_k(\Omega)$ uniformly bounded above among bounded domains $\Omega\subset N$? For the first non-zero eigenvalue, this question has been studied by many authors. See [@payne; @hps] for early results in the planar case. The series of paper by J. Escobar [@esco1; @esco2; @esco3] is influencial. For more recent results, see [@wangxia2; @fraschoen]. For higher eigenvalues in the planar situation, see [@gp; @gp2]. The main result of this paper (Theorem \[maindegree\]) is an upper bound for the eigenvalues of the Dirichlet-to-Neumann map on a domain in a complete Riemannian manifold satisfying a growth and a packing condition in terms of its isoperimetric ratio. We list here some applications. ### Domains in space forms {#domains-in-space-forms .unnumbered} The case of simply connected planar domain is well understood. See [@wein; @hps] and especially [@gp2] for a survey of this problem. If a bounded domain $\Omega \subset {\mathbb R}^2$ is simply connected, then $$\begin{gathered} \label{ineqPlanarDomain} \bar\sigma_k(\Omega)\le 2\pi k.\end{gathered}$$ This inequality is optimal. In higher dimensions, only few results about the first non-zero eigenvalue are known. See [@brock] for a different normalization. Our first result is a generalization of the above to the case of arbitrary[^1] domains in space forms. \[thmSpaceform\] There exists a constant $C_n$ depending only on the dimension $n$ such that, for each bounded domain $\Omega$ in a space form ${\mathbb R}^n$, $\mathbb H^n$ or in an hemisphere of $\mathbb S^n$, we have $$\begin{gathered} \label{ineqSpaceForm} \bar\sigma_k(\Omega)\leq C_n k^{2/n}. \end{gathered}$$ This result follows from a more general result allowing control of the Steklov spectrum of a domain in a complete manifold in terms of its isoperimetric ratio. ### Domains in a complete manifold {#domains-in-a-complete-manifold .unnumbered} The following theorem shows that under an additional assumption on Ricci curvature, we can control the normalized Steklov eigenvalue $\overline{\sigma}_k$ of a domain in terms of its isoperimetric ratio. \[ricpos\] Let $N$ be a complete manifold of dimension $n+1$. If $N$ is conformally equivalent to a complete manifold with non-negative Ricci curvature, then for each domain $\Omega\subset N$, we have $$\begin{gathered} \label{ineqricpos} \bar\sigma_k(\Omega) \le \frac{\gamma(n)}{I(\Omega)^{\frac{n-1}{n}}}k^{2/(n+1)}, \end{gathered}$$ where $I(\Omega)$ is the classical isoperimetric ratio of $\Omega$, namely $$\begin{gathered} I(\Omega) = \frac{\vert \Sigma \vert}{\vert \Omega \vert^{n/(n+1)}}. \end{gathered}$$ A surprising corollary of this theorem is that if $\dim N\geq 3$ then a large isoperimetric ratio $I(\Omega)$ implies that the normalized eigenvalue $\bar\sigma_k(\Omega)$ is small. This is false for surfaces ($n=1$), see Example \[exampleSurface\]. Since there exists a constant $c_n$ such that $$\overline{\sigma}_k(\Omega)\sim c_n k^{1/n}\mbox{ as }k\rightarrow\infty,$$ one may expect that a bound such as (\[ineqricpos\]) should hold with exponents $1/n$. In fact, for $n\geq 2$, this is impossible because it would imply an upper bound on $I(\Omega)$. Naturally, if we remove $I(\Omega)$, such a bound might still be possible. For instance, we do not know if inequality (\[ineqSpaceForm\]) holds with exponent improved to $1/n$. ### Large eigenvalues {#large-eigenvalues .unnumbered} The assumption of non-negative Ricci curvature is essential. In section \[large\] we will construct for each $n\geq 2$ and each $\kappa<0$ a complete manifold $N$ of dimension $n+1$ with Ricci curvature bounded below by $\kappa$ admitting a sequence $\Omega_j$ of domains such that the normalized eigenvalues $\bar\sigma_2(\Omega_j)\to\infty$ and the isoperimetric ratio $I(\Omega_j)\rightarrow\infty$. Under the assumption of non-negative Ricci curvature, we do not know if the presence of the isoperimetric ratio is essential. Namely, is there a constant $C(n,k)$ such that for each domain $\Omega\subset N$, $\bar\sigma_k(\Omega) \le C(n,k)$ ? Of course, this will be the case if we can give uniform lower bound on the isoperimetric ratio $I(\Omega)$. This situation will be discussed in Proposition \[croke\] and in Corollary \[domric\]. Surfaces -------- If $N$ is two-dimensional the isoperimetric ratio disappears from inequality (\[ineqricpos\]). This means that for any domain in a complete surface with conformally non-negative curvature we get a uniform bound similar to (\[ineqPlanarDomain\]) : $$\begin{gathered} \bar\sigma_k(\Omega) \le \gamma(2)k.\end{gathered}$$ In fact, in the case of surfaces, we don’t need to assume our compact manifold to be a domain in a complete manifold with non-negative Ricci curvature. Let $M$ be a compact surface with smooth boundary $\Sigma$. The Steklov spectrum of $M$ is defined exactly as in the case of a domain. \[surfaces\] There exists a constant $C$ such that for any compact orientable Riemannian surface $M$ of genus $\gamma$ with non-empty smooth boundary, $$\begin{aligned} \bar\sigma_k(M)\leq C\left\lfloor\frac{\gamma+3}{2}\right\rfloor k\label{ineqsurfaces} \end{aligned}$$ where $\lfloor.\rfloor$ is the integer part. This result is in the spirit of Korevaar [@kvr] and generalizes a recent result of Fraser-Schoen [@fraschoen]. Relationships with the spectrum of the Laplacian for Euclidean hypersurface --------------------------------------------------------------------------- In section \[sectionrelations\], we will use the result of our paper [@ceg1] to establish a relation between the spectrum of the Dirichlet-to-Neumann map and the spectrum of the Laplacian acting on smooth function of the boundary $\Sigma$. The main consequence of this estimate is that for a manifold embedded as hypersurface in Euclidean space, the presence of large normalized eigenvalue of the Laplacian will force the normalized eigenvalues $\overline{\sigma}_k$ to be small. Statement and proof of the main theorem {#sectionmainthm} ======================================== We consider a slightly more general eigenvalue problem than that of the introduction. Let $M$ be a sufficiently regular compact Riemannian manifold of dimension $n+1$ with boundary $\Sigma$. Let $\delta$ is a smooth non-negative and non identically zero function on $\Sigma$. The Steklov eigenvalue problem is $$\begin{gathered} \Delta f=0 \ \mbox{ in } M,\\ \partial_nf=\sigma\, \delta f \mbox{ on } \Sigma.\end{gathered}$$ It has positive and discrete spectrum [@band p. 95]: $$\begin{gathered} 0 = \sigma_1\le \sigma_2(M,\delta) \leq \sigma_3(M,\delta) \leq \cdots \nearrow\infty.\end{gathered}$$ Because the eigenvalues are not invariant under scaling of the Riemannian metric or of the mass density $\delta$, we consider the normalized eigenvalues $$\bar\sigma_k(M,\delta):=\sigma_k(M,\delta) {m(\Sigma,\delta)} |\Sigma|^{\frac 1 n},$$ with $$|\Sigma|=\int_{\Sigma}dv_{\Sigma} \quad\mbox{ and } \quad m(\Sigma,\delta) =\frac 1 {|\Sigma|}\int_{\Sigma}\delta\,dv_{\Sigma}.$$ Let $(N,g_0)$ be a complete Riemannian manifold of dimension $(n+1)$. We consider the Riemannian distance $d_0$ induced by $g_0$ and we assume: (**P1**) There exists a constant $C$ depending on $d_0$ such that each ball of radius $2r$ in $N_0$ may be covered by at most $C$ balls of radius $r$. (**P2**) There exists a constant $\omega$ depending only on $g_0$ such that, for each $x\in N_0$, and $r\ge 0$, $\vert B(x,r)\vert \le \omega r^{n+1}$. \[control\] There is a large supply of complete Riemannian manifolds satisfying these conditions. 1. If $N$ is compact, then (P1) and (P2) are satisfied. In this case the constants $C$ and $\omega$ depend on $g_0$. 2. If the Ricci curvature of $g_0$ is non-negative then, by Bishop-Gromov comparison theorem, there exist constants $C$ and $\omega$ depending only on the dimension of $N$ such that (P1) and (P2) are satisfied. This is in particular the case of the Euclidean space $\mathbb R^{n+1}$, and we will use this in the proof of Theorem \[spaceforms\] and Theorem \[ricpos\]. It follows from the previous example that Theorem \[ricpos\] is a corollary of the following theorem. \[maindegree\] Let $(N,g_0)$ be a complete Riemannian manifold of dimension $(n+1)$ satisfying $(P1)$ and $(P2)$. Let $g\in[g_0]$ be a metric in the conformal class of $g_0$. Then, there exists a constant $\gamma(g_0)$ depending only on the constants $C$ and $\omega$ coming from $(P1)$ and $(P2)$ such that, for any bounded domain $\Omega\subset N$ and any density $\delta$ on $\Sigma=\partial\Omega$, we have $$\begin{gathered} \label{ineqmainthm} \bar\sigma_k(\Omega,\delta) \leq\frac{\gamma(g_0)}{I(\Omega)^{\frac{n-1}{n}}}k^{2/(n+1)}. \end{gathered}$$ The proof of Theorem \[maindegree\] is based on the construction of a family of disjointly supported functions with controlled Rayleigh quotient $$\begin{gathered} R(f)= \frac{\int_{\Omega}\vert \nabla_{g} f \vert^2 dv_g}{\int_{\Sigma}f^2\delta dv_{\Sigma}}.\end{gathered}$$ On $N$ we consider the Borel measure $\mu=\delta dv_{\Sigma}$. That is, the measure of an open set $\mathcal{O}\subset N$ is $$\begin{aligned} \label{measure} \mu(\mathcal{O})=\int_{\mathcal{O}\cap\Sigma}\delta\,dv_{\Sigma}.\end{aligned}$$ In particular, we have $$\mu(N)= \int_{\Sigma}\delta dv_{\Sigma}= \vert \Sigma \vert m(\Sigma,\delta).$$ Let $(X,d)$ be a metric space. An *annulus* $A\subset X$ is a subset of the form $\{x \in X: r<d(x,a)<R\}$ where $a\in X$ and $0 \le r < R<\infty$. The annulus $2A$ is the annulus $\{x \in X: r/2 < d(x,a)< 2R\}$. In particular, $A \subset 2A$. Theorem 1.1 and Corollary 3.12 of [@gyn] tell us that if a metric measured space $(X,d,\nu)$ satisfy property $(P1)$ and if the measure $\nu$ is non-atomic, then there is a constant $c>0$ such that, for each positive integer $k$, there exist a family of $2k$ annuli $\{A_i\}_{i=1}^{2k}$ in $X$ such that $$\mu(A_i) \ge c\frac{\nu(X)}{k}.$$ and the annuli $2A_i$ are disjoint. The constant $c$ depends only on the constant $C$ of property $(P1)$, that is only on the distance $d$ and not on the measure $\nu$. Consider the metric measured space $(N,d_0,\mu)$, where $d_0$ is the Riemannian distance associated to $g_0$ and $\mu$ is the measure induced by $g_0$ and the density $\delta$ as defined above in (\[measure\]). It follows from Theorem 1 and Corollary 3.12 of [@gyn] mentioned above that there exist $2k$ annuli $A_1,...,A_{2k} \subset N$ with $$\begin{gathered} \label{IneqMuAibounded} \mu(A_i) \ge \frac{\mu(N)}{ck},\quad c=c(g_0)>0. \end{gathered}$$ The annuli $B_i=2A_i$ are mutually disjoint. We can reorder them so that the first $k$ of them satisfy $$\begin{gathered} \label{firstk} \vert B_i \cap \Omega \vert_g \leq\frac{\vert \Omega \vert_g }{k}\ \ \ (i=1,\cdots,k). \end{gathered}$$ Let $A=\{x \in N: r<d(x,a)<R\}$ be one of these first $k$ annuli and let $h$ a function supported in $2A$. Taking (\[firstk\]) into account, it follows from Hölder’s inequality and the conformal invariance of the generalized Dirichlet energy that $$\begin{aligned} \int_{B \cap \Omega} \vert \nabla_g h\vert^2\,dv_g&\leq \left(\int_{B \cap \Omega} \vert \nabla_g h\vert^{n+1}\,dv_g\right)^{2/(n+1)} \vert B \cap \Omega\vert_g^{1-2/(n+1)}\\ &\leq \left(\int_{2A} \vert \nabla_{g_0} h \vert^{n+1}\,dv_{g_0}\right)^{2/(n+1)} \left(\frac{\vert \Omega \vert_g}{k}\right)^{1-2/(n+1)} \end{aligned}$$ Choosing the function $h$ that is identically $1$ on $A$ and proportional to the distance to $A$ on $2A \setminus A$, we have $$\begin{gathered} \vert \nabla_{g_0} h \vert^{n+1}\leq \begin{cases} \frac{2^{n+1}}{r^{n+1}}& \mbox{ on } B(a,r)\setminus B(a,r/2),\\ \frac{1}{R^{n+1}}& \mbox{ on } B(a,2R)\setminus B(a,R). \end{cases} \end{gathered}$$ It follows from $(P_2)$ that $$\begin{gathered} \int_{2A} \vert \nabla_{g_0} h \vert^{n+1}\,dv_{g_0} \leq 2^{n+2}\omega. \end{gathered}$$ This leads to $$\begin{gathered} \int_{B \cap \Omega} \vert \nabla_g h\vert^2\,dv_g \leq (2^{n+2}\omega)^{2/(n+1)}\left(\frac{\vert \Omega \vert_g }{k}\right)^{(n-1)/(n+1)} \end{gathered}$$ Moreover, using  we get $$\begin{gathered} \int_{\Sigma} h^2\,\delta dv_{\Sigma} \ge \mu(A) \ge \frac{\mu(N)}{ck}\end{gathered}$$ By considering the Rayleigh quotient, this leads to $$\begin{gathered} \sigma_k(\Omega,\delta)\leq \frac{2^{n+2}\omega ck}{\mu(N)} \left(\frac{\vert \Omega \vert_g}{k}\right)^{(n-1)/(n+1)}.\end{gathered}$$ Using $\mu(N)= \vert \Sigma \vert m(\Sigma,\delta)$, we conclude $$\begin{gathered} \bar\sigma_k(\Omega,\delta)= \sigma_k(\Omega,\delta)m(\Sigma,\delta)\vert \Sigma \vert^{1/n} \leq \frac{\gamma(g_0)}{I(\Omega)^{\frac{n-1}{n}}}k^{2/(n+1)},\end{gathered}$$ with $\gamma(g_0)=2^{n+2}c \omega$. Applications of Theorem \[maindegree\] {#applications} ====================================== In this section, we prove most of the results announced in the introduction as consequence of our Theorem \[maindegree\]. Domains in a manifold with conformally non-negative Ricci curvature ------------------------------------------------------------------- It is difficult to estimate the packing constant $C$ and the growth constant $\omega$ of a general Riemannian manifold. Nevertheless, as was observed in Example \[control\], in the special situation where $\Omega$ is a domain $\Omega$ in a complete Riemannian manifold $N$ with non-negative Ricci curvature, it follows from the Bishop-Gromov inequality that these constants can be estimated in terms of the dimension. \[ricposDensity\] Let $(N,g)$ be a complete Riemannian manifold of dimension $(n+1)$ and assume that the metric $g$ is conformally equivalent to a metric $g_0$ with $Ric(g_0) \ge 0$. Then, for any bounded domain $\Omega \subset N$, and for any density $\delta$ on $\partial \Omega$, we have $$\begin{aligned} \bar\sigma_k(\Omega,\delta) &\leq\frac{\gamma(n)}{I(\Omega)^{\frac{n-1}{n}}}k^{2/(n+1)},\label{thmSteklovGeneral} \end{aligned}$$ where $\gamma(n)$ is a constant depending only on $n$. This theorem is a direct consequence of Theorem \[maindegree\] and of Example \[control\]. Theorem \[ricpos\] is the special case when $\delta\equiv 1$. If $n \ge 2$, large isoperimetric ratio $I(\Omega)$ implies small eigenvalues $\bar\sigma_k(\Omega,\delta)$. \[corLargeIso\] Under the assumptions of Theorem \[ricposDensity\], if a family of domains $\{\Omega_t\}_{0<t<1}$ is such that $\displaystyle\lim_{t\rightarrow 0}I(\Omega_t)=\infty,$ then, if $n \ge 2$ and for each density $\delta_t$ on $\partial \Omega_t$, we have $$\lim_{t\rightarrow 0}\bar\sigma_k(\Omega_t,\delta_t)\rightarrow 0.$$ This is false for $n=2$. See Example \[exampleSurface\]. Control of the isoperimetric ratio ---------------------------------- In general, it is difficult to estimate the isoperimetric ratio $I(\Omega)$. We give two special situations where we have a uniform lower estimate on it. This will be a consequence of the inequality of Croke [@croke] as presented by Chavel [@cha2 p.136]. \[croke\] Let $N$ be a complete Riemannian manifold. For each $x \in N$, let $\mbox{inj}(x)$ the injectivity radius of $N$ at $x$. Given $p \in N$ and $\rho >0$, consider $$\begin{gathered} \label{rCroke} r < \frac{1}{2}\left(inf_{x \in B(p,\rho)}\mbox{inj}(x)\right). \end{gathered}$$ Then, for a each domain $\Omega \subset B(p,r)$, we have $I(\Omega)\ge C(n)$ for a constant $C(n)$ depending only on the dimension. If the injectivity radius of $N$ is strictly positive, we can choose any $r<\frac{\mbox{inj}(N)}{2}$. ### Domains in space forms A special but very important case is when the ambient space $N$ is a space form, that is the Euclidean space $\mathbb R^{n+1}$, the hyperbolic space $\mathbb H^{n+1}$ or the sphere $\mathbb S^{n+1}$ with their natural metric of curvature $0,-1$ and $1$ respectively. \[spaceforms\] For any bounded domain $\Omega$ with smooth boundary $\Sigma=\partial\Omega$ in ${\mathbb R}^{n+1}$, $\mathbb H^{n+1}$ or on an hemisphere of the sphere $\mathbb{S}^{n+1}$ and any $k\ge 1$, we have $$\bar\sigma_k(\Omega,\delta) \leq \frac{\gamma(n)}{I(\Omega)^{\frac{n-1}{n}}}k^{2/(n+1)} \leq C_nk^{2/(n+1)}$$ where $C_n$ and $\gamma_n$ are constants depending only on $n$. The standard metrics on Euclidean space and on the sphere have non-negative Ricci curvature. The standard metric on the hyperbolic space is conformally equivalent to the Euclidean one. We can therefore apply Theorem \[maindegree\], with $g_0=g$ one of these standard metric. The injectivity radii of Euclidean and hyperbolic space are infinite. That of the unit sphere is $\pi$. The proof is completed by using Proposition \[croke\]. In particular, this proves Theorem \[thmSpaceform\]. It is also classically known that any domain $\Omega$ in Euclidean space, the hyperbolic space or an hemisphere, isoperimetric ratio bounded from below by a constant depending on the dimension. This can be used instead of Croke’s result in the above proof. ### Domains inside a ball In the case where the Ricci curvature of $N$ is non-negative, we deduce the following \[domric\] If the Ricci curvature of $N$ is non-negative and if $\Omega \subset B(p,r)$, where $r$ satisfy , then $$\begin{gathered} \bar\sigma_k(\Omega,\delta) \leq C_n k^{2/(n+1)} \end{gathered}$$ for some constant $C_n$ depending only on $n$. Relation between the spectrum of the Dirichlet-to-Neumann operator and the spectrum of the Laplacian. {#sectionrelations} ====================================================================================================== Let $\Delta_\Sigma$ be the Laplacian acting on smooth functions of the boundary $\Sigma=\partial M$ of a compact Riemannian manifold with boundary. Let $0=\lambda_1\leq\lambda_2(\Sigma)\leq\cdots\nearrow\infty$ be the spectrum of $\Delta_\Sigma$. It is well known that $\Lambda$ is a first order pseudodifferential operator and that its principal symbol is the square root of the principal symbol of $\Delta_\Sigma$. It follows that $\sigma_k\sim\sqrt{\lambda_k}$ as $k\rightarrow\infty$. See for instance ([@taylorPDEII p. 38 and p. 453],  [@shamma]). Can the eigenvalues $\sigma_k$ and $\lambda_l$ be compared to each other ? Recently, Wang and Xia studied this question [@wangxia2] for the first non-zero eigenvalues of both operators. Under the assumption that Ricci curvature of $M$ is non-negative and that the principal curvatures of $\partial M$ are bounded below by a positive constant $c$, they proved that $$\begin{aligned} \sigma_2 \le \frac{\sqrt{\lambda_2}}{nc}(\sqrt{\lambda_2}+\sqrt{\lambda_2-nc^2})\label{ineqWangXia}\end{aligned}$$ Note that Xia had previously proved [@xia], under the same hypothesis, that $\lambda_2 \ge nc^2$. In [@ceg1], we study the control of the spectrum of the Laplacian on a closed hypersurface by the isoperimetric ratio. The following is a particular case of one of our results. \[laplacienCEG\] Let $N$ be a complete Riemannian manifold with non-negative Ricci curvature. Let $\Omega \subset N$ be a bounded domain with smooth boundary $\Sigma = \partial \Omega$ contained in a ball of radius $r<\frac{\mbox{inj}(N)}{2}$. There is a constant $B_n$ depending only on dimension such that for any $k\geq 0$, $$\begin{gathered} \bar\lambda_k(\Sigma)\leq B_n I(\Omega)^{(n+2)/n}{k}^{2/n} \end{gathered}$$ where $\bar\lambda_k(\Sigma)=\lambda_k(\Sigma) {\vert \Sigma \vert}^{2/n}$ are the normalized eigenvalues of the Laplacian. Combining Theorem \[laplacienCEG\] and Corollary \[spaceforms\], we get \[comparison\] Let $N$ be a complete Riemannian manifold of dimension $(n+1)$ with non-negative Ricci curvature. There exists a constant $\kappa_n$ depending only on dimension such that for any bounded domain $\Omega \subset N$ with boundary $\Sigma = \partial\Omega$ contained in a ball of radius $r<\frac{\mbox{inj}(N)}{2}$ the following holds: $$\begin{gathered} \label{comp1} \bar\lambda_k(\Sigma)\bar \sigma_l(\Omega)\leq \kappa_n\left(\frac{\vert \Sigma\vert}{\vert\Omega\vert}\right)^{3/n}k^{2/n}l^{2/(n+1)}. \end{gathered}$$ In the special case where $N$ is the Euclidean space $\mathbb R^{n+1}$, the injectivity radius in each point is $\infty$, so that that is no further restrictions on $\Omega$, and Inequalities (\[comp1\]) and (\[comp2\]) are true for all bounded domains. Without the normalization, we have $$\begin{gathered} \label{comp2} \lambda_k(\Sigma)\sigma_l(\Omega) m(\Sigma,\delta) \leq \kappa_n\frac{k^{2/n}l^{2/(n+1)}}{\vert\Omega\vert^{3/(n+1)}}. \end{gathered}$$ In comparison with [@wangxia2], we make no assumption on the convexity of $\Omega$. We also have comparison for all eigenvalues. Note however that our method does not give any sharpness. A remarkable feature of this inequality is that large eigenvalues of the Laplacian are seen to impose small eigenvalues of the Dirichlet-to-Neumann map. Under the assumptions of Theorem \[comparison\], if a family of domains $\{\Omega_t\}_{0<t<1}$ of volume one with boundaries $\Sigma_t=\partial\Omega_t$ is such that $\displaystyle\lim_{t\rightarrow 0}\lambda_k(\Omega_t)=\infty,$ then, if $n\geq 2$ we have for each $l\geq 1$ $$\lim_{t\rightarrow 0}\bar\sigma_l(\Omega_t)\rightarrow 0.$$ Surfaces ======== The situation for surface is special. We begin by a proof of the upper bound of $\sigma_k$ in term of the genus. This is a modification of the proof of Theorem \[maindegree\]. By gluing a disk on each boundary components of $M$, we can see $M$ as a domain in a a compact surface $S$ of genus $\gamma$. This closed surface can be represented as a branched cover over $\mathbb{S}^2$ with degree $d=\lfloor\frac{\gamma+3}{2}\rfloor$ (See [@gunning] for instance). On $\mathbb{S}^2$ we consider the usual spherical distance $d$ and we define a Borel measure $\mu=\psi_* \left(\delta dv_{\Sigma}\right)$. That is, the measure of an open set $\mathcal{O}\subset\mathbb{S}^2$ is $$\begin{aligned} \mu(\mathcal{O})=\int_{\psi^{-1}(\mathcal{O})\cap\Sigma}\delta\,dv_{\Sigma}. \end{aligned}$$ In particular, $$\mu(\mathbb{S}^2)=\vert \Sigma \vert m(\Sigma,\delta).$$ It follows from Theorem 1 and Corollary 3.12 of [@gyn] applied to the metric measured space $(\mathbb{S}^2,d,\mu)$ that there exist $2k$ annuli $A_1,...,A_{2k} \subset \mathbb{S}^2$ with $$\begin{gathered} \label{IneqMuAiboundedS} \mu(A_i) \ge \frac{\mu(\mathbb{S}^2)}{ck}. \end{gathered}$$ Because the annuli $2A_i$ are mutually disjoint, so are the sets $B_i=\psi^{-1}(2A_i)$. These sets can be reordered so that the first $k$ of them satisfy $$\begin{gathered} \label{firstkS} \vert B_i \vert_g \leq\frac{\vert M \vert_g }{k}\ \ \ (i=1,\cdots,k). \end{gathered}$$ Let $A=\{x \in \mathbb{S}^2: r<d(x,a)<R\}$ be one of the above annuli and let $h$ be a function supported in $2A$. Let $f=h\circ\psi$ be the lift of this function to $M$. It is supported in the set $B=\psi^{-1}(2A)$. Taking (\[firstkS\]) into account, it follows from conformal invariance of the Dirichlet energy that $$\begin{aligned} \int_{B \cap \Omega} \vert \nabla_g f\vert^2\,dv_g &\leq \left(\mbox{deg}(\psi)\int_{2A} \vert \nabla_{g_0} h \vert^{n+1}\,dv_{g_0}\right)^{2/(n+1)} \left(\frac{\vert \Omega \vert_g}{k}\right)^{1-2/(n+1)}. \end{aligned}$$ The rest of the proof is almost identical to that of Theorem \[maindegree\] and is left to the reader. In Corollary \[corLargeIso\] it was mentioned that for manifold of dimension at least three, a large isoperimetric ratio implies small Steklov eigenvalues. The next example shows that this is false for surfaces. \[exampleSurface\] Let $M$ be a compact Riemannian manifold with metric $g$. Let $f\in C^{\infty}(\bar M)$ be a smooth function vanishing on the boundary $\partial M$. Consider a conformal perturbation $\tilde{g}=e^{f}g$ of the original metric. It is well known that the Laplacian is conformally invariant in dimension two. Moreover, because $\tilde{g}=g$ on $\partial M$, the normal derivative is also preserved. It follows that the the Dirichlet-to-Neumann map induced by $\tilde{g}$ is the same as that induced by by $g$. In particular, they have the same spectrum. On the other hand the measure of the surface is given by $$\begin{gathered} |M|_{\tilde{g}}=\int_Me^f\,dg. \end{gathered}$$ By taking a function $f$ that decays fast away from the boundary, we can make this quantity as small as we want. In other words, the isoperimetric ratio $I(M)=\frac{|\partial M|}{\sqrt{|M|_g}}$ will become very large. Construction of large eigenvalues {#large} ================================= The behavior of the Steklov spectrum depends on the interior of the domain in an essential way. For a closed Riemannian manifold $\Sigma$ with large eigenvalue $\lambda_k$ of the Laplacian, embedding as an hypersurface in Euclidean space forces very small Steklov eigenvalues. This comes from the fact that, by [@ceg1] the isoperimetric ratio $I(\Omega)$ has to be big, with $\Sigma =\partial \Omega$, and this implies the presence of small eigenvalues. If we embed $\Sigma$ as the cross-section of a cylinder $\Sigma\times\mathbb{R}$ with its product metric, we will see that exactly the opposite will happen. This shows that our geometric assumptions are necessary. \[calculation\] Let $\Sigma$ be a closed Riemannian manifold of volume one. Let the spectrum of its Laplace operator $\Delta_{\Sigma}$ be $$0=\lambda_1<\lambda_2\leq\lambda_3\cdots\nearrow\infty$$ and let $(u_k)$ be an orthonormal basis of $L^2(\Sigma)$ such that $$\Delta_{\Sigma} u_k=\lambda_ku_k.$$ Let $N=\mathbb{R}\times\Sigma$. On the domain $\Omega=[-L,L]\times\Sigma \subset N$, a complete system of orthogonal eigenfunctions of the Dirichlet-to-Neumann map is given by $$\begin{gathered} 1,\ t,\ \cosh(\sqrt{\lambda_k}t)f_k(x),\ \sinh(\sqrt{\lambda_k}t)f_k(x) \end{gathered}$$ with eigenvalues $$\begin{gathered} 0, 1/L,\ \sqrt{\lambda_k}\tanh (\sqrt{\lambda_k}L)< \sqrt{\lambda_k}\coth (\sqrt{\lambda_k}L). \end{gathered}$$ It is enough to check that these functions are Steklov eigenfunctions since their restriction to the boundary form a basis $L^2$. Let $\Sigma$ be a closed manifold of dimension $\geq 3$. On the product manifold $N=\Sigma\times\mathbb{R}$ there exists a complete Riemannian metric $g$ and a sequence of bounded domains $\Omega_i$ such that $$\begin{gathered} \lim_{i\rightarrow\infty}\overline{\sigma}_2(\Omega_i)=\infty, \mbox{ and }\lim_{i\rightarrow\infty}I(\Omega_i)=\infty. \end{gathered}$$ Let $\Sigma$ be a closed manifold of dimension $\geq 3$. The first author and Dodziuk [@cd] proved the existence of a sequence $h_i$ of Riemannian metrics of volume one such that $\lim_{i\rightarrow\infty}\lambda_2(\Sigma,h_i)=\infty.$ Without loss of generality, we assume for each $i$ that $\lambda_2(\Sigma,h_i)>1.$ Consider the cylinder $\Omega_i=\Sigma\times [i,i+L_i]$ with $$1>L_i= \frac{1}{\sqrt{\lambda_2(\Sigma,h_i)}}\rightarrow 0\mbox{ as } i\rightarrow\infty.$$ Let $g$ be a complete Riemannian metric on $\Sigma\times\mathbb{R}$ such that the restriction of $g$ to $\Omega_i$ is the product of $h_i$ with the Euclidean metric on $\mathbb{R}$. It follows from Lemma \[calculation\] that $$\begin{gathered} \sigma_2(\Omega_i)= \min\left(\sqrt{\lambda_2(\Sigma,h_i)},\sqrt{\lambda_2(\Sigma,h_i)}\tanh (1)\right) =\sqrt{\lambda_2(\Sigma,h_i)}\tanh (1). \end{gathered}$$ In particular $$\lim_{i\rightarrow\infty}\bar\sigma_2(\Omega_i)=\infty.$$ There exists a complete three-dimensional Riemannian manifold $N$ admitting a sequence of bounded domains $\Omega_i\subset N$ such that $$\begin{gathered} \lim_{i\rightarrow\infty}\overline{\sigma}_2(\Omega_i)=\infty, \mbox{ and }\lim_{i\rightarrow\infty}I(\Omega_i)=\infty. \end{gathered}$$ It is well known that there exists a sequence of Riemann surfaces of volume one $\Sigma_i$ such that $\lambda_2(\Sigma_i)\rightarrow\infty$ (see [@colboisElSoufi]). We consider the complete Riemannian manifold $N_i= \Sigma_i\times \mathbb R$ (with the product Riemannian metric) and the subset $$\Omega_i=\Sigma_i \times [0,L_i]$$ with $L_i= \frac{1}{\sqrt{\lambda_2(\Sigma_i)}}$. As before, we see that $\lim_{i\rightarrow\infty}\bar\sigma_2(\Omega_i) =\infty.$ The manifold $N$ is obtained by joining the $N_i$’s by tubes. Let $M$ be a compact manifold of dimension $\geq 4$. There exists a sequence of Riemannian metric $g_i$ and a domain $\Omega\subset M$ such that $$\begin{gathered} \lim_{i\rightarrow\infty}\overline{\sigma}_2(\Omega,g_i)=\infty, \mbox{ and }\lim_{i\rightarrow\infty}I(\Omega,g_i)=\infty. \end{gathered}$$ Let $\Omega$ be any domain of $M$ that is diffeomorphic to the cylinder $(0,1)\times\mathbb{S}^n$. Because $n\geq 3$, there exists a sequence of Riemannian metric $h_i$ on $\mathbb{S}^n$ such that $\lim_{i\rightarrow}\lambda_2(\mathbb{S}^n,h_i)=\infty$. Let $g_i$ be a Riemannian metric on $M$ such that the restriction of $g_i$ to $\Omega$ is isometric to product $\mathbb{S}^n\times(0,L_i)$ with $L_i= \frac{1}{\sqrt{\lambda_2(\mathbb{S}^n,h_i)}}$. It follows from scaling invariance of the normalized eigenvalues that in each of the three previous examples, the Riemannian metrics can be chosen to have Ricci curvature arbitrarily close to zero. [^1]: i.e. not necessarily simply connected.

--- abstract: 'This paper introduces an end-to-end fine-tuning method to improve hand-eye coordination in modular deep visuo-motor policies (modular networks) where each module is trained independently. Benefiting from weighted losses, the fine-tuning method significantly improves the performance of the policies for a robotic planar reaching task.' author: - 'Fangyi Zhang, Jürgen Leitner, Michael Milford, Peter I. Corke [^1] [^2]' bibliography: - 'deep\_manipulation.bib' - 'rl.bib' - 'nc.bib' title: 'Tuning Modular Networks with Weighted Losses for Hand-Eye Coordination' --- [^1]: FZ, JL, MM, PIC are with the Australian Centre for Robotic Vision (ACRV), Queensland University of Technology (QUT), Brisbane, Australia. [[email protected]]{} [^2]: This research was conducted by the Australian Research Council Centre of Excellence for Robotic Vision (project number CE140100016). Additional computational resources and services were provided by the HPC and Research Support Group at QUT.

[**On Liouville Type of Theorems to the 3-D** ]{}\ \[3mm\] [**Incompressible Axisymmetric Navier-Stokes Equations**]{} [[^1]]{} *School of Mathematical Sciences,* Capital Normal University, Beijing 100048,P. R. China [[^2]]{} *IMS and Department of Mathematics,* The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Introduction ============ We consider the Cauchy problem for the three-dimensional (3-D) incompressible Navier-Stokes equations $$\left\{ \begin{array}{ll}\label{1.1} &\partial_tu-\Delta u+(u\cdot\nabla)u+\nabla p=0, \quad (x,t)\in \mathbb{R}^3\times (0,T),\\ [3mm] &{\rm div}\ u=0 , \end{array} \right.$$ with the initial conditions $$\label{1.3} u(x,t)\mid_{t=0}=u_0(x).$$ Here the unknown functions are the velocity vector $u=(u_1(x,t), u_2(x,t), u_3(x,t))$ and the pressure $p(x,t)$ with $x\in \mathbb{R}^3, t\in [0,T]$, where $T>0$ is a constant. In (\[1.1\]), $ {\rm div}\ u=0$ means that the fluid is incompressible. The global existence of the Leray-Hopf weak solutions to the Cauchy problem or the initial-boundary problem of the three-dimensional Navier-Stokes equations has been proved long ago (see [@Le],[@Ho]). However, the uniqueness and regularity of the weak solutions remain completely open. Up to now, the weak solutions will be regular and unique provided that the Serrin-type conditions $u\in L^p([0,T);L^q(\mathbb{R}^3))$ hold, where $2/p+3/q\le 1$, $p\ge 2$ and $ q\le 3$ (see [@Se], [@Str], [@ESS]). The strong (or smooth) solution of the three-dimensional Navier-Stokes equations was proved to be unique but local in time (see [@Kat; @KT; @La; @MB; @Te]). On the other hand, Scheffer [@Sch] introduced and began to study the partial regularity of suitable weak solutions. The significant results, due to Caffarelli-Kohn-Nirenberg [@CKN] show that, for any suitable weak solutions, one-dimensional Hausdorff measure of the singular set is zero. The simplified proofs and further studies are referred to [@Lin], [@TX]. For the three-dimensional axisymmetric Navier-Stokes equations, if the angular component of the velocity $u_\theta= 0$, the global existence and uniqueness of the strong (or smooth) solution have been successfully obtained ([@La], [@UY]). In the presence of swirls, that is, $u_\theta\not\equiv 0$, the global well-posedness of the solution is still open. Recently, using DeGeogi-Nash-Moser iterations and a blow-up approach respectively, Chen-Strain-Tsai-Yau [@CSTY1; @CSTY2] and Koch-Nadirashvili-Seregin-$\breve{S}$ver$\acute{a}k$ [@KNSS] obtained an interesting and important development on this problem. Roughly speaking, they proved that if the solution satisfies $(1) \ |ru(x,t)|\le C$ or $(2)\ |u(x,t)|\le \frac{C}{\sqrt{T^*-t}}$ for $0<t<T^*$, where $C>0$ is an arbitrary and absolute constant and $(0,T^*)$ is the maximal existence interval of the solution, then there exists a constant $M>0$ such that $|u(x,t)|\le M$ for $0<t\le T^*$ which implies that the solution is globally regular on time. It should be remarked that these conditions are scaling invariant and imply the possible blow-up rate of the solution. Moreover, the singularity satisfying (2) is usually called type I singularity in the sense of [@Ham]. Thus, if an axisymmetric solution develops a singularity, it can only be a singularity of type II (any singularity which is not type I). The other regularity criteria and recent studies can be seen in [@Chae; @JX; @KPZ; @LZ; @Pan] and references therein. The basic idea of the blow-up approach is that if the solution would blow up at some space-time point, then making scaling transformation of the solution and enlarging the region near the possibly singular point, one can look into the equations satisfied by some suitably scaled solutions. In particular, after taking the limit, if the solution of the limit equation is trivial, which is so called a Liouville type of theorem, then one will obtain a contradiction and the blow-up will not happen. To the three-dimensional axisymmetric Navier-Stokes equations, the possible singularity of the solution may only appear on the symmetry axis due to the partial regularity theory in [@CKN]. Therefore it suffices to study the possible singularity of the solution on the symmetry axis. In this paper, we are concerned with Liouville type of theorems. First, we prove a Liouville type of theorem by assuming that $$\begin{aligned} \label{Oct1} \displaystyle\limsup_{\delta\to 0+}\|ru_\theta(x,t)\|_{L^\infty(\{x|r\le \delta\}\times (-1,0))}=0.\end{aligned}$$ It is shown that, under the assumption , there exists a bounded and continuous function $s(T)$ defined on $(-\infty,0]$ such that the ancient solution $\bar u=(0,0, s(T))$. It should be remarked that the assumption of is natural since $ru_\theta$ satisfies the maximum principle and if the initial data satisfies $|ru_{\theta0}|\le C$ then the solution will keep the bound $|ru_\theta|\le C$ for some constant $C>0$. This implies that the singularity of $u_\theta$ near the symmetry axis, if exists, may be of the rate $\frac{O(1)}{r}$ as $r\to 0$, where $O(1)$ means a finite constant. While the condition implies that the singularity which we impose on $u_\theta$ near the symmetry axis is of the rate $\frac{o(1)}{r}$ with $o(1)\to 0$ as $r\to 0$. Our approaches are based on [@CSTY1; @CSTY2] and [@KNSS]. In particular, we will use the integral expression on $\frac{\omega_\theta^{(k)}}{R}$ which is the scaled quantity of $\frac{\omega_\theta}{r}$ to prove a Liouville type of theorem to the ancient solution, where $\omega_\theta=\partial_ru_3-\partial_3u_r$ is the angular component of the vorticity. This is different from [@KNSS] in which the authors established a Liouville type of theorem by making full use of the strong maximum principle of the scalar equation of $\Gamma=ru_\theta$, under the assumption that $|ru|\le C$. Moereover, in comparison with the global regularity results in [@CSTY1; @CSTY2] and [@KNSS], we need but do not require the condition on the radial component of the velocity $u_r$. It should be noted that in the process of proving the Liouville type of theorem, we only need the condition . How to remove the condition will be very interesting and challenging. Second, we prove a Liouville type of theorem under weighted estimates of smooth solutions to the three-dimensional axisymmetric Navier-Stokes equations, which are also scaling invariant. This is motivated by weighted estimates in [@CKN] which can be carried out under assumptions of the initial data. Finally, in the end of the paper, we give further remarks on the possibility to rule out the singularity of the solution by using the Liouvile type of theorems established in this paper. The organization of this paper is as follows. In Section 2, we will present some preliminaries and establish a Liouville type of theorem under . In Section 3, we prove a Liouville type of theorem under weighted estimates of the solution. In Section 4, we give further remarks on how to use the Liouville type of theorem in future works. A Liouville Type of Theorem Under ================================== By an axisymmetric solution $(u, p)$ of , we mean that, in the cylindrical coordinate systems, the solution takes the form $p(x,t)=p(r,x_3,t)$ and $$u(x,t)=u_r(r,x_3,t)e_r+u_\theta(r,x_3,t)e_\theta+u_3(r,x_3,t)e_3,$$ where $$e_r=(\cos\theta, \sin\theta, 0),\quad e_\theta=(-\sin\theta, \cos\theta, 0),\quad e_3=(0, 0, 1).$$ Here $u_\theta(r,x_3,t)$ and $u_r(r,x_3,t)$ are the angular and radial components of $u(x,t)$ respectively. For the axisymmetric velocity field $u$, the corresponding vorticity $\omega=\nabla\times u$ is $$\omega=\omega_r e_r+\omega_\theta e_\theta+\omega_3 e_3,$$ where $$\omega_r=\t1_3 u_\theta,\ \omega_\theta=\t1_r u_3-\t1_3 u_r, \ \omega_3=-\frac 1r\t1_r(ru_\theta).$$ The 3-D axisymmetric Navier-Stokes equations read as $$\begin{aligned} &&\frac{\tilde Du_r }{Dt}-(\partial_r^2+\partial_3^2+\frac{1}{r}\partial_r)u_r+\frac{1}{r^2}u_r-\frac{1}{r}(u_\theta)^2+\partial_rp=0, \label{1.4} \\[3mm] &&\frac{\tilde Du_\theta}{Dt}-(\partial_r^2+\partial_3^2+\frac{1}{r}\partial_r)u_\theta+\frac{1}{r^2}u_\theta +\frac{1}{r}u_\theta u_r=0, \label{1.5} \\[3mm] &&\frac{\tilde Du_3}{Dt}-(\partial_r^2+\partial_3^2+\frac{1}{r}\partial_r)u_3+\t1_3p=0, \label{1.6} \\[3mm] &&\t1_r(ru_r)+\t1_3(ru_3)=0, \label{1.7}\end{aligned}$$ where $$\frac{\tilde D}{Dt}=\t1_t+u_r\t1_r+u_3\t1_3, \quad r=(x_1^2+x_2^2)^{1/2}.$$ In the following, we set $$\tilde\nabla=(\partial_r, \partial_3)$$ and use $C$ to denote an absolute constant which may be different from line to line. Without loss of generality, after translation on the time variable, $u(x,t)$ is assumed to be a smooth axisymmetric solution to -, defined in $\mathbb{R}^3\times (-1,0)$ with $u\in L^\infty(\mathbb{R}^3\times (-1,t')$ for any $-1<t'<0$. Let $$\begin{aligned} \label{P1} h(t)=\sup_{x\in \mathbb{R}^3}|u(x,t)|, \ \ H(t)=\sup_{-1\le s\le t<0} h(s).\end{aligned}$$ Suppose that the first singularity time for the solution $u(x,t)$ is at time $t=0$. Then it is clear that $\lim_{t\to 0-} H(t)=\infty$. In fact, by a classical result of Leray [@Le], if $u$ develops a singularity at $t=0$, then $$\begin{aligned} \label{P2} h(t)=\sup_{x\in \mathbb{R}^3}|u(x,t)|\ge \frac{\varepsilon_1}{\sqrt{-t}}\end{aligned}$$ for some $\varepsilon_1>0$. There exist $t_k\nearrow 0$ as $k\to\infty$ such that $H(t_k)=h(t_k)$. Denote $N_k=H(t_k)$. Then there exists a sequence of numbers $\gamma_k\searrow 1$ as $k\to\infty$ and $x_k\in \mathbb{R}^3$ such that $M_k=|u(x_k,t_k)|\ge N_k/\gamma_k, k=1,2\cdots$, satisfying $M_k\to \infty$ as $k\to\infty$. Define $$\begin{aligned} \label{P5} u^{(k)}(X,T)=\frac{1}{M_k}u(\frac{X_1}{M_k}, \frac{X_2}{M_k}, x_{k3}+\frac{X_3}{M_k}, t_k+\frac{T}{M^2_k}), k=1,2,\cdots\end{aligned}$$ In the cylindrical coordinate system, set $$u^{(k)}(X,T)=b^{(k)}(X,T)+u_\theta^{(k)}e_\theta,$$ where $b^{(k)}(X,T)=u^{(k)}_Re_R+u^{(k)}_Ze_Z, R=\sqrt{X_1^2+X_2^2}$. Then $u^{(k)}(X,T)$ are smooth solutions of the 3D Navier-Stokes equations, which are defined in $\mathbb{R}^3\times (A_k,B_k)$ with $$\begin{aligned} \label{P60} A_k=-M_k^2-M_k^2t_k, \ B_k=-M_k^2t_k.\end{aligned}$$ Note that $B_k=-M_k^2t_k\ge (\frac{N_k}{\gamma_k})^2(-t_k)\ge \frac{\varepsilon_1}{\gamma_k^2}$. Moreover, it holds that $$\begin{aligned} \label{P7} |u^{(k)}(X,T)|\le \gamma_k, X\in \mathbb{R}^3, T\in (A_k,0),\end{aligned}$$ and $$\begin{aligned} \label{P8} |u^{(k)}(M_kx_{k1},M_kx_{k2},0,0)|=1.\end{aligned}$$ It follows from the regularity theorem of the Navier-Stokes equations that $$\begin{aligned} \label{P8+} |\partial_Tu^{(k)}|+|D^lu^{(k)}|\le C_l, X\in \mathbb{R}^3, T\in (A_k,0]\end{aligned}$$ for $k=1, 2,\cdots$ and $|l|=0, 1, 2,\cdots$, where $l=(l_1,l_2,l_3)$ is a multi-index satisfying $l_1+l_2+l_3=|l|$ and $D^l=\frac{\partial^{|l|}}{\partial x_1^{l_1}\partial x_2^{l_2}\partial x_3^{l_3}}$. $C_l$ is a constant depending on $l$ but not on $k$. Then there exists a smooth function $\bar u(X,T)$ defined in $\mathbb{R}^3\times (-\infty,0)$ such that, for any $|l|=0,1,2 \cdots$, $$\begin{aligned} \label{Oct26} D^l u^{(k)}\longrightarrow D^l \bar u, \ k\to \infty, \\end{aligned}$$ uniformly in $C(\bar Q)$ for any compact subset $Q\subset\subset \mathbb{R}^3\times (-\infty,0]$. Denote $\bar \omega(X,T)=\bar\omega_\theta e_\theta+\bar\omega_re_r+\bar\omega_ze_z$ the voricity of $\bar u(X,T)$. Our main result of this section is a Liouville type of theorem as follows. [**Theorem 2.1**]{} Let $u(x,t)$ be an axisymmetric vector field defined in $\mathbb{R}^3\times(-1,0)$ which belongs to $L^\infty(\mathbb{R}^3\times(-1,t')$ for each $-1<t'<0$. Assume that $u$ satisfies $$\begin{aligned} \label{C10-1} |ru_\theta(x,t)|\le C, \ \ (x,t)\in \mathbb{R}^3\times (-1,0),\end{aligned}$$ and $$\begin{aligned} \label{Oct30} &\displaystyle\limsup_{\delta\to 0+}\|ru_\theta(x,t)\|_{L^\infty(\{x|r\le \delta\}\times (-1,0))}=0,\label{C1+}\end{aligned}$$ where $C>0$ is any finite constant. Then either $$\begin{aligned} \label{A-10} |u(x,t)|\le M, \ \ x\in \mathbb{R}^3,\ \ t\in [-1,0],\end{aligned}$$ where $M>0$ is an absolute constant depending on $C$, or $\bar\omega=0$ and $\bar u=(0,0,s(T))$, where $\bar u$ is same as in and $s(T): (-\infty, 0]\to \mathbb{R}$ is a bounded and continuous function. [**Remark 2.1**]{} The condition (\[C10-1\]) can be removed if the initial data satisfies $\|ru_{0\theta}\|_{L^\infty}<\infty$ (see [@Chae; @JX]). The condition means that the singularity of $u_\theta$ near the symmetry axis is of the rate $\frac{o(1)}{r}$ with $o(1)\to 0$ as $r\to 0$. [**Proof.**]{} Suppose that is false. Then one can rescale the solution as in -. It will be shown that $\bar\omega(X,T)=0$ and $\bar u(X,T)=(0,0,s(T))$ with $s(T): (-\infty, 0)\to \mathbb{R}$ a bounded and continuous function. Let $C_0>0$ be any fixed constant. For any $X\in \mathbb{R}^3$ with $R\le C_0$ and $T\in (A_k,0]$, it follows from that $$\begin{aligned} \label{P14} &&|\Gamma^{(k)}(X,T)|\equiv|Ru_\theta^{(k)}|=|\frac{R}{M_k}u_\theta(\frac{X_1}{M_k}, \frac{X_2}{M_k}, x_{k3}+\frac{X_3}{M_k}, t_k+\frac{T}{M^2_k})|\nonumber\\[3mm] &&\le \tilde F(k,C_0)\to 0\end{aligned}$$ as $k\to \infty$. Set $$F(k,C_0)=\max(\tilde F(k,C_0),\frac1k), \quad k=1,2,3\cdots.$$ It follows that $$\begin{aligned} \label{P15} |u_\theta^{(k)}(X,T)|\le R^{-1}F(k,C_0), 0<R\le C_0, T\in (A_k,0].\end{aligned}$$ It follows from (\[C10-1\]) that $$|u_\theta^{(k)}(X,T)|\le \frac{C}{R}, \ \ R>0, T\in (A_k,0]$$ Using (\[P8+\]) and the fact that $u_\theta^{(k)}|_{R=0}=0$, one has $$\begin{aligned} &|u_\theta^{(k)}(X,T)|\le C \min(R, R^{-1}), R>0, T\in (A_k,0],\label{P16}\\[3mm] & |\partial_Zu_\theta^{(k)}(X,T)|\le C \min(R, 1), R>0, T\in (A_k,0].\label{P17}\end{aligned}$$ Consequently, for any $T\in (A_k,0]$, $$\begin{aligned} \label{June-2-1} |u_\theta^{(k)}(X,T)|\le \left\{ \begin{array}{lll} &CR, \ & R<\sqrt{F(k,C_0)}, \\[3mm] &\frac{F(k,C_0)}{R}, \ & \sqrt{F(k,C_0)}\le R\le C_0, \\[3mm] &\frac{C}{R}, \ &R>C_0. \end{array} \right.\end{aligned}$$ $$\begin{aligned} \label{June-2-2} |\frac{\partial_Z(u_\theta^{(k)})^2(R,Z,T)}{R^2}|\le \left\{ \begin{array}{lll} &C, \ & R<\sqrt{F(k,C_0)}, \\[3mm] &C\frac{F(k,C_0)}{R^2}, \ &\sqrt{F(k,C_0)}\le R<1,\\[3mm] &C\frac{F(k,C_0)}{{R}^3}, \ & 1\le R\le C_0, \\[3mm] &\frac{C}{{R}^3}, \ &R\ge C_0. \end{array} \right.\end{aligned}$$ Let $\Omega=\frac{\omega_\theta(x,t)}{r}$ and $f^{(k)}=\Omega^{(k)}(X,T)=\frac{\omega^{(k)}_\theta(X,T)}{R}$. Then it holds that $$\begin{aligned} \label{20} |f^{(k)}(X,T)|\le C(1+R)^{-1},\ \ X\in \mathbb{R}^3, T\in (A_k,0).\end{aligned}$$ It follows from the equation of $\omega_\theta$ that $$\begin{aligned} \label{21} (\partial_T-L)f^{(k)}=g^{(k)}, \ \ L=\Delta+\frac2R\partial_R-b^{(k)}\cdot\nabla_X,\end{aligned}$$ where $g^{(k)}=R^{-2}\partial_Z(u_\theta^{(k)})^2$ and $b^{(k)}=u_R^{(k)}e_R+u_Z^{(k)}e_Z$. Regarding $f^{(k)}(X,T)=f^{(k)}(R,Z,T)$ as a 5-dimensional axisymmetric function by denoting $X=(\tilde X, X_5)=(X_1,\cdots,X_4,X_5)$, $R=|\tilde X|=\sqrt{X_1^2+X_2^2+X_3^2+X_4^2}$ and $Z=X_5$, we obtain $$\begin{aligned} \label{21+} (\partial_T+\tilde b^{(k)}\cdot\tilde \nabla_X-\Delta_5)f^{(k)}=g^{(k)}, \ \\end{aligned}$$ where $\tilde b^{(k)}=u_R^{(k)} \tilde e_R+u_Z^{(k)}\tilde e_Z$ with $\tilde e_R=(\frac{X_1}{R}, \frac{X_2}{R}, \frac{X_3}{R}, \frac{X_4}{R}, 0)$ and $\tilde e_Z=(0,0,0,0,1)$. The scaling can be rewritten as $$\begin{aligned} \label{Jan2} u_\theta^{(k)}(Y,T)=\frac{1}{M_k}u(\frac{R}{M_k}, z_k+\frac{Y_5}{M_k},t_k+\frac{T}{M_k^2}),\end{aligned}$$ where $z_k=x_{3k}$. Denote $P(T,X;S,Y)$ the kernel for $\partial_T+\tilde b^{(k)}\cdot\tilde \nabla_X-\Delta_5$ and $Y=(\tilde Y, Y_5)=(Y_1, \dots, Y_5)$. By the Duhamel’s formula, $$\begin{aligned} \label{22} &&f^{(k)}(X,T)=\int P(T,X;S,Y)f^{(k)}(Y,S) dY+\int_S^T\int P(T,X;\tau,Y)g^{(k)}(Y,\tau) dYd\tau\nonumber\\[3mm] &&=:I+II.\end{aligned}$$ Due to Carlen-Loss [@CL] and Chen-Strain-Tsai-Yau [@CSTY2], the kernel $P$ satisfies $P\ge 0, \int P(T,X;S,Y) dY=1$ and $$P(T,X;S,Y)\le C(T-S)^{-\frac52}e^{-h(|X-Y|,T-S)}, \ h(a,T)=C\frac{a^2}{T}[(1-\frac{T}{a})_+]^2,$$ where $f_+=\max \{0, f\}$ and we have used the fact that $\|\tilde b^{(k)}\|_\infty\le \gamma_k\le 2$ for $k\ge N$. It can be verified that the function $e^{-h(a,T)}, a\ge 0, T\ge 0,$ has the following properties: When $T\ge T_0>0$ for any fixed (but may be small) $T_0>0$, one has $$\begin{aligned} \label{23-0} e^{-h(a,T)}\le Ce^{-Ca/T}\end{aligned}$$ holds for some constant $C>0$ which may depend on $T_0$. When $T\ge \frac a2$, one has $$\begin{aligned} \label{23-1} e^{-h(a,T)}\le Ce^{-Ca/T}.\end{aligned}$$ When $0\le T\le \frac a2$, it is easy to get $$\begin{aligned} \label{23-2} e^{-h(a,T)}=e^{-C\frac{a^2}{T}(1-\frac{T}{a})^2}\le e^{-Ca^2/T}.\end{aligned}$$ In and , $C>0$ is some uniform constant. It follows from (\[23-0\]) and Hölder inequality that $$\begin{aligned} \label{24} &&|I|\le [\int P(T,X;S,Y)|f^{(k)}(Y,S)|^5 dY]^{\frac15}\nonumber\\[3mm] &&\le [C(T-S)^{-\frac52}\int e^{-C\frac{|X_5-Y_5|}{T-S}}\frac{\mathbb{R}^3 dR}{(1+R)^5} dY_5]^\frac15\nonumber\\[3mm] &&\le C(T-S)^{-\frac{3}{10}}\end{aligned}$$ for all $X\in \mathbb{R}^5, T, S\in (A_k,0)$ satisfying $T-S>0$. Let $L=\{\tau\in [S,T]: T-\tau\ge \frac{|X-Y|}{2}\}$ and $L^c=\{\tau\in [S,T]: T-\tau\le \frac{|X-Y|}{2}\}$. For any $C_0>0$, with help of (\[23-1\]) and (\[23-2\]), we have $$\begin{aligned} \label{25} &&|II|\le \int_L\int C(T-\tau)^{-\frac52} e^{-C\frac{|X-Y|}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\nonumber\\[3mm] &&+\int_{L^c} \int C(T-\tau)^{-\frac52} e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\nonumber\\[3mm] &&\le C\int_S^T (T-\tau)^{-\frac52} d\tau\int e^{-C\frac{|X_5-Y_5|}{T-\tau}}dY_5(\int_{\{R\le C_0\}}e^{-C\frac{|\tilde X-\tilde Y|}{T-\tau}} d\tilde Y)^{\frac 12}(\int_{\{ R\le C_0\}} |g^{(k)}(Y,\tau)|^2 d\tilde Y)^{\frac 12}\nonumber\\[3mm] && +C\int_S^T (T-\tau)^{-\frac52} d\tau\int e^{-C\frac{|X_5-Y_5|}{T-\tau}}dY_5(\int_{\{R\ge C_0\}}e^{-C\frac{|\tilde X-\tilde Y|}{T-\tau}} d\tilde Y)^{\frac 12}(\int_{\{ R\ge C_0\}} |g^{(k)}(Y,\tau)|^2 d\tilde Y)^{\frac 12}\nonumber\\[3mm] &&+C\int_S^T (T-\tau)^{-\frac52}d\tau\int e^{-C\frac{|X_5-Y_5|^2}{T-\tau}} dY_5(\int_{\{R\le C_0\}} e^{-C\frac{|\tilde X-\tilde Y|^2}{T-\tau}} d\tilde Y)^{\frac23}(\int_{\{ R\le C_0\}} |g^{(k)}(Y,\tau)|^3 d\tilde Y)^{\frac 13}\nonumber\\[3mm] &&+C\int_S^T (T-\tau)^{-\frac52}d\tau\int e^{-C\frac{|X_5-Y_5|^2}{T-\tau}} dY_5(\int_{\{R\ge C_0\}} e^{-C\frac{|\tilde X-\tilde Y|^2}{T-\tau}} d\tilde Y)^{\frac23}(\int_{\{ R\ge C_0\}} |g^{(k)}(Y,\tau)|^3 d\tilde Y)^{\frac 13}.\nonumber\\[3mm]\end{aligned}$$ It follows from (\[June-2-2\]) that $$\begin{aligned} &&\int_{\{ R\le C_0\}} |g^{(k)}(Y,\tau)|^2 d\tilde Y\\[3mm] &&=[\int_{\{ R<\sqrt{F(k,C_0)}\}}+\int_{\{ \sqrt{F(k,C_0)}\le R<1\}}+\int_{\{ 1\le R<C_0\}}] |g^{(k)}(Y,\tau)|^2 d\tilde Y\\[3mm] &&\le C[(\sqrt{F(k,C_0)})^4+F^2(k,C_0)(-\ln \sqrt{F(k,C_0)})+F^2(k,C_0)].\end{aligned}$$ and $$\begin{aligned} &&\int_{\{ R\le C_0\}} |g^{(k)}(Y,\tau)|^3 d\tilde Y\\[3mm] && \le C[(\sqrt{F(k,C_0)})^4+F^3(k,C_0)( \sqrt{F(k,C_0)})^{-2}+F^3(k,C_0)].\end{aligned}$$ It concludes that $$\begin{aligned} |II|&&\le C(T-S)^\frac32[(\sqrt{F(k,C_0)})^4+F^2(k,C_0)(-\ln \sqrt{F(k,C_0)})+F^2(k,C_0)+C_0^{-1}]\nonumber\\[3mm] &&+C(T-S)^\frac 13[(\sqrt{F(k,C_0)})^4+F^3(k,C_0)( \sqrt{F(k,C_0)})^{-2}+F^3(k,C_0)+C_0^{-\frac 53}].\end{aligned}$$ Letting $T-S=C_0^\frac13$, we obtain $$\begin{aligned} \label{25+} && |I|+|II|\le C(C_0^{-\frac{1}{10}}+C_0^{-\frac12}+C_0^{-\frac{14}{9}})\\[3mm] &&+C[(\sqrt{F(k,C_0)})^4+F^2(k,C_0)(-\ln \sqrt{F(k,C_0)})+F^2(k,C_0)]^\frac12\nonumber\\[3mm] &&+C[(\sqrt{F(k,C_0)})^4+F^3(k,C_0)( \sqrt{F(k,C_0)})^{-2}+F^3(k,C_0)]^\frac13.\end{aligned}$$ Taking the limit $k\to\infty$ first and then letting $C_0\to \infty$, we have $$\begin{aligned} \label{26+} |f^{(k)}(X,T)| \to 0,\end{aligned}$$ and hence $$\begin{aligned} \label{26+} |\omega_\theta^{(k)}(X,T)| \to 0,\end{aligned}$$ which implies that $\bar\omega_\theta(R,Z,T)=0$. Consequently, we obtain that $\bar \omega_\theta(X,T)=0$ for $X\in \mathbb{R}^3$ and $T\in (-\infty,0)$. Denote $\bar b(X,T)=\bar u_Re_R+\bar u_Z$. It follows from that $$\begin{aligned} \label{27} u^{(k)}(X,T)\to \bar u (X,T),\ b^{(k)}(X,T)\to \bar b (X,T), \ {\rm as}\ \ k\to \infty,\end{aligned}$$ uniformly in $C(\bar Q)$ with any compact subset $Q\subset\subset \mathbb{R}^3\times (-\infty,0]$. As a consequence of $curl_X b^{(k)}(X,T)=\omega_\theta^{(k)}e_\theta$, $div_X b^{(k)}(X,T)=0$ and , $\bar b(X,T)$ is a harmonic and bounded function. That is, $\Delta_X \bar b=0$ and $\bar b(X,T)$ is bounded. Since $\bar u_R(0,Z,T)=0$, there exists a continuous and bounded function $s(T): (-\infty,0]\to \mathbb{R}$ such that $\bar b(X,T)=(0, s(T))$. Moreover, since $|Ru_\theta^{(k)}|\to 0$ as $k\to \infty$ for $0<R\le C_0$, where $C_0>0$ is arbitrary, it follows that $\bar u_\theta(X,T)=0$ for all $X\in \mathbb{R}^3, T\in (-\infty, 0)$. Therefore we have proved that $\bar u(X,T)=(0, 0, s(T))$ and furthermore $\bar\omega(X,T)=0$. The proof of the theorem is finished. A Liouville Type Theorem Under Weighted Estimates ================================================= In [@CKN], some weighted estimates were obtained for suitable weak solutions $(u(x,t),p(x,t))$ of the three-dimensional Navier-Stokes equations . More precisely, suppose that there exists a small number $L_0>0$ such that if $$\begin{aligned} \label{Feb22-1} \int_{\mathbb{R}^3} \frac{|u_0|^2}{|x|} dx \le L\end{aligned}$$ with $0<L<L_0$, then $$\begin{aligned} \label{Feb22-2} \int_{\mathbb{R}^3} \frac{|u|^2}{|x|} dx+(L_0-L)exp\{\frac{1}{L_0}\int_0^t\int_{\mathbb{R}^3} \frac{|\nabla u|^2}{|x|}\} dxdt\le L_0, \ \ t\in [0,T].\end{aligned}$$ For the three-dimensional axisymmetric Navier-Stokes equations, if $$\begin{aligned} \label{Oct26-3} \int_{\mathbb{R}^3} \frac{|u_0|^2}{r} dx \le L,\end{aligned}$$ where $L>0$ is same as in , then it is clear that $$\begin{aligned} \label{N-7-1} \int_{\mathbb{R}^3} \frac{|u_0|^2}{\sqrt{r^2+(z-z_0)^2}} dx \le L\end{aligned}$$ for any $z_0\in R$. By the translation with respect to $z(=x_3)$, one can prove in a similar way as in [@CKN] that if holds, then $$\begin{aligned} \label{N-7-2} &&\int_{\mathbb{R}^3} \frac{|u|^2}{\sqrt{r^2+(z-z_0)^2}} dx\nonumber\\ &&~~~~~~~+(L_0-L)exp\{\frac{1}{L_0}\int_0^t\int_{\mathbb{R}^3} \frac{|\nabla u|^2}{\sqrt{r^2+(z-z_0)^2}}\} dxdt\le L_0\end{aligned}$$ for any $z_0\in \mathbb{R}$, where $L$ and $L_0$ are same as in . Note that the quantities on the left hand side of are scaling invariant. Let $B_{\bar R}=\{x\in \mathbb{R}^3| |x|<R\}$ be a ball with radius $\bar R>0$. Motivated by the weighted estimates , we have [**Theorem 3.1**]{} Let $u(x,t)$ be an axisymmetric vector field defined in $\mathbb{R}^3\times(-1,0)$ which belongs to $L^\infty(\mathbb{R}^3\times(-1,t')$ for each $-1<t'<0$ and $L^\infty (B_{\bar R}^c\times (-1,0))$ for some $\bar R>0$, where $B_{\bar R}^c=\mathbb{R}^3\backslash B_{\bar R}$. Moreover, assume that $u$ satisfies $$\begin{aligned} && (1)\ \ |ru_\theta(x,t)|\le C, \ \ (x,t)\in \mathbb{R}^3\times (-1,0), \label{C1}\\[3mm] && (2)\ \ \int_{\mathbb{R}^3} \frac{|u|^2}{\sqrt{r^2+(z-z_0)^2}} dx+\int_{-1}^0\int_{\mathbb{R}^3} \frac{|\omega_r|^2}{\sqrt{r^2+(z-z_0)^2}} r drdzdt\le C \ \label{Oct26-2}\end{aligned}$$ for any $z_0\in \mathbb{R}$. Then either $$\begin{aligned} \label{A-1} |u(x,t)|\le M, \ \ x\in \mathbb{R}^3,\ \ t\in (-1,0],\end{aligned}$$ where $M>0$ is an absolute constant depending on $C$, or $\bar u=0$, where $\bar u$ is defined in . [**Remark 3.1**]{} The assumption that $u(x,t)\in L^\infty (B_{\bar R}^c\times (-1,0))$ for some $R>0$ can be easily satisfied if the solution decays at far fields for all $t\in (-1,0)$. In particular, it follows from [@CKN] that [**Proposition 3.2**]{} Suppose that $u_0\in L^2(\mathbb{R}^3)$ and $$\int_{\mathbb{R}^3} |\nabla u_0|^2 dx <\infty.$$ Given a suitable weak solution $u(x,t)$ of the three-dimensional Navier-Stokes equations with initial data $u_0$. Then $u(x,t)$ is regular in the region $B^c_{\bar R}$ for some $\bar R>0$. [**Remark 3.2**]{} The second term in can be replaced by $$\begin{aligned} \label{Oct26-4} \int_0^t\int_{\mathbb{R}^3} \frac{|\omega_r|^2}{r} dxdt\le C.\end{aligned}$$ Note that $\omega_r=\partial_zu_\theta$. The second term in the condition is a weighted estimate on $\partial_zu_\theta$. Now we give a proof of Theorem 3.1. [**Proof of Theorem 3.1**]{} Suppose that the solution $u(x,t)$ has singularity at $t=0$. Then similar to the proof of Theorem 2.1, there exist $t_k\nearrow 0$ as $k\to\infty$ such that $H(t_k)=h(t_k)$, where $H(t)$ and $h(t)$ are same as in . Denote $N_k=H(t_k)$. Since $u(x,t)\in L^\infty (B_{\bar R}^c\times (-1,0))$ for some $\bar R>0$ by the assumption, we can choose a sequence of numbers $\gamma_k\searrow 1$ as $k\to\infty$ and $x_k\in B_{\bar R}$ such that $M_k=|u(x_k,t_k)|\ge N_k/\gamma_k, k=1,2\cdots$, satisfying $M_k\to \infty$ as $k\to\infty$. Using the same scaling , one can prove the theorem as for Theorem 2.1. To be more precise, we continue the proof based on . The first term $I$ in can be estimated as in . Thus, one needs to focus on the estimate of the second term $II$ in . Note that $$\begin{aligned} \label{251} &&|II|\le \int_L\int C(T-\tau)^{-\frac52} e^{-C\frac{|X-Y|}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\nonumber\\[3mm] &&+\int_{L^c} \int C(T-\tau)^{-\frac52} e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\equiv II_1+II_2,\end{aligned}$$ where $L=\{\tau\in [S,T]: T-\tau\ge \frac{|X-Y|}{2}\}$ and $L^c=\{\tau\in [S,T]: T-\tau\le \frac{|X-Y|}{2}\}$. Here we are estimating in 5-dimensional space, $X=(\tilde X,X_5)=(X_1,\cdots,X_4,X_5), Y=(\tilde Y,Y_5)=(Y_1,\cdots,Y_4,Y_5)$ and $R=|\tilde X|=\sqrt{X_1^2+\cdots+X_4^2}$. First, $II_2$ can be estimated as follows. $$\begin{aligned} \label{Feb22-5} &&|II_2|\le\int_S^T\int_{R^5} (T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\nonumber\\ &&=\int_S^T [\int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\le C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\le C_0\}}+\int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\ge C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\le C_0\}}\nonumber\\ &&+\int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\le C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\ge C_0\}}+\int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\ge C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\ge C_0\}}](T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\nonumber\\ &&\equiv J_1+J_2+J_3+J_4.\end{aligned}$$ For $0<\alpha<1$, direct estimates lead to $$\begin{aligned} \label{Feb22-6} &&J_1\le \int_S^T \int_{\{|Y|\le 2\sqrt{C_0(T-S)}+2|X|\}}(T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}\frac{|u_\theta^{(k)}\partial_Zu_\theta^{(k)}|}{R^2} dYd\tau\nonumber\\ &&\le \int_S^T\int_{\{|Y|\le 2\sqrt{C_0(T-S)}+2|X|\}}(T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}\frac{|\partial_Zu_\theta^{(k)}|^\alpha}{R^\alpha} \frac{|\partial_Zu_\theta^{(k)}|^{1-\alpha}}{(R^2|Y|)^{\frac{1-\alpha}{2}}}|Y|^{\frac{1-\alpha}{2}} \frac{|u_\theta^{(k)}|}{R}dYd\tau\nonumber\\ &&\le( 2\sqrt{C_0(T-S)}+2|X|)^{\frac{1-\alpha}{2}}(\int_S^T\int_{\{|Y|\le 2\sqrt{C_0(T-S)}+2|X|\}} \frac{|\partial_Zu_\theta^{(k)}|^2}{R^2|Y|} dYd\tau)^{\frac{1-\alpha}{2}}\nonumber\\ &&\times(\int_S^T\int_{\{|Y|\le 2\sqrt{C_0(T-S)}+2|X|\}} (T-\tau)^{-\frac{5}{1+\alpha}}e^{-C\frac{|X-Y|^2}{T-\tau}}dYd\tau)^{\frac{1+\alpha}{2}}\nonumber\\ &&\le F(C_0,T-S, |X|,\alpha,k)(\int_S^T (T-\tau)^{-\frac{5}{1+\alpha}+\frac52}d\tau)^{\frac{1+\alpha}{2}}\nonumber\\ &&\le F(C_0,T-S, |X|,\alpha,k),\end{aligned}$$ where $$F(C_0,T-S, |X|,\alpha,k)=F_1(C_0,T-S, |X|,\alpha) (\int_S^T\int_{\{|Y|\le 2\sqrt{C_0(T-S)}+2|X|\}} \frac{|\partial_Zu_\theta^{(k)}|^2}{R^2|Y|} dYd\tau)^{\frac{1-\alpha}{2}}$$ and $F_1(C_0,T-S, |X|,\alpha)$ is a constant depending on $ C_0,T-S, |X|$ and $\alpha$. Note that the angular component of the velocity in is $$u_\theta^{(k)}(Y,T)=\frac{1}{M_k}u_\theta(\frac{R}{M_k}, z_k+\frac{Y_5}{M_k},t_k+\frac{T}{M_k^2}),$$ where $z_k=x_{3k}$. Letting $$y_1=\frac{Y_1}{M_k}, \cdots, y_4=\frac{Y_4}{M_k},z=z_k+\frac{Y_5}{M_k}, t=t_k+\frac{\tau}{M_k^2}, r=\sqrt{y_1^2+\cdots+y_4^2},$$ one can get that $$\begin{aligned} \label{Nov30-2} && \int_S^T\int_{\{|Y|\le 2\sqrt{C_0(T-S)}+2|X|\}} \frac{|\partial_Zu_\theta^{(k)}|^2}{R^2|Y|} dYd\tau\nonumber\\ &&=\int_{t_k+\frac{S}{M_k^2}}^{t_k+\frac{T}{M_k^2}}\int_{\{\sqrt{r^2+(z-z_k)^2}\le \frac{2\sqrt{C_0(T-S)}+2|X|}{M_k}\}} \frac{|\partial_zu_\theta|^2}{\sqrt{r^2+(z-z_k)^2}} rdrdzdt\end{aligned}$$ Note that $|z_k|=|x_{k3}|\le \bar R$ is bounded. There exists a subsequence of $\{(z_k, t_k)\}$, still denoted by itself, and $\bar z\in [-\bar R, \bar R]$ such that $t_k\to 0, z_k\to \bar{z}$ as $k\to \infty$. By the assumption , for any $\varepsilon>0$, there exists a $\delta>0$ such that $$\begin{aligned} \label{Nov-2-0} \int_{-2\delta}^{0}\int_0^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta}\frac{|\partial_zu_\theta|^2}{\sqrt{r^2+(z-\bar z)^2}} r drdzdt \le \varepsilon.\end{aligned}$$ Using again leads to $$\begin{aligned} \label{Nov-30-0} \int_{-2\delta}^{0}\int_0^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta}\frac{|\partial_zu_\theta|^2}{\sqrt{r^2+(z-\bar z_k)^2}} r drdzdt \le C.\end{aligned}$$ Taking $k>K$ large enough such that $(t_k+\frac{S}{M_k},t_k+\frac{T}{M_k})\subset (-\delta, 0)$ and $\{(r,z)\in (0,\infty)\times(-\infty,\infty)|\sqrt{r^2+(z-\bar z)^2}\le \frac{2\sqrt{C_0(T-S)}+2|X|}{M_k}\}\subset (0,\delta)\times (\bar z-\delta,\bar z+\delta)$. Let $$h_k(r,z,t)=\frac{|\partial_zu_\theta|^2}{\sqrt{r^2+(z-z_k)^2}}, h(r,z,t)=\frac{|\partial_zu_\theta|^2}{\sqrt{r^2+(z-\bar z)^2}},$$ where $(r,z)\in (0,2\delta)\times(\bar z-2\delta,\bar z+2\delta)$. For any fixed $0<\delta_0<\delta$, we choose $0\le\varphi(r,z,t)\le 1$ to be a smooth function defined in $(0,\infty)\times(-\infty,\infty)\times (-1,0)$ satisfying $\varphi(r,z,t)\equiv 1$ if $(r,z,t)\in (\delta_0,\delta)\times(\bar z-\delta,\bar z+\delta)\times(-\delta,-\delta_0)$ and $\varphi(x,t)\equiv 0$ if $(r,z,t)\not\in Q_{\delta,\delta_0}\equiv (\frac{\delta_0}{2},2\delta)\times(\bar z-2\delta,\bar z+2\delta)\times (-2\delta,-\frac{\delta_0}{2})$. Then, there exists a subsequence (still denoted by itself) such that $$\begin{aligned} &&h_k\rightharpoonup \tilde h\quad {\rm in}\quad \mathcal{M}(Q_{\delta,\delta_0})\quad\quad (by\ \eqref{Nov-30-0})\\[3mm] &&h_k\to h\quad {\rm a.e.\ on} \ Q_{\delta,\delta_0},\end{aligned}$$ as $k\to \infty$, where $\tilde h\in \mathcal{M}(Q_{\delta,\delta_0})$ which is the finite Radon measure space restricted on $Q_{\delta,\delta_0}$. In particular, it concludes that $\tilde h=h$ and $$\begin{aligned} \label{Nov-2-1} \int_{Q_{\delta,\delta_0}} h_k\varphi r drdzdt\to \int_{Q_{\delta,\delta_0}} h\varphi r drdzdt,\end{aligned}$$ as $k\to \infty$. That is $$\begin{aligned} \label{Nov-30-1} \int_{-2\delta}^{-\frac{\delta_0}{2}}\int_{\frac{\delta_0}{2}}^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta} h_k\varphi r drdzdt\to \int_{-2\delta}^{-\frac{\delta_0}{2}}\int_{\frac{\delta_0}{2}}^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta} h\varphi r drdzdt,\end{aligned}$$ as $k\to \infty$. Using and , one obtains, for any $0<\delta_0<\delta$, that $$\begin{aligned} &&\limsup_{k\to\infty}\int_{-\delta}^{-\delta_0}\int_{\delta_0}^{\delta}\int_{\bar z-\delta}^{\bar z+\delta} h_k r drdzdt\le \limsup_{k\to\infty}\int_{-2\delta}^{-\frac{\delta_0}{2}}\int_{\frac{\delta_0}{2}}^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta} h_k\varphi r drdzdt\\ &&=\int_{-2\delta}^{-\frac{\delta_0}{2}}\int_{\frac{\delta_0}{2}}^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta} h\varphi r drdzdt \le \int_{-2\delta}^{0}\int_0^{2\delta}\int_{\bar z-2\delta}^{\bar z+2\delta} h r drdzdt\le \varepsilon.\end{aligned}$$ Due to the arbitrariness of $\delta_0$ and $\varepsilon>0$, and thanks to and , one obtains that $$\begin{aligned} \label{N-6-1} |J_1|\le F_1(C_0,T-S, |X|,\alpha)\int_{-\delta}^0\int_0^{\delta}\int_{\bar z-\delta}^{\bar z+\delta} h_k r drdzdt\to 0, \quad k\to \infty.\end{aligned}$$ Now we continue estimating $J_2-J_4$. $$\begin{aligned} &&J_2=\int_S^T \int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\ge C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\le C_0\}}(T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\\ &&\le \int_S^T (T-\tau)^{-\frac52} d\tau\int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\ge C_0\}}e^{-C\frac{|X_5-Y_5|^2}{T-\tau}}dY_5\int_{\{\frac{|\tilde X-\tilde Y|^2}{T-\tau}\le C_0\}}e^{-C\frac{|\tilde X-\tilde Y|^2}{T-\tau}} \frac{|u_\theta^{(k)}\partial_Zu_\theta^{(k)}|}{R^2} d\tilde Y\\ &&\le \int_S^T (T-\tau)^{-2} d\tau \int_{|\xi|\ge C_0} e^{-C|\xi|^2} d\xi\int_{\{|\tilde Y|\le \sqrt{C_0(T-S)}+|\tilde X|\}}e^{-C\frac{|\tilde X-\tilde Y|^2}{T-\tau}} \frac{|u_\theta^{(k)}\partial_Zu_\theta^{(k)}|}{R^2} d\tilde Y\\ &&\le C\int_{|\xi|\ge C_0}e^{-C|\xi|^2} d\xi (T-S),\end{aligned}$$ where one has used the transformation $\xi=\frac{X_5-Y_5}{T-\tau}$. $$\begin{aligned} &&J_3=\int_S^T \int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\le C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\ge C_0\}}(T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\\ &&\le \int_S^T (T-\tau)^{-\frac52} d\tau\int_{\frac{|X_5-Y_5|^2}{T-\tau}\le C_0}e^{-C\frac{|X_5-Y_5|^2}{T-\tau}}dY_5\int_{\{\frac{|\tilde X-\tilde Y|^2}{T-\tau}\ge C_0\}}e^{-C\frac{|\tilde X-\tilde Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dY\\ &&\le C(T-S)\int_{|\tilde\xi|\ge C_0} e^{-C|\tilde\xi|^2} d\tilde\xi,\end{aligned}$$ where $\tilde\xi=\frac{\tilde X-\tilde Y}{T-\tau}$. Similarly, one can estimate $$\begin{aligned} &&J_4=\int_S^T \int_{\{\frac{|X_5-Y_5|^2}{T-\tau}\ge C_0,\frac{|\tilde X-\tilde Y|^2}{T-\tau}\ge C_0\}}(T-\tau)^{-\frac52}e^{-C\frac{|X-Y|^2}{T-\tau}}|g^{(k)}(Y,\tau)| dYd\tau\\ &&\le C(T-S)\int_{|\xi|\ge C_0} e^{-C|\xi|^2} d\xi\int_{|\tilde\xi|\ge C_0} e^{-C|\tilde\xi|^2} d\tilde\xi.\end{aligned}$$ Putting estimates of $J_1-J_4$ into gives $$\begin{aligned} \label{Nov30-3} |II_2|&&\le F(C_0,T-S, |X|,\alpha,k)\nonumber\\[3mm] &&+C[(T-S)+1](\int_{|\xi|\ge C_0} e^{-C|\xi|^2} d\xi +\int_{|\tilde\xi|\ge C_0} e^{-C|\tilde\xi|^2} d\tilde\xi).\end{aligned}$$ The term $II_1$ can be treated as for $J_1$ so that $$\begin{aligned} \label{Nov30-51} |II_1|\le F(C_0,T-S, |X|,\alpha,k)\to 0, k\to\infty,\end{aligned}$$ for $C_0\ge 2$. Now taking $T-S=C_0\ge 2$ and using ,, and , one gets $$\begin{aligned} \label{N-6-2} |f^{(k)}(X,T)|&&\le C[C_0^{-\frac{3}{10}}+F(C_0,C_0, |X|,\alpha,k)\nonumber\\ &&+C_0(\int_{|\xi|\ge C_0} e^{-C|\xi|^2} d\xi+\int_{|\tilde\xi|\ge C_0} e^{-C|\tilde\xi|^2} d\tilde\xi)].\end{aligned}$$ Passing to the limit $k\to \infty$ first and then letting $C_0\to \infty$ in , one obtains that, for any $X\in \mathbb{R}^3, T\in (-\infty,0)$, $$\begin{aligned} \label{26+} |f^{(k)}(X,T)| \to 0, \ k\to \infty,\end{aligned}$$ and hence $$\begin{aligned} \label{N-6-3} |\omega_\theta^{(k)}(X,T)| \to 0, \ k\to\infty,\end{aligned}$$ which implies that $\bar\omega_\theta(R,Z,T)=0$. Since $curl_X b^{(k)}(X,T)=\omega_\theta^{(k)}e_\theta$ and $div_X b^{(k)}(X,T)=0$ so $\bar b(X,T)$ is a harmonic and bounded function defined on $\mathbb{R}^3\times (-\infty,0)$. Since $\bar u_R(0,Z,T)=0$, thus there exists a continuous and bounded function $s(T): (-\infty,0]\to \mathbb{R}$ such that $\bar b(X,T)=s(T)e_Z$. Using , one obtains that $\bar b(X,T)=0$. To prove that $\bar u_\theta=0$, one needs the following lemma due to [@KNSS]. [**Lemma 3.3**]{} Let $u(x,t)$ be a bounded weak solution of the Navier-Stokes equations in $\mathbb{R}^3\times(-\infty,0)$. Assume that $u(x,t)$ is axisymmetric and satisfies $$|u(x,t)\le \frac{C}{\sqrt{x_1^2+x_2^2}} \quad {\rm in}\quad \mathbb{R}^3\times (-\infty,0).$$ Then $u=0$ in $\mathbb{R}^3\times (-\infty,0)$. Applying the facts that $\bar b(X,T)=0$ and $ |R\bar u_\theta|\le C$, the proof of Theorem 3.1 follows from Lemma 3.3. Further Remarks =============== To rule out the possible singularity of the solution, one should make further investigations. As it is known, Liouville type of theorems play an important role in blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. How to apply Theorems 2.1 and Theorem 3.1 to study the global regularity of the three-dimensional axisymmetric Navier-Stokes equations would be interesting. As mentioned in the proof of Theorems 2.1 and 3.1, if the solution is not bounded, then there exist $t_k\nearrow 0$ as $k\to\infty$ such that $H(t_k)=h(t_k)$, where $h(t)$ and $H(t)$ are defined as . Denote $N_k=H(t_k)$. Then one can choose a sequence of numbers $\gamma_k\searrow 1$ as $k\to\infty$ and $x_k\in \mathbb{R}^3$ such that $M_k=|u(x_k,t_k)|\ge N_k/\gamma_k, k=1,2\cdots$, satisfying $M_k\to \infty$ as $k\to\infty$. Denote $r_k=\sqrt{(x_{k1})^2+(x_{k2})^2}$. We consider the following two cases. [**Case I.**]{} $\{r_kM_k\} (k=1,2,\cdots)$ is uniformly bounded. In this case, there exists a constant $C>0$ such that $$\begin{aligned} \label{P3} r_kM_k\le C.\end{aligned}$$ We will use the scaling as in . In view of (\[P3\]), there exists a point $(x_{\infty_1}, x_{\infty2})\in \mathbb{R}^2$ such that, up to a subsequence, $(M_kx_{k1},M_kx_{k2})\to (x_{\infty_1}, x_{\infty2})$ as $k\to\infty$. Here $\sqrt{(x_{\infty_1})^2+(x_{\infty_2})^2}\le C<\infty$. It follows from (\[P8\]) and (\[27\]) that $\bar u(x_{\infty_1},x_{\infty_2},0)=1$. Theorem 2.1 implies that $\bar u(X,T)=(0,0, s(T))$, where $s(T)$ is a bounded and continuous function defined in $(-\infty,0]$. Hence, in this case, to rule out the singularity of the solution, it suffices to prove that $s(T)=0$. Since otherwise, one will obtain a contradiction to the fact that $\bar u(x_{\infty_1},x_{\infty_2},0)=1$. Meanwhile, under assumptions of Theorem 3.1, one has $\bar u(X,T)=0$ which is a contradiction to the fact that $\bar u(x_{\infty_1},x_{\infty_2},0)=1$ and hence the singularity can be ruled out. [**Case II.**]{} $r_kM_k (k=1,2,\cdots)$ is not uniformly bounded. In this case, one can rescale the solution as $$\begin{aligned} \label{P1+} u^{(k)}(X,T)=\frac{1}{M_k}u(x_k+\frac{X}{M_k}, t_k+\frac{T}{M^2_k}).\end{aligned}$$ Then, similar to Case I, $u^{(k)}(X,T) (k=1,2,\cdots)$ are smooth solutions of the 3D Navier-Stokes equations, which are defined in $\mathbb{R}^3\times (A_k,B_k)$ with $$\begin{aligned} \label{P6-} A_k=-M_k^2-M_k^2t_k, \ B_k=-M_k^2t_k.\end{aligned}$$ Note that $B_k=-M_k^2t_k\ge (\frac{N_k}{\gamma_k})^2(-t_k)\ge \frac{\varepsilon_1^2}{\gamma_k^2}$ for some $\varepsilon_1>0$. Moreover, it is clear that $$\begin{aligned} \label{P7-} |u^{(k)}(X,T)|\le \gamma_k, X\in \mathbb{R}^3, T\in (A_k,0],\end{aligned}$$ and $$\begin{aligned} \label{P8-} |u^{(k)}(0,0)|=1.\end{aligned}$$ In this case, there exists a subsequence of $\{M_k\}$ (still denoted by itself) such that $r_kM_k\to \infty$ as $k\to\infty$. Due to the axis symmetry of $u$, $x_k$ can be chosen so that $\theta(x_k)\to \theta_\infty$ for some $\theta_\infty\in [0,2\pi]$. Then there exists an unit vector $\nu=(\nu_1,\nu_2,0)$ such that $e_r(x_k)\to \nu$ and $e_\theta(x_k)\to \nu^\perp=(-\nu_2,\nu_1,0)$. Moreover, it holds that $$x_k+\frac{X}{ M_k}\in B(x_k, \frac{r_k}{\sqrt{r_kM_k}})\ \ {\rm for}\ \ X\in B(0,\sqrt{r_k M_k}),$$ and $$t_k-(\frac{r_k}{\sqrt{r_k M_k}})^2<t_k+\frac{T}{ M_k^2}\le t_k<0 \ \ {\rm for }\ \ -M_kr_k<T\le 0.$$ By (\[C10-1\]), $$|u_\theta(y,t)|\le \frac{C}{r_k} \ {\rm for}\ \ y\in B( x_k, \frac{r_k}{2}), t<0,$$ which implies that $$\begin{aligned} \label{33-} |u^{(k)}(X,T)e_\theta(x_k+\frac{X}{M_k})|=\frac{1}{M_k}|u_\theta( x_k+\frac{X}{M_k}, t_k+\frac{T}{M_k^2})|\le \frac{C}{M_kr_k}\end{aligned}$$ for $(X,T)\in B(0,\sqrt{r_kM_k})\times (-r_kM_k,0]$. Since the flow is axisymmetric, thus, on $B(0,\sqrt{r_kM_k})\times (-r_kM_k,0]$, $e_R(x_k+\frac{X}{M_k})\to \nu$ and $e_\theta(x_k+\frac{X}{ M_k})\to \nu^\perp$ as $k\to\infty$. Moreover, for each $k$, $u^{(k)}$ is a bounded and smooth solution to the 3D Navier-Stokes equations. There exists a subsequence of $u^{(k)}$ (still denoted by itself) and a bounded ancient solution $\tilde u(X,T)$ to the 3D Navier-Stokes equations on $\mathbb{R}^3\times (-\infty,0]$, such that $$\begin{aligned} &&u^{(k)}(X,T)=\frac{1}{M_k}u_R(x_k+\frac{X}{ M_k}, t_k+\frac{T}{M_k^2})e_R(x_k+\frac{X}{M_k})\nonumber\\[3mm] &&+\frac{1}{M_k}u_\theta(x_k+\frac{X}{M_k}, t_k+\frac{T}{M_k^2})e_\theta(x_k+\frac{X}{M_k})\nonumber\\ &&+\frac{1}{M_k}u_Z(x_k+\frac{X}{M_k},\frac{T}{M_k^2})e_Z(x_k+\frac{X}{M_k})\nonumber\\[3mm] &&\rightarrow \tilde u(X,T)=\tilde u_R\nu+\tilde u_\theta\nu^\perp+\tilde u_Ze_Z\nonumber\end{aligned}$$ in $C(\bar Q)$ for any compact subset $Q$ of $\mathbb{R}^3\times (-\infty,0]$. In view of , one has $$\begin{aligned} \label{Oct29} |\tilde u(0,0)|=1.\end{aligned}$$ In this case, Lei and Zhang [@LZ] obtained a Liouville type of theorem which can be stated as follows. [**Proposition 4.1**]{} If $r_kM_k (k=1,2,\cdots)$ is not uniformly bounded, then $\tilde u(X,T)=(0, s_1(T), s_2(T))$, where $s_1(T)$ and $s_2(T)$ are bounded and continuous functions depending on $T\in (-\infty,0]$. The proof is referred to [@LZ] and we give a sketch of the proof here. [**Sketch of Proof**]{}. implies $\tilde u(X,T)\cdot \nu^\perp=0$. Hence $$\begin{aligned} \label{34-} \tilde u(X,T)=\tilde u_R(X,T)\nu+\tilde u_Z(X,T)e_Z.\end{aligned}$$ On the other hand, for $(y,s)\in B(x_k, \frac{r_k}{\sqrt{r_k M_k}})\times [t_k-(\frac{r_k}{\sqrt{r_kM_k}})^2, t_k]$, one has that $$\begin{aligned} &&\frac{1}{M_k}[u_r(y,s)e_\theta(y)-u_\theta(y,s)e_r(y)]\\[3mm] &&=\frac{1}{M_k}\partial_\theta[u_r(y,s)e_r(y)+u_\theta(y,s)e_\theta(y)]\\[3mm] &&=\frac{1}{M_k}\partial_\theta[u_r(y,s)e_r(y)+u_\theta(y,s)e_\theta(y)+u_z(y,s)e_z(y)]\\[3mm] &&=\partial_\theta[u^{(k)}(M_k(y-x_k), M_k^2(s-t_k))]\\[3mm] &&=M_k(\partial_\theta y\cdot\nabla)u^{(k)}(M_k(y-x_k),M_k^2(s-t_k))\\[3mm] &&=M_k|y|(e_\theta(y)\cdot\nabla)u^{(k)}(M_k(y-x_k), M_k^2(s-t_k)),\end{aligned}$$ which shows that $$\begin{aligned} &&\frac{1}{M_k}[u_r(x_k+\frac{X}{M_k}, t_k+\frac{T}{M_k^2})e_\theta(x_k+\frac{X}{M_k})\nonumber\\[3mm] &&-u_\theta(x_k+\frac{X}{M_k}, t_k+\frac{T}{M_k^2})e_R(x_k+\frac{X}{M_k}))]\nonumber\\[3mm] &&=M_k|x_k+\frac{X}{M_k}|(e_\theta(x_k+\frac{X}{M_k})\cdot\nabla)u^{(k)}(X,T)\label{35-}\end{aligned}$$ for $(X,T)\in B(0,\sqrt{r_kM_k})\times (-r_k M_k,0]$. Since $r_k M_k\to \infty$, so $M_k|x_k+\frac{X}{M_k}|\to \infty$ for any fixed $X\in B(0,\sqrt{r_k M_k})$. But the left hand side of (\[35-\]) is bounded. Hence, letting $k\to \infty$, one gets that $$\begin{aligned} \label{36} (\nu^\perp\cdot\nabla)\tilde u(X,T)=0.\end{aligned}$$ Note that the Navier-Stokes equations are invariant under rotation. Without loss of generality, we set $\nu=e_1, \nu^\perp=e_2$. Consequently, the limit function $$\tilde u(X,T)=\tilde u_R(X_1,Z,T)e_1+\tilde u_Z(X_1,Z,T)e_Z,$$ is a bounded ancient solution to the 2D Navier-Stokes equations. It follows from Theorem 5.1 and Remark 6.1 in [@KNSS] that $\nabla \tilde u_R=\nabla \tilde u_Z=0$. Hence, $\tilde u_R$ and $\tilde u_Z$ are bounded and continuous functions depending only on time variable $T\in(-\infty,0]$, denoted by $s_1(T)$ and $s_2(T)$ respectively. The proof of the proposition is finished. We finally remark that in case II, to role out the singularity of the solution, it suffices to prove that $s_1(T)=s_2(T)=0$. Since otherwise, one can obtain a contradiction to . [100]{} L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solution of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-837, 1982. E. A. Carlen, M. Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to 2-D Navier-Stokes equations, A celebration of John F. Nash Jr., Duke Mah. J. 81, 135-157, 1995. D. Chae, J. Lee, On the Regularity of Axisymmetric solutions of the Navier-Stokes Equations, Math. Z., 239 (4), 645-671, 2002. C. C. Chen, R. M. Strain, H. T. Yau, T. P. Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations, International Mathematics Reseach Notices, Vol. 2008: rnn016. C. C. Chen, R. M. Strain, T. P. Tsai, H. T. Yau, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations II, Communications in Partial Differential Equations, 34, 203-232, 2009. L. Escauriaza, G. Seregin, V. $\breve{S}$ver$\acute{a}k$, $L_{3,\infty}$-solutions to the Navier-Stokes equations and backward uniqueness, Uspekhi Matematicheskih Nauk, 58(2), 3-44. Enlish translation in Russian Mathematical Sueveys, 58(2), 211-250, 2003. E. Hopf, Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4, 213-231, 1954. R. S. Hamilton, The formation of singualrities in the Ricci flow, Surveys in Differential Geometry, Vol II, PP. 7-136, Int. Press, Cambridge, MA, 1995. T. Y. Hou, Z. Lei, C. M. Li, Global regularity of the 3D axisymmetric Navier-Stokes equations with anisotropic data, Comm. Partial Diff. Eqns., 33, 1622-1637, 2008. T. Y. Hou, C. M. Li, Dynamic stability of the 3D axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math. 61(5), 661-697, 2008. Q. S. Jiu, Z. P. Xin, Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations, 119-139, New Studies in Advanced Mathematics, 2, International Press, Somerville, MA 2003. T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $R^m$, with applications to weak solutions, Math. Z., 187(4), 471-480, 1984. G. Koch, N. Nadirashvili, G. Seregin, V. $\breve{S}$ver$\acute{a}k$, Liouville theorems for the Navier-Stokes equations and applications, Acta Math.,203, 83-106,2009. H. Koch, D. Tataru, Well-posedness for the Navier-Stokes eqautions, Adv. math., 157(1), 22-35, 2001. A.Kubica, M. Pokorny, W. Zajaczkowski, Remarks on regularity criteria for axially symmetric weak solutions to the Navier-Stokes equations, Math. Methods Appl. Sci., 35, 360-371, 2012. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach, 2nd. ed., 1969. O. A. Ladyzhenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leninggrad. Otdel. Math. Inst. Steklov. (LOMI) 7, 155-177, 1968 (Russian). Z. Lei, Q. S. Zhang, A Liouville theorem for the axially-symmetric Navier-Stokes equations, J. Funct. Anal., 261(8), 2323-2345, 2011. J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math., 63, 193-248, 1934. F. H. Lin, A New Proof of the Caffarelli-Kohn-Nirenberg Theorem, Communications on Pure and Applied Mathematica, Vol. LI, 0241-0257, 1998. A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27, Cambridge University Press, Cambride, 2002. J. Ne$\breve{c}$as, M. Ru$\breve{z}$i$\breve{c}$ka, V. $\breve{S}$ver$\acute{a}k$, On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math. 176 (2), 283-294, 1996. X. H. Pan, Regularity of solution to axi-symmetric Navier-Stokes equations with a slightly supercritical condition, arXiv:1410.6260v1, 2014. V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66, 535-552, 1976. G. Seregin, V. $\breve{S}$ver$\acute{a}k$, On type I sigularities of the local axisymmetric solutions of the Navier-Stokes equations, Comm. Partial Diff. Eqns., 34, 171-201, 2009. J. Serrin, On the interior regularity of weak solutions of the time-dependent Navier-Stokes equations, Arch. Rat. Mech.Anal., 9, 187-195, 1962. M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. on Pure and Appl. Math., 41, 437-458, 1988. R. Temam, Navier-Stokes equations, North-Holland, Amsterdan-New York-Oxford, 1977. G. Tian, Z.P. Xin, Gradient Estimation on Navier-Stokes Equations, Communications in Analysis and Geometry, 7(2), 221-257, 1999. T. P. Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rat. Mech. Anal., 143, 29-51, 1998. M. R. Uchovskii, B. I. Yudovich, Axially Symmetric Flows of an Ideal and Viscous Fluid, J. Appl. Math. Mech. 32, 52-61, 1968. [^1]: The research is partially supported by National Natural Sciences Foundation of China (No. 11171229, No.11231006 and No.11228102) and Project of Beijing Chang Cheng Xue Zhe. e-mail: [email protected] [^2]: The research is partially supported by Zheng Ge Ru Funds, Hong Kong RGC Earmarked Research Grants CUHK4041/11P and CUHK4048/13P, NSFC/RGC Joint Research Scheme Grant CUHK443/14, a Focus Area Grant at The Chinese University of Hong Kong, and a grant from the Croucher Foundation. e-mail: [email protected]

--- abstract: 'We calculate orbits, tidal radii, and bulge-bar and disk shocking destruction rates for 63 globular clusters in our Galaxy. Orbits are integrated in both an axisymmetric and a non-axisymmetric Galactic potential that includes a bar and a 3D model for the spiral arms. With the use of a Monte Carlo scheme, we consider in our simulations observational uncertainties in the kinematical data of the clusters. In the analysis of destruction rates due to the bulge-bar, we consider the rigorous treatment of using the real Galactic cluster orbit, instead of the usual linear trajectory employed in previous studies. We compare results in both treatments. We find that the theoretical tidal radius computed in the non-axisymmetric Galactic potential compares better with the observed tidal radius than that obtained in the axisymmetric potential. In both Galactic potentials, bulge-shocking destruction rates computed with a linear trajectory of a cluster at its perigalacticons give a good approximation to the result obtained with the real trajectory of the cluster. Bulge-shocking destruction rates for clusters with perigalacticons in the inner Galactic region are $smaller$ in the non-axisymmetric potential, as compared with those in the axisymmetric potential. For the majority of clusters with high orbital eccentricities ($e > 0.5$), their total bulge+disk destruction rates are $smaller$ in the non-axisymmetric potential.' author: - 'Edmundo Moreno, Bárbara Pichardo and Héctor Velázquez' title: 'Tidal radii and destruction rates of globular clusters in the Milky Way due to bulge-bar and disk shocking' --- Introduction {#introd} ============ In two previous papers [@AMP06; @AMP08 hereafter Papers I and II] tidal radii and destruction rates due to bulge and disk shocking were computed for 54 globular clusters in our Galaxy, using axisymmetric and non-axisymmetric Galactic potentials. In Paper I the non-axisymmetric Galactic potential included the Galactic bar, and in Paper II the additional effect of three dimensional (3D) spiral arms was also analyzed. The models for these non-axisymmetric components given by @PMME03 [@PMM04] were employed in those computations. The absolute proper motion data needed to compute the Galactic orbits of the globular clusters were obtained from the extensive studies of @D97 [@D99a; @D99b; @DI00; @D01; @D03] and @CD07, who have computed the proper motions for a good fraction of the total number of globular clusters; for other clusters, @D99b have compiled the proper motion data from various sources. Lately, @CD10 have given the absolute proper motions of other nine globular clusters, and @CD13 present new absolute proper motions of NGC 6397, NGC 6626, and NGC 6656; thus now we dispose of absolute proper motion data for a total of 63 globular clusters in our Galaxy. For the new sample of 63 globular clusters, we compute again their tidal radii and destruction rates due to bulge and disk shocking, now with some improvements. We use axisymmetric and non-axisymmetric Galactic potentials, the later including both the spiral arms and the Galactic bar models of @PMME03 [@PMM04], as in Paper II. A first part of our improvements has to do with this Galactic potential, the initial orbital conditions of the globular clusters, and the uncertainties in the computed quantities: (a) the Galactic potential is now rescaled to recent values of the galactocentric distance and rotation velocity of the local standard of rest, as found by @BRet11, (b) we use the solar velocity obtained by @SBD10, (c) the late compilation of clusters properties given by @H10 is used to update other parameters employed in our computations, and (d) we make Monte Carlo simulations to estimate the uncertainties in the tidal radii and destruction rates, and compare with estimates in Papers I and II. The second part of our improvements refers to the procedure to compute destruction rates. In Papers I, II and in previous studies of tidal heating due to the interaction with the Galactic bulge and heating by disk shocking [e.g., @AHO88; @GO97; @GO99], the impulse approximation and a straight-path cluster trajectory have been employed. Here we relax the straight-path approximation and follow the more rigorous treatment given by @G99a, who employ a fit to the tidal acceleration along the true Galactic orbit of the cluster. In this paper this procedure is undertaken in our non-spherical Galactic potentials, as opposed to the spherical potential used by @G99a. The results are compared with those obtained using the usual straight-path aproximation. In $\S$ \[datos\] we give the globular cluster data employed in our study. The Galactic potential and its parameters are presented in $\S$ \[gpot\]. In $\S$ \[galorb\] some properties of the Galactic orbits in both the axisymmetric and non-axisymmetric potentials are tabulated, and for some clusters we show their meridional orbits. The tidal radii are analyzed in $\S$ \[radmar\]. The needed formulism of destruction rates using the real trajectories of globular clusters is summarized in $\S$ \[tdestr\], and our results are presented in $\S$ \[destr\]. In $\S$ \[concl\] we present our conclusions. Employed data for the globular clusters {#datos} ======================================= In Table \[tbl-1\] we list the cluster parameters employed in our study. Equatorial coordinates $(\alpha,\delta)$, are given in columns 2 and 3. The distance $r$ and radial velocity $v_r$, in columns 4 and 5, are taken from the recent compilation by @H10. The absolute proper motions, ${\mu}_x$ = ${\mu}_{\alpha}\cos{\delta}$, ${\mu}_y$ = ${\mu}_{\delta}$, in columns 6 and 7, are the values given by @D97 [@D99a; @D99b; @DI00; @D01; @D03] and @CD07 [@CD10; @CD13], except for 47 Tuc (NGC 104) and M4 (NGC 6121) whose values are taken from @AK03 and @BPKA03, respectively. As in Papers I and II, the mass of a cluster, $M_c$, given in column 8, is computed using a mass-to-light ratio $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$. **In $\S$ \[radmar\] we also employ for some clusters their $M_c$ computed with dynamical mass-to-light ratios given by @MM05 . The observed tidal radius $r_{td}$ (we call $r_{td}$=$r_K$ if this radius is computed with a King model [@K62]) is not listed by @H10, but as he points out, it can be computed with his listed values for the concentration, $c$, and core radius, $r_c$, only for those clusters with a non-collapsed core. For clusters with a collapsed core, Harris suggests to take $r_{td}$ estimated by @MM05 and @PK75. For this type of clusters, and listed in Table \[tbl-1\], @MM05 give $r_{td}$ for NGC 362, NGC 1904, NGC 6266, and NGC 6723; we take their $r_{td}$ for a King model, which is the model we use in our computations. For other clusters in Table \[tbl-1\], @PK75 estimate $r_{td}$ in NGC 6397, NGC 6752, NGC 7078, and NGC 7099. These values of $r_{td}$ from @MM05 and @PK75 are transformed according to the distances $r$ given by @H10. For NGC 6284, NGC 6293, NGC 6342, and NGC 6522, we take $r_{td}$ from Harris’ previous compilation, transformed with his new listed distances. Column 9 gives the final $r_{td}$=$r_K$ employed values, and column 10 the half-mass radius, $r_h$, in each cluster.** The Galactic potential {#gpot} ====================== In our analysis we employ axisymmetric and non-axisymmetric models for the Galactic gravitational potential. The axisymmetric model is based on the Galactic model of @AS91, which gives a circular rotation speed on the Galactic plane ${\Theta}_0$ $\approx$ 220 km/s at its assumed Sun’s Galactocentric distance $R_0$ = 8.5 kpc. This model is scaled to the new Galactic parameters ${\Theta}_0$, $R_0$ given by @BRet11: ${\Theta}_0$ = 239$\pm$7 km/s, $R_0$ = 8.3$\pm$0.23 kpc. The non-axisymmetric Galactic model is built from the scaled axisymmetric model. First, all the mass in the spherical bulge component in this axisymmetric model is employed to built the Galactic bar. In Papers I and II, where only 70$\%$ of the bulge mass was employed to built the bar, we have mentioned some properties of the model used for this bar component. We use the third bar model given by @PMM04 (the model of superposition of ellipsoids), which approximates the boxy COBE/DIRBE brightness profiles shown by @F98. We also consider a 3D gravitational potential to represent the spiral arms. The model used for these arms, called PERLAS, is given by @PMME03, and has already been employed in Paper II. The total mass of the 3D spiral arms is taken as a small fraction of the mass in the disk component of the scaled axisymmetric model. We take $M_{\rm arms}/M_{\rm disk}$ = 0.04$\pm$0.01, considered by @PMA12 in their analysis of the maximum value on the Galactic plane of the parameter $Q_T$ [@ST80; @CS81], which, as a function of Galactocentric distance, is the ratio of the maximum azimuthal force of the spiral arms to the radial axisymmetric force at a given distance. The mass density at the center of the spiral arms falls exponentially with Galactocentric distance, and we take its corresponding radial scale length equal to the one of the Galactic exponential disk modeled by @BCHet05: $H$ = 3.9$\pm$0.6 kpc, using $R_0$ = 8.5 kpc, scaled now with the new value of $R_0$. Other properties of the Galactic bar and the Galactic spiral arms, have been collected in @PMA12. We use the following parameters in our computations: a) the present angle between the bar’s major axis and the Sun-Galactic center line is taken as 20$^\circ$, b) the angular velocity of the bar is in the range $\approx$ 55$\pm$5 $\kmskpc$, c) we consider the pitch angle of the spiral arms in the range $\approx$ 15.5$\pm$3.5$^\circ$, d) the range for the angular velocity of the spiral arms is $\approx$ 24$\pm$6 $\kmskpc$. Table \[tbl-2\] summarizes all the parameters employed in our Galactic models, along with the Solar velocity $(U,V,W)_{\odot}$ obtained by @SBD10 (here $U$ is taken negative towards the Galactic center) and its uncertainties estimated by @BRet11. Properties of the Galactic orbits {#galorb} ================================= For the computation of the Galactic orbits we have employed the Bulirsch-Stoer algorithm given by @P92, and also the Runge-Kutta algorithm of seventh-eight order elaborated by @F68. In our problem both algorithms give practically the same results, and due to the complicated mathematical forms of the gravitational potentials of the non-axisymmetric Galactic components (bar and 3D spiral arms), we have favored the Runge-Kutta algorithm to reduce the computing time, specially in the Monte Carlo calculations. The Bulirsch-Stoer algorithm is employed mainly in the computations with the axisymmetric potential. In Table \[tbl-3\] we give for each cluster some orbital parameters obtained with the non-axisymmetric (first line) and axisymmetric (second line) potentials. Except for the data given in columns 8 and 9, whose associated time interval is commented in the next section, the data presented in this table correspond to a backward time integration of 5 $\times$ $10^9$ yr in the axisymmetric case, and from $10^9$ to 3 $\times$ $10^9$ yr in the non-axisymmetric case, depending on the cluster. The second, third, and fourth columns show the average perigalactic distance, the average apogalactic distance, and the average maximum distance from the Galactic plane, respectively. The fifth column gives the average orbital eccentricity, this eccentricity defined as $e=(r_{max}-r_{min})/(r_{max}+r_{min})$, with $r_{min}$ and $r_{max}$ successive perigalactic and apogalactic distances. Columns 6 and 7 give the orbital energy per unit mass, $E$, and the z-component of angular momentum per unit mass, $h$, only in the axisymmetric potential, where these two quantities are constants of motion. Columns 8 and 9 list tidal radii, which are discussed in the next section. In Figures \[fig1\] and \[fig2\] we show meridional orbits for some clusters, whose NGC number is given. In each pair of columns the orbit in the axisymmetric potential is shown in the left frame, and that in the non-axisymmetric potential in the right frame. We choose this sample of clusters to illustrate how strong the effect of the non-axisymmetric Galactic components can be. The most conspicuous difference between the computations in both potentials is the orbital radial extent. This has important consequences in the clusters tidal radii, as shown in the next section. **Figures \[fig3\], \[fig4\], and \[fig5\] show the comparison in both Galactic potentials of the average perigalactic distance, average apogalactic distance, and average maximum distance from the Galactic plane, given respectively in the second, third, and fourth columns of Table \[tbl-3\]. Values in the axisymmetric potential (with a subindex ’ax’) and non-axisymmetric potential (with a subindex ’nax’) are given in the horizontal and vertical axes, respectively. The uncertainties shown in these figures are estimated with the differences from corresponding quantities obtained in the minimum and maximum energy orbits in each cluster, according to the uncertainties in the cluster radial velocity, distance, and proper motions.** Tidal radii {#radmar} =========== Comparison of theoretical and observed tidal radii in the axisymmetric and non-axisymmetric Galactic potentials {#compar} --------------------------------------------------------------------------------------------------------------- ### Comparison with observed King tidal radii $r_K$ {#king} As in Papers I and II, for each globular cluster, and in both axisymmetric and non-axisymmetric Galactic potentials employed in our analysis, we compute a theoretical tidal radius using two expressions. The first is King’s formula [@K62] $$r_{K_t}= \left [ \frac{M_c}{M_g(3+e)} \right ]^{1/3}r_{min}, \label{rKt}$$ where $M_c$ is the mass of the cluster, $M_g$ is an effective galactic mass, $e$ is the orbital eccentricity as defined in the previous section, and $r_{min}$ is the perigalactic distance. The mass $M_g$ is taken as the equivalent central mass point which gives an acceleration at the given perigalactic position with a magnitude equal to the magnitude of the actual acceleration at this point in the corresponding Galactic potential. The second expression is the one proposed in Paper I, computed at the perigalactic position $$r_{\ast} = \left [ \frac{GM_c}{\left (\frac{\partial F_{x'}} {\partial x'} \right )_{{\bf r'}= 0}+ \dot{\theta}^2 + \dot{\varphi}^2\sin^2 {\theta}} \right ]^{1/3}, \label{rast}$$ with $F_{x'}$ the component of the Galactic acceleration along the line $x'$ joining the cluster with the Galactic center, and its partial drivative evaluated at the given perigalactic point. The angles ${\varphi}$ and ${\theta}$ are angular spherical coordinates of the cluster in an inertial galactic frame. In Paper I these two expressions for a theoretical tidal radius gave similar values. For a given Galactic potential, this result is maintained in the present computations. Columns 8 and 9 in Table \[tbl-3\] give the average values $<$$r_{K_t}$$>$, $<$$r_{\ast}$$>$ of $r_{K_t}$ and $r_{\ast}$, over the last $10^9$ yr (in some clusters this time interval is extended to have a few perigalactic points). In this **and next sections of $\S$ \[radmar\] we make some comparisons using $r_{K_t}$ given by Eq. (\[rKt\]). The first comparison is $r_{K_t}$ with the observed tidal radius (also called the limiting radius) $r_{td}$=$r_K$ listed in Table \[tbl-1\], estimated with a King model [@K62]. In Figures \[fig6\] and \[fig7\] we show with big filled squares this comparison in the axisymmetric and non-axisymmetric Galactic potentials. These points have two marks: clusters in which the tidal radius $<$$r_{K_t}$$>$ computed with the non-axisymmetric potential is greater than $<$$r_{K_t}$$>$ computed with the axisymmetric potential, are marked with encircled points; crossed points correspond to clusters in which $<$$r_{K_t}$$>$ computed with the non-axisymmetric potential is less than $<$$r_{K_t}$$>$ computed with the axisymmetric potential. These marks are shown in both figures. Thus, encircled and crossed points in Figure \[fig6\] will move upwards and downwards, respectively, to give the corresponding Figure \[fig7\]. As in Papers I and II, the uncertainty in $<$$r_{K_t}$$>$, or $<$$r_{\ast}$$>$, is estimated in each cluster by computing $<$$r_{K_t}$$>$ in the minimum and maximum energy orbits, according to the uncertainties in the cluster radial velocity, distance, and proper motions. The small empty squares and empty triangles in Figures \[fig6\] and \[fig7\] show the values of $<$$r_{K_t}$$>$ in these minimum and maximum energy orbits, respectively.** From these figures we note that $<$$r_{K_t}$$>$ computed in the non-axisymmetric potential compares better with $r_K$: many clusters whose points lie below the line of coincidence in Figure \[fig6\], are closer to this line in Figure \[fig7\]; these are the encircled points. Likewise, several clusters with points (now the crossed points) above the line of coincidence in Figure \[fig6\], are closer to this line in Figure \[fig7\]. The rearrangement of the encircled points is the most conspicuous. In Figure \[fig6\] a sample of eight clusters has been selected, represented by encircled points numbered from 1 to 8, and correspond to NGC 362, NGC 5139, NGC 5897, NGC 5986, NGC 6287, NGC 6293, NGC 6342, and NGC 6584, respectively. **For these clusters, in Figure \[fig8\] we give the values of their perigalactic distance, $r_{min}$, as a function of time, over the last $10^9$ yr; black dots joined by black lines show the values in the axisymmetric potential, and the dots and lines in red correspond to the non-axisymmetric potential. The black and red horizontal dotted lines show the corresponding average value of $r_{min}$ in the given interval of time. Each frame shows the cluster name and also the identification number in Figure \[fig6\]. Except for NGC 6293, Figure \[fig8\] partly explains why in the case of encircled points, $<$$r_{K_t}$$>$ increases using the non-axisymmetric potential: in these clusters, the average value of $r_{min}$ obtained with the non-axisymmetric potential (red horizontal dotted lines) is greater than the corresponding average in the axisymmetric potential (black horizontal dotted lines).** The other factor which helps to understand the rearrangement of encircled points from Figure \[fig6\] to Figure \[fig7\] (at least those in the considered sample) is the value of the effective galactic mass $M_g$ employed in King’s formula Eq. (\[rKt\]). As stated in $\S$ \[gpot\], the original concentrated spherical bulge in the axisymmetric potential was employed to built the bar; thus, due to the less mass concentration of the non-axisymmetric potential in the inner Galactic region (see upper frame in figure 7 in Paper I), the contribution of the bar to $M_g$ computed at a given perigalactic distance in this inner region is expected to decrease compared with the contribution of the spherical bulge in the axisymmetric potential. In general, the value of $M_g$ in the non-axisymmetric potential will depend on the position of the perigalactic point relative to the axes of the Galactic bar, because this bar generates a non-axisymmetric force field. In addition, $M_g$ has also the effect of the spiral arms with their relative orientation to the axes of the bar at the time of occurrence of the perigalactic point. Thus, the value of $M_g$ depends on the specific perigalactic point. To illustrate these comments, in Figure \[fig9\] we give values of $M_g$ computed on the Galactic plane as a function of distance to the Galactic center. The black line corresponds to the axisymmetric potential; the continuous red and blue lines show values of $M_g$ due to the axisymmetric background (i.e. disk and spherical dark halo) plus the Galactic bar, along the major and minor axes of the bar, respectively. The dashed red and blue lines show values of $M_g$ along these major and minor axes with the addition of the spiral arms, i.e. considering all the mass components in the non-axisymmetric potential, taking in particular the major axis of the bar as the line where the spiral arms originate in the inner Galactic region. Thus, note the smaller values that $M_g$ can take in the inner Galactic region in the non-axisymmetric potential. Figure \[fig10\] shows the values of $M_g$ at the perigalactic points in Figure \[fig8\]. The correspondence of colors in this figure is that given in Figure \[fig8\]. The horizontal dotted lines show the average values of $M_g$ over the last $10^9$ yr; these lines are not plotted in NGC 6293 to avoid confusion in the lower continuous red line. The average values of $M_g$ in the axisymmetric and non-axisymmetric potentials almost coincide in the clusters NGC 5897, NGC 5986, NGC 6287, NGC 6342, and NGC 6584; thus in these clusters the increase of the average value of $r_{min}$ in the non-axisymmetric potential explains the corresponding increase of $<$$r_{K_t}$$>$. For the remaining clusters in this sample, NGC 362, NGC 5139, and NGC 6293, the average value of $M_g$ is sensibly smaller in the non-axisymmetric potential, specially in NGC 6293. This result combined with the increase of $r_{min}$, gives a net increase of $<$$r_{K_t}$$>$ for NGC 362 and NGC 5139 in the non-axisymmetric potential. On the other hand, in NGC 6293 the strong decrease of the average value of $M_g$ in the non-axisymmetric potential compared with that in the axisymmetric potential, counteracts the corresponding slight decrease of the average value of $r_{min}$ shown in Figure \[fig8\], giving a net increase of $<$$r_{K_t}$$>$ in the non-axisymmetric potential. ### Comparison with improved observed limiting radii {#kingwilson} Recently @MLF13 have derived the radial stellar density profiles of 26 Galactic globular clusters from resolved star counts, using high-resolution $Hubble Space Telescope$ observations. In particular, they derive the limiting radius $r_l$, what we call the observed tidal radius, employing King and Wilson [@W75] models. Considering the clusters in common between our sample and their 26 clusters, we have taken their best fits given by the least value of the reduced ${\chi}^2$ in the third column of their table 2, and compare our theoretical tidal radii $r_{K_t}$ with their corresponding limiting radii $r_l$. We give this comparison in Figure \[fig11\], in particular in the non-axisymmetric potential. The black points with their uncertainties are points already plotted in Figure \[fig7\] with corresponding King tidal radii $r_K$ employed in that figure. The red points are the comparison between $r_{K_t}$ with $r_l$; the uncertainties in $r_l$ are computed with data in table 2 of @MLF13 using distances given in our Table \[tbl-1\]. Thus these points are displaced in the horizontal axis with respect to the black points. The displacements are shown with dotted blue lines. Red crossed points correspond to clusters in which a Wilson model gives the best fit to the density profile. The comparison between $r_{K_t}$ with $r_l$ is almost the same as $r_{K_t}$ vs $r_K$ in Figure \[fig7\], except for the five red crossed points, where $r_l$ is about a factor of 2-3 greater than $r_K$ and $r_{K_t}$. These five points correspond to NGC 288, NGC 5024 (M53), NGC 5272 (M3), NGC 5466, and NGC 5904 (M5). As commented in Paper I, the last three clusters appear to be dissolving [@LMC00; @OG04; @B06]; also, NGC 288 has extended tails [@GJH04], and NGC 5024 is possibly an accreted cluster [@MG04]. Thus in these five clusters $r_l$ will be an upper bound for the tidal radius; some stars contributing to $r_l$ may be already escaping from the cluster. ### Comparison with improved cluster masses {#mldinamico} In their study of structural properties of massive star clusters, @MM05 have obtained dynamical mass-to-light ratios $(M/L)_V$ for 57 Galactic globular clusters. In this part we compute $r_{K_t}$ for clusters in common between our sample and those listed in their table 13, now with the cluster mass $M_c$ computed with their $(M/L)_V$, instead of $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$ employed in our study. With these new values of $r_{K_t}$, we compare $r_{K_t}$ vs $r_K$ in the non-axisymmetric potential. Figure \[fig12\] shows this comparison. The black points are points from Figure \[fig7\], using $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$; the red points employ the $(M/L)_V$ values of @MM05 with their uncertainties. For clarity in the figure, these red points are slightly displaced to the right of the black points. Thus, there is no much difference with respect to the comparison made in Figure \[fig7\], and the standard $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$ is a convenient test in our analysis. Tidal radii with Monte Carlo computations {#monte} ----------------------------------------- The comparison of theoretical and observed tidal radii made in the last section was repeated, now using Monte Carlo simulations. The uncertainties in the cluster distance, radial velocity, and proper motions, listed in Table \[tbl-1\], plus the uncertainties of the Galactic parameters listed in Table \[tbl-2\], were considered as 1$\sigma$ variations in a Gaussian Monte Carlo sampling. For each cluster we computed a few hundreds of orbits. In each sampled cluster orbit computed backward in time, we found the average $<$$r_{K_t}$$>$ over the last $10^9$ yr, and in turn, with all these values in a given cluster, determined its corresponding global average, denoted by $\langle r_{K_t} \rangle$, and its 1$\sigma$ variation. Figures \[fig13\] and \[fig14\] show the comparison of $\langle r_{K_t} \rangle$ with $r_K$ in the axisymmetric and non-axisymmetric Galactic potentials. The error bars in both figures correspond to the computed 1$\sigma$ variation. With these Monte Carlo calculations we obtain nearly the same comparisons shown in Figures \[fig6\] and \[fig7\]; thus, again our conclusion is that $\langle r_{K_t} \rangle$ compares better with $r_K$ employing the non-axisymmetric potential. These Figures \[fig13\] and \[fig14\] also show that the estimate of the uncertainty in $<$$r_{K_t}$$>$ done in $\S$ \[compar\] using the minimum and maximum energy orbits, is acceptable. An additional plotted point in Figures \[fig13\] and \[fig14\], which does not appear in Figures \[fig6\] and \[fig7\], is the one corresponding to the cluster Pal 3, the upper point in these figures. This cluster has an unbounded orbit computed with its mean distance, radial velocity, and proper motions listed in Table \[tbl-1\], and the mean Galactic parameters in Table \[tbl-2\]. In the Monte Carlo simulations we have picked out its bounded orbits. Overfilling excess of predicted tidal radius? {#exceso} --------------------------------------------- @WHS13 have considered observed limiting radii of Galactic globular clusters, given by a King model, $r_K$, and their theoretical tidal radius, $r_t$, computed at their perigalactic distance in the axisymmetric Galactic potential used by @JSH, but $r_t$ obtained as if the Galactic potential were spherically symmetric. They take the ratio of the difference ($r_K$$-$$r_t$) to the average ($r_K$+$r_t$)/2, and plot this ratio against the perigalactic distance. Clusters with this ratio greater than zero, overfill their predicted theoretical tidal radius; underfilling occurs in clusters which have this ratio less than zero. @WHS13 show in their figure 1 that the majority of clusters are overfilling their predicted tidal radius. Through $N$-body simulations of star clusters moving on the plane of symmetry of a given axisymmetric Galactic potential, they find an analytical correction to be applied to $r_t$ computed at perigalacticon, to obtain a better estimate of the cluster limiting radius $r_L$. With this value of $r_L$ employed instead of $r_t$ in the computation of the ratio mentioned above, @WHS13 find that the overfilling excess disappears, and there is a stronger agreement between theory and observations. To compare with the results of @WHS13, in our axisymmetric and non-axisymmetric Galactic potentials we compute the theoretical tidal radius at perigalacticon with King’s formula Eq. (\[rKt\]) (or alternatively with $r_{\ast}$ in Eq. (\[rast\])), take the average values of $r_{K_t}$ over the last $10^9$ yr listed in column 8 of Table \[tbl-3\], take the ratio of the difference ($r_K$$-$$<$$r_{K_t}$$>$) to the average ($r_K$+$<$$r_{K_t}$$>$)/2 ($r_K$ is listed in Table \[tbl-1\]) and plot this ratio against the logarithm of the average perigalactic distance in these last $10^9$ yr. We do the same taking only the last perigalacticon, and its distance to the Galactic center. Figure \[fig15\] shows our results. The two upper frames (a),(c) correspond to the last $10^9$ yr, and the two lower frames (b),(d) to the last perigalacticon. The frames on the left (a),(b) give results in the axisymmetric Galactic potential, and the frames on the right (c),(d) in the non-axisymmetric Galactic potential. At first sight, there is no evident overfilling nor underfilling excess of predicted theoretical tidal radius; the points scatter approximately around zero value in the ratio 2($r_K$$-$$<$$r_{K_t}$$>$)/($r_K$+$<$$r_{K_t}$$>$). This holds in both the axisymmetric and non-axisymmetric Galactic potentials, thus the main point in the discrepance of our results with those of @WHS13 seems to be the different ways in which the theoretical tidal radius is computed at perigalactic distance. If we apply the correction given by @WHS13 in their equation 8 to our $r_{K_t}$ computed at perigalacticon, this leads to a new cluster limiting radius $r_{LK}$. To compute this limiting radius we take in each cluster an average of $r_{K_t}$, the orbital eccentricity, and the orbital phase, over the last $10^9$ yr. We determine the ratio of the difference ($r_K$$-$$r_{LK}$) to the average ($r_K$+$r_{LK}$)/2, and plot this ratio against the logarithm of the average perigalactic distance in these last $10^9$ yr. This is shown in Figure \[fig16\] only for the non-axisymmetric Galactic potential. The error bars shown in this figure are obtained using the the minimum and maximum energy orbits, mentioned in $\S$ \[compar\]. We find a strong underfilling excess; there is a systematic shifting towards negative values in comparison with the initial approximately zero excess found in Figure \[fig15\]. Then our conclusion is that in our computations we do not need to apply the correction given by @WHS13. This issue needs a further study. Bulge-shocking destruction rates taking the real trajectory of a cluster {#tdestr} ======================================================================== In Paper I we have listed some relations needed to compute destruction rates of globular clusters due to bulge and disk shocking. Those corresponding to bulge shocking employ the impulse approximation, along with adiabatic corrections, and a linear trajectory of the cluster when passing at a given perigalactic point. In this section we give relations to compute destruction rates due to bulge shocking employing the real trajectory of a globular cluster, maintaining the impulse approximation. Let [$\cal F$]{}([$r$]{}) be the Galactic gravitational acceleration at the position [$r$]{} in an inertial reference system with origin at the Galactic center. In particular, in the following we consider this acceleration due to the spherical bulge component in the used axisymmetric model, and due to the bar in the non-axisymmetric model, in which all the bulge is represented by this bar, as mentioned in $\S$ \[gpot\]. With [$r$]{}$_c$ the position of the center of a globular cluster, and [$r'$]{} the position of a star in the cluster with respect to the cluster center, then [$r$]{}=[$r$]{}$_c$+[$r'$]{} is the position of the star in the inertial frame, and we define [$F$]{}([$r'$]{}) with the relation [$\cal F$]{}([$r$]{})$\equiv$ [$\cal F$]{}([$r$]{}$_c$+[$r'$]{})= [$F$]{}([$r'$]{}). Then, up to linear terms in [$r'$]{} and at the cluster position [$r$]{}$_c$, the tidal acceleration [$M$]{}([$r'$]{}) on the star is (with a sum over a repeated index) $$\mbox{\boldmath $M$} \left (\mbox{\boldmath $r'$} \right ) = \mbox{\boldmath $F$} \left (\mbox{\boldmath $r'$} \right )-\mbox{\boldmath $F$} \left (\mbox{\boldmath $r'$} = 0 \right ) \simeq \mbox{\boldmath $J$} \cdot \mbox{\boldmath $r'$} = \mbox{\boldmath $e$}_i x'_j \left (\frac{\partial F_{x'_i}}{\partial x'_j} \right )_{{\bf r'}= 0}. \label{acelmar}$$ The coordinates $x'_i$, $i$ = 1, 2, 3, are Cartesian coordinates of [$r'$]{}; this vector written in the inertial base of unitary vectors ([$e$]{}$_1$, [$e$]{}$_2$, [$e$]{}$_3$). The matrix [$J$]{} is given by $$\mbox{\boldmath $J$} = \bordermatrix{ & & & \cr & \frac{\partial F_{x'}}{\partial x'} & \frac{\partial F_{x'}}{\partial y'} & \frac{\partial F_{x'}}{\partial z'} \cr & \frac{\partial F_{y'}}{\partial x'} & \frac{\partial F_{y'}}{\partial y'} & \frac{\partial F_{y'}}{\partial z'} \cr & \frac{\partial F_{z'}}{\partial x'} & \frac{\partial F_{z'}}{\partial y'} & \frac{\partial F_{z'}}{\partial z'} \cr}_{\bf r' = 0}. \label{ecj}$$ \ To obtain the stellar velocity change due to [$M$]{}, we integrate d[$v'$]{}/dt = [$M$]{} in the inertial frame, taking two successive apogalactic points in the cluster’s orbit. The stellar velocity [$v'$]{} is measured in the inertial frame. Using the impulse approximation, the change in the $ith$ component of the stellar velocity between these two successive apogalactic points is (there is a sum over the index $j$) $${\Delta}v'_i = x'_j \int_{t_{ap1}}^{t_{ap2}} \left (\frac{\partial F_{x'_i}}{\partial x'_j} \right )_{{\bf r'}= 0} dt = x'_j I_{ij}, \label{delv}$$ with $I_{ij}$ defined by the integral, and the integration done numerically along the real orbit of the cluster, under the whole used Galactic potential, between successive apogalactic points occurring at times $t_{ap1}$, $t_{ap2}$, i.e. an apogalactic period. The change of stellar energy per unit mass is $\Delta E$ = [$v'$]{}$\cdot$$\Delta$[$v'$]{}+ (1/2)($\Delta$[$v'$]{})$^2$. Thus, with Eq. (\[delv\]), assuming spherical symmetry in the cluster, and an isotropic stellar velocity distribution within the cluster depending only on distance $r'$ = $|$[$r'$]{}$|$ from the center of the cluster, we have the two local (i.e. averaged on a spherical surface of radius $r'$) diffusion coefficients due to the interaction with the bulge $$<\!(\Delta E)_b\!>_{loc} = \frac{1}{2}<\!(\Delta \mbox{\boldmath $v'$})^2\!>_{loc} = \frac{1}{6}r'^2 \sum_{i,j}^{}I^2_{ij}, \label{delE}$$ $$<\!(\Delta E)_b^2\!>_{loc} \approx <\!(\mbox{\boldmath $v'$}\cdot\Delta \mbox{\boldmath $v'$})^2\!>_{loc} = \frac{1}{9}r'^2v'^2(r')(1 + {\chi}_{r',v'}(r')) \sum_{i,j}^{}I^2_{ij}, \label{delE2}$$ with $v'(r')$ the rms velocity within the cluster at distance $r'$, and the position-velocity correlation function ${\chi}_{r',v'}(r')$ given by @GO99. The sum in these equations gives nine squared coefficients $I^2_{ij}$. To take into account the stellar motion within the cluster during its interaction with the Galactic bulge or bar in an apogalactic period, adiabatic correction factors ${\eta}_1 (x)$, ${\eta}_2 (x)$ are introduced in Eqs. (\[delE\]) and (\[delE2\]), having the forms [@GO99] $${\eta}_1 (x(r')) = (1 + x^2(r'))^{-{\gamma}_1}, \label{adiab1}$$ $${\eta}_2 (x(r')) = (1 + x^2(r'))^{-{\gamma}_2}, \label{adiab2}$$ with $x(r') = {\omega}(r'){\tau}$; $\omega (r')$ is angular velocity of stars inside the cluster at distance $r'$, and as in Paper I, the angular velocity in circular motion at distance $r'$ is considered to represent this $\omega (r')$. The factor $\tau$ is an effective interaction time with the Galactic bulge or bar in an apogalactic period. The exponents ${\gamma}_1$, ${\gamma}_2$ depend on the ratio between $\tau$ and the cluster’s inner dynamical time evaluated at the half-mass radius, $t_{dyn,h} = ({\pi}^2{r_h}^3/2GM_c)^{1/2}$ (the half-mass radius $r_h$ and the mass of the cluster $M_c$ are listed in Table \[tbl-1\]). The values of ${\gamma}_1$, ${\gamma}_2$ are those considered in Paper I, based on Table 2 of @GO99. In the usual procedure employed in the impulse approximation with a linear trajectory of the cluster passing at a given perigalactic point, the effective interaction time $\tau$ is estimated as $\tau = |$[$r$]{}$_p|/|$[$v$]{}$_p|$, with [$r$]{}$_p$, [$v$]{}$_p$ the position and velocity of the cluster at this point with respect to the Galactic inertial frame. For the real trajectory of the cluster considered in this section, we follow the treatment of @G99a to estimate $\tau$. @G99a consider a potential with spherical symmetry and estimate $\tau$ making a Gaussian fit of the form $e^{-t^2/{\tau}^2}$ to the tidal acceleration in the z-direction. In our Galactic potentials we make a similar fit to the rms $total$ tidal acceleration. From Eq. (\[acelmar\]), the local averaged square tidal acceleration is $$<\!\mbox{\boldmath $M$}^2 \!>_{loc} = \frac{1}{3}r'^2 \sum_{i,j}^{} \left (\frac{\partial F_{x'_i}}{\partial x'_j} \right )^2_{{\bf r'}= 0} \label{acelmar2}$$ Taking an average over the cluster, the Gaussian fit in an apogalactic period is made on the rms tidal acceleration given by $$\left ( <\!\mbox{\boldmath $M$}^2 \!> \right )^{1/2} = \left \{ \frac{1}{3} <\!r_c^2\!> \sum_{i,j}^{} \left (\frac{\partial F_{x'_i}}{\partial x'_j} \right )^2_{{\bf r'}= 0} \right \}^{1/2}, \label{rms}$$ where $<$$r_c^2$$>$ is the mean square cluster radius, computed below. The resulting value of $\tau$ given by the fit is employed in Eqs. (\[adiab1\]) and (\[adiab2\]). In the axisymmetric potential the tidal acceleration ($<$[$M$]{}$^2>$)$^{1/2}$ has one maximum in every apogalactic period. However, in the non-axisymmetric potential, and in some clusters, this acceleration may have a complicated behavior in some apogalactic periods, showing more than one maximum. Figure \[fig17\] shows as an example some apogalactic periods in a run in the cluster NGC 6266. The black dots give the positions in time of apogalactic points; the black curve is ($<$[$M$]{}$^2>$)$^{1/2}$ and the red curves show the typical approximate fits made to the main acceleration peaks in cases like this. Including the adiabatic correction factors, the averages of Eqs. (\[delE\]) and (\[delE2\]) over the cluster are $$<\!(\Delta E)_b\!> = \frac{1}{6}<\!r'^2 {\eta}_1 (x(r'))\!> \sum_{i,j}^{}I^2_{ij}, \label{delEtot}$$ $$<\!(\Delta E)_b^2\!> \approx \frac{1}{9}<\!r'^2v'^2(r')(1 + {\chi}_{r',v'}(r')) {\eta}_2 (x(r'))\!> \sum_{i,j}^{}I^2_{ij}, \label{delE2tot}$$ with $$<\!r'^2 {\eta}_1 (x(r'))\!> = \frac{4\pi}{M_c} \int_{0}^{r_K} {\rho}_c(r') {\eta}_1(x(r')) r'^4 dr', \label{prom1}$$ $$<\!r'^2v'^2(r')(1 + {\chi}_{r',v'}(r')){\eta}_2 (x(r'))\!> = \frac{4{\pi}G}{M_c} \int_{0}^{r_K} {\rho}_c(r') M_c(r') {\eta}_2(x(r')) (1+{\chi}_{r',v'}(r')) r'^3 dr', \label{prom2}$$ and $<$$r_c^2$$>$ in Eq. (\[rms\]) $$<\!r_c^2\!> = <\!r'^2 \!> = \frac{4\pi}{M_c} \int_{0}^{r_K} {\rho}_c(r') r'^4 dr', \label{promr}$$ $r_K$ is the tidal radius of the cluster (listed in Table \[tbl-1\]), $M_c$ is its total mass, ${\rho}_c(r')$ is its spatial density, obtained with a @K66 model, and $M_c(r')$ is the mass of the cluster within radius $r'$. As in Paper I, we approximate the rms velocity $v'(r')$ with the corresponding circular velocity at that $r'$. With $E_c \simeq -0.2GM_c/r_h$ the mean binding energy per unit mass of the cluster, and if the cluster has a dominant maximum of the tidal acceleration ($<$[$M$]{}$^2>$)$^{1/2}$ in a given apogalactic period, bulge shock timescales in this period are defined as [@GO97] $$t_{bulge,1} = \left ( \frac{-E_c}{<\!(\Delta E)_b\!>} \right )P_{orb}, \label{tb1}$$ $$t_{bulge,2} = \left ( \frac{E_c^2}{<\!(\Delta E)_b^2\!>} \right )P_{orb}, \label{tb2}$$ with $P_{orb}$ the apogalactic period. If the cluster has more than one maximum of ($<$[$M$]{}$^2>$)$^{1/2}$ in the given apogalactic period, as in Figure \[fig10\], instead of $P_{orb}$ we use in each main fitted peak the corresponding interval of time taken in the fit. In each case the total destruction rate due to bulge shocking is $$\frac{1}{t_{bulge}} = \frac{1}{t_{bulge,1}} + \frac{1}{t_{bulge,2}}. \label{tbtot}$$ Disk and spiral arms shocking {#disco} ============================= **The treatment for disk shocking remains the same as in Paper I. The corresponding expressions to Eqs. (\[delEtot\]) and (\[delE2tot\]) for disk shocking are obtained averaging equations (1) and (2) in @G99b, resulting in** $$<\!(\Delta E)_d\!> = \frac{2g_m^2}{3v_z^2} <\!r'^2 {\eta}_1 (x(r'))\!>, \label{delEd}$$ $$<\!(\Delta E)_d^2\!> = \frac{4g_m^2}{9v_z^2} <\!r'^2 v'^2 {\eta}_2 (x(r'))(1 + {\chi}_{r',v'}(r'))\!>. \label{delE2d}$$ In both, the axisymmetric and non-axisymmetric Galactic potentials, $|g_m|$ is the maximum acceleration produced by the corresponding axisymmetric disk component in its perpendicular z-direction, on the perpendicular line to the plane of the disk passing at the position where the cluster crosses the disk, and $|v_z|$ is the z-velocity of the cluster at this point. Here $\tau$ in $x(r') = {\omega}(r'){\tau}$ is given by $\tau = |z_m|/|v_z|$, with $|z_m|$ the z-distance at which $|g_m|$ is reached. With $n$ crossings of the cluster orbit with the Galactic plane, disk shock timescales and corresponding total destruction rate are given by $$t_{disk,1} = \left ( \frac{-E_c}{<\!(\Delta E)_d\!>} \right ) \frac{P_{orb}}{n}, \label{td1}$$ $$t_{disk,2} = \left ( \frac{E_c^2}{<\!(\Delta E)_d^2\!>} \right ) \frac{P_{orb}}{n}, \label{td2}$$ $$\frac{1}{t_{disk}} = \frac{1}{t_{disk,1}} + \frac{1}{t_{disk,2}}. \label{tdtot}$$ In the non-axisymmetric potential, the spiral arms represent a plane mass distribution, analogous to the axisymmetric disk component, and also produce a shock on a cluster crossing the Galactic plane, with corresponding averaged diffusion coefficients $<\!(\Delta E)_{arms}\!>$ and $<\!(\Delta E)_{arms}^2\!>$ . At a given crossing point, the ratios $<\!(\Delta E)_{arms}\!>/<\!(\Delta E)_d\!>$, $<\!(\Delta E)_{arms}^2\!>/<\!(\Delta E)_d^2\!>$ between averaged diffusion coefficients due to the spiral arms and axisymmetric disk, will depend on the squared ratio of corresponding maximum accelerations $(|g_m|_{arms}/|g_m|_{disk})^2$. The velocity $|v_z|$ has the same value in both type of diffusion coefficients, as this velocity, as well as the orbit itself, is computed under the whole Galactic potential, i.e. including the axisymmetric (disk and dark halo) and non-axisymmetric (bar and spiral arms) components. There will be also a dependence of these ratios on the corresponding $|z_m|$ given by the spiral arms, through the dependence of ${\eta}_1$ and ${\eta}_2$ on $\tau$. Figures \[fig18\] and \[fig19\] show the azimuth-averaged ratios $(|g_m|_{arms}/|g_m|_{disk})^2$ and $|z_m|_{arms}/|z_m|_{disk}$ as functions of the distance $R$ to the Galactic center of the point where a cluster orbit crosses the Galactic plane. The squared ratio $(|g_m|_{arms}/|g_m|_{disk})^2$ is important only in the region of the spiral arms (2-12 kpc) and of order $10^{-2}$-$10^{-3}$. In this region $|z_m|_{arms}/|z_m|_{disk}$ is close to unity. Thus, in our analysis we ignore the spiral arms shocking, which compared with the one of the disk will be two or three orders of magnitude lower. Destruction rates. Results {#destr} ========================== In this section we present bulge-shocking destruction rates obtained with the formulism given in the last section, using the real trajectories of globular clusters in the employed Galactic potentials, and compare with corresponding values obtained with the usual linear trajectory approximation used in Paper I. The disk-shocking destruction rates are also computed and compared in the axisymmetric and non-axisymmetric potentials. All the computations are done with Monte Carlo simulations. Table \[tbl-4\] shows our results. Bulge-shocking destruction rates averaged over the last $10^9$ yr in a cluster’s orbit (this time is increased in some clusters) and their lower, ${\sigma}_{-}$, and upper, ${\sigma}_{+}$, uncertainties, are listed according to the linear (columns 2-4) or real (columns 5-7) employed cluster’s trajectory. Columns 8-10 give the disk-shocking destruction rates. For each cluster, the first line gives values in the non-axisymmetric potential, and the second line in the axisymmetric potential. With the data given in Table \[tbl-4\], Figures \[fig20\] and \[fig21\] show separately the comparison of bulge-shocking total destruction rates in the axisymmetric and non-axisymmetric Galactic potentials. Values obtained with the real trajectory of the cluster are shown in the vertical axis, and in the horizontal axis those with the linear trajectory. The error bars in these and following figures are given by the ${\sigma}_{-}$ and ${\sigma}_{+}$ values in Table \[tbl-4\]. The conclusion from these two figures is that the use of the linear trajectory, along with the associated effective interaction time estimated as $\tau = |$[$r$]{}$_p|/|$[$v$]{}$_p|$, gives a good approximation to compute destruction rates due to the bulge, in the axisymmetric and non-axisymmetric Galactic potentials. **To see how much the computed destruction rates can change if we take the cluster mass-to-light ratio $(M/L)_V$ different from the assumed $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$, we consider, as in Figure \[fig12\], dynamical mass-to-light ratios $(M/L)_V$ obtained by @MM05. For clusters in common between our sample and in table 13 of @MM05, the new cluster mass $M_c$ is computed, and in Figure \[fig22\] we show the comparison of bulge-shocking total destruction rates in particular in the axisymmetric potential, comparing values using the real (vertical axis) and linear (horizontal axis) trajectory. The black points with their uncertainties are points from Figure \[fig20\], obtained with the assumed $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$. The red points are obtained with the dynamical mass-to-light ratios given by @MM05. The correspondig shifts between black and red points in a cluster are shown with blue lines. Thus, the destruction rates do not change too much, specially those with high values, and the standard $(M/L)_V$ = 2 $M_{\odot}/L_{\odot}$ is a convenient test value.** Taking the real trajectories, Figure \[fig23\] shows the comparison of bulge-shocking total destruction rates in the non-axisymmetric (vertical axis) and axisymmetric (horizontal axis) Galactic potentials. Here we note important differences between both potentials, specially in the region of high destruction rates, where values obtained with the non-axisymmetric potential are $smaller$ than those with the axisymmetric potential. As noted in $\S$ \[gpot\], in the non-axisymmetric potential all the bulge is represented by the Galactic bar; thus, there is no remnant of the original concentrated spherical bulge in the axisymmetric potential, whose mass is now distributed over the bar, with less central concentration (see upper frame in figure 7 in Paper I) and thus less dangerous for clusters crossing its region. This explains the behavior in the high destruction rate region in Figure \[fig23\]. **In $\S$ \[king\] we saw a related $decrease$ of tidal radii in the inner Galactic region in the non-axisymmetric potential; see discussion of Figure \[fig8\] in that section. Our conclusion is that the more appropriate non-axisymmetric Galactic potential employed in our computations, reduces the destruction rates due to the bulge (bar, in this case) for clusters with perigalacticons in the inner Galactic region.** Disk-shocking destruction rates in the non-axisymmetric and axisymmetric potentials are compared in Figure \[fig24\]. Practically these destruction rates are the same in both potentials. Comparing this figure with Figure \[fig23\], we note that the disk dominates the destruction rate for clusters in the low destruction rate region, i.e. clusters with perigalacticons relatively distant from the Galactic center. Adding the bulge-shocking total destruction rate obtained with the real trajectory of the cluster, and the disk-shocking destruction rate, results in the total bulge+disk destruction rate. Figure \[fig25\] shows the comparison of these total values in the non-axisymmetric (vertical axis) and axisymmetric (horizontal axis) Galactic potentials. In Figure \[fig26\] we show the same Figure \[fig25\] but now without the error bars. Empty and black squares correspond to clusters with orbital eccentricity $e \leq 0.5$ and $e > 0.5$, respectively. The points marked with a circle show the clusters whose mass is less than $10^5 M_{\odot}$. The position of some clusters are marked with their NGC and Pal numbers. As in Figure \[fig23\], in these last figures we note that in the region of high destruction rates, dominated by the bulge, the total destruction rates obtained with the non-axisymmetric potential are $smaller$ than those resulting with the axisymmetric potential. Figure \[fig26\] shows that the majority of clusters with high eccentricities ($e > 0.5$), have smaller destruction rates in the non-axisymmetric potential. With the non-axisymmetric Galactic potential employed in our analysis, along with the more appropriate Monte Carlo simulations, we see from Figure \[fig26\] that seven clusters have particularly high destruction rates at the present time, due to bulge and disk shocking: Pal 5, NGC 6144, NGC 6121, NGC 6342, NGC 5897, NGC 6293, and NGC 6522. In Paper I, using only the Galactic bar in the non-axisymmetric potential, we found that NGC 6528 had the greatest destruction rate; now with the present non-axisymmetric Galactic model and the Monte Carlo simulations, this cluster has a low destruction rate, as shown in Figure \[fig26\]. Conclusions {#concl} =========== We have employed the available 6-D data (positions and velocities) of 63 globular clusters in our Galaxy to analyze their Galactic orbits and compute their tidal radii, as well as their bulge and disk shocking destruction rates. This analysis has been made in axisymmetric and non-axisymmetric Galactic potentials; in particular, the used non-axisymmetric potential is a very detailed model which includes both the Galactic bar and a 3D model for the spiral arms. Our analysis is made using Monte Carlo simulations, to take into account the several uncertainties in the kinematical data of the clusters. For the computation of destruction rates due to the bulge in both Galactic potentials, we have employed the rigorous treatment of considering the real Galactic cluster orbit, instead of the usual linear trajectory employed in previous studies. Our first result is that the theoretical tidal radius computed in the non-axisymmetric Galactic potential compares better with the observed tidal radius than that computed in the axisymmetric potential. This result leaves an open question with a recent study made by @WHS13, who propose a correction to be applied to the theoretical tidal radius computed at perigalacticon, to have a better comparison with the observed tidal radius. In our computations we do not need to introduce this correction. The first conclusion from our results of bulge-shocking destruction rates is that the usual linear trajectory of the cluster considered at perigalacticon, gives a good approximation to the result obtained taking the real trajectory of the cluster. This conclusion holds in both the axisymmetric and non-axisymmetric potentials. Our second conclusion is that the bulge-shocking destruction rates for clusters with perigalacticons in the inner Galactic region, turn out to be $smaller$ in the non-axisymmetric potential, as compared with those in the axisymmetric one. The majority of clusters with high orbital eccentricities ($e > 0.5$) have $smaller$ total bulge+disk destruction rates in the non-axisymmetric potential. We acknowledge financial support from UNAM DGAPA-PAPIIT through grant IN114114. Aguilar, L., Hut, P., & Ostriker, J. P. 1988, , 335, 720 Allen, C., Moreno, E., & Pichardo, B. 2006, , 652, 1150 (Paper I) Allen, C., Moreno, E., & Pichardo, B. 2008, , 674, 237 (Paper II) Allen, C., & Santillán, A. 1991, , 22, 255 Anderson, J., & King, I. R. 2003, , 126, 772 Bedin, L. R., Piotto, G., King, I. R., & Anderson, J. 2003, , 126, 247 Belokurov, V., Evans, N. W., Irwin, M. J., Hewett, P. C., & Wilkinson, M. I. 2006, , 637, L29 Benjamin, R. A., et al. 2005, , 630, L149 Brunthaler, A., et al. 2011, AN, 332, No. 5, 461 Casetti-Dinescu, D. I., Girard, T. M., Herrera, D., van Altena, W. F., López, C. E., & Castillo, D. J. 2007, , 134, 195 Casetti-Dinescu, D. I., Girard, T. M., Korchagin, V. I., van Altena, W. F., & López, C. E. 2010, , 140, 1282 Casetti-Dinescu, D. I., Girard, T. M., Jílková, L., van Altena, W. F., Podestá, F., & López, C. E. 2013, , 146, 33 Combes, F., & Sanders, R. H. 1981, , 96, 164 Dinescu, D. I., Girard, T. M., van Altena, W. F., Méndez, R. A., & López, C. E. 1997, , 114, 1014 Dinescu, D. I., van Altena, W. F., Girard, T. M., & López, C. E. 1999a, , 117, 277 Dinescu, D. I., Girard, T. M., & van Altena, W. F. 1999b, , 117, 1792 Dinescu, D. I., Majewski, S. R., Girard, T. M., & Cudworth, K. M. 2000, , 120, 1892 Dinescu, D. I., Majewski, S. R., Girard, T. M., & Cudworth, K. M. 2001, , 122, 1916 Dinescu, D. I., Girard, T. M., van Altena, W. F., & López, C. E. 2003, , 125, 1373 Drimmel, R. 2000, , 358, L13 Fehlberg, E. 1968, NASA TR R-287 Freudenreich, H. T. 1998, , 492, 495 Gerhard, O. 2002, ASP Conf. Ser., 273, 73 Gerhard, O. 2011,   Suppl., Vol. 18, 185 Gnedin, O. Y., & Ostriker, J. P. 1997, , 474, 223 Gnedin, O. Y., & Ostriker, J. P. 1999, , 513, 626 Gnedin, O. Y., Hernquist, L., & Ostriker, J. P. 1999, , 514, 109 Gnedin, O. Y., Lee, H. M., & Ostriker, J. P. 1999, , 522, 935 Grillmair, C. J., Jarrett, T. H., & Ha, A. C. 2004, ASP Conf. Ser., 327, 276 Harris, W. E. 1996, , 112, 1487 Harris, W. E. 2010, arXiv:1012.3224v1 Johnston, K. V., Spergel, D. N., & Hernquist, L. 1995, , 451, 598 King, I. R. 1962, , 67, 471 King, I. R. 1966, , 71, 64 Leon, S., Meylan, G., & Combes, F. 2000, , 359, 907 Mackey, A. D., & Gilmore, G. F. 2004, , 355, 504 McLaughlin, D. E., & van der Marel, R. P. 2005, , 161, 304 Miocchi, P., Lanzoni, B., Ferraro, F. R., Dalessandro, E., Vesperini, E., Pasquato, M., Beccari, G., Pallanca, C., & Sanna, N. 2013, , 774, 151 Odenkirchen, M., & Grebel, E. K. 2004, ASP Conf. Ser., 327, 284 Peterson, C. J. & King, I. R. 1975, , 80, 427 Pichardo, B., Martos, M., & Moreno, E. 2004, , 582, 230 Pichardo, B., Martos, M., Moreno, E., & Espresate, J. 2003, , 582, 230 Pichardo, B., Moreno, E., Allen, C., Bedin, L. R., Bellini, A., Pasquini, L. 2012, , 143, 73 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, Numerical Recipes in Fortran 77: The Art of Scientific Computing (2nd ed.; Cambridge: Cambridge Univ. Press) Sanders, R. H., & Tubbs, A. 1980, , 235, 803 Schönrich, R., Binney, J., & Dehnen, W. 2010, , 403, 1829 Webb, J. J., Harris, W. E., Sills, A., & Hurley, J. R. 2013, , 764, 124 Wilson, C. P. 1975, , 80, 175 [crrrrrrcrr]{} NGC 104 & 6.02363 & $-$72.08128 & 4.5$\pm$0.45 & $-$18.0$\pm$0.1 & 5.64$\pm$0.20 & $-$2.02$\pm$0.20 & 0.10E+07 & 55.37 & 4.15\ NGC 288 & 13.18850 & $-$26.58261 & 8.9$\pm$0.89 & $-$45.4$\pm$0.2 & 4.67$\pm$0.42 & $-$5.62$\pm$0.23 & 0.86E+05 & 34.15 & 5.77\ NGC 362 & 15.80942 & $-$70.84878 & 8.6$\pm$0.86 & 223.5$\pm$0.5 & 5.07$\pm$0.71 & $-$2.55$\pm$0.72 & 0.40E+06 & 26.61 & 2.05\ NGC 1851 & 78.52817 & $-$40.04655 & 12.1$\pm$1.21 & 320.5$\pm$0.6 & 1.28$\pm$0.68 & 2.39$\pm$0.65 & 0.37E+06 & 22.95 & 1.80\ NGC 1904 & 81.04621 & $-$24.52472 & 12.9$\pm$1.29 & 205.8$\pm$0.4 & 2.12$\pm$0.64 & $-$0.02$\pm$0.64 & 0.24E+06 & 30.90 & 2.44\ NGC 2298 & 102.24754 & $-$36.00531 & 10.8$\pm$1.08 & 148.9$\pm$1.2 & 4.05$\pm$1.00 & $-$1.72$\pm$0.98 & 0.57E+05 & 23.36 & 3.08\ NGC 2808 & 138.01292 & $-$64.86350 & 9.6$\pm$0.96 & 101.6$\pm$0.7 & 0.58$\pm$0.45 & 2.06$\pm$0.46 & 0.97E+06 & 25.35 & 2.23\ Pal 3 & 151.38292 & 0.07167 & 92.5$\pm$9.25 & 83.4$\pm$8.4 & 0.33$\pm$0.23 & 0.30$\pm$0.31 & 0.32E+05 & 107.81 & 17.49\ NGC 3201 & 154.40342 & $-$46.41247 & 4.9$\pm$0.49 & 494.0$\pm$0.2 & 5.28$\pm$0.32 & $-$0.98$\pm$0.33 & 0.16E+06 & 36.13 & 4.42\ NGC 4147 & 182.52625 & 18.54264 & 19.3$\pm$1.93 & 183.2$\pm$0.7 & $-$1.85$\pm$0.82 & $-$1.30$\pm$0.82 & 0.50E+05 & 34.16 & 2.69\ NGC 4372 & 186.43917 & $-$72.65900 & 5.8$\pm$0.58 & 72.3$\pm$1.2 & $-$6.49$\pm$0.33 & 3.71$\pm$0.32 & 0.22E+06 & 58.91 & 6.60\ NGC 4590 & 189.86658 & $-$26.74406 & 10.3$\pm$1.03 & $-$94.7$\pm$0.2 & $-$3.76$\pm$0.66 & 1.79$\pm$0.62 & 0.15E+06 & 44.67 & 4.52\ NGC 4833 & 194.89133 & $-$70.87650 & 6.6$\pm$0.66 & 200.2$\pm$1.2 & $-$8.11$\pm$0.35 & $-$0.96$\pm$0.34 & 0.32E+06 & 34.14 & 4.63\ NGC 5024 & 198.23021 & 18.16817 & 17.9$\pm$1.79 & $-$62.9$\pm$0.3 & 0.50$\pm$1.00 & $-$0.10$\pm$1.00 & 0.52E+06 & 95.64 & 6.82\ NGC 5139 & 201.69683 & $-$47.47958 & 5.2$\pm$0.52 & 232.1$\pm$0.1 & $-$5.08$\pm$0.35 & $-$3.57$\pm$0.34 & 0.22E+07 & 73.19 & 7.56\ NGC 5272 & 205.54842 & 28.37728 & 10.2$\pm$1.02 & $-$147.6$\pm$0.2 & $-$1.10$\pm$0.51 & $-$2.30$\pm$0.54 & 0.61E+06 & 85.22 & 6.85\ NGC 5466 & 211.36371 & 28.53444 & 16.0$\pm$1.60 & 110.7$\pm$0.2 & $-$4.65$\pm$0.82 & 0.80$\pm$0.82 & 0.11E+06 & 72.98 & 10.70\ Pal 5 & 229.02188 & $-$0.11161 & 23.2$\pm$2.32 & $-$58.7$\pm$0.2 & $-$1.78$\pm$0.17 & $-$2.32$\pm$0.23 & 0.20E+05 & 51.17 & 18.42\ NGC 5897 & 229.35208 & $-$21.01028 & 12.5$\pm$1.25 & 101.5$\pm$1.0 & $-$4.93$\pm$0.86 & $-$2.33$\pm$0.84 & 0.13E+06 & 36.88 & 7.49\ NGC 5904 & 229.63842 & 2.08103 & 7.5$\pm$0.75 & 53.2$\pm$0.4 & 5.07$\pm$0.68 & $-$10.70$\pm$0.56 & 0.57E+06 & 51.55 & 3.86\ NGC 5927 & 232.00288 & $-$50.67303 & 7.7$\pm$0.77 & $-$107.5$\pm$0.9 & $-$5.72$\pm$0.39 & $-$2.61$\pm$0.40 & 0.23E+06 & 37.45 & 2.46\ NGC 5986 & 236.51250 & $-$37.78642 & 10.4$\pm$1.04 & 88.9$\pm$3.7 & $-$3.81$\pm$0.45 & $-$2.99$\pm$0.37 & 0.41E+06 & 24.15 & 2.96\ NGC 6093 & 244.26004 & $-$22.97608 & 10.0$\pm$1.00 & 8.1$\pm$1.5 & $-$3.31$\pm$0.58 & $-$7.20$\pm$0.67 & 0.33E+06 & 20.88 & 1.77\ NGC 6121 & 245.89675 & $-$26.52575 & 2.2$\pm$0.22 & 70.7$\pm$0.2 & $-$12.50$\pm$0.36 & $-$19.93$\pm$0.49 & 0.13E+06 & 33.16 & 2.77\ NGC 6144 & 246.80775 & $-$26.02350 & 8.9$\pm$0.89 & 193.8$\pm$0.6 & $-$3.06$\pm$0.64 & $-$5.11$\pm$0.72 & 0.94E+05 & 86.35 & 4.22\ NGC 6171 & 248.13275 & $-$13.05378 & 6.4$\pm$0.64 & $-$34.1$\pm$0.3 & $-$0.70$\pm$0.90 & $-$3.10$\pm$1.00 & 0.12E+06 & 35.33 & 3.22\ NGC 6205 & 250.42183 & 36.45986 & 7.1$\pm$0.71 & $-$244.2$\pm$0.2 & $-$0.90$\pm$0.71 & 5.50$\pm$1.12 & 0.45E+06 & 43.39 & 3.49\ NGC 6218 & 251.80908 & $-$1.94853 & 4.8$\pm$0.48 & $-$41.4$\pm$0.2 & 1.30$\pm$0.58 & $-$7.83$\pm$0.62 & 0.14E+06 & 24.13 & 2.47\ NGC 6254 & 254.28771 & $-$4.10031 & 4.4$\pm$0.44 & 75.2$\pm$0.7 & $-$6.00$\pm$1.00 & $-$3.30$\pm$1.00 & 0.17E+06 & 23.64 & 2.50\ NGC 6266 & 255.30333 & $-$30.11372 & 6.8$\pm$0.68 & $-$70.1$\pm$1.4 & $-$3.50$\pm$0.37 & $-$0.82$\pm$0.37 & 0.80E+06 & 23.10 & 1.82\ NGC 6273 & 255.65750 & $-$26.26797 & 8.8$\pm$0.88 & 135.0$\pm$4.1 & $-$2.86$\pm$0.49 & $-$0.45$\pm$0.51 & 0.77E+06 & 37.30 & 3.38\ NGC 6284 & 256.11879 & $-$24.76486 & 15.3$\pm$1.53 & 27.5$\pm$1.7 & $-$3.66$\pm$0.64 & $-$5.39$\pm$0.83 & 0.26E+06 & 102.72 & 2.94\ NGC 6287 & 256.28804 & $-$22.70836 & 9.4$\pm$0.94 & $-$288.7$\pm$3.5 & $-$3.68$\pm$0.88 & $-$3.54$\pm$0.69 & 0.15E+06 & 19.02 & 2.02\ NGC 6293 & 257.54250 & $-$26.58208 & 9.5$\pm$0.95 & $-$146.2$\pm$1.7 & 0.26$\pm$0.85 & $-$5.14$\pm$0.71 & 0.22E+06 & 39.32 & 2.46\ NGC 6304 & 258.63438 & $-$29.46203 & 5.9$\pm$0.59 & $-$107.3$\pm$3.6 & $-$2.59$\pm$0.29 & $-$1.56$\pm$0.29 & 0.14E+06 & 22.74 & 2.44\ NGC 6316 & 259.15542 & $-$28.14011 & 10.4$\pm$1.04 & 71.4$\pm$8.9 & $-$2.42$\pm$0.63 & $-$1.71$\pm$0.56 & 0.37E+06 & 22.97 & 1.97\ NGC 6333 & 259.79692 & $-$18.51594 & 7.9$\pm$0.79 & 229.1$\pm$7.0 & $-$0.57$\pm$0.57 & $-$3.70$\pm$0.50 & 0.26E+06 & 18.39 & 2.21\ NGC 6341 & 259.28079 & 43.13594 & 8.3$\pm$0.83 & $-$120.0$\pm$0.1 & $-$3.30$\pm$0.55 & $-$0.33$\pm$0.70 & 0.33E+06 & 30.05 & 2.46\ NGC 6342 & 260.29200 & $-$19.58742 & 8.5$\pm$0.85 & 115.7$\pm$1.4 & $-$2.77$\pm$0.71 & $-$5.84$\pm$0.65 & 0.63E+05 & 36.74 & 1.80\ NGC 6356 & 260.89554 & $-$17.81303 & 15.1$\pm$1.51 & 27.0$\pm$4.3 & $-$3.14$\pm$0.68 & $-$3.65$\pm$0.53 & 0.43E+06 & 41.01 & 3.56\ NGC 6362 & 262.97913 & $-$67.04833 & 7.6$\pm$0.76 & $-$13.1$\pm$0.6 & $-$3.09$\pm$0.46 & $-$3.83$\pm$0.46 & 0.10E+06 & 30.73 & 4.53\ NGC 6388 & 264.07179 & $-$44.73550 & 9.9$\pm$0.99 & 80.1$\pm$0.8 & $-$1.90$\pm$0.45 & $-$3.83$\pm$0.51 & 0.99E+06 & 19.43 & 1.50\ NGC 6397 & 265.17538 & $-$53.67433 & 2.3$\pm$0.23 & 18.8$\pm$0.1 & 3.69$\pm$0.29 & $-$14.88$\pm$0.26 & 0.77E+05 & 29.79 & 1.94\ NGC 6441 & 267.55442 & $-$37.05144 & 11.6$\pm$1.16 & 16.5$\pm$1.0 & $-$2.86$\pm$0.45 & $-$3.45$\pm$0.76 & 0.12E+07 & 24.11 & 1.92\ NGC 6522 & 270.89175 & $-$30.03397 & 7.7$\pm$0.77 & $-$21.1$\pm$3.4 & 6.08$\pm$0.20 & $-$1.83$\pm$0.20 & 0.20E+06 & 36.82 & 2.24\ NGC 6528 & 271.20683 & $-$30.05628 & 7.9$\pm$0.79 & 206.6$\pm$1.4 & $-$0.35$\pm$0.23 & 0.27$\pm$0.26 & 0.73E+05 & 9.45 & 0.87\ NGC 6553 & 272.32333 & $-$25.90869 & 6.0$\pm$0.60 & $-$3.2$\pm$1.5 & 2.50$\pm$0.07 & 5.35$\pm$0.08 & 0.22E+06 & 13.37 & 1.80\ NGC 6584 & 274.65667 & $-$52.21578 & 13.5$\pm$1.35 & 222.9$\pm$15.0 & $-$0.22$\pm$0.62 & $-$5.79$\pm$0.67 & 0.20E+06 & 30.13 & 2.87\ NGC 6626 & 276.13671 & $-$24.86978 & 5.5$\pm$0.55 & 17.0$\pm$1.0 & 0.63$\pm$0.67 & $-$8.46$\pm$0.67 & 0.31E+06 & 17.96 & 3.15\ NGC 6656 & 279.09975 & $-$23.90475 & 3.2$\pm$0.32 & $-$146.3$\pm$0.2 & 7.37$\pm$0.50 & $-$3.95$\pm$0.42 & 0.43E+06 & 29.70 & 3.13\ NGC 6712 & 283.26792 & $-$8.70611 & 6.9$\pm$0.69 & $-$107.6$\pm$0.5 & 4.20$\pm$0.40 & $-$2.00$\pm$0.40 & 0.17E+06 & 17.12 & 2.67\ NGC 6723 & 284.88813 & $-$36.63225 & 8.7$\pm$0.87 & $-$94.5$\pm$3.6 & $-$0.17$\pm$0.45 & $-$2.16$\pm$0.50 & 0.23E+06 & 30.20 & 3.87\ NGC 6752 & 287.71712 & $-$59.98456 & 4.0$\pm$0.40 & $-$26.7$\pm$0.2 & $-$0.69$\pm$0.42 & $-$2.85$\pm$0.45 & 0.21E+06 & 40.48 & 2.22\ NGC 6779 & 289.14821 & 30.18347 & 9.4$\pm$0.94 & $-$135.6$\pm$0.9 & 0.30$\pm$1.00 & 1.40$\pm$0.10 & 0.16E+06 & 28.86 & 3.01\ NGC 6809 & 294.99879 & $-$30.96475 & 5.4$\pm$0.54 & 174.7$\pm$0.3 & $-$1.42$\pm$0.62 & $-$10.25$\pm$0.64 & 0.18E+06 & 24.07 & 4.45\ NGC 6838 & 298.44371 & 18.77919 & 4.0$\pm$0.40 & $-$22.8$\pm$0.2 & $-$2.30$\pm$0.80 & $-$5.10$\pm$0.80 & 0.30E+05 & 10.35 & 1.94\ NGC 6934 & 308.54738 & 7.40447 & 15.6$\pm$1.56 & $-$411.4$\pm$1.6 & 1.20$\pm$1.00 & $-$5.10$\pm$1.00 & 0.16E+06 & 33.83 & 3.13\ NGC 7006 & 315.37242 & 16.18733 & 41.2$\pm$4.12 & $-$384.1$\pm$0.4 & $-$0.96$\pm$0.35 & $-$1.14$\pm$0.40 & 0.20E+06 & 52.37 & 5.27\ NGC 7078 & 322.49304 & 12.16700 & 10.4$\pm$1.04 & $-$107.0$\pm$0.2 & $-$0.95$\pm$0.51 & $-$5.63$\pm$0.50 & 0.81E+06 & 63.23 & 3.03\ NGC 7089 & 323.36258 & $-$0.82325 & 11.5$\pm$1.15 & $-$5.3$\pm$2.0 & 5.90$\pm$0.86 & $-$4.95$\pm$0.86 & 0.70E+06 & 41.65 & 3.55\ NGC 7099 & 325.09217 & $-$23.17986 & 8.1$\pm$0.81 & $-$184.2$\pm$0.2 & 1.42$\pm$0.69 & $-$7.71$\pm$0.65 & 0.16E+06 & 37.43 & 2.43\ Pal 12 & 326.66183 & $-$21.25261 & 19.0$\pm$1.90 & 27.8$\pm$1.5 & $-$1.20$\pm$0.30 & $-$4.21$\pm$0.29 & 0.10E+05 & 105.56 & 9.51\ Pal 13 & 346.68517 & 12.77200 & 26.0$\pm$2.60 & 25.2$\pm$0.3 & 2.30$\pm$0.26 & 0.27$\pm$0.25 & 0.54E+04 & 16.59 & 2.72\ [lcr]{} $R_0$ & 8.3$\pm$0.23 kpc & 1\ ${\Theta}_0$ & 239$\pm$7 km/s & 1\ $(U,V,W)_{\odot}$ & ($-11.1\pm$1.2,12.24$\pm$2.1,7.25$\pm$0.6) km s$^{-1}$ & 2,1\ position of major axis & 20$^{\circ}$ & 3\ angular velocity & 55$\pm$5 $\kmskpc$ & 4\ $M_{\rm arms}/M_{\rm disk}$ & 0.04$\pm$0.01 & 5\ scale length ($H$) & 3.9$\pm$0.6 kpc ($R_0$ = 8.5 kpc) & 6\ pitch angle & 15.5$\pm$3.5$^{\circ}$ & 7\ angular velocity & 24$\pm$6 $\kmskpc$ & 4 [cccccrccc]{} NGC 104 & 5.78 & 8.30 & 3.13 & 0.177 & & & 91.6 & 102.9\ & 6.25 & 7.57 & 3.16 & 0.095 & $-1482.12$ & 134.55 & 98.3 & 112.4\ NGC 288 & 2.60 & 12.38 & 6.40 & 0.654 & & & 23.6 & 22.8\ & 2.78 & 12.25 & 6.70 & 0.632 & $-1384.68$ & $-46.38$ & 24.5 & 23.5\ NGC 362 & 1.39 & 9.28 & 3.85 & 0.736 & & & 24.3 & 20.7\ & 0.74 & 11.09 & 2.16 & 0.877 & $-1492.42$ & $-12.56$ & 17.1 & 15.9\ NGC 1851 & 6.66 & 33.28 & 7.61 & 0.667 & & & 70.1 & 68.2\ & 6.73 & 31.75 & 7.59 & 0.650 & $-939.52$ & 238.99 & 71.5 & 69.5\ NGC 1904 & 5.25 & 18.85 & 5.54 & 0.563 & & & 51.4 & 51.4\ & 5.19 & 20.52 & 5.37 & 0.596 & $-1137.89$ & 173.68 & 52.0 & 51.6\ NGC 2298 & 3.18 & 20.38 & 11.49 & 0.731 & & & 22.0 & 20.8\ & 3.20 & 17.90 & 9.52 & 0.698 & $-1212.81$ & $-56.36$ & 22.7 & 21.3\ NGC 2808 & 2.27 & 10.74 & 2.39 & 0.649 & & & 46.0 & 43.4\ & 2.73 & 12.74 & 2.59 & 0.647 & $-1395.39$ & 94.19 & 53.3 & 51.4\ NGC 3201 & 9.00 & 16.86 & 4.54 & 0.304 & & & 67.5 & 72.3\ & 8.99 & 17.12 & 4.52 & 0.311 & $-1151.30$ & $-251.56$ & 67.8 & 72.5\ NGC 4147 & 3.89 & 27.33 & 13.97 & 0.750 & & & 25.0 & 22.8\ & 3.78 & 28.64 & 14.73 & 0.766 & $-1001.35$ & 66.83 & 25.4 & 22.8\ NGC 4372 & 2.39 & 5.30 & 1.57 & 0.386 & & & 30.4 & 33.2\ & 3.19 & 7.41 & 1.60 & 0.397 & $-1624.75$ & 95.34 & 37.7 & 39.9\ NGC 4590 & 9.60 & 30.81 & 11.86 & 0.525 & & & 67.5 & 68.6\ & 9.57 & 30.40 & 11.82 & 0.521 & $-932.42$ & 264.42 & 68.0 & 68.6\ NGC 4833 & 1.04 & 8.41 & 1.54 & 0.778 & & & 22.1 & 18.3\ & 0.98 & 7.43 & 1.93 & 0.767 & $-1685.97$ & 26.11 & 18.3 & 17.3\ NGC 5024 & 16.43 & 36.30 & 24.34 & 0.377 & & & 155.4 & 161.6\ & 16.44 & 36.46 & 24.44 & 0.379 & $-811.52$ & 143.90 & 155.7 & 161.9\ NGC 5139 & 1.49 & 5.81 & 1.69 & 0.592 & & & 47.5 & 47.4\ & 0.98 & 6.45 & 1.16 & 0.737 & $-1770.34$ & $-34.25$ & 36.7 & 35.5\ NGC 5272 & 5.61 & 13.28 & 8.77 & 0.404 & & & 76.3 & 80.2\ & 5.60 & 14.22 & 8.99 & 0.435 & $-1267.90$ & 79.36 & 76.7 & 79.4\ NGC 5466 & 6.81 & 60.45 & 36.45 & 0.797 & & & 48.8 & 44.1\ & 6.85 & 60.20 & 36.33 & 0.796 & $-663.05$ & $-32.56$ & 49.1 & 44.2\ Pal 5 & 3.80 & 18.74 & 10.93 & 0.663 & & & 19.2 & 18.5\ & 3.97 & 18.88 & 10.92 & 0.653 & $-1179.26$ & 54.78 & 19.6 & 18.7\ NGC 5897 & 1.89 & 7.95 & 5.09 & 0.621 & & & 23.4 & 22.5\ & 1.48 & 8.94 & 4.49 & 0.719 & $-1552.41$ & 23.18 & 17.5 & 16.6\ NGC 5904 & 2.65 & 36.80 & 17.89 & 0.866 & & & 46.5 & 40.4\ & 2.76 & 37.35 & 18.11 & 0.863 & $-888.65$ & 40.12 & 46.6 & 40.5\ NGC 5927 & 3.44 & 4.64 & 0.80 & 0.150 & & & 41.4 & 48.8\ & 4.50 & 5.45 & 0.79 & 0.095 & $-1693.39$ & 110.15 & 48.7 & 57.3\ NGC 5986 & 1.05 & 3.97 & 1.33 & 0.562 & & & 24.7 & 24.6\ & 0.46 & 4.90 & 1.31 & 0.831 & $-1904.82$ & 1.51 & 12.9 & 12.4\ NGC 6093 & 2.07 & 3.11 & 3.02 & 0.201 & & & 34.1 & 37.0\ & 2.01 & 3.78 & 3.14 & 0.311 & $-1867.55$ & 10.18 & 31.7 & 32.8\ \ NGC 6121 & 0.39 & 5.95 & 0.49 & 0.874 & & & 11.6 & 7.7\ & 0.55 & 5.47 & 2.11 & 0.827 & $-1824.59$ & $-1.58$ & 9.8 & 8.7\ NGC 6144 & 2.14 & 2.99 & 2.64 & 0.166 & & & 22.2 & 24.9\ & 2.08 & 2.66 & 2.33 & 0.123 & $-1981.75$ & $-20.51$ & 22.1 & 23.3\ NGC 6171 & 2.28 & 3.14 & 2.34 & 0.157 & & & 25.1 & 28.4\ & 2.70 & 3.31 & 2.41 & 0.104 & $-1886.70$ & 39.42 & 28.3 & 31.2\ NGC 6205 & 5.35 & 21.93 & 13.90 & 0.609 & & & 67.7 & 67.2\ & 5.30 & 22.61 & 14.22 & 0.621 & $-1087.59$ & $-30.31$ & 67.1 & 66.4\ NGC 6218 & 2.76 & 5.94 & 2.21 & 0.363 & & & 30.0 & 32.1\ & 2.73 & 5.32 & 2.56 & 0.323 & $-1744.69$ & 59.13 & 29.5 & 31.2\ NGC 6254 & 3.86 & 5.84 & 2.43 & 0.204 & & & 38.8 & 43.5\ & 3.46 & 4.93 & 2.40 & 0.175 & $-1737.04$ & 71.23 & 37.3 & 41.9\ NGC 6266 & 1.52 & 2.63 & 0.83 & 0.276 & & & 37.2 & 43.6\ & 1.41 & 2.22 & 0.85 & 0.223 & $-2167.78$ & 33.20 & 32.8 & 34.9\ NGC 6273 & 1.28 & 2.40 & 1.28 & 0.304 & & & 34.4 & 38.4\ & 1.35 & 1.83 & 1.60 & 0.153 & $-2169.73$ & $-11.81$ & 32.0 & 33.2\ NGC 6284 & 6.34 & 8.52 & 2.78 & 0.147 & & & 62.2 & 69.7\ & 6.40 & 8.10 & 2.68 & 0.117 & $-1461.17$ & 148.72 & 64.3 & 73.3\ NGC 6287 & 0.91 & 5.03 & 2.82 & 0.707 & & & 16.3 & 14.2\ & 0.87 & 4.30 & 2.44 & 0.671 & $-1895.79$ & $-3.07$ & 8.7 & 7.7\ NGC 6293 & 0.32 & 3.34 & 0.46 & 0.826 & & & 14.2 & 8.9\ & 0.37 & 2.67 & 1.19 & 0.756 & $-2168.38$ & $-3.58$ & 8.6 & 7.7\ NGC 6304 & 1.90 & 3.25 & 0.53 & 0.276 & & & 23.7 & 27.4\ & 1.84 & 3.09 & 0.57 & 0.253 & $-2054.85$ & 48.88 & 22.8 & 24.3\ NGC 6316 & 0.72 & 3.07 & 1.18 & 0.626 & & & 19.9 & 19.2\ & 0.96 & 2.59 & 0.83 & 0.460 & $-2170.75$ & $-26.10$ & 18.2 & 19.0\ NGC 6333 & 1.44 & 5.32 & 1.53 & 0.582 & & & 21.5 & 21.5\ & 1.02 & 4.37 & 1.34 & 0.623 & $-1937.01$ & 27.83 & 15.3 & 15.1\ NGC 6341 & 1.20 & 10.43 & 2.43 & 0.793 & & & 22.9 & 20.6\ & 1.30 & 10.86 & 2.59 & 0.786 & $-1496.54$ & 30.29 & 21.6 & 20.1\ NGC 6342 & 1.29 & 2.09 & 1.37 & 0.245 & & & 14.9 & 17.1\ & 0.73 & 1.68 & 1.13 & 0.401 & $-2304.73$ & 10.92 & 8.3 & 8.6\ NGC 6356 & 2.45 & 7.94 & 2.08 & 0.528 & & & 38.2 & 39.6\ & 2.30 & 7.74 & 1.98 & 0.542 & $-1637.01$ & 67.45 & 37.3 & 37.8\ NGC 6362 & 2.31 & 5.05 & 1.98 & 0.374 & & & 22.4 & 24.0\ & 2.28 & 5.80 & 1.70 & 0.436 & $-1760.08$ & 62.23 & 23.4 & 24.4\ NGC 6388 & 0.70 & 3.03 & 1.10 & 0.627 & & & 27.6 & 26.1\ & 0.53 & 2.95 & 0.88 & 0.696 & $-2148.65$ & $-13.57$ & 16.2 & 16.2\ NGC 6397 & 2.53 & 5.12 & 1.46 & 0.344 & & & 23.1 & 25.4\ & 3.33 & 6.42 & 1.66 & 0.317 & $-1672.42$ & 91.10 & 27.4 & 29.7\ NGC 6441 & 0.57 & 3.97 & 0.87 & 0.751 & & & 27.6 & 22.5\ & 0.48 & 3.15 & 1.45 & 0.742 & $-2082.24$ & $-2.77$ & 16.9 & 15.7\ NGC 6522 & 0.37 & 3.87 & 0.57 & 0.831 & & & 13.4 & 8.9\ & 0.81 & 2.23 & 1.16 & 0.471 & $-2199.97$ & 18.26 & 13.2 & 13.4\ \ NGC 6528 & 0.81 & 2.85 & 1.20 & 0.571 & & & 12.5 & 12.0\ & 0.57 & 1.51 & 0.77 & 0.454 & $-2399.18$ & 14.31 & 7.3 & 7.6\ NGC 6553 & 2.09 & 8.75 & 0.33 & 0.615 & & & 28.9 & 28.1\ & 2.31 & 12.02 & 0.52 & 0.677 & $-1445.87$ & 97.14 & 30.2 & 28.0\ NGC 6584 & 1.38 & 12.43 & 4.43 & 0.804 & & & 21.7 & 19.4\ & 1.06 & 12.20 & 3.11 & 0.843 & $-1434.63$ & 24.27 & 15.3 & 14.5\ NGC 6626 & 1.03 & 2.82 & 0.85 & 0.472 & & & 21.7 & 24.0\ & 0.74 & 3.09 & 0.88 & 0.613 & $-2111.49$ & 21.96 & 14.2 & 14.3\ NGC 6656 & 2.85 & 7.96 & 1.18 & 0.472 & & & 42.9 & 45.1\ & 3.10 & 9.18 & 1.28 & 0.495 & $-1542.68$ & 105.56 & 45.7 & 46.8\ NGC 6712 & 0.60 & 5.56 & 1.14 & 0.809 & & & 15.9 & 13.6\ & 0.91 & 6.34 & 1.86 & 0.749 & $-1767.49$ & 13.60 & 12.9 & 12.5\ NGC 6723 & 2.00 & 3.25 & 3.03 & 0.242 & & & 29.1 & 31.4\ & 2.06 & 2.66 & 2.65 & 0.127 & $-1969.12$ & $-0.19$ & 30.1 & 30.8\ NGC 6752 & 4.65 & 6.71 & 1.75 & 0.180 & & & 45.8 & 52.7\ & 4.74 & 5.81 & 1.72 & 0.102 & $-1640.42$ & 109.14 & 48.9 & 56.8\ NGC 6779 & 0.62 & 12.52 & 0.77 & 0.906 & & & 15.2 & 10.7\ & 0.84 & 12.49 & 2.44 & 0.875 & $-1428.17$ & $-24.15$ & 11.7 & 10.5\ NGC 6809 & 1.98 & 5.95 & 3.87 & 0.508 & & & 25.5 & 26.1\ & 1.78 & 5.61 & 3.59 & 0.526 & $-1741.44$ & 19.74 & 22.9 & 22.7\ NGC 6838 & 5.01 & 6.56 & 0.33 & 0.131 & & & 25.2 & 29.4\ & 4.88 & 6.98 & 0.29 & 0.177 & $-1599.94$ & 134.92 & 25.8 & 29.6\ NGC 6934 & 6.88 & 34.87 & 20.35 & 0.670 & & & 54.6 & 52.3\ & 6.83 & 35.12 & 20.60 & 0.674 & $-892.95$ & $-56.25$ & 54.6 & 52.4\ NGC 7006 & 17.89 & 79.15 & 26.93 & 0.631 & & & 115.4 & 111.1\ & 17.90 & 79.32 & 26.97 & 0.632 & $-514.26$ & 572.52 & 115.7 & 111.4\ NGC 7078 & 6.12 & 10.89 & 5.25 & 0.281 & & & 91.5 & 98.5\ & 6.48 & 11.00 & 5.45 & 0.259 & $-1347.69$ & 139.60 & 94.4 & 103.1\ NGC 7089 & 6.10 & 33.12 & 18.03 & 0.689 & & & 83.4 & 78.9\ & 6.14 & 34.19 & 18.77 & 0.695 & $-908.90$ & $-67.44$ & 84.1 & 79.3\ NGC 7099 & 3.16 & 6.91 & 4.20 & 0.373 & & & 32.6 & 35.0\ & 3.09 & 7.38 & 4.70 & 0.412 & $-1584.65$ & $-46.62$ & 32.9 & 34.0\ Pal 12 & 15.19 & 19.64 & 15.76 & 0.128 & & & 40.3 & 44.7\ & 15.25 & 19.86 & 15.90 & 0.131 & $-1010.08$ & 165.80 & 40.4 & 44.8\ Pal 13 & 11.84 & 88.01 & 38.03 & 0.763 & & & 25.4 & 23.5\ & 11.86 & 88.13 & 38.24 & 0.763 & $-485.90$ & $-329.53$ & 25.5 & 23.6\ [cccccccccc]{} NGC 104 & 0.321E-15 & 0.191E-15 & 0.962E-14 & 0.286E-14 & 0.225E-14 & 0.650E-14 & 0.146E-12 & 0.359E-13 & 0.616E-13\ & 0.806E-16 & 0.304E-16 & 0.497E-16 & 0.466E-15 & 0.104E-15 & 0.165E-15 & 0.148E-12 & 0.281E-13 & 0.352E-13\ NGC 288 & 0.750E-10 & 0.669E-10 & 0.289E-09 & 0.157E-09 & 0.129E-09 & 0.408E-09 & 0.809E-11 & 0.440E-11 & 0.563E-11\ & 0.621E-09 & 0.585E-09 & 0.538E-08 & 0.542E-09 & 0.516E-09 & 0.496E-08 & 0.899E-11 & 0.493E-11 & 0.842E-11\ NGC 362 & 0.365E-11 & 0.259E-11 & 0.445E-11 & 0.228E-11 & 0.144E-11 & 0.244E-11 & 0.128E-12 & 0.383E-13 & 0.439E-13\ & 0.767E-10 & 0.580E-10 & 0.134E-09 & 0.390E-10 & 0.301E-10 & 0.672E-10 & 0.170E-12 & 0.528E-13 & 0.614E-13\ NGC 1851 & 0.540E-16 & 0.495E-16 & 0.227E-14 & 0.248E-15 & 0.236E-15 & 0.703E-14 & 0.777E-15 & 0.642E-15 & 0.332E-14\ & 0.145E-15 & 0.138E-15 & 0.146E-13 & 0.488E-16 & 0.466E-16 & 0.523E-14 & 0.860E-15 & 0.723E-15 & 0.419E-14\ NGC 1904 & 0.127E-12 & 0.123E-12 & 0.306E-11 & 0.135E-12 & 0.126E-12 & 0.126E-11 & 0.628E-13 & 0.506E-13 & 0.164E-12\ & 0.105E-11 & 0.104E-11 & 0.866E-10 & 0.714E-12 & 0.704E-12 & 0.639E-10 & 0.635E-13 & 0.512E-13 & 0.176E-12\ NGC 2298 & 0.789E-11 & 0.741E-11 & 0.469E-10 & 0.983E-11 & 0.889E-11 & 0.465E-10 & 0.835E-12 & 0.668E-12 & 0.151E-11\ & 0.769E-10 & 0.743E-10 & 0.834E-09 & 0.562E-10 & 0.544E-10 & 0.645E-09 & 0.989E-12 & 0.768E-12 & 0.182E-11\ NGC 2808 & 0.498E-14 & 0.383E-14 & 0.201E-13 & 0.819E-14 & 0.666E-14 & 0.459E-13 & 0.727E-14 & 0.289E-14 & 0.406E-14\ & 0.225E-14 & 0.171E-14 & 0.210E-13 & 0.672E-15 & 0.510E-15 & 0.636E-14 & 0.537E-14 & 0.220E-14 & 0.328E-14\ Pal 3 & 0.506E-15 & 0.499E-15 & 0.830E-14 & 0.154E-14 & 0.149E-14 & 0.239E-13 & 0.132E-14 & 0.129E-14 & 0.203E-13\ & 0.163E-14 & 0.161E-14 & 0.826E-13 & 0.107E-14 & 0.106E-14 & 0.556E-13 & 0.490E-14 & 0.483E-14 & 0.241E-12\ NGC 3201 & 0.903E-16 & 0.221E-16 & 0.331E-16 & 0.168E-15 & 0.999E-16 & 0.174E-14 & 0.107E-13 & 0.504E-14 & 0.125E-13\ & 0.895E-16 & 0.212E-16 & 0.302E-16 & 0.801E-16 & 0.345E-16 & 0.779E-16 & 0.109E-13 & 0.505E-14 & 0.133E-13\ NGC 4147 & 0.320E-11 & 0.308E-11 & 0.300E-10 & 0.300E-11 & 0.276E-11 & 0.209E-10 & 0.466E-12 & 0.384E-12 & 0.984E-12\ & 0.219E-10 & 0.213E-10 & 0.669E-09 & 0.217E-10 & 0.213E-10 & 0.685E-09 & 0.519E-12 & 0.427E-12 & 0.164E-11\ NGC 4372 & 0.910E-10 & 0.697E-10 & 0.463E-09 & 0.670E-10 & 0.458E-10 & 0.234E-09 & 0.792E-10 & 0.327E-10 & 0.567E-10\ & 0.352E-11 & 0.167E-11 & 0.213E-11 & 0.238E-11 & 0.106E-11 & 0.136E-11 & 0.436E-10 & 0.137E-10 & 0.171E-10\ NGC 4590 & 0.299E-15 & 0.149E-15 & 0.337E-15 & 0.147E-15 & 0.655E-16 & 0.258E-15 & 0.151E-13 & 0.887E-14 & 0.223E-13\ & 0.295E-15 & 0.145E-15 & 0.351E-15 & 0.112E-15 & 0.590E-16 & 0.154E-15 & 0.162E-13 & 0.951E-14 & 0.247E-13\ NGC 4833 & 0.661E-10 & 0.408E-10 & 0.792E-10 & 0.160E-09 & 0.878E-10 & 0.144E-09 & 0.389E-11 & 0.107E-11 & 0.152E-11\ & 0.454E-09 & 0.326E-09 & 0.106E-08 & 0.264E-09 & 0.189E-09 & 0.633E-09 & 0.422E-11 & 0.100E-11 & 0.126E-11\ NGC 5024 & 0.188E-12 & 0.187E-12 & 0.415E-10 & 0.487E-12 & 0.484E-12 & 0.597E-10 & 0.491E-13 & 0.475E-13 & 0.138E-11\ & 0.307E-12 & 0.305E-12 & 0.937E-10 & 0.292E-12 & 0.291E-12 & 0.104E-09 & 0.561E-13 & 0.539E-13 & 0.151E-11\ NGC 5139 & 0.371E-10 & 0.187E-10 & 0.443E-10 & 0.958E-10 & 0.546E-10 & 0.105E-09 & 0.639E-11 & 0.157E-11 & 0.184E-11\ & 0.156E-09 & 0.992E-10 & 0.320E-09 & 0.108E-09 & 0.754E-10 & 0.241E-09 & 0.746E-11 & 0.151E-11 & 0.189E-11\ NGC 5272 & 0.438E-12 & 0.377E-12 & 0.718E-11 & 0.170E-11 & 0.161E-11 & 0.221E-10 & 0.118E-11 & 0.618E-12 & 0.124E-11\ & 0.854E-12 & 0.770E-12 & 0.441E-10 & 0.661E-12 & 0.609E-12 & 0.481E-10 & 0.127E-11 & 0.681E-12 & 0.132E-11\ NGC 5466 & 0.538E-11 & 0.503E-11 & 0.136E-09 & 0.171E-10 & 0.153E-10 & 0.247E-09 & 0.307E-11 & 0.245E-11 & 0.113E-10\ & 0.495E-11 & 0.462E-11 & 0.208E-09 & 0.417E-11 & 0.392E-11 & 0.221E-09 & 0.288E-11 & 0.227E-11 & 0.101E-10\ Pal 5 & 0.345E-08 & 0.312E-08 & 0.149E-07 & 0.110E-07 & 0.950E-08 & 0.445E-07 & 0.461E-09 & 0.317E-09 & 0.516E-09\ & 0.304E-07 & 0.289E-07 & 0.300E-06 & 0.399E-07 & 0.383E-07 & 0.406E-06 & 0.616E-09 & 0.428E-09 & 0.172E-08\ NGC 5897 & 0.250E-09 & 0.220E-09 & 0.625E-09 & 0.406E-09 & 0.366E-09 & 0.108E-08 & 0.196E-10 & 0.113E-10 & 0.154E-10\ & 0.434E-08 & 0.402E-08 & 0.218E-07 & 0.378E-08 & 0.354E-08 & 0.198E-07 & 0.246E-10 & 0.146E-10 & 0.280E-10\ NGC 5904 & 0.350E-12 & 0.244E-12 & 0.853E-12 & 0.499E-12 & 0.393E-12 & 0.170E-11 & 0.189E-12 & 0.976E-13 & 0.190E-12\ & 0.302E-12 & 0.208E-12 & 0.631E-12 & 0.149E-12 & 0.104E-12 & 0.342E-12 & 0.198E-12 & 0.102E-12 & 0.192E-12\ NGC 5927 & 0.535E-11 & 0.467E-11 & 0.473E-10 & 0.142E-11 & 0.118E-11 & 0.106E-10 & 0.483E-11 & 0.220E-11 & 0.324E-11\ & 0.255E-14 & 0.146E-14 & 0.506E-14 & 0.116E-13 & 0.395E-14 & 0.620E-14 & 0.156E-11 & 0.445E-12 & 0.564E-12\ NGC 5986 & 0.222E-10 & 0.121E-10 & 0.187E-10 & 0.233E-10 & 0.151E-10 & 0.204E-10 & 0.523E-12 & 0.144E-12 & 0.172E-12\ & 0.640E-09 & 0.437E-09 & 0.564E-09 & 0.228E-09 & 0.155E-09 & 0.174E-09 & 0.605E-12 & 0.147E-12 & 0.171E-12\ NGC 6093 & 0.194E-11 & 0.180E-11 & 0.735E-11 & 0.248E-11 & 0.232E-11 & 0.900E-11 & 0.128E-12 & 0.743E-13 & 0.872E-13\ & 0.140E-09 & 0.136E-09 & 0.332E-09 & 0.487E-10 & 0.474E-10 & 0.116E-09 & 0.156E-12 & 0.873E-13 & 0.112E-12\ NGC 6121 & 0.286E-09 & 0.144E-09 & 0.167E-09 & 0.676E-09 & 0.310E-09 & 0.786E-09 & 0.176E-10 & 0.416E-11 & 0.651E-11\ & 0.252E-08 & 0.147E-08 & 0.219E-08 & 0.215E-08 & 0.125E-08 & 0.180E-08 & 0.137E-10 & 0.288E-11 & 0.372E-11\ NGC 6144 & 0.126E-08 & 0.678E-09 & 0.206E-08 & 0.304E-08 & 0.161E-08 & 0.575E-08 & 0.325E-09 & 0.986E-10 & 0.862E-10\ & 0.226E-08 & 0.157E-08 & 0.182E-07 & 0.495E-08 & 0.249E-08 & 0.272E-07 & 0.472E-09 & 0.128E-09 & 0.176E-09\ NGC 6171 & 0.206E-10 & 0.135E-10 & 0.468E-10 & 0.229E-10 & 0.140E-10 & 0.840E-10 & 0.170E-10 & 0.450E-11 & 0.486E-11\ & 0.897E-10 & 0.840E-10 & 0.188E-08 & 0.866E-10 & 0.775E-10 & 0.189E-08 & 0.177E-10 & 0.470E-11 & 0.604E-11\ NGC 6205 & 0.325E-14 & 0.241E-14 & 0.300E-13 & 0.388E-13 & 0.342E-13 & 0.577E-12 & 0.611E-13 & 0.331E-13 & 0.931E-13\ & 0.277E-14 & 0.198E-14 & 0.156E-13 & 0.920E-15 & 0.662E-15 & 0.542E-14 & 0.657E-13 & 0.354E-13 & 0.813E-13\ NGC 6218 & 0.260E-11 & 0.219E-11 & 0.183E-10 & 0.140E-11 & 0.115E-11 & 0.141E-10 & 0.136E-11 & 0.549E-12 & 0.870E-12\ & 0.280E-12 & 0.216E-12 & 0.323E-11 & 0.126E-12 & 0.889E-13 & 0.142E-11 & 0.123E-11 & 0.332E-12 & 0.454E-12\ NGC 6254 & 0.821E-12 & 0.737E-12 & 0.714E-11 & 0.315E-12 & 0.258E-12 & 0.544E-11 & 0.777E-12 & 0.424E-12 & 0.679E-12\ & 0.127E-13 & 0.100E-13 & 0.859E-13 & 0.228E-13 & 0.157E-13 & 0.587E-13 & 0.584E-12 & 0.206E-12 & 0.308E-12\ NGC 6266 & 0.594E-12 & 0.470E-12 & 0.341E-11 & 0.245E-12 & 0.158E-12 & 0.268E-12 & 0.781E-13 & 0.290E-13 & 0.418E-13\ & 0.130E-11 & 0.118E-11 & 0.760E-10 & 0.865E-12 & 0.705E-12 & 0.136E-10 & 0.900E-13 & 0.421E-13 & 0.726E-13\ NGC 6273 & 0.214E-10 & 0.154E-10 & 0.553E-10 & 0.152E-10 & 0.863E-11 & 0.339E-10 & 0.208E-11 & 0.656E-12 & 0.765E-12\ & 0.148E-10 & 0.104E-10 & 0.108E-09 & 0.216E-10 & 0.153E-10 & 0.546E-10 & 0.262E-11 & 0.892E-12 & 0.994E-12\ NGC 6284 & 0.898E-10 & 0.871E-10 & 0.100E-08 & 0.144E-09 & 0.136E-09 & 0.212E-08 & 0.557E-10 & 0.451E-10 & 0.162E-09\ & 0.215E-09 & 0.209E-09 & 0.883E-08 & 0.342E-09 & 0.333E-09 & 0.172E-07 & 0.556E-10 & 0.450E-10 & 0.188E-09\ NGC 6287 & 0.188E-10 & 0.981E-11 & 0.141E-10 & 0.264E-10 & 0.136E-10 & 0.181E-10 & 0.493E-12 & 0.132E-12 & 0.195E-12\ & 0.721E-09 & 0.443E-09 & 0.553E-09 & 0.323E-09 & 0.195E-09 & 0.234E-09 & 0.758E-12 & 0.266E-12 & 0.325E-12\ NGC 6293 & 0.487E-09 & 0.242E-09 & 0.240E-09 & 0.334E-09 & 0.146E-09 & 0.319E-09 & 0.295E-10 & 0.106E-10 & 0.107E-10\ & 0.107E-07 & 0.684E-08 & 0.105E-07 & 0.118E-07 & 0.769E-08 & 0.122E-07 & 0.364E-10 & 0.140E-10 & 0.183E-10\ NGC 6304 & 0.130E-10 & 0.100E-10 & 0.428E-10 & 0.100E-10 & 0.779E-11 & 0.203E-09 & 0.349E-11 & 0.173E-11 & 0.220E-11\ & 0.945E-11 & 0.842E-11 & 0.247E-09 & 0.139E-10 & 0.126E-10 & 0.364E-09 & 0.358E-11 & 0.162E-11 & 0.316E-11\ NGC 6316 & 0.872E-11 & 0.687E-11 & 0.178E-10 & 0.396E-11 & 0.268E-11 & 0.826E-11 & 0.461E-12 & 0.174E-12 & 0.281E-12\ & 0.128E-09 & 0.113E-09 & 0.566E-09 & 0.818E-10 & 0.732E-10 & 0.399E-09 & 0.710E-12 & 0.309E-12 & 0.659E-12\ NGC 6333 & 0.243E-11 & 0.192E-11 & 0.785E-11 & 0.321E-11 & 0.273E-11 & 0.895E-11 & 0.170E-12 & 0.577E-13 & 0.931E-13\ & 0.154E-10 & 0.113E-10 & 0.878E-10 & 0.501E-11 & 0.370E-11 & 0.341E-10 & 0.222E-12 & 0.757E-13 & 0.929E-13\ NGC 6341 & 0.668E-11 & 0.449E-11 & 0.108E-10 & 0.627E-11 & 0.482E-11 & 0.135E-10 & 0.472E-12 & 0.114E-12 & 0.144E-12\ & 0.462E-10 & 0.395E-10 & 0.203E-09 & 0.280E-10 & 0.247E-10 & 0.126E-09 & 0.501E-12 & 0.107E-12 & 0.133E-12\ NGC 6342 & 0.586E-09 & 0.347E-09 & 0.675E-09 & 0.620E-09 & 0.324E-09 & 0.406E-09 & 0.109E-09 & 0.325E-10 & 0.344E-10\ & 0.445E-08 & 0.252E-08 & 0.321E-08 & 0.856E-08 & 0.491E-08 & 0.608E-08 & 0.144E-09 & 0.372E-10 & 0.363E-10\ NGC 6356 & 0.179E-10 & 0.157E-10 & 0.747E-10 & 0.244E-10 & 0.225E-10 & 0.153E-09 & 0.223E-11 & 0.154E-11 & 0.232E-11\ & 0.139E-09 & 0.132E-09 & 0.171E-08 & 0.101E-09 & 0.971E-10 & 0.130E-08 & 0.231E-11 & 0.154E-11 & 0.283E-11\ NGC 6362 & 0.417E-10 & 0.281E-10 & 0.145E-09 & 0.135E-10 & 0.104E-10 & 0.110E-09 & 0.212E-10 & 0.611E-11 & 0.782E-11\ & 0.683E-11 & 0.385E-11 & 0.906E-11 & 0.349E-11 & 0.183E-11 & 0.543E-11 & 0.178E-10 & 0.320E-11 & 0.396E-11\ NGC 6388 & 0.122E-11 & 0.883E-12 & 0.186E-11 & 0.166E-12 & 0.120E-12 & 0.512E-12 & 0.125E-13 & 0.366E-14 & 0.507E-14\ & 0.408E-10 & 0.299E-10 & 0.639E-10 & 0.572E-11 & 0.409E-11 & 0.780E-11 & 0.179E-13 & 0.457E-14 & 0.684E-14\ NGC 6397 & 0.131E-10 & 0.108E-10 & 0.699E-10 & 0.438E-11 & 0.345E-11 & 0.206E-10 & 0.115E-10 & 0.479E-11 & 0.714E-11\ & 0.237E-12 & 0.114E-12 & 0.279E-12 & 0.176E-12 & 0.724E-13 & 0.153E-12 & 0.667E-11 & 0.145E-11 & 0.185E-11\ NGC 6441 & 0.379E-11 & 0.270E-11 & 0.586E-11 & 0.584E-12 & 0.401E-12 & 0.115E-11 & 0.331E-13 & 0.113E-13 & 0.196E-13\ & 0.142E-09 & 0.107E-09 & 0.366E-09 & 0.274E-10 & 0.204E-10 & 0.694E-10 & 0.500E-13 & 0.185E-13 & 0.314E-13\ NGC 6522 & 0.211E-09 & 0.142E-09 & 0.213E-09 & 0.285E-09 & 0.200E-09 & 0.328E-09 & 0.154E-10 & 0.569E-11 & 0.842E-11\ & 0.450E-08 & 0.403E-08 & 0.145E-07 & 0.528E-08 & 0.480E-08 & 0.168E-07 & 0.277E-10 & 0.143E-10 & 0.203E-10\ NGC 6528 & 0.296E-11 & 0.213E-11 & 0.492E-11 & 0.355E-12 & 0.249E-12 & 0.144E-11 & 0.377E-13 & 0.130E-13 & 0.161E-13\ & 0.935E-10 & 0.800E-10 & 0.506E-09 & 0.221E-10 & 0.185E-10 & 0.782E-10 & 0.948E-13 & 0.557E-13 & 0.827E-13\ NGC 6553 & 0.103E-12 & 0.954E-13 & 0.210E-11 & 0.894E-13 & 0.792E-13 & 0.475E-12 & 0.643E-14 & 0.345E-14 & 0.853E-14\ & 0.247E-12 & 0.240E-12 & 0.405E-10 & 0.582E-13 & 0.563E-13 & 0.909E-11 & 0.422E-14 & 0.220E-14 & 0.776E-14\ NGC 6584 & 0.163E-10 & 0.139E-10 & 0.356E-10 & 0.252E-10 & 0.211E-10 & 0.545E-10 & 0.944E-12 & 0.554E-12 & 0.865E-12\ & 0.220E-09 & 0.195E-09 & 0.685E-09 & 0.145E-09 & 0.130E-09 & 0.440E-09 & 0.128E-11 & 0.793E-12 & 0.126E-11\ NGC 6626 & 0.640E-11 & 0.422E-11 & 0.152E-10 & 0.999E-12 & 0.594E-12 & 0.160E-11 & 0.290E-12 & 0.896E-13 & 0.139E-12\ & 0.113E-09 & 0.943E-10 & 0.569E-09 & 0.333E-10 & 0.273E-10 & 0.132E-09 & 0.328E-12 & 0.118E-12 & 0.173E-12\ NGC 6656 & 0.151E-12 & 0.133E-12 & 0.328E-11 & 0.972E-13 & 0.746E-13 & 0.344E-12 & 0.244E-12 & 0.906E-13 & 0.241E-12\ & 0.885E-14 & 0.399E-14 & 0.990E-14 & 0.322E-14 & 0.140E-14 & 0.345E-14 & 0.164E-12 & 0.488E-13 & 0.767E-13\ NGC 6712 & 0.140E-10 & 0.950E-11 & 0.199E-10 & 0.124E-10 & 0.904E-11 & 0.208E-10 & 0.423E-12 & 0.130E-12 & 0.191E-12\ & 0.886E-10 & 0.740E-10 & 0.591E-09 & 0.279E-10 & 0.232E-10 & 0.150E-09 & 0.475E-12 & 0.131E-12 & 0.155E-12\ NGC 6723 & 0.121E-10 & 0.823E-11 & 0.470E-10 & 0.238E-10 & 0.186E-10 & 0.853E-10 & 0.466E-11 & 0.126E-11 & 0.134E-11\ & 0.572E-11 & 0.433E-11 & 0.470E-09 & 0.890E-11 & 0.474E-11 & 0.167E-09 & 0.640E-11 & 0.137E-11 & 0.162E-11\ NGC 6752 & 0.241E-11 & 0.229E-11 & 0.776E-10 & 0.394E-12 & 0.311E-12 & 0.361E-11 & 0.415E-11 & 0.207E-11 & 0.581E-11\ & 0.535E-14 & 0.330E-14 & 0.113E-13 & 0.198E-13 & 0.834E-14 & 0.215E-13 & 0.272E-11 & 0.749E-12 & 0.101E-11\ NGC 6779 & 0.367E-10 & 0.244E-10 & 0.536E-10 & 0.722E-10 & 0.473E-10 & 0.868E-10 & 0.229E-11 & 0.809E-12 & 0.958E-12\ & 0.353E-09 & 0.272E-09 & 0.746E-09 & 0.233E-09 & 0.179E-09 & 0.485E-09 & 0.222E-11 & 0.666E-12 & 0.750E-12\ NGC 6809 & 0.245E-10 & 0.187E-10 & 0.638E-10 & 0.309E-10 & 0.232E-10 & 0.551E-10 & 0.228E-11 & 0.612E-12 & 0.969E-12\ & 0.756E-11 & 0.405E-11 & 0.153E-10 & 0.371E-11 & 0.228E-11 & 0.795E-11 & 0.231E-11 & 0.515E-12 & 0.642E-12\ NGC 6838 & 0.493E-13 & 0.487E-13 & 0.150E-11 & 0.653E-14 & 0.539E-14 & 0.367E-12 & 0.380E-13 & 0.182E-13 & 0.135E-12\ & 0.905E-16 & 0.276E-16 & 0.436E-16 & 0.198E-15 & 0.489E-16 & 0.662E-16 & 0.241E-13 & 0.633E-14 & 0.870E-14\ NGC 6934 & 0.415E-12 & 0.405E-12 & 0.161E-10 & 0.277E-12 & 0.264E-12 & 0.524E-11 & 0.948E-13 & 0.812E-13 & 0.436E-12\ & 0.148E-11 & 0.146E-11 & 0.112E-09 & 0.115E-11 & 0.114E-11 & 0.961E-10 & 0.895E-13 & 0.769E-13 & 0.439E-12\ NGC 7006 & 0.203E-12 & 0.201E-12 & 0.244E-10 & 0.641E-13 & 0.631E-13 & 0.271E-11 & 0.429E-13 & 0.418E-13 & 0.740E-12\ & 0.138E-11 & 0.138E-11 & 0.303E-09 & 0.131E-11 & 0.130E-11 & 0.305E-09 & 0.536E-13 & 0.523E-13 & 0.156E-11\ NGC 7078 & 0.175E-13 & 0.159E-13 & 0.734E-12 & 0.956E-14 & 0.832E-14 & 0.168E-12 & 0.246E-12 & 0.183E-12 & 0.689E-12\ & 0.562E-14 & 0.488E-14 & 0.656E-13 & 0.299E-14 & 0.245E-14 & 0.260E-13 & 0.225E-12 & 0.166E-12 & 0.472E-12\ NGC 7089 & 0.243E-13 & 0.235E-13 & 0.130E-11 & 0.324E-13 & 0.308E-13 & 0.636E-12 & 0.204E-13 & 0.168E-13 & 0.772E-13\ & 0.515E-12 & 0.510E-12 & 0.697E-10 & 0.330E-12 & 0.328E-12 & 0.484E-10 & 0.231E-13 & 0.191E-13 & 0.989E-13\ NGC 7099 & 0.322E-11 & 0.272E-11 & 0.133E-10 & 0.409E-11 & 0.315E-11 & 0.246E-10 & 0.296E-11 & 0.132E-11 & 0.144E-11\ & 0.296E-11 & 0.246E-11 & 0.172E-10 & 0.163E-11 & 0.136E-11 & 0.117E-10 & 0.312E-11 & 0.138E-11 & 0.155E-11\ Pal 12 & 0.205E-11 & 0.160E-11 & 0.218E-10 & 0.109E-10 & 0.782E-11 & 0.528E-10 & 0.270E-10 & 0.204E-10 & 0.982E-10\ & 0.214E-11 & 0.167E-11 & 0.423E-10 & 0.101E-10 & 0.785E-11 & 0.668E-10 & 0.281E-10 & 0.210E-10 & 0.115E-09\ Pal 13 & 0.135E-13 & 0.126E-13 & 0.764E-12 & 0.568E-13 & 0.519E-13 & 0.238E-11 & 0.718E-13 & 0.645E-13 & 0.705E-12\ & 0.982E-14 & 0.910E-14 & 0.542E-12 & 0.512E-14 & 0.481E-14 & 0.381E-12 & 0.679E-13 & 0.611E-13 & 0.714E-12\

--- abstract: 'Stratified turbulence shows scale- and direction-dependent anisotropy and the coexistence of weak turbulence of internal gravity waves and strong turbulence of eddies. Straightforward application of standard analyses developed in isotropic turbulence sometimes masks important aspects of the anisotropic turbulence. To capture detailed structures of the energy distribution in the wave-number space, it is indispensable to examine the energy distribution with non-integrated spectra by fixing the codimensional wave-number component or in the two-dimensional domain spanned by both the horizontal and vertical wave numbers. Indices which separate the range of the anisotropic weak-wave turbulence in the wave-number space are proposed based on the decomposed energies. In addition, the dominance of the waves in the range is also verified by the small frequency deviation from the linear dispersion relation. In the wave-dominant range, the linear wave periods given by the linear dispersion relation are smaller than approximately one third of the eddy-turnover time. The linear wave periods reflect the anisotropy of the system, while the isotropic Brunt-Väisälä period is used to evaluate the Ozmidov wave number, which is necessarily isotropic. It is found that the time scales in consideration of the anisotropy of the flow field must be appropriately selected to obtain the critical wave number separating the weak-wave turbulence.' author: - Naoto Yokoyama - Masanori Takaoka title: ' Energy-based analysis and anisotropic spectral distribution of internal gravity waves in strongly stratified turbulence ' --- Introduction ============ Turbulence in nature essentially has anisotropy especially in large scales. Theoretical approaches in turbulence researches originate from Kolmogorov’s local isotropy hypothesis, and have been extended to researches in anisotropic turbulence systems. Numerical simulations of high Reynolds-number turbulent flows and their analyses are developed also in homogeneous statistically-isotropic turbulence systems, and are often incorporated in anisotropic turbulence systems simply. It is essential to introduce appropriate analytical tools which do not diminish scale- and direction-dependent anisotropic properties in the anisotropic turbulence. Stratified turbulence is one of the most fundamental turbulence systems which have statistical anisotropy, and is observed in the oceans and the atmosphere. The gravity produces the density or thermal stratification, and makes statistical differences in energy distribution between the vertical direction and the horizontal direction. The breaking of the internal gravity waves affects the global climate and our lives; the upwelling due to breaking is an important part of the thermohaline circulation in the oceans [@MUNK1966707], and the breaking in the atmosphere sometimes causes clear-air turbulence that may expose aircraft flight to risk . The breaking corresponds the energy transfer from waves to vortices. Various kinds of energy spectra have been reported in observations, experiments, and simulations of stratified turbulence. The variety is derived from the physical mechanisms, the length scales, and other parameters. Such different energy spectra can coexist, and the coexistence is obtained in atmospheric observations and numerical simulations [@nastrom1984kinetic; @FLM:8539071; @FLM:409533]. For example, a kinetic-energy spectrum observed in atmospheric flows has a power law $K_{\perp}(k_{\perp}) \propto k_{\perp}^{-3}$ at large scales [@nastrom1984kinetic]. Here, $k_{\perp}$ is a horizontal wave number, and $K_{\perp}(k_{\perp})$ is the horizontal kinetic-energy spectrum as a function of $k_{\perp}$. Another power law $K_{\perp}(k_{\perp}) \propto k_{\perp}^{-5/3}$ is also observed at mesoscales, and the same power law is obtained analytically and numerically [@FLM:409533]. Observation and theoretical prediction also have a variety of the kinetic-energy spectrum as a function of the vertical wave number $k_{\|}$: the breaking of the internal gravity waves makes the total kinetic-energy spectrum $K(k_{\|}) \propto k_{\|}^{-3}$  for example. The Bolgiano-Obukhov phenomenology predicts coexistence of two power-laws in kinetic spectra: $K(k)\propto k^{-11/5}$ for $k<k_{\mathrm{B}}$ and $K(k)\propto k^{-5/3}$ for $k>k_{\mathrm{B}}$, where $k_{\mathrm{B}}$ is the Bolgiano wave number [@JGR:JGR897; @Obukhov1959]. The pioneering work for the two-dimensional energy spectrum of the internal gravity waves observed in the ocean is the Garrett-Munk spectrum, which has $K(k_{\perp}, k_{\|}) \propto k_{\perp}^{-2} k_{\|}^{-1}$ at relatively large wave numbers [@GM_ARF]. The weak turbulence theory predicts a variety of power laws including the Garrett-Munk spectrum [@iwthLPTN]. A spectral model that allows even variability was proposed [@JGRD:JGRD2467]. In this way, the kinetic-energy spectra as well as the potential-energy spectra are diverse, and the diversity may result from the boundary conditions and the magnitude relation between the horizontal wave number and the vertical wave number. On the other hand, when the stratification is relatively weak, the vortices are dominant in the flow, and the three-dimensional isotropic Kolmogorov turbulence appears. Then, the energy spectrum shows the Kolmogorov’s power law $K(k) \propto k^{-5/3}$. To elucidate the variability of the energy spectra at the small wave numbers and to consistently observe them, the dominant physical mechanism at a wave number is required to be evaluated. In this case, the one-dimensionalized energy spectra such as $K_{\perp}(k_{\perp})$ obtained by integration over $k_{\|}$ cannot properly reflect the energy distribution in the anisotropic turbulence. The wave-number range where one of the physical mechanisms framing the anisotropic turbulence is dominant should be identified in the $k_{\perp}$–$k_{\|}$ space. It is the general practice to focus on time scales to find a dominant mechanism in complex dynamical systems which have multiple physics [@kevorkian2012multiple]. In the three-dimensional isotropic Kolmogorov turbulence, for example, the eddies in the inertial subrange have the eddy-turnover time shorter than the dissipation time, while the dissipation time is shorter than the eddy-turnover time in the dissipation range. The Kolmogorov wave number, which separates the inertial subrange and the dissipation range, is defined so that the eddy-turnover time is equal to the dissipation time. The weak turbulence theory, which has been successfully applied to the statistical description of the nonlinear energy transfers among weakly-coupled dispersive waves, assumes that the linear time scale evaluated by the linear dispersion relation is much smaller than the nonlinear time scale of energy transfers. However, the linear time scale becomes comparable with the nonlinear time scale, and the assumption of the weak nonlinearity is violated either at small or large wave numbers in most of the wave turbulence systems [@Biven200128; @newell01; @Biven200398]. As a result, the weak-wave turbulence and the strong turbulence coexist in many wave turbulence systems such as stratified turbulence considered here, rotating turbulence [@PhysRevFluids.2.092602], magnetohydrodynamic turbulence [@PhysRevX.8.031066], elastic-wave turbulence [@PhysRevE.89.012909] and quantum turbulence [@Vinen2002]. In stratified turbulence, the Brunt-Väisälä period and the eddy-turnover time have respectively been used as the linear and nonlinear time scales. The Ozmidov wave number defined as the wave number at which these two time scales are comparable has been considered as the critical wave number that separates the strongly anisotropic turbulence and the isotropic Kolmogorov turbulence [@ozmidovscale]. In fact, the wave numbers much larger than the Ozmidov wave number, the stratification can be almost negligible, and the isotropic Kolmogorov turbulence appears. The buoyancy wave number, which is defined by the characteristic horizontal velocity and the Brunt-Väisälä frequency, gives the scale of the shear layers and the breaking of the internal gravity waves [@doi:10.1063/1.3599699]. On the other hand, the weak-wave turbulence does not appear at all the wave numbers smaller than the Ozmidov wave number or the buoyancy wave number. The anisotropic quasi-two-dimensional turbulence such as the layer-wise two-dimensional turbulence  and the pancake turbulence [@:/content/aip/journal/pof2/13/6/10.1063/1.1369125] exists at such small wave numbers. Neither the Ozmidov wave number nor the buoyancy wave number can identify the wave-number range where statistically-anisotropic gravity-wave turbulence is dominant because of the isotropy assumed in their derivations. The anisotropy of the time scales can be introduced by using the period given by the linear dispersion relation instead of the Brunt-Väisälä period as the linear time scale [@nazarenko3488critical]. The wave number at which the period given by the linear dispersion relation and the eddy-turnover time are comparable can separate the weak-wave turbulence and the isotropic or anisotropic strong turbulence in magnetohydrodynamic turbulence [@PhysRevX.8.031066; @goldreich1995toward; @PhysRevLett.110.145002; @PhysRevLett.116.105002]. However, it is not clear in rotating turbulence [@:/content/aip/journal/pof2/26/3/10.1063/1.4868280]. In this paper, direct numerical simulations of strongly stratified turbulence are performed, and anisotropic properties of internal gravity-wave turbulence are characterized by distribution and decomposition of energy. The organization of the paper is as follows. The numerical scheme of the direct numerical simulations and decomposition of the wave-number space and the flow field are shown in Sec. \[sec:formulation\], where some definitions of the energies to characterize the anisotropic weak-wave turbulence are provided. The numerical results are exhibited in Sec. \[sec:numericalresults\]. Indices to identify the range of the anisotropic internal gravity-wave turbulence are proposed, and the range is examined in the two-dimensional domain spanned by both of the horizontal and vertical wave numbers. The last section is devoted to the summary. Formulation {#sec:formulation} =========== Numerical scheme {#ssec:scheme} ---------------- Incompressible flows in stably stratified background flow in the $z$ direction is considered. Under the Boussinesq approximation, the governing equation for the velocity $\bm{u}$ and buoyancy $b$ is given as follows: $$\begin{aligned} & \frac{\partial}{\partial t} \bm{u} + (\bm{u} \cdot \nabla) \bm{u} = -\nabla p + b \bm{e}_z + \nu \nabla^2 \bm{u} + \bm{f} , \\ & \nabla \cdot \bm{u} = 0 , \\ & \frac{\partial}{\partial t} b + (\bm{u} \cdot \nabla) b = - N^2 \bm{u} \cdot \bm{e}_z + \kappa \nabla^2 b . \end{aligned}$$ \[eq:gov\] The buoyancy $b$ is given as $b = -g \theta^{\prime} / \theta_0$ in atmospheric flows, for example, where $g$, $\theta^{\prime}$, and $\theta_0$ are respectively the gravity acceleration, the temperature fluctuation, and the mean temperature. The Brunt-Väisälä frequency $N$ is assumed to be constant. The external force $\bm{f}$ is added to obtain the non-equilibrium statistically-steady state. The kinematic viscosity and the diffusion constant are respectively denoted by $\nu$ and $\kappa$. In this work, direct numerical simulations of Eq. (\[eq:gov\]) are performed in a periodic box with the side $2\pi$. The Fourier coefficients of the dependent variables appearing in Eq. , $\widetilde{\bm{u}}_{\bm{k}}$, $\widetilde{p}_{\bm{k}}$, and $\widetilde{b}_{\bm{k}}$, are used, and the tildes are omitted below. The pseudo-spectral method with aliasing removal due to the phase shift is employed to evaluate the nonlinear term. The Runge-Kutta-Gill method is used for the time integration. The external force is added in the wave-number space to the wave-number mode in $k_{\mathrm{f}}-1/2 \leq |\bm{k}| < k_{\mathrm{f}}+1/2$, where the forced wave number $k_{\mathrm{f}}$ is set to $4$. The external force is generated by the Ornstein-Uhlenbeck process [@FLM:8539071] as follows. The colored noise $\hat{\bm{f}}_{\bm{k}} = (\hat{f}_{x \bm{k}}, \hat{f}_{y \bm{k}}, 0)$, which consists of two spatial components each having a complex value, is obtained for each wave number according to the following stochastic differential equation: $$\begin{aligned} \begin{pmatrix} d \hat{\bm{f}}_{\bm{k}} \\ d \hat{\bm{g}}_{\bm{k}} \end{pmatrix} = dt \begin{pmatrix} -\alpha & 1 \\ 0 & -\alpha \end{pmatrix} \begin{pmatrix} \hat{\bm{f}}_{\bm{k}} \\ \hat{\bm{g}}_{\bm{k}} \end{pmatrix} + \begin{pmatrix} \bm{0} \\ \gamma_{\bm{k}} d\bm{W}_{\bm{k}} \end{pmatrix} , \end{aligned}$$where $d\bm{W}_{\bm{k}}$ represents the normal random variables with mean $0$ and variance $dt$ and has four independent components. The correlation time of $\hat{\bm{f}}_{\bm{k}}$ is $O(1/\alpha)$, and $\alpha$ is set to be $N$ in this paper. Because $\langle |\hat{\bm{f}}_{\bm{k}}|^2 \rangle = \gamma_{\bm{k}}^2/\alpha$, $\gamma_{\bm{k}}$ is used to control the amplitude of the external force. Finally, the Fourier coefficient of the external force is set as $\bm{f}_{\bm{k}} = \hat{\bm{f}}_{\bm{k}} - \bm{k} (\bm{k} \cdot\hat{\bm{f}}_{\bm{k}}) / k^2$ to satisfy the divergence-free condition. The number of the grid points used is up to $2048^3$. The low-resolution simulations with $1024^3$ grid points are also used to examine the parameter dependence. The corresponding largest wave number $k_{\mathrm{max}}$ is approximately $970$ or $480$. The Brunt-Väisälä frequency is set to $N=10$. The Prandtl number is set to be unity, i.e., $\nu=\kappa$, and $\nu$ is chosen so that $k_{\mathrm{max}} / k_{\eta} \approx 1.2$. Here, $k_{\eta} = (\overline{\varepsilon}/\nu^3)^{1/4}$ is the Kolmogorov wave number, and $\overline{\varepsilon}$ denotes the mean dissipation rate of the kinetic energy. The coefficient $\gamma_{\bm{k}}$ to control the amplitude of the external force is varied in the simulations with $1024^3$ grids. The parameters in the numerical simulations and their definitions which follow those in Ref. [@maffioli_davidson_2016] are summarized in Table \[tab:parameters\]. ----------------------- ------------------- --------------------------------------------- -------------------------------------- -------------------------------------------- ----------------------------------------- ------------------------------------- ---------------------------- number of grid points $\gamma_{\bm{k}}$ $Re$ $Re_{\mathrm{b}}$ $Fr_{\perp}$ $Fr_{\|}$ $k_{\mathrm{O}}$ $k_{\mathrm{b}}$ $u_{\perp \mathrm{rms}} \ell_{\perp} / \nu$ $\overline{\varepsilon} / (\nu N^2)$ $u_{\perp \mathrm{rms}} /(N \ell_{\perp})$ $u_{\perp \mathrm{rms}} /(N \ell_{\|})$ $\sqrt{N^3/\overline{\varepsilon}}$ $N/u_{\perp \mathrm{rms}}$ $2048^3$ $0.5$ $7.7\times 10^4$ $2.1$ $9.0\times 10^{-3}$ $0.80$ $490$ $27$ $1024^3$ $0.1$ $1.8 \times 10^4$ $9.4\times 10^{-2}$ $4.9 \times 10^{-3}$ $0.15$ $2300$ $76$ $1024^3$ $0.2$ $1.1 \times 10^4$ $0.24$ $9.4 \times 10^{-3}$ $0.17$ $1200$ $57$ $1024^3$ $0.5$ $2.5\times 10^4$ $0.80$ $9.6\times 10^{-3}$ $0.46$ $500$ $29$ $1024^3$ $1$ $4.1 \times 10^4$ $1.9$ $1.4 \times 10^{-2}$ $1.6$ $260$ $15$ $1024^3$ $2$ $3.7 \times 10^4$ $4.7$ $1.9 \times 10^{-2}$ $2.0$ $130$ $11$ $1024^3$ $5$ $2.2 \times 10^4$ $18$ $3.5 \times 10^{-2}$ $1.0$ $53$ $8.2$ ----------------------- ------------------- --------------------------------------------- -------------------------------------- -------------------------------------------- ----------------------------------------- ------------------------------------- ---------------------------- : Parameters in the numerical simulations. $Re$: horizontal Reynolds number, $Re_{\mathrm{b}}$: buoyancy Reynolds number, $Fr_{\perp}$: horizontal Froude number, $Fr_{\|}$: vertical Froude number, $k_{\mathrm{O}}$: Ozmidov wave number, $k_{\mathrm{b}}$: buoyancy wave number. The root-mean square of the horizontal velocity is denoted by $u_{\perp \mathrm{rms}}$. The horizontal and vertical integral length scales, $\ell_{\perp}$ and $\ell_{\|}$, are defined by transverse velocity correlations. []{data-label="tab:parameters"} The initial condition of a simulation is a statistically steady state of the lower-resolution simulation. Therefore, the small wave-number modes are numerically integrated over a long time as $N t = O(10^3)$. Because all the simulations relax to statistically steady states after some times depending on the amplitudes of the external force, the growth without stationarity reported in Ref. [@smith_waleffe_2002] was not observed in the simulations. The time averaging is performed to draw the spectra for $N t = 100$ with every $12.5$ in the high-resolution simulation. It might be short to remove the fluctuation at the small wave numbers, but the results shown in this paper are confirmed to be unchanged in the low-resolution simulations, where long-time averaging is performed. Ratios of time scales to find the dominant physical mechanism {#ssec:timeratio} ------------------------------------------------------------- The Ozmidov wave number $k_{\mathrm{O}}$ has been considered as a wave number which separates the strongly anisotropic range and the isotropic range in the wave-number space. The Ozmidov wave number is given as a wave number at which the Brunt-Väisälä period $1/N$ and the eddy-turnover time of the three-dimensional (3D) isotropic turbulence $\tau_{\bm{k}} = 1/(ku) = (k^2 \overline{\varepsilon})^{-1/3}$ are comparable, i.e., $k_{\mathrm{O}} = \sqrt{N^3/\overline{\varepsilon}}$. It should be noted that the Ozmidov wave number is independent of the direction of the wave number vector, i.e., isotropic. The 3D isotropic Kolmogorov turbulence is expected to dominate at the wave numbers larger than $k_{\mathrm{O}}$, but $k_{\mathrm{O}}$ does not necessarily determine the wave-number range where the weak gravity-wave turbulence is dominant because of the lack of the anisotropy. The buoyancy wave number $k_{\mathrm{b}} = N/u_{\perp \mathrm{rms}}$ is another wave number that characterizes the transition from the quasi-two-dimensional turbulence to the 3D isotropic turbulence. The buoyancy wave number is also isotropic. Owing to the anisotropy, the spectral structures in the wave-number space should be investigated in the $k_{\perp}$–$k_{\|}$ space. The theory of the critical balance states that the energy is transferred in the transitional wave-number range between the wave-dominant and vortex-dominant ranges [@nazarenko3488critical]. In this theory, the wave period of the gravity wave given by the linear dispersion relation is employed as the linear time scale instead of the Brunt-Väisälä period. Note that the linear dispersion relation is anisotropic. Because $k_{\perp} \ll k_{\|}$ and hence $|\bm{u}_{\perp}| \gg |u_{\|}|$ owing to the divergence-free condition were assumed in Refs. [@nazarenko3488critical; @nazarenkobook], the linear dispersion relation was rewritten as $\sigma_{\mathrm{2D}\bm{k}} = Nk_{\perp}/k_{\|}$, and the eddy-turnover time of the two-dimensional (2D) turbulence $\tau_{\mathrm{2D}\bm{k}} = 1/(k_{\perp}u_{\perp}) = (k_{\perp}^2 \overline{\varepsilon})^{-1/3}$ was used as the nonlinear time. In the present work, since the strong turbulence is not only 2D but also 3D and $k_{\perp} \ll k_{\|}$ does not necessarily hold, the general linear dispersion relation $\sigma_{\bm{k}} = Nk_{\perp}/k$ is used to evaluate the linear time. Moreover, the eddy-turnover time of the 3D turbulence $\tau_{\bm{k}} = (k^2 \overline{\varepsilon})^{-1/3}$ is used as the nonlinear time. Then, the nonlinearity is evaluated by $\chi_{\bm{k}} = 1/(\sigma_{\bm{k}} \tau_{\bm{k}})$. The ratio of the gravity-wave period to the 2D eddy-turnover time $\chi_{\mathrm{2D}\bm{k}} = 1/(\sigma_{\mathrm{2D}\bm{k}} \tau_{\mathrm{2D}\bm{k}})$ is also introduced for reference. Decomposition of turbulent flow {#ssec:decomposition} ------------------------------- To examine the idea of the critical balance, it is indispensable to identify the wave-dominant range. The Craya-Herring (Cartesian) decomposition and the helical-mode decomposition are used for the identification in this paper. In the Craya-Herring decomposition [@doi:10.1063/1.1694822; @FLM:8539071], an orthonormal basis, $\bm{e}_1 = \bm{k} \times \bm{e}_z/k_{\perp}$, $\bm{e}_2 = \bm{k} \times \bm{e}_1 /k$, and $\bm{e}_3 = \bm{k}/k$, is introduced. The two basis vectors $\bm{e}_1$ and $\bm{e}_2$ are defined only when $\bm{k}$ and $\bm{e}_z$ are not in parallel, that is, horizontal component of $\bm{k}$, $\bm{k}_{\perp}$, is non-zero. The orthogonal basis decomposes the velocity as $$\begin{aligned} \bm{u}_{\bm{k}} = \begin{cases} u_{\mathrm{v}} \bm{e}_1 + u_{\mathrm{w}} \bm{e}_2 & \text{for } k_{\perp} \neq 0 \\ \bm{u}_{\mathrm{s}} & \text{for } k_{\perp} = 0 \end{cases} . \end{aligned}$$ The Fourier component of the velocity is given by two components perpendicular to the wave-number vector $\bm{k}$ because of the incompressibility $\bm{k} \cdot \bm{u}_{\bm{k}} = 0$. When the wave numbers with $k_{\perp} = 0$ are included, such decomposition is called the Cartesian decomposition. When the viscosity and the diffusion are neglected for $k_{\perp} \neq 0$, the governing equation (\[eq:gov\]) can be linearized as $$\begin{aligned} \frac{\partial u_{\mathrm{v} \bm{k}}}{\partial t} = 0, \; \frac{\partial u_{\mathrm{w} \bm{k}}}{\partial t} = - \frac{k_{\perp}}{k} b_{\bm{k}}, \; \frac{\partial b_{\bm{k}}}{\partial t} = N^2 \frac{k_{\perp}}{k} u_{\mathrm{w} \bm{k}} . \label{eq:linearinviscidnondiffusive}% \end{aligned}$$This linear inviscid non-diffusive equation indicates that $u_{\mathrm{v}} = i \omega_z / k_{\perp}$ is a vortical mode that is not affected by the linear buoyancy term, and $u_{\mathrm{w}} = - k u_z / k_{\perp}$ is a wave mode. Here, $\omega_z$ denotes the $z$ component of the vorticity. The second equation and the third one in Eq. (\[eq:linearinviscidnondiffusive\]) give the linear dispersion relation of the gravity waves: $\sigma_{\bm{k}} = N k_{\perp} / k$. The velocity for $k_{\perp} = 0$, $\bm{u}_{\mathrm{s} k_z} = \bm{u}_{\perp}$, represents a vertically-sheared horizontal flow. Namely, the Cartesian decomposition simply represents the decomposition of the velocity into the vortices, the waves and the shear flows at the lowest order. The Cartesian decomposition is equivalent to the normal-mode decomposition [@waite_bartello_2006]. The helical-mode decomposition has also been used for the decomposition of the velocity. In the helical-mode decomposition, the basis $\bm{h}_{\pm} = (\bm{e}_2 \mp i \bm{e}_1)/\sqrt{2}$ is the eigen vector for the curl operation, $i \bm{k} \times \bm{h}_{\pm} = \pm k \bm{h}_{\pm}$. Then, the velocity is decomposed as $\bm{u} = \xi_+ \bm{h}_+ + \xi_- \bm{h}_-$, where $\xi_{\pm} = \bm{u} \cdot \bm{h}_{\mp}$ is the helical-mode intensity. Note that $\bm{h}_{\pm} \cdot \bm{h}_{\pm} = 0$ and $\bm{h}_{\pm} \cdot \bm{h}_{\mp} = 1$. A wave-number mode $\bm{k}$ has a total energy $E_{\bm{k}}$, which is sum of the kinetic energy $K_{\bm{k}} = \langle|\bm{u}_{\bm{k}}|^2\rangle/2$ and potential energy $V_{\bm{k}} = \langle|b_{\bm{k}}|^2\rangle/(2N^2)$. The kinetic energy can be given by horizontal kinetic energy $K_{\perp \bm{k}} = K_{x \bm{k}} + K_{y \bm{k}} =(\langle|u_{x \bm{k}}|^2\rangle + \langle|u_{y \bm{k}}|^2\rangle)/2$ and vertical kinetic energy $K_{\| \bm{k}} = K_{z \bm{k}} = \langle|u_{z \bm{k}}|^2\rangle/2$, focused on the direction of the velocity. Similarly, the Cartesian decomposition defines vortical kinetic energy $K_{\mathrm{v} \bm{k}} = \langle|u_{\mathrm{v} \bm{k}}|^2\rangle/2 = \langle|\omega_{z \bm{k}}|^2\rangle/(2 k_{\perp}^2)$, wave kinetic energy $K_{\mathrm{w} \bm{k}} = \langle|u_{\mathrm{w} \bm{k}}|^2\rangle/2 = k^2 \langle|u_{z \bm{k}}|^2\rangle/(2 k_{\perp}^2)$, and shear kinetic energy $K_{\mathrm{s} k_z} = \langle|\bm{u}_{\mathrm{s} k_z}|^2\rangle/2 = \langle|\bm{u}_{\perp k_z}|^2\rangle/2$. Because the shear flow is defined only for $k_{\perp} = 0$, it depends only on $k_z$. Moreover, according to the helical-mode decomposition, the kinetic energy in the $m$ direction, where $m=x,y,z$, can be written as $$\begin{aligned} K_{m \bm{k}} &= \frac{K(k)}{8\pi k^2} \left(1- \frac{k_{m}^2}{k^2}\right) + \frac{1}{2} \left(K_{\bm{k}} - \frac{K(k)}{4\pi k^2}\right) \left(1- \frac{k_{m}^2}{k^2}\right) + \mathrm{Re} [Z_{\bm{k}} h_{+ m \bm{k}}^2] . \label{eq:Kalphak} \end{aligned}$$ Here, $K(k)$ is the one-dimensionalized energy spectrum, and $Z_{\bm{k}} = \langle\xi_{+ \bm{k}} \xi_{- \bm{k}}^{\ast} \rangle = K_{\mathrm{w} \bm{k}} - K_{\mathrm{v} \bm{k}} + i \mathrm{Re} \langle u_{\mathrm{v} \bm{k}}^{\ast} u_{\mathrm{w} \bm{k}}\rangle$. The terms in the right-hand side of Eq. (\[eq:Kalphak\]) represent isotropic part, directional anisotropic part with respect to the direction of $\bm{k}$, and polarization anisotropic part with respect to the direction of $\bm{u}$ of the kinetic energy [@9780511546099]. In this work, the vertical kinetic energy $$\begin{aligned} K_{z\bm{k}} = K_{\| \bm{k}} = \frac{K(k)}{8\pi k^2} \left(\frac{k_{\perp}}{k^2}\right)^2 + \frac{1}{2} \left(K_{\bm{k}} - \frac{K(k)}{4\pi k^2}\right) \left(\frac{k_{\perp}}{k^2}\right)^2 + \frac{1}{2} (K_{\mathrm{w}\bm{k}}-K_{\mathrm{v}\bm{k}}) \left(\frac{k_{\perp}}{k^2}\right)^2 , \label{eq:Kzk} \end{aligned}$$ and its polarization anisotropic part, $K_{z \mathrm{PA}\bm{k}}$, which is the last term in the right-hand side of Eq. , are used to quantify the anisotropy of a wave-number mode. Numerical results {#sec:numericalresults} ================= Energy spectra {#ssec:energyspectra} -------------- ![ Integrated energy spectra: total kinetic energy, horizontal kinetic energy, vertical kinetic energy, and potential energy. (a) as functions of horizontal wave numbers integrated over the vertical wave numbers, and (b) as functions of vertical wave numbers integrated over the horizontal wave numbers. The green, blue and red vertical dashed lines respectively show the forced wave number $k_{\mathrm{f}}$, the buoyancy wave number $k_{\mathrm{b}}$, and the Ozmidov wave number $k_{\mathrm{O}}$. []{data-label="fig:spobs"}](spobs_5_1_psfrag.eps) Spectra of total kinetic energy $K$, horizontal kinetic energy $K_{\perp}$, vertical kinetic energy $K_{\|}$, and potential energy $V$ obtained in the numerical simulations with $2048^3$ grid points are shown in Fig. \[fig:spobs\]. The one-dimensional total kinetic-energy spectrum as a function of the horizontal wave numbers, for example, is defined as $$\begin{aligned} K(k_{\perp}) = \frac{1}{\Delta k_{\perp}} {\sum_{\bm{k}_{\perp}^{\prime}}}^{\prime} \sum_{k_{\|}^{\prime}} \frac{1}{2} \langle |\bm{u}_{\bm{k}_{\perp}^{\prime}, k_{\|}^{\prime}}|^2\rangle ,\end{aligned}$$where the summation $\sum_{\bm{k}_{\perp}^{\prime}}^{\prime}$ is taken over $||\bm{k}_{\perp}^{\prime}| - k_{\perp}| < \Delta k_{\perp}/2$, and $\Delta k_{\perp}$ is the bin width to obtain the spectrum. The summation $\sum_{k_{\|}^{\prime}}$ is taken over all the vertical wave number. These one-dimensional spectra are referred to as integrated spectra in this paper. Figure \[fig:spobs\](a) shows the energy spectra as functions of the horizontal wave numbers integrated over the vertical wave numbers, while Fig. \[fig:spobs\](b) shows those as functions of the vertical wave numbers integrated over the horizontal wave numbers. Note that although the forced wave number $k_{\mathrm{f}}$ is marked for reference in the figures, the forced wave numbers exist in the range $k_{\perp} < k_{\mathrm{f}}$ and $k_{\|} < k_{\mathrm{f}}$ because $|\bm{k}|=(k_{\perp}^2 + k_{\|}^2)^{1/2}$. The buoyancy wave number $k_{\mathrm{b}}$ and the Ozmidov wave number $k_{\mathrm{O}}$ have the same property. The integrated spectra show that the kinetic energy comes mostly from the horizontal component, and the potential energy spectra lies between the horizontal and vertical kinetic-energy spectra for all the wave numbers except for the horizontal wave-number spectra at the very large horizontal wave numbers. The horizontal wave-number spectra have a relatively steep spectrum close to $k_{\perp}^{-2}$ at the small horizontal wave numbers, and a less steep spectrum that is approximately $k_{\perp}^{-5/3}$ at the large horizontal wave numbers. The transition is observed approximately at the buoyancy wave number as reported in Ref. [@doi:10.1063/1.3599699]. However, the energy spectrum at the large $k_{\perp}$ in Fig. \[fig:spobs\](a) is much steeper than that reported in Ref. [@doi:10.1063/1.3599699], where the Kelvin-Helmholtz billows are supposed to generate the bump at the large horizontal wave numbers. It is worth pointing out that the computational box is flatter and that the hyper viscosity and the hyper diffusion are used in the simulation in Ref. [@doi:10.1063/1.3599699]. Because of the flat computational box, the bump consists of the large vertical wave-number modes. The less steep energy spectra appear near the dissipation range in the inertial subrange, and they are due to the so-called bottleneck effect. The hyper viscosity and the hyper diffusion are known to enhance the bottleneck effect. One may observe that this horizontal wave-number spectrum is proportional to $k_{\perp}^{-5/3}$ in all the inertial subrange without any transition, but there actually exists a transition as seen below. Similar transition was observed in Refs. [@brethouwer_billant_lindborg_chomaz_2007; @FLM:8539071]. Note that the range of the 3D Kolmogorov turbulence is too small to observe in the spectrum because the buoyancy Reynolds number is evaluated approximately as $2.1$. The vertical wave-number spectra are also non-uniform, and the power laws at the small wave numbers and the large wave numbers are respectively close to those in Ref. [@GM_ARF] and Ref. . Similarly to the horizontal wave-number spectra, the gradual transition is observed roughly at the buoyancy wave number. The steep spectra similar to $k_{\|}^{-3}$ in the range $k_{\mathrm{b}} < k_{\|} < k_{\mathrm{O}}$ are due to balance between the inertia and the buoyancy [@:/content/aip/journal/pof2/13/6/10.1063/1.1369125; @PhysRevFluids.2.104802]. It is evident in these integrated spectra that the energy distribution is not scale-invariant and the energy spectra in the 2D domain spanned by the horizontal and vertical wave numbers show the anisotropy. It must be emphasized that these power laws of the integrated spectra consisting of the various slopes do not necessarily reflect the spectral structures unaffected by the boundary conditions. In this paper, the anisotropic energy distribution will be directly investigated below. ![ (a) Horizontal wave-number spectra and (b) vertical wave-number spectra of vortical kinetic energy, wave kinetic energy and shear energy. The abscissa is scaled linearly for $k_{\perp}, k_{\|} \leq 1$ and logarithmically for $k_{\perp}, k_{\|} \geq 1$. See also the caption of Fig. \[fig:spobs\] for the vertical lines. []{data-label="fig:spkhkv"}](spkhkv_5_1_psfrag.eps) The coexistence of the different power-law exponents in the energy spectra, where the transition is observed approximately at the buoyancy wave number, is also observed in the horizontal wave-number spectra of the vortical kinetic energy and the wave kinetic energy (Fig. \[fig:spkhkv\](a)). While the vortical energy spectrum and the wave kinetic energy spectrum are respectively close to $k_{\perp}^{-3}$ and $k_{\perp}^{-2}$ at the small horizontal wave numbers, both energy spectra approximately have $k_{\perp}^{-5/3}$ at the large horizontal wave numbers. The vertical wave-number spectra in Fig. \[fig:spkhkv\](b) also exhibit the coexistence; the rather flat spectra appears at the small vertical wave numbers, and the steep spectra similar to the saturation spectrum $k_{\|}^{-3}$ does at the large vertical wave numbers. These energy spectra are similar to the ones in Ref. [@FLM:8539071]. The shear energy is defined only for $k_{\perp} = 0$, but it is large. In fact, the kinetic energies of the vortical, wave, and shear flows integrated over all the wave numbers are roughly $4 \times 10^{-2}$, $4 \times 10^{-2}$ and $8 \times 10^{-2}$, respectively. The largest energy appears at $k_{\perp}=0$ and $k_{\|}=4$, which can be directly excited by the external force, as the shear energy. Note that although the external force excites both waves and vortices as well as the shear flows at a wave-number mode, and their amplitudes depend on the wave-number mode as recognized from the energies at the forced wave numbers in Fig. \[fig:spkhkv\]. ![ Kinetic energy spectra (a) for each $k_{\|}$ as function of $k_{\perp}$ and (b) for each $k_{\perp}$ as function of $k_{\|}$. See also the caption of Fig. \[fig:spobs\] for the vertical lines. []{data-label="fig:spslice"}](spslice_5_1_psfrag.eps) The non-uniformity of the horizontal wave-number spectra of the energies shown in Fig. \[fig:spobs\](a) indicates the existence of the inner structure in the vertical wave-number spectra drawn in Fig. \[fig:spobs\](b) and vice versa. The same applies to the vortical kinetic energy and the wave kinetic energy in Fig. \[fig:spkhkv\]. The horizontal wave-number spectra of the energies shown in Fig. \[fig:spobs\](a) are obtained by integration over the vertical wave numbers, and the energy spectra without the integration are required to observe the inner structure. Such non-integrated kinetic-energy spectrum for each $k_{\|}$ as a function of $k_{\perp}$ is defined as $$\begin{aligned} K_{k_{\|}}(k_{\perp}) = \frac{1}{\Delta k_{\perp}} {\sum_{\bm{k}_{\perp}^{\prime}}}^{\prime} \frac{1}{\Delta k_{\|}} {\sum_{k_{\|}^{\prime}}}^{\prime} \frac{1}{2} \langle |\bm{u}_{\bm{k}_{\perp}^{\prime}, k_{\|}^{\prime}}|^2\rangle .\end{aligned}$$ The non-integrated kinetic-energy spectrum for each $k_{\perp}$ as a function of $k_{\|}$ is similarly defined. The non-integrated kinetic-energy spectra are drawn in Fig. \[fig:spslice\]. The kinetic-energy spectra as functions of $k_{\perp}$ for $k_{\|} \leq 32$ shown in Fig. \[fig:spslice\](a) are not so different from each other, since the vertical-energy spectra are the rather flat spectra as $k_{\|}^{-1}$ as shown in Fig. \[fig:spobs\](b). Nevertheless, we can observe that the energy spectra at small horizontal wave numbers become less steep roughly from $k_{\perp}^{-3}$ to $k_{\perp}^{-2}$. As $k_{\|}$ increases further, the maximal wave number moves to larger $k_{\perp}$. Most of the kinetic energy at small $k_{\|}$ exists in $k_{\perp} \leq 2$ as shown in Fig. \[fig:spslice\](b). The integrated energy spectra as functions of the vertical wave numbers shown in Fig. \[fig:spobs\](b) consist of the corresponding non-integrated energy spectra in $k_{\perp} \leq 2$. It is consistent with the fact that the horizontal wave-number spectra uniformly and rapidly decrease as shown in Fig. \[fig:spobs\](a). Moreover, in the range $30 \lessapprox k_{\|} \lessapprox 500$, the relatively flat spectrum close to $k_{\|}^{-1/2}$ extends to the large $k_{\|}$ as $k_{\perp}$ increases. Then, the large $k_{\perp}$ has larger energy at the large $k_{\|}$ than the small $k_{\perp}$ has [@PhysRevFluids.2.104802]. Thus, the integration over $k_{\perp}$ makes the saturation spectrum complex in the large $k_{\|}$ range. The saturation spectrum is considered to consist of the breaking of the internal gravity waves [@doi:10.1029/JD091iD02p02742]. Since the integrated spectra of the kinetic energy shown in Figs. \[fig:spobs\](a) and \[fig:spobs\](b) are respectively obtained by summation of the non-integrated spectra shown in Figs. \[fig:spslice\](a) and \[fig:spslice\](b), the integrated spectra are determined mostly by the non-integrated spectra in the few small codimensional wave numbers. In this sense, the integrated spectra cannot properly reflect the energy distribution at the moderate wave numbers unaffected by the boundary conditions. Moreover, the identification of the dominant physical mechanism by the integrated spectra requires a careful inspection. To observe the anisotropic structures of the energy spectra, the 2D spectra for total, vortical, wave kinetic, and potential energies in the horizontal and vertical wave-number domain are drawn in Fig. \[fig:spKvKwmap\], which provides an overview of the energy spectra. The 2D spectrum is defined as $$\begin{aligned} K(k_{\perp}, k_{\|}) = \frac{1}{2\pi k_{\perp}} \frac{1}{\Delta k_{\perp}} {\sum_{\bm{k}_{\perp}^{\prime}}}^{\prime} \frac{1}{\Delta k_{\|}} {\sum_{k_{\|}^{\prime}}}^{\prime} \frac{1}{2} \langle |\bm{u}_{\bm{k}_{\perp}^{\prime}, k_{\|}^{\prime}}|^2\rangle ,\end{aligned}$$ where the normalizing constant, $1/ (2\pi k_{\perp})$, is introduced for the contours of the energy spectra to be compared easily with the completely isotropic ones. ![ 2D spectra of (a) total kinetic energy, (b) vortical energy, (c) wave kinetic energy, and (d) potential energy. The contours are drawn for $10^{-12}, 10^{-10}, \cdots, 10^{-4}$. The critical wave number at which $\chi_{\bm{k}}=1/3$ and that at which $\chi_{\bm{k}}=1$ are represented by the thick and thin green curves, respectively. The 2D critical wave number at which $\chi_{\mathrm{2D}\bm{k}}=1/3$ and that at which $\chi_{\mathrm{2D}\bm{k}}=1$ are represented by the thick and thin yellow curves, respectively. The buoyancy wave number and Ozmidov wave number are respectively represented by the blue and magenta curves. []{data-label="fig:spKvKwmap"}](spE_map_psfrag.eps) All the energies shown in Fig. \[fig:spKvKwmap\] accumulate at small $k_{\perp}$. It is consistent with the large energies at small $k_{\perp}$ in the integrated and non-integrated spectra shown in Figs. \[fig:spobs\]–\[fig:spslice\]. The energies drawn as the 2D spectra obviously show the anisotropy in small $k_{\perp}$ and $k_{\|}$. As $k = \sqrt{k_{\perp}^2+k_{\|}^2}$ becomes large, the contours of each energy are more similar to the isotropic curves which show the buoyancy wave number and the Ozmidov wave number. Such fact indicates that the anisotropy that exists at the small $k$ gradually decreases and the flow at these scales is closer to the 3D isotropic Kolmogorov turbulence, as $k$ become large. Note that even at the Ozmidov wave number the energy in $k_{\perp} < k_{\|}$ is larger than that in $k_{\perp} > k_{\|}$, and the energy spectra are still weakly anisotropic. It is not clear in Fig. \[fig:spKvKwmap\] where the wave kinetic energy and the potential energy are larger than the vortical energy. Furthermore, the four 2D energy spectra may appear close enough. However, by careful observation, we can find that the spectra of the wave kinetic energy (Fig. \[fig:spKvKwmap\](c)) and the potential energy (Fig. \[fig:spKvKwmap\](d)) are similar, but the vortical-energy spectrum (Fig. \[fig:spKvKwmap\](b)) is different from these. Distribution of turbulence indices in wave-number space {#ssec:cbrange} ------------------------------------------------------- It is indispensable to separate the wave-number space based on the dominant physical mechanisms of turbulence. In particular, the theory of the critical balance needs the separation of the wave-dominant range. To quantitatively discuss whether the balance between linear and nonlinear time scales can identify the wave-dominant range, the energy decomposition written in Sec. \[ssec:decomposition\] is employed for the definition. ![ (a) ratio of the wave kinetic energy to the total kinetic energy $K_{\mathrm{w}\bm{k}}/K_{\bm{k}}$, (b) relative difference between the wave kinetic energy and the potential energy $(K_{\mathrm{w}\bm{k}} - V_{\bm{k}})/(K_{\mathrm{w}\bm{k}} + V_{\bm{k}})$, and (c) ratio of the polarization anisotropic part to the kinetic energy $K_{z \mathrm{PA}\bm{k}}/K_{\bm{k}}$. The contours are drawn for every $0.2$ in (a) and (c), and for every $0.1$ in (b). The critical wave number at which $\chi_{\bm{k}}=1/3$, the 2D critical wave number at which $\chi_{\mathrm{2D}\bm{k}}=1/3$, the buoyancy wave number, and the Ozmidov wave number are represented by the green, yellow, blue, and magenta curves, respectively. []{data-label="fig:Kpa_Kz"}](sppa_map_psfrag.eps) The difference of the vortical energy from the wave kinetic energy and the potential energy, and the similarity of the wave kinetic energy and the potential energy can be used to characterize the wave turbulence and the strong turbulence. In the wave-dominant range, the wave kinetic energy is postulated to be much larger than the vortical energy. The weak nonlinearity assumes that the wave kinetic energy is also expected to be close to the potential energy in the same range. Since the energies are not uniform in the wave-number space, a normalization of the energy is required to characterize each range; the ratios of the energies are drawn in Fig. \[fig:Kpa\_Kz\] to quantify the dominance of the weak-wave turbulence. For example, the ratio of the wave kinetic energy to the total kinetic energy is used instead of direct comparison between the wave kinetic energy and the vortical energy. The ratio of the wave kinetic energy to the total kinetic energy $$\begin{aligned} \frac{K_{\mathrm{w}\bm{k}}}{K_{\bm{k}}} = \frac{K_{\mathrm{w}\bm{k}}}{K_{\mathrm{v}\bm{k}}+K_{\mathrm{w}\bm{k}}+K_{\mathrm{s}\bm{k}}} \label{eq:wave2total} \end{aligned}$$ is drawn in Fig. \[fig:Kpa\_Kz\](a). Note that the shear kinetic energy is defined only on $k_{\perp}=0$, and it does not appear in Fig. \[fig:Kpa\_Kz\](a). The weak turbulence theory requires that the linear time scale is much shorter than the nonlinear time scale, and the ratios of the nonlinear time scale to the linear time scale $\chi_{\bm{k}}$ are usually $O(0.1)$. See Ref. [@PhysRevE.89.012909] for example. It was reported in magnetohydrodynamic turbulence that the wave numbers at which the ratio of the nonlinear time scale to the linear time scale $\chi_{\bm{k}}=1/3$ are the critical wave numbers separating the weak and strong turbulence [@PhysRevLett.116.105002]. Note that the value $1/3$ is introduced as a rough indication because the transition between the wave-dominant range and the vortex-dominant range is gradual. In the present numerical simulation, the contour of $K_{\mathrm{w}\bm{k}}/K_{\bm{k}}=0.6$ is close to the curve of $\chi_{\bm{k}} = 1/3$. The wave kinetic energy is dominant in the total kinetic energy over the vortical energy at the wave numbers where $\chi_{\bm{k}} \lessapprox 1/3$. Note that the range of $k_{\perp}, k_{\|} < 5$ is directly affected by the external force, and is not considered here. The dominance of the wave-kinetic energy does not always results in the weak-wave turbulence [@kafiabad_bartello_2018]. In the weak-wave turbulence, the wave-number modes must have the wave kinetic energy close to the potential energy. The relative difference between the wave kinetic energy and the potential energy $$\begin{aligned} \frac{K_{\mathrm{w}\bm{k}} - V_{\bm{k}}}{K_{\mathrm{w}\bm{k}} + V_{\bm{k}}} \label{eq:weaknonlinear} \end{aligned}$$ is drawn in Fig. \[fig:Kpa\_Kz\](b). In the weak-wave turbulence, $K_{\mathrm{w}} \approx V$, i.e., it is anticipated that the relative difference is close to $0$ because of the weak nonlinearity. In fact, $-0.2 < (K_{\mathrm{w}\bm{k}} - V_{\bm{k}})/(K_{\mathrm{w}\bm{k}} + V_{\bm{k}}) < 0.1$ in the range where $\chi_{\bm{k}} \lessapprox 1/3$. Therefore, the wave-number modes where the wave-kinetic energy is dominant over the vortical energy coincide with the modes which have the relative difference between the wave kinetic energy and the potential energy close to $0$. Namely, the wave-number modes where $\chi_{\bm{k}} \lessapprox 1/3$ is in the weak-wave turbulence. Moreover, in Fig. \[fig:Kpa\_Kz\](c), the ratio of the polarization anisotropic part to the total kinetic energy $$\begin{aligned} \frac{K_{z \mathrm{PA}\bm{k}}}{K_{\bm{k}}} = \frac{K_{\mathrm{w}\bm{k}}-K_{\mathrm{v}\bm{k}}}{2K_{\bm{k}}} \left(\frac{k_{\perp}}{k}\right)^2 \label{eq:kzpa} \end{aligned}$$ is drawn. Here, $K_{z \mathrm{PA}\bm{k}} = \mathrm{Re} [Z_{\bm{k}} h_{+ z \bm{k}}^2] = (k_{\perp}/k)^2 (K_{\mathrm{w}\bm{k}}-K_{\mathrm{v}\bm{k}}) / 2$ represents the polarization anisotropic part of the vertical kinetic energy according to the helical-mode decomposition. Equation (\[eq:kzpa\]) indicates the direct relation between the anisotropy and the dominance of the wave-kinetic energy over the vortical energy given by Eq. (\[eq:wave2total\]). In fact, the wave-number modes in the weak-wave turbulence, where $\chi_{\bm{k}} \lessapprox 1/3$, has $K_{z \mathrm{PA}\bm{k}}/K_{\bm{k}} > 0.2$. The weak-wave turbulence of internal gravity waves has strong anisotropy. The ratio of the gravity-wave period to the eddy-turnover time $\chi_{\bm{k}}$ well separates the weak-wave turbulence also from the horizontally long waves $k_{\perp} \approx 1$ and $k_{\|} \sim O(10)$. The 2D ratio $\chi_{\mathrm{2D}\bm{k}}$ also does it if $\chi_{\mathrm{2D}\bm{k}} = 1/3$ is selected as a threshold, though $\chi_{\mathrm{2D}\bm{k}}$ cannot separate the weak-wave turbulence from the 3D isotropic Kolmogorov turbulence by definition. The wave-number range of the anisotropic weak-wave turbulence is smaller than the inner range of the Ozmidov wave number. The transient wave-number range from the anisotropic weak-wave turbulence to the 3D isotropic Kolmogorov turbulence appears in the middle of the two turbulence range, where the quasi-2D turbulence is dominant. In this transient range, the eddy-turnover time of the wave-number mode is larger than the Brunt-Väisälä period and is smaller than $1/3$ of the linear wave period of the mode. i.e., $1/N \lessapprox \tau_{\bm{k}} \lessapprox 3/\sigma_{\bm{k}}$, and the range is noticeable at the small horizontal and large vertical wave numbers. The wave-breaking is known to occur mainly at the small horizontal and large vertical wave numbers . The saturation spectrum $K(k_{\|}) \propto k_{\|}^{-3}$ is observed in this range as shown in Fig. \[fig:spobs\](b). In the wave-number range $k_{\perp} \gg k_{\|}$, $K_{\mathrm{w}} = (k/ k_{\perp})^2 K_{\|} \approx K_{\|}$, and $K_{\mathrm{v}} \approx K_{\perp}$. Therefore, the horizontal energy spectrum $K_{\perp} \propto k_{\perp}^{-5/3}$ shown in Fig. \[fig:spobs\](a) results mainly from the vortical mode. The fact that $K_{\mathrm{w}} > K_{\mathrm{v}}$ indicates that $K_{\|} > K_{\perp}$ in the wave-number range, which is confirmed by drawing $K_{\|} / K$ though the figure is omitted. The weak-wave turbulence is stronger than the quasi-2D turbulence in the range where $\chi_{\bm{k}} \lessapprox 1/3$. In addition, the quasi-2D turbulence, i.e., the pancake turbulence [@:/content/aip/journal/pof2/13/6/10.1063/1.1369125] is dominant in the small $k_{\perp}$ and large $k_{\|}$ range where $\chi_{\bm{k}} \gtrapprox 1/3$ and $k<k_{\mathrm{O}}$. ![ Ratio of the wave kinetic energy to the total kinetic energy $K_{\mathrm{w}\bm{k}}/K_{\bm{k}}$. (a) $\gamma_{\bm{k}}=0.1$, (b) $0.2$, (c) $0.5$, (d) $1$, (e) $2$, and (f) $5$. See also the caption of Fig. \[fig:Kpa\_Kz\] for the curves. []{data-label="fig:parameter"}](sppa_small_psfrag.eps) The wave period given by the linear dispersion relation characterizes the weak-wave turbulence better than the the Brunt-Väisälä period as seen in Fig. \[fig:Kpa\_Kz\]. To confirm it, the ratios of the wave kinetic energy to the total kinetic energy for different amplitudes of the external force are drawn in Fig. \[fig:parameter\]. The numerical simulations to draw Fig. \[fig:parameter\] are performed by using $1024^3$ grid points. The amplitude of the external force $\gamma_{\bm{k}}$ is varied from $0.1$ to $5$ in the low-resolution simulations for comparison with $\gamma_{\bm{k}}=0.5$, which is used to draw Figs. \[fig:spobs\]–\[fig:Kpa\_Kz\]. The range of the weak-wave turbulence is the largest when the external force is the smallest (Fig. \[fig:parameter\](a)), and the range becomes smaller as the external force is larger. (Figs. \[fig:parameter\](b)–\[fig:parameter\](e)) It results from the fact that the eddy-turnover time becomes smaller as the turbulent fluctuation is more excited. The threshold $\chi_{\bm{k}} = 1/3$ well separates the weak-wave turbulence independently of the buoyancy Reynolds number and the vertical Froude numbers considered here. For $\gamma_{\bm{k}} = 1$, the wave-number range of $\chi_{\bm{k}} \lessapprox 1/3$ and hence the number of the wave-number modes are small. (Fig. \[fig:parameter\](f)) Then, the weak-wave turbulence cannot be organized because the resonant interactions are rare. Such divergence in the simulation with this large external force is consistent with the break in the monotonicity of the Reynolds number and the vertical Froude number in Table \[tab:parameters\]. It is derived from the limitation of numerical simulations due to the discretization and the periodic boundary condition. The wave-dominant range should exist even for this buoyancy Reynolds number and the Froude number, if the simulations in a much larger computational domain, which provides denser grid points in the wave-number space, were performed. Deviation from linear dispersion relation in wave-number space -------------------------------------------------------------- It has been exhibited in the previous subsection that the ratios of the nonlinear time scale to the linear time scale $\chi_{\bm{k}}$ i.e., the characteristic times can successfully separate the wave-dominant range by using the Cartesian decomposition and the helical-mode decomposition. To observe that the dominance of the waves in the range where $\chi_{\bm{k}} \lessapprox 1/3$ in another way, a frequency deviation from the linear dispersion relation is evaluated. It is convenient to introduce a complex amplitude used in the weak turbulence theory [@zak_book]. The complex amplitude in the present system is defined as $$\begin{aligned} a_{\bm{k}} = \frac{1}{\sqrt{2 \sigma_{\bm{k}}}} \left(u_{z\bm{k}} - \frac{i}{N} b_{\bm{k}}\right).\end{aligned}$$ Because the linear inviscid non-diffusive equation (\[eq:linearinviscidnondiffusive\]) can be rewritten as $\partial a_{\bm{k}}/ \partial t = - i \sigma_{\bm{k}} a_{\bm{k}}$, the frequency spectrum of $a_{\bm{k}}$ has a value only at $-\sigma_{\bm{k}} = -Nk_{\perp}/k$ in the linear inviscid non-diffusive limit. The minus sign in front of the frequency comes from the conventional expression of the canonical equation in the weak turbulence theory. A frequency deviation is defined as $$\begin{aligned} \delta \sigma_{\bm{k}} = \left( \frac{\displaystyle \sum_{\sigma} (\sigma + \sigma_{\bm{k}})^2 |\widetilde{a}_{\bm{k},\sigma}|^2}{\displaystyle \sum_{\sigma} |\widetilde{a}_{\bm{k},\sigma}|^2} \right)^{\frac{1}{2}} ,\end{aligned}$$ where $\widetilde{a}_{\bm{k},\sigma}$ denotes the Fourier coefficient obtained from the time series of $a_{\bm{k}}(t)$. The relative frequency deviation, $\delta \sigma_{\bm{k}} / \sigma_{\bm{k}}$, is employed for the measure of the wave nature of a wave-number mode in this paper. When the weakly nonlinear wave mode is dominant at a wave-number mode, the frequency spectrum is narrow-band and it has a peak at the frequency given by the linear dispersion relation, and the relative frequency deviation of the wave-number mode is small. Conversely, when the nonlinearity is not weak owing to the vortical mode and/or other wave-number modes, the frequency spectrum is broad-band or it has peaks away from the linear frequency [@kafiabad_bartello_2018], and the relative frequency deviation is large. Note that the nonlinearity changes the frequency spectrum in two ways: one is the excitation of frequencies which do not satisfy the dispersion relation due to the nonlinear interactions among wave-number modes, and the other is the frequency shift due to the small-wave-number flows such as the Doppler effect. ![ Relative frequency deviation $\delta \sigma_{\bm{k}} / \sigma_{\bm{k}}$ for $\bm{k} = (k_x,k_y,k_z) = (0, 2^p, 2^q)$ where $p,q=0,1,2,\cdots$. The contours are drawn for $1$ and $10$. See also the caption of Fig. \[fig:Kpa\_Kz\] for the curves. []{data-label="fig:freqsp"}](freqsp.eps) The relative frequency deviation is drawn in Fig. \[fig:freqsp\]. The frequency spectra are obtained from the time series of $a_{\bm{k}}$, where $\bm{k} = (k_x,k_y,k_z) = (0, 2^p, 2^q)$ and $p,q=0,1,2,\cdots$, in the high-resolution simulation. The relative frequency deviation is small in the range where $\chi_{\bm{k}} \lessapprox 1/3$, and shows similarity to $\chi$, becoming large as $\chi$ increases. This results from the increase of the band width of the frequency spectrum due to the nonlinearity. One may notice that the difference between the contours of the relative frequency deviation and $\chi$ at the large horizontal wave numbers where $k_y=32, 64$ and $k_z\leq 16$ is relatively large. The difference can be interpreted by the Doppler shift due to the horizontal flows with the small horizontal wave numbers including the vertically-sheared horizontal flows having most of the total energy in the flow field as recognized from Figs. \[fig:spobs\]–\[fig:spslice\]. Then, the dominance of the weakly nonlinear wave mode in the range where $\chi_{\bm{k}} \lessapprox 1/3$ is supported by the frequency deviation of wave-number modes. Concluding remark {#sec:summary} ================= In this paper, direct numerical simulations of strongly stratified turbulence where the internal gravity-wave turbulence and the strong turbulence coexist were performed. The energies accumulate at the small horizontal wave numbers, and the energies at the small vertical wave numbers are also large. Then, the one-dimensional spectra, which are obtained by the integration over the horizontal or vertical wave numbers, or by using the norm of the wave-number vector, mask the inner structures, and do not appropriately represent the critical wave numbers separating the wave-number range of the weak-wave turbulence. The non-integrated spectra and the two-dimensional spectra drawn in the horizontal and vertical wave-number domain reveal the inner structures of the anisotropic turbulence. The results show that the power laws observed in the one-dimensional spectra are superposition of various distributions of the spectral amplitude. Therefore, much care should be taken when the spectra are compared with the experimentally observed spectra, which are mostly obtained from one-dimensional time series. Following the premise that the wave kinetic energy is much larger than the vortical energy, and is close to the potential energy in the range of the weak-wave turbulence, non-dimensional indices based on the energies, Eqs. (\[eq:wave2total\])–(\[eq:weaknonlinear\]), were proposed to determine the range in the wave-number space. It was also clarified by another non-dimensional index based on the energies, Eq. (\[eq:kzpa\]), that the polarization anisotropy in the range is large, resulting from the wave kinetic energy being larger than the vortical energy. These non-dimensional indices proposed in this paper show the similar distribution, which confirms the appropriateness of the indices for the identification of the range of the anisotropic weak-wave turbulence. The dominance of the waves in the range is also verified by the frequency spectra having peaks at the frequency given by the linear dispersion relation. From the distributions of the non-dimensional indices in the horizontal and vertical wave-number domain, it was found that the range, which emerges at the small horizontal and vertical wave numbers, is anisotropic and smaller than the inner range of the Ozmidov wave number. The wave-number modes in the weak-wave turbulence have the linear time scale given by the linear dispersion relation smaller than $1/3$ of the nonlinear time scale, which is the eddy-turnover time. In other words, the critical wave number which separates the weak-wave turbulence has the ratio of the linear time scale to the nonlinear one being $1/3$. In most anisotropic turbulence systems, we have some options for linear and nonlinear time scales. The present results show that the range of the anisotropic weak-wave turbulence in the wave-number space can be identified when the appropriate time scales are selected in consideration of the anisotropy of the flow field. The difference between the linear period given by the linear dispersion relation and the isotropic Brunt-Väisälä period is large in the range where the horizontal wave numbers are small and the vertical wave numbers are relatively large. The dynamics in the wave-number range is determined neither by the weak-wave turbulence nor by the three-dimensional isotropic Kolmogorov turbulence. The wave breaking is dominant in this wave-number range [@mccomas-1981-11], and it is consistent with the saturation spectrum in Fig. \[fig:spobs\](b). The critical balance states the energy transfer from the waves to the eddies in this range. In this sense, the coexistence of the waves and eddies might play an important role in the energy spectrum [@waite_bartello_2004; @waite_bartello_2006]. The critical balance is the energy transfer in such transitional wave-number range between the wave-dominant and vortex-dominant ranges. The separation of the weak-wave turbulence in the present paper suggests that the critical balance should appear in the wave-number range $1/N \lessapprox \tau_{\bm{k}} \lessapprox 3/\sigma_{\bm{k}}$. The energy transfer in the transitional wave-number range will be reported elsewhere. Numerical computation in this work was carried out at the Yukawa Institute Computer Facility, Kyoto University and Research Institute for Information Technology, Kyushu University. This work was partially supported by JSPS KAKENHI Grant No. 15K17971, No. 16K05490, No. 17H02860, and No. 18K03927. 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addtoreset[equation]{}[section]{} \ We consider compactifications of type I supergravity on manifolds with $SU(3)$ structure, in the presence of RR fluxes and magnetized D9-branes, and analyze the generalized Dirac and Laplace-Beltrami operators associated to the D9-brane worldvolume fields. These compactifications are T-dual to standard type IIB toroidal orientifolds with NSNS and RR 3-form fluxes and D3/D7 branes. By using techniques of representation theory and harmonic analysis, the spectrum of open string wavefunctions can be computed for Lie groups and their quotients, as we illustrate with explicit twisted tori examples. We find a correspondence between irreducible unitary representations of the Kaloper-Myers algebra and families of Kaluza-Klein excitations. We perform the computation of 2- and 3-point couplings for matter fields in the above flux compactifications, and compare our results with those of 4d effective supergravity. Introduction ============ Realizing that background fluxes have a non-trivial effect on the spectrum of a string compactification has been an important step towards constructing realistic 4d string vacua. This is particularly manifest in those vacua that admit a 10d supergravity description, where compactifications with fluxes [@review1; @review2; @review3] have been shown to provide a powerful framework to address moduli stabilization and supersymmetry breaking. Indeed, in the regime of weak fluxes and constant warp factor, the effect of fluxes on the light string modes can be summarized by adding a superpotential to the 4d effective theory that arises in the fluxless limit [@gvw]. This superpotential has then the effect of lifting a non-trivial set of moduli and producing $\cn=0$ vacua at tree level [@sethi; @gkp]. While the above observation has mainly been exploited for the gravity sector of the theory, it is easy to see that it also applies to the gauge sector. In particular, in the context of type II compactifications with D-branes, it has been shown that fluxes induce supersymmetric and soft term masses on the light open string degrees of freedom of the theory. This can be seen both from a microscopic [@ciu03; @ggjl03; @ciu04] and from a 4d effective field theory viewpoint [@lrs04; @geosoft]. In fact, in this particular case it turns out that the 4d effective sugra approach is somehow more complete that the higher dimensional results, since it allows to compute soft term masses for certain open strings modes that the analysis in terms of D-brane actions has yet not been able to deal with. These modes are nothing but open strings with twisted boundary conditions, and more precisely those arising between two stacks of intersecting and/or magnetized D-branes. Generically, these open string modes are the ones giving rise to the chiral content of the 4d effective theory [@review3; @reviews]. Hence, analyzing these modes is crucial to describe the effect of fluxes on the visible sector of a realistic string compactification. Here we would like to improve the current situation by considering a string theory limit where the coupling between open string modes and open and closed background fluxes is well-defined. More precisely, we consider type I supergravity compactifications in the presence of gauge bundles, torsion and non-trivial RR 3-form fluxes. Due to the closed string fluxes and the torsion, the internal manifold is not Calabi-Yau, but possesses an $SU(3)$-structure. One can then analyze the effect of the closed string background fluxes on open strings by directly looking at how their presence modifies the 10d equations of motion for the fluctuations of the gauge sector of the theory. Such modification will affect the spectrum of open string modes, which in this approach are described as eigenfunctions of the flux-modified Laplace and Dirac operators. These new open string wavefunctions, together with the new couplings induced by the background fluxes, will dictate the effect of fluxes on the 4d effective action upon dimensional reduction of the 10d supergravity background. Note that this approach of computing explicit wavefunctions and using them in the dimensional reduction is essentially the one used in [@yukawa] to compute Yukawa couplings in toroidal models with magnetized D9-branes (see also [@quevedo; @ako08; @diveccia; @akp09]). In this sense, this work can be seen as an extension of [@yukawa] to compactifications with non-vanishing closed string fluxes. Moreover, here we will analyze the full spectrum of Kaluza-Klein modes, which in fact can also be seen as open strings with twisted boundary conditions.[^1] Finally note that, unlike in the fluxless case, the CFT techniques of [@cim03; @cp03; @ao03; @tkahler2; @lerda1; @drs07] can no longer be used and supergravity is the only available tool. As pointed out in the literature, dimensional reduction in a fluxed closed string background presents several subtleties that need to be addressed. In fact, a concrete prescription for performing a consistent 4d truncation of the theory in twisted tori (and more generical, in manifolds with SU(3) structure) is missing.[^2] The common practice is then to use instead the harmonic expansion of a standard torsionless manifold. This indeed produces the right results for the light modes in the 4d supergravity regime. Here we will follow an alternative, more controlled strategy and use techniques of non-commutative harmonic analysis to explicitly solve for the spectrum of eigenmodes of the flux-modified Dirac and Laplace operators. In this way, we perform the computation of wavefunctions for massless and massive Kaluza-Klein modes of vector bosons, scalars, fermions and matter fields for magnetized D-branes in simple type I flux compactifications. Interestingly, we find that the resulting spectrum can be classified in terms of irreducible unitary representations of the Kaloper-Myers gauge algebra [@kaloper]. The computation of the above wavefunctions carries a lot of information, that can be used for several phenomenological applications. First, by means of this formalism we can show explicitly that some wavefunctions in flux compactifications are insensitive to the flux background. Thus, if those are the lightest modes of the spectrum (as is indeed the case for weak fluxes), it is justified to expand the fluctuations in fluxless harmonics. We can also compute physical observables in the 4d effective theory, such as Yukawa couplings, in terms of overlap integrals of the corresponding wavefunctions. As a last application, one may consider integrating the spectrum of massive charged excitations in order to compute threshold corrections to the physical gauge couplings. This will however be addressed in a separate publication [@wip]. The above techniques are applied to three different classes of vacua: $\mathcal{N}=2$ vacua without flux-induced masses in the open string sector, $\mathcal{N}=1$ vacua with flux-induced $\mu$-terms and $\mathcal{N}=0$ vacua, and more precisely to explicit examples based on twisted tori. These examples are T-dual to type IIB flux compactifications with D3/D7-branes [@sethi; @gkp] and S-dual to heterotic compactifications with torsion [@intrinsic; @becker]. It is then easy to see that our analysis can be easily extended to other families of flux compactifications. The outline of the paper is as follows. In Section \[sec2\] we identify the class of type I flux vacua that we consider in the paper, and compute the modified Dirac and Laplace operators for their open string modes. We also provide two explicit supersymmetric examples of such vacua, to which we will apply our techniques in the sections to follow. Indeed, in Section \[sec:wgauge\] we address the computation of the wavefunctions for gauge bosons and introduce the necessary tools to solve for the spectrum of the Laplace-Beltrami operator in arbitrary twisted tori. Sections \[sec:scalars\] and \[sec:fermions\] are devoted respectively to the computation of wavefunctions for neutral scalars and fermions and, finally, matter field wavefunctions are considered in Section \[sec:wmatter\]. In Section \[sec:app\] we summarize the structure of massive excitations previously obtained, and then compare our results to those obtained from a 4d supergravity approach. We also translate our results to the more familiar context of type IIB flux compactifications. Section \[sec:conclu\] contains our conclusions, while the most technical material has been left for the appendices. In particular, in Appendix \[ap:N=0\] we show that our approach can also be applied to $\cn=0$ vacua. Dirac and Laplace equations in type I flux vacua {#sec2} ================================================ Type I Dirac and Laplace equations {#diraclap} ---------------------------------- A simple way to construct a theory of gravity and non-Abelian gauge interactions is to consider the low-energy limit of either heterotic or type I superstring theories. Indeed, in such limit we obtain a 10d $\cn=1$ supergravity whose bosonic and fermionic degrees of freedom are contained in a gravity and a vector multiplet as [ccc]{} & &\ & g\_[MN]{}, C\_[MN]{}, & \_M,\ & A\_M\^& \^ The gravitational content is then given by the 10d metric $g$, the two-form $C_2$, the dilaton $\phi$ and the Majorana-Weyl fermions $\psi$ and $\lam$, respectively dubbed gravitino and dilatino. The vector multiplet is that of 10d $\cn =1$ Yang-Mills theory, with both the gauge vector $A$ and the gaugino $\chi$ transforming in the adjoint of the gauge group $G_{gauge}$. Both multiplets couple to each other via a relatively simple 10d $\cn =1$ action which, in the Einstein frame, is given by [@cham1; @cham2; @cham3] $$\begin{gathered} S=-\int dx^{10}(\textrm{det }g)^{1/2}\textrm{Tr}\left[ \frac{e^{\phi/2}}{4}F^{\alpha}_{MN}F^{\alpha,MN} + \bar\chi^\a\Gamma^{M}D_M\chi^\a+\frac{e^{\phi}}{24}F_{MNP}F^{MNP} \right. \\ \left. + \frac{1}{24}e^{\phi/2}F_{MNP}\bar\chi^\a\Gamma^{MNP}\chi^\a -\frac12e^{\phi/4}F_{MN}^\a\bar \chi^\a\Gamma^Q\Gamma^{MN}(\psi_Q+\frac{\sqrt{2}}{12}\Gamma_Q\lambda)+\dots \right] \label{accion}\end{gathered}$$ where all terms not involving $A$ or $\chi$ have been dropped. Here $F_{MN}$ and $F_{MNP}$ are gauge-invariant field strengths $$\begin{aligned} &F^\alpha_{MN}\, =\, \partial_{M}A_N^\alpha- \partial_NA_M^{\alpha}+g^\alpha_{\beta\gamma}A^\beta_MA^\gamma_N \\ &F_{MNP}\, =\, 3! \partial_{[M}C_{NP]}+3!\ A^\alpha_{[M}\partial_NA_{P]}^\alpha +2g_{\alpha\beta\g}A^{\a}_MA^\b_NA^\g_P\end{aligned}$$ that will be respectively written as $F_2$ and $F_3$ when expressed in $p$-form language. Finally the gauge-covariant derivative $D_M$ acts on the gaugino as D\_M \^ = \_M\^+ g\^\_ A\^\_M \^with $g^{\a}_{\b\g}$ the structure constant of $G_{gauge}$. In bosonic backgrounds $\langle \psi \rangle = \langle \lam \rangle \equiv 0$, and so the last piece of (\[accion\]) does not contribute to the equations of motion for the components of the vector multiplet. Applying the Euler-Lagrange equations, it is easy to see that those read $$\begin{aligned} \label{gauginoeq} &\left( \slashed{D}+\frac{1}{4}e^{\phi/2}\slashed{F}_{3}\right)\chi\, =\, 0 \\ &\nabla_KF^{KP}-i[A_K,F^{KP}]-\frac{e^{\phi/2}}{2}F_{MN}F^{MNP}=0 \label{gaugeeq}\end{aligned}$$ where we have introduced the slashed notation $\slashed{A}_n\equiv \frac{1}{n!}A_{i_1\ldots i_n}\Gamma^{i_1\ldots i_n}$, and we have made use of the equation of motion for $F_3$ to discard terms proportional to $\nabla_n F^{nkp}$ in (\[gaugeeq\]). In the spirit of [@intrinsic], let us consider 4d vacua with non-trivial $F_3$. In order to preserve 4d Poincaré invariance one imposes an Einstein frame ansatz of the form ds\^2 = Z\^[-1/2]{} ds\^2\_[\^[1,3]{}]{} + ds\^2\_[\_6]{} \[metric10\] where the warp factor $Z$ only depends on $\cam_6$, as well as all $F_3$ indices lie along $\cam_6$. In general, vacua of this kind are such that $\cam_6$ admits an SU(3) structure, specified in terms of two globally well-defined SU(3) invariant forms $J$ and $\Om$. In particular, we consider backgrounds where the following relations are satisfied \[rel1\] Z e\^ & & g\_s =\ g\_s\^[1/2]{} e\^[/2]{} F\_3 & = & \*\_[\_6]{} e\^[-3/2]{} d( e\^[3/2]{} J) \[jj\]\ d( e\^J J )&= & 0 \[rel2\]Note that these equations are less restrictive than those obtained in [@intrinsic].[^3] As discussed in [@Schulz04; @geosoft; @dwsb], these are necessary conditions to construct a 4d vacuum of no-scale type. Sufficiency conditions also involve a constraint on $d\Omega$, which for supersymmetric vacua reads $d(Z^{-5/4}\Om) = 0$ and implies that $\cam_6$ is a complex manifold. Due to the presence of $F_3$, the compactification manifold $\cam_6$ has intrinsic torsion and it is not Calabi-Yau. As a result, the usual Dirac and Laplace equations of Calabi-Yau compactifications are also modified. Let us then compute the new equations via a general dimensional reduction of eqs.(\[gauginoeq\]) and (\[gaugeeq\]) to 4d, closely following [@yukawa]. For simplicity, we will consider a $U(N)$ gauge field $A$.[^4] It can then be expanded as $$A_M=B_M^\alpha U_\alpha+W^{\alpha\beta}_M e_{\alpha\beta} \label{splita}$$ with $B^\alpha_M$ real and $(W^{\alpha\beta}_M)^*=W^{\beta\alpha}_{M}$. The $U(N)$ generators $U_\alpha$ and $e_{\alpha\beta}$ are given by $$(U_\alpha)_{ij}=\delta_{\alpha i}\delta_{\alpha j}\quad \quad (e_{\alpha\beta})_{ij}=\delta_{\alpha i}\delta_{\beta j}\quad \quad \alpha\neq \beta \label{generators}$$ In general, when performing a dimensional reduction on an SU(3)-structure manifold several subtleties arise.[^5] The first and most important one concerns the identification of a suitable basis to expand the four dimensional fluctuations [@minasian], since different choices should be related by highly non-trivial field redefinitions in the 4d effective theory. In our computations below, we find convenient to expand the vector fields in terms of vielbein 1-forms $e^m$ of $\cam_6$[^6] $$\begin{aligned} B(x^\mu,x^i) &\ =\ b_\mu(x^\mu)\ B(x^i)\ dx^\mu\ +\ \sum_{m}b^m(x^\mu)\ [\langle B^m\rangle+ \xi^m](x^i) \ e^m \label{splitboson}\\ W(x^\mu,x^i) &\ =\ w_\mu(x^\mu)\ W(x^i)\ dx^\mu\ +\ \sum_{m}w^m(x^\mu)\ \Phi^m(x^i) \ e^m \label{splitboson2}\end{aligned}$$ where $x^\mu$, $x^i$ denote respectively the 4d Minkowski and 6d internal coordinates. Here, as in [@yukawa], we have set $\langle W\rangle = 0$ and allowed for a non-trivial internal vev for $B$, which breaks the initial $U(N)$ gauge group into a subgroup $G_{unbr} = \prod_i U(n_i) \subset U(N)$. The modes $b_\mu(x^\mu)$, $w_\mu(x^\mu)$, and $b^m(x^\mu)$, $w^m(x^\mu)$ transform respectively as 4d Lorentz vector and scalar fields, while from the point of view of $G_{unbr}$ the $b$’s transform in the adjoint and the $w$’s in the bifundamental representation. Finally, these modes satisfy standard equations of motion for 4d gauge bosons $$\nabla_\mu F^{\mu\nu}-i[A_\mu,F^{\mu\nu}]=m^2_A A^\nu \label{4dgaugeom}$$ and Klein-Gordon fields $$\begin{aligned} \nabla^2_{\IR^{1,3}} b^{m}&=m^2_\xi\ b^{m} \\ \nabla^2_{\IR^{1,3}} w^{m}&=m^2_\Phi\ w^{m}\end{aligned}$$ where in (\[4dgaugeom\]) $A_\mu=b_\mu+w_\mu$ and $m^2_A=m^2_B+m^2_W$. Similarly, the 10d Majorana-Weyl spinor $\chi$ can be decomposed as = + \^\* \^\* = \_4 \_6 \[splitgaug\] where $\chi_6$ is a 6d Weyl spinor of negative chirality, $\mathcal{B} = \mathcal{B}_4 \otimes \mathcal{B}_6$ a Majorana matrix and $\chi_4$ is a 4d Weyl spinor of positive chirality satisfying $$\gamma_{(4)} \slashed{\p}_{\IR^{1,3}} \mathcal{B}_4^* \chi_4^* = - m_\chi\, \chi_4\label{ferm4d}$$ where the 4d fermionic modes will arise from. Just as in eqs.(\[splitboson\]), (\[splitboson2\]), in the decomposition (\[splitgaug\]) there is a choice of basis for the 4d fluctuation modes, now implicit in the definition of $\chi_6$. Such choice of basis is given in Appendix \[ap:ferm\], where the fermion conventions used in this paper are specified. As one can check explicitly in the examples below, the choices performed in the bosonic and fermionic sectors are related to each other via the 10d supersymmetry variation \_A\_M = | \_M \[10dSUSYt\] where $\eps$ is the 10d Killing spinor of the background.[^7] As a result, the effective theory obtained from the above dimensional reduction scheme will inherit a 4d SUSY structure that can be obtained directly from reducing (\[10dSUSYt\]). In general, in order to fully specify the 4d couplings of the effective action one first needs to compute internal wavefunctions of the fields $B(y)$, $W(y)$, $\xi^m(y)$, $\Phi^m(y)$ and $\chi_6(y)$ that appear in eqs.(\[splitboson\]), (\[splitboson2\]) and (\[splitgaug\]). Such wavefunctions can be obtained by solving the corresponding internal 6d Dirac and Laplace equations for a type I background with fluxes. One can compute these equations by plugging (\[splitboson\])-(\[ferm4d\]) into (\[gauginoeq\])-(\[gaugeeq\]) and the ansatz (\[metric10\]). We obtain[^8] \[Beq\] \^[\_6]{}\^m\^[\_6]{}\_m B - (\_mZ)\^[\_6]{}\^[m]{} B = -Z\^[1/2]{} m\_B\^2 B\ \[Weq\] \^m\_m W -2(\_mZ)\^[m]{}W = -Z\^[1/2]{} m\_W\^2 W $$\begin{gathered} \nabla^{\mathcal{M}_6}{}^m\nabla^{\mathcal{M}_6}_m\xi^{p,\alpha}-[\nabla_m^{\mathcal{M}_6},\nabla^{\mathcal{M}_6}{}^p]\xi^{m,\alpha}-2(\partial_k\textrm{log }Z)\nabla^{\mathcal{M}_6}{}^{[k}\xi^{p],\alpha}+\\ +e^{\phi/2}(\nabla^{\mathcal{M}_6}_m\xi^{n,\alpha})F_n{}^{mp}=-Z^{1/2}m_\xi^2\xi^{p,\alpha} \label{xieq}\end{gathered}$$ $$\begin{gathered} \tilde{D}^m\tilde{D}_m\Phi^{p,\alpha\beta}-[\nabla_m^{\mathcal{M}_6},\nabla^{\mathcal{M}_6}{}^p]\Phi^{m,\alpha\beta}-2(\partial_k\textrm{log }Z)\tilde{D}^{[k}\Phi^{p],\alpha\beta}+2i\Phi^{m,\alpha\beta}\langle G_m{}^{p,\alpha\beta}\rangle+\\ +e^{\phi/2}(\tilde{D}_m\Phi^{n,\alpha\beta})F_n{}^{mp}=-Z^{1/2}m_\Phi^2\Phi^{p,\alpha\beta} \label{phieq}\end{gathered}$$ for the bosonic wavefunctions and $$\G_{(4)} \left( \slashed{D}^{\cam_6} + \frac{1}{4} e^{\phi/2} \slashed{F}_3 - \frac{1}{2} \slashed{\p} \ln Z \right) \chi_6 \, =\, Z^{1/4} m_\chi\, \mathcal{B}_6^* \chi_6^* \label{dirac6d}$$ for the fermionic wavefunctions, where $\nabla^{\mathcal{M}_6}_m$ and $\slashed{D}^{\cam_6} = \G^m D_m$ are the bosonic and fermionic covariant derivatives in $\cam_6$ and we have introduced the notation $$\begin{aligned} &\tilde D_m\Phi^{\alpha\beta}_n=\nabla_m^{\mathcal{M}_6}\Phi^{\alpha\beta}_n-i(\langle B^\alpha_m\rangle-\langle B^\beta_m\rangle)\Phi^{\alpha\beta}_n \label{prot1}\\ &\langle G^{\alpha\beta}_{mn}\rangle=2\nabla_{[m}^{\mathcal{M}_6}\langle B^\alpha_{n]}\rangle-2\nabla_{[m}^{\mathcal{M}_6}\langle B^\beta_{n]}\rangle\label{prot2}\end{aligned}$$ Finally, note that if we expand the fermionic wavefunction as $\chi_6 = \lam^\a U_\a + \Psi^{\a\b}e_{\a\b}$, we have that $\slashed{D}^{\cam_6} \lam^\a = \slashed{\nabla}^{\cam_6} \lam^\a$ and $\slashed{D}^{\cam_6} \Psi^{\a\b} = \tilde{\slashed{D}} \Psi^{\a\b}$. Elliptic fibrations {#subsec:elliptic} ------------------- A simple way to find solutions to the equations (\[rel1\])-(\[rel2\]) is to consider the particular case where $\cam_6$ is an elliptic fibration of fiber $\Pi_2$ over a four dimensional base $B_4$ [@sethi; @Schulz04; @geosoft; @dwsb]. In particular, we consider a metric ansatz of the form ds\^2\_[\_6]{} = Z\^[-1/2]{} \_[a \_2]{} (e\^a)\^2 + Z\^[3/2]{} ds\^2\_[B\_4]{} \[mansatz\] where neither the base metric $ds^2_{B_4}$ nor the vielbein 1-forms of the fiber $e^a$ depend on the warp factor $Z$, which in turn only depends on the $B_4$ coordinates. This will be indeed the case if $Z$ is sourced by background fluxes and/or D5-branes/O5-planes wrapped on $\Pi_2$ (see e.g. [@Schulz04; @dwsb] for explicit examples of this kind). The structure of the (unwarped) fibration can then be parameterized as $$de^a=\frac{1}{2}f^a_{mn}e^m\wedge e^n\in H^2(B_4,\IR) \label{defviel}$$ with $f^a_{mn}$ some structure constants.[^9] In general, $\nabla^{\mathcal{M}_6}_m$, $\slashed{D}^{\cam_6}$ and $e^{\phi/2} \slashed{F}$ will depend on the warp factor $Z$, that will enter eqs.(\[Beq\])-(\[dirac6d\]) in a rather non-trivial way. Even if as shown in Appendix \[ap:warp\] the on-shell relations (\[rel1\])-(\[rel2\]) simplify such dependence, we would like to simplify the problem by taking a limit of constant warp factor. In practice, one can achieve such limit via the non-isotropic fibration $\textrm{Vol}_{B_4}^{1/2} \gg \textrm{Vol}_{\Pi_2}$, that in terms of mass scales translates into the hierarchy $m^{\text{KK}}_{\text{fib}} \gg m^{\text{KK}}_{\text{base}} \gg m_{\text{flux}}$ [@Schulz04]. Here $m_{\text{flux}}$ (denoted $\varepsilon$ in the following sections) is the mass scale introduced by the presence of background fluxes, and in particular the mass scale of closed and open string lifted moduli. As a result, this hierarchy of scales is essential to understand the process of moduli stabilization in terms of a 4d $\cn=1$ effective theory where all KK modes have been integrated out. In addition, as discussed in section \[sugra\] the condition $m^{\text{KK}}_{\text{base}} \gg m_{\text{flux}}$ also ensures that the warp factor can be taken to be constant, which is the approximation that we would like to consider in the following.[^10] Finally, imposing $\textrm{Vol}_{B_4}^{1/2}, \textrm{Vol}_{\Pi_2} \gg \a'$ guarantees that the supergravity approximation in which we are working remains valid. Splitting the 2-form $J$ as $J = J_{\Pi_2} + J_{B_4}$ as in [@geosoft], introducing the projectors, P\_\^[\_2]{} = (1 i\_[\_2]{} \_[(6)]{} ) \[ex1proj\] and taking $Z$ constant eq.(\[dirac6d\]) becomes (see Appendix \[ap:warp\]) (\^[\_2]{} + \^[B\_4]{} + P\_+\^[\_2]{} ) \_6 = m\_ \_6\^\* \_6\^\* \[dirac6duw\] where we have absorbed the operator $\G_{(4)}$ in the definition of slashed contraction. Indeed, in the expression above all slashed quantities are constructed from the set of $\G$-matrices defined in (\[commG\]), a convention that we will take from now on. Finally, we have defined the antisymmetrized geometric flux f\_[mnp]{} = 3 \_[r\[m]{}f\^r\_[np\]]{} \[metricflux\] The projector $P_+^{\Pi_2}$ corresponds to the chirality projector of the 4d base $B_4$. One can then split the internal 6d fermion as \_6 = \_[\_2]{} + \_[B\_4]{} \[split4+2\] where $\chi_{\Pi_2,B_4}$ satisfy $P_+^{\Pi_2} \chi_{\Pi_2} = \chi_{\Pi_2}$ and $P_+^{\Pi_2} \chi_{B_4} = 0$.[^11] Since $\mathcal{B}_6$ changes the fiber chirality but not the base chirality, we can split the Dirac equation as \[6dsplit1\] \^[\_2]{}\_[B\_4]{} + \^[B\_4]{} \_[\_2]{} & = & m\_\_6\^\* \_[B\_4]{}\^\*\ \[6dsplit2\] \^[\_2]{}\_[\_2]{} + \^[B\_4]{} \_[B\_4]{} + \_[\_2]{} & = & m\_\_6\^\* \_[\_2]{}\^\* A similar analysis can be carried out for the scalar wavefunctions, governed by eqs.(\[xieq\]) and (\[phieq\]). Distinguishing between scalars corresponding to the base and to the fiber, the equations of motion (\[xieq\]) and (\[phieq\]) read $$\begin{aligned} \hat\partial_m\hat\partial^m\xi^{p}_{\Pi_2} -(f^{p}_{mn}-e^{\phi/2}F_{nm}{}^{p})\hat\partial^m\xi^n_{B_4} -\frac12f^{a}_{mn}(f^{p}_{mn}-e^{\phi/2}F_{nm}{}^{p})\xi^{a}_{\Pi_2}=-m_\xi^2\xi^p_{\Pi_2} \label{xi+}\\ \hat\partial_m\hat\partial^m\xi^{p}_{B_4}-(f^{m}_{pn}-e^{\phi/2}F_{n{m}}{}^p) \hat\partial^{m}\xi^n_{B_4}+(f_{mp}^{n}+e^{\phi/2}F_{{n}m}{}^p)\hat\partial^m\xi^{n}_{\Pi_2} =-m_\xi^2\xi_{B_4}^p \label{xi-}\end{aligned}$$ and $$\begin{aligned} & \hat D_m\hat D^m\Phi^{p}_{\Pi_2}-(f^{p}_{mn}-e^{\phi/2}F_{nm}{}^{p}) \hat D^m\Phi^n_{B_4}-\frac12f^{a}_{mn}(f^{p}_{mn}-e^{\phi/2}F_{nm}{}^{p}) \Phi^{a}_{\Pi_2}=-m_\Phi^2\Phi^p_{\Pi_2} \label{lapfin1}\\ & \hat D_m\hat D^m\Phi^{p}_{B_4}-(f^{m}_{pn}-e^{\phi/2}F_{n{m}}{}^p) \hat D^{m}\Phi^n_{B_4}+(f_{mp}^{n}+e^{\phi/2}F_{{n}m}{}^p) \hat D^m\Phi^{n}_{\Pi_2}+2i\Phi_{\Pi_2}^m\langle\hat G_m{}^p\rangle\nonumber\\ &\hspace{11cm}=-m_\Phi^2\Phi_{B_4}^p \label{lapfin2}\end{aligned}$$ where $\hat D_m\Phi_n^{\alpha\beta}$ and $\langle \hat G^{\alpha\beta}_{mn}\rangle$ are respectively defined as in (\[prot1\]) and (\[prot2\]), but replacing the covariant derivative $\nabla_m^{\mathcal{M}_6}$ by twisted derivatives defined in terms of the vielbein as \_ae\_a\^(x) \_[x\^]{} \[hatted\] Finally, we have assumed that $\langle B_m^\a\rangle$ is constant along the fiber, as dictated by cancelation of Freed-Witten anomalies [@fw1; @kaloper]. Twisted tori examples {#subsec:twisted} --------------------- In order to provide explicit examples of the metric ansatz (\[mansatz\]) one may consider the simple case where the base of the fibration $B_4$ corresponds to a flat four-torus $T^4$. This basically implies that, up to warp factors, $\cam_6$ lies within a particular class of twisted tori, which are in fact the simplest non-trivial examples of SU(3) structure manifold. A very interesting feature of twisted tori, and which will be crucial in the discussion of next section, is that they can be defined in a group theoretic way, and more precisely as a left quotient of groups $\cam_6 = \G \backslash G$. Indeed, let us consider a $d$-dimensional group manifold $G$ and its Lie algebra $\mathfrak{g}=\textrm{Lie}(G)$. The latter is specified by a set of structure constants $f^a_{bc}$ that satisfy the Jacobi identity $f^a_{[bc}f^g_{d]a}=0$. In terms of a matrix representation of the Lie Group $g_G \in GL(n)$, one can easily compute the vielbein left-invariant 1-forms as $g_G^{-1}dg_G = e^a \mathfrak{t}_a$, with $\mathfrak{t}_a \in \mathfrak{g}$ the algebra generators, and hence the structure constants via $$de^a=\frac{1}{2}f^a_{bc}e^b\wedge e^c \quad \quad \Leftrightarrow \quad \quad [\hat \partial_b,\hat \partial_c]=-f^a_{bc}\hat \partial_a \label{torsion}$$ with $\hat{\p}_a$ defined as in (\[hatted\]). These twisted derivatives can then be identified with $\mathfrak{t}_a$. While in general $G$ may not be a compact manifold, one can construct such manifold by left-quotienting $G$ by a discrete, cocompact subgroup $\G \subset G$.[^12] The resulting twisted torus $\cam_d = \G \backslash G$ is no longer a group, but it is a parallelizable manifold since the left-invariant 1-forms are still globally well-defined. Given a set of structure constants $f^a_{bc}$, constructing a compact manifold $\cam_d = \G \backslash G$ is usually a non-trivial problem. This is however greatly simplified if we restrict ourselves to the case where $\mathfrak{g}$ is a nilpotent Lie algebra.[^13] That is, we consider the case where the series $\{\mathfrak{g}_s\equiv [\mathfrak{g}_{s-1},\mathfrak{g}_0]\}$, with $\mathfrak{g}_0\equiv \mathfrak{g}=\textrm{Lie}(G)$, has $k$ non-vanishing elements, in which case $\mathfrak{g}$ is said to be $k$-step nilpotent. Then, in order for a cocompact $\G$ to exist, we only need to require that $f^a_{ab} =0$ and that the structure constants are integers in some particular basis [@Malcev]. The resulting nilmanifold is a non-flat, compact (usually iterated) fibration of tori. In particular, we will obtain elliptic fibrations that fit into our metric ansatz (\[mansatz\]). If in particular we consider an elliptic fibration over $T^{d-2}$, then $\mathfrak{g}$ should be 2-step nilpotent. The associated Lie group has then the following faithful representation $$g_G= \begin{pmatrix} \Id_{d} - \oh ad_{\vec X} & \vec X\\ 0& 1 \end{pmatrix} \quad \quad \quad [ad_{\vec X}]^i_j\, =\, X^k f^i_{kj} \label{magicdisney}$$ in terms of $GL(d+1,\mathbb{R})$ matrices. Here $\vec X$ is a $d$-dimensional coordinate vector parameterizing $\mathfrak{g}$ and $ad$ is the adjoint representation of the algebra, which due to 2-step nilpotency satisfies $ad^2 =0$. Note that this implies that $e^i = dX^i + \oh f^i_{kj} X^kdX^j$. A classical example of this construction is given by the $(2p+1)$-dimensional Heisenberg manifold $\mathcal{H}_{2p+1}$, the canonical example of nilpotent Lie group. Here we can split $\vec X^t = (z, \vec x ^{\, t}, \vec y^{\, t})$, $\vec x, \vec y \in \IR^p$ and express the algebra as $$[\mathfrak{t}_{x^i}, \mathfrak{t}_{y^j}]\, =\, \d_{ij}\,\mathfrak{t}_{z} \label{heisalg}$$ so that (\[magicdisney\]) reads $$g_{\mathcal{H}_{2p+1}}\, =\, \begin{pmatrix}1 & -\frac12\vec{y}^{\, t} & \frac12\vec{x}^{\, t} & z\\ 0 & 1 & 0 & \vec x \\ 0 & 0 & 1 & \vec y\\ 0 & 0 & 0 & 1\end{pmatrix} \label{heis}$$ In this case, a suitable choice for $\Gamma$ is the lattice $\G_{\ch_{2p+1}} = \{(\vec x,\vec y,z) = M (\vec{n}_x, \vec{n}_y, n_z)\}$ with $M, n_z \in \IZ$ and $\vec{n}_x, \vec{n}_y \in \IZ^n$.[^14] One can then normalize the generators as $\tilde{\mathfrak{t}}_{a} = M \mathfrak{t}_a$, so that the algebra becomes $[\tilde{\mathfrak{t}}_{x^i}, \tilde{\mathfrak{t}}_{y^j}]\, =\, \d_{ij}M\, \tilde{\mathfrak{t}}_{z}$ and the invariant 1-forms read \^z = dz - (x\^[ t]{}dy - y\^[ t]{} dx) \^[x\^i]{} = dx\^i \^[y\^i]{} = dy\^i \[1formheis\] The nilmanifold $\G \backslash G$ then corresponds to an $S^1$ fibration (whose fiber is parameterized by $z$) over a $T^{2p}$ (parameterized by $(\vec x,\vec y)$) and of Chern class $F_2 = M\sum_i dy^i \wedge dx^i$. Such U(1)-bundle structure will become manifest below, when analyzing the spectrum of the Laplace and Dirac operators in the (compactified) Heisenberg manifold. Finally, a rescaling of the form $\tilde{\mathfrak{t}}_a \raw (2 \pi R_a)^{-1} \tilde{\mathfrak{t}}_a$, $X^a \raw 2\pi R_a\, X^a$ will take us to a moduli-dependent set of structure constants, which are those that correspond to the set of vielbein left-invariant 1-forms in (\[defviel\]) and (\[torsion\]). It follows from the above discussion that a good starting point to construct explicit solutions to eqs.(\[rel1\])-(\[rel2\]) is to consider $\cam_6$ to be either a nilmanifold or a product like $S^1 \times \G_{\ch_5} \backslash \ch_5$. In the following we will provide two different type I backgrounds based on such strategy, for which we will later on explicitly solve the Laplace and Dirac equations (see Sections \[sec:wgauge\] to \[sec:wmatter\]). According to the open string spectrum, we can roughly classify nilmanifold type I flux vacua in two different classes. The first one is that where the spectrum of massless open string adjoint scalars $b^m$ (see (\[splitboson\])) remains identical with respect to a toroidal (or toroidal orientifold), fluxless compactification. The second class is that where, because of the presence of the flux, some of these adjoint scalars develop up a mass of the order of $m_{\text{flux}}$, just like the closed string moduli of the compactification. We will dub such classes of vacua as vacua with vanishing and non-vanishing flux-generated $\mu$-term, respectively, and present a supersymmetric example for each of them below. A non-supersymmetric type I flux vacua will be considered in Appendix \[ap:N=0\]. ### Example with vanishing $\mu$-terms {#vmu} Let us consider the following type I flux background, displayed in the ten dimensional Einstein frame and $\alpha'$ units \[bg1\] $$\begin{aligned} \label{bg11} &ds^2=Z^{-1/2}(ds^2_{\IR^{1,3}}+ds^2_{\Pi_2})+Z^{3/2}ds^2_{T^4} \\ \label{bg12} &ds^2_{T^4}=(2\pi)^2 \sum_{m=1,2,4,5}(R_mdx^m)^2 \\ \label{bg13} &ds^2_{\Pi_2}= (2\pi)^2 \left[(R_3 dx^3)^2+(R_6\tilde e^{6})^2\right] \\ \label{bg14} &F_3=-(2\pi)^2 N ( dx^1\wedge dx^2+ dx^4\wedge dx^5)\wedge \tilde e^6-g_s^{-1}*_{T^4}dZ^2 \\ &e^{\phi} Z=g_s = \text{const.}\end{aligned}$$ where we have included the warp factor dependence, as well as provisionally set $F_2=0$. Let us first focus on the metric background (\[bg11\])-(\[bg13\]), parameterized by the six compactification radii $R_i$. Here $\tilde e^6$ stands for a left-invariant 1-form satisfying[^15] $$d \tilde e^6=M( dx^1\wedge dx^2+ dx^4 \wedge dx^5)\ . \label{ejtwist}$$ so from (\[heisalg\]) it is easy to see that (up to warp factors) $\cam_6$ looks locally like $\IR \times \mathcal{H}_5$, and that $\tilde e^6$ is associated to the center of the 5-dimensional Heisenberg group $\mathcal{H}_5$. Following our general discussion above, we can easily integrate eq.(\[ejtwist\]) to obtain $$\tilde e^6=dx^6+\frac{M}{2}(x^1dx^2-x^2dx^1+x^4dx^5-x^5dx^4) \label{integrated1}$$ as well as the vielbein 1-forms $e^a$. From the latter, we obtain the twisted derivatives \[hatted1\] $$\begin{aligned} \hat\partial_1&=(2\pi R_1)^{-1}\left(\partial_{x^1}+\frac{M}{2}x^2\partial_{x^6}\right)& \hat\partial_4&=(2\pi R_4)^{-1}\left(\partial_{x^4}+\frac{M}{2}x^5\partial_{x^6}\right)\\ \hat\partial_2&=(2\pi R_2)^{-1}\left(\partial_{x^2}-\frac{M}{2}x^1\partial_{x^6}\right)& \hat\partial_5&=(2\pi R_5)^{-1}\left(\partial_{x^5}-\frac{M}{2}x^4\partial_{x^6}\right)\\ \hat\partial_3&=(2\pi R_3)^{-1}\partial_{x^3}&\hat\partial_6&=(2\pi R_6)^{-1}\partial_{x^6}\end{aligned}$$ Finally, the global structure of $\cam_6$ is not $\IR \times \mathcal H_5$ but rather the compact manifold $\cam_6 = \G \backslash (\IR \times \mathcal H_5)$, where $\G$ is a cocompact subgroup of $\IR \times \mathcal H_5$, which we take to be $\IZ \times \G_{\ch_5}$. Such quotient requires $M \in 2\IZ$ and produces the identifications \[reli\] $$\begin{aligned} &x^1\to x^1+1 \quad \quad x^6\to x^6-\frac{Mx^2}{2} \label{reli1}\\ &x^2\to x^2+1 \quad \quad x^6\to x^6+\frac{Mx^1}{2} \\ &x^3\to x^3+1 \\ &x^4\to x^4+1 \quad \quad x^6\to x^6-\frac{Mx^5}{2} \\ &x^5\to x^5+1 \quad \quad x^6\to x^6+\frac{Mx^4}{2} \\ &x^6\to x^6+1\label{reli2}\end{aligned}$$ which by construction leave (\[integrated1\]) and (\[hatted1\]) invariant. Taking now into account the RR flux (\[bg14\]) it is easy to see that eqs.(\[rel1\])-(\[rel2\]) are satisfied provided that the on-shell relations $g_s N = M R_6^2$ and $R_1R_2=R_4R_5$ are imposed. This implies that $d(Z^{-5/4}\Om) = 0$ for some suitable choice of $\Omega$ (see below), which in turn implies that our compactification manifold $\cam_6$ is complex and our 4d theory supersymmetric. Finally, we should also impose $N \in \IZ$ by standard Dirac quantization arguments. Since $\cam_6$ is a compact manifold, we should check that both NSNS and RR tadpoles are canceled globally. Before that, let us include in our background an open string field strength of the form F\_2 = F\_[14]{} dx\^1 dx\^4 + F\_[25]{} dx\^2 dx\^5 as well as D5-branes and O5-planes wrapping $\Pi_2$. The Bianchi identity for $F_3$ then reads dF\_3 & =&- ( (2)\^2 + g\_s\^[-1]{} \^2\_[T\^4]{}Z\^2 ) e\^1e\^2e\^3e\^4\ & =& (2)\^2 \_jq\_j\_[T\^4]{}(x - x\_j)e\^1e\^2e\^3e\^4 \[BI1\] where in the second line we have made use of the warp factor equation -g\_s\^[-1]{} \^2\_[T\^4]{}Z\^2 = (2)\^2 ( +\_jq\_j\_[T\^4]{}(x - x\_j)) \[NSNSt1\] where $q_j=1$ for D5-branes and $q_j = -2$ for O5-planes. Note that (\[BI1\]) does not imply that $\sum_j q_j = 0$, as it would for $\cam_6 = T^6$, but rather that $\sum_j q_j = 0\, \text{mod}\, M$, due to the torsional cohomology of $\cam_6$ [@Schulz04; @torsion]. On the other hand, the r.h.s. of (\[NSNSt1\]) must vanish upon integration on $B_4$. Since the background BPS conditions imply that $NM > 0$ and that $\tr (F_{14}F_{25}) > 0$, this is only possible if O5-planes are present on the compactification. We will thus implement their presence via the additional orbifold quotient ${\cal R}: x^m \mapsto - x^m$, where $x^m$ is a $B_4$ coordinate. Finally, let us discuss the amount of supersymmetry preserved by this background. The fact we are compactifying type I string theory sets the maximal amount of supersymmetry to 4d $\cn=4$, which would be the case if we were compactifying in $T^6$. Adding the orbifold quotient ${\cal R}$ above (or equivalently adding the induced O5-planes) halves the amount of SUSY to 4d $\cn =2$. These two generators of supersymmetry can be associated with two different choices of complex structure, $(z^1, z^2, z^3)$ and $(\bar{z}^1, \bar{z}^2, z^3)$, with $z^i = x^i + i \tau_i x^{i+3}$ and $\tau_i = R_{i+3}/R_i $, that preserve the orientation of $T^6$ and of the 2-cycle $\Pi_2$ wrapped by the O5-plane. If as a last ingredient we add the background flux (RR and geometric) with the above choice of dilaton and compactification radii ($g_sN = MR_6^2$ and $R_1R_2 = R_4R_5$) we see that no further supersymmetries are broken. Indeed, taking for simplicity the $Z=1$ limit, this can be checked by noting that $g_s F_3 - i dJ$ is a (2,1)-form for both choices of complex structure, or by the fact that both choices of 3-form $\Omega = e^{z^1} \wedge e^{z^2} \wedge e^{z^3}$ and $\Omega' = e^{\bar{z}^1} \wedge e^{\bar{z}^2} \wedge e^{z^3}$ are closed, and so define a good complex structure even in the presence of the geometric flux. ### Example with non-vanishing $\mu$-terms {#nvmu} Let us now consider a slightly more involved solution to the equations (\[rel1\])-(\[rel2\]), this time yielding supersymmetric mass terms ($\mu$-terms) for some of the 4d adjoint multiplets. Such background is given by \[bg2\] $$\begin{aligned} &ds^2=Z^{-1/2}(ds^2_{\IR^{1,3}}+ds^2_{\Pi_2})+Z^{3/2}ds^2_{T^4} \\ &ds^2_{T^4}=(2\pi)^2 \sum_{m=1,2,4,5}(R_mdx^m)^2 \\ &ds^2_{\Pi_2}=(2\pi)^2\left[ (R_3\tilde e^3)^2+\left(R_6 \tilde e^{6}\right)^2\right] \\ &F_3=(2\pi)^2(N_6\, dx^2\wedge \tilde e^6 -N_3\, dx^5\wedge \tilde e^3)\wedge dx^4 -g_s^{-1}*_{T^4}dZ^2\end{aligned}$$ and again $e^\phi Z = g_s$. This time the left-invariant 1-forms satisfy $$d \tilde e^3=M_3 dx^1\wedge dx^2 \quad \quad \text{and} \quad \quad d\tilde e^6=M_6 dx^1 \wedge dx^5\label{nil2}$$ which again corresponds to a nilpotent Lie algebra. The twisted derivatives now read $$\begin{aligned} \hat\partial_1&=(2\pi R_1)^{-1} \left(\partial_{x^1}+\frac{M_3}{2}x^2\partial_{x^3}+\frac{M_6}{2}x^5\partial_{x^6}\right)&\hat \partial_4&=(2\pi R_4)^{-1}\partial_{x^4}\\ \hat\partial_2&=(2\pi R_2)^{-1}\left(\partial_{x^2}-\frac{M_3}{2}x^1\partial_{x^3}\right)&\hat\partial_5&=(2\pi R_5)^{-1}\left(\partial_{x^5}-\frac{M_6}{2}x^1\partial_{x^6}\right)\\ \hat\partial_3&=(2\pi R_3)^{-1}\partial_{x^3}&\hat\partial_6&=(2\pi R_6)^{-1}\partial_{x^6}\end{aligned}$$ and the quotient by $\G$ produces the identifications \[relii\] $$\begin{aligned} &x^1 \to x^1 + 1 \quad \quad x^3 \to x^3 - \frac{M_3}{2}x^2 \quad \quad x^6 \to x^6 - \frac{M_6}{2}x^5 \label{relii1}\\ &x^2 \to x^2 + 1 \quad \quad x^3 \to x^3 + \frac{M_3}{2}x^1 \\ &x^3 \to x^3 + 1 \\ &x^4 \to x^4 + 1 \\ &x^5 \to x^5 + 1 \quad \quad x^6 \to x^6 + \frac{M_6}{2}x^1 \\ &x^6 \to x^6 + 1\label{relii2}\end{aligned}$$ so that the resulting nilmanifold can be seen as the simultaneous fibration of two $S^1$’s along a $T^4$ base. The equations of motion for this background now require the on-shell relations $M_3 R_3^2 R_4 R_5 = g_s N_3 R_1R_2$ and $M_6 R_6^2 R_2R_4 = g_s N_6 R_1R_5$, with $N_3, N_6, M_3, M_6 \in \IZ$. In addition, tadpole cancelation will need of the presence of O5-planes wrapping the $\Pi_2$ fiber, that again will be introduced via the orbifold quotient ${\cal R}: x^m \mapsto - x^m$ on the base coordinates. As before the presence of O5-planes will reduce the amount of supersymmetry as $\cn =4 \raw \cn = 2$, while the background fluxes will further break the amount of SUSY. More precisely, if we impose $M_3N_3 = M_6 N_6$, we will satisfy the supersymmetry condition $d(Z^{-5/4} \Om) = 0$ for the choice $\Omega = e^{z^1} \wedge e^{z^2} \wedge e^{z^3}$, this being the only choice of closed SU(3)-invariant 3-form. Hence, in general the fluxes will break the 4d supersymmetry as $\cn =2 \raw \cn =0$, while they will do as $\cn =2 \raw \cn =1$ if we impose that $M_3N_3 = M_6 N_6$. For simplicity, we will assume the latter constraint to hold for the rest of the paper. Wavefunctions for gauge bosons {#sec:wgauge} ============================== The simplest family of wavefunctions that one may analyze in type I flux vacua correspond to the gauge bosons of the 4d gauge group $G_{unbr}$ and their massive Kaluza-Klein excitations, transforming in the adjoint representation of $G_{unbr}$. Indeed, all these modes arise from the term $b_\mu (x^\mu) B(x^i)$ in the expansion (\[splitboson\]) and, as (\[Beq\]) shows, their internal Laplace equation for $B$ does not involve the flux $F_3$. In fact, in the limit of constant warp factor (\[Beq\]) reduces to the standard Laplace-Beltrami equation in the manifold $\cam_6$. In the notation of Sections \[subsec:elliptic\] and \[subsec:twisted\] such equation can be written as $$\Delta B\, =\, \hat \partial_a \hat{\partial}^{a}B=-m_B^2B \label{lap}$$ where $B$ is a complex wavefunction describing two real d.o.f. of the 4d gauge boson,[^16] while $\hat{\p}_a$ are the twisted derivatives defined by (\[hatted\]). From (\[lap\]) it is easy to see that, as expected, gauge boson zero modes are given by constant internal wavefunctions $B= const$. Computing the internal wavefunction of massive KK modes is however more involved, and in general requires the explicit knowledge of the Laplace-Beltrami operator. As shown in the previous section, twisted tori provide simple examples of compactification manifolds where $\hat{\p}_a$ have a simple, globally well-defined expression, which allows to compute analytically the full spectrum of KK masses and wavefunctions of $\Delta$. Indeed, in this section we will compute such spectrum for the explicit twisted tori examples described in Section \[subsec:twisted\]. As we will see, in simple twisted tori like that of subsection \[vmu\] the spectrum of wavefunctions is analogous to that of open strings in magnetized D-brane models, and so it can be easily computed using the results of [@yukawa]. On the other hand, for more involved nilmanifolds such analogy becomes less fruitful, and one is led to apply group theoretic techniques as well as tools of non-commutative harmonic analysis to compute the spectrum of $\Delta$. We will present below a general description of the latter method, and apply it to the computation of wavefunctions in the twisted torus background of subsection \[nvmu\]. Vanishing $\mu$-terms {#nili} --------------------- Let us then consider the Laplace-Beltrami equation for the type I vacuum of subsection \[vmu\]. As discussed above, in the limit of constant warp factor this equation reduces to (\[lap\]), where the twisted derivatives are given by (\[hatted1\]). Solving (\[lap\]), however, does still not guarantee that our wavefunction is well-defined globally, as the twisted derivatives only see the local geometry $\IR \times \mathcal H_5$ of the twisted torus $\cam_6 = \Gamma\backslash (\IR \times \mathcal H_5)$. Hence, proper wavefunctions will also be invariant under the left action of the discrete subgroup $\G$, and more precisely under the identifications (\[reli\]). Following a similar strategy to [@yukawa], we will first impose $\G$-invariance via the ansatz $$B_{k_3,k_6}(\vec x)=\sum_{k_1,k_4}f_{k_1,k_3,k_4,k_6}(x^2,x^5)\, {e}^{2\pi i (k_1 x^1+k_3 x^3+k_4 x^4+k_6\dot{x}^6)} \quad \quad \quad k_i \in \IZ \label{ansz}$$ with $$f_{k_1,k_3,k_4,k_6}(x^2+\ell_2,x^5+\ell_5)=f_{k_1+Mk_6\ell_2,k_3,k_4+Mk_6\ell_5,k_6}(x^2,x^5) \quad \quad \quad \ell_2,\ell_5\in \mathbb{Z} \label{period}$$ and where we have performed the change of variables $$\dot x^6\equiv x^6+\frac{M}{2}(x^1x^2+x^4x^5) \label{change}$$ Then, substituting into eq.(\[lap\]) and proceeding by separation of variables $$f_{k_1,k_3,k_4,k_6}(x^2,x^5)\equiv f_{k_1,k_3,k_6}(x^2)f_{k_3,k_4,k_6}(x^5) \label{sep}$$ one can see that (\[lap\]) is equivalent to a system of Weber differential equations [@whitaker] $$\begin{aligned} &\left[(\partial_{\dot x^2})^2-\frac{1}{4}(\dot x^2)^2+\nu-\alpha\right]f_{k_1,k_3,k_6}(\dot x^2)=0 \label{web1}\\ &\left[(\partial_{\dot x^5})^2-\frac{1}{4}(\dot x^5)^2+\alpha\right]f_{k_3,k_4,k_6}(\dot x^5)=0 \label{web2}\end{aligned}$$ for some constant $\alpha$, where we have made the following definitions: $$\begin{aligned} \dot x^2&\equiv\frac{2}{R_1} a^{-1/2}(k_1+k_6Mx^2) \label{x2}\\ \dot x^5&\equiv\frac{2}{R_4} a^{-1/2}(k_4+k_6Mx^5) \\ \nu&\equiv\frac{1}{a}\left(m_B^2 - \left[\left(\frac{k_6}{R_6}\right)^2+\left(\frac{k_3}{R_3}\right)^2\right]\right) \label{x5}\\ a&\equiv \frac{|k_6 M|}{\pi R_1R_2} \label{a}\end{aligned}$$ The general solution is then given in terms of Hermite functions $\psi_n(x)$ as[^17] $$\begin{aligned} \label{wfunc1} f_{k_1,k_3,k_6}(\dot x^2)&=\psi_{\nu-\alpha-\oh}\left(\frac{\dot x^2}{\sqrt{2}}\right)\\ f_{k_3,k_4,k_6}(\dot x^5)&=\psi_{\a -\oh}\left(\frac{\dot x^5}{\sqrt{2}}\right) \label{wfunc2}\end{aligned}$$ where $$\psi_n(x) \equiv \frac{1}{\sqrt{n!2^n\pi^{1/2}}}e^{-x^2/2}H_n(x),\label{hermite}$$ and $H_n(x)$ stands for the Hermite polynomial of degree $n$. Note that this requires that the Hermite functions in (\[wfunc1\]) and (\[wfunc2\]) have subindices $\nu - \a -1/2,\ \a - 1/2 \in \IN$ and, in particular, that $\nu -1 = n \in \IN$. This turns out to fix the mass eigenvalues, obtaining the following KK mass spectrum $$m_B^2=\frac{|k_6 M|}{\pi R_1R_2}(n+1)+ \left(\frac{k_6}{R_6}\right)^2+\left(\frac{k_3}{R_3}\right)^2 \label{eigen}$$ Plugging back these solutions into (\[sep\]) and (\[ansz\]) and defining $k_{1,4}=\delta_{1,4}+k_6Ms_{1,4}$ with $s_i \in \IZ$, we obtain the set of eigenfunctions $$\begin{gathered} B^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}\, =\, \cn_B \sum_{s_1,s_4}\psi_{n-k}\left(\frac{\dot x^2}{\sqrt{2}}\right)\ \psi_{k}\left(\frac{\dot x^5}{\sqrt{2}}\right) \, e^{2\pi i \left[(\delta_{1}+k_6Ms_{1}) x^1+k_3 x^3+(\delta_{4}+k_6Ms_{4}) x^4+k_6\dot{x}^6\right]} \\ \cn_B\, =\, \left(\frac{2\pi |k_6M|}{\textrm{Vol}_{\cam_6}}\frac{R_5}{R_1}\right)^{1/2} \quad \quad \quad \textrm{Vol}_{\cam_6}=\prod_{i=1}^6 (2\pi R_i) \quad \quad \quad \quad \quad \quad \label{set1}\end{gathered}$$ where the indices run as $k=\a -1/2 = 0,\ldots, n$ and $\delta_{1,4}=0, \ldots, k_6M-1$. As in [@yukawa], the fact that different choices of $\d_1, \d_4$ give independent wavefunctions is related to the recurrence relation (\[period\]). Finally, the normalization has been fixed so that $$\langle {B}^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}\ , \ B^{(k',\delta_1',\delta_4')}_{n',k_3',k_6'}\rangle =\prod_{i=n,k,k_3,k_6,\delta_1,\delta_4}\delta_{ii'}$$ where $\langle\, , \rangle$ stands for the usual inner product of complex functions. Besides the set of wavefunctions (\[set1\]) there is a different family of solutions to (\[lap\]). Indeed, simple inspection shows that these are given by $$\begin{aligned} &B_{k_1,k_2,k_3,k_4,k_5}(\vec x)=\textrm{exp} [2\pi i(k_1x^1+k_2x^2+k_3x^3+k_4x^4+k_5x^5)] \label{set2}\\ &m_B^2=\sum_{i=1}^5\left(\frac{k_i}{R_i}\right)^2\end{aligned}$$ so that, in terms of the ansatz (\[ansz\]), correspond to the choice $k_6 = 0$. We then find that there are two families of Kaluza-Klein excitations for each 4d massless gauge boson, and that KK modes enter in one family or the other depending on whether they have KK momentum along the fiber coordinate $x^6$ or not. The spectrum of KK modes which are not excited along $x^6$, given by the wavefunctions (\[set2\]), is the same than we would find in an ordinary $T^5$. On the other hand, Kaluza-Klein modes excited along $x^6$, given by the wavefunctions (\[set1\]), present an interesting Landau degeneracy. For each energy level there are exactly $(k_6M)^2(n+1)$ degenerate modes, labeled by the triplet $(k,\delta_1,\delta_4)$. We have represented in figure \[fig0\] the resulting spectrum of particles associated to the gauge boson, in the regime $R_1R_2\gg M R_6$, and we have compared with the spectrum resulting in the fluxless case. As discussed in Section \[subsec:twisted\], in this regime the KK excitations along the base are much lighter than the excitations along the fiber, and the mass scale induced by the fluxes is much smaller than any KK scale. Hence, in analogy with standard type IIB flux compactifications with large volumes and diluted fluxes, the effect of the flux can be understood as a perturbation from the fluxless toroidal setup. ![\[fig0\] Spectra of massive gauge bosons in a fluxless toroidal compactification (left) and in the fluxed example at hand (right), in the regime $R_1R_2\gg M R_6$. The mass scale introduced by the fluxes is given by $\varepsilon = M R_6/\pi R_1R_2$.](espectroscopia2.eps){width="9cm"} Note that, even if we consider diluted fluxes, there are some qualitative differences in the KK open string spectrum with respect to the fluxless case. In particular, for $k_6\neq 0$ the masses of all the excitations along the base $B_4$ scale linearly with respect to their KK quantum numbers, whereas in the toroidal case these scale quadratically. In addition, the wavefunctions $|B^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}|^2$ have a non-constant profile only along two directions, $x^1$ and $x^4$, as depicted in figure \[fig1\] for the first energy levels, reflecting the localization (independently of the warping) of these Kaluza-Klein modes along those directions. Note that the localization of Kaluza-Klein excitations may affect in an interesting way the effective supergravity description, leading to suppressions in the couplings of these modes to the low energy effective theory. ![\[fig1\] $|B^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}|^2$ for $k=0,1$, $k_6M=0,1,2$, $n=k$ and arbitrary $\delta_1, \delta_4,$ and $k_3$, in the plane $x^i=0, \ \ i=3\ldots 6$. The normalization has been left unfixed.](gauginossmall.eps){width="14cm"} Interestingly, the family of wavefunctions (\[set1\]) can be easily understood in terms of ordinary theta functions as follows. First note that for $n=0$ and $k_6M >0$ we have $$\begin{gathered} B^{(0,\delta_1,\delta_4)}_{0,k_3,k_6}= \left(\frac{2 |k_6M|R_5} {R_1\ \textrm{Vol}_{\cam_6}}\right)^{1/2} \vartheta\left[{-\frac{\delta_1}{k_6M} \atop 0}\right]\left( k_6M \tilde z_1;\ k_6M\tilde \tau_1\right)\ \vartheta\left[{-\frac{\delta_4}{k_6M} \atop 0}\right]\left(k_6M \tilde z_2;\ k_6M \tilde \tau_2\right) \\ \times \textrm{exp}\left[i\pi k_6M\left(\frac{\tilde z_1\textrm{Im }\tilde z_1}{\textrm{Im }\tilde \tau_1}+ \frac{\tilde z_2\textrm{Im }\tilde z_2}{\textrm{Im }\tilde \tau_2}\right)\right] \textrm{exp}\left[2\pi i(k_6x^6+k_3x^3)\right] \label{thetawave}\end{gathered}$$ where we have defined a non-standard complex structure [ccc]{} z\_1=x\^1+\_1x\^2& & \_1=iR\_2/R\_1\ z\_2=x\^4+\_2x\^5 & & \_2=iR\_5/R\_4 \[zeff\] The higher energy levels corresponding to $n>0$ can then be built by acting with the following raising operators $$a^\dagger_1\equiv \hat\partial_1-i\hat\partial_2 \qquad a^\dagger_2\equiv \hat\partial_4-i\hat\partial_5 \label{raising}$$ which act on the wavefunctions (\[set1\]) as $$\begin{aligned} &a^\dagger_1B^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}= i\sqrt{\frac{k_6 M(n-k+1)}{\pi R_1R_2}}B^{(k,\delta_1,\delta_4)}_{n+1,k_3,k_6} \\ &a^\dagger_2B^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}= i\sqrt{\frac{k_6 M(k+1)}{\pi R_1R_2}}B^{(k+1,\delta_1,\delta_4)}_{n+1,k_3,k_6}\end{aligned}$$ Similarly, for $k_6M < 0$ we should complex conjugate $\tilde z_k, \tilde \tau_k$ in (\[thetawave\]) and $a^\dagger_k$ in (\[raising\]). Note that the kind of wavefunctions (\[thetawave\]) are precisely those arising from open string zero modes charged under a constant $U(1)$ field strength $F_2$ in toroidal magnetized compactifications [@yukawa]. This was indeed expected, as nilmanifolds $\G\backslash\ch_{2p+1}$ based on the Heisenberg manifold are standard examples of $S^1\simeq U(1)$ bundles, and so both kind of wavefunctions can be understood mathematically in terms of sections of the same vector bundle. It is amusing, however, to note that the physical origin of the bundle geometry is quite different for these two cases. Indeed, while in [@yukawa] the bundle arises from an open string flux $F_2$ and the $U(1)$ fiber is not a physical dimension, in the present case the bundle geometry is sourced entirely form closed string fluxes, and all the coordinates of the fibration correspond to the background geometry. This multiple interpretation of the wavefunctions (\[thetawave\]) could presumably be understood as a particular case of open/closed string duality, where the closed string background (\[bg1\]) is dual to a background of magnetized D9-branes. More precisely, one can build a dictionary between both classes of backgrounds as ------------------- ------------------- ------------ $e^6$ $\leftrightarrow$ $A$ $x^6$ $\leftrightarrow$ $\Lambda$ $F_3^{\text{cl}}$ $\leftrightarrow$ $\omega_3$ ------------------- ------------------- ------------ where $F_3 = F_3^{\text{cl}} + \om_3$, $\omega_3$ is the Chern-Simons 3-form for the open string gauge bundle and $\Lambda$ the gauge transformation parameter. To finish our discussion let us comment on the uniqueness of the above solutions. Note in particular that the change of variables in (\[change\]) is not unique, and one can check that taking different choices for $\dot{x}^6$ leads to wavefunctions that are localized along different directions. Again, this fact is not totally unexpected, since similar effects occur in the context of magnetized D-branes in toroidal compactifications [@yukawa]. Let us then consider the following change of coordinates $${\dot x}^6\equiv x^6 + \frac{M}{2}(\epsilon_ax^1x^2+\epsilon_bx^4x^5) \label{xgen}$$ with $\epsilon_a, \ \epsilon_b=\pm 1$. From a group theoretical point of view, this choice of signs are nothing but the four possible [manifold polarizations]{}[^18] of the 5-dimensional Heisenberg group $\ch_5$. Proceeding as we did in the previous sections, we obtain the following set of wavefunctions $$\begin{gathered} B^{(k,\delta_a,\delta_b)_{\epsilon_a,\epsilon_b}}_{n,k_3,k_6}= \cn_{\eps_a\eps_b}\sum_{s_a, s_b\in\mathbb{Z}}\psi_{n-k}\left(\frac{\dot{x}^a}{\sqrt{2}}\right)\ \psi_{k}\left(\frac{\dot{x}^b}{\sqrt{2}}\right) \, e^{2\pi i \left((\delta_a+k_6Ms_a) x^a+k_3 x^3+(\delta_b+k_6Ms_b) x^b+k_6\dot x^6\right)} \\ \cn_{\eps_a\eps_b} \, =\, \left(\frac{2\pi |k_6M|}{\textrm{Vol}_{\cam_6}}\frac{R_1R_2}{R_aR_b}\right)^{1/2} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \label{generic}\end{gathered}$$ with $$\begin{aligned} \dot{x}^a&\equiv\begin{cases}\frac{2}{R_a} a^{-1/2}(\delta_a+k_6M(x^2+s_a)) & \textrm{for } \epsilon_a=+1\\ \frac{2}{R_a} a^{-1/2}(\delta_a-k_6M(x^1-s_a)) & \textrm{for } \epsilon_a=-1 \end{cases}\\ \dot{x}^b&\equiv\begin{cases}\frac{2}{R_b} a^{-1/2}(\delta_b+k_6M(x^5+s_b)) & \textrm{for } \epsilon_b=+1 \\ \frac{2}{R_b} a^{-1/2}(\delta_b-k_6M(x^4-s_b)) & \textrm{for } \epsilon_b=-1 \end{cases}\end{aligned}$$ $$x^a\equiv \begin{cases}x^1 &\textrm{for } \epsilon_a=+1\\ x^2 &\textrm{for } \epsilon_a=-1\end{cases} \qquad x^b\equiv \begin{cases}x^4 &\textrm{for } \epsilon_b=+1\\ x^5 &\textrm{for } \epsilon_b=-1\end{cases} \label{xab}$$ and an analogous definition to (\[xab\]) for $R_{a,b}$. Note that $\epsilon_a=+1\ (-1)$ leads to wavefunctions localized in $x^1\ (x^2)$, whereas $\epsilon_b=+1\ (-1)$ leads to wavefunctions localized in $x^4\ (x^5)$. Each choice of polarization, however, leads to a complete set of wavefunctions. Therefore any wavefunction within a given polarization can be expressed as a linear combination of wavefunctions in a different polarization through a discrete Fourier transform [@yukawa]. See Appendix \[kirillov\] for a more general, formal presentation of manifold polarizations for the case of nilmanifolds. Laplace-Beltrami operators for group manifolds {#gener} ---------------------------------------------- When finding solutions to the equation (\[lap\]) in our previous example, a key ingredient was to impose $\G$-invariance via the ansatz (\[ansz\]). While such ansatz is easy to guess either from the identifications (\[reli\]) or from the magnetized D-brane literature, it is a priori not obvious how to formulate such an ansatz for arbitrary twisted tori. In the following we would like to systematize the procedure above and generalize it to solve the Laplace-Beltrami equation in arbitrary manifolds of the form $\cam_6 = \G\backslash G$. As we will see, the method described below not only leads automatically to the two families of KK towers (\[set1\]) and (\[set2\]) that we found for $\cam_6 = \G\backslash(\IR \times \ch_5)$, but also gives a simple group theoretical understanding of their existence in terms of the irreducible representations of $\IR \times \ch_5$. In fact, the relation between families of KK modes on $\cam_6 = \G\backslash G$ and irreducible representations of a group $G$ can be traced back to the mathematical literature that analyzes the spectrum of Laplace-Beltrami operators in group manifolds. Particularly useful for our purposes will be the tools developed in the context of non-commutative harmonic analysis (see e.g. [@taylor; @heis]), a field aiming to extend the results of Fourier analysis to non-commutative topological groups. In order to motivate this approach let us first consider the Laplace eigenvalue problem in the Abelian case $\cam_n = \IZ^n \backslash \IR^n = T^n$. Here the twisted derivative operators $\hat\partial_m$ are nothing but ordinary derivatives, so (\[lap\]) reduces to $$\partial_{x^i}\partial^{x^i}B=-m_B^2B \label{laptorito}$$ and the underlying algebra of isometries is Abelian. A standard approach to solve this Laplace equation is to apply Fourier analysis. More precisely, we can apply the Fourier transform $$\hat{f}_{\vec \omega}=\int_{\mathbb{R}^n}B(\vec x)e^{i\vec\omega\cdot\vec x}d\vec x \label{fourierTn}$$ to rewrite (\[laptorito\]) in the dual space of momenta. We then obtain $$\int_{\IR^n} |\vec \omega|^2\hat{f}_{\vec \omega}\,=\, \int_{\IR^n}m_B^2\hat{f}_{\vec \omega}$$ which easily gives $\hat{f}_{\vec \omega} = \d(\vec \omega - \vec \omega_0)$ and $|\vec \omega_0|^2 = m_B^2$. Hence, the eigenfunctions of the Laplace operator correspond to Kaluza-Klein excitations with constant momentum of norm $m_B$. Applying the inverse Fourier transform we find that these are given by $B_{\vec \omega}(\vec x)=e^{-i\vec \omega_0\cdot\vec x}$. The eigenfunctions of $\Delta$ are then nothing but the irreducible unitary representations $e^{i\vec\omega\cdot\vec x}$ of the group $\IR^n$, which are also the “coefficients" entering the Fourier transform (\[fourierTn\]). Finally, imposing invariance under the compactification lattice $\G = \IZ^n$ restricts $\vec \omega$ to the dual sublattice $2\pi \IZ$. So one interesting observation that we can extract from this example is that the irreducible unitary representations $\pi_{\vec \omega}(\vec x) = e^{i\vec\omega\cdot\vec x}$ of the Abelian group $G= \IR^n$ correspond to the eigenfunctions of the Laplace operator. In particular, those which are invariant under the subgroup $\G = \IZ^n$ are well-defined in the compact quotient $\G\backslash G$, and so describe the KK wavefunctions of $T^n$. Naively, we would expect that some sort of analogous statement can be made for $G$ a non-Abelian group. Again, a good starting point is to consider the non-commutative version of (\[fourierTn\]),[^19] which reads [@taylor; @heis] $$\hat f_{\vec \omega}\, \varphi(\vec s)=\int_{G}B(g)\pi_{\vec \omega}(g)\varphi(\vec s)dg \label{fourier}$$ with $\pi_{\vec \omega}(g)$ a complete set of inequivalent irreducible unitary representations of $G$. An important difference with respect to the Abelian case is that the irreducible representations $\pi_{\vec \omega}(g)$ are no longer simple functions, but rather operators acting on a Hilbert space of functions, $\varphi(\vec s)\in L^2(\mathbb{R}^{p(\pi)})$ with $p(\pi)\in \mathbb{N}$, and so is $\hat f_{\vec \omega}$. Remarkably, the set $\pi_{\vec \omega}$ can be computed systematically by means of the so-called orbit method, mainly developed by A. Kirillov [@kirillovv], and which we briefly summarize in Appendix \[kirillov\]. In principle, one could follow the standard strategy of the Abelian case and make use of (\[fourier\]) to write down eq.(\[lap\]) in the space of momenta, and then apply the inverse Fourier transform to obtain our wavefunction $B$. An alternative approach, which we will adopt here, is to start with an educated ansatz for $\G$-invariant wavefunctions, based on the close relation between Laplace-Beltrami eigenfunctions and unitary irreps of $G$. Indeed, consider a complex valued function $B_{\vec \omega} :G \raw \IC$, defined as B\^[, ]{}\_ (g) = ( \_(g) , ) \_[\^[p()]{}]{} |(s) ds where $(\, , \, )$ is the usual $L^2(\IC^{p(\pi)})$ norm. If $\cl$ is a differential operator acting on the space of wavefunctions $L^2(G)$ that can be expressed as a polynomial $P(\{\mathfrak{t}_a\})$ of the algebra generators, then it is easy to see that ( \_(g) , ) = ( \_(g) \_() , ) where $\pi_{\vec \omega}(\cl)$ is defined in the obvious way [@taylor; @heis]. Hence, finding eigenfunctions of $\cl$ reduces to finding eigenfunctions of $\pi_{\vec \omega}(\cl)$ in the auxiliary space $L^2(\mathbb{R}^{p(\pi)})$, since $\pi_{\vec \omega}(\cl) \vphi = \lam \vphi \Raw \cl B^{\vphi, \psi}_{\vec \omega} = \lam B^{\vphi, \psi}_{\vec \omega}$. Note that this is independent of our choice of $\psi$, which we can take to be, e.g., a delta function $\delta(\vec s - \vec s_0)$. A suitable set of eigenfunctions of $\cl$ is then given by B\^[\_]{}\_ (g) = \_(g) \_(s\_0) \[ansatzLB1\] where $\vphi_\a$ is an eigenfunction of $\pi_{\vec \omega}(\cl)$. In particular, this result applies to the Laplace-Beltrami operator $\Delta$, which can be written as a quadratic form on $\{\mathfrak{t}_a\}$. Hence, (\[ansatzLB1\]) provides a clear correspondence between unitary irreps of $G$ and families of eigenfunctions of its Laplace-Beltrami operator. As stressed before, we also need to impose that our wavefunctions are well-defined in the quotient space $\cam = \G \backslash G$. A simple way to proceed is to consider the sum B\_ (g) = \_ \_(g) (s\_0) \_\^(g) (s\_0) \[ansatzLB2\] keeping only the wavefunctions $B_{\vec \omega}$ belonging to $L^2(\cam)$.[^20] Again, if $\vphi$ is an eigenfunction of $\pi_{\vec \omega}(\cl)$ then (\[ansatzLB2\]) is automatically an eigenfunction of $\Delta$. Alternatively, one may consider $\vphi$ an unknown function and the expression (\[ansatzLB2\]) an educated ansatz to be plugged into the Laplace-Beltrami equation (\[lap\]). In order to illustrate how this ansatz works, let us again consider the $(2p+1)$ dimensional Heisenberg manifold $\mathcal{H}_{2p+1}$, discussed in Section \[subsec:twisted\]. The Stone-von Neumann theorem [@taylor; @heis] states that the irreducible unitary representations for $\mathcal{H}_{2p+1}$ are given by two inequivalent sets[^21] $$\begin{aligned} &\pi_{k_z^\prime} (\vec X)\, u(\vec s)\,=\, e^{2\pi ik_z^\prime[z + \vec x\cdot\vec y/2 + \vec y\cdot \vec s ]}\, u(\vec s+\vec x)\ & u(\vec s)&\in L^2(\mathbb{R}^p) \label{rep1}\\ &\pi_{\vec k_x^\prime,\vec k_y^\prime} (\vec X) \,=\, e^{2\pi i(\vec k_x^\prime\cdot \vec x\ +\ \vec k_y^\prime\cdot\vec y)}& & \label{rep2}\end{aligned}$$ where we are taking the same parameterization $\vec X^t = (z, \vec x ^{\, t}, \vec y^{\, t})$ of $\mathcal{H}_{2p+1}$ as in (\[heis\]). Considering the cocompact subgroup $\G_{\ch_{2p+1}} = \{(\vec x,\vec y,z) = M (\vec{n}_x, \vec{n}_y, n_z) \in M\IZ^{2p+1}\}$, $M \in 2\IZ$, and the $\G_{\ch_{2p+1}}$-invariant representations $\pi^\G$ we obtain $$\begin{aligned} &\pi_{k_z}^\G (\vec X)\, u(\vec s)= \hspace*{-.35cm}\sum_{\vec s_x, \vec s_y \in \IZ^p} \hspace*{-.35cm} e^{2\pi i k_z [z + \frac{M}{2} \vec x \cdot \vec y + (\vec y + \vec s_y)\cdot (\vec s + M\vec s_x)]} \, u(\vec s+ M(\vec s_x + \vec x)) \quad \quad \quad k_z\in \mathbb{Z} \label{rep1inv}\\ &\pi_{\vec k_x,\vec k_y}^\G (\vec X) = e^{2\pi i(\vec k_x\cdot \vec x\ +\ \vec k_y\cdot\vec y)} \hspace*{7.25cm} \vec k_x, \vec k_y \in \IZ^p \label{rep2inv}\end{aligned}$$ where as before we have normalized the generators of the algebra as $\tilde{\mathfrak{t}}_\a = M\mathfrak{t}_\a$, and in addition we have relabeled the unirreps as $\vec k_a = M \vec k_a^\prime$, $a = x, y, z$. An interesting effect of considering the invariant unirreps $\pi^\G$ is that the allowed choices for $\vec s \in \IR^p$ become discrete. Indeed, note that (\[rep1inv\]) vanishes unless $k_z \vec s \in \IZ^p$, and that if we impose the latter condition we no longer need to sum over $\vec s_y$ to produce an invariant unirrep. Hence, we can identify our set of $\G$-invariant unirreps producing our ansatz (\[ansatzLB2\]) as $$\begin{aligned} &\pi_{k_z}^\G (\vec X)\, \vphi_{\vec \d}= \hspace*{-.2cm} \sum_{\vec s_x \in \IZ^p} e^{2\pi i k_z \left(z + \frac{M}{2} \vec x \cdot \vec y\right)} e^{2\pi i \left(\vec y \cdot (\vec \d + k_zM\vec s_x)\right)} \vphi (\vec \d + k_zM(\vec s_x + \vec x)) \quad \quad k_z\in \mathbb{Z} \label{rep1invb}\\ &\pi_{\vec k_x,\vec k_y}^\G (\vec X) = e^{2\pi i(\vec k_x\cdot \vec x\ +\ \vec k_y\cdot\vec y)} \hspace*{7.25cm} \vec k_x, \vec k_y \in \IZ^p \label{rep2invb}\end{aligned}$$ where $\vphi(\vec s) = u(k_z^{-1} \vec s)$ and $\vec \d \in \IZ^p$. Note that because of the sum over $\vec s_x$, for fixed $k_z$ there are only $|k_z M|^p$ independent choices of $\vec \d$ that we can take. Moreover, all these choices can be related via a redefinition of $\vec x$, so if we find a solution to the Laplace equation via the ansatz (\[rep1invb\]) in general we will have $|k_z M|^p$ independent solutions. To be more concrete, let us go back to the twisted torus example of subsection \[vmu\]. Recall that there the internal geometry is given by $\cam_6 = \G \backslash (\IR \times \ch_5) = S^1 \times \G_{\ch_5}\backslash \ch_5$, and that in (\[rep1\]) and (\[rep2\]) we should take $p=2$ and identify $z\equiv x^6$, $\vec x\equiv(x^2,x^5)$ and $\vec y\equiv(x^1,x^4)$. The ansatz (\[ansatzLB2\]) then amounts to take the invariant unirreps (\[rep1invb\]) and (\[rep2invb\]) with the same identifications, and tensored with the unitary irreps of $S^1\simeq U(1)$, given by $e^{2\pi i k_3x^3}$. More precisely we obtain $$\begin{aligned} &B^{(\delta_1,\delta_4)}_{k_3,k_6}(\vec x)\, =\, \sum_{k_1,k_4} \vphi \left(k_1+k_6Mx^2, k_4+k_6Mx^5\right) e^{2\pi i \left( k_1 x^1 + k_3 x^3 + k_4 x^4 + k_6\dot x^6\right)} \label{nili1} \\ & \hspace*{5cm} k_i = \d_i + k_6 M s_i \quad \quad \quad n_i \in \IZ \nonumber\\ &B_{k_1,k_2,k_3,k_4,k_5}(\vec x)=\textrm{exp}[2\pi i(k_1x^1+k_2x^2+k_3x^3+k_4x^4+k_5x^5)] \label{nili2}\end{aligned}$$ with $\vphi(x,y)$ a function to be determined. Eq.(\[nili1\]) is indeed the ansatz considered in eq.(\[ansz\]), while (\[nili2\]) gives the set of wavefunctions (\[set2\]) obtained by inspection. Finally, plugging (\[nili1\]) into (\[lap\]), directly leads to $\vphi(x,y)=\psi_k(\mu_1x)\psi_{n-k}(\mu_2y)$, with $\mu_{1}^2=4\pi k_6 R_{2}/R_{1}$ and $\mu_{2}^2=4\pi k_6 R_{5}/R_{4}$ reproducing the results of the previous section. As promised, the ansatz (\[ansatzLB2\]) gives a direct relation between families of KK modes on $\cam_6 = \G\backslash G$ and invariant unirreps of $G$. In this respect, note that the inequivalent unirreps of $G = {\rm exp\, } \mathfrak{g}$ can be extracted from its Lie algebra $\mathfrak{g}$, given by (\[torsion\]). Now, from the 4d effective theory point of view $\mathfrak{g}$ is nothing but the 4d gauge algebra resulting from dimensional reduction of the 10d metric [@kaloper]. Hence, we can establish a correspondence between inequivalent unirreps of the 4d gauged isometry algebra and families of eigenfunctions of the internal Laplace-Beltrami operator. Note also that $\mathfrak{g}$ is only part of the full 4d $\mathcal{N}=4$ gauged supergravity algebra, as there are further gauge symmetries arising from dimensional reduction of the 10d $p$-forms. As we will argue below, by making use of the global $SL(2)\times SO(6,6+n)$ symmetry one should be able to extend such correspondence to the full 4d gauged algebra and the full set of massive modes of the untwisted D9-brane sector. Non-vanishing $\mu$-terms {#conmu} ------------------------- Let us now apply the ansatz (\[ansatzLB2\]) to a more involved background, namely the twisted torus compactification with flux-generated $\mu$-terms of subsection \[nvmu\]. Again, the wavefunctions for the 4d gauge boson are given by the eigenfunctions of the Laplace-Beltrami operator $\Delta$, and more precisely by the solutions to eq.(\[lap\]), with the twisted derivatives given by (\[hatted1\]). As before, the first step of the ansatz is to find the set of inequivalent unirreps of the Lie group $G$. This can be done via the orbit method, as shown in Appendix \[kirillov\]. We then find four families of irreducible unitary representations associated to the Lie algebra defined by eq.(\[nil2\]), given in eqs.(\[pi1\])-(\[pi6\]). As a second step, we need to impose $\G$-invariance on these unirreps. For this purpose it is useful to introduce the variables $$\dot x^3=x^3-\frac{M_3}{2}x^1x^2 \quad \text{and} \quad \dot x^6=x^6-\frac{M_6}{2}x^5x^1$$ so that the action of $\G$, given by (\[reli\]), now reads $$x^1\to x^1+1 \quad \quad \dot x^3\to \dot x^3-M_3x^2 \quad \quad \dot x^6 \to \dot x^6 - M_6x^5 \label{lat}$$ with all the other coordinates being periodic, $x^i\to x^i+1$ for $i=2,4,5$, and $\dot x^i \to \dot x^i+1$ for $i=3,6$. Imposing invariance of (\[pi1\])-(\[pi6\]) under (\[lat\]) and plugging the result into (\[lap\]), leads to the following $6\times 4=24$ towers of KK gauge boson wavefunctions:\ \ These are given by standard toroidal wavefunctions in the base $$\begin{aligned} B_{k_1,k_2,k_4,k_5}=e^{2\pi i(k_1 x^1+k_2x^2+k_4x^4+k_5x^5)} \label{mod1}\end{aligned}$$ with mass eigenvalue $$m_B^2=\sum_{a=1,2,4,5}\left(\frac{k_a}{R_a}\right)^2\label{mod1m}$$ In particular, this includes the massless gauge boson.\ \ Their wavefunction is given by B\^[()]{}\_[k\_r,k\_4,k\_[8-r]{},n]{} = \_[s]{}\_n() e\^[2i(k\_rx\^r+k\_4x\^4+k\_[8-r]{}x\^[8-r]{}+(+sk\_rM\_r)x\^[r-1]{})]{} \[rr\] with $\delta=0\ldots k_rM_r-1$ and where $\varepsilon_\mu \equiv {M_3R_3}/{2\pi R_1R_2}$ stands for the mass scale of the flux $$\cn = \left(\frac{2\pi R_1}{\textrm{Vol}_{\cam_6}}\sqrt{\frac{|k_r\varepsilon_\mu|}{R_r}}\right)^{1/2} \quad \quad \dot x^1\equiv 2\pi R_1 \left(\frac{2|\varepsilon_\mu k_r|}{R_r}\right)^{1/2}\left(x^1-s-\frac{\delta}{k_rM_r}\right)$$ The corresponding mass eigenvalues are $$m_B^2=\frac{|\varepsilon_\mu k_r|}{R_r}(2n+1)\, +\sum_{a=r,4,8-r}\left(\frac{k_a}{R_a}\right)^2\label{rrm}$$ \ The wavefunctions for these modes are B\^[(\_2,\_5)]{}\_[k\_3,k\_4,k\_6,n]{}= \_[s]{}\_n() e\^[2i]{} \[mod2\] with $$\cn = \left(\frac{2\pi R_1\sqrt{\Delta_{k_3,k_6}|\varepsilon_\mu|}}{\textrm{Vol}_{\cam_6}}\right)^{1/2} \quad \quad \dot x^1\equiv 2\pi(2\Delta_{k_3,k_6}|\varepsilon_\mu|)^{1/2}R_1\left(x^1-s-\frac{\delta_2}{k_3M_3}\right)$$ \_[k\_3,k\_6]{}\^2()\^2+()\^2 \[delta\] and where $\delta_2,\delta_5\in \mathbb{Z}$ are related through the constraint $k_6\delta_2M_6=\delta_5k_3M_3$. Finally, the mass eigenvalues are $$m_B^2=\Delta_{k_3,k_6}^2+\left(\frac{k_4}{R_4}\right)^2+|\varepsilon_\mu|\Delta_{k_3,k_6}(2n+1) \label{mod2m}$$ Scalar wavefunctions {#sec:scalars} ==================== In this section we proceed with the computation of the wavefunctions for the 4d scalar modes transforming in the adjoint representation of the gauge group $G_{unbr}$. These modes arise from the term $b^m (x^\mu) \xi^m(x^i)$ in the dimensional reduction (\[splitboson\]) of the 10d gauge boson, so they can be thought as Wilson line moduli of the compactification plus their KK replicas. Note that the choice of expansion (\[splitboson\]) in terms of the left-invariant 1-forms $e^m$ indeed simplifies the computation of the wavefunctions $\xi^m(x^i)$ which, just as the previous gauge boson wavefunction $B(x^i)$, are invariant under the action of the subgroup $\G$ in $\cam_6 = \G \backslash G$. In fact, we will see that having computed the spectrum of $B(x^i)$’s, the computation of $\xi^m(x^i)$’s reduces to a purely algebraic problem. This problem is easily solved in the case of our compactification with vanishing $\mu$-terms, since it basically amounts to diagonalize a $3 \times 3$ matrix with commuting entries. The case with non-vanishing $\mu$-term, on the other hand, turns out to be more involved, as the entries of this $3 \times 3$ matrix become non-commutative.[^22] Vanishing $\mu$-terms {#vanish} --------------------- As discussed in Section \[subsec:elliptic\], for elliptic fibrations of the form (\[mansatz\]) the internal profiles $\xi^p_{\Pi_2,B_4}$ of the 4d scalars in the adjoint of $G_{unbr}$ are real functions satisfying eqs.(\[xi+\]) and (\[xi-\]). These equations of motion can be summarized in matrix notation as $$\left[\mathbb{M}+m_b^2\ \mathbb{I}_{6}\right]\mathbb{V}=0 \label{eigenval}$$ where = \^1\ \^2\ \^3\ \^[\* 1]{}\ \^[\* 2]{}\ \^[\* 3]{} [c]{} \^1 \_[B\_4]{}\^1+i\_[B\_4]{}\^4\ \^2 \_[B\_4]{}\^2+i\_[B\_4]{}\^5\ \^3 \_[\_2]{}\^3+i\_[\_2]{}\^6 [c]{} \^[\* 1]{} \_[B\_4]{}\^1-i\_[B\_4]{}\^4\ \^[\* 2]{} \_[B\_4]{}\^2-i\_[B\_4]{}\^5\ \^[\* 3]{} \_[\_2]{}\^3-i\_[\_2]{}\^6 \[standcom\] and the matrix $\mathbb{M}$ has as entries differential operators whose general expression is given in Appendix \[ap:matrix\]. For the type I vacuum of subsection \[vmu\], one can check that $\mathbb{M}$ reduces to $$\mathbb{M}\,=\, \begin{pmatrix} \hat\partial_m\hat\partial^m& -\varepsilon\hat\partial_6&0&0&0&0\\ \varepsilon\hat\partial_6&\hat\partial_m\hat\partial^m&0&0&0&0\\ 0&0&\hat\partial_m\hat\partial^m&0&0&0\\ 0&0&0&\hat\partial_m\hat\partial^m&-\varepsilon\hat\partial_6&0\\ 0&0&0&\varepsilon\hat\partial_6&\hat\partial_m\hat\partial^m&0\\ 0&0&0&0&0&\hat\partial_m\hat\partial^m \end{pmatrix} \label{system1}$$ where as before $\varepsilon = M R_6/\pi R_1R_2$ is the flux scale. Note that all the entries of the matrix $\mathbb{M}$ commute, and so (\[eigenval\]) can be treated as an ordinary eigenvalue problem. Moreover, $\mathbb{M}$ is block diagonal, with no entries mixing holomorphic and antiholomorphic states. This can be traced back to the fact that our compactification manifold $\cam_6$ is complex, as required by $\mathcal{N}=1$ supersymmetry. Therefore, it is enough to solve for one of the $3\times 3$ blocks in (\[system1\]). In order to do so let us distinguish again between states which are excited along the fiber coordinate $x^6$ and states which are not excited along it. For the latter the wavefunction should not depend on $x^6$, and so they are annihilated by $\hat \partial_6$. Therefore, for those states $\mathbb{M}$ is proportional to the Laplace-Beltrami operator, whose eigenvalues were solved for in Section \[nili\]. It is then straightforward to verify that the wavefunctions associated with these modes are given by the same functions $B_{k_1,k_2,k_3,k_4,k_5}$ defined in equation (\[set2\]). Similarly, the mass eigenvalues are $$m_\xi^2=\sum_{i=a}^5\left(\frac{k_a}{R_a}\right)^2\nonumber$$ so the wavefunction $B_{0,0,0,0,0} = const.$ corresponds to the six real Wilson line moduli. On the other hand, $\hat \partial_6$ does not act trivially on modes excited along the fiber, as they depend on $x^6$. Note however that $\hat\partial_6$ belongs to the center of the Lie algebra $\mathfrak{g}$ of our twisted torus $\cam_6 = \G \backslash G$. Hence, $\hat \partial_6$ commutes with the Laplace-Beltrami operator $\hat\partial_m\hat\partial^m$, and so they can be simultaneously diagonalized. In fact, it turns out that the family of wavefunctions (\[set1\]) obtained above are not only eigenfunctions of $\hat\partial_m\hat\partial^m$ but also of $\hat \partial_6$, their eigenvalue for the latter being $i k_6/R_6$. This allows to diagonalize the upper $3 \times 3$ block of (\[system1\]) for the fiber KK modes as $$(\xi_\pm)_{n,k_3,k_6}^{(k,\delta_1,\delta_4)}\equiv \begin{pmatrix}1\\ \pm i\\ 0\end{pmatrix}B_{n,k_3,k_6}^{(k,\delta_1,\delta_4)} \label{neutpm}$$ with mass eigenvalue $$m^2_{\xi_{\pm}}\, = \, \frac{|\varepsilon k_6|}{R_6} \left(n + 1 \mp s_{k_6M} \right) +\Delta^2_{k_3,k_6}$$ where $s_{k_6M} = \text{sign}(k_6M)$ and $\Delta_{k_3,k_6}$ is given by (\[delta\]). The effect of the off-diagonal entries in (\[system1\]) is then to shift up or down the mass eigenvalues with respect to the ones computed in Section \[nili\] for the gauge bosons. In figure \[splitlevel\] we have represented the splitting of the Laplace-Beltrami energy levels due to this mass shift effect. The remaining eigenvector is $$(\xi_3)_{n,k_3,k_6}^{(k,\delta_1,\delta_4)}\equiv\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}B_{n,k_3,k_6}^{(k,\delta_1,\delta_4)} \label{extrapol}$$ with mass eigenvalue $$m^2_{\xi_{3}}\, =\, \frac{|\varepsilon k_6|}{R_6} (n+1) + \Delta^2_{k_3,k_6}$$ identical to the KK masses of the corresponding massive gauge boson. In fact, the degrees of freedom coming from (\[extrapol\]) should be seen as the extra polarizations that massive gauge bosons have with respect to massless ones. ![\[splitlevel\] Mass spectra for the complex scalar modes $\xi_3$ and $\xi_{\pm}$ excited along the fiber with same momentum $|k_6|$ in the example with vanishing $\mu$-terms. Continuous red lines relate states with same $n$ and $k_6<0$, whereas dashed blue lines relate states with same $n$ and $k_6>0$. We have labeled the energy levels by $n^{s_{k_6M}}$. The spectrum of gauge boson excitations coincide with the one of $\xi_3$. The flux mass scale is given by $\varepsilon=\frac{MR_6}{\pi R_1R_2}$, whereas $\Delta_{k_3,k_6}^2$ is defined in (\[delta\]). We have also indicated the number of real scalars at each energy level, for fixed $s_{k_6}$, $s_{k_3}$.](espectroscopia.eps){width="14cm"} Putting these results together with the spectrum of gauge bosons computed in Section \[nili\], and the fermionic spectrum (to be computed in next section), one can observe that the content of massive Kaluza-Klein replicas can be arranged into $\mathcal{N}=4$ vector multiplets, except for the levels $k_6\neq 0$, $n=0$, which only fit into ultrashort $\mathcal{N}=2$ hypermultiplets. See Section \[susyspect\] for a more detailed discussion. Non-vanishing $\mu$-terms {#nonvanish} ------------------------- Let us now turn to the type I vacuum of subsection \[nvmu\], where the background induces a non-vanishing mass term for one of the chiral multiplets. The internal profiles of the adjoint scalars must again satisfy the eigenvalue problem (\[eigenval\]), now with $\mathbb{M}$ given by $$\mathbb{M}= \begin{pmatrix} \hat\partial_m\hat\partial^m&-\varepsilon_\mu\hat\partial_{z^3}&-\varepsilon_\mu\hat\partial_{z^2}&0&0&0\\ \varepsilon_\mu\hat\partial_{\bar z^3}&\hat\partial_m\hat\partial^m&\varepsilon_\mu\hat\partial_{z^1}&0&0&0\\ \varepsilon_\mu\hat\partial_{\bar z^2}&-\varepsilon_\mu\hat\partial_{\bar z^1}&\hat\partial_m\hat\partial^m-\varepsilon_\mu^2&0&0&0\\ 0&0&0&\hat\partial_m\hat\partial^m&-\varepsilon_\mu\hat\partial_{\bar z^3}&-\varepsilon_\mu\hat\partial_{\bar z^2}\\ 0&0&0&\varepsilon_\mu\hat\partial_{z^3}&\hat\partial_m\hat\partial^m&\varepsilon_\mu\hat\partial_{\bar z^1}\\ 0&0&0&\varepsilon_\mu\hat\partial_{z^2}&-\varepsilon_\mu\hat\partial_{z^1}&\hat\partial_m\hat\partial^m-\varepsilon_\mu^2 \end{pmatrix} \label{system2}$$ with $\varepsilon_\mu\equiv M_3R_3/2\pi R_1R_2$ and where the complexification $$\hat\partial_{z^k}\equiv \hat\partial_k-i\hat\partial_{k+3} \label{comphat}$$ is related to the standard choice of complex structure $z^k = x^k + i(R_{k+3}/R_k)x^{k+3}$. Note that again the mass matrix is block diagonal, as expected for a 4d SUSY vacuum. We will thus solve (\[eigenval\]) for the upper block and obtain the other eigenfunctions by complex conjugation. An important qualitative difference with the case of vanishing $\mu$-term (\[system1\]), is that the entries of the matrix $\mathbb{M}$ are operators that no longer commute. However, using the following commutation relations $$\begin{aligned} &[\hat{\p}_{z^1}, \hat{\p}_{z^2}] \, =\, [\hat{\p}_{\bar{z}^1}, \hat{\p}_{z^2}]\, =\, - \varepsilon_\mu \hat \p_{z^3} \qquad \quad [\hat{\p}_{z^1}, \hat{\p}_{\bar{z}^2}] \, =\, [\hat{\p}_{\bar{z}^1}, \hat{\p}_{\bar{z}^2}]\, =\, - \varepsilon_\mu \hat \p_{\bar{z}^3} \label{com}\\ &[\hat{\p}_m\hat{\p}^m, \hat{\p}_{z^2}]\, =\,-\varepsilon_\mu\hat{\p}_{z^3}(\hat{\p}_{z^1}+\hat{\p}_{\bar z^1})\qquad [\hat{\p}_m\hat{\p}^m, \hat{\p}_{\bar z^2}]\, =\,-\varepsilon_\mu\hat{\p}_{\bar z^3}(\hat{\p}_{z^1}+\hat{\p}_{\bar z^1}) \nonumber\\ &[\hat{\p}_m\hat{\p}^m, \hat{\p}_{z^1}]\, =\, [\hat{\p}_m\hat{\p}^m, \hat{\p}_{\bar z^1}]\, = \, \varepsilon_\mu\left(\hat{\p}_{\bar z^2}\hat{\p}_{z^3}+\hat{\p}_{z^2}\hat{\p}_{\bar z^3}\right) \nonumber\end{aligned}$$ one can still diagonalize this matrix. Indeed, after some little effort one can check that the above system have a complex eigenvector $$\xi_3 \equiv \begin{pmatrix} \hat\partial_{\bar z^1}\\ \hat\partial_{\bar z^2}\\\hat\partial_{\bar z^3} \end{pmatrix} B(\vec x) \label{muxi3}$$ with mass eigenvalue $m^2_{\xi_3}=m_B^2$, and two complex eigenvectors $$\xi_{\pm}\equiv \begin{pmatrix} \hat\partial_{z^3}\hat\partial_{\bar z^1}+m_{\xi_\pm}\hat\partial_{z^2}\\ \hat\partial_{z^3}\hat\partial_{\bar z^2}-m_{\xi_\pm}\hat\partial_{z^1}\\\hat\partial_{z^3}\hat\partial_{\bar z^3}+m_{\xi_\pm}^2 \end{pmatrix}B(\vec x) \label{muxipm}$$ with mass eigenvalues $$m_{\xi_\pm}^2-\varepsilon_\mu m_{\xi_\pm}-m_B^2=0\quad \Longrightarrow \quad m_{\xi_\pm}^2=\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2+4m_B^2}\right)^2 \label{cuadr}$$ Here $B(\vec x)$ is any of the gauge boson wavefunctions (\[mod1\]), (\[rr\]) or (\[mod2\]) with mass $m_B^2$ given respectively by eqs.(\[mod1m\]), (\[rrm\]) and (\[mod2m\]). Hence, for each Kaluza-Klein boson with mass $m_B^2$, there is one complex scalar with the same mass (eaten by the massive gauge boson via a Higgs mechanism) and two complex scalars whose masses are solutions to the quadratic equation in (\[cuadr\]). Note that for the lowest modes of the neutral gauge boson, $B=\textrm{const.}$, the eigenvector parametrization (\[muxi3\]) and (\[muxipm\]) breaks down, and does not constitute a good representation of the lightest modes for the scalar fields. Instead, these states correspond to the constant eigenvectors $$(\xi_\pm)_{0}\equiv \begin{pmatrix}1\\ \pm i\\ 0\end{pmatrix} \times \text{const.} \qquad (\xi_3)_{0}\equiv\begin{pmatrix}0\\ 0\\ 1 \end{pmatrix}\times \text{const.}$$ with masses $m_{\xi_\pm}^2=0$ and $m_{\xi_3}^2=\varepsilon_\mu^2$, respectively, recovering in this way the low energy effective supergravity result [@geosoft]. We will come back to this point in Section \[sugra\]. Fermionic wavefunctions {#sec:fermions} ======================= Let us now turn to the equation (\[dirac6duw\]) describing the wavefunctions of fermionic eigenmodes. As in the two previous sections, we will consider those modes transforming in the adjoint representation of the unbroken gauge group $G_{unbr}$, computing them explicitly for the two examples of Section \[subsec:twisted\]. In general, for compactifications preserving 4d $\cn=1$ supersymmetry, one expects all those modes belonging to the same supermultiplet to share the same internal wavefunction. This should in particular apply to the two type I vacua examples analyzed above, and so the eigenvalue problem for fermionic modes should reduce to the one already solved in Sections \[sec:wgauge\] and \[sec:scalars\]. We will see that this is indeed the case. Let us however stress that, as our approach treats bosons and fermions independently, the method below could also be applied to type I backgrounds where the flux breaks 4d supersymmetry and so wavefunctions no longer match. An example of such $\cn=0$ flux vacuum is discussed in Appendix \[ap:N=0\], where both classes of open string wavefunctions are computed. Following the conventions of Appendix \[ap:ferm\], we can take our wavefunction as a linear combination of the fermionic basis (\[basisMW\]). Defining the vector = ( [c]{} \^0\ \^ 1\ \^2\ \^3 ) \[fvector\] it is then easy to see that (\[dirac6duw\]) can be expressed as i ([**D**]{} + [**F**]{}) = m\_\^\* \[6db\] where = ( [cccc]{} 0 & \_[[z]{}\^1]{} & \_[[z]{}\^2]{} & \_[[z]{}\^3]{}\ - \_[[z]{}\^1]{} & 0 & - \_[|[z]{}\^3]{} & \_[|[z]{}\^2]{}\ - \_[[z]{}\^2]{} & \_[|[z]{}\^3]{} & 0 & - \_[|[z]{}\^1]{}\ - \_[[z]{}\^3]{} & - \_[|[z]{}\^2]{} & \_[|[z]{}\^1]{} & 0 ) and ${\bf F}$ contains the contribution of the term proportional to $\slashed{f}$ in eq.(\[dirac6duw\]). In particular, we have that ${\bf F} = 0$ for vanishing $\mu$-terms. Eq.(\[6db\]) implies that ([**D**]{} + [**F**]{})\^\* ([**D**]{} + [**F**]{}) = |m\_|\^2 \[6dsq\] which is the fermionic equivalent to (\[eigenval\]). Vanishing $\mu$-terms {#vanishing-mu-terms} --------------------- Let us then consider the internal Dirac equation in the vanishing $\mu$-term background of subsection \[vmu\]. First, given the choice of fibration and the conventions of Appendix \[ap:ferm\], the splitting (\[split4+2\]) reads \_[\_2]{} = \^0 \_[—]{} + \^3 \_[++-]{}\ \_[B\_4]{} = \^1 \_[-++]{} + \^2 \_[+-+]{} Second, recall that the contraction of indices in (\[dirac6d\]) and (\[dirac6duw\]) is performed with the internal gamma matrices in (\[commG\]), which are essentially the 6d matrices in (\[tilgamma\]). Then, the contribution of the geometric flux to the Dirac equation (\[dirac6duw\]) reads = ( + ) = (2)\^[-1]{} ( \^[126]{} + \^[456]{}) where we have used the condition $R_1R_2 = R_4R_5$. In addition we have that \[SUSYf1\] \^[126]{} + \^[456]{} = -i( \_1 \_2 - \_2 \_1) \_2 = (\_[z]{} \_[|[z]{}]{} - \_[|[z]{}]{} \_[z]{}) \_2 where \_z = ( [cc]{} 0 & 2\ 0 & 0 )\_[|[z]{}]{} = ( [cc]{} 0 & 0\ 2 & 0 ) Hence, we see that $\slashed{f} \chi_{\Pi_2} \equiv \slashed{f} P_+^{\Pi_2}\chi = 0$, and so ${\bf F} = 0$, as expected from the fact that in this background no $\mu$-term is generated for D9-brane moduli. In order to solve the squared Dirac equation (\[6dsq\]) we just need to compute the action of ${\bf D}^*{\bf D}$, which in general reads [c]{} -[**D**]{}\^\* [**D**]{} = \_m\^m \_4 +\ \_a ( [cccc]{} (f\_[1|[1]{}]{}\^a + f\_[2|[2]{}]{}\^a + f\_[3|[3]{}]{}\^a) & f\_[|[2]{}|[3]{}]{}\^a & f\_[|[3]{}|[1]{}]{}\^a & f\_[|[1]{}|[2]{}]{}\^a\ f\_[[3]{}[2]{}]{}\^a & (f\_[1|[1]{}]{}\^a - f\_[2|[2]{}]{}\^a - f\_[3|[3]{}]{}\^a) & f\_[[2]{}|[1]{}]{}\^a & f\_[[3]{}|[1]{}]{}\^a\ f\_[[1]{}[3]{}]{}\^a & f\_[[1]{}|[2]{}]{}\^a & (-f\_[1|[1]{}]{}\^a + f\_[2|[2]{}]{}\^a - f\_[3|[3]{}]{}\^a) & f\_[[3]{}|[2]{}]{}\^a\ f\_[[2]{}[1]{}]{}\^a & f\_[[1]{}|[3]{}]{}\^a & f\_[[2]{}|[3]{}]{}\^a & ( - f\_[1|[1]{}]{}\^a - f\_[2|[2]{}]{}\^a +f\_[2|[3]{}]{}\^a)\ ) \_a \[6dsqgen\] and that for the case at hand reduces to -[**D**]{}\^\* [**D**]{} = ( [cccc]{} \_m\^m & 0 & 0 & 0\ 0 & \_m\^m & -\_6 & 0\ 0 & \_6 & \_m\^m & 0\ 0 & 0 & 0 & \_m\^m ) \[6dsq1\] This operator matrix is block diagonal, and it is easy to see that the upper $1\times 1$ box, containing the Laplace-Beltrami operator, corresponds to the eigenvalue problem for the 4d gaugino and its KK replicas, arising from $\psi_0$ in (\[fvector\]). The lower $3\times 3$ block, on the other hand, corresponds to the squared Dirac operator for the fermionic superpartners of the 4d scalars, since it exactly matches the $3\times 3$ blocks of (\[system1\]). As the diagonalization of (\[6dsq1\]) proceeds exactly as in Section \[vanish\], we will not repeat it here. Non-vanishing $\mu$-terms {#scalarmu} ------------------------- Let us now turn to the type I vacuum with $\mu$-term of subsection \[nvmu\]. An obvious difference with respect to the case without $\mu$-term is the contribution of the background fluxes to the internal Dirac equation, which now reads = (2)\^[-1]{} ( \^[123]{} + \^[156]{}) =\_( \^[123]{} + \^[156]{}) where we have again used the condition $M_3R_3/R_2 = M_6R_6/R_5$. Hence now we have that \[SUSYf2\] \^[123]{} + \^[156]{} = \_1 \_1 i\_2 - \_1 i\_2 \_1 = \_1 ( \_[|[z]{}]{} \_[z]{} - \_[z]{} \_[|[z]{}]{}) and so $\slashed{f}$ does not kill $\chi_{\Pi_2}$, as expected from a compactification with non-trivial $\mu$-terms. As a result, ${\bf F}$ does not vanish, and we have that + [**F**]{} = ( [cccc]{} 0 & \_[[z]{}\^1]{} & \_[[z]{}\^2]{} & \_[[z]{}\^3]{}\ - \_[[z]{}\^1]{} & 0 & - \_[|[z]{}\^3]{} & \_[|[z]{}\^2]{}\ - \_[[z]{}\^2]{} & \_[|[z]{}\^3]{} & 0 & - \_[|[z]{}\^1]{}\ - \_[[z]{}\^3]{} & -\_[|[z]{}\^2]{} & \_[|[z]{}\^1]{} & \_ ) where $\varepsilon_\mu$ is now defined as in (\[system2\]). The r.h.s. of eq.(\[6dsq\]) then reads -([**D**]{} + [**F**]{})\^\* ([**D**]{} + [**F**]{}) = ( [cccc]{} \_m\^m & 0 & 0 & 0\ 0 & \_m\^m & -\_\_[[z]{}\^3]{} & - \_\_[[z]{}\^2]{}\ 0 & \_\_[|[z]{}\^3]{} & \_m\^m & \_\_[[z]{}\^1]{}\ 0 & \_\_[|[z]{}\^2]{} & -\_\_[|[z]{}\^1]{} & \_m\^m - \_\^2 ) \[6dsq2\] Note that even in this more involved case, where ${\bf F} \neq 0$, the operator matrix (\[6dsq2\]) is block diagonal, as expected from 4d supersymmetry. Again, we can identify the upper block with the gaugino + KK modes eigenvalue equation and the lower one with that for the 4d holomorphic scalars of Section \[nonvanish\].[^23] Hence, the diagonalization of (\[6dsq2\]) proceeds exactly as for the bosonic sector of the theory. Matter field wavefunctions {#sec:wmatter} ========================== Recall that in our general discussion of Section \[sec2\], we considered a gauge subsector $U(N) \subset G_{gauge}$ and a $U(N)$ gauge field (\[splita\]) whose vev broke this gauge symmetry as $U(N) \raw \prod_i U(n_i) \equiv G_{unbr}$. Just like in the more familiar fluxless case [@yukawa], from this gauge breaking pattern we obtain 4d fields transforming in the adjoint representation of each $U(n_i)$ factor, arising from the fluctuations contained in (\[splitboson\]) and their fermionic partners, as well as 4d fields in the bifundamental representations $({n}_i, \bar{n}_j)$, arising from those in (\[splitboson2\]).[^24] Up to now we have focused on those open string modes that correspond to $U(n_i)$ adjoint representations or, otherwise said, on those wavefunctions arising from (\[splitboson\]). As we have seen, both the mass spectrum and the internal wavefunctions of these modes are directly modified by the closed string background flux $F_3$ and by the torsional metric of the compactification manifold $\cam_6$. While this sector of adjoint representation modes already gives us a lot of information on the interplay between open strings and background fluxes, for phenomenological purposes it is clearly not the most interesting one. Indeed, from our recent experience with D-brane model building (see, e.g., [@review3; @reviews; @denef08]) we know that the bifundamental modes arising from (\[splitboson2\]) and their fermionic partners can in principle reproduce the matter content of the MSSM from their lightest modes. Since these light matter fields wavefunctions are crucial to compute effective theory quantities like Yukawa couplings and soft terms, an essential question to be answered is how they are affected in the presence of background fluxes. We will devote this section to obtain the spectrum of bifundamental eigenmodes and eigenfunctions arising from the expansion (\[splitboson2\]), leaving the discussion in terms of 4d effective theory for the next section. As can be guessed from the magnetized D-brane literature, matter field wavefunctions will not only be affected by closed string fluxes like $F_3$, but also by the open string magnetic flux $F_2 = dA$ under which they are charged. As we will see, the resulting wavefunction can be understood as an open string mode charged under an effective closed + open string magnetic flux, with the relative densities of both kind of fluxes entering the wavefunction in a rather interesting way. Let us be more precise and let us consider the gauge symmetry breaking $U(N)\to U(p_\alpha)\times U(p_\beta)$. In the twisted tori examples of Section \[subsec:twisted\], this breaking will be induced by an open string flux $F_2$ with indices on the $T^2\times T^2$ base of $\cam_6$ and of the form[^25] $$F_2=2\pi \sum_{k=1,2}\begin{pmatrix}m^k_\alpha\mathbb{I}_{n_\alpha}& 0\\ 0&m^k_\beta\mathbb{I}_{n_\beta}\end{pmatrix}\ dx^k\wedge dx^{k+3} \label{f2}$$ with $n_\a + n_\b = N$ and $p_\Lambda \equiv g.c.d.(n_\Lambda, n_\Lambda m_\Lambda^1, n_\Lambda m_\Lambda^2, n_\Lambda m_\Lambda^1 m_\Lambda^2)$, $\Lambda = \a,\b$. For simplicity, we will assume that $n_\Lambda, m_\Lambda^k \in \IZ$, which in the language of [@yukawa] corresponds to a compactification with Abelian Wilson lines. Given this particular choice of open string flux, we can proceed with our dimensional reduction scheme of eqs.(\[splitboson\]) and (\[splitboson2\]). Here $\langle B \rangle = \langle B^\Lambda \rangle U_\Lambda$, where $U_\Lambda$ is defined by (\[generators\]) and $ \langle B^\Lambda \rangle$ can be chosen to be $$\langle B^\Lambda\rangle = \pi \sum_{k=1,2}m_\Lambda^k[x^kdx^{k+3}-x^{k+3}dx^{k}] \qquad \qquad \Lambda=\alpha,\beta$$ where, again for simplicity, we have set all Wilson lines to zero. The gauge transformation of a $U(N)$ adjoint field along a non-trivial closed path $\gamma$ is then $$W \ \to \ \textrm{exp}\left[i\oint_\gamma \langle B^\Lambda\rangle U_\Lambda\right]\cdot W \cdot \textrm{exp}\left[- i\oint_\gamma \langle B^\Lambda\rangle U_\Lambda\right] \label{wilson}$$ so for a $(\bar{n}_\a, {n}_\b)$ representation we have $$\begin{aligned} x^k\to x^k+1\ , \ \ldots \ : \qquad &W^{\alpha\beta}\to e^{i\pi I^k_{\alpha\beta} x^{k+3}}W^{\alpha\beta} \label{boundary}\\ x^{k+3}\to x^{k+3}+1\ , \ \ldots \ : \qquad &W^{\alpha\beta}\to e^{-i\pi I^k_{\alpha\beta}x^k}W^{\alpha\beta} \nonumber\end{aligned}$$ where the dots in the l.h.s. indicate a possible accompanying action on the fiber, as dictated by the structure of our twisted torus $\cam_6$ (see e.g. (\[reli\]) or (\[relii\])), and $k=1,2$. We have also defined $I^k_{\alpha\beta}=m^k_\alpha-m^k_\beta$, following the conventions in [@yukawa]. Finally, consistency with the equations of motion for $F_2$ requires that $I^1_{\alpha\beta}I^2_{\alpha\beta}<0$. Let us in particular assume that $I^2_{\alpha\beta}>0>I^1_{\alpha\beta}$, and introduce the quantities $$\sigma_{\pm}=\frac{1}{2\pi}\left(\frac{I^2_{\alpha\beta}}{R_2R_5}\pm\frac{I^1_{\alpha\beta}}{R_1R_4}\right)$$ so that $\sigma_-$ is the total density of flux $F_2$, whereas $\sigma_+$ is proportional to the D-term induced by $F_2$ [@cim02]. One can check that the SUSY conditions for $F_2$ amount to [@mmms99; @raul01] $$J^2 \wedge F_2 - \frac{1}{3}F_2^3 = J^2 \wedge F_2 = 0 \quad \Leftrightarrow \quad \sigma_+=0 \label{susy}$$ W bosons -------- Let us start considering the 4d vector bosons $w_\mu$ in (\[splitboson2\]), transforming in the bifundamental representation of $G_{unbr} = U(n_\a) \times U(n_\b)$. The internal profile of such open string mode is given by the scalar wavefunction $W = W^{\a\b}e_{\a\b}$, with components $W^{\a\b}$ satisfying the equation of motion $$\hat{D}^m\hat{D}_m W^{\alpha\beta} = -m_{W}^2 W^{\alpha\beta} \label{gaugebos}$$ with $$\hat D_m W^{\alpha\beta}=\hat\partial_m W^{\alpha\beta} -i(\langle B_m^\alpha\rangle-\langle B_m^\beta\rangle)W^{\alpha\beta} \label{covariant}$$ in agreement with our notation in eqs.(\[lapfin1\]) and (\[lapfin2\]). Note that (\[gaugebos\]) reduces to (\[lap\]) if we set $\langle B^\Lambda \rangle = 0$, so it is reasonable to expect a structure of KK modes similar to the one found in Section \[sec:wgauge\]. In particular, for bosons in the adjoint representation we have seen that KK modes not excited along the fiber do not feel the closed string fluxes at all, and so they present the spectrum of a standard, fluxless toroidal compactification. The same result applies to W bosons, in the sense that if $W^{\alpha\beta}$ does not depend on the coordinates of the fiber $\hat{\p}$ becomes the standard partial derivative. As a result, (\[gaugebos\]) becomes in this case the equation of motion for a $W$ boson in a magnetized $T^2\times T^2$, and their spectrum follows from the results in [@yukawa]. Indeed, the lightest mode, of mass $m^2_W=|\sigma_-|$, is given by $$\begin{gathered} W^{\alpha\beta, \ (0,j_1,j_2)}_{0}(\tilde z_1', \tilde z_2') =\cn \prod_{k=1,2} e^{i\pi |I^k_{\alpha\beta}| \tilde z_k' \textrm{Im }\tilde z_k' / \textrm{Im }\tilde \tau_k'}\ \vartheta\left[{\frac{j_k}{|I^k_{\alpha\beta}|} \atop 0}\right](|I^k_{\alpha\beta}|\tilde z_k'\, {\Large ;} \, |I^k_{\alpha\beta}|\tilde \tau_k')\\ \cn \, =\, \left(\frac{2}{\textrm{Vol}_{\cam_6}}\right)^{1/2} \prod_{k=1,2}(|I^k_{\alpha\beta}|\textrm{Im }\tilde \tau_k')^{1/4} \qquad \begin{array}{ccc} \tilde{z}_1' =x^4+\tilde \tau_1' x^1 & \quad & \tilde \tau_1'=iR_1/R_4 \\ \tilde z_2' =x^2+\tilde \tau_2' x^5 & \quad & \tilde \tau_2' =iR_5/R_2 \end{array} \label{magtheta}\end{gathered}$$ where again we have defined a non-standard choice of complex structure. As in [@yukawa], a full KK tower can be constructed from (\[magtheta\]) by applying appropriate raising operators. On the other hand, for modes with non-vanishing Kaluza-Klein momentum along the fiber, some subtleties arise. Let us for concreteness focus on the example without flux-generated $\mu$-terms of subsection \[vmu\], whose gauge boson spectrum was analyzed in Section \[nili\]. There, we saw that in practice one can trade the effect of a closed string flux on $\cam_6$ by an appropriate magnetic flux $F_2^{\text{cl}} = 2\pi k_6M (dx^1 \wedge dx^2 + dx^4 \wedge dx^5)$ on the $T^2\times T^2$ base of the fibration. Since now our W boson also feels the genuine open string flux $(F_2^{\a\b})^\text{op} = 2\pi (I^1_{\a\b} dx^1 \wedge dx^4 + I^1_{\a\b} dx^2 \wedge dx^5)$, it is natural to consider a total, effective magnetic flux defined as $(F_2)_{\text{eff}} = F_2^\text{op} + F_2^\text{cl}$, which in this case reads $$(F^{\alpha\beta}_2)_{\rm eff}\,=\,2\pi\, dx^1\wedge (k_6M dx^2+I^1_{\alpha\beta} dx^4)+2\pi\, (k_6Mdx^4+ I^2_{\alpha\beta}dx^2)\wedge dx^5 \label{totalflux}$$ and to expect that our open string modes behave as particles charged under $(F_2)_{\rm eff}$.[^26] In our example, the choice of $T^2\times T^2$ metric (\[bg12\]), guarantees that both fluxes $F_2^{\text{cl}}$ and $F_2^\text{op}$ are factorizable, in the sense that they can be decomposed as $F_2 = F_2|_{(T^2)_i} + F_2|_{(T^2)_j}$, with $(T^2)_i$ and $(T^2)_j$ two orthogonal two-tori. In turn, this property implies that their associated lowest KK mode can be written as a product of two theta functions, which in the case at hand are given by (\[thetawave\]) for $F_2^\text{cl}$ and (\[magtheta\]) for $F_2^\text{op}$. Note, however, that $(F_2)_{\text{eff}} = F_2^\text{op} + F_2^\text{cl}$ will in general not be factorizable, and so we cannot expect the associated lowest KK mode to be again a product of two Jacobi theta functions, but rather a Riemann $\vartheta$-function [@yukawa]. Hence, for matter modes excited along the fiber we would expect a lowest KK mode wavefunction of the form $$\begin{gathered} W^{\alpha\beta,\ (j_1,j_2)}_{0,0,k_3,k_6}= \mathcal{N}\ e^{i\pi(\mathbf{N}\cdot\vec z)\cdot (\textrm{Im }\mathbf{\Omega}_{\bf U})^{-1}\cdot \textrm{Im }\vec z}\ \vartheta\left[{\vec j \atop 0}\right](\mathbf{N}\cdot \vec z\, ;\, \mathbf{N}\cdot \mathbf{\Omega}_{\bf U})\ e^{2\pi i(k_3x^3+k_6x^6)} \label{wavematter}\end{gathered}$$ where $\vec z\in \mathbb{C}^2$, and $\mathbf{N}$ and $\mathbf{\Omega}_{\bf U}$ are $2\times 2$ real and complex matrices, respectively. The definition of the Riemann $\vartheta$-function and its properties can be found in Appendix \[riem\]. Indeed, one can check that the ansatz (\[wavematter\]) is a solution of (\[gaugebos\]), (\[reli\]), (\[boundary\]), with mass eigenvalue $m^2_W=\Delta^2_{k_3,k_6}+\rho$, if we set[^27] $$\vec{z}=\begin{pmatrix}x^4\\ x^2\end{pmatrix} +\mathbf{\Omega}_{\bf U}\cdot\begin{pmatrix}x^1\\ x^5\end{pmatrix} \quad \qquad \mathbf{\Omega}_{\bf U}=\bar{\bf B}^{-1}\cdot \bar{\bf U}\cdot\bar{\bf B}\cdot\bf{\Omega} \label{rotcpxst}$$ and $$\begin{aligned} \mathbf{N}&=\begin{pmatrix}-I^1_{\alpha\beta}&-k_6M\\ k_6M&I^2_{\alpha\beta}\end{pmatrix} & \mathbf{\Omega}&=i\begin{pmatrix}\frac{R_1}{R_4}& 0\\ 0& \frac{R_5}{R_2}\end{pmatrix}\label{nomega}\\ \mathbf{B}&= \sqrt{2} \pi \begin{pmatrix}R_4& 0\\ 0&R_2\end{pmatrix} & \mathbf{U}&=\begin{pmatrix}\textrm{cos }\phi &\textrm{sin }\phi\\ -\textrm{sin }\phi&\textrm{cos }\phi\end{pmatrix} \\ \mathcal{N}&=\left(\frac{2R_5\, {\rm det\, } {\bf N}}{R_2 \textrm{Vol}_{\cam_6}} \right)^{1/2} & & \vec j^{\, t} {\bf N} \in \IZ^2\end{aligned}$$ where we have defined the effective flux density $\rho$ and the interpolation angle $\phi$ as[^28] $$\rho=\sqrt{\rho_{\text{op}}^2 + \rho_{\text{cl}}^2} = \sqrt{\sigma_-^2+ \left(\frac{k_6 \varepsilon}{R_6}\right)^2}\quad \qquad \textrm{tan }\phi= \frac{\rho_{\text{cl}}}{\rho_{\text{op}}}= \frac{k_6\varepsilon}{R_6 \sigma_-}$$ Note that ${\bf N}$ and ${\bf \Omega}_{\bf U}$ satisfy the convergence conditions (\[conv\]) that allow (\[wavematter\]) to be well-defined, and that the degeneracy of each level is given by $\textrm{det }\mathbf{N}$. Moreover, under the lattice transformations (\[reli1\])-(\[reli2\]), $W^{\alpha\beta}$ transforms as dictated by (\[boundary\]). Finally, $\phi$ interpolates between the two choices of complex structures (\[zeff\]) and (\[magtheta\]). In the limit $\phi \raw \pi/2$ we recover from (\[rotcpxst\]) $z_k = \tilde z_k$ and the factorized wavefunctions (\[thetawave\]) for neutral bosons, while in the limit $\phi \raw 0$ we obtain $z_k = \tilde z_k'$ and the wavefunctions for charged bosons without KK momentum along the fiber, given by (\[magtheta\]). In order to build the full tower of Kaluza-Klein excitations for the charged bosons, we can systematically act on (\[wavematter\]) with the holomorphic covariant derivatives defined in Appendix \[riem\], which for the case at hand read $$\begin{aligned} a^\dagger_1& \equiv \hat D_2+i\ \textrm{sin }\phi\, \hat D_1-i\ \textrm{cos }\phi\, \hat D_5 \\ a^\dagger_2& \equiv \hat D_4-i\ \textrm{cos }\phi\, \hat D_1-i\ \textrm{sin }\phi\, \hat D_5\end{aligned}$$ Indeed, note that the deformation angle $\phi$ is such that $$\textrm{Im }{\bf \Omega}_{\bf U}^{-1}\cdot {\bf N}^t= (\textrm{Im }{\bf \Omega}^{-1}_{\bf U})^t\cdot {\bf N}$$ and as a result $[a_1^\dagger,a_2^\dagger]=0$. This allows us to write a number operator $$N=\hat{D}_m\hat{D}^m+\rho$$ and to build the full KK tower of states by applying $(a_1^\dagger)^{n-k} (a_2^\dagger)^{k}$ to (\[wavematter\]). The resulting spectrum of masses is given by $$m_W^2=\left(\frac{k_3}{R_3}\right)^2+\left(\frac{k_6}{R_6}\right)^2+(n+1)\rho + (2k-n)\frac{\sigma_+\sigma_-}{\rho} \quad \qquad k, n \in \IZ \label{masquasi}$$ where $k=0, \ldots, n$. Interestingly, for a vanishing D-term $\sigma_+=0$ the effective flux (\[totalflux\]) factorizes, and so the Riemann $\vartheta$-function in (\[wavematter\]) becomes the product of two ordinary $\vartheta$-functions. In that case, the complete tower of wavefunctions is given by \[setmagnetico\] & & W\^[, (k,\_1,\_2)]{}\_[n,k\_3,k\_6]{} = ()\^[1/2]{}\_[s\_1,s\_2]{}\_[n-k]{}() \_[k]{}()\ & & with $\delta_{k}=0\ldots \textrm{g.c.d}(k_6M,I^k_{\alpha\beta})-1$. Note that these wavefunctions have the same structure as in (\[set1\]), but now they localize along the tilted coordinates $$\begin{aligned} &\dot x^a\equiv \frac{1}{R_2}\sqrt{\frac{4\pi}{\rho}}[\delta_2+k_6M(x^1+s_1)-I^2_{\alpha\beta}(x^5+s_2)] \\ &\dot x^b\equiv \frac{1}{R_4}\sqrt{\frac{4\pi}{\rho}}[\delta_1-I^1_{\alpha\beta}(x^1+s_1)+k_6M(x^5+s_2)]\end{aligned}$$ Alternatively, we could have derived all the above results by considering an extended version of the algebra (\[torsion\]), accounting for the D9-brane gauge generators, and then making use of the representation theory methods described in Section \[gener\]. More precisely, we know that the algebra (\[torsion\]) is part of the four dimensional gauge algebra, corresponding to the gauge symmetries which arise from dimensional reduction of the metric tensor. This, however, is not the full 4d gauge algebra. In particular, in the presence of D9-branes, we should also include the generators of the $U(1)$ gauge symmetries arising from such open string sector [@kaloper; @algebras] $$\begin{aligned} [\hat D_m,\hat D_n] &=-f^p_{mn}\hat D_p+i F^\alpha_{mn}U_\alpha \label{extalgebra}\\ [\hat D_m,U_\alpha] &=[U_\alpha,U_\beta]=0 \nonumber\end{aligned}$$ where the covariant twisted derivatives $\hat D_m$ are defined as in (\[covariant\]) and the Abelian gauge generators $U_\alpha$ by (\[generators\]). Given such extended algebra, it is straightforward to apply the methods of Section \[gener\] and Appendix \[kirillov\] to compute its irreducible unitary representations. For the case at hand, we find the following two sets of irreducible unitary representations[^29] $$\begin{aligned} &\pi_{k_1,k_2,k_3,k_4,k_5}=\prod_{r=1}^5\textrm{exp}[2\pi ik_rx^r] \label{generirr}\\ &\pi_{k_3,k_6,k_q}=\textrm{exp}\left[2\pi i\left(k_3x^3+k^{\alpha\beta} \left(\textrm{Tr }\Lambda_{\alpha\beta}+I^2_{\alpha\beta}x^2\left(s_5+\frac{x^5}{2}\right)- I^1_{\alpha\beta}x^4\left(s_1+\frac{x^1}{2}\right)\right)\right)\right.\nonumber\\ &\hspace*{.5cm} \left. + k_6\left(x^6-Mx^2\left(s_1+\frac{x^1}{2}\right)+Mx^4\left(s_5+\frac{x^5}{2}\right)\right)\right] u(s_1+x^1,s_5+x^5) \label{generirr2}\end{aligned}$$ where $u(\vec s)\in L^2(\mathbb{R}^2)$ and $\textrm{Tr }\Lambda_{\alpha\beta}$ is the trace of the gauge parameter (i.e. the unphysical coordinate in the $U(1)\simeq S^1$ D9-brane gauge fibers). Note that there is a new natural quantum number $k^{\alpha\beta}$ which we did not find in our previous analysis. Plugging now (\[generirr\]) and (\[generirr2\]) into (\[gaugebos\]) we find that, indeed, the unirreps (\[generirr2\]) with $k^{\alpha\beta}=1$ lead to the matter wavefunction solutions (\[wavematter\]), as well as the more massive replicas produced by acting with $a_1^\dag$, $a_2^\dag$. It would then seem that those unirreps in (\[generirr\]) with $k^{\alpha\beta} \neq 1$ would not correspond to any physical modes, somehow against the general philosophy of Section \[gener\]. Let us try to argue that such modes do exist. First, let us consider the meaning of $k^{\a\b} \in \IZ$. If we set $k^{\a\b} = 0$, then from (\[generirr\]) and (\[generirr2\]) we recover the vector boson adjoint modes (\[set1\]) and (\[set2\]) of Section \[sec:wgauge\]. Indeed, (\[generirr\]) directly correspond to adjoint bosons without Kaluza-Klein momentum along the fiber, given by (\[set2\]), while (\[generirr2\]) with $k^{\alpha\beta}=0$ correspond to the adjoint bosons with Kaluza-Klein momentum along the fiber, given by (\[set1\]). This is not a surprise since, after all, the KK modes of Section \[sec:wgauge\] arose from the irreducible unitary representations of a subalgebra of (\[extalgebra\]). What is perhaps more illuminating is the fact that neither of the above subset of modes satisfy eq.(\[gaugebos\]), but rather the Laplace-Beltrami equation (\[lap\]) for a neutral boson. This clearly suggest that the internal differential equation that should be satisfied by an arbitrary wavefunction arising from (\[generirr2\]) is given by (\[gaugebos\]), but with the gauge covariant derivative defined as $$\hat D_m W^{\alpha\beta}=\hat\partial_m W^{\alpha\beta} -ik^{\a\b} (\langle B_m^\alpha\rangle-\langle B_m^\beta\rangle)W^{\alpha\beta} \quad \quad k^{\a\b} \in \IZ \label{covariantk}$$ instead of (\[covariant\]). In this sense, the massive modes corresponding to $k^{\a\b} \neq 1$ should be understood as states with $U(1)$ charges $k^{\a\b}(-n_\a, n_\b)$, which hence undergo the gauge transformations $$\begin{aligned} x^k\to x^k+1\ , \ \ldots \ : \qquad &W^{\alpha\beta}\to e^{i\pi k^{\a\b} I^k_{\alpha\beta} x^{k+3}}W^{\alpha\beta} \label{boundaryk}\\ x^{k+3}\to x^{k+3}+1\ , \ \ldots \ : \qquad &W^{\alpha\beta}\to e^{-i\pi k^{\a\b} I^k_{\alpha\beta}x^k}W^{\alpha\beta} \nonumber\end{aligned}$$ In particular, those states with $k^{\a\b} = -1$ correspond to the bifundamental representation $(n_\a, \bar{n}_\b)$, whose wavefunction can be obtained by complex conjugation of (\[wavematter\]). Finally, those modes with $|k^{\a\b}| > 1$ should be non-perturbative in nature, as they cannot arise from the perturbative open string spectrum. Note that the existence of these exotic non-perturbative charged vector states is not only suggested by the spectrum of unirreps (\[generirr2\]), but also required by global symmetry arguments. Indeed, the 4d effective action of the untwisted D9-brane sector is given by a $\mathcal{N}=4$ gauged supergravity, whose global symmetry group is $SL(2)\times SO(6,6+N)$, and where $N=n_\alpha+n_\beta$ is the number of extra vector multiplets coming from $D9$-brane gauge symmetries. The spectrum of 4d particles is therefore naturally arranged in multiplets of this global symmetry. In the particular example at hand, the global symmetry group includes a $\mathbb{Z}_2$ generator corresponding to the open/closed string correspondence discussed in Section \[nili\]. This generator maps neutral bosons with Kaluza-Klein momentum $|k_6| > 1$ along the fiber to non-perturbative charged bosons with $U(1)$ charge $|k^{\a\b}| > 1$. Making use of this global symmetry, we would expect the following masses for the non-perturbative modes $$m_W^2=\left(\frac{k_3}{R_3}\right)^2+\left(\frac{k_6}{R_6}\right)^2+(n+1)\rho_{\rm n.p.} + (2k -n) (k^{\a\b})^2 \frac{\sigma_+\sigma_-}{\rho_{\rm n.p.}} \label{masquasi2}$$ where $$\rho_{\rm n.p.}=\sqrt{(k^{\alpha\beta}\sigma_-)^2+\left(\frac{k_6\varepsilon}{R_6}\right)^2}$$ Finally, note that the algebra (\[extalgebra\]) is still not the full four dimensional gauge algebra. There are further gauge symmetries which arise from dimensional reduction of e.g. the RR 2-form. In particular, the RR 3-form fluxes enter as structure constants of the complete 4d algebra [@algebras]. We expect the irreducible unitary representations of the complete four dimensional algebra to encode further untwisted states of the higher dimensional string theory. We leave the exploration of these issues for future work. Similarly, we can work out the wavefunctions for the charged bosons in the example with non-vanishing $\mu$-term of subsection \[nvmu\]. In that case, the total effective magnetic flux is given by $$(F_2^{\alpha\beta})_{\rm eff}=2\pi\left(I^1_{\alpha\beta}dx^1\wedge dx^4+I^2_{\alpha\beta}dx^2\wedge dx^5+k_3M_3dx^1\wedge dx^2+k_6M_6dx^1\wedge dx^5\right) \label{totalflux2}$$ Recall that for this vacuum we should distinguish between bosons with no Kaluza-Klein momentum along the $S^1$ fibers, bosons with Kaluza-Klein momentum along only one of the fibers, and bosons with momentum along both of the fibers. We can easily adapt our previous discussion in this section to describe the wavefunctions for the first two types of bosons. Indeed, it is not difficult to see that the wavefunctions for charged bosons without momentum along any fiber are given again by eq.(\[magtheta\]), whereas the wavefunctions for charged bosons with Kaluza-Klein momentum along only one of the fibers, e.g. $k_3\neq 0, \ k_6=0$, are given by eq.(\[wavematter\]) with the same parameters $\vec z$, ${\bf \Omega}$ and ${\bf B}$, but with charge matrix, deformation angle and effective flux density given by $${\bf N}=\begin{pmatrix}-I^1_{\alpha\beta}& -k_3M_3\\ 0&I^2_{\alpha\beta}\end{pmatrix} \quad \quad \quad \textrm{tan }\phi=\frac{k_3\varepsilon_\mu}{R_3\sigma_-} \quad \quad \quad \rho=\sqrt{\sigma_-^2+\left(\frac{k_3\varepsilon_\mu}{R_3}\right)^2}$$ The set of charged bosons excited along both fibers, with arbitrary $k_3$ and $k_6$, is however a more involved sector, and in particular does not fall into the class of functions (\[wavematter\]). This basically comes from the fact that $F_2$ has then all the possible components of the form $dx^1 \wedge dx^\alpha$. We refer to the reader to Appendix \[riem\] for a more precise statement as well as a more detailed discussion of this point. Bifundamental scalars and fermions {#bifund} ---------------------------------- Just like for adjoint KK modes, the wavefunctions for bifundamental scalars and fermions are easily worked out once that the 4d vector boson wavefunctions are known. Note that these bifundamental KK modes are particularly interesting in semi-realistic flux compactification vacua, since they correspond to the MSSM matter fields and their KK replicas. As before, let us start our analysis by considering the scalars in the bifundamental. From eqs.(\[lapfin1\])-(\[lapfin2\]), we see that the corresponding mass matrix can be obtained from the one for adjoint scalars analyzed in Appendix \[ap:matrix\], by simply replacing twisted derivatives $\hat \p_m$ by covariant twisted derivatives $\hat D_m$, and adding a term proportional to $\langle G_{mp}^{\alpha\beta}\rangle$. In particular, for the example without $\mu$-terms discussed in subsection \[vmu\] we obtain the mass matrix = D\_mD\^m \_6 + -& -D\_6&0&0&0&0\ D\_6&-&0&0&0&0\ 0&0&0&0&0&0\ 0&0&0&&-D\_6&0\ 0&0&0&D\_6&&0\ 0&0&0&0&0&0 \[chargscalar\] where again $\varepsilon = MR_6/\pi R_1R_2$ and we are using the conventions of (\[standcom\]), with $\xi^p_{\Pi_2,B_4}$ now complex functions. Like in the case of adjoint scalars, this matrix is block diagonal, and so it will be enough to diagonalize the upper $3\times 3$ block. Note that both blocks are related by an $\mathcal{N}=2$ R-symmetry transformation. However, it is important to notice that, since we are dealing with charged modes, this transformation takes $\sigma_+\to-\sigma_+$. For the upper $3\times 3$ block we find the eigenvectors $$\Phi_3^{\alpha\beta}\equiv\begin{pmatrix}0\\ 0\\ 1\end{pmatrix}W^{\alpha\beta}(\vec x) \label{phi3}$$ with mass eigenvalue $m^2_{\Phi_3}=m^2_W$, and with $W^{\alpha\beta}(\vec x)$ the wavefunction of a charged boson. In addition, we find $$\Phi_{\pm}^{\alpha\beta}\equiv\begin{pmatrix}\sigma_-\mp\rho\\ i\varepsilon \frac{k_6}{R_6}\\ 0\end{pmatrix}W^{\alpha\beta}(\vec x) \label{phipm}$$ with mass eigenvalues $m^2_{\Phi_\pm}=m^2_W+\sigma_+\pm\rho$. The R-symmetry conjugates $\bar \Phi_\pm$ has then mass eigenvalues $m^2_{\bar{\Phi}_\pm}=m^2_W-\sigma_+\pm\rho$. Thus, $\Phi_\pm$ and $\bar \Phi_\pm$ lead to scalars with masses $$\begin{aligned} m^2_{\Phi_+}&=\left(\frac{k_3}{R_3}\right)^2+\left(\frac{k_6}{R_6}\right)^2+ (n+2)\rho+(2k-n)\sigma_+\sigma_-\rho^{-1} \\ m^2_{\Phi_-}&=\left(\frac{k_3}{R_3}\right)^2+\left(\frac{k_6}{R_6}\right)^2+n\rho+(2k-n) \sigma_+\sigma_-\rho^{-1} \\ m^2_{\bar{\Phi}_+}&=\left(\frac{k_3}{R_3}\right)^2+\left(\frac{k_6}{R_6}\right)^2+(n+2)\rho -(2k-n)\sigma_+\sigma_-\rho^{-1} \\ m^2_{\bar{\Phi}_-}&=\left(\frac{k_3}{R_3}\right)^2+\left(\frac{k_6}{R_6}\right)^2+n\rho -(2k-n)\sigma_+\sigma_-\rho^{-1}\end{aligned}$$ Then, as expected, for supersymmetry preserving open string fluxes we observe two massless modes, whereas for generic fluxes there is always a single tachyonic mode. Similarly, if we analyze the charged scalars in the example with non-vanishing $\mu$-term of subsection \[nvmu\] we have to diagonalize the following mass matrix = D\_mD\^m \_6 + -&-\_D\_[z\^3]{}& -\_D\_[z\^2]{}&0&0&0\ \_D\_[|z\^3]{}&-& \_D\_[z\^1]{}&0&0&0\ \_D\_[|z\^2]{}&-\_D\_[|z\^1]{}&-|\_|\^2&0&0&0\ 0&0&0&&-\_D\_[|z\^3]{}& -\_D\_[|z\^2]{}\ 0&0&0&\_D\_[z\^3]{}&& \_D\_[|z\^1]{}\ 0&0&0&\_D\_[z\^2]{}&-\_D\_[z\^1]{}&-|\_|\^2 \[chargscalarm\] with $\varepsilon_\mu = M_3R_3/2\pi R_1R_2$. This is again a non-commutative eigenvalue problem, that can be solved with the aid of the commutation relations $$\begin{aligned} &[\hat{D}_{z^1}, \hat{D}_{z^2}] \, =\, [\hat{D}_{\bar{z}^1}, \hat{D}_{z^2}] \, =\, - \varepsilon_\mu \hat D_{z^3} \label{comuta1} \\ & [\hat{D}_{z^1}, \hat{D}_{\bar{z}^2}] \, =\, [\hat{D}_{\bar{z}^1}, \hat{D}_{\bar{z}^2}]\, = \, - \varepsilon_\mu \hat D_{\bar{z}^3} \nonumber \\ &[\hat{D}_m\hat{D}^m, \hat{D}_{z^2}]\, =\,-\varepsilon_\mu\hat{D}_{z^3}(\hat{D}_{z^1}+ \hat{D}_{\bar z^1}) -\frac{I^2_{\alpha\beta}}{\pi R_2R_5}\hat{D}_{z^2} \nonumber\\ & [\hat{D}_m\hat{D}^m, \hat{D}_{\bar z^2}]\, =\,-\varepsilon_\mu\hat{D}_{\bar z^3}(\hat{D}_{z^1} +\hat{D}_{\bar z^1}) + \frac{I^2_{\alpha\beta}}{\pi R_2R_5}\hat{D}_{\bar z^2}\nonumber\\ &[\hat{D}_m\hat{D}^m, \hat{D}_{z^1}]\, =\, \varepsilon_\mu\left(\hat{D}_{\bar z^2}\hat{D}_{z^3} +\hat{D}_{z^2}\hat{D}_{\bar z^3}\right) -\frac{I^1_{\alpha\beta}}{\pi R_1R_4} \hat{D}_{z^1} \nonumber\\ & [\hat{D}_m\hat{D}^m, \hat{D}_{\bar z^1}]\, = \, \varepsilon_\mu\left(\hat{D}_{\bar z^2}\hat{D}_{z^3} +\hat{D}_{z^2}\hat{D}_{\bar z^3}\right) +\frac{I^1_{\alpha\beta}}{\pi R_1R_4} \hat{D}_{\bar z^1}\nonumber\end{aligned}$$ in close analogy with what we did for the neutral scalars in Section \[scalarmu\]. For the upper $3\times 3$ block in $\mathbb{M}$, we obtain the eigenvectors $$\Phi_3=\begin{pmatrix}\hat D_{\bar z^1}\\ \hat D_{\bar z^2}\\ \hat D_{\bar z^3} \end{pmatrix}W^{\alpha\beta}(\vec x) \label{eig1}$$ with mass eigenvalue $m^2_{\Phi_3}=m_W^2$ and $$\Phi_\pm=\begin{pmatrix}\hat D_{z^3}\hat D_{\bar z^1}+\tilde m_\pm\hat D_{z^2}\\ \hat D_{z^3}\hat D_{\bar z^2}-\tilde m_\pm \hat D_{z^1}\\ \hat D_{z^3}\hat D_{\bar z^3}+\tilde m^2_ \pm-2\varepsilon_\mu^{-1}\tilde m_\pm\sigma_+\end{pmatrix}W^{\alpha\beta}(\vec x) \label{eig2}$$ with mass eigenvalues $m_{\Phi_{\pm}}^2=m_W^2+\varepsilon_\mu\tilde m_\pm$, and $\tilde m_\pm$ given by the quadratic equation $$-m_{W}^2\varepsilon_\mu+\varepsilon_\mu \tilde m^2_\pm- \tilde m_\pm(\varepsilon_\mu^2\pm 2\sigma_+) \pm \varepsilon_\mu\sigma_+=0$$ so that $$m^2_{\Phi_\pm}=\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2+ 4m^2_W+(\varepsilon_\mu^{-1}\sigma_+)^2}\right)^2 -(\varepsilon_\mu^{-1}\sigma_+)^2+\sigma_+$$ Analogously, the lower $3\times 3$ block in $\mathbb{M}$ leads to the conjugate scalars $$\bar \Phi_3= \begin{pmatrix}\hat D_{z^1}\\ \hat D_{z^2}\\ \hat D_{z^3}\end{pmatrix} W^{\alpha\beta}(\vec x) \qquad \bar \Phi_\pm= \begin{pmatrix} \hat D_{\bar z^3}\hat D_{z^1}+\tilde m_{\mp}\hat D_{\bar z^2}\\ \hat D_{\bar z^3}\hat D_{z^2}-\tilde m_{\mp} \hat D_{\bar z^1}\\ \hat D_{\bar z^3}\hat D_{z^3}+\tilde m_{\mp}^2+2\varepsilon_\mu^{-1}\tilde m_{\mp}\sigma_+ \end{pmatrix}W^{\alpha\beta}(\vec x) \label{eig3}$$ with mass eigenvalues $$m^2_{\bar \Phi_3}=m_W^2\quad \text{and} \quad m^2_{\bar \Phi_\pm}=\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2+4m^2_W+(\varepsilon_\mu^{-1}\sigma_+)^2}\right)^2 -(\varepsilon_\mu^{-1}\sigma_+)^2-\sigma_+$$ As in Section \[nonvanish\], special care has to be taken with the zero modes. The vectors (\[eig1\])-(\[eig3\]) break down for the lightest modes, and the latter have to be taken apart. After some thinking, it is not difficult to see that the vectors (\[eig1\])-(\[eig3\]) have to be supplemented with the lightest modes $$(\Phi^{\alpha\beta})_0=\begin{pmatrix} 1\\ 0 \\ 0 \end{pmatrix}W^{\alpha\beta}_{0} \quad \quad ({\bar \Phi}^{\alpha\beta})_0=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}W^{\alpha\beta}_{0}$$ where $W^{\alpha\beta}_{0} \equiv W^{\alpha\beta, \ (0,j_1,j_2)}_{0}(\tilde z_1', \tilde z_2')$ is given by eq.(\[magtheta\]). The mass eigenvalues are respectively $m^2_{\Phi_0}=\sigma_+$ and $m^2_{\bar \Phi_0}=-\sigma_+$. Finally, let us compute the wavefunctions for the bifundamental fermions. These again satisfy an equation of the form (\[6db\]), where now the covariant derivative must be incorporated [**D\_A**]{} = ( [cccc]{} 0 & \_[[z]{}\^1]{} & \_[[z]{}\^2]{} & \_[[z]{}\^3]{}\ -\_[[z]{}\^1]{} & 0 & - \_[|[z]{}\^3]{} & \_[|[z]{}\^2]{}\ - \_[[z]{}\^2]{} & \_[|[z]{}\^3]{} & 0 & - \_[|[z]{}\^1]{}\ -\_[[z]{}\^3]{} & -\_[|[z]{}\^2]{} & \_[|[z]{}\^1]{} & 0 ) \[repcov\] Taking into account that the commutation relation for these operators is given by (\[extalgebra\]), that $f^i_{k\bar{k}} = 0$ and that the only non-vanishing components of the open string magnetic flux are $F_{1\bar{1}}$ and $F_{2\bar{2}}$, we have that [$$\begin{array}{c} -{\bf D_A}^* {\bf D_A} \, =\, \hat{D}_m\hat{D}^m\, \mathbb{I}_4 \, + \sum_a \left( \begin{array}{cccc} - \sigma_+ & f_{\bar{2}\bar{3}}^a\, \hat{D}_a & f_{\bar{3}\bar{1}}^a \,\hat{D}_a & f_{\bar{1}\bar{2}}^a\, \hat{D}_a \\ f_{{3}{2}}^a\, \hat{D}_a& \sigma_- & f_{{2}\bar{1}}^a\,\hat{D}_a & f_{{3}\bar{1}}^a\,\hat{D}_a \\ f_{{1}{3}}^a\,\hat{D}_a & f_{{1}\bar{2}}^a\,\hat{D}_a & - \sigma_- & f_{{3}\bar{2}}^a\,\hat{D}_a \\ f_{{2}{1}}^a\,\hat{D}_a & f_{{1}\bar{3}}^a\, \hat{D}_a& f_{{2}\bar{3}}^a\,\hat{D}_a & \sigma_+ \\ \end{array} \right) \end{array} \label{6dsqgenA}$$]{} Hence, in our example without $\mu$-term -[**D\_A**]{}\^\* [**D\_A**]{} = \_m\^m \_4 + ( [cccc]{} - \_+ & 0 & 0 & 0\ 0 & \_- & -\_6 & 0\ 0 & \_6 & - \_- & 0\ 0 & 0 & 0 & \_+ ) \[6dsq1A\] which again contains the upper $3\times 3$ block of the scalar mass matrix (\[chargscalar\]), with the diagonal shifted by $\sigma_+$. Therefore we obtain the same eigenvectors (\[phi3\]) and (\[phipm\]), but now with masses $$\Psi_\pm\, \raw\, m^2_{\Phi_\pm} - \sigma_+\qquad \bar \Psi_\pm \, \raw \, m^2_{\bar \Phi_\pm} + \sigma_+\label{dterm}$$ and similarly for $\Psi_W$ and $\Psi_3$. This indeed reflects the D-term breaking of the charged $\mathcal{N}=2$ supermultiplets caused by an open string flux with $\sigma_+\neq 0$. Similar considerations apply also for the charged fermions in the example with non-vanishing $\mu$-term. Indeed, in that case we have that $$\begin{array}{c} -({\bf D_A} + {\bf F})^* ({\bf D_A} + {\bf F}) \, =\, \hat{D}_m\hat{D}^m\, \mathbb{I}_4 \, + \left( \begin{array}{cccc} - \sigma_+ & 0 & 0 & 0 \\ 0 & \sigma_- & -\varepsilon_\mu \hat{D}_{{z}^3} & - \varepsilon_\mu \hat{D}_{{z}^2} \\ 0 & \varepsilon_\mu \hat{D}_{\bar{z}^3} & - \sigma_- & \varepsilon_\mu\hat{D}_{{z}^1} \\ 0 & \varepsilon_\mu \hat{D}_{\bar{z}^2} & -\varepsilon_\mu \hat{D}_{\bar{z}^1} & \sigma_+ - \varepsilon_\mu^2 \end{array} \right) \end{array} \label{6dsq2A}$$ so again the eigenvalue problem is already solved by the knowledge of the bosonic sector. Indeed, comparing with (\[chargscalarm\]) we see that these states have the same eigenvectors than their scalar superpartners (\[eig1\]) and (\[eig2\]), with their masses given again by eq.(\[dterm\]). Applications {#sec:app} ============ Having computed the open string spectrum in several type I flux vacua,[^30] we now would like to apply these results to understand better the effect of fluxes on open strings. First we will consider the effect of fluxes on the open string massive spectrum, and in particular how they may break the degeneracies present in fluxless compactifications. Second, we will focus on the light spectrum of the theory, and compare our results with those derived from a 4d effective supergravity analysis. Finally, we will consider a type IIB T-dual setup, where the open strings arise from a stack of D7-branes in the presence of $G_3$ fluxes, and translate the effect of fluxes on open strings to this more familiar picture. Further applications of the above results will be explored in [@wip]. Supersymmetric spectrum {#susyspect} ----------------------- As emphasized in the literature, flux vacua based on twisted tori are special in the sense that they are directly related to 4d $\cn=4$ gauged supergravity. Moreover, in the vanishing flux limit ($\varepsilon \raw 0$ for the vacua of Section \[subsec:twisted\]) one should recover the $\cn =4$ spectrum of a toroidal compactification. Hence, in general one would expect that the flux lifts the mass degeneracies of the $\cn = 4$ spectrum by an amount directly related to $\varepsilon$, so that the previous 4d $\cn=4$ supermultiplets split into smaller ones. In particular, for the type I flux vacua of subsections \[vmu\] and \[nvmu\] the flux breaks the bulk $\cn=4$ supersymmetry down to $\cn=2$ and $\cn=1$, respectively, so the neutral open string modes of Sections \[sec:wgauge\], \[sec:scalars\] and \[sec:fermions\] should feel such kind of splitting.[^31] On the other hand, the open string flux $F_2$ already breaks $\cn=4 \raw \cn=2$,[^32] and so the charged, bifundamental modes of Section \[sec:wmatter\] could feel the effects of fluxes in a rather different way. Finally, let us recall that for the vacua of subsection \[vmu\], no multiplet splitting occurs at the massless level, while for the vacua of subsection \[nvmu\] this is clearly the case. It is then natural to wonder how these facts will translate in terms of the full massive spectrum of the theory. In order to classify our spectrum let us recall the content of massless and massive 4d $\cn=1$ vector and chiral multiplets. Following a notation similar to that of Sections \[sec:wgauge\] to \[sec:wmatter\], we have for the massless $\mathcal{N}=1$ multiplets ---------------- ---------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [neutral]{} [charged]{} [**vector**]{} $({\mathcal{A}}^\alpha)_0=(B^\alpha, \Psi_B^\alpha)$ $(\mathcal{A}^{\alpha\beta})_0=(W^{\alpha\beta}, \Psi_W^{\alpha\beta})$ $(\mathcal{\bar{A}}^{\alpha\beta})_0=(\bar{W}^{\alpha\beta}, \bar{\Psi}_W^{\alpha\beta})$ [**chiral**]{} $(\mathcal{C}^\alpha_p)_0=(\xi^\alpha_p, \Psi^\alpha_p)$ $(\mathcal{C}^{\alpha\beta}_p)_0=(\Phi^{\alpha\beta}_p, \Psi^{\alpha\beta}_p)$ $(\mathcal{\bar{C}}^{\alpha\beta}_p)_0=(\bar{\Phi}^{\alpha\beta}_p, \bar{\Psi}^{\alpha\beta}_p)$ ---------------- ---------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- where neutral multiplets contain particles in a real (in our case adjoint) representation of the gauge group $G_{unbr}$, while charged multiplets transform in complex representations (in our case the bifundamental rep. of Section \[sec:wmatter\]). The index $\a$ runs over the factors of $G_{unbr} = \prod_\a U(n_\a)$, and the same applies for $\b$. The index $p$ labels instead different chiral multiplets inside the same representation, and in our case takes the three different values $p = \pm, 3$, as in (\[neutpm\]) and (\[extrapol\]). Finally, $\mathcal{A}^{\alpha\beta}$ and $\mathcal{C}^\alpha_p$ contain 4d spinors of positive chirality and $\mathcal{\bar{A}}^{\alpha\beta}$ and $\mathcal{\bar{C}}^\alpha_p$ of negative chirality, and the above degrees of freedom should be completed with their CPT conjugates. For massive $\cn=1$ multiplets the above picture has to be slightly modified. In particular, gauge bosons eat extra degrees of freedom in order to become massive through the standard Higgs mechanism, whereas chiral fields group into vector-like combinations. We can thus express their field content as ---------------- -------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- [neutral]{} [charged]{} [**vector**]{} ${\mathcal{A}}^\alpha=({\mathcal{A}}^\alpha)_0+(\mathcal{C}_3^\alpha)_0$ ${\mathcal{A}}^{\alpha\beta}=(\mathcal{A}^{\alpha\beta})_0+(\mathcal{\bar{A}}^{\alpha\beta})_0+(\mathcal{C}_3^{\alpha\beta})_0+(\mathcal{\bar{C}}_3^{\alpha\beta})_0$ [**chiral**]{} ${\mathcal{C}}^\alpha_p=(\mathcal{C}^\alpha_p)_0$ ${\mathcal{C}}^{\alpha\beta}_\pm=(\mathcal{C}^{\alpha\beta}_\pm)_0+(\mathcal{\bar{C}}^{\alpha\beta}_\pm)_0$ ---------------- -------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- where we have taken $\mathcal{C}_3$ to contain the degrees of freedom eaten by the gauge bosons, in agreement with the notation in Sections \[nonvanish\], \[scalarmu\] and \[bifund\]. On the other hand, massless $\mathcal{N}=2$ vector and hyper multiplets are given by ---------------- --------------------------------------------------- ----------------------------------------------------------------- [neutral]{} [charged]{} [**vector**]{} $\mathcal{B}^\alpha={\mathcal{A}}^\alpha$ $\mathcal{B}^{\alpha\beta} = {\mathcal{A}}^{\alpha\beta}$ [**hyper**]{} $\mathcal{H}^\alpha_\pm={\mathcal{C}}_\pm^\alpha$ $\mathcal{H}^{\alpha\beta}_\pm={\mathcal{C}}_\pm^{\alpha\beta}$ ---------------- --------------------------------------------------- ----------------------------------------------------------------- where $\mathcal{H}^\alpha_p$ are in fact half-hypermultiplets. For $\cn=2$ massive multiplets we have ------------------------------------------------------------------------------------- $\mathcal{V}^\alpha=\mathcal{B}^\alpha+\mathcal{H}^\alpha_+ + \mathcal{H}^\alpha_-$ ------------------------------------------------------------------------------------- and similarly for $\cv^{\a\b}$, looking like $\mathcal{N}=4$ vector multiplets. Finally, we may also have ultrashort $\cn=2$ massive multiplets, containing the same particle content as massless $\cn=2$ multiplets $\cb$ and $\ch$ and corresponding to $\oh$-BPS objects of the theory. Let us now go back to the two main families of flux vacua analyzed in the previous sections. In Tables \[table0\] and \[table1\] we summarize, respectively, the resulting neutral and charged spectrum for the class of $\mathcal{N}=2$ compactifications with vanishing $\mu$-term introduced in subsection \[vmu\]. We have taken a supersymmetric configuration of the open string flux (i.e., $\sigma_+=0$) and we have introduced the shorthand notation $$\Delta_{k_{i_1},k_{i_2},\ldots}^2\equiv \sum_{r=i_1,i_2,\ldots}\left(\frac{k_r}{R_r}\right)^2$$ for the squared mass of a fluxless, toroidal KK mode. The open string field content in this class of compactifications can be arranged into different 4d $\mathcal{N}=2$ multiplets. More precisely, for the neutral sector of the open string spectrum there is a tower of standard $\mathcal{N}=2$ massive multiplets $\cv^\a$ associated to each irreducible unitary representation of the closed string algebra (\[torsion\]), plus an extra tower of ultrashort $\mathcal{N}=2$ hypers $\ch^\a$. Since in principle the multiplets $\cv^\a$ can be identified with vector $\cn=4$ multiplets and $\ch^\a$ cannot, the latter can be seen as a clear effect of the $\cn=4 \raw \cn=2$ supersymmetry breaking induced by the closed string fluxes into the open string sector. Multiplets $(\textrm{Mass})^2$ Degeneracy ------------------------------------------------------------------- -------------------------------------------------------- ----------------- $(\mathcal{V}^\alpha)_{k_1,k_2,k_3,k_4,k_5}$ $\Delta^2_{k_1,k_2,k_3,k_4,k_5}$ $1$ $(\mathcal{V}^\alpha)_{n,k_3,k_6}^{(k,\delta_1,\delta_4)}$ $\Delta^2_{k_3,k_6} + |\varepsilon|\Delta_{k_6} (n+1)$ $(k_6M)^2(n+1)$ $(\mathcal{H}^\alpha_{s_{k_6M}})_{k_3,k_6}^{(\delta_1,\delta_4)}$ $\Delta^2_{k_3,k_6}$ $(k_6M)^2$ : Spectrum of neutral $\mathcal{N}=2$ multiplets for D9-brane fields in the model with vanishing $\mu$-terms of subsection \[vmu\].[]{data-label="table0"} At the massless level the theory contains of a single neutral $\mathcal{N}=4$ vector multiplet $(\mathcal{V}^\alpha)_{0,0,0,0,0}$ for each adjoint representation of $G_{unbr} = \prod_\a U(n_\a)$, and $|I^1_{\alpha\beta}I^2_{\alpha\beta}|$ charged $\mathcal{N}=2$ hypermultiplets $(\mathcal{H}^{\alpha\beta})_{0}^{(j_1,j_2)}$ in the bifundamental representation of $U(n_\alpha)\times U(n_\beta)$. Therefore the massless open string spectrum is the same than in flat space, and the same applies to the open string wavefunctions. In fact, in the limit of diluted closed string fluxes, on which the size of the fiber is much smaller than any other size ($R_{6}\ll R_k$ with $k\neq 6$) the lightest Kaluza-Klein modes (which correspond to the modes $(\mathcal{V}^\alpha)_{k_1,k_2,k_3,k_4,k_5}$ in Table \[table0\]) also match with the ones in the fluxless case. ------------------------------------------------------------------------------------------------------------------------------------------------- Multiplets $(\textrm{Mass})^2$ Degeneracy ------------------------------------------------------------------- ---------------------- ------------------------------------------------------ $(\mathcal{V}^{\alpha\beta})_{n,k_3,k_6}^{(k,\delta_1,\delta_4)}$ $\rho(n+1) $[(k_6M)^2-I^1_{\alpha\beta}I^2_{\alpha\beta}](n+1)$ +\Delta^2_{k_3,k_6}$ $(\mathcal{H}^{\alpha\beta}_-)_{k_3,k_6}^{(j_1,j_2)}$ $\Delta^2_{k_3,k_6}$ $(k_6M)^2-I^1_{\alpha\beta}I^2_{\alpha\beta}$ $(\mathcal{H}^{\alpha\beta}_-)_{0}^{(j_1,j_2)}$ $0$ $|I^1_{\alpha\beta}I^2_{\alpha\beta}|$ ------------------------------------------------------------------------------------------------------------------------------------------------- : Spectrum of charged $\mathcal{N}=2$ multiplets for D9-brane fields in the vanishing $\mu$-term model of subsection \[vmu\], for supersymmetric open string fluxes.[]{data-label="table1"} For the class of $\mathcal{N}=1$ compactifications with non-vanishing $\mu$-term introduced in subsection \[nvmu\], the further breaking of the supersymmetry to $\mathcal{N}=1$ and the presence of the $\mu$-term makes the spectrum slightly more complicated. We have summarized in Tables \[table2\] and \[table3\] the resulting neutral and charged spectra.[^33] The field content again corresponds to a tower of $\mathcal{N}=4$ vector multiplets $\cv$ for each set of unirreps of the closed string algebra, but now with a mass mass splitting on their $\mathcal{N}=1$ constituents induced by the fluxes. Indeed, in terms of $\mathcal{N}=1$ representations, each multiplet $\cv$ leads to one massive vector multiplet and two chiral multiplets. For compactifications with vanishing $\mu$-terms all these multiplets have degenerate Dirac mass $m_{\mathcal{B}}$, thus assembling into a $\mathcal{N}=4$ vector representation. For compactifications with non-vanishing $\mu$-term, however, the closed string background induces a Majorana mass $\varepsilon_\mu$ for one of the two chiral multiplets, leading to a mass-matrix which is of the form[^34] $$\begin{pmatrix}{\mathcal{C}}_1& {\mathcal{C}}_2\end{pmatrix}\begin{pmatrix}\varepsilon_\mu & m_{\mathcal{B}}\\ m_{\mathcal{B}}& 0\end{pmatrix}\begin{pmatrix}{\mathcal{C}}_1\\ {\mathcal{C}}_2\end{pmatrix}$$ The mass eigenvalues for this matrix are then given by $$m_{{\mathcal{C}}_\pm}^2-\varepsilon_\mu m_{{\mathcal{C}}_\pm} -m_{\mathcal{B}}^2=0\quad \Longrightarrow \quad m_{{\mathcal{C}}_\pm}^2 =\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2+4m_{\mathcal{B}}^2}\right)^2 \label{cuadr2}$$ reproducing the result we obtained in (\[cuadr\]). Hence, after the breaking to $\cn=1$, one can associate to each set of irreducible unitary representations a tower of massive $\mathcal{N}=1$ vector multiplets and two towers of $\mathcal{N}=1$ chiral multiplets, with their masses given by eq.(\[cuadr2\]). Regarding the massless modes, for each stack of magnetized branes we have two neutral $\mathcal{N}=1$ chiral multiplets $(\mathcal{C}^\alpha_\pm)_{0}$ and one $\mathcal{N}=1$ vector multiplet $(\mathcal{A}^\alpha)_0$, while for each pair of factors $U(n_\alpha)\times U(n_\beta) \subset G_{unbr}$ we have $|I^1_{\alpha\beta}I^2_{\alpha\beta}|$ charged $\mathcal{N}=2$ hypermultiplets $(\mathcal{H}^{\alpha\beta})_{0}^{(j_1,j_2)} = (\cc^{\a\b})_0^{(j_1,j_2)} + (\bar{\cc}^{\a\b})_0^{(j_1,j_2)}$ in the bifundamental representation. Thus, of the three originally present neutral $\mathcal{N}=1$ chiral multiplets in flat space, we see that only two remain massless in the presence of the closed string fluxes, whereas $\mathcal{C}^\alpha_3$ gets a mass equal to $\varepsilon_\mu^2$. As we will see in the next section, this is also what is expected from the four dimensional effective supergravity analysis. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Multiplets $(\textrm{Mass})^2$ Degeneracy -------------------------------------------------------------------- ------------------------------------------------------------------------------------------------- -------------------------------------- $({\mathcal{A}}^\alpha)_{k_1,k_2,k_4,k_5}$ $\Delta^2_{k_1,k_2,k_4,k_5}$ $1$ $({\mathcal{A}}^\alpha)_{n,k_3,k_4,k_5}^{(\delta)}$ $|\varepsilon_\mu|\Delta_{k_3}(2n+1)+\Delta^2_{k_3,k_4,k_5}$ $|k_3M_3|$ $({\mathcal{A}}^\alpha)_{n,k_2,k_4,k_6}^{(\delta)}$ $|\varepsilon_\mu| \Delta_{k_6} (2n+1)+\Delta^2_{k_2,k_4,k_6}$ $|k_6M_6|$ $({\mathcal{A}}^\alpha)_{n,k_3,k_4,k_6}^{(\delta_2,\delta_5)}$ $|\varepsilon_\mu| \Delta_{k_3,k_6}(2n+1)+\Delta_{k_3,k_6}^2$ $\textrm{l.c.m.}(|k_3M_3|,|k_6M_6|)$ $({\mathcal{C}}^\alpha_\pm)_{k_1,k_2,k_4,k_5}$ $\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2 $1$ +4\Delta^2_{k_1,k_2,k_4,k_5}}\right)^2$ $({\mathcal{C}}^\alpha_\pm)_{n,k_3,k_4,k_5}^{(\delta)}$ $\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2+\ 4|\varepsilon_\mu| \Delta_{k_3}(2n+1) $|k_3M_3|$ +4\Delta^2_{k_3,k_4,k_5}}\right)^2$ $({\mathcal{C}}^\alpha_\pm)_{n,k_2,k_4,k_6}^{(\delta)}$ $\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2 $|k_6M_6|$ + 4|\varepsilon_\mu| \Delta_{k_6} (2n+1)+4\Delta^2_{k_2,k_4,k_6}}\right)^2$ $({\mathcal{C}}^\alpha_\pm)_{n,k_3,k_4,k_6}^{(\delta_2,\delta_5)}$ $\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2 $\textrm{l.c.m.}(|k_3M_3|,|k_6M_6|)$ +4\varepsilon_\mu \Delta_{k_3,k_6}(2n+1)+4\Delta_{k_3,k_6}^2}\right)^2$ $({\mathcal{A}}^\alpha)_0$ $0$ $1$ $(\mathcal{C}^\alpha_\pm)_0$ $0$ $1$ ${\mathcal{C}}^\alpha_3$ $\varepsilon_\mu^2$ $1$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Spectrum of neutral $\mathcal{N}=1$ multiplets for D9-brane fields in the model with non-vanishing $\mu$-term of subsection \[nvmu\].[]{data-label="table2"} Multiplets $(\textrm{Mass})^2$ Degeneracy -------------------------------------------------------- ----------------------------------------------------------------------------------------------- ---------------------------------------- $({\mathcal{A}}^{\alpha\beta})_{n,k_3}^{(\delta)}$ $\rho(n+1)+\Delta^2_{k_3}$ $|I^1_{\alpha\beta}I^2_{\alpha\beta}|$ $({\mathcal{C}}^{\alpha\beta}_\pm)_{n,k_3}^{(\delta)}$ $\frac14\left(\varepsilon_\mu\pm\sqrt{\varepsilon_\mu^2+4\rho(n+1)+4\Delta^2_{k_3}}\right)^2$ $|I^1_{\alpha\beta}I^2_{\alpha\beta}|$ $(\mathcal{H}^{\alpha\beta})_{0}^{(j_1,j_2)}$ $0$ $|I^1_{\alpha\beta}I^2_{\alpha\beta}|$ : Partial spectrum of charged $\mathcal{N}=2$ and $\mathcal{N}=1$ multiplets for D9-brane fields in the non-vanishing $\mu$-term model of subsection \[nvmu\], for SUSY open string fluxes ($\sigma_+=0$).[]{data-label="table3"} Finally, let us point out that in the above discussion we have not included the effect of the $\IZ_{2n}$ orbifold needed for the consistency of the construction. In principle, this effect could partially project out the spectrum above, as it is known to happen for the massless sector. This projection will however depend on the particular choice of orbifold action,[^35] and it can be implemented in our framework along the lines of [@ako08]. We defer a more detailed analysis of the different possibilities to [@wip]. Comparison with 4d effective supergravity {#sugra} ----------------------------------------- When analyzing the 4d effective theory of type I flux vacua we only need to keep a small set of light modes in order to describe the low energy dynamics. Such dynamics can then be encoded in terms of a 4d effective Kähler potential and a superpotential which, at least at tree-level, can be expressed as integrals over the internal space $\cam_6$. As a result, finding vacua in the 4d effective theory can be translated into certain 10d conditions which, if our effective theory is accurate, should describe 10d vacua. The main caveat in the above approach is whether the appropriate set of light modes has been chosen. Since in the presence of closed string fluxes the internal manifold $\cam_6$ is no-longer Calabi-Yau, it is in general not known how to perform the light mode truncation. A popular ansatz is to take the set of massless modes of the Calabi-Yau $\cam_6^{\tiny \text{CY}}$ that is obtained from $\cam_6$ by ‘turning off’ the background fluxes. This procedure is well-defined when the fluxes are weak compared to the KK scales in $\cam_6^{\tiny \text{CY}}$, but far from reliable beyond this regime. For instance, considering type IIB flux vacua on warped Calabi-Yau manifolds, non-dilute fluxes in general lead to strong warping effects, which could in principle lower the mass of an $\a'$ state below the flux scale. Clearly, the same kind of observations apply to open strings and, in particular, to the type I spectra analyzed above. Since we have followed a well-defined prescription when dimensionally reducing our flux vacua, comparing the 10d approach with the standard 4d effective supergravity analysis can be made manifest, and it can be checked explicitly under which circumstances both approaches agree. This will be the purpose of the present subsection. ### 10d versus 4d approach For SU(3)-structure compactifications with O9/O5-planes, one can write the 4d Kähler potential and superpotential in terms of integrals over the internal manifold as [@kahler1; @kahler2] $$\begin{aligned} \hat{K} &=-\textrm{log}\left[-i\int_{\mathcal{M}_6}\Omega\wedge\Omega^*\right]-\textrm{log}[2e^{-\phi}]-2\textrm{log}\left[\int_{\mathcal{M}_6} J\wedge J\wedge J\right] \label{kahler}\\ W&=\int_{\mathcal{M}_6} \Omega\wedge (F_3+ie^{-\phi/2}dJ) \label{super}\end{aligned}$$ with $J$ and $\Omega$ the SU(3)-invariant 2-form and 3-forms of $\cam_6$, respectively. In addition we can write $F_3 = F_3^{\text{cl}} + \om_3$, where $F_3^{\text{cl}}$ depends on the RR closed string fields and $$\omega_3=\textrm{Tr}\left(A\wedge dA+\frac23 A\wedge A\wedge A\right)$$ is the 10d Chern-Simons 3-form, containing the open string degrees of freedom. Now, when the internal manifold is not Calabi-Yau, as occurs in the presence of closed string background fluxes, a prescription to expand $J$ and $\Omega$, and $\om_3$ in terms of closed and open string light fields is in general not known. In that case, one usually proceeds by expanding them in a base of harmonics for the Calabi-Yau manifold $\cam_6^{\tiny \text{CY}}$ which results in the limit of vanishing fluxes. In our case, this prescription amounts to take either $\cam_6^{\tiny \text{CY}} = T^6$ or a toroidal orbifold, and so the wavefunctions used in our dimensional reduction should look like those that arise from an unwarped $T^6$. From our results on open string wavefunctions, it is clear that this will be the case as long as ${\it i)}$ the warping can be neglected and ${\it ii)}$ the light modes of the compactification do not contain any KK mode excited along the fiber. Whether neglecting the warping is a good approximation can be read from eq.(\[NSNSt1\]). Using the conditions it can be rewritten as \^2\_[T\^4]{}Z\^2 = - \^2 + …where $\varepsilon$ is the flux mass scale of our compactification, and the dots stand for $F_2$ and $\d$-function contributions. Thus, away from localized sources and setting $F_2 =0$ for simplicity, the warp factor can be taken constant for $m^{\text{KK}}_{\text{base}} \gg \varepsilon$. It is easy to see [@Schulz04] that this is guaranteed if we take $\textrm{Vol}_{B_4}^{1/2} \gg \textrm{Vol}_{\Pi_2}$, which in turn implies that $m^{\text{KK}}_{\text{fib}} \gg m^{\text{KK}}_{\text{base}}$ and hence that no fiber KK mode will be a light field of the theory. Indeed, as we will show below, under the assumption $\textrm{Vol}_{B_4}^{1/2} \gg \textrm{Vol}_{\Pi_2}$ the 4d effective supergravity succeeds in describing the spectrum of light modes that we have obtained by dimensional reduction. On the contrary, in the regime where the volume of the fiber is of the same order of magnitude than the volume of the base, the mass of the fiber KK modes will be comparable to the mass of the base modes and lifted open string moduli, and they cannot be omitted from the 4d effective supergravity description. As discussed around figure \[fig1\], the wavefunctions of these fiber KK modes present interesting localization properties, which should be added to the standard localization effects due to the strong warping effects. It would be very interesting to see how their combined effect may affect standard dimensional reduction. Let us then take the limit $R_{\text{base}} \gg R_{\text{fib}}$ and truncate the theory to the lightest neutral and charged modes, denoted in the following by $\varphi^{\alpha,k}$ and $\varphi^{\alpha\beta,k}$, respectively. In terms of the notation of Section \[susyspect\], the scalar component of these fields are $$(\xi_\pm^\alpha)_{0}\equiv \varphi^{\alpha,1}\pm i\varphi^{\alpha,2}\quad \quad (\xi_3^\alpha)_{0}\equiv \varphi^{\alpha,3}\quad \quad (\Phi_\pm^{\alpha\beta})_0\equiv \varphi^{\alpha\beta,1}\pm i\varphi^{\alpha\beta,2} \label{lights}$$ where the subscript $0$ denotes the lightest KK mode of each tower. In the following we will analyze the two and three-point couplings for this set of light fields. ### 2-point couplings In supersymmetric compactifications to 4d Minkowski, the only source for scalar masses are $\mu$-terms in the superpotential. In terms of these, the 2-point couplings in the 4d effective action read[^36] $$-S=Z_{i\bar j}(M,M^*)\partial_\mu \varphi^i \partial^\mu (\varphi^i)^*+e^{\hat K(M,M^*)}\mu_{ik}\bar\mu_{\bar l\bar j}Z^{k\bar l}(M,M^*)\varphi^i(\varphi^j)^* + \ldots\label{ssugra}$$ where we have expanded the effective superpotential and the full Kähler potential in powers of the light open string fields $\varphi^i$ as $$\begin{aligned} K(M,M^*,\varphi,\varphi^*)&=\hat K(M,M^*)+Z_{i\bar j}(M,M^*)\varphi^i(\varphi^{\bar j})^*+\ldots \\ W(M,\varphi)&=\hat W(M)+\frac12 \mu_{ij}(M)\varphi^i\varphi^j+\frac{1}{3!}\tilde Y_{ijk}\varphi^i\varphi^j\varphi^k+\ldots\end{aligned}$$ and $M$ stands for the full set of closed string moduli/light fields, whose Kähler potential $\hat{K}$ is given by (\[kahler\]). The standard procedure in the 4d supergravity approach is then to approximate (\[kahler\]) by the Kähler potential of a factorizable $T^6$. For $\mathcal{N}=2$ configurations of the open string flux $F_2$, this is given to quadratic order in the fields by [@tkahler1; @tkahler2; @lerda1; @lerda2; @diveccia] $$K=-\textrm{log }(2s)+\sum_{k=1}^3\left[-\textrm{log}(4t_k u_k)+\sum_\alpha \frac{|\varphi^{\alpha,k}|^2}{4t_ku_k}\right]+\sum_{\alpha,\beta}\frac{|\varphi^{\alpha\beta,1}|^2+|\varphi^{\alpha\beta,2}|^2}{16(t_1u_1t_2u_2)^{1/2}}$$ where $$2s=g_s^{1/2} \textrm{Vol}_{\cam_6}\quad \qquad 2t_a=4\pi^2g_s^{-1/2}R_aR_{a+3} \quad \qquad 2u_a=\frac{R_{a+3}}{R_a}\quad \qquad a=1,2,3 \label{moduli}$$ are the real parts of the moduli in a toroidal orientifold with O5/O9-planes [@louisiib]. Under these assumptions, the integration of the superpotential (\[super\]) was performed in [@geosoft] for toroidal compactifications, obtaining the following expressions for the gravitino mass and for the effective $\mu$-term of the lightest neutral modes[^37] $$m_{3/2}=e^{\hat K/2}\langle \hat W\rangle=\frac{3}{4\sqrt{2s}}f^k_{\bar i\bar j}\quad \qquad \mu_{kk}=\frac{e^{-\hat K/2}Z_{k\bar k}}{\sqrt{2s}}f^{\bar k}_{\bar i\bar j}$$ where $f^{\bar i}_{\bar j\bar k}$ are the (moduli dependent) structure constants of the algebra (\[torsion\]) expressed in the complex basis. These equations, which depend only on the NSNS part of the background, assume that the on-shell conditions (\[rel1\])-(\[rel2\]) are satisfied. Note that when the manifold is complex $f^k_{\bar i\bar j}=0$, the gravitino is massless and the background preserves $\mathcal{N}\geq 1$ supersymmetry in four dimensions [@geosoft; @lawrence]. In that case, from (\[ssugra\]) we get $$\begin{gathered} -S=\frac{1}{4u_it_i}\partial_\mu\varphi^{\alpha,i}\partial^\mu(\varphi^{\alpha,i})^* +\frac{1}{16(t_1u_1t_2u_2)^{1/2}}(\partial_\mu\varphi^{\alpha\beta,1} \partial^\mu(\varphi^{\alpha\beta,1})^*+\partial_\mu\varphi^{\alpha\beta,2} \partial^\mu(\varphi^{\alpha\beta,2})^*)\\ -\sum_{i\neq k\neq j}\frac{1}{8st_ku_k} |f^{\bar k}_{\bar i\bar j}|^2|\varphi^{k,\alpha}|^2\end{gathered}$$ and so, making use of the moduli definitions (\[moduli\]) we have $$\begin{gathered} -S=\frac{g_s^{1/2}}{(2\pi R_{i+3})^2}\partial_\mu\varphi^{\alpha,i}\partial^\mu(\varphi^{\alpha,i})^* +\frac{g_s^{1/2}}{16\pi^2R_4R_5}(\partial_\mu\varphi^{\alpha\beta,1}\partial^\mu (\varphi^{\alpha\beta,1})^*+\partial_\mu\varphi^{\alpha\beta,2}\partial^\mu (\varphi^{\alpha\beta,2})^*)\\ -\sum_{i\neq k\neq j}\frac{1}{\textrm{Vol}_{\cam_6}} \frac{1}{(2\pi R_{k+3})^2}|f^{\bar k}_{\bar i\bar j}|^2|\varphi^{k,\alpha}|^2 \label{2point}\end{gathered}$$ Let us see how this expression applies to the two classes of type I flux vacua that have been analyzed in this paper. First, note that in the example of subsection \[vmu\] with vanishing $\mu$-terms, the structure constants $f^{\bar k}_{\bar i\bar j}$ are all zero. From (\[2point\]) we see that then all the lightest scalars remain massless, in agreement with the 10d result that there are no flux-generated $\mu$-terms in this case. The open string massless content is therefore the same than in a fluxless toroidal (or toroidal orbifold) compactification, as we have also concluded from direct dimensional reduction of the 10d supergravity background. On the other hand, for the example of subsection \[nvmu\] we see from (\[com\]) that the only non-vanishing structure constant whose all indices are anti-holomorphic is given by $f^{\bar 3}_{\bar 1\bar 2}=\varepsilon_\mu$. Hence, as expected from the 10d analysis all the light scalars (\[lights\]) are massless except for $\varphi^{3,\alpha}$. Moreover, after the rescaling $\varphi^{3,\alpha}\to 2\pi R_6 g_s^{-1/4}\varphi^{3,\alpha}$ in order to have canonically normalized kinetic terms, one obtains a 4d mass given by $$m^2_{\varphi^{3,k}}=(g_{YM}\varepsilon_\mu)^2 \label{massmu}$$ where $g_{YM}=(g_s^{1/2}\textrm{Vol}_{\cam_6})^{-1/2}$ is the gauge coupling constant. Again, this matches the result obtained in Section \[nonvanish\] by means of dimensional reduction.[^38] ### 3-point couplings Let us now turn to the 3-point couplings between the lightest modes and compare again with the effective supergravity results. We will focus on those Yukawa couplings of the form $$S=\int dx^4\, Y_{ijk} \bar \psi^{\alpha\beta,i}\psi^{\beta\alpha,j}\varphi^k \label{yuki}$$ where $\psi^{\alpha\beta,i}$ is some massless fermion in the bifundamental representation of $U(n_\alpha)\times U(n_\beta)$, and $\varphi^k$ a complex scalar in the adjoint representation. Recall that in the specific closed string background at hand, one cannot turn on a magnetic flux $F_2$ such that $\int_{\cam_6} F_2^3 \neq 0$ since, in particular, such $F_2$ cannot be turned on the elliptic fiber $\Pi_2$ wrapped by the D5-branes. As a result, (\[yuki\]) is the only possible class of Yukawa couplings involving the light modes of these constructions. As usual, the coupling $Y_{ijk}$ can be obtained by dimensional reduction of the kinetic term of the 10d gaugino, given in eq.(\[accion\]), resulting in the expression [@yukawa; @diveccia] $$Y_{ijk}\,=\, g_s^{-1/4}\int_{\mathcal{M}_6}(\Psi^{\beta\alpha}_i)^\dagger \tilde \gamma^m \Psi^{\beta\alpha}_j(\xi_k)_m$$ with $\Psi^{\alpha\beta,i}$ and $\xi^k$ the corresponding wavefunctions for the 4d modes $\psi^{\alpha\beta,i}$ and $\varphi^k$, respectively. More precisely, in the two examples of flux compactifications considered above, the only non-vanishing Yukawa coupling involving the two fermionic superpartners of $\varphi^{\alpha\beta,1}$ and $\varphi^{\alpha\beta,2}$, denoted as $\psi^{\alpha\beta,1}$ and $\psi^{\alpha\beta,2}$ respectively, are given by $$Y_{123}\,=\, =\, \frac{1}{g_s^{1/4}\textrm{Vol}_{\cam_6}^{1/2}} \int_{\mathcal{M}_6}(\Psi^{\beta\alpha,(j_1,j_2)}_1)^\dagger \tilde \gamma^3 \Psi^{\beta\alpha,(j_1',j_2')}_2= -ig_{YM}\delta_{j_1j_1'}\delta_{j_2j_2'}$$ where we have normalized the wavefunction of $\varphi^3$ such that $$\int_{\mathcal{M}_6}(\xi_3)^\dagger \xi_3=1$$ The computation then exactly follows the one carried out in [@diveccia] for fluxless toroidal compactifications. In terms of the moduli definitions (\[moduli\]) we have $$Y_{123}=-\frac{i\delta_{j_1j_1'}\delta_{j_2j_2'}}{\sqrt{2s}}$$ which can be compared with the standard expression for the physical Yukawa couplings in 4d effective supergravity $$Y_{ijk}=e^{\hat K/2}\tilde Y_{ijk}(Z_{i\bar i}Z_{j\bar j}Z_{k\bar k})^{-1/2}$$ where $\tilde Y_{ijk}$ is the holomorphic Yukawa coupling appearing in the superpotential. We then obtain $\tilde Y_{123}=-i\delta_{j_1j_1'}\delta_{j_2j_2'}$, as in standard toroidal compactifications. Comparison with T-dual type IIB vacua {#D7dual} ------------------------------------- An interesting feature of the type I flux vacua analyzed in this paper is that they have a simple dual description in terms of standard type IIB flux compactifications. Indeed, if we take type I theory in an elliptically fibered manifold of the form (\[mansatz\]) and we perform two T-dualities along the fiber coordinates $a \in \Pi_2$, we will obtain type IIB string theory compactified on the direct product $\cam_6' = B_4 \times \Pi_2$ (up to an overall warp factor) and threaded by an NSNS 3-from flux $H_3$. Regarding the open string sector, the type I gauge theory analyzed in Section \[diraclap\] will be mapped to a set of O7-planes and D7-branes wrapped on $B_4$, while O5-planes and D5-branes wrapped on $\Pi_2$ will be taken to O3-planes and D3-branes, respectively. This fact applies, in particular, to the twisted tori examples of Section \[subsec:twisted\], for which $B_4 = T^4/\IZ_{2n}$. Following [@kstt02] and ignoring the presence of the orbifold for simplicity, we have that the type IIB T-dual of these twisted tori is given by the following closed string background \[bgiib\] $$\begin{aligned} \label{bgiib1} &ds^2=Z^{-1}ds^2_{\IR^{1,3}}+Z\, ds^2_{T^4 \times T^2} \\ \label{bgiib2} &ds^2_{T^4 \times T^2}=(2\pi)^2 \left[ \sum_{m=1,2,4,5}(R_mdx^m)^2 + \sum_{m=3,6} \left(\frac{dx^m}{R_m}\right)^2 \right] \\ \label{bgiib3} & F_5 = (1 + *_{10})\, d{\rm vol}_{M_4} \wedge dh \\ \label{bgiib4} & \tau = ie^{-\phi_0} = \text{const.}\end{aligned}$$ with $e^{\phi_0} = g_s/R_3R_6$, and $h-Z^{-2} e^{-\phi_0}$= const. In addition, the internal $T^6$ will be threaded by RR and NSNS 3-form fluxes, which depend on the particular choice of T-dual type I flux vacuum. In particular, the type IIB NSNS flux $H_3$ is related to the choice of structure constants in the type I elliptic fibration, while the RR flux $F_3$ comes from the type I quantity $F_3^{\text{bg}}$ defined in Appendix \[ap:warp\]. In particular, the type IIB duals of the vacua in subsection \[vmu\] contain the fluxes \[iibflux1\] $$\begin{aligned} & H_3 = (2\pi)^2\, N\, (dx^1 \wedge dx^2 + dx^4 \wedge dx^5) \wedge dx^6 \\ & F_3 = - (2\pi)^2\, M\, (dx^1 \wedge dx^2 + dx^4 \wedge dx^5) \wedge dx^3\end{aligned}$$ that impose the supersymmetry conditions $NR_6 = M R_3 e^{\phi_0}$ and $R_1R_2 = R_4 R_5$, on the closed string moduli of the compactification, identical to the ones obtained in the type I side.[^39] The type IIB duals to the vacua in subsection \[nvmu\] contain, on the other hand, the 3-form fluxes \[iibflux2\] $$\begin{aligned} & H_3 = (2\pi)^2\, (M_3\, dx^2 \wedge dx^3 + M_6\, dx^5 \wedge dx^6) \wedge dx^1 \\ & F_3 = (2\pi)^2\, (N_6\, dx^2 \wedge dx^3 + N_3\, dx^5 \wedge dx^6) \wedge dx^4\end{aligned}$$ that impose the SUSY conditions $N_6 R_1 e^{\phi_0} = M_3 R_4$, $N_3 R_1 e^{\phi_0} = M_6 R_4$ and $M_3 R_3R_5 = M_6 R_2R_6$, again identical to the dual type I conditions. Rather than analyzing the closed string sector of these type IIB vacua, we would like to understand the dynamics governing the open string sector. In particular, we would like to translate the type I open string spectrum to the present picture, and interpret the open string wavefunctions of Sections \[sec:wgauge\] to \[sec:wmatter\] in terms of type IIB quantities. In this sense, note that the initial $G_{gauge} = U(N)$ gauge theory considered in Section \[diraclap\] will now arise from a stack of $N$ D7-branes, and that the gauge group will be broken to $G_{unbr} = \prod_i U(n_i) \subset U(N)$ via the presence of a magnetic open string flux $F_2$ on them. The analysis of the open string Dirac and Laplace equations could then in principle be carried out via a dimensional reduction of the D7-brane 8D U(N) twisted SYM theory, along the lines of [@quevedo]. Extracting our wavefunction information from the type I T-dual setup, however, has the advantage of automatically including the coupling of the D7-brane open strings to the warp factor and to the background fluxes, which is in general only known for $U(1)$ theories [@dirac; @fershiu]. In the set of type IIB vacua at hand, the stack of N D7-branes under analysis will wrap $T^4 = (T^2)_1 \times (T^2)_2 = \{x^1, x^4, x^2, x^5\}$ and sit at a particular point in the transverse space $(T^2)_3$. Setting $F_2=0$ and neglecting the effect of closed string fluxes, we obtain at the massless level three 4d $\cn=1$ chiral multiplets $\Phi^i$ in the U(N) adjoint representation, which are nothing but the D7-brane moduli and modulini. More precisely, the bosonic components of these multiplets are given by two complex Wilson line moduli $\phi^i$ arising from dimensional reduction of the 8D gauge boson $A_M$ on $(T^2)_i$, $i=1,2$, and by the D7-brane geometric modulus $\phi^3$ in the $(T^2)_3$ transverse space. In the absence of background fluxes it is easy to see that these D7-brane moduli are mapped to the type I Wilson line moduli via the dictionary ---------------------------------------- -- ----------------------------------------------------------- [D7-brane]{} [D9-brane]{} [**Wilson line **]{} $\phi^1$ $\phi^2$ [**Wilson line **]{} $(\xi^{1,2})_0 \equiv \varphi^{1,2}$ [**Geom. modulus **]{} $\phi^3$ [**Wilson line **]{} $(\xi^3)_0 \equiv \varphi^3$ ---------------------------------------- -- ----------------------------------------------------------- where we are defining our type I fields as in (\[standcom\]) and (\[lights\]). Turning on the closed string background fluxes, it is easy to see that the same dictionary will still apply. Indeed, using the results of [@ciu04; @lustf; @osl] one expects the D7-brane Wilson line moduli $\phi^i$ to remain massless in the presence of background fluxes, and the geometric modulus $\phi^3$ to generically gain a mass. This latter point will of course depend on the choice of background fluxes and, by construction, we expect it to differ for both set of fluxes (\[iibflux1\]) and (\[iibflux2\]). Indeed, applying the analysis of [@ciu04] to the background fluxes above, it is easy to check that for the choice (\[iibflux1\]) $\phi^3$ remains massless, while for (\[iibflux2\]) a $\mu$-term is generated which exactly reproduces (\[massmu\]). In terms of wavefunctions, a more interesting sector is given by massive open string modes. Again, in the absence of closed string fluxes one has the dictionary --------------------------------------------- -- -------------------------------------------------------- [D7-brane]{} [D9-brane]{} [**KK mode on**]{} $(T^2)_1 \times (T^2)_2$ [**KK mode on**]{} $B_4 \simeq (T^2)_1 \times (T^2)_2$ [**Winding mode on**]{} $(T^2)_3$ [**KK mode on**]{} $\Pi_2 \simeq (T^2)_3$ --------------------------------------------- -- -------------------------------------------------------- between D7-brane and D9-brane massive modes. Let us now turn on background fluxes and translate our type I open string wavefunctions to the type IIB setup via the above dictionary. For simplicity, we will first focus on the gauge boson wavefunctions of Section \[sec:wgauge\]. A general result is then that a D9-brane KK mode along the base $B_4$ will never feel the effect of the fluxes, while the KK modes along the fiber $\Pi_2$ could indeed have a distorted wavefunction. More precisely, a KK mode on the fiber will behave as an open string charged under a magnetic flux $F_2^{\text{cl}}$ that depends on the $\Pi_2$ KK momenta. In terms of D7-brane modes, we thus obtain that KK modes are unaffected by the presence of type IIB $G_3$ fluxes, while winding modes behave as magnetized open strings. Indeed, it is not hard to convince oneself that a D7-D7 string winded around the closed path $\g \subset (T^2)_3$ can in principle feel different B-fields on both ends, and that their difference is given by B|\_ = \_H\_3 \[Bdif\] as illustrated in figure \[figw\]. Moreover, for a closed $H_3$ (\[Bdif\]) will only depend on the winding numbers of $\g$, which upon T-duality translate into the KK-modes $(k_3, k_6)$ on the elliptic fiber $\Pi_2$. Finally, one can check that computing (\[Bdif\]) for the examples (\[iibflux1\]) and (\[iibflux2\]) and mapping the result to the T-dual type I setup one indeed obtains the closed string magnetic flux $F_2^{\text{cl}}$. Hence, we can summarize the D7-brane winding mode wavefunction as \^ = \^[B]{}([x\_[B\_4]{}]{}) e\^[2i (k\_3 x\^3 + k\_6x\^6)]{} \[wwinding\] where $k_3, k_6$ are the winding modes of $\g$ in $(T^2)_3$, $\vec x_{B_4} = \{x^1,x^4, x^2,x^5\}$, and $\psi^{\Delta B}$ is the wavefunction of an open string in a magnetized D7-brane wrapping $B_4$, and whose magnetic flux is given by (\[Bdif\]). This clearly matches our type I T-dual results. ![\[figw\] Open string wavefunction for a D7-brane winding mode in the T-dual type IIB flux picture. Even if both ends of the open string sit on the same point in the internal space, they feel a different B-field due to the presence of the NSNS flux $H_3$ and the extended nature of the winding mode. As a result, D7-brane winding modes behave as open strings that end on D7-branes with different magnetizations, and so do their wavefunctions.](winding.eps){width="15cm"} Turning now to the wavefunctions for fermions and 4d scalars, it is easy to see that D7-brane KK modes should be insensitive to the presence of the flux. Winding modes, on the other hand, should feel the background flux in a more involved way than their gauge boson counterparts, as it is manifest from the matrix $\mathbb{M}$ that appears in their equation of motion in the type I picture, and which contains off diagonal terms proportional to the components of $F_2^{\text{cl}}$. In the case of the example (\[iibflux1\]) with vanishing $\mu$-terms on the D7, the off-diagonal terms should correspond to those of (\[system1\]), and they may be understood as the mixing terms $G_m^{\ \, p}$ that usually appear in the equations of motion for magnetized D-branes (see e.g., eq.(\[phieq\])), with the substitution $F_2 \raw F_2^{\text{cl}}$. The interpretation of these off-diagonal terms for the example with non-vanishing $\mu$-term (\[iibflux2\]) (given by those of (\[system2\])) remain however more obscure from the type IIB viewpoint. Note in particular that, according to our first dictionary above, the eigenfunctions (\[muxi3\]) and (\[muxipm\]) obtained in the type I side, should correspond to a bound state of winding modes of D7-brane Wilson lines and moduli. It would be interesting to understand how these eigenstates arise from the type IIB side of the duality.[^40] Finally, let us consider those matter field wavefunctions analyzed in Section \[sec:wmatter\]. From the type IIB side, the exotic W boson wavefunction (\[setmagnetico\]) and its generalization to non-vanishing D-term should arise from a D7-brane winding mode which also feels a difference on the open string magnetic flux $\Delta F_2 = (F_2^{\a\b})^{\text{op}}$. Hence, in this picture the total difference in flux felt by such a D7$_{\a}$-D7$_{\b}$ string is given by the gauge invariant quantity = B|\_ + 2’ F\_2 = 2F\_2\^ + 2(F\_2\^)\^ = 2(F\_2\^)\_ which is nothing but the open + closed effective flux entering the definition of the wavefunction (\[wavematter\]) and the more massive modes of this sector. Hence, we find that the open string wavefunctions obtained in the type I flux vacua studied in this paper fit nicely into our understanding of the D7-brane wavefunctions physics in the type IIB T-dual setup. Conclusions and outlook {#sec:conclu} ======================= In this work we have given a concrete prescription for performing dimensional reduction in flux compactifications. The procedure relies on the observation that in presence of closed string fluxes it is still possible to define some modified Dirac and Laplace-Beltrami operators in the internal manifold which account for the effect of the fluxes on the open string fluctuations. These operators are extracted from the type I supergravity action in the limit on which closed string fluctuations are frozen and the warping can be neglected. To analyze the spectrum of eigenmodes of these operators, we have found very helpful some of the tools of non-commutative harmonic analysis and representation theory, which we have summarized in Section \[gener\] and Appendix \[kirillov\]. This formalism seems to point out towards a deep connection between the 4d spectrum of massive excitations, symplectic geometry and 4d gauged supergravity algebras. In particular, we have found that the spectrum of Kaluza-Klein excitations for neutral and charged modes in a stack of magnetized D9-branes is classified by irreducible unitary representations of the Kaloper-Myers gauge algebra [@kaloper].[^41] Notice that for sectors of the theory which preserve enough number of supersymmetries, one can in addition consider the global symmetries of the effective action and compute other massive excitations such as winding modes. Indeed, notice that the Kaloper-Myers algebra is only a portion of the full $\mathcal{N}=4$ gauged supergravity algebra. It is therefore natural to conjecture that irreducible unitary representations of the full algebra classify not only Kaluza-Klein modes, but also winding and non-perturbative modes associated to the $\mathcal{N}=4$ sectors of the theory. Following this philosophy we have conjectured the presence of some massive non-perturbative charged modes in the worldvolume of magnetized D9-branes. We can extract several conclusions from the results of this paper. First, notice that generically there is always a set of fields which is insensitive to the background fluxes, and therefore their wavefunctions are the same than in a fluxless compactification. Moreover, the on-shell conditions usually ensure that these are the lightest modes in the limit of diluted fluxes and constant warping, which has two important consequences. On the one side, the lightest sector is usually not affected by the fluxes, up to possible flux induced mass terms. On the other, if one considers only this sector of the theory, it is enough to dimensionally reduce as if being in a fluxless compactification.[^42] Thus, we find that fluxes mainly affect the structure of massive Kaluza-Klein replicas. In particular, for the class of vacua that we have considered, the resulting spectrum can be understood in terms of Landau degeneracies, mass shifts and mixings induced by the fluxes. We therefore expect that fluxes change in an important way the threshold corrections to the 4d low energy effective theory. The computation of gauge threshold corrections in flux compactifications will be addressed in a future publication [@wip]. We have also observed that wavefunctions in the presence of closed string fluxes are not very different from wavefunctions in compactifications with only magnetized branes. This has been interpreted in the light of open/closed string duality, showing that in many cases the closed string fluxes can be interpreted as non diagonal magnetic fluxes in a dual background. There are several possible further directions to explore, apart from the ones already mentioned. For example, it would be interesting to see how the warping fits in this picture, and in particular to try to combine these results with the ones e.g. in [@fershiu]. This is particularly important for applying these methods in the context of the AdS/CFT correspondence. Some recent applications of wavefunctions in this context include models of holographic gauge mediation [@holographic], where Kaluza-Klein modes mediate the transmission of supersymmetry breaking between the hidden and visible sectors, and models for meson spectroscopy (see [@meson] for a review and references), where meson resonances are identified with Kaluza-Klein modes in a dual supergravity theory. We expect that the techniques introduced here will result useful in these contexts, once they are extended conveniently to account for the strong warping. Also, one could similarly consider other vacua different than the no-scale solutions that we have analyzed. For instance, we could make use of the same methods for dimensionally reduce type IIA $\textrm{AdS}_4$ compactifications on nearly Kähler manifolds, in the same spirit than in [@kashani1; @nearly]. This would be particularly relevant for computing the structure of massive modes in these backgrounds. Finally, from the phenomenological point of view, the vacua considered here are not very appealing, since they are non-chiral. In this sense, it would be desirable to extend this computation to models including magnetized D5-branes and more realistic matter content. In particular, the T-duals of the chiral flux compactifications considered in [@marchesashiu] fall into this class. With that same aim, it would be also desirable to extend these techniques to general, non-parallelizable SU(3)-structure manifolds. Acknowledgments {#acknowledgments .unnumbered} =============== [We would like to thank L. Alvarez-Gaumé, E. Dudas and A. Uranga for useful discussions and comments. The work of P.G.C. is supported by the European Union through an Individual Marie-Curie IEF. Additional support comes from the contracts ANR-05-BLAN-0079-02, MRTN-CT-2004-005104, MRTN-CT-2004-503369 and CNRS PICS \# 4172, 3747. Finally, we would like to thank the Ecole Polytechnique, CERN and the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work.]{} Fermion conventions {#ap:ferm} =================== In order to describe explicitly fermionic wavefunctions we take the following representation for $\G$-matrices in flat 10d space \^ = \^\_2 \_2 \_2 \^[[m]{}]{} = \_[(4)]{} \^[m-3]{} \[ulG:ap\] where $\mu = 0, \dots, 3$, labels the 4d Minkowski coordinates, whose gamma matrices are \^0 = ( [cc]{} 0 & -\_2\ \_2 & 0 ) \^i = ( [cc]{} 0 & \_i\ \_i & 0 ) $m = 4, \dots, 9$ labels the extra $\R^6$ coordinates [lll]{} \^[1]{} = \_1 \_2 \_2 & & \^[4]{} = \_2 \_2 \_2\ \^[2]{} = \_3 \_1 \_2 & & \^[5]{} = \_3 \_2 \_2\ \^[3]{} = \_3 \_3 \_1 & & \^[6]{} = \_3 \_3 \_2 \[tilgamma\] and $\sig_i$ indicate the usual Pauli matrices. The 4d chirality operator is then given by \_[(4)]{} = \_[(4)]{} \_2 \_2 \_2 where $\g_{(4)} = i \g^0\g^1\g^2\g^3$, and the 10d chirality operator by \_[(10)]{} = \_[(4)]{} \_[(6)]{} = ( [cc]{} \_2 & 0\ 0 & -\_2 ) \_3 \_3 \_3 with $\g_{(6)} = -i \tilde{\g}^1\tilde{\g}^2\tilde{\g}^3\tilde{\g}^4\tilde{\g}^5\tilde{\g}^6$. Finally, in this choice of representation a Majorana matrix is given by \[ap:Maj\] = \^[[2]{}]{}\^[[7]{}]{}\^[[8]{}]{}\^[[9]{}]{} = ( [cc]{} 0 & \_2\ -\_2 & 0 ) \_2 i\_1 \_2 = \_4 \_6 which indeed satisfies the conditions $\mathcal{B}\mathcal{B}^* = \Id$ and $\mathcal{B}\, \G^{{\underline}{M}} \mathcal{B}^* = \G^{{\underline}{M}*}$. Notice that the 4d and 6d Majorana matrices $\mathcal{B}_4 \equiv \g^2 \g_{(4)}$ and $\mathcal{B}_6 \equiv \tilde{\g}^4 \tilde{\g}^5 \tilde{\g}^6$ satisfy analogous conditions $\mathcal{B}_4\mathcal{B}_4^* = \mathcal{B}_6\mathcal{B}_6^* = \Id$ and $\mathcal{B}_4\, \g^{\mu} \mathcal{B}_4^* = \g^{\mu*}$, $\mathcal{B}_6\, \g^{m} \mathcal{B}_6^* = - \g^{m*}$. In the text we mainly work with 10d Majorana-Weyl spinors of negative chirality, meaning those spinors $\theta$ satisfying $\theta = - \G_{(10)} \theta = \mathcal{B}^*\theta^*$. In the conventions above this means that we have spinors of the form \[basisMW\] $$\begin{aligned} \theta^0\, =\, \psi^0 \, \left( \begin{array}{c} \xi_+ \\ 0 \end{array} \right) \otimes \chi_{---} + i (\psi^0)^*\, \left( \begin{array}{c} 0 \\ \sig_2\xi_+^* \end{array} \right) \otimes \chi_{+++}\\ \theta^1\, =\, \psi^1 \, \left( \begin{array}{c} \xi_+ \\ 0 \end{array} \right) \otimes \chi_{-++} - i (\psi^1)^*\, \left( \begin{array}{c} 0\\ \sig_2\xi_+^* \end{array} \right) \otimes \chi_{+--}\\ \theta^2\, =\, \psi^2 \, \left( \begin{array}{c} \xi_+ \\ 0 \end{array} \right) \otimes \chi_{+-+} + i (\psi^2)^*\, \left( \begin{array}{c} 0 \\ \sig_2\xi_+^* \end{array} \right) \otimes \chi_{-+-}\\ \theta^3\, =\, \psi^3 \, \left( \begin{array}{c} \xi_+ \\ 0 \end{array} \right) \otimes \chi_{++-} - i (\psi^3)^*\, \left( \begin{array}{c} 0 \\ \sig_2\xi_+^* \end{array} \right) \otimes \chi_{--+}\end{aligned}$$ where $\psi^j$ is the spinor wavefunction, $(\xi_+ \ 0)^t$ is a 4d spinor of positive chirality and $\chi_{\epsilon_1\epsilon_2\epsilon_3}$ is a basis of 6d spinors of such that \_[—]{} = ( [c]{} 0\ 1 ) ( [c]{} 0\ 1 ) ( [c]{} 0\ 1 ) \_[+++]{} = ( [c]{} 1\ 0 ) ( [c]{} 1\ 0 ) ( [c]{} 1\ 0 ) \[spinorbasis:ap\] etc. Note that these basis elements are eigenstates of the 6d chirality operator $\g_{(6)}$, with eigenvalues $\epsilon_1\epsilon_2\epsilon_3$. Finally, let us recall that to dimensionally reduce a 10d fermionic action, one has to simultaneously diagonalize two Dirac operators: $\slashed{\p}_{\IR^{1,3}}$ and $\slashed{D}^{\text{int}}$, built from $\G^{{\underline}{\mu}}$ and $\G^{{\underline}{m}}$, respectively. However, as these two set of $\G$-matrices do not commute, nor will $\slashed{\p}_{\IR^{1,3}}$ and $\slashed{D}^{\text{int}}$, and so we need instead to construct these Dirac operators from the alternative $\G$-matrices \^ = \_[(4)]{} \^ = \_[(4)]{}\^\_2 \_2 \_2 \^[[m]{}]{} = \_[(4)]{} \^[[m]{}]{} = \_4 \^[m-3]{} \[commG\] following the common practice in the literature. Warped Dirac equation {#ap:warp} ===================== Let us consider the 6d Dirac equation deduced in eq.(\[dirac6d\]) ( \^[\_6]{} + e\^[/2]{} \_3 - Z ) \_6 = Z\^[1/4]{} m\_ \_6\^\* \_6\^\* \[ap:dirac6d\] where now all slashed quantities are constructed from the set of $\G$-matrices defined in (\[commG\]). Let us also consider a compactification ansatz of the form (\[mansatz\]), where again $Z$ only depends on the coordinates of the base $B_4$. Then, as in [@geosoft], the 2-form $J$ splits as $J = J_{\Pi_2} + J_{B_4}$, and we can split $F_3$ accordingly. Indeed, let us define e\^[/2]{} F\_3\^[bg]{} e\^[/2]{} F\_3 - 2 \*\_[\_6]{} (d J\_[\_2]{}) \[F3bg\] so that eq.(\[ap:dirac6d\]) becomes ( \^[\_6]{} + e\^[/2]{} \_3\^[bg]{} - Z P\_+\^[\_2]{} ) \_6 = Z\^[-1/4]{} m\_4 \_6\^\* \_6\^\* where we have introduced the projectors $P_\pm^{\Pi_2}$ defined in (\[ex1proj\]). In addition, we have that the covariant derivative reads \^[\_6]{}\_m = \_m + \^[B\_4]{}\_m - (\_mZ -\_[m]{}Z ) - \_m\^[n]{} (\_nZ -\_[n]{}Z - \_[n]{} ) where $\om^{B_4}$ is the spin connection of $B_4$, $f_{mnp}$ is defined by (\[metricflux\]) and $\Lambda$ is a block-diagonal matrix specified by \_[mn]{} = g\_[mn]{} -2 e\^[[a]{}]{}\_m e\_[[a]{}n]{},a \_2 \[Lam\] Finally, (\[F3bg\]) implies that e\^[/2]{} F\_3\^[bg]{} = \*\_[\_6]{} and this, if the $B_4$ base is symplectic, implies that $e^{\phi/2} \slashed{F}_3^{\text{bg}} = i \slashed{f} \slashed{J}_{\Pi_2} \g_{(6)}$. We thus obtain a 6d Dirac equation of the form (\^[\_2]{} + \^[B\_4]{} + P\_+\^[\_2]{} - Z ( P\_+\^[\_2]{} -)) \_6 = Z\^[1/4]{} m\_4 \_6\^\* \_6\^\* \[ap:dirac6dsf\] containing the coupling of fermions to the warping. Note that by taking $Z = 1$ we recover the unwarped equation (\[dirac6duw\]) used in the main text. Now, if we normalize the internal spinor as $\chi_6^\dagger\chi_6 =1$, then the warp factor dependence of the metric ansatz (\[mansatz\]) will induce a non-standard 4d kinetic terms for $\chi_4$. In order to recover a canonical kinetic terms upon dimensional reduction we need instead to consider the rescaled Weyl fermion Z\^[-7/8]{} \_6 in terms of which the warped 6d Dirac equation reads (\^[\_2]{} + \^[B\_4]{} + P\_+\^[\_2]{} - Z P\_+\^[\_2]{} ) = Z\^[1/4]{} m\_4 \_6\^\* \^\* \[ap:6dnorm\] Note that the projector $P_+^{\Pi_2}$ is basically the chirality projector of the 4d base $B_4$. As in (\[split4+2\]), let us split $\eta$ as = \_[\_2]{} + \_[B\_4]{} \[ap:split4+2\] where $P_+^{\Pi_2} \eta_{\Pi_2} = \eta_{\Pi_2}$, $P_+^{\Pi_2} \eta_{B_4} = 0$. We can then split the Dirac equation (\[ap:6dnorm\]) as \[ap:6dsplit1\] \^[\_2]{}\_\_[B\_4]{} + \^[B\_4]{}\_ Z\^[-1]{} \_[\_2]{} & = & m\_4 \_6\^\* \_[B\_4]{}\^\*\ \[ap:6dsplit2\] \^[\_2]{}\_\_[\_2]{} + Z\^[-1]{} \^[B\_4]{}\_ \_[B\_4]{} + Z\^[-2]{} \_ \_[\_2]{} & = & m\_4 \_6\^\* \_[\_2]{}\^\* where we have extracted the warp factor dependence from the $\Gamma$-matrices contractions. Note that a simple set of solutions is obtained by setting $\eta_{\Pi_2} = 0$, $m_4 = 0$ and $\slashed{D}^{B_4} \eta_{B_4} = 0$, since neither the warp factor nor the fluxes play any role in this case. This simple zero mode equation does not come as a surprise if one compares eq.(\[ap:6dnorm\]) with the Dirac equation for D7-branes in type IIB warped Calabi-Yau flux backgrounds. Indeed, by the results of [@fershiu] it is easy to identify the modes $\eta_{\Pi_2}$ in (\[ap:split4+2\]) as those containing the gaugino and geometric modulini of a T-dual D7-brane, as well as their KK replicas, whereas $\eta_{B_4}$ are T-dual to the D7-brane Wilsonini.[^43] Now, since the Wilson line zero modes of a D7-brane do not feel the effect of the background fluxes [@ciu04; @osl] nor that of the warping [@fershiu], the same statement must apply to the open string zero modes arising from $\eta_{B_4}$, as is indeed the case. A non-supersymmetric example {#ap:N=0} ============================ On the main text we have analyzed examples where the closed string background fluxes preserve at least $\mathcal{N}=1$ supersymmetry in four dimensions. However, as we have treated bosons and fermions independently, our techniques apply equally well to ${\cn = 0}$ vacua of the theory. To illustrate this fact, in this appendix we apply them to one of such examples, based on a compactification on the Heisenberg manifold. Let us then consider the background $$\begin{aligned} &ds^2=Z^{-1/2}(ds^2_{\mathbb{R}^{1,3}}+ds^2_{\Pi_2})+Z^{3/2}ds^2_{T^4} \\ &ds^2_{T^4}=(2\pi)^2\sum_{m=1,2,4,5}(R_m dx^m)^2 \\ &ds^2_{\Pi_2}=( 2\pi)^2[ (R_3 dx^3)^2+(R_6 \tilde{e}^{6})^2] \\ &F_3=-(2\pi)^2N dx^1\wedge dx^2\wedge \tilde e^6-g_s^{-1}*_{T^4}dZ^2 \\ &e^\phi Z=g_s=\textrm{const.}\end{aligned}$$ which is almost identical to (\[bg1\]). In the present case, however, $\tilde e^6$ stands for the left-invariant 1-form satisfying $$d \tilde{e}^6=M dx^4\wedge dx^5$$ so that $\mathcal{M}_6$ is given locally by $\mathbb{R}\times \mathcal{H}_3$. The twisted derivatives are then $$\begin{aligned} \hat\partial_1&=(2\pi R_1)^{-1}\partial_{x^1} & \hat\partial_4&=(2\pi R_4)^{-1}(\partial_{x^4}+\frac{M}{2}x^5\partial_{x^6})\\ \hat\partial_2&=(2\pi R_2)^{-1}\partial_{x^2} & \hat\partial_5&=(2\pi R_5)^{-1}(\partial_{x^5}-\frac{M}{2}x^4\partial_{x^6})\\ \hat\partial_3&=(2\pi R_3)^{-1}\partial_{x^3} & \hat\partial_6&=(2\pi R_6)^{-1}\partial_{x^6}\end{aligned}$$ Finally, the compact structure of $\mathcal{M}_6$ is produced by the following identifications which result from quotienting by $\Gamma = \G_{\ch_3} \times \IZ^3$ $$\begin{aligned} & x^4\to x^4+1 \qquad x^6\to x^6 - \frac{M}{2}x^5\\ & x^5\to x^5+1 \qquad x^6\to x^6 + \frac{M}{2}x^4\\ & x^i \to x^i + 1 \qquad \qquad \textrm{for }i\neq 4,5\end{aligned}$$ In addition, the equations of motion require the conditions $R_4R_5=4\pi^2 R_6^2R_1R_2$ and $g_sN=M$, with $N,\ M \in \mathbb{Z}$. This in particular ensures that the first torsion class $\mathcal{W}_1$, defined as $J\wedge d\Omega = \mathcal{W}_1 J\wedge J\wedge J$, is non-vanishing and, hence, $\mathcal{M}_6$ is not a complex manifold. As the gravitino mass is proportional to $\mathcal{W}_1$ [@geosoft; @lawrence; @dwsb], this reflects the fact that the background does not preserves any supersymmetry in 4d. As in our previous examples, in order to cancel the RR charges and tensions, O5-planes (and maybe also D5-branes) wrapping $\Pi_2$ are required, which again will be introduced via the orbifold quotient $\mathcal{R}: x^m \mapsto -x^m$ on the $T^4$ base coordinates. To simplify our discussion, in this section we will assume $F_2=0$, although one can easily add the effect of a non-trivial $F_2$ along the lines of Section \[sec:wmatter\]. Bosonic wavefunctions --------------------- As usual, the wavefunction for the four dimensional neutral gauge bosons is given by the eigenfunctions of the corresponding Laplace-Beltrami operator of the manifold $$\hat\partial_m\hat\partial^m B=-m_B^2B$$ The solutions to this equations can be found using the techniques described in Section \[sec:wgauge\]. More precisely, we find two towers of KK modes associated to the four dimensional gauge boson, which in a suitable polarization read $$B_{k_1,k_2,k_3,k_4,k_5}(\vec x)=\textrm{exp}[2\pi i(k_1x^1+k_2x^2+k_3x^3+k_4x^4+k_5x^5)]\label{nosusy1}$$ for the first tower, with mass eigenvalue $$m_B^2=\sum_{i=1}^5\left(\frac{k_i}{R_i}\right)^2$$ while for the second tower $$\begin{gathered} B^{\delta}_{n,k_1,k_2,k_3,k_6}=\left(\frac{2\pi^2R_5|k_6 M|}{R_4\textrm{Vol}_{\mathcal{M}_6}}\right)^{1/4} \sum_{k_4\in \delta+k_6M\mathbb{N}} \psi_{n}\left(\frac{\dot x^5}{\sqrt{2}}\right) e^{2\pi i \left(k_1x^1+k_2x^2+k_3 x^3+k_4 x^4+k_6 \dot x^6\right)} \label{nosusy2}\end{gathered}$$ with eigenvalue $$m_B^2=\frac{ |k_6\varepsilon|}{R_6}(2n+1)+\sum_{m=1,2,3,6} \left(\frac{k_m}{R_m}\right)^2$$ with $\delta=0\ldots k_6M-1$, $\varepsilon=MR_6/2\pi R_4R_5$ and $$\dot x^5=\left(\frac{4\pi R_5}{R_4|k_6M|}\right)^{1/2}(k_4+k_6M)\quad \qquad \dot x^6\equiv x^6+\frac{M}{2}x^4x^5$$ Similarly, we can work out the wavefunctions for the four dimensional scalars. Plugging the background into eqs.(\[xi+\])-(\[xi-\]) leads to an equation of the form (\[eigenval\]) where, in complex coordinates (\[standcom\]), the mass matrix now reads $$\mathbb{M}=\begin{pmatrix}\hat\partial_m\hat\partial^m&-\varepsilon\hat\partial_6&-\frac{i\varepsilon}{2}\hat\partial_{z^2}&0&0&\frac{i\varepsilon}{2}\hat\partial_{z^2}\\ \varepsilon\hat\partial_6&\hat\partial_m\hat\partial^m&\frac{i\varepsilon}{2}\hat\partial_{z^1}&0&0&-\frac{i\varepsilon}{2}\hat\partial_{z^1}\\ -\frac{i\varepsilon}{2}\hat\partial_{\bar z^2}&\frac{i\varepsilon}{2}\hat\partial_{\bar z^1}&\hat\partial_m\hat\partial^m-\frac{\varepsilon^2}{2}&-\frac{i\varepsilon}{2}\hat\partial_{z^2}&\frac{i\varepsilon}{2}\hat\partial_{z^1}&\frac{\varepsilon^2}{2}\\ 0&0&-\frac{i\varepsilon}{2}\hat\partial_{\bar z^2}&\hat\partial_m\hat\partial^m&-\varepsilon\hat\partial_6&\frac{i\varepsilon}{2}\hat\partial_{\bar z^2}\\ 0&0&\frac{i\varepsilon}{2}\hat\partial_{\bar z^1}&\varepsilon\hat\partial_6&\hat\partial_m\hat\partial^m&-\frac{i\varepsilon}{2}\hat\partial_{\bar z^1}\\ \frac{i\varepsilon}{2}\hat\partial_{\bar z^2}&-\frac{i\varepsilon}{2}\hat\partial_{\bar z^1}&\frac{\varepsilon^2}{2}&\frac{i\varepsilon}{2}\hat\partial_{z^2}&-\frac{i\varepsilon}{2}\hat\partial_{z^1}&\hat\partial_m\hat\partial^m-\frac{\varepsilon^2}{2} \end{pmatrix}\label{scalarnosusy}$$ This is a non-commutative eigenvalue problem similar to the one found in Section \[nonvanish\]. Notice, however, that in the present case the mass matrix is not block diagonal, reflecting the fact that the background does not preserves the complex structure of $T^6$, and in particular the complex structure given by the choice (\[standcom\]). As in the supersymmetric case, the eigenvalues and eigenfunctions of (\[scalarnosusy\]) can be found we the aid of the commutation relations of the twisted derivatives and the Laplacian, which in the present case read $$\begin{aligned} & [ \hat{\p}_{{z}^1}, \hat{\p}_{{z}^2}] \, =\, [ \hat{\p}_{\bar{z}^1}, \hat{\p}_{\bar{z}^2}] \, =\, \varepsilon \hat{\p}_{6} \quad \quad [ \hat{\p}_{{z}^1}, \hat{\p}_{\bar{z}^2}] \, =\, [ \hat{\p}_{\bar{z}^1}, \hat{\p}_{{z}^2}] \, =\, - \varepsilon \hat{\p}_{6}\\ &- [ \hat{\p}_m\hat{\p}^m, \hat{\p}_{{z}^1}] \, =\, [ \hat{\p}_m\hat{\p}^m, \hat{\p}_{\bar{z}^1}] \, =\,\varepsilon \hat{\p}_{6} (\hat{\p}_{\bar{z}^2}-\hat{\p}_{z^2})\\ &[ \hat{\p}_m\hat{\p}^m, \hat{\p}_{{z}^2}] \, =\, - [ \hat{\p}_m\hat{\p}^m, \hat{\p}_{\bar{z}^2}] \, =\,\varepsilon \hat{\p}_{6} (\hat{\p}_{\bar{z}^1}-\hat{\p}_{z^1})\end{aligned}$$ After some work, we find that the resulting spectrum is given by the two eigenvectors $$\xi_3(\vec x)\equiv\begin{pmatrix}0\\ 0\\ 1\\ 0\\ 0\\ 1\end{pmatrix}B(\vec x)\quad \qquad \xi_{3}^*(\vec x)\equiv \begin{pmatrix}\hat{\p}_{\bar{z}^1}\\ \hat{\p}_{\bar{z}^2}\\ 2i\hat{\p}_6\\ \hat{\p}_{{z}^1}\\ \hat{\p}_{{z}^2}\\ 0\end{pmatrix}B(\vec x)$$ with mass eigenvalues $m_{\xi_{3}}^2=m_{\xi_{3}^*}^2=m_B^2$, the two eigenvectors $$\xi_+(\vec x)\equiv\begin{pmatrix}\hat{\p}_{\bar{z}^1}-i\hat{\p}_{{z}^2}\\ \hat{\p}_{\bar{z}^2}+i\hat{\p}_{{z}^1}\\ 0\\ -\hat{\p}_{{z}^1}+i\hat{\p}_{\bar{z}^2}\\ -\hat{\p}_{{z}^2}-i\hat{\p}_{\bar{z}^1}\\ 0\end{pmatrix}(\hat{\p}_4-i\hat{\p}_5)B(\vec x) \quad \qquad \xi_{-}(\vec x)\equiv \begin{pmatrix}-\hat{\p}_{\bar{z}^1}-i\hat{\p}_{{z}^2}\\ -\hat{\p}_{\bar{z}^2}+i\hat{\p}_{{z}^1}\\ 0\\ \hat{\p}_{{z}^1}+i\hat{\p}_{\bar{z}^2}\\ \hat{\p}_{{z}^2}-i\hat{\p}_{\bar{z}^1}\\ 0\end{pmatrix}(\hat{\p}_4+i\hat{\p}_5)B(\vec x)$$ with mass eigenvalues $m_{\xi_{\pm}}^2=m_B^2\pm (\varepsilon k_6/R_6)$, and the two eigenvectors $$\xi_{\pm}^*\equiv\begin{pmatrix}-(m^2_{\xi_{\pm}^*}-m^2_B)\hat{\p}_{z^2}+ \varepsilon\hat{\p}_{\bar{z}^1}\hat{\p}_6\\ (m^2_{\xi_{\pm}^*}-m^2_B)\hat{\p}_{z^1}+ \varepsilon\hat{\p}_{\bar{z}^2}\hat{\p}_6\\ i\frac{(m^2_{\xi_{\pm}^*}-m^2_B)^2}{\varepsilon} +i\varepsilon\hat{\p}_6\\ -(m^2_{\xi_{\pm}^*}-m^2_B)\hat{\p}_{\bar{z}^2}+\varepsilon\hat{\p}_{{z}^1}\hat{\p}_6\\ (m^2_{\xi_{\pm}^*}-m^2_B)\hat{\p}_{\bar{z}^1}+\varepsilon\hat{\p}_{{z}^2}\hat{\p}_6\\ -i\frac{(m^2_{\xi_{\pm}^*}-m^2_B)^2}{\mu}+i\varepsilon\hat{\p}_6 \end{pmatrix}B(\vec x)$$ with mass eigenvalue $$m^2_{\xi_{\pm}^*}=\frac14 \left(\varepsilon\pm\sqrt{\varepsilon^2+4m_B^2-4\left(\frac{k_3}{R_3}\right)^2}\right)^2 +\left(\frac{k_3}{R_3}\right)^2$$ where in all these expressions $B(\vec x)$ is a gauge boson wavefunction, (\[nosusy1\])-(\[nosusy2\]), with mass eigenvalue $m_B$. Fermionic wavefunctions {#fermionic-wavefunctions} ----------------------- Regarding the fermionic wavefunctions, in this case we have that = (2)\^[-1]{} \^[456]{} = (2)\^[-1]{} i \_2 \_1 \_2 = \_6 and so the Dirac operator reads + [**F**]{} = ( [cccc]{} i & \_[[z]{}\^1]{} & \_[[z]{}\^2]{} & \_[[z]{}\^3]{}\ - \_[[z]{}\^1]{} & 0 & - \_[|[z]{}\^3]{} & \_[|[z]{}\^2]{}\ - \_[[z]{}\^2]{} & \_[|[z]{}\^3]{} & 0 & - \_[|[z]{}\^1]{}\ - \_[[z]{}\^3]{} & - \_[|[z]{}\^2]{} & \_[|[z]{}\^1]{} & i ) from which we extract the following mass matrix -([**D**]{} + [**F**]{})\^\* ([**D**]{} + [**F**]{}) = ( [cccc]{} \_m\^m - & i \_[[z]{}\^1]{} & i \_[[z]{}\^2]{} & 0\ i \_[|[z]{}\^1]{} & \_m\^m & - \_[6]{} & - i \_[[z]{}\^2]{}\ i \_[|[z]{}\^2]{} & \_[6]{} & \_m\^m & i \_[[z]{}\^1]{}\ 0 & -i \_[z\^2]{} & i\_[z\^1]{} & \_m\^m - ) Since the background does not preserve any supersymmetry, it is natural to expect the eigenfunctions and eigenvalues of this matrix to be rather different from their bosonic counterparts. Indeed, after some algebra one can show that the fermionic wavefunctions are given by the eigenvectors [$$\Psi_{\pm}(\vec x)\equiv\begin{pmatrix} i\varepsilon(m^2_{\Psi_\pm}+\hat{\p}_{z^3}\hat{\p}_{\bar{z}^3})\\ 2(\hat{\p}_{\bar z^1}+i\hat{\p}_{z^2})(m^2_{\Psi_\pm}-m_B^2)\\ 2(\hat{\p}_{\bar z^2}-i\hat{\p}_{z^1})(m^2_{\Psi_\pm}-m_B^2)\\ \varepsilon(m^2_{\Psi_\pm}+\hat{\p}_{z^3}\hat{\p}_{\bar{z}^3}) \end{pmatrix}B(\vec x)\quad \qquad \Psi'_{\pm}(\vec x)\equiv\begin{pmatrix} -i\varepsilon(m^2_{\Psi'_{\pm}}+\hat{\p}_{z^3}\hat{\p}_{\bar{z}^3})\\ -2(\hat{\p}_{\bar z^1}-i\hat{\p}_{z^2})(m^2_{\Psi'_{\pm}}-m_B^2)\\ -2(\hat{\p}_{\bar z^2}+i\hat{\p}_{z^1})(m^2_{\Psi'_{\pm}}-m_B^2)\\ \varepsilon(m^2_{\Psi'_{\pm}}+\hat{\p}_{z^3}\hat{\p}_{\bar{z}^3}) \end{pmatrix}B(\vec x)$$]{} with mass eigenvalues $$\begin{aligned} m^2_{\Psi_\pm}&=\frac{1}{16}\left(\varepsilon\pm \sqrt{16m_B^2+\varepsilon^2-16\left(\frac{k_3}{R_3}\right)^2 -\frac{\varepsilon k_6}{2R_6}}\right)^2+\left(\frac{k_3}{R_3}\right)^2\\ m^2_{\Psi'_{\pm}}&=\frac{1}{16}\left(\varepsilon\pm \sqrt{16m_B^2+\varepsilon^2-16\left(\frac{k_3}{R_3}\right)^2 +\frac{\varepsilon k_6}{2R_6}}\right)^2+\left(\frac{k_3}{R_3}\right)^2\end{aligned}$$ The orbit method {#kirillov} ================ In this appendix we summarize the notions of representation theory required for solving the generalized Dirac and Laplace equations in parallelizable manifolds. More precisely we consider the orbit method developed mostly by A. Kirillov in the 60’s, applied to nilmanifolds.[^44] Basically, the method relies the existence of a connection between harmonic analysis and symplectic geometry. The main objects are the orbits of a coadjoint action, which we will define in brief. These orbits turn out to be in one to one correspondence with the irreducible unitary representations of the group. More precisely, consider a compact nilmanifold given by $\mathcal{M}=G/\Gamma$, with $G$ a nilpotent group and $\Gamma$ a discrete subgroup. For matrix groups, we can introduce the hermitian product $$\langle A,B \rangle \equiv \textrm{Tr}(AB)$$ for $A,B\in \textrm{Mat}_n(\mathbb{R})$. We can then introduce the algebra $\mathfrak{g}^*$, dual to the Lie algebra of $G$, $\mathfrak{g}=\textrm{Lie}(G)$, through the partition $$\textrm{Mat}_n(\mathbb{R})=\mathfrak{g}^*\ \oplus\ \mathfrak{g}^\perp$$ where $$\mathfrak{g}^\perp=\{A\in \textrm{Mat}_n(\mathbb{R})\ |\ \langle A,B\rangle = 0 \ \forall \ B\in \mathfrak{g} \}$$ The coadjoint representation $K$ of $\mathfrak{g}^*$ is then defined as $$K(g): \ \mathfrak{g}^* \to \mathfrak{g}^*, \qquad K(g)F=p_{\mathfrak{g}^*}(gFg^{-1})$$ for $g\in G$ and $p_{\mathfrak{g}^*}$ the projector of $\textrm{Mat}_n(\mathbb{R})$ onto $\mathfrak{g}^*$. The central idea underlying the orbit method then states that there is a one to one correspondence between the orbits $\Omega$ of the coadjoint action $K$, and the irreducible unitary representations of $\mathfrak{g}$ acting on $L^2(\mathbb{R}^{\frac{\textrm{dim }\Omega}{2}})$, given by $$\pi_\Omega(g)u(\vec s)=e^{2\pi i \langle F,\textrm{log }h(\vec s,g)\rangle}u(\vec s\cdot g) \label{form}$$ acting on $L^2(\mathbb{R}^{\frac{\textrm{dim }\Omega}{2}})$. This equation needs some explanation. Here, $F$ is an arbitrary point in $\Omega$, whereas $\textrm{log }h(\vec s,g)$ represents the Lie algebra element corresponding to the group element $h(\vec s,g)$. The latter is a solution of the master equation $$S(\vec s)g=h(\vec s,g)S(\vec s\cdot g)$$ with $S$ a section $G/H \to G$, and $H\in G$ the subgroup corresponding to a subalgebra $\mathfrak{h}\in\mathfrak{g}$ of dimension dim $\mathfrak{h}=\textrm{dim }\mathfrak{g}-\frac12\textrm{dim }\Omega$ [^45] such that $$\langle F,[\mathfrak{h},\mathfrak{h}]\rangle=0$$ Each subalgebra of the right dimension satisfying this equation leads to a different manifold polarization of the representation associated to an orbit $\Omega$, and different polarizations are related among themselves by generalizations of the Abelian Fourier transform. In order to illustrate this powerful procedure, in what follows we consider a couple of examples relevant for the material presented in the main text.\ [**Example 1. Irreducible unitary representations of $\mathcal{H}_{2p+1}$**]{} Consider the $2p+1$ dimensional Heisenberg group. As already mentioned in Section \[nili\], a suitable matrix representation for the group is given by (\[heis\]) $$G=\begin{pmatrix}1 & -\frac12\vec{y}^t & \frac12\vec{x}^t & z\\ 0 & 1 & 0 & \vec x \\ 0 & 0 & 1 & \vec y\\ 0 & 0 & 0 & 1\end{pmatrix}$$ From here the matrix representations for $\mathfrak{g}$ and $\mathfrak{g}^*$ are easily worked out $${g}=\begin{pmatrix}0 & -\frac12\vec{y}^t & \frac12\vec{x}^t & z\\ 0 & 0 & 0 & \vec x \\ 0 & 0 & 0 & \vec y\\ 0 & 0 & 0 & 0 \end{pmatrix}\quad \qquad {g}^*=\begin{pmatrix}0&0&0&0\\ -\vec g_y&0&0&0\\ \vec g_x&0&0&0\\ g_z&\frac12\vec{g}_x^t&\frac12\vec{g}_y^t&0 \end{pmatrix}\label{gg}$$ where $\vec g_x$ and $\vec g_y$ are $p$-dimensional vectors. The coadjoint representation then reads $$K(G)(\vec{g}_x,\vec{g}_y,g_z)=(\vec{g}_x+\vec y\cdot\vec{g}_z,\ \vec{g}_y-\vec x\cdot\vec{g}_z,\ g_z)$$ Observe that there are only two types of orbits: zero dimensional orbits given by the points $\Omega_{\mu,\nu}\equiv(\vec{\mu},\vec{\nu},0)$ with $\vec{\mu}$ and $\vec{\nu}$ constant vectors, and two dimensional orbits given by the hyperplanes $\Omega_\lambda\equiv(*,*,\lambda)$, with $\lambda\neq 0$. The irreducible unitary representations associated to zero dimensional orbits, $\Omega_{\mu,\nu}$, can be worked out very easily. The corresponding subalgebra is $(2p+1)$-dimensional, and therefore it is the full Heisenberg algebra. The master equation becomes trivial, and the corresponding irreducible unitary representations are given by $$\pi_{\mu,\nu}=e^{2\pi i\langle\left.\mathfrak{g}^*\right|_{\Omega_{\mu,\nu}},\mathfrak{g}\rangle}=e^{2\pi i(\vec \mu\cdot \vec x+\vec \nu\cdot \vec y)}$$ For the irreducible unitary representations associated to the $2p$-dimensional orbits, $\Omega_\lambda$, we have to select a $p+1$-dimensional subalgebra $\mathfrak{h}$ such that $$\langle\left.\mathfrak{g}^*\right|_{\Omega_\lambda}, \ [\mathfrak{h},\mathfrak{h}]\rangle=0$$ Different choices correspond to different manifold polarizations. Here, for concreteness, we focus in the subalgebra generated by $\vec x=0$ in (\[gg\]). A suitable section in $G/H$ is then given by $$S(\vec s)=\begin{pmatrix}1 & 0 & -\frac12\vec{s}^t & 0\\ 0 & 1 & 0 & \vec s \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}$$ and the solution to the master equation reads $$h(\vec s, g)=\begin{pmatrix}1&-\frac12 \vec y&0&z+\frac12\vec s\cdot \vec y\\ 0&1&0&0\\ 0&0&1&\vec y\\ 0&0&0&1\end{pmatrix} \qquad \vec s\cdot g = (\vec x+\vec s, \vec y, z)$$ Plugging into eq.(\[form\]) we finally get the irreducible unitary representations associated to the orbits $\Omega_\lambda$ $$\pi_\lambda u(\vec s)=e^{2\pi i\lambda [z+\vec y\cdot \vec s+\vec x\cdot\vec y/2]}u(\vec x+\vec s)$$ In this way we have rederived the Stone - von Neumann theorem, discussed in eqs.(\[rep1\])-(\[rep2\]), by means of the orbit method. Let us now consider a more involved example.\ [**Example 2. Irreducible unitary representations of the algebra (\[nil2\])**]{} Consider the nilpotent group associated to the nilmanifold defined by eq.(\[nil2\]). Matrix representations for the group, the algebra, and the dual algebra, can be easily worked out, resulting in $$G=\begin{pmatrix}1&-\frac{M_6x^1}{2}&0&0&0&\frac{M_6x^5}{2}&x^6\\ 0&1&0&0&0&0&x^5\\ 0&0&1&0&0&0&x^4\\ 0&0&0&1&-\frac{M_3x^1}{2}&\frac{M_3x^2}{2}&x^3\\ 0&0&0&0&1&0&x^2\\ 0&0&0&0&0&1&x^1\\ 0&0&0&0&0&0&1 \end{pmatrix}$$ $$\mathfrak{g}=\begin{pmatrix}0&-\frac{M_6x^1}{2}&0&0&0&\frac{M_6x^5}{2}&x^6\\ 0&0&0&0&0&0&x^5\\ 0&0&0&0&0&0&x^4\\ 0&0&0&0&-\frac{M_3x^1}{2}&\frac{M_3x^2}{2}&x^3\\ 0&0&0&0&0&0&x^2\\ 0&0&0&0&0&0&x^1\\ 0&0&0&0&0&0&0 \end{pmatrix}\quad \quad \mathfrak{g}^*=\begin{pmatrix}0&0&0&0&0&0&0\\ -\frac{2g_1}{M_6}&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ \frac{2g_5}{M_6}&0&0&\frac{2g_2}{M_3}&0&0&0\\ g_6&g_5&g_4&g_3&g_2&g_1&0 \end{pmatrix}$$ The projector into $\mathfrak{g}^*$ is given by $$P_{\mathfrak{g}^*}(A)=\begin{pmatrix}0&0&0&0&0&0&0\\ \frac{A_{21}}{2}-\frac{A_{76}}{M_6}&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ \frac{A_{61}}{2}+\frac{A_{72}}{M_6}&0&0&\frac{A_{64}}{2}+\frac{A_{75}}{M_3}&0&0&0\\ A_{71}&\frac{A_{72}}{2}+\frac{M_6A_{61}}{4}&A_{73}&A_{74}&\frac{A_{75}}{2}+\frac{M_3A_{64}}{4}&\frac{A_{76}}{2}-\frac{M_6A_{21}}{4}&0 \end{pmatrix}$$ From these expressions, the coadjoint representation reads $$\begin{gathered} K(G)(g_1,g_2,g_3,g_4,g_5,g_6)=\left(g_1-\frac14 g_3M_3x^2-\frac12 g_6M_6x^5,\ g_2 +\frac12M_3g_3x^1,\ g_3,\right.\\ \left.g_4,\ g_5+\frac12M_6g_6x^1,\ g_6\right)\end{gathered}$$ We observe, therefore, four classes of orbits, one 0-dimensional, and three 2-dimensional, given by $$\begin{aligned} \Omega_{\mu,\nu,\sigma,\rho}&=(\mu,\nu,0,\sigma,\rho,0)\\ \Omega_{\mu,r,p}&=(*_1,M_3r*_2,r,\mu,M_6p*_2,p)\\ \Omega_{\mu,\nu,p}&=(*_1,\mu,0,\nu,*_2,p)\\ \Omega_{\mu,r,\nu}&=(*_1,*_2,r,\mu,\nu,0)\end{aligned}$$ with $p,r\neq 0$. Proceeding as in the previous example, we arrive to the following set of irreducible unitary representations $$\begin{aligned} \pi_{\mu,\nu,\sigma,\rho}&=e^{2\pi i(\mu x^1+\nu x^2+\sigma x^4+\rho x^5)} \label{pi1}\\ \pi_{\mu,r,p}u(s_1)&=e^{2\pi i(\mu x^4+ra_3+pa_6)]}u(s_1+x^1)\\ \pi_{\mu,\nu,p}u(s_1)&=e^{2\pi i(\mu x^2+\nu x^4+pa_6)}u(s_1+x^1) \\ \pi_{\mu,r,\nu}u(s_1)&=e^{2\pi i(\mu x^4+\nu x^5+ra_3)}u(s_1+x^1) \label{pi6}\end{aligned}$$ where $$a_3\equiv x^3-M_3x^2\left(s_1+\frac{x^1}{2}\right)\ ,\qquad a_6\equiv x^6-M_6x^5\left(s_1+\frac{x^1}{2}\right)$$ and for simplicity we have taken the same polarization for all the representations. Scalar wavefunction matrix {#ap:matrix} ========================== Let us rewrite eqs.(\[xi+\]) and (\[xi-\]) in matrix notation, and more precisely as $$\left[\mathbb{M}+m_b^2\ \mathbb{I}_{6\times 6}\right]\mathbb{V}=0 \label{ap:eigenval}$$ where = \^1\ \^2\ \^3\ \^[\*1]{}\ \^[\*2]{}\ \^[\*3]{} [c]{} \^1 \_[B\_4]{}\^1+i\_[B\_4]{}\^4\ \^2 \_[B\_4]{}\^2+i\_[B\_4]{}\^5\ \^3 \_[\_2]{}\^3+i\_[\_2]{}\^6 [c]{} \^[\*1]{} \_[B\_4]{}\^1-i\_[B\_4]{}\^4\ \^[\*2]{} \_[B\_4]{}\^2-i\_[B\_4]{}\^5\ \^[\*3]{} \_[\_2]{}\^3-i\_[\_2]{}\^6 \[ap:standcom\] and = \_m\^m \_6 + ( [cc]{} A & B\ -B\^& A\^\* ) with A = ( [ccc]{} 0 & - (G\_-)\^3\_[|1 2]{} \_3 - (G\_-)\^[|3]{}\_[|1 2]{} \_[|3]{} & - (G\_+)\^[|3]{}\_[|1 |2]{} \_[2]{} - (G\_+)\^[|3]{}\_[|1 2]{} \_[|2]{}\ (G\_-)\^3\_[1 |2]{} \_3 + (G\_-)\^[|3]{}\_[1 |2]{} \_[|3]{} & 0 & (G\_+)\^[|3]{}\_[|1 |2]{} \_[1]{} + (G\_+)\^[|3]{}\_[1 |2]{} \_[|1]{}\ (G\_+)\^[3]{}\_[1 |2]{} \_[2]{} + (G\_+)\^[3]{}\_[12]{} \_[|2]{} & - (G\_+)\^[3]{}\_[|1 2]{} \_[1]{} - (G\_+)\^[3]{}\_[12]{} \_[|1]{} & - a ) B = ( [ccc]{} 0 & - (G\_-)\^3\_[|1 |2]{} \_3 - (G\_-)\^[|3]{}\_[|1 |2]{} \_[|3]{} & - (G\_+)\^[3]{}\_[|1 |2]{} \_[2]{} - (G\_+)\^[3]{}\_[|1 2]{} \_[|2]{}\ (G\_-)\^3\_[|1 |2]{} \_3 + (G\_-)\^[|3]{}\_[|1 |2]{} \_[|3]{} & 0 & (G\_+)\^[3]{}\_[|1 |2]{} \_[1]{} + (G\_+)\^[3]{}\_[1 |2]{} \_[|1]{}\ (G\_+)\^[3]{}\_[|1 |2]{} \_[2]{} + (G\_+)\^[3]{}\_[|1 2]{} \_[|2]{} & - (G\_+)\^[3]{}\_[|1 |2]{} \_[1]{} - (G\_+)\^[3]{}\_[1|2]{} \_[|1]{} & - b ) and where (G\_)\^a\_[bc]{} f\^a\_[bc]{} g\^[a|[a]{}]{} F\_[|a bc]{} is constructed without imposing the on-shell condition (\[jj\]). Finally, we have defined a & = & (G\_+)\^3\_[12]{}f\^[|3]{}\_[|1 |2]{} + (G\_+)\^3\_[|1 |2]{} f\^[|3]{}\_[12]{} + (G\_+)\^3\_[|1 2]{} f\^[|3]{}\_[1 |2]{} + (G\_+)\^3\_[1|2]{}f\^[|3]{}\_[|1 2]{}\ b & = & (G\_+)\^3\_[12]{}f\^[3]{}\_[|1 |2]{} + (G\_+)\^3\_[|1 |2]{} f\^[3]{}\_[12]{} + (G\_+)\^3\_[|1 2]{} f\^[3]{}\_[1 |2]{} + (G\_+)\^3\_[1|2]{}f\^[3]{}\_[|1 2]{} In general, for non-supersymmetric backgrounds the matrix $B$ is different from zero, reflecting the fact that in that case the internal manifold is not complex. In that case, holomorphic and anti-holomorphic indices label different elements in a complex basis of 1-forms. The fact that the internal manifold is not complex manifests in a spectrum of wavefunctions for which some of the “holomorphic” scalars have different mass eigenvalues than their “anti-holomorphic” counterparts. General magnetic fluxes and Riemann $\vartheta$-function {#riem} ======================================================== As emphasized in [@yukawa], in the presence of general magnetic fluxes $F_2$ on a $T^{2n}$, the zero modes of the Dirac and Laplace operators are given in terms of Riemann $\vartheta$-functions, instead of the more familiar Jacobi $\vartheta$-functions that appear for the factorizable case of a $(T^2)^n$ with a magnetic flux $F_2 = \sum_i F_2|_{(T^2)_i}$. As we have seen in the main text, for type I open string wavefunctions in flux compactifications this non-factorizable case is quite natural, and in particular for those matter field wavefunctions analyzed in Section \[sec:wmatter\] that feel closed and open string fluxes simultaneously. The purpose of this appendix is thus to extend the discussion of [@yukawa] on Riemann $\vartheta$-functions and non-factorizable magnetic fluxes, in order to accommodate the wavefunctions of Section \[sec:wmatter\] into the general scheme of [@yukawa]. See also [@akp09] for some recent similar results on this topic. Let us then consider a general ${T}^{2n}$ and a magnetic $U(1)$ flux $F_2 = dA$ of the form F\_2 = \_[ij]{} q\_[ij]{} dx\^i dx\^j \[flux2\] where $q_{ij} \in \IZ$ and $x^i \in [0,1]$ label the $T^{2n}$ coordinates. This means that we can write the vector potential as A = \_[ij]{} q\_[ij]{} x\^i dx\^j = \^[ t]{} [**Q**]{} d \[potential\] with ${\bf Q}^t = - {\bf Q}$. Let us now define some complex coordinates in $T^{2n}$ as = + [****]{} \[cpx\] with $\vec{\xi}, \vec{\eta}$, two $n$-dimensional real vectors in which we split the components of $\vec x$. The matrix [**Q**]{} then splits as = ( [cc]{} [**Q\^**]{} & [**Q\^**]{}\ [**Q\^**]{} & [**Q\^**]{} ) \[blocks\] In practice, computing open string wavefunctions greatly simplifies if the magnetic flux $F_2$ can be written as a (1,1)-form for some choice of complex structure (\[cpx\]). In that case we can express (\[potential\]) as A = ( \^[ t]{} [**C**]{} d ) \[ansatz\] Direct comparison reveals that the matrices [**C**]{} and [**Q**]{} are related as & = & \[integerxx\]\ [**Q\^**]{} & = & + \[integerxy\]\ [**Q\^**]{} & = & \^t - \^t +\ & & \^t + \^t \[integeryy\] and that ${\bf Q}^t = - {\bf Q}$ implies ${\bf C}^\dag = {\bf C}$. It turns out that the Dirac and Laplace zero mode wavefunctions can be easily expressed as Riemann $\vartheta$-functions if we also impose the constraint ${\bf Q}^{\xi\xi} = \im {\bf C} = 0$. Indeed, the system (\[integerxx\])-(\[integeryy\]) is then solved by taking & = & [**Q\^**]{} ()\^[-1]{} \[solfxy1\]\ [**Q\^**]{} & = & \^t [**Q\^**]{} - [**Q\^**]{}\^[ t]{} \[solfyy2\] where we have assumed that $\im {\bf \Om}$ is invertible. If we now define the $n\times n$ integer matrix ${\bf N} = {\bf Q^{\xi\eta}}^{\, t}$, we can express the above solution as & = & [**N**]{}\^t ()\^[-1]{} \[solfxya\]\ - [**Q\^**]{} & = & [**N**]{} - ([**N**]{} )\^t \[solfyyb\] In terms of [**N**]{}, the antiholomorphic covariant derivative reads $$\hat{D}_{\bar a}= \frac{1}{2\pi R_a} \left( \nabla - i A\right)_{\bar{a}} = \left(\nabla+\frac{\pi}{2}[{\bf N}\cdot \vec z]^t\cdot(\textrm{Im }{\bf \Omega})^{-1}\right)_{\bar a}$$ and it is easy to check that it annihilates the wavefunction [@yukawa] $$\psi^{\vec{j}, {\bf N}}(\vec z, {\bf \Om})\, =\, \mathcal{N}\ e^{i\pi [{\bf N}\, \vec{z}]^t (\text{Im}\, {\bf \Om})^{-1} \text{Im}\, \vec{z}}\, \vartheta \left[ \begin{array}{c} \vec{j} \\ 0 \end{array} \right] \left({\bf N} \cdot \vec{z} \ ; {\bf N} \cdot {\bf \Om} \right) \label{ap:wavematter}$$ which is of the form (\[wavematter\]) up to fiber-dependent phases. The normalization constant is given by $$\mathcal{N}=\left(2^n|\textrm{det}({\bf N}\, \im {\bf \Om})|\,\textrm{Vol}^{-2}_{T^{2n}}\right)^{1/4}$$ whereas $\vartheta$ stands for the Riemann $\vartheta$-function, defined as $$\vartheta\left[{\vec a \atop \vec b}\right](\vec \nu \, ; \, \mathbf{\Omega})= \sum_{\vec m\in\mathbb{Z}^n}e^{i\pi(\vec m-\vec a)^t \mathbf{\Omega}(\vec m-\vec a)}e^{2\pi i(\vec m-\vec a)\cdot(\vec\nu-\vec b)} \label{thetariem}$$ with $\vec a,\ \vec b\in \mathbb{R}^n$. Under lattice shifts $\vec n\in\mathbb{Z}^n$, $\vartheta$ undergoes the transformations $$\begin{aligned} \vartheta\left[{\vec a \atop \vec b}\right](\vec \nu+\vec n\, ; \, \mathbf{\Omega})&= e^{-2\pi i\vec a\cdot\vec n}\cdot\vartheta\left[{\vec a\atop \vec b}\right](\vec\nu\ ; \ \mathbf{\Omega}) \\ \vartheta\left[{\vec a \atop \vec b}\right](\vec \nu+\mathbf{\Omega}\, \vec n\ ; \ \mathbf{\Omega})&= e^{-i\pi\vec n^t\mathbf{\Omega} \vec n-2\pi i\vec n\cdot (\vec \nu-\vec b)}\cdot \vartheta\left[{\vec a\atop \vec b}\right](\vec\nu\, ; \, \mathbf{\Omega})\end{aligned}$$ which implies that the wavefunction (\[ap:wavematter\]) transforms as \^[j, [**N**]{}]{} (z + n, [****]{}) & = & e\^[i n\^t [**Q**]{}\^ ]{} \^[j, [**N**]{}]{} (z, [****]{}) \[trans1\]\ \^[j, [**N**]{}]{} (z + [****]{} n, [****]{}) & = & e\^[i n\^t ([**Q**]{}\^ + [**Q**]{}\^ )]{} \^[j, [**N**]{}]{} (z, [****]{}) \[trans2\] provided that ${\bf N}^t \vec j = {\bf Q}^{\xi\eta} \vec j \in \IZ^n$. Here we have used the fact that ${\bf N}\, \im \Om$ is symmetric, which is implied by (\[solfxy1\]) and that ${\bf C}$ is Hermitian. The transformations (\[trans1\]) and (\[trans2\]) are indeed those of a particle coupled with unit charge to a vector potential (\[potential\]) that satisfies ${\bf Q}^{\xi\xi} =0$. The above result, however, does not imply that for any potential (\[potential\]) such that ${\bf Q}^{\xi\xi} =0$ for some choice of $\vec \xi$, $\vec \eta$, we can find a zero mode wavefunction of the form (\[ap:wavematter\]). First, recall that $F_2 = dA$ should correspond to a (1,1) form for a choice of ${\bf \Om}$ compatible with the $T^{2n}$ metric, and second we should guarantee the convergence of the $\vartheta$-function in (\[ap:wavematter\]), which requires the positive definiteness condition $$\mathbf{N}\cdot\textrm{Im }\mathbf{\Omega} >0 \label{conv}$$ In Section \[sec:wmatter\] we have provided some examples of wavefunctions satisfying all these constraints for certain families of non-factorizable fluxes $F_2$ on $T^4$, more precisely for (\[totalflux\]) and (\[totalflux2\]). One can there check that the complex structure ${\bf \Omega_U}$ is rotated by an $SO(2)$ matrix ${\bf U}$. Let us see how these kind of solutions arise in the context of the above discussion. For that aim, let us write the $T^{2n}$ metric as ds\^2 = (d\^t d\^t) ( [c]{}\ ) = ( [cc]{} d\^[ t]{} & d\^[ t]{} ) ( [cc]{} 0 & [**h**]{}\ [**|[h]{}**]{} & 0 ) ( [c]{} d\ d ) \[metricm\] with ${\bf h}$ an hermitian matrix. We then have that = 2 ( [cc]{} & ([**|[h]{}** ]{})\ ([**|[h]{}** ]{})\^t & ([****]{}\^t ) ) \[G\] and so we would like to characterize those deformations of ${\bf \Omega}$ that leave [**h**]{} and [**G**]{} invariant. Note that since ${\bf h}$ is Hermitian we can write it as ${\bf h} \, = \, {\bf B}^\dag {\bf B}$, with ${\bf B}$ invertible. This allows to parameterize a deformation of ${\bf \Om}$ as = [**|[B]{}**]{}\^[-1]{} with ${\bf U}$ an arbitrary matrix. Then we have that ([**\_U**]{}\^t [**h**]{} [**|\_U**]{}) = ([****]{}\^t [**B**]{}\^\^) so this term remains invariant if ${\bf U} \in U(n)$. The off-diagonal terms of (\[G\]), on the other hand, remain invariant if ([**B**]{}\^) = ([**B**]{}\^) which, for ${\bf B}$ real and ${\bf \Om}$ pure imaginary, is satisfied by simply imposing that ${\bf U}$ is also real. Together with the above constraint this implies that ${\bf U} \in O(n)$. The wavefunctions of Section \[sec:wmatter\] precisely fall in the category of wavefunctions (\[ap:wavematter\]) with rotated complex structure ${\bf \Om_U}$. Indeed, note that for the factorized $T^4$ metric of the form (\[bg12\]), ${\bf \Om}$ is indeed pure imaginary and so, by the discussion above, ${\bf U}$ is an orthogonal matrix. In addition, if we take ${\bf N}$ definite positive (as we do in the examples of Section \[sec:wmatter\]) we need to constrain ${\bf U} \in SO(n)$ as a requirement for the convergence condition (\[conv\]). The precise choice of ${\bf U}$ is then given by the condition that $F_2$ is a (1,1)-form for the complex structure ${\bf \Om_U}$. Note, however, that the above setup clashes with the degree of freedom ${\bf Q}^{\eta\eta} \neq 0$ which in principle we have for our magnetic flux $F_2$. Indeed, (\[solfyy2\]) requires that $\re {\bf \Om_U} \neq 0$ if ${\bf Q}^{\eta\eta} \neq 0$, while $\re {\bf \Om_U} \neq 0$ is not allowed by a rotation ${\bf U} \in SO(n)$. Hence, at least naively, the wavefunctions (\[ap:wavematter\]) apply directly to those magnetic fluxes (\[flux2\]) such that ${\bf Q}^{\xi\xi} = {\bf Q}^{\eta\eta} = 0$ for some choices of $\vec \xi, \vec \eta$. Note that this is not the case for the flux (\[totalflux2\]) in the more general situation $k_3, k_6 \neq 0$, and this is the reason why in Section \[sec:wmatter\] no explicit wavefunctions have been provided for such sector of the theory. [99]{} M. Graña, [*“Flux compactifications in string theory: A comprehensive review,”*]{} Phys. Rept.  [**423**]{}, 91 (2006) \[arXiv:hep-th/0509003\]. M. R. Douglas and S. Kachru, [*“Flux compactification,”*]{} Rev. Mod. Phys.  [**79**]{}, 733 (2007) \[arXiv:hep-th/0610102\]. R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, [*“Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes,”*]{} Phys. Rept.  [**445**]{}, 1 (2007) \[arXiv:hep-th/0610327\]. S. Gukov, C. Vafa and E. Witten, [*“CFT’s from Calabi-Yau four-folds,”*]{} Nucl. Phys.  B [**584**]{}, 69 (2000) \[Erratum-ibid.  B [**608**]{}, 477 (2001)\] \[arXiv:hep-th/9906070\]. K. Dasgupta, G. Rajesh and S. Sethi, [*“M theory, orientifolds and G-flux,”*]{} JHEP [**9908**]{}, 023 (1999) \[arXiv:hep-th/9908088\]. S. B. Giddings, S. Kachru and J. Polchinski, [*“Hierarchies from fluxes in string compactifications,”*]{} Phys. Rev.  D [**66**]{}, 106006 (2002) \[arXiv:hep-th/0105097\]. P. G. Cámara, L. E. Ibánez and A. M. Uranga, [*“Flux-induced SUSY-breaking soft terms,”*]{} Nucl. Phys.  B [**689**]{}, 195 (2004) \[arXiv:hep-th/0311241\]. M. Graña, T. W. Grimm, H. Jockers and J. Louis, [*“Soft Supersymmetry Breaking in Calabi-Yau Orientifolds with D-branes and Fluxes,”*]{} Nucl. Phys.  B [**690**]{}, 21 (2004) \[arXiv:hep-th/0312232\]. P. G. Cámara, L. E. Ibáñez and A. M. Uranga, [*“Flux-induced SUSY-breaking soft terms on D7-D3 brane systems,”*]{} Nucl. Phys.  B [**708**]{}, 268 (2005) \[arXiv:hep-th/0408036\]. D. Lüst, S. Reffert and S. Stieberger, [*“Flux-induced Soft Supersymmetry Breaking in Chiral Type IIB Orientifolds with D3/D7-Branes,”*]{} Nucl. Phys.  B [**706**]{}, 3 (2005) \[arXiv:hep-th/0406092\]. Nucl. Phys.  B [**727**]{}, 264 (2005) \[arXiv:hep-th/0410074\].\ A. Font and L. E. Ibáñez, [*“SUSY-breaking Soft Terms in a MSSM Magnetized D7-brane Model,”*]{} JHEP [**0503**]{}, 040 (2005) \[arXiv:hep-th/0412150\]. P. G. Cámara and M. Graña, [*“No-scale supersymmetry breaking vacua and soft terms with torsion,”*]{} JHEP [**0802**]{}, 017 (2008) \[arXiv:0710.4577 \[hep-th\]\]. F. Marchesano, [*“Progress in D-brane model building,”*]{} Fortsch. Phys.  [**55**]{}, 491 (2007) \[arXiv:hep-th/0702094\]. D. Cremades, L. E. Ibáñez and F. Marchesano, [*“Computing Yukawa couplings from magnetized extra dimensions,”*]{} JHEP [**0405**]{}, 079 (2004) \[arXiv:hep-th/0404229\]. J. P. Conlon, A. Maharana and F. Quevedo, [*“Wave Functions and Yukawa Couplings in Local String Compactifications,”*]{} JHEP [**0809**]{}, 104 (2008) \[arXiv:0807.0789 \[hep-th\]\]. H. Abe, T. Kobayashi and H. Ohki, [*“Magnetized orbifold models,”*]{} JHEP [**0809**]{}, 043 (2008) \[arXiv:0806.4748 \[hep-th\]\]. P. Di Vecchia, A. Liccardo, R. Marotta and F. Pezzella, [*“Kähler Metrics and Yukawa Couplings in Magnetized Brane Models,”*]{} JHEP [**0903**]{}, 029 (2009) \[arXiv:0810.5509 \[hep-th\]\]. I. Antoniadis, A. Kumar and B. Panda, [*“Fermion Wavefunctions in Magnetized branes: Theta identities and Yukawa couplings,”*]{} arXiv:0904.0910 \[hep-th\]. D. Cremades, L. E. Ibáñez and F. Marchesano, [*“Yukawa couplings in intersecting D-brane models,”*]{} JHEP [**0307**]{}, 038 (2003) \[arXiv:hep-th/0302105\]. M. Cvetič and I. Papadimitriou, [*“Conformal field theory couplings for intersecting D-branes on orientifolds,”*]{} Phys. Rev.  D [**68**]{}, 046001 (2003) \[Erratum-ibid.  D [**70**]{}, 029903 (2004)\] \[arXiv:hep-th/0303083\]. S. A. Abel and A. W. Owen, [*“Interactions in intersecting brane models,”*]{} Nucl. Phys.  B [**663**]{}, 197 (2003) \[arXiv:hep-th/0303124\]; [*“N-point amplitudes in intersecting brane models,”*]{} Nucl. Phys.  B [**682**]{}, 183 (2004) \[arXiv:hep-th/0310257\]. D. Lüst, P. Mayr, R. Richter and S. Stieberger, [*“Scattering of gauge, matter, and moduli fields from intersecting branes,”*]{} Nucl. Phys.  B [**696**]{}, 205 (2004) \[arXiv:hep-th/0404134\]. M. Bertolini, M. Billò, A. Lerda, J. F. Morales and R. Russo, [*“Brane world effective actions for D-branes with fluxes,”*]{} Nucl. Phys.  B [**743**]{}, 1 (2006) \[arXiv:hep-th/0512067\]. D. Duò, R. Russo and S. Sciuto, [*“New twist field couplings from the partition function for multiply wrapped D-branes,”*]{} JHEP [**0712**]{}, 042 (2007) \[arXiv:0709.1805 \[hep-th\]\]. A. K. Kashani-Poor and R. Minasian, [*“Towards reduction of type II theories on SU(3) structure manifolds,”*]{} JHEP [**0703**]{}, 109 (2007) \[arXiv:hep-th/0611106\]. C. Caviezel, P. Koerber, S. Körs, D. Lüst, D. Tsimpis and M. Zagermann, [*“The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets,”*]{} Class. Quant. Grav.  [**26**]{}, 025014 (2009) \[arXiv:0806.3458 \[hep-th\]\]. D. Cassani and A. K. Kashani-Poor, [*“Exploiting N=2 in consistent coset reductions of type IIA,”*]{} Nucl. Phys.  B [**817**]{}, 25 (2009) \[arXiv:0901.4251 \[hep-th\]\]. N. Kaloper and R. C. Myers, [*“The O(dd) story of massive supergravity,”*]{} JHEP [**9905**]{}, 010 (1999) \[arXiv:hep-th/9901045\]. C. M. Hull, [*“Superstring Compactifications With Torsion And Space-Time Supersymmetry,”*]{} Print-86-0251 (CAMBRIDGE)\ A. Strominger, [*“Superstrings With Torsion,”*]{} Nucl. Phys. B [**274**]{}, 253 (1986). K. Becker and K. Dasgupta, [*“Heterotic strings with torsion,”*]{} JHEP [**0211**]{}, 006 (2002) \[arXiv:hep-th/0209077\]. A. H. Chamseddine, [*“N=4 Supergravity Coupled To N=4 Matter,”*]{} Nucl. Phys.  B [**185**]{}, 403 (1981); [*“Interacting Supergravity In Ten-Dimensions: The Role Of The Six-Index Gauge Field,”*]{} Phys. Rev.  D [**24**]{}, 3065 (1981). E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, [*“Ten-Dimensional Maxwell-Einstein Supergravity, Its Currents, And The Issue Of Its Auxiliary Fields,”*]{} Nucl. Phys.  B [**195**]{}, 97 (1982). G. F. Chapline and N. S. Manton, [*“Unification Of Yang-Mills Theory And Supergravity In Ten-Dimensions,”*]{} Phys. Lett.  B [**120**]{}, 105 (1983). M. B. Schulz, [*“Superstring orientifolds with torsion: O5 orientifolds of torus fibrations and their massless spectra,”*]{} Fortsch. Phys.  [**52**]{}, 963 (2004) \[arXiv:hep-th/0406001\]. D. Lüst, F. Marchesano, L. Martucci and D. Tsimpis, [*“Generalized non-supersymmetric flux vacua,”*]{} JHEP [**0811**]{}, 021 (2008) \[arXiv:0807.4540 \[hep-th\]\]. S. B. Giddings and A. Maharana, [*“Dynamics of warped compactifications and the shape of the warped landscape,”*]{} Phys. Rev.  D [**73**]{}, 126003 (2006) \[arXiv:hep-th/0507158\]. C. P. Burgess, P. G. Cámara, S. P. de Alwis, S. B. Giddings, A. Maharana, F. Quevedo and K. Suruliz, [*“Warped supersymmetry breaking,”*]{} JHEP [**0804**]{}, 053 (2008) \[arXiv:hep-th/0610255\]. G. Shiu, G. Torroba, B. Underwood and M. R. Douglas, [*“Dynamics of Warped Flux Compactifications,”*]{} JHEP [**0806**]{}, 024 (2008) \[arXiv:0803.3068 \[hep-th\]\].\ M. R. Douglas and G. Torroba, [*“Kinetic terms in warped compactifications,”*]{} arXiv:0805.3700 \[hep-th\]. F. Marchesano, P. McGuirk and G. Shiu, [*“Open String Wavefunctions in Warped Compactifications,”*]{} JHEP [**0904**]{}, 095 (2009) \[arXiv:0812.2247 \[hep-th\]\]. L. Martucci, [*“On moduli and effective theory of N=1 warped flux compactifications,”*]{} JHEP [**0905**]{}, 027 (2009) \[arXiv:0902.4031 \[hep-th\]\]. D. S. Freed and E. Witten, [*“Anomalies in string theory with D-branes,”*]{} arXiv:hep-th/9907189. L. Charlap, [*“Bieberbach groups and flat manifolds,”*]{} Springer, New York, 1986 C. Cid and T. Schulz, [*“Computation of Five- and Six-Dimensional Bieberbach Groups,”*]{} Experiment. Math. 10 (2001), no. 1, 109–115 K. Dekimpe, [*“Almost-Bieberbach Groups: Affine and Polynomial Structures,”*]{} Lecture Notes in Mathematics 1639, Springer, Berlin, 2009 M. Graña, R. Minasian, M. Petrini and A. Tomasiello, [*“A scan for new N=1 vacua on twisted tori,”*]{} JHEP [**0705**]{}, 031 (2007) \[arXiv:hep-th/0609124\]. A. I. Malc’ev, [*“On a class of homogeneous spaces,”*]{} Izv. Akad. Nauk SSSR Ser. Mat. [**13**]{}, 9-32 (1949), English translation, Amer. Math. Soc. Transl. [**39**]{} (1962). A. R. Frey and J. Polchinski, [*“N = 3 warped compactifications,”*]{} Phys. Rev.  D [**65**]{}, 126009 (2002) \[arXiv:hep-th/0201029\]. F. Marchesano, [*“D6-branes and torsion,”*]{} JHEP [**0605**]{}, 019 (2006) \[arXiv:hep-th/0603210\]. E.T. Whittaker and G.N. Watson, [*“The Parabolic Cylinder Function.”* ]{} §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348, 1990. M.E. Taylor, [*“Noncommutative harmonic analysis”*]{}, American Mathematical Society, 1986. S. Thangavelu, [*“Harmonic analysis on the Heisenberg group”,*]{} Springer, 1998. B. Pioline, [*“The automorphic NS5-brane,”*]{} arXiv:0902.3274 \[hep-th\]. A. Kirillov, [*“Lectures on the orbit method’,’*]{} American Mathematical Society, 2004. F. Denef, [*“Les Houches Lectures on Constructing String Vacua,”*]{} arXiv:0803.1194 \[hep-th\]. D. Cremades, L. E. Ibáñez and F. Marchesano, [*“SUSY quivers, intersecting branes and the modest hierarchy problem,”*]{} JHEP [**0207**]{}, 009 (2002) \[arXiv:hep-th/0201205\]. M. Mariño, R. Minasian, G. W. Moore and A. Strominger, [*“Nonlinear instantons from supersymmetric p-branes,”*]{} JHEP [**0001**]{}, 005 (2000) \[arXiv:hep-th/9911206\]. R. Rabadán, [*“Branes at angles, torons, stability and supersymmetry,”*]{} Nucl. Phys.  B [**620**]{}, 152 (2002) \[arXiv:hep-th/0107036\]. G. Aldazábal, P. G. Cámara and J. A. Rosabal, [*“Flux algebra, Bianchi identities and Freed-Witten anomalies in F-theory compactifications,”*]{} Nucl. Phys.  B [**814**]{}, 21 (2009) \[arXiv:0811.2900 \[hep-th\]\]. E. G. Gimon and J. Polchinski, [*“Consistency Conditions for Orientifolds and D-Manifolds,”*]{} Phys. Rev.  D [**54**]{}, 1667 (1996) \[arXiv:hep-th/9601038\]. M. Graña, J. Louis and D. Waldram, [*“Hitchin functionals in N = 2 supergravity,”*]{} JHEP [**0601**]{}, 008 (2006) \[arXiv:hep-th/0505264\]. I. Benmachiche and T. W. Grimm, [*“Generalized N = 1 orientifold compactifications and the Hitchin functionals,”*]{} Nucl. Phys.  B [**748**]{}, 200 (2006) \[arXiv:hep-th/0602241\]. L. E. Ibáñez, C. Muñoz and S. Rigolin, [*“Aspects of type I string phenomenology,”*]{} Nucl. Phys.  B [**553**]{}, 43 (1999) \[arXiv:hep-ph/9812397\]. M. Billò, M. Frau, I. Pesando, P. Di Vecchia, A. Lerda and R. Marotta, [*“Instantons in N=2 magnetized D-brane worlds,”*]{} JHEP [**0710**]{}, 091 (2007) \[arXiv:0708.3806 \[hep-th\]\]. T. W. Grimm and J. Louis, [*“The effective action of N = 1 Calabi-Yau orientifolds,”*]{} Nucl. Phys.  B [**699**]{}, 387 (2004) \[arXiv:hep-th/0403067\]. A. Lawrence, T. Sander, M. B. Schulz and B. Wecht, [*“Torsion and Supersymmetry Breaking,”*]{} JHEP [**0807**]{}, 042 (2008) \[arXiv:0711.4787 \[hep-th\]\]. S. Kachru, M. B. Schulz, P. K. Tripathy and S. P. Trivedi, [*“New supersymmetric string compactifications,”*]{} JHEP [**0303**]{}, 061 (2003) \[arXiv:hep-th/0211182\]. M. Graña and J. Polchinski, [*“Gauge / gravity duals with holomorphic dilaton,”*]{} Phys. Rev.  D [**65**]{}, 126005 (2002) \[arXiv:hep-th/0106014\]. L. Martucci, J. Rosseel, D. Van den Bleeken and A. Van Proeyen, [*“Dirac actions for D-branes on backgrounds with fluxes,”*]{} Class. Quant. Grav.  [**22**]{}, 2745 (2005) \[arXiv:hep-th/0504041\]. D. Lüst, P. Mayr, S. Reffert and S. Stieberger, [*“F-theory flux, destabilization of orientifolds and soft terms on D7-branes,”*]{} Nucl. Phys.  B [**732**]{}, 243 (2006) \[arXiv:hep-th/0501139\]. J. Gomis, F. Marchesano and D. Mateos, [*“An open string landscape,”*]{} JHEP [**0511**]{}, 021 (2005) \[arXiv:hep-th/0506179\]. V. Braun, T. Brelidze, M. R. Douglas and B. A. Ovrut, [*“Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds,”*]{} JHEP [**0807**]{}, 120 (2008) \[arXiv:0805.3689 \[hep-th\]\]. F. Benini, A. Dymarsky, S. Franco, S. Kachru, D. Simič and H. Verlinde, [*“Holographic Gauge Mediation,”*]{} arXiv:0903.0619 \[hep-th\]. J. Erdmenger, N. Evans, I. Kirsch and E. Threlfall, [*“Mesons in Gauge/Gravity Duals - A Review,”*]{} Eur. Phys. J.  A [**35**]{}, 81 (2008) \[arXiv:0711.4467 \[hep-th\]\]. A. K. Kashani-Poor, [*“Nearly K"ahler Reduction,”*]{} JHEP [**0711**]{}, 026 (2007) \[arXiv:0709.4482 \[hep-th\]\]. F. Marchesano and G. Shiu, [*“MSSM vacua from flux compactifications,”*]{} Phys. Rev.  D [**71**]{}, 011701 (2005) \[arXiv:hep-th/0408059\]. JHEP [**0411**]{}, 041 (2004) \[arXiv:hep-th/0409132\]. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, [*“Infinite conformal symmetry in two-dimensional quantum field theory,”*]{} Nucl. Phys.  B [**241**]{}, 333 (1984).\ E. Witten, [*“Coadjoint Orbits of the Virasoro Group,”*]{} Commun. Math. Phys.  [**114**]{}, 1 (1988).\ W. Taylor, [*“Coadjoint orbits and conformal field theory,”*]{} arXiv:hep-th/9310040. [^1]: Indeed, in our examples the wavefunctions are remarkably similar to the ones obtained in models with only open string fluxes, which can be interpreted as some sort of open/closed string duality. As we will see, this in turn leads to conjecture the existence of extra non-perturbative charged states. [^2]: See however [@minasian; @nearly; @kashani2] for progress in this direction. [^3]: In order to compare to the results in [@intrinsic] and related heterotic literature, one has to replace $\phi \raw -\phi$, $H_3 \raw F_3$ and then convert all quantities to the string frame. [^4]: For $\cam_6$ a smooth manifold, one should in principle take $G_{gauge} = Spin(32)/ \IZ_2$. In this sense, $A_M \in U(N)$ lies in a gauge subsector of the full theory. [^5]: Familiar examples are conformal CY manifolds, arising in the context of warped compactifications. Dimensional reduction in those backgrounds has been studied in detail in, e.g., [@giddings; @warping; @douglas; @fershiu; @luca09]. [^6]: More precisely, $e^m$ stand for left-invariant 1-forms of a group manifold related to $\cam_6$, as in [@nearly]. [^7]: In $\cn =0$ no-scale models, $\eps$ should be seen as an approximate supersymmetry generator that nevertheless specifies an SU(3) structure in $\cam_6$ [@geosoft; @dwsb]. [^8]: In order to derive these equations we have neglected the 3- and 4-point interactions and we have taken the gauge fixing conditions, $\nabla^{\mathcal{M}_6}_m\xi^{m,\alpha}=0$ and $\tilde{D}_m\Phi^{m,\alpha\beta}=0$ as in [@quevedo]. [^9]: Note that these are not the usual integer-valued structure constants used in, e.g., the twisted-tori literature, as they also include some dependence on the compactification moduli. See below. [^10]: In our analysis below we will not be interested in closed strings dynamics and moduli stabilization, and so the limit $\textrm{Vol}_{B_4}^{1/2} \gg \textrm{Vol}_{\Pi_2}$ is in fact not essential for our purposes. We will however take it for technical purposes, as it greatly simplifies the open strings equations of motion. [^11]: This splitting has a simple geometric interpretation in the type IIB T-dual setup of Section \[D7dual\]. [^12]: One could actually be more general and quotient $G$ by a discrete subgroup of its affine group, $\pi\subset \textrm{Aff}(G)$, obtaining a freely-acting orbifold of a twisted torus. Indeed, these kind of constructions are well-known for $G\simeq \mathbb{R}^d$ and $\pi \in \text{Aff}(\IR^d)$ a torsion-free crystallographic group (a.k.a. Bieberbach groups [@Bieber1; @Bieber2]), that lead to standard freely-acting orbifolds of $T^n$. Analogously, for $G$ a nilpotent Lie group and $\pi$ an almost-Bieberbach group one obtains the so-called infra-nilmanifolds [@dekimpe]. [^13]: See [@scan] for a discussion of this problem in the more general context of solvable Lie algebras. [^14]: In fact, we need $M \in 2\IZ$ if we want $\G$ to be a subgroup. Interestingly, the same condition is required by the presence of orientifold planes [@fp03]. [^15]: Recall that according to our definition (\[defviel\]) the [*vielbein*]{} left-invariant 1-form is not given by $\tilde e^6$, but rather by the moduli-dependent 1-form $e^6 \equiv 2\pi R_6 \tilde e^6$. [^16]: For massive gauge bosons there is a third d.o.f. showing up as a scalar mode. See Section \[susyspect\]. [^17]: There exist additional solutions given by general parabolic cylinder functions. However it can be checked that these do not lead to convergent sums when plugged into (\[sep\]) and (\[ansz\]). [^18]: Not to be confused with the gauge boson polarization to be discussed below. [^19]: For a recent application of this Fourier transform in a different physical context see [@pioline]. [^20]: This procedure may present some subtleties. For instance, if $\pi_{\vec \om}(\vec x) = e^{i \vec \om \cdot \vec x}$ and $\vec \om \in 2\pi \IZ$, then the sum over $\G = \IZ^n$ does not converge. In those cases, one should rather think of (\[ansatzLB2\]) as a way of replacing $\pi_{\vec \om}$ with $\G$-invariant irreps $\pi_{\vec \omega}^\G$ in (\[ansatzLB1\]). We have followed this philosophy in eqs.(\[rep1\]) and (\[rep2\]) below. [^21]: See Appendix \[kirillov\] for an alternative derivation of this result. [^22]: The fact that we have to diagonalize a $3 \times 3$ matrix instead of a general $6 \times 6$ mass matrix is due to the fact that the 4d vacua considered in this section are supersymmetric, and to the exact pairing between bosonic and fermionic wavefunction that this implies. See Appendix \[ap:N=0\] for a non-supersymmetric example where bosonic and fermionic wavefunctions are no longer the same. [^23]: Had we chosen to write eq.(\[6dsq\]) in terms of $\Psi^*$, we would have obtained the lower $3 \times 3$ block of (\[system2\]) instead of the upper one. [^24]: In a more complete discussion of these type I flux vacua one should consider [*i)*]{} The full gauge sector $G_{gauge} = Spin(32)/\IZ_2$, that could in principle give rise to $(n_i, n_j)$, symmetric and antisymmetric representations of $G_{unbr}$. [*ii)*]{} The spectrum arising from the inclusion of D5-branes. [*iii)*]{} The action of the orbifold on the open string sector of the theory. None of these points will be essential for the computations of this section, so in order to simplify our discussion we will not consider them for the time being. A more detailed analysis will be carried on [@wip]. [^25]: Note that the Bianchi identity, $dF_2=0$, does not allow to turn on a magnetic flux along the fiber of $\cam_6$. As a result, for the examples at hand $\int _{\cam_6} F_2^3 = 0$ and the resulting 4d spectrum will be non-chiral. We nevertheless expect that the general results for matter field wavefunctions obtained below remain valid for more involved, chiral flux vacua. [^26]: In the next section we will see that, in the T-dual setup of type IIB flux compactifications, (\[totalflux\]) translates into the gauge invariant field strength $\cf = F_2 + B$ in the worldvolume of a D7-brane. [^27]: See [@akp09] for a similar set of wavefunctions recently derived in the context of magnetized D9-brane models without closed string background fluxes. [^28]: In terms of the open/closed string correspondence of Section \[nili\], we have the relation $\rho_{\rm cl} = g\, q\, m_\text{flux}$, with $g = R_6^{-1}$ a coupling constant, $q = k_6$ an integer charge and $m_{\text{flux}} = \varepsilon$ the flux mass scale. [^29]: For completeness, let us present the coadjoint action of the algebra: $$\begin{gathered} K(G)(g_1,g_2,g_3,g_4,g_5,g_6,g_\Lambda)=(g_1,\ g_2,\ g_3+ \frac12(Mg_2x^4+I^2_{\alpha\beta}g_1x^2),\ \\ g_4-\frac12(g_2Mx^5-g_1I^1_{\alpha\beta}x^1),\ g_5,\ g_6 +\frac12(g_2Mx^1-g_1I^2_{\alpha\beta}x^5),\ g_\Lambda -\frac12(g_2Mx^2+g_1I^1_{\alpha\beta}x^4))\end{gathered}$$ [^30]: It should be noted that our discussion misses those open string modes which are genuine stringy oscillations and therefore cannot be captured by a supergravity analysis. [^31]: In fact, recall that for consistency we need to add a $\IZ_{2}$ (or $\IZ_{2n}$) orbifold that induces O5-planes wrapping the twisted torus fiber, and that this already breaks $\cn=4 \raw \cn=2$ at the string scale. However, for those open string sectors that are untwisted (i.e., not fixed by the orbifold action) neutral, and not related to D5-branes, the tree-level fluxless spectrum arranges indeed into $\cn=4$ multiplets, and the present discussion applies. For the twisted open string spectrum one just needs to take into account the effect of the orbifold on the wavefunctions, along the lines of [@ako08]. [^32]: For simplicity, we will assume a supersymmetric ($\sigma_+ = 0$) open string flux $F_2$. [^33]: Actually, we present only that part of the charged spectrum computed in Section \[sec:wmatter\]. [^34]: We thank E. Dudas for pointing out this structure to us. [^35]: Indeed, for some choices of, e.g., $\IZ_2$ orbifold the massless chiral multiplets $(\cc^\a_\pm)_0$ in Table \[table2\] are projected out, while for some other choices like in [@gp96] it remains in the spectrum. [^36]: In this section we will be working in 4d Planck mass units. [^37]: We have corrected a normalization factor $t_I$ in eq.(3.46) of [@geosoft] and expressed the result in terms of the conventions used in this paper. [^38]: There is a factor $g_{YM}^2$ with respect of the expressions in Section \[conmu\] which can be explained from the fact that the results in the previous sections have been obtained in the 10d Einstein frame, whereas in this section we are working in the 4d Einstein frame. [^39]: In the type IIB picture these conditions come from imposing that $G_3 = F_3 - \tau H_3$ is a (2,1)-form [@granapolcho]. For general choices of complex structure $dz^i = dx^i + \tau_idx^{i+3}$ in $\prod_i (T^2)_i$ they read $M\tau^3 = N\tau$ and $\tau_1\tau_2 = -1$. [^40]: In view of the non-commutative nature of (\[system2\]), this could perhaps be naturally explained in terms of a non-commutative field theory in the internal D7-brane coordinates. [^41]: A similar observation has been made in [@douglas2] in the context of fluxless Calabi-Yau compactifications. [^42]: The same result was found in [@nearly] for the closed string sector of type IIA AdS$_4$ vacua. [^43]: In fact, such mapping can be made explicit by simply T-dualizing our type I compactification along the two fiber coordinates $a \in \Pi_2$ of the elliptic fibration (\[mansatz\]), as done in Section \[D7dual\]. [^44]: See [@kirillovv] for a more rigorous introduction to the orbit method and its application to general compact group manifolds, as well as [@orbitst] and references therein for earlier applications of this method in the context of CFT and string theory. [^45]: A general feature of coadjoint orbits, related to their symplectic structure, is that they are always even dimensional.

--- abstract: 'Surface codes are the leading family of quantum error-correcting codes. Here, we explore the properties of the 3D surface code. We develop a new picture for visualising 3D surface codes which can be used to analyse the properties of stacks of three 3D surface codes. We then use our new picture to prove that the $CCZ$ gate is transversal in 3D surface codes. We also generalise the techniques of lattice surgery to 3D surface codes. Finally, we introduce a hybrid 2D/3D surface code architecture which supports universal quantum computation without magic state distillation.' author: - Michael Vasmer - 'Dan E. Browne' bibliography: - 'references.bib' title: Universal Quantum Computing with 3D Surface Codes --- The authors would like to thank Hussain Anwar, Earl Campbell, Alex Kubica and Paul Webster for helpful discussions. MV is supported by the EPSRC (grant number EP/L015242/1).

--- abstract: 'The Skorokhod reflection of a continuous semimartingale is unfolded, in a possibly skewed manner, into another continuous semimartingale on an enlarged probability space according to the excursion-theoretic methodology of Prokaj (2009). This is done in terms of a skew version of the Tanaka equation, whose properties are studied in some detail. The result is used to construct a system of two diffusive particles with rank-based characteristics and skew-elastic collisions. Unfoldings of conventional reflections are also discussed, as are examples involving skew Brownian Motions and skew Bessel processes.' author: - 'TOMOYUKI ICHIBA [^1]' - 'IOANNIS KARATZAS [^2]' date: 'July 2, 2014' nocite: - '[@FIK12]' - '[@FIKP]' - '[@IKP]' title: ' <span style="font-variant:small-caps;">Skew-Unfolding the Skorokhod Reflection of a Continuous Semimartingale</span> [^3] ' --- *Dedicated to Terry Lyons on the occasion of his 60th birthday* [*Keywords and Phrases:*]{} Skorokhod and conventional reflections; skew and perturbed Tanaka equations; skew Brownian and Bessel processes; pure and Ocone martingales; local time; competing particle systems; asymmetric collisions. [*AMS 2000 Subject Classifications:*]{} Primary, 60G42; secondary, 60H10. amssym.def amssym The Result {#Res} ========== On a filtered probability space $({\Omega}, \mathcal{F}, \mathbb{P}), \,\mathbb{F} = \{ \mathcal{F} (t) \}_{0 \le t < \infty}$ satisfying the so-called “usual conditions" of right continuity and augmentation by null sets, we consider a real-valued continuous semimartingale $U(\cdot)$ of the form $$\label{1} U (t) \, =\, M(t) + A(t)\,, \qquad 0 \le t < \infty$$ with $M(\cdot) $ a continuous local martingale and $A(\cdot)$ a process of finite first variation on compact intervals. We assume $M(0)=A(0)=0$ for concreteness. There are two ways to “fold", or reflect, this semimartingale about the origin. One is the [*conventional reflection*]{} $$\label{2a} R (t) \, :=\, | U(t) |\,, \qquad 0 \le t < \infty\,;$$ the other is the [*<span style="font-variant:small-caps;">Skorokhod</span> reflection*]{} $$\label{2} S(t) \,:=\, U(t) + \max_{0 \le s \le t} \big( - U(s) \big)\,, \qquad 0 \le t < \infty\,.$$ The following result, inspired by <span style="font-variant:small-caps;">Prokaj</span> (2009), shows how the first can be obtained from the second, by suitably unfolding the <span style="font-variant:small-caps;">Skorokhod</span> reflection in a possibly “skewed" manner. \[Thm1\] Fix a constant $\alpha \in (0,1)$. There exists an enlargement $\,\big(\widetilde{{\Omega}}, \widetilde{\mathcal{F}}, \widetilde{\mathbb{P}}\big),\, \widetilde{\mathbb{F}} = \{ \widetilde{\mathcal{F}} (t) \}_{0 \le t < \infty}\,$ of the filtered probability space $({\Omega}, \mathcal{F}, \mathbb{P}), \,\mathbb{F} = \{ \mathcal{F} (t) \}_{0 \le t < \infty}$ with a measure-preserving map $\,\pi : {\Omega}{\rightarrow}\widetilde{{\Omega}}\,$, and on this enlarged space a continuous semimartingale $X(\cdot)$ that satisfies $$\label{3} \big|X(\cdot) \big| = S(\cdot)\,,\qquad L^X (\cdot) ={\alpha}\, L^S (\cdot)\,, \qquad X(\cdot) \,=\int_0^{\, \cdot} \overline{\text{sgn}} \big( X(t) \big)\, {\mathrm{d}}U(t) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^X (\cdot)\,.$$ Here and throughout this note, we use the notation $$\label{LT} L^U (\cdot) \, :=\, \lim_{{\varepsilon}{\downarrow}0} \, { 1 \over \, 2 \, {\varepsilon}\,} \int_0^{\, \cdot} 1_{ \{0 \le U(t) < {\varepsilon}\} }\, {\mathrm{d}}\langle U \rangle (t)\,, \qquad \widehat{L}^{U}(\cdot) \, :=\, { 1 \over \,2\,}\, \left(L^{ U}(\cdot) + L^{- U}(\cdot) \right)$$ respectively for the [*right*]{} and the [*symmetric*]{} local time at the origin of a continuous semimartingale as in (\[1\]), and the conventions $$\overline{\text{sgn}} (x)\,:=\, \mathbf{ 1}_{(0, \infty)}(x)- \mathbf{ 1}_{(-\infty, 0)} (x)\,, \qquad \text{sgn} (x) \,:=\, \mathbf{ 1}_{(0, \infty)}(x)- \mathbf{ 1}_{(-\infty, 0]} (x) \,, \qquad x \in {\mathbb{R}}$$ for the symmetric and the left-continuous versions, respectively, of the signum function. We also denote by $\, \mathbb{F}^U = \{ \mathcal{F}^U (t) \}_{0 \le t < \infty}$ the “natural filtration" of $\, U(\cdot)\,$, that is, the smallest filtration that satisfies the usual conditions and with respect to which $\, U(\cdot)\,$ is adapted; we set $\, \mathcal{F}^U (\infty) := \sigma \big( \bigcup_{\,0 \le t < \infty} \mathcal{F}^U (t) \big)\,$. Equalities between stochastic processes, such as in (\[3\]), are to be understood throughout in the almost sure sense. Theorem \[Thm1\] constructs a continuous semimartingale $X(\cdot)$ whose conventional reflection coincides with the <span style="font-variant:small-caps;">Skorokhod</span> reflection of the given semimartingale $U(\cdot)$, and which satisfies the stochastic integral equation in (\[3\]). We think of this equation as a [*skew version*]{} of the celebrated <span style="font-variant:small-caps;">Tanaka</span> equation driven by the continuous semimartingale $U(\cdot)$, whose “skew-unfolding" it produces via the parameter $ \, {\alpha}\, $. When there is no skewness, i.e., with ${\alpha}= 1/2\,$, the integral equation of (\[3\]) reduces to the classical <span style="font-variant:small-caps;">Tanaka</span> equation; in this case Theorem \[Thm1\] is just the main result in the paper by <span style="font-variant:small-caps;">Prokaj</span> (2009), which inspired our work. We shall prove Theorem \[Thm1\] in section \[Pf\], then use it in section \[Appl\] to construct a system of two diffusive particles with rank-based characteristics and skew-elastic collisions. Section \[CR\] discusses a similar skew-unfolding of the conventional reflection $\,R(\cdot) = |U(\cdot)|\,$ of $ \,U(\cdot) \,$. In the section that follows we discuss briefly some properties of the [*skew <span style="font-variant:small-caps;">Tanaka</span> equation*]{} in (\[3\]). The Skew Tanaka Equation {#ST} ======================== A first question that arises regarding the stochastic integral equation in (\[3\]), is whether it can be written in the more conventional form $$\label{3a} X(\cdot) \,=\int_0^{\, \cdot} \text{sgn} \big( X(t) \big)\, {\mathrm{d}}U(t) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^X (\cdot)\,,$$ in terms of the asymmetric (left-continuous) version of the signum function. For this, it is necessary and sufficient to have $$\label{3b} \int_0^{\, \cdot} \mathbf{ 1}_{ \{ X(t) =0\} }\, {\mathrm{d}}U(t)\equiv 0\,, \qquad \text{or equivalently} \qquad \int_0^{\, \cdot} \mathbf{ 1}_{ \{ S(t) =0\} }\, {\mathrm{d}}U(t)\equiv 0$$ in the context of Theorem \[Thm1\]. Now from (\[1\]), (\[2\]) it is clear that $M(\cdot)$ is the local martingale part of the continuous semimartingale $S(\cdot)$, so we have $\, \langle S \rangle (\cdot) = \langle U \rangle (\cdot) = \langle M \rangle (\cdot) \,$ and $$\label{3c} \int_0^{\infty} \mathbf{ 1}_{ \{ S(t) =0\} }\, {\mathrm{d}}\langle M \rangle (t)\,=\, 0$$ (e.g., <span style="font-variant:small-caps;">Karatzas & Shreve</span>, Exercise 3.7.10). This gives $\,\int_0^{\, \cdot} \mathbf{ 1}_{ \{ S(t) =0\} }\, {\mathrm{d}}M(t)\equiv 0\,$, so (\[3b\]) will follow if and only if $$\label{3cc} \int_0^{\, \cdot} \mathbf{ 1}_{ \{ S(t) =0\} }\, {\mathrm{d}}A(t)\,\equiv\, 0$$ holds; and on the strength of (\[3c\]), a sufficient condition for (\[3cc\]) is that $A(\cdot)$ be absolutely continuous with respect to the quadratic variation process $\, \langle M \rangle (\cdot)$. We have the following result. \[Prop1\] For a given continuous semimartingale $U(\cdot)$ of the form (\[1\]) the stochastic integral equation of (\[3\]) can be cast equivalently in the form (\[3a\]), if and only if (\[3cc\]) holds; and in this case we have the identification $\, L^S (t) = \max_{\,0 \le s \le t} \big( - U(s) \big)\,$ and the filtration comparisons $$\label{FiltComp} {\cal F}^{|X|} (t)= {\cal F}^U (t) \subseteq {\cal F}^{X} (t)\,, \quad 0 \le t < \infty\,.$$ Whereas, a sufficient condition for (\[3cc\]) to hold, is that there exist an $\mathbb{F}-$progressively measurable process $\,p(\cdot)\,$, locally integrable with respect to $\, \langle M \rangle (\cdot)\,$ and such that $$\label{3d} A(\cdot)\,=\int_0^{\, \cdot} p(t)\, {\mathrm{d}}\langle M \rangle (t)\,.$$ [*Proof:*]{} The first and third claims have already been argued. As for the second, we observe that the <span style="font-variant:small-caps;">Itô-Tanaka</span> formula applied to (\[3a\]) gives $$S(\cdot) = \big| X(\cdot) \big| =\int_0^{\, \cdot} \text{sgn} \big( X(t) \big)\, {\mathrm{d}}X(t) + 2 \, L^X (\cdot) =U(\cdot) - {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^X (\cdot)+ 2 \, L^X (\cdot) = U(\cdot) + L^S (\cdot)$$ on the strength of the second equality in (\[3\]). It is clear from this expression that the filtration comparison $\, {\cal F}^U (t) \subseteq {\cal F}^{S} (t)\,$ holds for all $\, 0 \le t < \infty\,$; whereas the reverse inclusion and the claimed identification are direct consequences of (\[2\]). [*Remark:*]{} More generally (that is, in the absence of condition (\[3cc\])), the local time at the origin of the <span style="font-variant:small-caps;">Skorokhod</span> reflection $\,S(\cdot)\,$ is $\, L^S (t) = \max_{\,0 \le s \le t} \, ( - U(s) )+ \int_0^{ t} \mathbf{ 1}_{ \{ S(u) =0\} }\, {\mathrm{d}}A(u)\,, ~~0 \le t < \infty\,.$ Uniqueness in Distribution for the Skew Tanaka Equation {#Uni} ------------------------------------------------------- A second question that arises regarding the skew-<span style="font-variant:small-caps;">Tanaka</span> equation of (\[3\]), is whether it can be solved uniquely. It is well-known that we cannot expect pathwise uniqueness or strength to hold for this equation. Such strong existence and uniqueness fail already with $\, {\alpha}= 1/2\,$ and $\, U(\cdot)\,$ a standard Brownian motion, in which case we have in (\[FiltComp\]) also the strict inclusion $\, {\cal F}^U (t) \subsetneqq {\cal F}^X (t)\,$ for all $\, t \in (0, \infty)\,$ (e.g., <span style="font-variant:small-caps;">Karatzas & Shreve</span> (1991), Example 5.3.5). The <span style="font-variant:small-caps;">Skorokhod</span> reflection of $\,U(\cdot)\,$ can then be “unfolded" into a Brownian motion $X(\cdot)$, whose filtration is strictly finer than that of the original Brownian motion $\,U(\cdot)$: the unfolding cannot be accomplished without the help of some additional randomness. The issue, therefore, is whether [*uniqueness in distribution*]{} holds for the skew-<span style="font-variant:small-caps;">Tanaka</span> equation of (\[3\]), under appropriate conditions. We shall address this question in the case of a continuous local martingale $\,U(\cdot)\,$ with $\, U(0)=0\,$ and $\,\langle U \rangle (\infty) = \infty\,$. Let us recall a few notions and facts about such a process, starting with its <span style="font-variant:small-caps;">Dambis-Dubins-Schwarz</span> representation $$\label{Ocone} U( t)= B \big( \langle U \rangle ( t) \big)\,, \qquad 0 \le t < \infty$$ (cf.$\,$<span style="font-variant:small-caps;">Karatzas & Shreve</span> (1991), Theorem 3.4.6); here $\, B (\theta) = U (Q (\theta)), \, 0 \le \theta <\infty\,$ is standard Brownian motion, and $\, Q (\cdot)\,$ the right-continuous inverse of the continuous, increasing process $\, \langle U \rangle (\cdot)$. We say that this $\,U(\cdot)\,$ is [*pure,*]{} if each $\, \langle U \rangle (t)\,$ is $\, \mathcal{F}^{B}(\infty)-$measurable; we say that it is an [*<span style="font-variant:small-caps;">Ocone</span> martingale,*]{} if the processes $\, B(\cdot)\,$ and $\, \langle U \rangle (\cdot)$ are independent (cf.$\,$<span style="font-variant:small-caps;">Ocone</span> (1993) and <span style="font-variant:small-caps;">Dubins et al.</span>$\,$(1993), Appendix). As discussed in <span style="font-variant:small-caps;">Vostrikova & Yor</span> (2000), a pure <span style="font-variant:small-caps;">Ocone</span> martingale is a Gaussian process. \[Prop2\] Suppose that $\, U(\cdot)\,$ is a continuous local martingale with $\, U(0)=0\,$ and $\,\langle U \rangle (\infty) = \infty\,$. Then uniqueness in distribution holds for the skew-<span style="font-variant:small-caps;">Tanaka</span> equation of (\[3\]), or equivalently of (\[3a\]), provided that either \(i) $\, U(\cdot)\,$ is pure; or that \(ii) the quadratic variation process $\, \langle U \rangle (\cdot)\,$ is adapted to a Brownian motion $\, \Gamma (\cdot) := ( \Gamma_1 (\cdot), \cdots, \Gamma_n (\cdot) )' $, with values in some Euclidean space $\, {\mathbb{R}}^n\,$ and independent of the real-valued Brownian motion $\, B(\cdot)\,$ in the representation (\[Ocone\]). [*Proof:*]{} Let us consider a continuous local martingale $ U(\cdot) $ with $U(0)=0$, and any continuous semimartingale $X(\cdot)$ that satisfies the stochastic integral equation in (\[3\]). Then $X(\cdot)$ also satisfies the equation of (\[3a\]), as the condition (\[3d\]) holds in this case trivially with $\, p (\cdot) \equiv 0$. In fact, the equation (\[3a\]) can be written then in the form $$X(Q (s)) \,=\int_0^{s} \text{sgn} \big( X(Q (\theta)) \big)\, {\mathrm{d}}B(\theta) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^X (Q (s))\,, \qquad 0 \le s < \infty\,,$$ with $\, Q (\cdot)\,$ the right-continuous inverse of the continuous, increasing process $\, \langle U \rangle (\cdot)$; cf. Proposition 3.4.8 in <span style="font-variant:small-caps;">Karatzas & Shreve</span> (1991). Setting $$\widetilde{X} (s) := X \big( Q (s) \big)\,, \quad \text{it is straightforward to check} \quad L^{\widetilde{X}} (s) = L^X \big( Q (s)\big)\,, \quad 0 \le s < \infty\,;$$ for this, one uses the representation (\[LT\]) for the local time at the origin, along with the fact that the local martingale part of the continuous seminartingale $\, X(\cdot)\,$ in (\[3a\]) has quadratic variation process $\, \langle U \rangle (\cdot)\,$. Thus, the time-changed process $\, \widetilde{X} (\cdot)\,$ satisfies the stochastic integral equation $$\label{HS0} \widetilde{X}(s) \,=\int_0^{s} \text{sgn} \big( \widetilde{X} (\theta) \big)\, {\mathrm{d}}B(\theta) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{\widetilde{X}} (s)\,, \qquad 0 \le s < \infty\,.$$ This can be cast as the <span style="font-variant:small-caps;">Harrison-Shepp</span> (1981) equation $$\label{HS} \widetilde{X}(\cdot) \,=\, \widetilde{W}(\cdot) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{\widetilde{X}} (\cdot)$$ for the skew Brownian motion, driven by the standard Brownian motion $$\label{tildeW} \widetilde{W}(\cdot) := \int_0^{\, \cdot} \text{sgn} \big( \widetilde{X} (\theta) \big)\, {\mathrm{d}}B(\theta)\,.$$ It is well-known from the theory of <span style="font-variant:small-caps;">Harrison & Shepp</span> (1981) that the equation (\[HS\]) has a pathwise unique, strong solution; in fact, the skew Brownian motion $ \widetilde{X}(\cdot)$ and the Brownian motion $ \,\widetilde{W}(\cdot)$ generate the same filtration. Since $$\label{Rep} X(t) \,=\, \widetilde{X} \big( \langle U \rangle ( t) \big)\,, \qquad 0 \le t < \infty$$ holds with $\, \widetilde{X} (\cdot)\,$ adapted to $\, \mathbb{F}^{\,\widetilde{W}}$, the distribution of $\, X(\cdot)\,$ is uniquely determined whenever $$\label{claim} \text{the Brownian motion}~ \widetilde{W}(\cdot)~ \text{of (\ref{tildeW}) is independent of the process} ~\langle U \rangle (\cdot)\,,$$ or whenever $$\label{"pure"} \langle U \rangle (t) ~\,~\text{is}~\,\, \mathcal{F}^{\,\widetilde{W}}(\infty)-\text{measurable, for every} ~ ~t \in [0, \infty)\,.$$ But (\[“pure”\]) holds when $\, U(\cdot)\,$ is pure (case [*(i)*]{} of the Proposition); this is because from (\[tildeW\]) we have $\, B (\cdot) = \int_0^{\, \cdot} \text{sgn} \big( \widetilde{X} (\theta) \big)\, {\mathrm{d}}\widetilde{W}(\theta)\,$, therefore $ \, {\cal F}^B (t) \subseteq {\cal F}^{\widetilde{W}} (t) $ for all $\, t \in [0, \infty)$ and thus $ \,{\cal F}^B (\infty) \subseteq {\cal F}^{\widetilde{W}} (\infty) $. On the other hand, (\[claim\]) holds under the condition of case [*(ii)*]{} in the Proposition, as $\, \langle U \rangle (\cdot)\,$ is then adapted to the filtration generated by the $n-$dimensional Brownian motion $\, \Gamma (\cdot)\,$; this, in turn, is independent of $ \,\widetilde{W}(\cdot)\,$ on the strength of the <span style="font-variant:small-caps;">P. Lévy</span> Theorem (e.g., <span style="font-variant:small-caps;">Karatzas & Shreve</span>, Theorem 3.3.16), since $$\langle \widetilde{W}, \Gamma_j \rangle (\cdot)\,=\int_0^{\, \cdot} \text{sgn} \big( \widetilde{X} (\theta) \big)\, {\mathrm{d}}\langle B, \Gamma_j \rangle (\theta)\, \equiv \,0\,, \qquad \forall ~~ j=1, \cdots, n\,.$$ The proof of the proposition is complete. It would be interesting to obtain sufficient conditions for either (\[claim\]) or (\[“pure”\]) to hold, which are weaker than those of Proposition \[Prop2\]. As Example \[Counter\] shows, however – and contrary to our own initial guess – we cannot expect the conclusions of Proposition \[Prop2\] to remain true for general <span style="font-variant:small-caps;">Ocone</span> martingales. \[Skew\^1\] [*From Brownian Motion to Skew Brownian Motion:*]{} Suppose that $\, U(\cdot)\,$ is standard, real valued Brownian motion. Then the conditions of Propositions \[Prop1\] and \[Prop2\] are satisfied rather trivially; uniqueness in distribution holds for the skew-<span style="font-variant:small-caps;">Tanaka</span> equation of (\[3a\]) (equivalently, of (\[3\])); and every continuous semimartingale $\,X(\cdot)\,$ that satisfies (\[3a\]) is of the form $$X (\cdot) \,=\, W (\cdot) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{ X } (\cdot)\qquad \text{with} \qquad W(\cdot)\,:=\int_0^{\, \cdot} \text{sgn} \big( X(t) \big) \, {\mathrm{d}}U(t)\,,$$ or equivalently $$X (\cdot) \,=\, W (\cdot) + 2\, \big( 2 \, {\alpha}- 1\big)\, \widehat{L}^{ X } (\cdot)$$ in terms of the symmetric local time as in (\[LT\]). Of course $\,W(\cdot)\,$ is standard Brownian motion by the <span style="font-variant:small-caps;">P. Lévy</span> theorem, and the <span style="font-variant:small-caps;">Harrison-Shepp</span> (1981) theory once again characterizes $\,X(\cdot)\,$ as skew Brownian motion with parameter $\, {\alpha}\,$. The processes $\, W(\cdot)\,$ and $\, X(\cdot)\,$ generate the same filtration, which is strictly finer than the filtration generated by the original Brownian motion $\, U(\cdot)=\int_0^{\, \cdot} \text{sgn} \big( X(t) \big) \, {\mathrm{d}}W(t)\,$. \[Counter\] [*Failure of Uniqueness in Distribution for General <span style="font-variant:small-caps;">Ocone</span> Martingales:*]{} We adapt to our setting a construction from page 131 of <span style="font-variant:small-caps;">Dubins et al.</span>$\,$(1993). We start with a filtered probability space $({\Omega}, \mathcal{F}, \mathbb{P}), \,\mathbb{F}^B = \{ \mathcal{F}^B (t) \}_{0 \le t < \infty}\,$ where $\, B(\cdot)\,$ is standard Brownian motion with $B(0)=0$, and define the adapted, continuous and strictly increasing process $$\label{eq: ex3} A({t}) \, :=\, t \cdot {\bf 1}_{\{ t \le 1\}} + \big \{ 1 + \big( u \cdot {\bf 1}_{ \{ B(1) > 0\}} + v \cdot {\bf 1}_{\{B(1) \le 0\}} \big)(t-1) \big\} \cdot {\bf 1}_{ \{ t > 1\}} \, , \quad 0 \le t < \infty$$ where $u >0$ and $v>0$ are given real numbers with $u \neq v$, as well as the processes $$\label{eq: ex1} X(\cdot) \, :=\, B(A(\cdot)) \, , \qquad \Xi(\cdot) \,:=\, -X(\cdot)\,.$$ The <span style="font-variant:small-caps;">Lévy</span> transform $$\betab (\cdot) \, :=\, \int^{\,\cdot}_{0} \text{sgn} (B(t))\, {\mathrm d} B(t)$$ of $B(\cdot)$ is a standard Brownian motion adapted to the filtration $ \,\mathbb{F}^{|B|} = \{ \mathcal{F}^{|B|} (t) \}_{0 \le t < \infty}\,$, which is strictly coarser than $\,\mathbb{F}^B \,$; in particular, it can be seen that $\, \betab (\cdot) \, $ is independent of sgn$(B(1))= 2\, \mathbf{ 1}_{ \{ B(1) >0\} } -1\,$, and thus of the process $\, A(\cdot)\,$ as well. On the other hand, the process $\,X(\cdot)\,$ is a martingale of its natural filtration $ \,\mathbb{F}^X = \{ \mathcal{F}^B \big( A (t) \big)\}_{0 \le t < \infty}\,$; therefore, so is its “mirror image" $\, \Xi (\cdot)\,$, and more importantly its <span style="font-variant:small-caps;">Lévy</span> transform $$U (\cdot) \, :=\, \int^{\,\cdot}_{0} \text{sgn} \big(X(t)\big)\, {\mathrm d} X(t) \,=\, \betab \big( A(\cdot)\big) \qquad \text{with} \qquad \langle U \rangle (\cdot) \, =\, A(\cdot)\,,$$ which is thus seen to be an <span style="font-variant:small-caps;">Ocone</span> martingale. Now clearly, both $\, X(\cdot)\,$ and $\, \Xi(\cdot)\,$ satisfy the equation (\[3a\]) with $\, \alpha = 1/2\,$ driven by $\, U(\cdot)$, so pathwise uniqueness fails for this equation. We also note that the conditions of Proposition \[Prop2\] fail too in this case. [*We claim that uniqueness in distribution fails as well.*]{} In a manner similar to the treatment in <span style="font-variant:small-caps;">Dubins et al.</span> (1993), we shall argue that the distributions of $\,X(\cdot) \,$ and $\, \Xi(\cdot)\,$ at time $\,t \, =\, 2\,$ are different. Now if the random variables $$X(2) \, =\, B(1+u)\cdot {\bf 1}_{\{B(1) > 0\}} + B(1+v) \cdot {\bf 1}_{\{B(1) \le 0\}} \qquad \text{and} \qquad \Xi (2) \, =\, - X(2)$$ had the same probability distributions, that is, if the distribution of the random variable $\, X(2)\,$ were symmetric about the origin, we would have $\, \mathbb{E} [ (X(2))^3] =0\,$. However, let us note the decomposition $$X(2) \, =\, B(1) + \big( B(1+u) - B(1) \big) \cdot {\bf 1}_{\{B(1) > 0\}} + \big( B(1+v) - B(1) \big) \cdot {\bf 1}_{\{B(1)\le 0\}}\,,$$ which gives $$\mathbb{E} \big[ (X(2))^3\big] \,=\, 3\, \mathbb{E} \left[ B(1) \, \big( B(1+u) - B(1) \big)^2 \, {\bf 1}_{\{B(1) > 0\}} \right] + 3\, \mathbb{E} \left[ B(1) \, \big( B(1+v) - B(1) \big)^2 \, {\bf 1}_{\{B(1) \le 0\}} \right]$$ $$=\, 3\, \mathbb{E} \left[ \big( B(1) \big)^+\,\right] \, \big(u-v \big) \, \neq \, 0\,.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ This contradiction establishes the claim. The Perturbed Skew-Tanaka Equation is Strongly Solvable {#Str} ------------------------------------------------------- The addition of some independent noise can restore pathwise uniqueness, thus also strength, to weak solutions of the stochastic equation in (\[3\]) or (\[3a\]). In the spirit of <span style="font-variant:small-caps;">Prokaj</span> (2013) or <span style="font-variant:small-caps;">Fernholz, Ichiba, Karatzas & Prokaj</span> (2013), hereafter referred to as \[FIKP\], we have the following result. \[Prop5\] Suppose that the continuous semimartingale $U(\cdot)$ as in (\[1\]) satisfies the conditions of Proposition \[Prop1\], where now the $\mathbb{F}-$progressively measurable process $\,p(\cdot)\,$ of (\[3d\]) is locally square-integrable with respect to $\, \langle M \rangle (\cdot)\,;$ and that $$\, V(\cdot) \,=\, N(\cdot) + \Delta (\cdot)\,$$ is another continuous semimartingale, with continuous local martingale part $N(\cdot)$ and finite variation part $\Delta(\cdot)$ which satisfy $N(0)= \Delta(0)=0\,$ and $$\langle M, N \rangle (\cdot) \equiv 0\,, \qquad \langle M \rangle (\cdot) \, =\int_0^{\, \cdot} q (t) \, {\mathrm{d}}\langle N \rangle ( t)$$ for some $\mathbb{F}-$progressively measurable process $\,q(\cdot)\,$ with values in a compact interval $\,[0, b]\,$. Then pathwise uniqueness holds for the perturbed skew-<span style="font-variant:small-caps;">Tanaka</span> equation $$\label{skewProk} X(\cdot) \,=\int_0^{\, \cdot} \text{sgn} \big( X(t) \big)\, {\mathrm{d}}U(t)+ V(\cdot) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^X (\cdot)\,,$$ provided that either\ (i) $~\,\alpha = 1/2\,$, or that\ (ii) $\,U(\cdot)$ and $V(\cdot)$ are independent, standard Brownian motions. In this case a weak solution to (\[skewProk\]) exists, and is thus strong. The claim of case [*(i)*]{} is proved in Theorem 8.1 of \[FIKP\], and the claim of case [*(ii)*]{} in an Appendix, section \[PfProp\]. In case [*(ii)*]{} of Proposition \[Prop5\] the equation can be written equivalently as $$X(\cdot) \,= \int_0^{\, \cdot} \mathbf{ 1}_{ \{ X(t) >0\} } \, {\mathrm{d}}W_+(t) \,+ \int_0^{\, \cdot} \mathbf{ 1}_{ \{ X(t) < 0\} } \, {\mathrm{d}}W_-(t) \, + \,{\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^X (\cdot) \,.$$ Here $\, W_\pm (\cdot) := V(\cdot)\pm U(\cdot)\,$ are independent Brownian motions with local variance 2; one of them governs the motion of $\, X(\cdot)\,$ during its positive excursions, the other during the negative ones, whereas these excursions get skewed when $\, \alpha \neq 1/2\,$. Proof of Theorem \[Thm1\] {#Pf} ========================= We shall follow very closely the methodology of <span style="font-variant:small-caps;">Prokaj</span> (2009), with some necessary modifications related to the skewness. The enlargement of the filtered probability space $({\Omega}, \mathcal{F}, \mathbb{P}), \,\mathbb{F} = \{ \mathcal{F} (t) \}_{0 \le t < \infty}$ is done in terms of a sequence $\, \{ \xi_k\}_{k \in {\mathbb{N}}}\,$ of independent random variables with common <span style="font-variant:small-caps;">Bernoulli</span> distribution $$\label{4} \mathbb{P} \big( \xi_1 = +1 \big) = {\alpha}\,, \qquad \mathbb{P} \big( \xi_1 = -1 \big) = 1-{\alpha}$$ (thus with expectation $\, \mathbb{E} ( \xi_1) = 2 {\alpha}-1$), which is independent of $\, \mathcal{F} (\infty) = \sigma \big( \bigcup_{\,0 \le t < \infty} {\cal F}(t) \big)\,$. On the enlarged probability space $\, \big(\widetilde{{\Omega}}, \widetilde{\mathcal{F}}, \widetilde{\mathbb{P}}\big)\,$ we have all the objects of the original space, so we keep the same notation for them. We denote by $$\label{5} \mathfrak{Z} \,:=\, \big\{ t \ge 0 : S(t) =0 \big\}$$ the zero set of the <span style="font-variant:small-caps;">Skorokhod</span> reflection $\, S(\cdot) $ in (\[2\]), and enumerate as $\, \{ \mathcal{C}_k\}_{k \in {\mathbb{N}}}\,$ the disjoint components of $\, [0, \infty) \setminus \mathfrak{Z}\,$, that is, the countably-many excursion intervals of the process $\, S(\cdot)\,$ away from the origin. This we do in a measurable manner, so that $$\big\{ t \in \mathcal{C}_k \big\} \in \mathcal{F} (\infty)\,, \qquad \forall ~~ t \ge 0\,, ~ k \in {\mathbb{N}}\,.$$ In order to simplify notation, we set $$\label{6} \mathcal{C}_0 \,:=\, \mathfrak{Z} \,, \quad \xi_0 \,:=\,0\,.$$ We define now $$\label{7} Z(t) \,:=\, \sum_{k \in {\mathbb{N}}_0} \, \xi_k\, \mathbf{ 1}_{ \mathcal{C}_k } (t)\,, \qquad \widetilde{\mathcal{F}} (t) := \mathcal{F} (t) \vee \mathcal{F}^Z (t)$$ for all $ \, t \in [0, \infty)\,$; this gives the enlarged filtration $ \,\widetilde{\mathbb{F}} = \big\{ \widetilde{\mathcal{F}} (t) \big\}_{0 \le t < \infty}\,$. We posit the following two claims. \[Prop3\] The process $\, M(\cdot)\,$ of (\[1\]) is a continuous local martingale of the enlarged filtration $\, \widetilde{\mathbb{F}}\,$. Consequently, both $\, U(\cdot)\,$ and $\, S(\cdot)\,$ are continuous $ \,\widetilde{\mathbb{F}}-$semimartingales. \[Prop4\] In the notation of (\[2\]) and (\[7\]), we have $$\label{8} Z(\cdot) \, S(\cdot)\,=\, \int_0^{\, \cdot} Z(t)\, {\mathrm{d}}S(t) + \big( 2 {\alpha}-1\big)\, L^S (\cdot)\,.$$ Taking the claims of these two propositions at face-value for a moment, we can proceed with the proof of Theorem \[Thm1\] as follows. We define the process $$\label{9} X(\cdot) \,:=\, Z (\cdot) \, S(\cdot)$$ and note $$\label{10} Z(\cdot) \,=\, \overline{\text{sgn}} \big( X(\cdot) \big)\,, \qquad \big| X(\cdot) \big| \,=\, S(\cdot)$$ thanks to (\[6\]) and (\[7\]), as well as $$\label{11} X(\cdot) - \int_0^{\, \cdot} \overline{\text{sgn}} \big( X( t) \big)\,{\mathrm{d}}S(t)\,=\, Z (\cdot) \, S(\cdot) - \int_0^{\, \cdot}Z( t) \,{\mathrm{d}}S(t)\,=\, \big( 2 {\alpha}-1 \big)\, L^S (\cdot)$$ thanks to (\[9\]), (\[8\]). In particular, $X(\cdot)\,$ is an $\, \widetilde{\mathbb{F}}-$semimartingale, and we note the property $$2\, L^X (\cdot) - L^S (\cdot) \,=\,2\, L^X (\cdot) - L^{|X|} (\cdot) \,=\, \int_0^{\, \cdot} \mathbf{ 1}_{ \{ X(t) =0 \} }\, {\mathrm{d}}X(t)$$ of its local time at the origin (cf. section 2.1 in <span style="font-variant:small-caps;">Ichiba et al.</span> (2013)). In conjunction with (\[11\]) and the fact that $X(\cdot)$, $S(\cdot)$, and $Z(\cdot)$ all have the same zero set $\, \mathfrak{Z}\,$ as in (\[5\]), (\[6\]), we get from this last equation $$\label{11a} 2\, L^X (\cdot) - L^S (\cdot) \,= \int_0^{\, \cdot} \mathbf{ 1}_{ \{ X(t) =0 \} }\,\big[ \, \overline{\text{sgn}} \big( X( t) \big) \,{\mathrm{d}}S(t) + \big( 2 {\alpha}-1 \big)\, L^S ( t) \, \big]= \big( 2 {\alpha}-1 \big)\, L^S (\cdot)\,,~~$$ thus $$\label{11b} L^X (\cdot) \,=\, {\alpha}\, L^S(\cdot)\,,$$ establishing the second equality in (\[3\]). Back in (\[11\]), this leads to $$\label{12} X(\cdot) \,=\, \int_0^{\, \cdot} \overline{\text{sgn}} \big( X( t) \big)\, \big[\, {\mathrm{d}}U(t) + {\mathrm{d}}C(t) \big] + \big( 2 {\alpha}-1 \big)\, L^S (\cdot)\,,$$ where $C(\cdot)$ is the continuous, adapted and increasing process $$C(t) \,:=\, S(t) - U(t) \,=\, \max_{0 \le s \le t} \big( - U(s) \big)\,, \qquad 0 \le t < \infty\,.$$ From the theory of the <span style="font-variant:small-caps;">Skorokhod</span> reflection problem we know that this process $\,C(\cdot)\,$ is flat off the set $ \{ t \ge 0 : S(t) =0\} = \mathfrak{Z}\,$, so the skew-<span style="font-variant:small-caps;">Tanaka</span> equation of (\[3\]) follows now from (\[12\]), (\[11b\]). The proof of Theorem \[Thm1\] is complete. [*Proof of Proposition \[Prop3\]:*]{} By localization of necessary, it suffices to show that if $M(\cdot)$ is an $\,\mathbb{F}-$martingale, then it is also an $\,\widetilde{\mathbb{F}}-$martingale; that is, for any given $\, 0 < \theta <t<\infty$ and $\, A \in \widetilde{\mathcal{F}} (\theta)\,$ we have $$\label{13} \mathbb{E} \, \big[\, \big( M(t) - M(\theta) \big) \, \mathbf{ 1}_A \, \big] \,=\,0\,.$$ It is clear from (\[7\]) that we need to consider only sets of the form $\, A = B \cap D\,$, where $\, B \in \mathcal{F} (\theta)\,$ and $$\label{14} D \,=\, \bigcap_{j=1}^n \big\{ Z(t_j) = {\varepsilon}_j \big\} \,=\, \bigcap_{j=1}^n \big\{ \xi_{\,\kappa (t_j)} = {\varepsilon}_j \big\}$$ for $\, n \in {\mathbb{N}}\,$, $\, 0 < t_1 < t_2 < \cdots < t_n <\theta <t\,$ and $\, {\varepsilon}\in \{ -1, 0 , 1\}\,$. Here we have denoted by $\, \kappa (u)\,$ the (random) index of the excursion interval $\, \mathcal{C}_k\,$ to which a given $\, u \in [0, \infty)\,$ belongs. For such choices, and because $$\mathbb{E} \, \big[\, \big( M(t) - M(\theta) \big) \, \mathbf{ 1}_A \, \big] \,=\,\mathbb{E} \, \big[\, \big( M(t) - M(\theta) \big) \, \mathbf{ 1}_B \cdot \mathbb{E} \, \big( \mathbf{ 1}_D \,|\, \mathcal{F} (\infty) \big) \, \big] \,,$$ we see that, in order to prove (\[13\]), it is enough to argue that $$\label{15} \mathbb{E} \, \big( \mathbf{ 1}_D \,|\, \mathcal{F} (\infty) \big) \quad\text{is}~~\mathcal{F} (\theta)-\text{measurable.}$$ But the random variables $\, \kappa (t_j)\,$ in (\[14\]) are measurable with respect to $\, \mathcal{F} (\infty)\,$, whereas the random variables $\, \xi_1\,, \, \xi_2\,, \cdots \,$ are independent of this $\sigma-$algebra. Therefore, we have $$\label{eq: quant} \mathbb{E} \, \big( \mathbf{ 1}_D \,|\, \mathcal{F} (\infty) \big)= \mathbb{P} \left[\, \bigcap_{j=1}^n \big\{ \xi_{\,\kappa (t_j)} = {\varepsilon}_j \big\} \, \Big| \, \mathcal{F} (\infty)\, \right]=\mathbb{P} \big( \, \xi_{k_1} = {\varepsilon}_1, \cdots , \xi_{k_n} = {\varepsilon}_n\, \big) \Big|_{k_1 = \kappa (t_1), \cdots, k_n = \kappa (t_n)}.$$ For given indices $\, ( k_1, \cdots, k_n)\,$ and $\, ( {\varepsilon}_1, \cdots, {\varepsilon}_n)\,$, let us denote by $m$ the number of distinct non-zero indices in $\, ( k_1, \cdots, k_n)\,$, by $\, \lambda \,$ the number from among those distinct indices of the corresponding $\, {\varepsilon}_j$’s that are equal to 1, and observe $$\mathbb{P} \big( \, \xi_{k_1} = {\varepsilon}_1, \cdots , \xi_{k_n} = {\varepsilon}_n\, \big) \,=\,0\,, ~\text{if}~ ~ ( {\varepsilon}_1, \cdots, {\varepsilon}_n) ~~ \text{contradicts}~~ ( k_1, \cdots, k_n)\,;$$ $$\label{quant} ~~~~~~~~~~~~~~~\,\,~~~~=\, {\alpha}^{\,\lambda} \, \big( 1-{\alpha}\big)^{m-\lambda}\,, ~~\text{otherwise}\,.$$ Here “$ ( {\varepsilon}_1, \cdots, {\varepsilon}_n)\,$ contradicts $\, ( k_1, \cdots, k_n) $" means that we have either\ $.~ k_i =k_j$ but ${\varepsilon}_i \neq {\varepsilon}_j\,$ for some $\, i \neq j\,$; or\ $.~ k_i=0$ but $\, {\varepsilon}_i \neq 0\,$, for some $\,i\,$; or\ $.~ k_i \neq 0$ but $\, {\varepsilon}_i = 0\,$, for some $\,i\,$. We note now that when $\,{k_1 = \kappa (t_1)\, , \, \cdots \, , \, k_n = \kappa (t_n)}\,$, the value of $\,m\,$ (that is, the number of excursion intervals in $\, [0,s] \setminus \mathfrak{Z} \,$ that contain some $\,t_i$), the value of $\lambda$ (i.e., the number of such excursion intervals that are positive) and the statement “$\, ( {\varepsilon}_1, \cdots, {\varepsilon}_n)\,$ contradicts $\, ( k_1, \cdots, k_n)\,$", can all be determined on the basis of the trajectory $\, S(u),\, 0 \le u \le \theta\,$; that is, the quantity on the right-hand side of (\[eq: quant\]) is $\, \mathcal{F}^S(\theta)-$measurable. As a consequence, the property (\[15\]) holds. [*Proof of Proposition \[Prop4\]:*]{} For any $\, {\varepsilon}\in (0,1)\,$ we define recursively, starting with $\, \tau^{\varepsilon}_0 :=0\,$, a sequence of stopping times $$\tau^{\varepsilon}_{2 \ell +1 }\,:=\, \inf \big\{ t > \tau^{\varepsilon}_{2 \ell }\,:\, S(t) > {\varepsilon}\big\}\,, \qquad \tau^{\varepsilon}_{2 \ell +2 }\,:=\, \inf \big\{ t > \tau^{\varepsilon}_{2 \ell +1 }\,:\, S(t) =0 \big\}$$ for $\, \ell \in {\mathbb{N}}_0\,$. We use this sequence to approximate the process $\, Z(\cdot)\,$ of (\[7\]) by $$Z^{\varepsilon}(t)\,:=\, \sum_{\ell \in {\mathbb{N}}_0} \, Z(t)\, \mathbf{ 1}_{ \,( \tau^{\varepsilon}_{2 \ell +1 }, \tau^{\varepsilon}_{2 \ell +2 }]} (t)\,, \qquad 0 \le t < \infty\,.$$ Let us note that the resulting process $\, Z^{\varepsilon}(\cdot)\,$ is constant on each of the indicated intervals; that the sequence of stopping times just defined does not accumulate on any bounded time-interval, on account of the fact that $\, S(\cdot)\,$ has continuous paths; and that the process $\, Z^{\varepsilon}(\cdot)\,$ is of finite first variation over compact intervals. We deduce $$\label{16} Z^{\varepsilon}(T)\, S(T) \,=\, \int_0^{T} Z^{\varepsilon}( t)\, {\mathrm{d}}S( t) + \int_0^{T} S( t) \, {\mathrm{d}}Z^{\varepsilon}( t)\,, \qquad 0 \le T <\infty \,.$$ The piecewise-constant process $\, Z^{\varepsilon}(\cdot)\,$ tends to $\, Z(\cdot)\,$ pointwise as $\, {\varepsilon}{\downarrow}0\,$, and we have $$\label{17} \lim_{{\varepsilon}{\downarrow}0} \, \int_0^{T} Z^{\varepsilon}( t)\, {\mathrm{d}}S( t)\,=\, \int_0^{T} Z (t)\, {\mathrm{d}}S( t)\,, \quad \text{in probability}$$ for any given $\, T \in [0, \infty)\,$; all the while, $\, |Z^{\varepsilon}(\cdot)| \le 1\,$. On the other hand, the second integral in (\[16\]) can be written as $$\int_0^{T} S( t) \, {\mathrm{d}}Z^{\varepsilon}( t)\,= \sum_{ \{ \ell \,: \,\tau^{\varepsilon}_{2 \ell +1 } < T \}} S\big( \tau^{\varepsilon}_{2 \ell +1 } \big) \, Z\big( \tau^{\varepsilon}_{2 \ell +1 } \big)= \, {\varepsilon}\sum_{ \{ \ell \,:\, \tau^{\varepsilon}_{2 \ell +1 } < T \}} Z\big( \tau^{\varepsilon}_{2 \ell +1 } \big)$$ $$~~~~~~~~~~~~~~= \, {\varepsilon}\sum_{j =1}^{N(T,{\varepsilon})} \xi_{\, \ell_j}\,=\, {\varepsilon}\, N(T,{\varepsilon}) \cdot { 1 \over \,N(T,{\varepsilon})\,} \sum_{j =1}^{N(T,{\varepsilon})} \xi_{\, \ell_j}\,,$$ where $\, \big\{ \xi_{\ell_j} \big\}_{j=1}^{N(T, {\varepsilon})}\,$ is an enumeration of the values $\,Z\big( \tau^{\varepsilon}_{2 \ell +1 } \big)\,$ and $$N(T, {\varepsilon}) \,:= \,\# \, \big\{ \ell \,: \,\tau^{\varepsilon}_{2 \ell +1 } < T \big\}\,$$ is the number of upcrossings of the interval $\, (0, {\varepsilon})\,$ that the process $\, S(\cdot)\,$ has completed by time $T$. From Theorem VI.1.10 in <span style="font-variant:small-caps;">Revuz & Yor</span> (1999), we have the representation of local time $\, \lim_{{\varepsilon}{\downarrow}0} \, \,{\varepsilon}\, N (T, {\varepsilon}) = L^S(T)\,$; whereas the strong law of large numbers gives $$\lim_{{\varepsilon}{\downarrow}0} \, { 1 \over \,N(T,{\varepsilon})\,} \sum_{j =1}^{N(T,{\varepsilon})} \xi_{\, \ell_j}\,=\, \mathbb{E} \big( \xi_1\big)\,.$$ Back into (\[16\]) and with the help of (\[17\]), these considerations give $$Z(T) \, S(T)\,=\, \int_0^{T} Z(t)\, {\mathrm{d}}S(t) +\mathbb{E} \big( \xi_1\big)\cdot L^S (T)\,, \qquad 0 \le T <\infty\,,$$ that is, (\[8\]). Conventional Reflection {#CR} ======================= In a similar manner one can establish the following analogue of Theorem \[Thm1\], which uses the conventional reflection in place of the <span style="font-variant:small-caps;">Skorokhod</span> reflection. \[Thm2\] Fix a constant $\alpha \in (0,1)$. There exists an enlargement $\,\big(\widehat{{\Omega}}, \widehat{\mathcal{F}}, \widehat{\mathbb{P}}\big),\, \widehat{\mathbb{F}} = \{ \widehat{\mathcal{F}} (t) \}_{0 \le t < \infty}$ of the filtered probability space $({\Omega}, \mathcal{F}, \mathbb{P}), \,\mathbb{F} = \{ \mathcal{F} (t) \}_{0 \le t < \infty}\,$, with a measure-preserving map $\,\pi : {\Omega}{\rightarrow}\widehat{{\Omega}}\,$, and on this enlarged space a continuous semimartingale $\widehat{X}(\cdot)$ that satisfies $$\label{3f} \big|\widehat{X}(\cdot) \big| = \big|U(\cdot) \big|\,,\qquad L^{\widehat{X}} (\cdot) = {\alpha}\, L^{|U|}(\cdot)\,, \qquad \widehat{X}(\cdot) \,=\int_0^{\, \cdot} \overline{\text{sgn}} \big( \widehat{X}(t) \big)\, {\mathrm{d}}\widehat{U}(t) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{\widehat{X}} (\cdot)\,.$$ Here $$\label{18} \widehat{U}(\cdot)\,:=\, \int_0^{\, \cdot} \overline{\text{sgn}} \big( U(t) \big)\, {\mathrm{d}}U (t)$$ is the <span style="font-variant:small-caps;">Lévy</span> transform of the semimartingale $\, U(\cdot)\,$, and the classical reflection $\, R(\cdot)= | U (\cdot)|\,$ of $\, U (\cdot)\,$ coincides with the <span style="font-variant:small-caps;">Skorokhod</span> reflection of the process $\, \widehat{U}(\cdot)\,$ in (\[18\]), namely $$\,\widehat{S}(t)\, :=\, \widehat{U}(t) + \max_{0 \le s \le t} \big( - \widehat{U}(s) \big)\,, ~~~~~~ 0 \le t < \infty\,.$$ Indeed, most of the argument of the proof in section \[Pf\] goes through verbatim, with $\, S(\cdot), \, X(\cdot)\,$ replaced here by $\, R(\cdot), \, \widehat{X}(\cdot)\,$, up to and including the display (\[11b\]). But now we have $$\label{18a} R(\cdot)= | U(\cdot)| = \int_0^{\, \cdot}\overline{\text{sgn}} \big( U(t) \big)\, {\mathrm{d}}U (t) + L^{|U|} (\cdot)\, =\, \widehat{U}(\cdot) + L^{R} (\cdot)$$ from the <span style="font-variant:small-caps;">It\^ o-Tanaka</span> formula, so (\[12\]) is replaced by $$\widehat{X}(\cdot) \,=\int_0^{\, \cdot}\overline{\text{sgn}} \big( \widehat{X}(t) \big)\,\big[ {\mathrm{d}}\widehat{U} (t) + {\mathrm{d}}L^{R} ( t) \big] + \big( 2 {\alpha}-1 \big) L^R (\cdot)\,.$$ The property $\,L^{\widehat{X}} (\cdot) = {\alpha}\, L^R(\cdot)\,$ is established exactly as in (\[11b\]), so the stochastic integral equation in (\[3f\]) follows from this last display. On the other hand, since the local time $\, L^R(\cdot)\,$ grows only on the set $\, \{ t \ge 0: R(t) =0\} = \{ t \ge 0: \widehat{X} (t)=0\}$, the equality of the first and last terms in (\[18a\]) identifies $\, R(\cdot)\,$ as the <span style="font-variant:small-caps;">Skorokhod</span> reflection $\,\widehat{S}(\cdot)\,$ of the <span style="font-variant:small-caps;">Lévy</span> transform $\,\widehat{U}(\cdot)$, as claimed in the last sentence of Theorem \[Thm2\]. It is well-known (see, for instance, <span style="font-variant:small-caps;">Chaleyat-Maurel & Yor</span> (1978)) that the processes $\, | U(\cdot)|\,$ and $\,\widehat{U}(\cdot)\,$ generate the same filtration. Let us note that the stochastic integral equation in (\[3f\]) can always be written in the more conventional form $$\label{3g} \widehat{X}(\cdot) \,=\int_0^{\, \cdot} \text{sgn} \big( \widehat{X}(t) \big)\, {\mathrm{d}}\widehat{U}(t) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{\widehat{X}} (\cdot)\,,$$ [*without any additional conditions on*]{} $U(\cdot)$. This is because the analogue $\,\int_0^{\, \cdot} \mathbf{ 1}_{ \{ \widehat{X}(t) =0\} }\, {\mathrm{d}}\widehat{U}(t)\equiv 0\,$ of the property in (\[3b\]) is now satisfied trivially, on account of (\[18\]). \[Skew\^2\] [*From One Skew Brownian Motion to Another:*]{} Suppose that $\, U(\cdot)\,$ is a skew Brownian motion with parameter $\, {\gamma}\in (0,1) $, i.e., $$U(\cdot ) \,=\, B(\cdot) + { \, 2 \,{\gamma}-1\, \over {\gamma}}\, L^U (\cdot)$$ for some standard, real-valued Brownian motion $B(\cdot)$. We have in this case $\,\int_0^{\infty} \mathbf{ 1}_{ \{ U(t) =0 \} }\, {\mathrm{d}}t=0\,$ as well as the local time property $$\, 2\, L^U (\cdot) - L^{|U|} (\cdot) = \int_0^{\, \cdot} \mathbf{ 1}_{ \{ U(t) =0 \} }\, {\mathrm{d}}U(t)= \frac{\,2 \,{\gamma}- 1\,}{ {\gamma}}\, L^U(\cdot)\,,$$ thus $\, L^U (\cdot) = {\gamma}\,L^{|U|} (\cdot)\,$ and therefore $ R(\cdot)\,=\, \big| U(\cdot) \big| \,=\, \int_0^{\, \cdot} \overline{\text{sgn}} \big( U(t) \big) \, {\mathrm{d}}U(t) + L^{|U|} (\cdot)\,=\, W(\cdot) + L^{|U|} (\cdot)\,$. Here we have denoted the <span style="font-variant:small-caps;">Lévy</span> transform of (\[18\]) as $$W(\cdot)\,:=\, \widehat{U}(\cdot)\,=\int_0^{\, \cdot} \overline{\text{sgn}} \big( U(t) \big) \left( {\mathrm{d}}B (t)+ { \, 2 \,{\gamma}-1\, \over {\gamma}}\, {\mathrm{d}}L^U ( t) \right) \,=\int_0^{\, \cdot} \text{sgn} \big( U(t) \big) \, {\mathrm{d}}B(t)\,,$$ and observed that it is another standard Brownian motion. Thus, the stochastic integral equation of (\[3g\]) becomes $$\widehat{X}(\cdot) \,=\int_0^{\, \cdot} \text{sgn} \big( \widehat{X}(t) \big)\, {\mathrm{d}}W (t) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{\widehat{X}} (\cdot)\,=\, \widehat{W} (\cdot) + {\, 2 \, {\alpha}- 1\, \over {\alpha}}\, L^{\widehat{X}} (\cdot)$$ with $\, \widehat{W} (\cdot)=\int_0^{\, \cdot} \text{sgn} \big( \widehat{X}(t) \big)\, {\mathrm{d}}W (t)\,$ yet another standard Brownian motion. The <span style="font-variant:small-caps;">Harrison-Shepp</span> (1981) theory characterizes now $\widehat{X} (\cdot)$ as skew Brownian motion with skewness parameter ${\alpha}\,$. The processes $\widehat{X} (\cdot)$ and $\widehat{W} (\cdot)$ generate the same filtration, as do the processes $$\, \widehat{U}(\cdot) \,=\int_0^{\, \cdot} \text{sgn} \big( \widehat{X}(t) \big)\, {\mathrm{d}}\widehat{W} (t) =W (\cdot)\, \qquad \text{and } \qquad \,R(\cdot)= | U(\cdot)| \, ;$$ and the first filtration is finer than the second. Skew Bessel Processes {#BESd} --------------------- In this subsection suppose that $\,U^{2}(\cdot)\,$ is a squared <span style="font-variant:small-caps;">Bessel</span> process with dimension $\,\delta \in (1, 2)\,$, i.e., $\,U^{2}(\cdot)\,$ is the unique strong solution of the equation $$U^{2}(t) \, =\, \delta\, t + 2 \int^{t}_{0} \sqrt{U^{2}(t)} \,{\mathrm d} B(t) \, , \quad 0 \le t < \infty \,$$ for some standard, real-valued Brownian motion $\,B(\cdot)\,$. When $\, \delta \in (1, 2)\,$, the square root $\, R(\cdot) := \lvert U(\cdot)\rvert \ge 0\,$ of this process is a semimartingale that keeps visiting the origin almost surely, and can be decomposed as $$\label{eq: BESd} R(\cdot) \, =\, \int^{\cdot}_{0}\frac{\, \delta - 1\, }{2 \, R(t) \, } \cdot {\bf 1}_{\{R(t) \neq 0\}} {\mathrm d} t + B(\cdot) \, \quad \text{ with } \quad L^{R}(\cdot) \equiv 0\, , \quad \int^{\cdot}_{0} {\bf 1}_{\{R(t) \, =\, 0\}} {\mathrm d} t \, \equiv\, 0 \, .$$ For the study of the stochastic differential equation (\[eq: BESd\]) with $\, \delta \in (1, 2) \,$ see, for example, <span style="font-variant:small-caps;">Cherny</span> (2000). Given $\,\alpha \in (0, 1) \,$, following again the argument of the proof in section 3 through verbatim, with $\,S(\cdot)\,$, $\,X(\cdot)\,$ replaced respectively by $\,R(\cdot)\,$, $\, \widehat{X}(\cdot)\,$, we unfold the nonnegative <span style="font-variant:small-caps;">Bessel</span> process $\,R(\cdot) \,$ to obtain $$\label{eq: sBESd} \widehat{X}(\cdot) \, =\, Z(\cdot) R(\cdot) \, =\, \int^{\cdot}_{0} Z(t) {\mathrm d} R(t) + (2 \alpha - 1) L^{R}(\cdot) \, =\, \int^{\cdot}_{0}\frac{\, \delta - 1\, }{2 \, \widehat{X} (t) \, } \cdot {\bf 1}_{\{ \widehat{X}(t) \neq 0\}} {\mathrm d} t + \widehat{{\bm \beta}} (\cdot) \,,$$ with $\, Z(\cdot) \:= \text{sgn} ( \widehat{X}(\cdot)) \,$ and with $\, \widehat{{\bm \beta}}(\cdot) :=\int^{\cdot}_{0} Z(t) {\mathrm d} B(t) \,$ another standard Brownian motion on an extended probability space, as a consequence of Theorem 4.1 and of the properties in (\[eq: BESd\]). We note that the semimartingale $\, \widehat{X}(\cdot)\,$ does not accumulate local time at the origin, because of $\, L^{R}(\cdot) \equiv 0\,$. We claim that the process $\, \widehat{X}(\cdot) \,$ constructed here in (\[eq: sBESd\]) is the [*$\,\delta\,$-dimensional skew Bessel process with skewness parameter $\,\alpha \,$*]{}. This process was introduced and studied in <span style="font-variant:small-caps;">Blei</span> (2012). Indeed, let us consider the functions $\,g(x) \, :=\, \lvert x \rvert^{2-\delta} / (2-\delta)\,$ and $\,G(x) \, :=\, \text{sgn} (x) \cdot g(x) \,$ for $\, x \in \mathbb R \,$, and examine $\, g( \widehat{X}(\cdot))\,$ and $\, G( \widehat{X}(\cdot))\,$. This scaling is a right choice to measure the boundary behavior of $\, \widehat{X}(\cdot) \,$ around the origin. By substituting $\,q = 2-\delta\,$, $\,p = (2-\delta)\, / \, (1-\delta)\,$, $\,\nu = - 1\, / \, 2 \,$ in Proposition XI.1.11 of <span style="font-variant:small-caps;">Revuz & Yor</span> (2005), we find there exists a (nonnegative) one-dimensional <span style="font-variant:small-caps;">Bessel</span> process $\, {\bm \rho} (\cdot) \,$ on the same probability space such that $\, {\bm \rho} (0) \, =\, (2-\delta)^{\delta-1} g( \widehat{X}(0))\,$ and $$g( \widehat{X}(t)) \, =\, \frac{1}{\, 2-\delta\, } \, \big| \widehat{X}(t)\big|^{2-\delta} \, =\, \frac{1}{\, (2-\delta)^{\delta - 1}} \, {\bm \rho} \big(\Lambda (t)\big) \, , \qquad 0 \le t < \infty\,,$$ where $$\Lambda (t) \, :=\, \inf\{ s \ge 0: K(s) \ge t \}\, , \quad K(s) \, :=\, \int^{s}_{0} \big({\bm \rho}(u)\big)^{ \frac{2\delta - 2}{2 - \delta}} {\mathrm d} u \, ,$$ that is, $\,g( \widehat{X}(\cdot))\,$ is a time-changed, conventionally reflected Brownian motion with the stochastic clock $\, \Lambda (\cdot)\,$. Thus the local time of $\,g (\widehat{X}(\cdot))\,$ accumulates at the origin with this clock $\, \Lambda (\cdot)\,$. In the same manner as in the construction of $\, Z(\cdot) R(\cdot) \,$ in Theorem 4.1, we obtain here $$G( \widehat{X}(T)) \, =\, \text{sgn} ( \widehat{X}(T)) g( \widehat{X}(T)) \, =\, \int^{T}_{0} \text{sgn} ( \widehat{X}(t) ) {\mathrm d} \big( g( \widehat{X}(t))\big) + (2 \alpha - 1) \, L^{g(\widehat{X})}(T) \,$$ as well as $$\label{eq: skBESlt} L^{G( \widehat{X})}(\cdot) - L^{-G( \widehat{X})} (\cdot) \, =\, (2 \alpha - 1) \big( L^{G (\widehat{X})}(\cdot) + L^{-G( \widehat{X})}(\cdot)\big)$$ and $$(1-\alpha) \, L^{G( \widehat{X})}(\cdot) \, =\, \alpha \, L^{-G( \widehat{X})}(\cdot) \, , \quad L^{g( \widehat{X})}(\cdot) \, =\, \frac{\, 1\, }{\, 2\, } \big( L^{G (\widehat{X})}(\cdot) + L^{-G( \widehat{X})}(\cdot)\big) \, .$$ in the notation of (\[LT\]). From these relationships (\[eq: skBESlt\]), and on the strength of Theorem 2.22 of <span style="font-variant:small-caps;">Blei</span> (2012), we identify the process of (\[eq: sBESd\]) as the $\,\delta\,$-dimensional skew <span style="font-variant:small-caps;">Bessel</span> process. Here the process $\,G ( \widehat{X}(\cdot)) \,$ and its local time $\,L^{G( \widehat{X})}(\cdot)\,$ correspond to $\, Y(\cdot)\,$ and $\,L^{X}_{m}(\cdot)\,$, respectively, in the notation of <span style="font-variant:small-caps;">Blei</span> (2012). For various properties and representations of this process, we refer the study of <span style="font-variant:small-caps;">Blei</span> (2012), in particular, Remark 2.26 there. An Application: Two Diffusive Particles with Asymmetric Collisions {#Appl} ================================================================== In the paper \[FIKP\], the authors construct a planar continuous semimartingale $\,\mathcal X (\cdot)= (X_{1}(\cdot), X_{2}(\cdot))\,$ with dynamics $$\label{eq: 2D1} {\mathrm d} X_{1}(t) = \big( g {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}} - h {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}} \big) {\mathrm d} t + \big( \rho {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}} + \sigma {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}}\big) {\mathrm d} B_{1}(t)\, , ~~~$$ $$\label{eq: 2D2} {\mathrm d} X_{2}(t) = \big( g {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}} - h {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}} \big) {\mathrm d} t + \big( \rho {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}} + \sigma {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}}\big) {\mathrm d} B_{2}(t) \, , ~~~$$ for arbitrary real constants $\,g, \, h$ and $ \, \rho > 0\,, \, \sigma > 0\,$ with $ \, \rho^2 + \sigma^2=1 $. They show that, for an arbitrary initial condition $\, (X_{1}(0), X_{2}(0)) =(x_1, x_2) \in {\mathbb{R}}^2\,$ and with $\, (B_1 (\cdot), \, B_2(\cdot))\,$ a planar Brownian motion, the system of (\[eq: 2D1\]), (\[eq: 2D2\]) has a pathwise unique, strong solution. This is a model for two “competing" Brownian particles, with diffusive motions whose drift and dispersion characteristics are assigned according to their ranks. $\bullet~$ In another recent paper <span style="font-variant:small-caps;">Fernholz, Ichiba & Karatzas</span> (2013), hereafter referred to as \[FIK\], a planar continuous semimartingale $\, \widetilde{\mathcal X } (\cdot) \, =\, ( \widetilde{X}_{1}(\cdot), \widetilde{X}_{2}(\cdot)) \,$ is constructed according to the dynamics $${\mathrm d} \widetilde{X}_{1}(t) \, =\, \big( g {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} - h {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} \big) {\mathrm d} t + \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{1}(t) \,$$ $$\label{eq: 2Dskew1} + \,\frac{1-\zeta_{1}}{2}\, {\mathrm d} L^{ \widetilde{X}_{1} - \widetilde{X}_{2}}(t) + \frac{1-\eta_{1}}{2}\, {\mathrm d} L^{ \widetilde{X}_{2}- \widetilde{X}_{1}}(t) \, ,$$ $${\mathrm d} \widetilde{X}_{2}(t) \, =\, \big( g {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} - h {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} \big) {\mathrm d} t + \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{2}(t) \,$$ $$\label{eq: 2Dskew2} +\, \frac{1-\zeta_{2}}{2}\, {\mathrm d} L^{ \widetilde{X}_{1} - \widetilde{X}_{2}}(t) + \frac{1-\eta_{2}}{2} \,{\mathrm d} L^{ \widetilde{X}_{2}- \widetilde{X}_{1}}(t) \, ,$$ Here again $\,g, \, h$ are arbitrary real constants, $ \, \rho > 0\,$ and $ \, \sigma > 0\,$ satisfy $ \, \rho^2 + \sigma^2=1 \,$, whereas $\,\zeta_{i}, \eta_{i}\,$ are real constants satisfying $$0 \le \alpha \, :=\, \frac{\eta}{\, \eta + \zeta\, } \le 1\, , \quad \zeta \, :=\, 1+ \frac{\, \zeta_{1} - \zeta_{2}\, }{2} \, , \quad \eta \, :=\, 1 - \frac{\, \eta_{1} - \eta_{2}\, }{2} \, , \quad \zeta + \eta \neq 0 \, .$$ This new system is a version of the previous competing Brownian particle system, but now with [*elastic and asymmetric collisions*]{} whose effect is modeled by the local time terms $\, L^{ \widetilde{X}_{2}- \widetilde{X}_{1}}(\cdot)\,$ and $\, L^{ \widetilde{X}_{2}- \widetilde{X}_{1}}(\cdot)\,$. Every time the two particles collide, their trajectories feel a “drag" proportional to these local time terms, whose presence makes the analysis of the system (\[eq: 2Dskew1\]), (\[eq: 2Dskew2\]) considerable more involved than that of (\[eq: 2D1\]), (\[eq: 2D2\]). It is shown in \[FIK\] under the above conditions that, for an arbitrary initial condition $\, (\widetilde{X}_{1}(0), \widetilde{X}_{2}(0) ) =(x_1, x_2) \in {\mathbb{R}}^2\,$, and with $\, (\widetilde{B}_1 (\cdot), \, \widetilde{B}_2(\cdot))\,$ a planar Brownian motion, the system of (\[eq: 2Dskew1\]), (\[eq: 2Dskew2\]) has a pathwise unique, strong solution. $\bullet~$ We shall show how to use the unfolding of Theorem \[Thm1\], in order to construct the planar process $\, \widetilde{\mathcal X } (\cdot) = ( \widetilde{X}_{1}(\cdot), \widetilde{X}_{2}(\cdot)) \,$ of (\[eq: 2Dskew1\]), (\[eq: 2Dskew2\]) with skew-elastic collisions, starting from the planar diffusion $\,\mathcal X (\cdot)= (X_{1}(\cdot),$ $ X_{2}(\cdot))\,$ of (\[eq: 2D1\]), (\[eq: 2D2\]). For simplicity, we shall take the initial condition $\, (x_1, x_2) =(0,0)\,$ from now on. \[Prop7\] Suppose we are given a planar continuous semimartingale $\,\mathcal X (\cdot)=(X_{1}(\cdot), X_{2}(\cdot))\,$ that satisfies the system of (\[eq: 2D1\]), (\[eq: 2D2\]) on some filtered probability space $\,\big(\Omega, \mathcal F, \mathbb P \big),\, \mathbb{F} = \{ \mathcal{F} (t) \}_{0 \le t < \infty}\,$ with a planar Brownian motion $\, (B_1 (\cdot), \, B_2(\cdot))\,$. There exists then an enlargement $\, \big(\widetilde{\Omega}, \widetilde{\mathcal F}, \widetilde{\mathbb P} \big),\, \widetilde{\mathbb{F}} = \{ \widetilde{\mathcal{F}} (t) \}_{0 \le t < \infty} \,$ of this filtered probability space, with a planar Brownian motion $\, \big(\widetilde{B}_1 (\cdot), \, \widetilde{B}_2(\cdot)\big)\,$, and on it a planar continuous semimartingale $\,\widetilde{\mathcal X } (\cdot) = (\widetilde{X}_{1}(\cdot), \widetilde{X}_{2}(\cdot))\,$ that satisfies the system of (\[eq: 2Dskew1\]), (\[eq: 2Dskew2\]) with skew-elastic collisions, as well as $$\big(X_{1}( t) - X_{2}( t)\big) + \sup_{0 \le s \le t } \big(X_{1}(s) - X_{2}(s)\big)^{+} \, =\, \big| \widetilde{X}_{1}( t) - \widetilde{X}_{2}( t) \big| \,, \qquad 0 \le t < \infty\, .$$ In other words, the size of the gap between the new processes $\, \widetilde{X}_{1}(\cdot)\,,\, \widetilde{X}_{2}(\cdot)\,$ coincides with the <span style="font-variant:small-caps;">Skorokhod</span> reflection of the difference $\, X_{1}(\cdot)- X_{2}(\cdot)\,$ of the original processes about the origin. We devote the remainder of this section to the proof of this result. Reduction to symmetric local times ---------------------------------- First, some preparatory steps. We define the averages $\, \overline{\zeta} \, :=\, (\zeta_{1}+ \zeta_{2}) \, / \, 2\,$, $\,~ \overline{\eta} \, :=\, (\eta_{1} + \eta_{2}) \, / \, 2\,$, and introduce yet another parameter $$\label{beta} \beta \, :=\, \alpha \cdot \frac{\, \zeta_{1} + \zeta_{2}\, }{2} + (1-\alpha) \cdot \frac{\, \eta_{1} + \eta_{2}\, }{2} \, =\, \alpha \, \overline{\zeta} + (1-\alpha) \, \overline{\eta} \, .$$ For notational simplicity we shall write all the processes related to the skew collisions with a tilde, e.g., $\, \widetilde{Y}(\cdot) \, :=\, \widetilde{X}_{1}(\cdot) - \widetilde{X}_{2}(\cdot) \,$. From the relation between the [*right*]{} local time $\,L^{ \widetilde{Y}} (\cdot)\,$ and the [*symmetric local time*]{} $ \, \widehat{L}^{\, \widetilde{Y}}(\cdot)\,$ as in (\[LT\]), we obtain the relations $$\label{eq: 2Dlt} \zeta L^{ \widetilde{Y}}(\cdot) \, =\, \eta L^{- \widetilde{Y}}(\cdot)\, , \quad L^{ \widetilde{Y}}(\cdot) \, =\, 2 \, \alpha \,\widehat{L}^{ \, \widetilde{Y}}(\cdot) \, , \quad L^{ \widetilde{Y}}_{-}(\cdot) \, :=\, L^{ - \widetilde{Y}}(\cdot) \, =\, 2\, (1-\alpha) \,\widehat{L}^{\, \widetilde{Y}}(\cdot) \,$$ as in \[FIK\]. This way, the system (\[eq: 2Dskew1\])-(\[eq: 2Dskew2\]) can be re-cast as $${\mathrm d} \widetilde{X}_{1}(t) \, =\, \big( g {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} - h {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} \big) {\mathrm d} t + \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}}\big) \,{\mathrm d} \widetilde{B}_{1}(t) \,$$ $$\label{eq: 2Dskew3} {} + (2\alpha - \beta) \,{\mathrm d} \widehat{L}^{ \,\widetilde{Y}}(t) \, ,$$ $${\mathrm d} \widetilde{X}_{2}(t) \, =\, \big( g {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} - h {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} \big) {\mathrm d} t + \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{2}(t) \,$$ $$\label{eq: 2Dskew4} {} + ( 2- 2\alpha - \beta) {\mathrm d} \widehat{L}^{ \, \widetilde{Y}}(t) \, .$$ We shall construct the system (\[eq: 2Dskew3\])-(\[eq: 2Dskew4\]) first, and then obtain from it the system (\[eq: 2Dskew1\])-(\[eq: 2Dskew2\]). Proof of Theorem \[Prop7\] -------------------------- By applying a <span style="font-variant:small-caps;">Girsanov</span> change of measure twice, we can remove the drifts from both of the systems (\[eq: 2D1\])-(\[eq: 2D2\]) and (\[eq: 2Dskew3\])-(\[eq: 2Dskew4\]). Then, in the following, let us construct the two-dimensional Brownian motion with rank-based dispersions and skew-elastic collisions $$\label{eq: noDskew} \begin{split} {\mathrm d} \widetilde{X}_{1}(t) \, &=\, \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{1}(t) \, + (2\alpha - \beta)\, {\mathrm d} \widehat{L}^{\, \widetilde{Y}}(t) \, , \\ {\mathrm d} \widetilde{X}_{2}(t) \, &=\, \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{2}(t) + ( 2- 2\alpha - \beta) \,{\mathrm d} \widehat{L}^{\, \widetilde{Y}}(t) \, \end{split}$$ from the solution $\,((X_{1}(\cdot), X_{2}(\cdot)), (B_{1}(\cdot), B_{2}(\cdot)))\,$ of the system $$\label{eq: noD} \begin{split} {\mathrm d} X_{1}(t) \, &=\, \big( \rho {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}} + \sigma {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}}\big) \,{\mathrm d} B_{1}(t) \, , \\ {\mathrm d} X_{2}(t) \, &=\, \big( \rho {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}} + \sigma {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}}\big) \,{\mathrm d} B_{2}(t) \, , \end{split}$$ which is known from \[FIKP\] to be strongly solvable. Since there is no drift in these last equations, the difference $\,Y(\cdot) \, :=\, X_{1}(\cdot) - X_{2}(\cdot)\,$ between the two components of the system (\[eq: noD\]) is given by the real-valued Brownian motion $$\label{eq: Y} Y(\cdot) \, =\, W(\cdot) \, :=\, \rho W_{1}(\cdot) + \sigma W_{2}(\cdot) \, . $$ Here $$W_{1}(\cdot) \, :=\, \int_0^{\,\cdot} {\bf 1}_{\{X_{1}(t) > X_{2}(t) \}} {\mathrm d} B_{1}(t) - \int_0^{\,\cdot} {\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}} {\mathrm d} B_{2}(t) \, ,$$ $$W_{2}(t) \, :=\, \int_0^{\,\cdot} {\bf 1}_{\{X_{1}(t) \le X_{2}(t) \}} {\mathrm d} B_{1}(t) - \int_0^{\,\cdot} {\bf 1}_{\{X_{1}(t) > X_{2}(t)\}} {\mathrm d} B_{2}(t)$$ are independent Brownian motions. As in \[FIKP\], let us recall also the Brownian motion $$V(\cdot) \, :=\, \rho V_{1}(\cdot) + \sigma V_{2}(\cdot) \, ,$$ where again $$V_{1}(\cdot) \, :=\, \int_0^{\,\cdot}{\bf 1}_{\{X_{1}(t) > X_{2}(t) \}} {\mathrm d} B_{1}(t) + \int_0^{\,\cdot}{\bf 1}_{\{X_{1}(t) \le X_{2}(t)\}} {\mathrm d} B_{2}(t) \, ,$$ $$V_{2}(\cdot) \, :=\, \int_0^{\,\cdot}{\bf 1}_{\{X_{1}(t) \le X_{2}(t) \}} {\mathrm d} B_{1}(t) + \int_0^{\,\cdot}{\bf 1}_{\{X_{1}(t) > X_{2}(t)\}} {\mathrm d} B_{2}(t)$$ are independent Brownian motions. For a given number $ \alpha \in (0, 1) $, there exists by Theorem \[Thm1\] an adapted, continuous process $\, \widetilde{Y}(\cdot) \,$ which satisfies $$Y(t) + \sup_{0 \le s \le t} (- Y(s))^{+} \, =\, \big| \widetilde{Y}(t)\big| \, , \qquad 0 \le t < \infty\,$$ as well as $$\label{eq: 2Dunfold} \widetilde{Y}(\cdot) \, = \int_0^{\, \cdot} \overline{\text{sgn}} \big(\widetilde{Y}(t)\big)\, {\mathrm d} Y(t) + \frac{\,2\alpha - 1\,}{ \alpha} \, L^{ \widetilde{Y}}(\cdot) \, = \int_0^{\, \cdot} {\text{sgn}} \big(\widetilde{Y}(t)\big) \,{\mathrm d} W(t) + 2(2\alpha - 1) \, \widehat{L}^{ \,\widetilde{Y}}(\cdot)\, ,$$ where the last equality follows from Proposition \[Prop1\] and (\[eq: 2Dlt\]). Thus, the “unfolded process" $\,\widetilde{Y}(\cdot)\,$ is a skew Brownian motion, with skewness parameter $\,\alpha\,$. Now let us define the new planar Brownian motion $\, \big(\widetilde{B}_{1}(\cdot), \widetilde{B}_{2}(\cdot)\big)\,$ as $${\mathrm d}\widetilde{B}_{1}(\cdot) \, :=\, \big( {\bf 1}_{\{ Y(\cdot) > 0 , \widetilde{Y}(\cdot) > 0\}} - {\bf 1}_{\{ Y(\cdot) \le 0 , \widetilde{Y}(\cdot) \le 0\}} \big){\mathrm d}B_{1}(\cdot) + \big( {\bf 1}_{\{ Y(\cdot) > 0 , \widetilde{Y}(\cdot) \le 0\}} - {\bf 1}_{\{ Y(\cdot) \le 0 , \widetilde{Y}(\cdot) > 0\}} \big){\mathrm d}B_{2}(\cdot)\, ,$$ $${\mathrm d}\widetilde{B}_{2}(\cdot) \, :=\, \big( {\bf 1}_{\{ Y(\cdot) > 0 , \widetilde{Y}(\cdot) \le 0\}} - {\bf 1}_{\{ Y(\cdot) \le 0 , \widetilde{Y}(\cdot) > 0\}} \big){\mathrm d}B_{1}(\cdot) + \big( {\bf 1}_{\{ Y(\cdot) > 0 , \widetilde{Y}(\cdot) > 0\}} - {\bf 1}_{\{ Y(\cdot) \le 0 , \widetilde{Y}(\cdot) \le 0\}} \big){\mathrm d}B_{2}(\cdot) \, ,$$ and, with the number $\, \beta \in \mathbb R\,$ as in (\[beta\]), the processes $\, \widetilde{\Xi}(\cdot)\,$, $\, \big( \widetilde{X}_{1}(\cdot), \widetilde{X}_{2}(\cdot)\big)\,$ and $\, \big(\widetilde{V}(\cdot), \widetilde{W}(\cdot)\big)\,$ by $$\label{eq: 2DnewX} \, \widetilde{\Xi}(\cdot)\, :=\, \widetilde{V}(\cdot) + 2( 1- \beta) \widehat{L}^{ \widetilde{Y}}(\cdot) \,, \quad \widetilde{X}_{1}(\cdot)\, :=\, \frac{ \, \widetilde{\Xi}(\cdot) + \widetilde{Y}(\cdot) \,}{2}\, , \quad \widetilde{X}_{2}(\cdot)\, :=\, \frac{ \, \widetilde{\Xi}(\cdot) - \widetilde{Y}(\cdot) \,}{2}\, ,$$ $${\mathrm d} \widetilde{V}(\cdot) \, :=\, \big( \rho {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} \big) {\mathrm d} \widetilde{B}_{1}(\cdot) + \big( \rho {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} \big) {\mathrm d} \widetilde{B}_{2}(\cdot) \, ,$$ $${\mathrm d} \widetilde{W}(\cdot) \, :=\, \big( \rho {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} \big) {\mathrm d} \widetilde{B}_{1}(\cdot) - \big( \rho {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} \big) {\mathrm d} \widetilde{B}_{2}(\cdot) \, ,$$ Then by (\[eq: Y\]) and (\[eq: 2DnewX\]) we obtain $$\begin{split} {\text{sgn}} ( \widetilde{Y}( \cdot)) {\mathrm d} W(\cdot) \, &=\, {\text{sgn}} (\widetilde{Y}( \cdot)) \Big[ \big( \rho {\bf 1}_{\{X_{1}(\cdot) > X_{2}(\cdot)\}} + \sigma {\bf 1}_{\{X_{1}(\cdot) \le X_{2}(\cdot)\}}\big) {\mathrm d} B_{1}(\cdot) \\ & \hspace{5cm} {} - \big( \rho {\bf 1}_{\{X_{1}(\cdot) \le X_{2}(\cdot)\}} + \sigma {\bf 1}_{\{X_{1}(\cdot) > X_{2}(\cdot)\}}\big) {\mathrm d} B_{2}(\cdot) \Big] \\ \, &= \, {\text{sgn}} (\widetilde{Y}(\cdot)) \Big[ \big( \rho {\bf 1}_{\{Y(\cdot) > 0\}} + \sigma {\bf 1}_{\{Y(\cdot) \le 0\}}\big) {\mathrm d} B_{1}(\cdot) - \big( \rho {\bf 1}_{\{Y(\cdot) \le 0\}} + \sigma {\bf 1}_{\{Y(\cdot) > 0\}}\big) {\mathrm d} B_{2}(\cdot) \Big] \,, \\ {\mathrm d} \widetilde{W}(\cdot) \, &= \, \big( \rho {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} \big) {\mathrm d} \widetilde{B}_{1}(\cdot) - \big( \rho {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} \big) {\mathrm d} \widetilde{B}_{2}(\cdot) \, \\ &= {\bf 1}_{\{ \widetilde{Y}(\cdot) > 0\}} \big( \rho\, {\mathrm d} \widetilde{B}_{1}(\cdot) - \sigma {\mathrm d} \widetilde{B}_{2}(\cdot)\big) + {\bf 1}_{\{ \widetilde{Y}(\cdot) \le 0\}} \big( \sigma {\mathrm d} \widetilde{B}_{1}(\cdot) - \rho\, {\mathrm d} \widetilde{B}_{2}(\cdot)\big) \, . \end{split}$$ Because of the relationship between $\,(B_{1}(\cdot), B_{2}(\cdot))\,$ and $\, (\widetilde{B}_{1}(\cdot), \widetilde{B}_{2}(\cdot))\,$, it can be shown that $$\label{eq: intertwine} {\mathrm d} \widetilde{W}(\cdot) \, =\, {\text{sgn}} \big( \widetilde{Y}( \cdot)\big)\, {\mathrm d} W(\cdot) \, .$$ In fact, these identities can be verified formally via the following table: signs of $\, (Y(\cdot), \widetilde{Y}(\cdot)) \,$ $\, {\mathrm d} \widetilde{B}_{1}(\cdot)\,$ $\, {\mathrm d} \widetilde{B}_{2}(\cdot) \,$ $\, {\mathrm d} \widetilde{W}(\cdot) \, $ $\, = \,$ $\, {\text{sgn}} ( \widetilde{Y}( \cdot)) {\mathrm d} W(\cdot)\, $ --------------------------------------------------- --------------------------------------------- ---------------------------------------------- ------------------------------------------------------------------------------------------------------- ------------- ------------------------------------------------------------------------------- $\, (+, + )\,$ $\, {\mathrm d} B_{1}(\cdot) \,$ $\, {\mathrm d}B_{2}(\cdot)\,$ $\, \rho\, {\mathrm d} \widetilde{B}_{1}(\cdot) - \sigma \,{\mathrm d} \widetilde{B}_{2}(\cdot)\,$ $\, =\, $ $\, \rho \,{\mathrm d} B_{1}(\cdot) - \sigma\, {\mathrm d} B_{2}(\cdot)\,$ $\,(-, +)\,$ $\, - {\mathrm d} B_{2}(\cdot)\,$ $\, - {\mathrm d} B_{1}(\cdot)\, $ $\,\rho\, {\mathrm d} \widetilde{B}_{1}(\cdot) - \sigma\, {\mathrm d} \widetilde{B}_{2}(\cdot) \,$ $\, =\, $ $\, \sigma \,{\mathrm d} B_{1}(\cdot) - \rho \,{\mathrm d} B_{2}(\cdot) \,$ $\,(+, -)\,$ $\, {\mathrm d} B_{2}(\cdot) \,$ $\, {\mathrm d} B_{1}(\cdot)\,$ $\, \sigma \,{\mathrm d} \widetilde{B}_{1}(\cdot) - \rho \,{\mathrm d} \widetilde{B}_{2}(\cdot) \,$ $\, =\, $ $\,- \rho \,{\mathrm d} B_{1}(\cdot) + \sigma \,{\mathrm d} B_{2}(\cdot) \,$ $\,(-, -)\,$ $\, - {\mathrm d} B_{1}(\cdot)\,$ $\, - {\mathrm d} B_{2}(\cdot)\,$ $\, \sigma\, {\mathrm d} \widetilde{B}_{1}(\cdot) - \rho \,{\mathrm d} \widetilde{B}_{2}(\cdot) \, $ $\, =\, $ $\, - \sigma \,{\mathrm d} B_{1}(\cdot) + \rho\, {\mathrm d} B_{2}(\cdot) \,$ Substituting this relation (\[eq: intertwine\]) into (\[eq: 2Dunfold\]) and recalling (\[eq: 2DnewX\]), we obtain $$\label{eq: 2DYt} {\mathrm d} \big( \widetilde{X}_{1}( t) - \widetilde{X}_{2}( t)\big) \, =\, {\mathrm d} \widetilde{Y}(t) \, =\, {\mathrm d} \widetilde{W}(t) + 2 (2 \alpha - 1)\, {\mathrm d} \widehat{L}^{\, \widetilde{Y}}( t) \, .$$ Moreover, because of the correspondence between $\,(\widetilde{V}(\cdot), \widetilde{W}(\cdot))\,$ and $\,(V(\cdot), W(\cdot))\,$ and the relation (\[eq: 2DnewX\]), we obtain $$\label{eq: sumdiffVW} \frac{1}{\,2\,} \, {\mathrm d} \big( \widetilde{V}( t) + \widetilde{W}( t) \big) \, =\, \big( \rho {\bf 1}_{\{ \widetilde{Y}( t) > 0\}} + \sigma {\bf 1}_{\{ \widetilde{Y}( t) \le 0\}} \big)\, {\mathrm d} \widetilde{B}_{1}( t) \, ,$$ $$\frac{1}{\,2\,} \, {\mathrm d} \big( \widetilde{V}( t) - \widetilde{W}( t) \big) \, =\, \big( \sigma {\bf 1}_{\{ \widetilde{Y}( t) > 0\}} + \rho {\bf 1}_{\{ \widetilde{Y}( t) \le 0\}}\big) \, {\mathrm d} \widetilde{B}_{2}( t) \, .$$ Therefore, by calculating the coefficients in front of the local time terms and by combining (\[eq: 2DnewX\]), (\[eq: 2DYt\]) and (\[eq: sumdiffVW\]), we can verify that $\, (\widetilde{X}_{1}(\cdot), \widetilde{X}_{2}(\cdot))\, $ satisfies $$\label{eq: noDskew2} \begin{split} {\mathrm d} \widetilde{X}_{1}(t) \, &=\, \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{1}(t) \, + (2\alpha - \beta) {\mathrm d} \widehat{L}^{\, \widetilde{Y}}(t) \, , \\ {\mathrm d} \widetilde{X}_{2}(t) \, &=\, \big( \rho {\bf 1}_{\{ \widetilde{X}_{1}(t) \le \widetilde{X}_{2}(t)\}} + \sigma {\bf 1}_{\{ \widetilde{X}_{1}(t) > \widetilde{X}_{2}(t)\}}\big) {\mathrm d} \widetilde{B}_{2}(t) + ( 2- 2\alpha - \beta) {\mathrm d} \widehat{L}^{\, \widetilde{Y}}(t) \, \end{split}$$ that is, (\[eq: noDskew\]) with the new Brownian motion $\,(\widetilde{B}_{1}(\cdot), \widetilde{B}_{2}(\cdot))\,$. By the <span style="font-variant:small-caps;">Girsanov</span> theorem, we obtain (\[eq: 2Dskew3\])-(\[eq: 2Dskew4\]); whereas the relationship (\[eq: 2Dlt\]) between the left local time $\,L^{- \widetilde{Y}}(\cdot)\,$ and the right local time $\,L^{ \widetilde{Y}}(\cdot)\,$ allows us now to recover the dynamics of (\[eq: 2Dskew1\])-(\[eq: 2Dskew2\]) from those of (\[eq: 2D1\])-(\[eq: 2D2\]). Appendix: Proof of Proposition \[Prop5\] {#PfProp} ======================================== Given a planar Brownian motion $\,(B_{1}(\cdot), B_{2}(\cdot))\,$ on a probability space $\,(\Omega, \mathcal F, \mathbb P) \,$ and real constants $\,\alpha \in (0, 1)\,$, $\,x_{0} \in {\mathbb{R}}\,$, we shall construct a process $\,X(\cdot) := \mathfrak q(Y(\cdot))\,$ from the solution $\,Y(\cdot) \,$ of the stochastic differential equation $$\label{eq: strong 1} Y(\cdot) = \mathfrak p(x_{0}) + \int^{\, \cdot}_{0} \mathfrak s\big(Y(t)\big) \,\mathrm d \big(B_{1}(t) + B_{2}(t)\big)\, ,$$ where $\,\mathfrak p(\cdot)\,$, $\,\mathfrak q(\cdot)\,$ and $\,\mathfrak s(\cdot)\,$ are defined by $$\mathfrak p(x) \, :=\, (1-\alpha)\, x \, {\bf 1}_{(0, \infty)}(x) + \alpha \, x \, {\bf 1}_{(-\infty, 0]}(x)\, , \quad \mathfrak q(x) \, :=\, \frac{1}{\, 1-\alpha\, } \, {\bf 1}_{(0, \infty)}(x) + \frac{1}{\, \alpha\, } \, {\bf 1}_{(-\infty, 0)}(x) \, ,$$ $$\mathfrak s(x) \, :=\, (1-\alpha)\, {\bf 1}_{(0, \infty)}(x) + \alpha\, {\bf 1}_{(-\infty, 0]}(x)\, ; \quad x \in {\mathbb{R}}\, .$$ From the work on <span style="font-variant:small-caps;">Nakao</span> (1972) we know that the equation (\[eq: strong 1\]) has a pathwise unique, strong solution. Since $\,\mathfrak q (\mathfrak p(x)) = x\, $, $\,x \in {\mathbb{R}}\,$, by applying the <span style="font-variant:small-caps;">Itô-Tanaka</span> formula to the process $\,X(\cdot) = \mathfrak q(Y(\cdot))\,$ we identify the dynamics of $\,X(\cdot)\,$ as those of the skew Brownian motion (<span style="font-variant:small-caps;">Harrison & Shepp</span> (1981)), namely $$\label{B1_B2} X(\cdot) = x_0 + \big(B_{1}(\cdot) + B_{2}(\cdot)\big) + \frac{\, 2 \alpha - 1\, }{\alpha} L^{X}(\cdot) \, ,$$ driven by the Brownian motion $\,B_{1}(\cdot) + B_{2}(\cdot)\,$. We rewrite this equation in the form $$X(\cdot) - x_{0} - \int^{\, \cdot}_{0} \text{sgn}\big(X(t)\big) \mathrm d U(t) - V(\cdot) \,=\, \frac{ 2\alpha - 1 }{\alpha} L^{X}(\cdot) \,=\, 2 \big(2\alpha - 1\big) \widehat{L}^{X}(\cdot)$$ of (\[skewProk\]), driven by a new planar Brownian motion $\,( U(\cdot), V(\cdot))\,$ with components $$\label{U} U(\cdot) \, := \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) > 0\}} \mathrm d B_{1}(t) - \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) \le 0\}} \mathrm d B_{2}(t) \, ,$$ $$\label{Q} V(\cdot) \, := \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) \le 0\}} \mathrm d B_{1}(t) + \int^{t}_{0} {\bf 1}_{\{X(t) > 0\}} \mathrm d B_{2}(t) \, .$$ Therefore, the perturbed skew <span style="font-variant:small-caps;">Tanaka</span> equation (\[skewProk\]) has the weak solution $\,(X(\cdot), (U(\cdot), V(\cdot)))\,$ just constructed. Conversely, suppose we start with an arbitrary weak solution $\,(X(\cdot), (U(\cdot), V(\cdot)))\,$ of the equation (\[skewProk\]), with $\,( U(\cdot), V(\cdot))\,$ a planar Brownian motion. Then we can cast this equation in the form (\[B1\_B2\]) in terms of the planar Brownian motion $\,( B_1(\cdot), B_2(\cdot))\,$ whose components are given by “disentangling" in (\[U\]), (\[Q\]), namely $$B_1(\cdot) \, = \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) > 0\}} \mathrm d U(t) + \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) \le 0\}} \mathrm d V(t) \, ,$$ $$B_2(\cdot) \, = \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) > 0\}} \mathrm d V(t) - \int^{\, \cdot}_{0} {\bf 1}_{\{X(t) \le 0\}} \mathrm d U(t) \,.$$ But this shows that $\, X(\cdot)\,$ is skew Brownian motion, so its probability distribution is determined uniquely. In other words, the equation of (\[skewProk\]) admits a weak solution, and this solution in unique in the sense of the probability distribution. $\bullet~$ [ *Now we shall see that we have not just uniqueness in distribution, but also pathwise uniqueness, for the equation (\[skewProk\]) driven by the planar Brownian motion $\,( U(\cdot), V(\cdot))\,$.*]{} The argument that follows is based on Lemma 1 of <span style="font-variant:small-caps;">Le Gall</span> (1983), and is almost identical to the proof of Theorem 8.1 of \[FIKP\] except for the evaluation of the additional local times. Note that <span style="font-variant:small-caps;">Le Gall</span>’s Lemma 1 works for continuous semimartingales, in general. Suppose that there are two solutions $\,X_{1}(\cdot)\,$ and $\,X_{2}(\cdot)\,$ of (\[skewProk\]), defined on the same probability space as the driving planar Brownian motion $\,( U(\cdot), V(\cdot))\,$. We shall check their difference $\,D(\cdot) \, :=\, X_{1}(\cdot) - X_{2}(\cdot)\,$ satisfies (c.f. (8.4) in <span style="font-variant:small-caps;">Fernholz et al.</span> (2011)) : $$\label{eq: Lemma1 LeGall} \mathbb E \Big[ \int^{T}_{0} \frac{\mathrm d \langle D\rangle(s)}{D(s)} \,{\bf 1}_{\{D(s) > 0 \}} \Big] < \infty \, , \quad 0 < T < \infty \, ,$$ where $$\langle D\rangle(\cdot) \, =\, \int^{\cdot}_{0} \big( \text{sgn}(X_{1}(t)) - \text{sgn}(X_{2}(t)) \big)^{2} \mathrm d t \, \le \, 2 \int^{\cdot}_{0} \big| \text{sgn}(X_{1}(t)) - \text{sgn}(X_{2}(t)) \big|\, \mathrm d t \, .$$ We approximate the signum function by a sequence $\,\{ f_{k} \}_{k \in \mathbb N} \subset C^{1}(\mathbb R)\,$ which converges to the function $\,f_{\infty}(\cdot) \, =\, \text{sgn}(\cdot)\,$ pointwise and satisfies $\,\lim_{k\to \infty} \lVert f_{k} \rVert_{TV} = \lVert f_{\infty} \rVert_{TV}\,$. Now the parametrized process $$\,Z^{(u)}(t) \, :=\, (1-u) X_{1}(t) + u X_{2}(t)\, , \qquad \,0 \le u \le 1\, , ~~~\,0 \le t < \infty \,$$ takes the form of $$Z^{(u)}(\cdot) \,= \,x_{0} + \int^{\, \cdot}_{0} \big((1-u) \text{sgn}(X_{1}(t)) + u \, \text{sgn}(X_{2}(t)) \big) \,\mathrm d U(t)$$ $$~~~~~~~~~~~~~~~~~~~~+ \,V(\cdot) + \frac{\, 2 \alpha - 1\, }{\alpha} \big( u L^{X_{1}}(\cdot) + (1-u) L^{X_{2}}(\cdot) \big) \, .$$ The local times in the last term do not affect the size of $\,\langle Z^{(u)}\rangle(\cdot)\,$, for which we have the estimate $\, \mathbb{E} \big(\langle Z^{(u)}\rangle(T) \big)\le 2 T\,$. Proceeding as in \[FIKP\] we obtain for every $\,\delta > 0\,$ the bound $$\mathbb E \Big[ \int^{T}_{0} \frac{\, \lvert f_{k}(X_{1}(s)) - f_{k}(X_{2}(s)) \rvert\, }{X_{1}(s) - X_{2}(s)} {\bf 1}_{\{ X_{1}(s) - X_{2}(s) > \delta\}} \mathrm d t \Big] \, \le\, c \, \lVert f_{k} \rVert_{TV} \cdot \sup_{a, u} \mathbb E \big( 2 L^{(u)} (T, a) \big) \, ,$$ where $\,L^{(u)}(T, a)\,$ is the right local time of the continuous semimartingale $\, Z^{(u)}(\cdot)\,$ accumulated at $\,a \in {\mathbb{R}}\,$ and $\,c\,$ is a constant chosen independently of $\, k, u, \delta \,$. Letting $\,k \uparrow \infty\,$ and $\,\delta \downarrow 0\,$, we estimate $$\mathbb E \Big[ \int^{T}_{0} \frac{\mathrm d \langle D\rangle(s)}{D(s)} {\bf 1}_{\{D(s) > 0 \}} \Big] \,<\, 2 \,c \, \lVert f_{\infty} \rVert_{TV} \cdot \sup_{a, u} \mathbb E \big( 2 L^{(u)} (T, a) \big) \, .$$ Finally, we estimate $\,\mathbb E(L^{(u)}(T, a)) \,$ using <span style="font-variant:small-caps;">Tanaka</span>’s formula $$\lvert Z^{(u)}(T) - a \rvert = \lvert Z^{(u)}(0) - a \rvert + \int^{T}_{0} \text{sgn} \big(Z^{(u)}(t) - a\big) \mathrm d Z^{(u)}(t) + 2 L^{(u)}(T, a) \, ,$$ and a combination of the <span style="font-variant:small-caps;">Cauchy-Schwartz</span> inequality and the <span style="font-variant:small-caps;">Itô</span>’s isometry: $$\hspace{-6cm} \mathbb E \big(2 L^{(u)}(T, a) \big) \le \mathbb E \lvert Z^{(u)}(T) - Z^{(u)}(0) \rvert + \big\{ \mathbb E ( \langle Z^{(u)}\rangle (T) ) \big\}^{1/2}$$ $$\hspace{4cm} + \frac{\, 2 \alpha - 1\, }{\alpha} \big( u \, \mathbb E(L^{X_{1}}(T) ) + (1-u )\, \mathbb E( L^{X_{2}}(T)) \big)$$ $$\hspace{0.5cm} \le 2 \Big[ \big\{ \mathbb E \big( \langle Z^{(u)}\rangle (T) \big) \big\}^{1/2} + \frac{\, 2\alpha -1 \, }{\alpha } \big( u \, \mathbb E(L^{X_{1}}(T) ) + (1-u ) \,\mathbb E( L^{X_{2}}(T)) \big) \Big] \, .$$ The last term $\,\mathbb E(L^{X_{i}}(T))\,$ is evaluated by the same procedure: by <span style="font-variant:small-caps;">Tanaka</span>’s formula $$\frac{1}{\, \alpha\, } L^{X_{i}}(T) \, =\, \lvert X_{i}(T) \rvert - \lvert X_{1}(0) \rvert - \int^{T}_{0} \text{sgn}\big(X_{i}(t)\big)\, \mathrm d V(t) - U(T) \, ,$$ and hence $$\mathbb E\big( L^{X_{i}}(T)\big) \le 2 \alpha \big\{ \mathbb E ( \langle X_{i} \rangle (T) ) \big\}^{1/2} \le 2^{3/2} \, \alpha \, T^{1/2} \, , \quad i = 1, 2\, .$$ Therefore, we obtain (\[eq: Lemma1 LeGall\]), and by Lemma 1 of <span style="font-variant:small-caps;">Le Gall</span> (1983) we verify $\,L^{D}(\cdot)= L^{X_{1} - X_{2}}(\cdot) \equiv 0\,$. $\bullet~$ [*Final step*]{}: By exchanging the rôles of $\,X_{1}(\cdot)\,$ and $\,X_{2}(\cdot)\,$, we obtain $\,L^{-D}(\cdot) = L^{X_{2} - X_{1}}(\cdot) \equiv 0\,$ as well as $\, \widehat{L}^{D}(\cdot) \equiv 0\,$. Furthermore, by Corollary 2.6 of <span style="font-variant:small-caps;">Ouknine & Rutkowski</span> (1995), we obtain $$\widehat{L}^{X_{1} \vee X_{2}}(t) \, =\, \int^{t}_{0} {\bf 1}_{\{X_{2}(s) \le 0\}}\, {\mathrm d} \widehat{L}^{X_{1}}(s) + \int^{t}_{0} {\bf 1}_{\{X_{1}(s) < 0\}} \,{\mathrm d} \widehat{L}^{X_{2}}(s) \, ; \quad 0 \le t < \infty \, .$$ Combining these results with <span style="font-variant:small-caps;">Tanaka</span>’s formula, we obtain the dynamics of $\,M(\cdot) \, :=\, X_{1}(\cdot) \vee X_{2}(\cdot)\,$: $${\mathrm d} M(t) \, =\, {\bf 1}_{\{X_{1} (t) \ge X_{2}(t)\}}{\mathrm d} X_{1}(t) + {\bf 1}_{\{X_{1}(t) < X_{2}(t)} {\mathrm d} X_{2} (t) + {\mathrm d} L^{X_{1} - X_{2}}(t) ~~~~~~~~~~~~~~~~~~$$ $$~~\, =\, {\bf 1}_{\{X_{1} (t) \ge X_{2}(t)\}} \Big( \text{sgn} (X_{1}(t)) {\mathrm d} U(t) + {\mathrm d} V(t) + 2(2\alpha - 1) {\mathrm d} \widehat{L}^{X_{1}}(t) \Big)$$ $$\hspace{3cm} {}+ {\bf 1}_{\{X_{1}(t) < X_{2}(t)}\Big( \text{sgn} (X_{2}(t)) {\mathrm d} U(t) + {\mathrm d} V(t) + 2(2\alpha - 1) {\mathrm d} \widehat{L}^{X_{2}}(t) \Big) \,$$ $$\, =\, \text{sgn} (M(t)) {\mathrm d} U(t) + {\mathrm d} V(t) + 2(2\alpha - 1) {\mathrm d} \widehat{L}^{M}(t) \, ; \quad 0 \le t < \infty \, .$$ In other words, each of the continuous semimartingales $\,X_{1}(\cdot)\,$, $\,X_{2}(\cdot)\,$ and $\,M(\cdot) = X_{1}(\cdot) \vee X_{2}(\cdot)\,$ satisfies the equation (\[skewProk\]); but uniqueness in the sense of the probability distribution holds for this equation, so all three processes have the same distribution. Since $\, M(\cdot)\ge X_i (\cdot)\,$, this forces $\, M(\cdot)= X_i (\cdot)\,$, $\,i=1, 2\,$, thus pathwise uniqueness. By the theory of <span style="font-variant:small-caps;">Yamada</span> and <span style="font-variant:small-caps;">Watanabe</span> (e.g., subsection 5.3.D in <span style="font-variant:small-caps;">Karatzas & Shreve</span> (1991)), the solution to (\[skewProk\]) is therefore strong. The proof of Proposition \[Prop5\] is complete. Bibliography {#bibliography .unnumbered} ============ <span style="font-variant:small-caps;">Blei, S.</span> (2012) On symmetric and skew Bessel processes. [*Stochastic Processes and Their Applications*]{} [**122**]{}, no. 9, 3262–3287. <span style="font-variant:small-caps;">Chaleyat-Maurel, M. & Yor, M.</span> (1978) Les filtrations de $\,|X|\,$ et $\,X^+\,$, lorsque $\,X\,$ est une semi-martingale continue. In “Temps Locaux". [*Astérisque*]{} [**52-53**]{}, 193-196. <span style="font-variant:small-caps;">Cherny, A. S.</span> (2000) On the strong and weak solutions of stochastic differential equations governing Bessel processes, [*Stochastics & Stochastics Reports*]{} [**70**]{}, no. 3-4, 213–219. <span style="font-variant:small-caps;">Dubins, L.E., Émery, M. & Yor, M.</span> (1993) On the Lévy transformation of Brownian motions and continuous martingales. In “Séminaire de Probabilités XXVII ". [*Lecture Notes in Mathematics*]{} [**1577**]{}, 122-132. Springer Verlag, New York. <span style="font-variant:small-caps;">Fernholz, E.R., Ichiba, T. & Karatzas, I.</span> (2013) \[FIK\] [Two [B]{}rownian particles with rank-based characteristics and skew-elastic collisions.]{} [*Stochastic Processes and Their Applications* ]{} [**123**]{}, 2999-3026. <span style="font-variant:small-caps;">Fernholz, E.R., Ichiba, T., Karatzas, I. & Prokaj, V.</span> (2013) \[FIKP\] [A planar diffusion with rank-based characteristics, and perturbed Tanaka equations.]{} [*Probability Theory and Related Fields*]{} [**156**]{}, 343-374. <span style="font-variant:small-caps;">Harrison, J.M. & Shepp, L.A.</span> (1981). On skew Brownian motion. [*Annals of Probability*]{} [**9**]{}, 309–313. <span style="font-variant:small-caps;">Ichiba, T., Karatzas, I. & Prokaj, V.</span> (2013) [Diffusions with rank-based characteristics and values in the nonnegative orthant.]{} [*Bernoulli*]{} [**19**]{}, 2455-2493. <span style="font-variant:small-caps;">Karatzas, I. & Shreve, S.E.</span> (1991) [*Brownian Motion and Stochastic Calculus.*]{} Second Edition, Springer Verlag, New York. <span style="font-variant:small-caps;">Le Gall, J-F.</span> (1983) Applications du temps local aux équations différentielles stochastiques unidimensionnelles. In “Séminaire de Probabilités XVII". [*Lecture Notes in Mathematics*]{} [**986**]{}, 15-31. Springer Verlag, New York. <span style="font-variant:small-caps;">Nakao, S.</span> (1972) On pathwise uniqueness of solutions of stochastic differential equations. [*Osaka Journal of Mathematics*]{} [**9**]{}, 513-518. <span style="font-variant:small-caps;">Ocone, D.</span> (1993) A symmetry characterization of conditionally independent increment martingales. [*Proceedings of the San Felice Workshop on Stochastic Analysis*]{} (D.$\,$Nualart & M.$\,$Sanz, editors), 147-167. Birkhäuser-Verlag, Basel and Boston. <span style="font-variant:small-caps;">Ouknine, Y. & Rutkowski, M.</span> (1995) Local times of functions of continuous semimartingales. [*Stochastic Analysis and Applications*]{} [**13**]{}, 211-231. <span style="font-variant:small-caps;">Prokaj, V.</span> (2009) Unfolding the Skorokhod reflection of a semimartingale. [*Statistics and Probability Letters*]{} [**79**]{}, 534-536. <span style="font-variant:small-caps;">Prokaj, V.</span> (2013) The solution of the perturbed Tanaka equation is pathwise unique. [*Annals of Probability*]{} [**41**]{}, 2376-2400. <span style="font-variant:small-caps;">Revuz, D. and Yor, M.</span> (1999) [*Continuous Martingales and Brownian Motion*]{}. Third edition, Springer Verlag, New York. <span style="font-variant:small-caps;">Vostrikova, L. & Yor, M.</span> (2000) Some invariance properties of the laws of Ocone martingales. In “Séminaire de Probabilités XXXIV ". [*Lecture Notes in Mathematics*]{} [**1729**]{}, 417-431. Springer Verlag, New York. [^1]:   Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA 93106, USA (E-mail: [*[email protected]*]{}). Research supported in part by the National Science Foundation under grant NSF-DMS-13-13373. [^2]:   Department of Mathematics, Columbia University (E-mail: [*[email protected]*]{}), and <span style="font-variant:small-caps;">Intech</span> Investment Management, One Palmer Square, Suite 441, Princeton, NJ 08542, USA (E-mail: [*[email protected]*]{}). Research supported in part by the National Science Foundation under grants NSF-DMS-09-05754 and NSF-DMS-14-05210. [^3]:   We are indebted to Marcel Nutz and Daniel Ocone for stimulating discussions, to Vilmos Prokaj and Johannes Ruf for their careful reading of the manuscript and for their suggestions, and to the referee for simplifying the last part of our argument in Example \[Counter\].

--- author: - 'J. Clausen [^1], M. Dakna, L. Knöll, D.-G. Welsch' title: | CONDITIONAL QUANTUM STATE ENGINEERING\ AT BEAM SPLITTER ARRAYS --- =1 =6 [ Friedrich-Schiller-Universität Jena,\ Theoretisch-Physikalisches Institut,\ Max-Wien-Platz 1, D-07743 Jena, Germany]{} Introduction ============ If two traveling (pulse-shaped) modes of the radiation field are mixed at a beam splitter, then the two outgoing modes are in an entangled state in general. Therefore, the reduced state of one of them depends on the result of a measurement performed on the other. By mixing a signal pulse with a reference pulse prepared in a known state and discriminating from all pulses leaving the signal output port those corresponding to a particular measurement result in the other output port, quantum state engineering can be realized [@DaknaJacobi; @pap2; @pap3]. On the other hand, the overlap of a signal with a chosen quantum state may be obtained by mixing the signal mode with a reference mode prepared in a quantum state that is specific to the overlap and performing a measurement on the outgoing field [@Barnett]. Hence, novel possibilities of direct quantum state generation and measurement are offered, provided that the designed reference states can be prepared and the required measurements can be realized. In what follows we present a scheme for the generation of arbitrary quantum states of traveling fields, in which coherent and 1-photon Fock states are fed into an array of beam splitters and zero-photon measurements are performed. We then show how the scheme can be modified in order to measure the overlap of an unknown quantum state of a signal mode with an arbitrarily chosen quantum state. Whereas in the former case a source for 1-photon Fock states should be available, in the latter case only 1-photon Fock state detection is required. In Section 2 the underlying formalism is outlined and the basic formulas are given. The problem of the generation of arbitrary quantum states is considered in Section 3, and Section 4 is devoted to the problem of overlap measurements. In order to give an example, we consider in Section 5 the generation of and measurement of overlap with Schrödinger-cat-like states. A summary and some concluding remarks are given in Section 6. Conditional quantum state transformation ======================================== Let us consider the state transformation at a beam splitter array. As outlined in Fig.1, [ Fig. 1. Conditional quantum state manipulation at an array of beam splitters ${\rm B}_k$. The incoming signal state $\hat{\varrho}_{{\rm in}}$ is combined with states $|\Psi_{{\rm in}_k}\rangle$, and $\hat{\varrho}_{{\rm out}}$ is the signal state generated under the condition that the outgoing reference modes have been measured to be in states $|\Psi_{{\rm out}_k}\rangle$ by means of the measuring devices ${\rm D}_k$. ]{} the incoming signal prepared in a state $\hat{\varrho}_{{\rm in}}$ passes an array of $N$ beam splitters at which it is mixed with reference input modes in states . When the measuring devices ${\rm D}_1,$ $\!\ldots,$ $\!{\rm D}_N$ detect the reference output modes in states , then the conditional signal output state reads \_[[out]{}]{}= \_[[in]{}]{} \^, where p=[Tr]{}(\_[[in]{}]{}\^) is the probability of generating $\hat{\varrho}_{{\rm out}}$. The non-unitary transformation operator =\_N\_2\_1 is the product of the individual conditional operators \_k=\_[[out]{}\_k]{}|\_k| \_[[in]{}\_k]{} , \[Yk\] where \_k = e\^[[i]{}(\_T + \_R)\_z]{} e\^[2[i]{}\_y]{} e\^[[i]{}(\_T -\_R)\_z]{} = T\^e\^[-R\^\*\^\_k ]{}e\^[R\^\_k]{}T\^[-\_k]{} is the unitary transformation operator of the beam splitter ${\rm B}_k$ in Fig. 1, with $\hat{L}_y$ $\!=$ $\!{\rm i}(\hat{a}_k^\dagger\hat{a}$ $\!-$ $\!\hat{a}^\dagger\hat{a}_k)/2$ and $\hat{L}_z$ $\!=$ $\!(\hat{n}$ $\!-$ $\!\hat{n}_k)/2$ [@Campos; @Daknaadded]. Here, $T$ $\!=$ $\!\cos\vartheta e^{{\rm i}\varphi_T}$ and $R$ $\!=$ $\!\sin\vartheta e^{{\rm i}\varphi_R}$ are the transmittance and reflectance, respectively. The operators $\hat{Y}_k$ can be represented as $s$-ordered operator products [@pap3] \_k = {(R\^) \^(-\^) }\_sT\^, \[order\] where the operators $\hat{F}$ and $\hat{G}$, respectively, generate $|\Psi_{{\rm in}_k}\rangle$ and $|\Psi_{{\rm out}_k}\rangle$ from the vacuum, |\_[[in]{}\_k]{}=(\_k\^)|0\_k, |\_[[out]{}\_k]{}=(\_k\^)|0\_k, and the ordering parameter $s$ is determined by the absolute value of the beam splitter reflectance as s=-1 . Note that the ordering procedure in (\[order\]) can be omitted if $|\Psi_{{\rm in}_k}\rangle$ or $|\Psi_{{\rm out}_k}\rangle$ is a coherent state [@pap3], since for |\_[[in]{}\_k]{}=\_k()(\_k\^) |0\_k, |\_[[out]{}\_k]{}=\_k()(\_k\^) |0\_k we have \_k = () \_k(\^[=0]{}\_[=0]{}) (). \[noorder\] Generation of truncated quantum states $|\Psi\rangle$ ===================================================== Each quantum state $|\Psi\rangle$ that is composed of a finite number of Fock states $|n\rangle$ can be written as |=\_[n=0]{}\^N\_n|n =\_[k=1]{}\^N (\^-\_k\^\*)|0, \[state\] where $\beta_1,$ $\!\ldots,$ $\!\beta_N$ denote the $N$ solutions of the equation $\langle\Psi|\beta\rangle\equiv\langle\Psi|\hat{D}(\beta)|0\rangle=0$. This reveals that $|\Psi\rangle$ can be generated from the vacuum state by alternate displacement and photon adding, as outlined in Fig. 2 [@pap2]. [ Fig. 2. Generation of truncated states $|\Psi\rangle$ from the vacuum by alternate displacement and photon adding. The coherent displacements are achieved by combination with highly]{} \ [ excited coherent states $|\alpha/R^\prime\rangle$ at highly transmitting beam splitters ($T^\prime\rightarrow1$), and photon adding is realized by combination with one-photon Fock states $|1\rangle$ and measuring zero photons in the outgoing reference modes with photodetectors D$_k$. ]{} A coherent displacement $\hat{D}(\alpha)$ can be realized by mixing the mode with a reference mode in a strong coherent state $|\alpha/R^\prime\rangle$ at a highly transmitting beam splitter [@Paris], and photon adding is achieved by mixing the mode with a reference mode in a Fock state $|1\rangle$ and measuring zero photons in the output detection channel (detectors ${\rm D}_k$ in Fig. 2) [@Daknaadded]. We assume that all the beam splitters used for photon adding have the same transmittance $T$ and reflectance $R$. The non-unitary transformation operator then reads as =(\_[N+1]{})\_N(\_N) \_1(\_1), where \_k=R\^T\^, \[Ykgen\] and the complex parameters $\alpha_1$,…, $\alpha_{N+1}$ are determined from the equation = || as $\alpha_k$ $\!=$ $\!T^{*N+1-k}(\beta_{k-1}$ $\!-$ $\!\beta_k)$ for $k$ $\!=$ $\!2,\ldots,N+1$ ($\beta_{N+1}$ $\!=$ $\!0$), and $\alpha_1$ $\!=$ $\!-\sum_{l=1}^NT^{-l} \alpha_{l+1}$. The probability of generating the desired state $\hat{\varrho}_{{\rm out}}=|\Psi\rangle\langle\Psi|$, i.e. the probability that all $N$ detectors register zero photons, is given by |0\^2= \[prob\] and decreases rapidly with increasing $N$. Nevertheless, for small $N$ this scheme offers a way to generate specific traveling quantum states, given the possibility to prepare 1-photon Fock states. Measuring arbitrary overlaps ============================ One fundamental task in quantum mechanics is to find the probability that a specific quantum state $|\Psi\rangle$ is contained in a state $\hat{\varrho}_{{\rm in}}$ of a given system. With regard to traveling waves, this overlap $\langle\Psi|\hat{\varrho}_{{\rm in}}|\Psi\rangle$ can be measured for a given state $|\Psi\rangle$ of the type (\[state\]) as outlined in Fig. 3 [@pap4]. [ Fig. 3. Measurement of the overlap $\langle\Psi|\hat{\varrho}_{{\rm in}}|\Psi\rangle$ of a signal state $\hat{\varrho}_{{\rm in}}$ with a desired state $|\Psi\rangle$ by successive combination with appropriately chosen coherent states $|\alpha_1\rangle,$ $\!\ldots,$ $\!|\alpha_N\rangle$ at beam splitters and measurement of the relative frequency of detecting simultaneously $1$ photon at the photodetectors ${\rm D}_1,$ $\!\ldots,$ ${\rm D}_N$ and 0 photons at ${\rm D}_{N+1}$. ]{} The signal mode in state $\hat{\varrho}_{{\rm in}}$ is mixed with reference modes in coherent states $|\alpha_1\rangle,$ $\!\ldots,$ $\!|\alpha_N\rangle$ at an array of $N$ beam splitters. Photodetectors ${\rm D}_1,$ $\!\ldots,$ $\!{\rm D}_{N+1}$ perform photon number measurements at the output modes. The joint probability $p(1,1;2,1;\ldots;N,1;N+1,0)$ that each of the detectors registers one photon and ${\rm D}_{N+1}$ none is then given by p(1,1;2,1;…;N,1;N+1,0)= 0|\_[[in]{}]{}\^|0 \[joint\] with =\_N\_2\_1, where for identical beam splitters \_k=-R\^\*()T\^ (-\_k), \[Ykpro\] see (\[Yk\]) as well as (\[order\]) and (\[noorder\]). This expression reveals that the signal is manipulated by alternate displacement and photon subtraction. If we now choose the arguments $\alpha_k$ ($k$ $\!=$ $\!1,\ldots,N$) of the coherent states as $\alpha_k$ $\!=$ $\!(R^*/T^{*k})\sum_{l=1}^k|T|^{2l-1}(\beta_l$ $\!-$ $\!\beta_{l-1})$, with , so that for a chosen $|\Psi\rangle$ the relation =|| is valid, then from (\[joint\]) it is seen that the sought overlap can be obtained from the joint probability as |\_[[in]{}]{}|= . The denominator $\|\hat{Y}^\dagger|0\rangle\|^2$ is given by (\[prob\]), with $(\beta_{N+2-l}$ $\!-$ $\!\beta_{N+1-l})$ being replaced with $(\beta_l$ $\!-$ $\!\beta_{l-1})$. It may be regarded as being the “fidelity” of the measurement, since it is a measure of the maximal occurence of the detection coincidences. Obviously, it is equal to $p(1,1;2,1;\ldots;N,1;N+1,0)$ in the particular case when signal and measured state coincide, $\hat{\varrho}_{{\rm in}}$ $\!=$ $\!|\Psi\rangle\langle\Psi|$. Schrödinger-cat-like states =========================== The schemes in Figs. 2 and 3 can be simplified if some of the $\beta_k$ in (\[state\]) are equal: |=\_[l=1]{}\^M (\^-\_l\^\*)\^[d\_l]{}|0with $M$ $\!<$ $\!N$. In this case it is possible to add $d_k$ photons at once in Fig. 2 by using $M$ detectors and combining with Fock states $|d_k\rangle$ or to subtract $d_k$ photons at each beam splitter in Fig. 3 by using $M$ $\!+$ $\!1$ detectors and measuring the relative frequency of the event $(1,d_1;\ldots;M,d_M;M+1,0)$. All calculations are analogous if $R\hat{a}^\dagger$ in (\[Ykgen\]) is replaced with $(R\hat{a}^\dagger)^{d_k}/\sqrt{d_k!}$ and $-R^*\hat{a}$ in (\[Ykpro\]) is replaced with $(-R^*\hat{a})^{d_k}/\sqrt{d_k!}$. As an example let us consider the states |\_n\^[,]{}=(\_3) \^[n]{}(\_2)\^[n]{}(\_1) |0, where $\gamma_1$ $\!=$ $\!{\rm i}(\beta$ $\!-$ $\!\alpha)/2$, $\gamma_2$ $\!=$ $\!{\rm i}(\alpha-\beta)$, $\gamma_3$ $\!=$ $\![(1-{\rm i})\alpha$ $\!+$ $\!(1+{\rm i})\beta]/2$, and the normalization factor is ${\cal N}$ $\!=$ $\!(4^nn!/\sqrt{\pi})$ $\!\mbox{$\Gamma(n+1/2)$}$ $\!_1F_2[-n,1/2-n,1,|\alpha-\beta|^4/64]$. From the above considerations it is clear that these states can be generated using 2 detectors, and the overlap of a signal state with such states can be measured using 3 detectors. The states $|\Psi_n^{\alpha,\beta}\rangle$ reveal the interesting property that for increasing $n=|\alpha-\beta|^2/4$ they approach superpositions of coherent states [@pap4], $|\Psi_\infty^{\alpha,\beta}\rangle\langle\Psi_\infty^{\alpha,\beta}|= |\Psi^{\alpha,\beta}\rangle\langle\Psi^{\alpha,\beta}|$ with |\^[,]{}=( |+|) (note that $|\langle\Psi_{n=3}^{\alpha,\beta}|\Psi^{\alpha,\beta}\rangle|^2$ $\!>$ $\!0.95$). This offers the possibility of generating Schrödinger-cat-like states, provided that two $n$-photon Fock states are available. On the other hand, measuring overlaps with Schrödinger-cat-like states allows one, in principle, to reconstruct the signal state [@Freyberger]. It is worth noting that choosing a squeezed coherent signal state offers the possibility of measuring the field strength statistics of a Schrödinger cat without need to generate the cat state. Note that squeezed coherent states approach field strength states for sufficiently strong squeezing. Finally, it should be pointed out that the probability of generation of the states $|\Psi_n^{\alpha,\beta}\rangle$ and the fidelity of measurement of the overlap with them are equal. For large $n$, they decrease exponentially with increasing $n$: p=. \[p\] Conclusion ========== We have discussed conditional quantum state engineering at beam splitter arrays. We have presented a scheme for the generation of arbitrary quantum states which requires coherent states and 1-photon Fock states and zero-photon detection. Further, we have given a scheme for the measurement of the overlap of an unknown signal state with an arbitrary quantum state which requires coherent states and $0$- and $1$-photon detections. For the two schemes we have calculated the probabilities of state generation and overlap measurement. Finally, we have shown how the schemes can be simplified under special conditions. As an example we have considered the generation of (and overlap measurement with) states which approach superpositions of two arbitrary coherent states. [**Acknowledgements**]{}\ This work was supported by the Deutsche Forschungsgemeinschaft. [99]{} J. Clausen, M. Dakna, L. Knöll, D.G. Welsch: [*Quant. Semiclass. Opt.*]{}, in press; J. Clausen, M. Dakna, L. Knöll, D.G. Welsch: in preparation; D.G. Fischer, M. Freyberger: Opt. Commun. [**159**]{} (1999) 158. [^1]:

--- abstract: 'A quantization procedure for the Yang-Mills equations for the Minkowski space $\mathbf{R}^{1,3}$ is carried out in such a way that field maps satisfying Wightman’s axioms of Constructive Quantum Field Theory can be obtained. Moreover, the spectrum of the corresponding Hamilton operator is proven to be positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level. The particles corresponding to all solution fields are bosons.' author: - 'Simone Farinelli [^1]' title: 'Four Dimensional Quantum Yang-Mills Theory and Mass Gap' --- \[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[section\] \[section\] Introduction ============ Yang-Mills fields, which are also called gauge fields, are used in modern physics to describe physical fields that play the role of carriers of an interaction (cf. [@EoM02]). Thus, the electromagnetic field in electrodynamics, the field of vector bosons, carriers of the weak interaction in the Weinberg-Salam theory of electrically weak interactions, and finally, the gluon field, the carrier of the strong interaction, are described by Yang-Mills fields. The gravitational field can also be interpreted as a Yang-Mills field (see [@DP75]). The idea of a connection as a field was first developed by H. Weyl (1917), who also attempted to describe the electromagnetic field in terms of a connection. In 1954, C.N. Yang and R.L. Mills (cf. [@MY54]) suggested that the space of intrinsic degrees of freedom of elementary particles (for example, the isotropic space describing the two degrees of freedom of a nucleon that correspond to its two pure states, proton and neutron) depends on the points of space-time, and the intrinsic spaces corresponding to different points are not canonically isomorphic. In geometrical terms, the suggestion of Yang and Mills was that the space of intrinsic degrees of freedom is a vector bundle over space-time that does not have a canonical trivialization, and physical fields are described by cross-sections of this bundle. To describe the differential evolution equation of a field one has to define a connection in the bundle, that is, a trivialization of the bundle along the curves in the base. Such a connection with a fixed holonomy group describes a physical field, usually called a Yang-Mills field. The equations for a free Yang-Mills field can be deduced from a variational principle. They are a natural non-linear generalization of Maxwell’s equations. Field theory does not give the complete picture. Since the early part of the 20th century, it has been understood that the description of nature at the subatomic scale requires quantum mechanics, where classical observables correspond to typically non commuting self-adjoint operators on a Hilbert space and classic notions as “the trajectory of a particle” do not apply. Since fields interact with particles, it became clear by the late 1920s that an internally coherent account of nature must incorporate quantum concepts for fields as well as for particles. When doing this components of fields at different points in space-time become non-commuting operators. The most important Quantum Field Theories describing elementary particle physics are gauge theories formulated in terms of a principal fibre bundle over the Minkowskian space-time with particular choices of the structure group. They are depicted in Table \[GT\]. **Gauge Theory** **Fundamental Forces** **Structure Group** ----------------------------- ------------------------ --------------------------------------------- Quantum Electrodynamics Electromagnetism $U(1)$ (QED) Electroweak Theory Electromagnetism $\text{SU}(2)\times U(1)$ (Glashow-Salam-Weinberg) and weak force Quantum Chromodynamics Strong force $\text{SU}(3)$ (QCD) Standard Model Strong, weak forces $\text{SU}(3)\times\text{SU}(2)\times U(1)$ and electromagnetism Georgi-Glashow Grand Strong, weak forces $\text{SU}(5)$ Unified Theory (GUT1) and electromagnetism Fritzsch-Georgi-Minkowski Strong, weak forces $\text{SO}(10)$ Grand Unified Theory (GUT2) and electromagnetism Grand Unified Strong, weak forces $\text{SU}(8)$ Theory (GUT3) and electromagnetism Grand Unified Strong, weak forces $\text{O}(16)$ Theory (GUT4) and electromagnetism : Gauge Theories[]{data-label="GT"} In order for Quantum Chromodynamics to completely explain the observed world of strong interactions, the theory must imply: - **Mass gap:** There must exist some positive constant $\Delta$ such that the excitation of the vacuum state has energy at least $\Delta$. This would explain why the nuclear force is strong but short-ranged. - **Quark confinement:** The physical particle states corresponding to proton, neutron and pion must be $\text{SU}(3)$-invariant. This would explain why individual quarks are never observed. - **Chiral symmetry breaking:** In the limit for vanishing quark-bare masses the vacuum is invariant under a certain subgroup of the full symmetry group that acts on the quark fields. This is needed to account for the “current algebra” theory of soft pions. The seventh CMI-Millenium prize problem is the following conjecture. \[CMI\] For any compact simple Lie group $G$ there exists a nontrivial Yang-Mills theory on the Minkowskian $\mathbf{R}^{1,3}$, whose quantization satisfies Wightman’s axiomatic properties of Constructive Quantum Field Theory and has a mass gap $\eta>0$. The conjecture is explained in [@JW04] and commented in [@Do04] and in [@Fad05]. To our knowledge this conjecture is unproved. There have been serious attempts by Jormakka([@Jo10]) and Dynin ([@Dy13]). See also the work of Frasca ([@Fr07]). It is important at what stage quantization is performed. In Jormakka([@Jo10]) the classical Yang-Mills equations are solved first, and the solution quantized afterwards. But Quantum Field Theory needed to prove Conjecture \[CMI\] requires the quantization of the Yang-Mills equations first and the solution of the quantized problem, then. Moreover, Jormakka, whose original aim was to tackle Conjecture \[CMI\], constructs solutions of the classical Yang-Mills equations but does not completely verify all Wightman’s axioms and concludes by an argument requiring generalized eigenvectors and eigenvalues that there is no positive spectral gap. We believe this last conclusion is wrong. Dynin provides a context for non perturbative quantum field theory in nuclear Kree-Gelfand triples, solves the Yang-Mills equation on Euclidean balls and presents the anti-normal quantization of the Yang-Mills energy-mass functional in Kree-Gelfand triples proving the existence of a positive spectral gap. Frasca proves by means of perturbation theory that in the limit of the coupling constant going to infinity a quantized Yang-Mills theory is equivalent to $\lambda\phi^4$ theory with the dynamics ruled just by a homogeneous equation. He therefore proves that there exist a spectral gap in the strong coupling limit. This paper is organized as follows. Section 2 presents the Yang-Mills equations and their Hamiltonian formulation for the Minkowskian $\mathbf{R}^{1,3}$. Section 3 depicts Wightman’s axioms of Constructive Quantum Field Theory, which are verified in Sections 4, where Yang-Mills Equations are quantized and the existence of a positive mass gap proven. Section 5 concludes. Yang-Mills Connections ====================== Definitions, Existence and Uniqueness {#ptrel} ------------------------------------- A Yang-Mills connection is a connection in a principal fibre bundle over a (pseudo-)Riemannian manifold whose curvature satisfies the harmonicity condition i.e. the Yang-Mills equation. Let $P$ be a principal $G$-fibre bundle over a pseudoriemannian $m$-dimensional manifold $(M,h)$, and let $V$ be the vector bundle associated with $P$ and $\mathbf{C}^K$, induced by the representation $\rho:G\rightarrow\text{GL}(\mathbf{C}^K)$. A connection on the principal fibre bundle $P$ is Lie-algebra $\mathcal{G}$ valued one form $A$ on $M$. It defines a connection $\nabla$ for the vector bundle $V$, i.e. an operator acting on the space of cross sections of $V$. The vector bundle connection $\nabla$ can be extended to an operator $d:\Gamma(\bigwedge^p(M)\bigotimes V)\rightarrow \Gamma(\bigwedge^{p+1}(M)\bigotimes V)$, by the formula $$d^{\nabla}(\eta\otimes v):= d\eta\otimes v + (-1)^{p}\eta\otimes \nabla v.$$ The operator $\delta^{\nabla}:\Gamma(\bigwedge^{p+1}(M)\bigotimes V)\rightarrow \Gamma(\bigwedge^{p}(M)\bigotimes V)$, defined as the formal adjoint to $d$ is equal to $$\delta^{\nabla}\eta= (-1)^{p+1}\ast d^{\nabla} \ast,$$ where $\ast$ denotes the Hodge-star operator. A connection $A$ in a principal fibre bundle $P$ is called a Yang-Mills field if the curvature $F:=dA+A\wedge A$, considered as a $2$-form with values in the vector bundle $\mathcal{G}$, satisfies the Yang-Mills equations $$\label{YME} \delta^{\nabla}F=0,$$ or, equivalently $$\delta^{\nabla}R^{\nabla}=0,$$ where $R^{\nabla}(X,Y):=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$ denotes the curvature of the vector bundle $V$ and is a $2$-form with values in $V$. Since the connection $A$ is a Lie algebra $\mathcal{G}$ valued $1$-form on $M$, the fields $A_j(x):=A(x)e_j=\sum_{k=1}^KA_j^k(x)t_k$ define by means of the tangential map $T_e\rho:\mathcal{G}\rightarrow\mathcal{L}(\mathbf{C}^K)$ of the representation $\rho:G\rightarrow\text{GL}(\mathbf{C}^K)$, fields of endomorphisms $T_e\rho A_1,\dots,T_e\rho A_m\in\mathcal{L}(V_x)$ for the bundle $V$. Given a basis of the Lie algebra $\mathcal{G}$ denoted by $\{t_1,\dots,t_K\}$, the endomorphisms $\{w_s:=T_e\rho.t_s\}_{s=1,\dots,K}$ in $\mathcal{L}(\mathbf{C}^K)$ have matrix representations with respect to a local basis $\{v_s(x)\}_{s=1,\dots,K}$ denoted by $[w_s]_{\{v_s(x)\}}$. Being $\rho$ a representation, $T_e\rho$ has maximal rank and the endomorphism are linearly independent. Given a local basis $\{e_j(x)\}_{j=1,\dots,m}$ for $x\in U\subset M$, the Christoffel symbols of the connection $\nabla$ are locally defined by the equation $$\nabla_{e_j}v_s=\sum_{r=1}^K\Gamma_{j,s}^rv_r,$$ holding on $U$ and satisfy the equalities $$\Gamma_{j,s}^r=\sum_{a=1}^K[w_a]_s^rA^a_j.$$ Given a local vector field $v=\sum_{s=1}^Kf^sv_s$ in $V|_U$ and a local vector field $e$ in $TM|_U$ the connection $\nabla$ has a local representation $$\nabla_ev=\sum_{s=1}^K(df^s(e).v_s+f^s\omega(e).v_s),$$ where $\omega$ is an element of $T^*U|_U\bigotimes \mathcal{L}(V|_U)$ ie. an endomorphism valued $1-$form satisfying $$\omega(e_j)v_s=\sum_{r=1}^K\Gamma_{j,s}^rv_r.$$ The curvature two form reads in local coordinates as $$F=\sum_{1\le i<j\le M}\sum_{k=1}^KF_{i,j}^k\,t_k\,dx_i\wedge dx_j=\frac{1}{2}\sum_{i,j=1}^M\sum_{k=1}^K\left(\partial_j A^k_i-\partial_i A_j^k-\sum_{a,b=1}^KC^k_{a,b}A^a_iA_j^b\right)t_k<,dx_i\wedge dx_j,$$ where $C=[C^c_{a,b}]_{a,b,c=1,\dots,K}$ are the *structure constants* of the Lie algebra $\mathcal{G}$ corresponding to the basis $\{t_1,\dots,t_K\}$, which means that for any $a,b$ $$[t_a,t_b]=\sum_{c=1}^KC_{a,b}^ct_c.$$ The existence and uniqueness of solutions of the Yang-Mills equations in the Minkowski space have been first established by Segal (cf. [@Se78] and [@Se79]), who proves that the corresponding Cauchy problem encoding initial regular data has always a unique local and global regular solution. He proves as well that the temporal gauge ($A_0=0$) chosen to express the solution does not affect generality, because any solution of the Yang-Mills equation can be carried to one satisfying the temporal gauge. This subject has been undergoing intensive research, improving the original results. For example in [@EM82] and in [@EM82Bis] the Yang-Mills-Higgs equations, which generalize (\[YME\]) and are non linear order two PDEs, have been reformulated in the temporal gauge as a non linear order one PDE satisfying a constraint equation. This PDE can be written as integral equation solving (always and uniquely, locally and globally) the Cauchy data problem with improved regularity results. Existence, uniqueness and regularity of the Yang-Mills-Higgs equations under the MIT Bag boundary conditions have been investigated in [@ScSn94] and [@ScSn95]. Hamiltonian Formulation for the Minkowski Space ----------------------------------------------- The Hamilton function describes the dynamics of a physical system in classical mechanics by means of Hamilton’s equations. Therefore, we have to reformulate the Yang-Mills equations in Hamiltonian mechanical terms. We focus our attention on the Minskowski $\mathbb{R}^4$ with the pseudoriemannian structure of special relativity $h=dx^0\otimes dx^0-dx^1\otimes dx^1-dx^2\otimes dx^2-dx^3\otimes dx^3$. The coordinate $x^0$ represents the time $t$, while $x^1,x^2,x^3$ are the space coordinates. We introduce Einstein’s summation notation and adopt the convention that indices for coordinate variables from the greek alphabet vary over $\{0,1,2,3\}$ and those from the latin alphabet vary over the space indices $\{1,2,3\}$. For a generic field $F=[F_\mu]_{\mu=0,1,2,3}$ let $\mathbf{F}:=[F_i]_{i=1,2,3}$ denote the “space” component. The color indices lie in $\{1,\dots,K\}$. Let $$\varepsilon^{a,b,c}:=\left\{ \begin{array}{ll} +1 & \hbox{($\pi$ is even)} \\ -1 & \hbox{($\pi$ is odd)} \\ \;\;\; 0 & \hbox{(two indices are equal),} \end{array} \right.$$and any other choice of lower and upper indices, be the Levi-Civita symbol, defined by mean of the permutation $\pi:=\left( \begin{array}{ccc} 1 & 2 & 3 \\ a & b & c \\ \end{array} \right)$ in $\mathfrak{S}^3$. If the Lie group $G$ is simple, then the Lie-Algebra is simple and the structure constants can be written as $$C_{a,b}^c = g\varepsilon^c_{a,b},$$ for a positive constant$g$ called *coupling constant*, (see f.i. [@We05] Chapter 15, Appendix A). The components of the curvature read then $$F_{\mu,\nu}^k=\frac{1}{2}(\partial_\nu A^k_\mu-\partial_\mu A_\nu^k-g\varepsilon^k_{a,b}A^a_\mu A^b_\nu).$$ We will consider only simple Lie groups. We need to introduce an appropriate gauge for the connections we are considering. A connection $A$ over the Minkowski space satisfies the *Coulomb gauge* if and only if $$\partial_jA_j^a=0$$ for all $a=1,\dots,K$. Let $\mathcal{F}$ be the Fourier transform on functions in $L^2(\mathbf{R}^3,\mathbf{R})$. The transverse projector $T:L^2(\mathbf{R}^3,\mathbf{R}^3)\rightarrow L^2(\mathbf{R}^3,\mathbf{R}^3)$ is defined as $$(Tv)_i:=\mathcal{F}^{-1}\left(\left[\delta_{i,j}-\frac{p_ip_j}{|p|^2}\right]\mathcal{F}(v_j)\right),$$ and the vector field $v$ decomposes as sum of a *transversal* ($v^{\perp}$) and *longitudinal* ($v^{\parallel}$) component: $$v_i=v_i^{\perp}+v_i^{\parallel},\quad v_i^{\perp}:=(Tv)_i,\quad v_i^{\parallel}:= v_i-(Tv)_i.$$ The Coulomb gauge condition for a connection $A$ is equivalent with the vanishing of its longitudinal component: $${A_i^a}^{\parallel}(t,\cdot)=0$$ for all $i=1,2,3$, all $a=1,\dots K$ and any $t\in\mathbf{R}$. \[ModGreen\] For a simple Lie group as structure group let $A$ be a connection over the Minkowskian $\mathbf{R}^4$ satisfying the Coulomb gauge and assume that $A_i^a(t,\cdot)\in C^{\infty}(\mathbf{R}^3,\mathbf{R})\cap L^2(\mathbf{R}^3,\mathbf{R})$ for all $i=1,2,3$, all $a=1,\dots K$ and any $t\in\mathbf{R}$. The operator $L$ on the real Hilbert space $L^2(\mathbf{R}^3,\mathbf{R}^K)$ defined as $$L=L(\mathbf{A};x)=[L^{a,b}(\mathbf{A};x)]:=[\delta^{a,b}\Delta_x^{\mathbf{R}^3}+g\varepsilon^{a,c,b}A_k^c(t,x)\partial_k]$$ is essentially self adjoint and elliptic for any time parameter $t\in\mathbf{R}$. Its spectrum lies on the real line and decomposes into discrete $\operatorname{spec}_d(L)$ and continuous spectrum $\operatorname{spec}_c(L)$. If $0$ is an eigenvalue, then it has finite multiplicity, i.e. $\ker(L)$ is always finite dimensional . The modified Green’s function $G=G(\mathbf{A};x,y)=[G^{a,b}(\mathbf{A};x,y)]\in\mathcal{S}^{\prime}(\mathbf{R}^3,\mathbf{R}^{K\times K})$ for the operator $L$ is the distributional solution to the equation $$L^{a,b}(\mathbf{A};x)G^{b,d}(\mathbf{A};x,y)=\delta^{a,d}\delta(x-y)-\sum_{n=1}^{N}\psi_n^a(x,\mathbf{A})\psi_n^d(y,\mathbf{A}),$$ where $\{\psi_n(\mathbf{A}; \cdot)\}_n$ is an o.n. $L^2$-basis of $N$-dimensional $\ker(L)$. This modified Green’s function can be written as Riemann-Stielties integral: for any $\varphi\in\mathcal{S}(\mathbf{R}^3,\mathbf{R}^K)\cap L^2(\mathbf{R}^3,\mathbf{R}^K)$ $$\label{gen} G(\mathbf{A};x,\cdot)(\varphi)=\int_{\lambda\neq0}\frac{1}{\lambda}d(E_{\lambda}\varphi)(x),$$ where $(E_{\lambda})_{\lambda\in\mathbf{R}}$ is the resolution of the identity corresponding to $L$. In [@Br03] and [@Pe78] the modified Green’s function is constructed assuming that the operator $L$ has a discrete spectral resolution $(\psi_n(\mathbf{A};\cdot),\lambda_n)_{n\ge0}$ as $$\label{part} G(\mathbf{A};x,y)=\sum_{n:\lambda_n\neq0}\frac{1}{\lambda_n}\psi_n(\mathbf{A};x)\psi_n^{\dagger}(\mathbf{A};y).$$ Since the discontinuity points of the spectral resolution $(E_{\lambda})_{\lambda\in\mathbf{R}}$ are the eigenvalues, i. e. the elements of $\operatorname{spec}_d(L)$ (cf. [@Ri85], Chapter 9), the solution (\[gen\]) extends (\[part\]) to the general case. Rigged Hilbert spaces have been introduced in mathematical physics to utilize Dirac calculus for the spectral theory of operators appearing in quantum mechanics (see [@Ro66], [@ScTw98] and [@Ma08]). Within that framework the role of distributions to provide a rigorous foundation to generalized eigenvectors and eigenvalues is highlighted by the Gel’fand-Kostyuchenko spectral theorem (see [@Ze09] Chapter 12.2.4 and [@GS64] Section I.4). \[GK\] Let $A:\mathcal{S}(\mathbf{R}^N)\rightarrow\mathcal{S}(\mathbf{R}^N)$ be a linear sequentially continuous operator, which is essentially selfadjoint in the Hilbert Space $L^2(\mathbf{R}^N)$. Then, the operator $A$ has a complete system $(F_m)_{m\in M}$ of eigendistributions. This means that there exists a non empty index set $M$ such that $F_M\in\mathcal{S}^{\prime}(\mathbf{R}^N)$ for all $m\in M$ and 1. Eigendistributions: there exist a function $\lambda:M\rightarrow\mathbf{R}$ such that $$F_m(A\varphi)=\lambda(m)F_m(\varphi),$$ for all $m\in M$ and all test functions $\varphi\in\mathcal{S}(\mathbf{R}^N)$. 2. Completeness: if $F_m(\varphi)=0$ for all $m\in M$ and fixed test function $\varphi\in\mathcal{S}(\mathbf{R}^N)$, then $\varphi=0$. The spectrum of $A$ reads then $\operatorname{spec}(A)=\lambda(M)$, where for all $m\in M$ the real number $\lambda(m)$ is either an eigenvalue (i.e. $F_m\in L^2(\mathbf{R}^N)$) or an element of the continuous spectrum (i.e. $F_m\notin L^2(\mathbf{R}^N)$). As long as the connection satisfies the Coulomb gauge condition, the operator $L$ is symmetric and essentially selfadjoint on the appropriate domain, as a direct computation involving integration by parts can show. Being its leading symbol elliptic, the operator $L$ is elliptic and restricted to $[-\frac{R}{2}, +\frac{R}{2}]^3$ under the Dirichlet boundary conditions it has a discrete spectral resolution (cf. [@Gi95], Chapter 1.11.3). Every eigenvalue has finite multiplicity. The dimension of the eigenspaces is an integer valued continuous and hence constant function of $R$. For $R\rightarrow+\infty$ the discrete Dirichlet spectrum of $L$ on $[-\frac{R}{2}, +\frac{R}{2}]^3$ clusters in the spectrum of $L$ on $\mathbf{R}^3$ which decomposes into discrete and continuous spectrum. Therefore, the eigenvalues must have finite multiplicity and, in particular $\ker(L)$ is finite dimensional.\ Equation (\[gen\]) gives the modified Green’s function as it can be verified by the following computation which holds true for any $\varphi\in\mathcal{S}(\mathbf{R}^3,\mathbf{R}^K)\cap L^2(\mathbf{R}^3,\mathbf{R}^K)$: $$\begin{split} L(\mathbf{A};x)G(\mathbf{A};x,\cdot)(\varphi)&=L\int_{\lambda\neq0}\frac{1}{\lambda}d(E_{\lambda}\varphi)(x)=\int_{\lambda\neq0}\frac{1}{\lambda}Ld(E_{\lambda}\varphi)(x)=\\ &=\int_{\mathbf{R}}d(E_{\lambda}\varphi)(x)-\int_{0^-}^{0^+}d(E_{\lambda}\varphi)(x)=\phi(x)-P_{\ker(L)}\varphi(x)=\\ &=\delta(x-\cdot)(\varphi)-\sum_{n=1}^{N}\psi_n(x,\mathbf{A})\psi_n(\cdot,\mathbf{A})(\varphi). \end{split}$$ The existence of eigenvalues of $L$ depends on the additive perturbation to the Laplacian given by $g\varepsilon^{a,c,b}A_k^c(t,x)\partial_k$. For example, if $g=0$ or $\mathbf{A}=0$, the operator $L$ has no eigenvalues and $\operatorname{spec}(L)=\operatorname{spec}_c(L)=]-\infty,0]$. In general the spectrum depends on the choice of the connection $A$. In [@BEP78] and in [@Pe78] special cases comprising pure gauges and Wu-Yang monopoles are computed explicitly. We are interested in a reformulation of the general solution (\[gen\]), where the dependence on the connection becomes explicit. Inspired by [@Ha97] we find \[prop\] For a simple Lie group as structure group, if we assume that the coupling constant $g<1$, then a Green’s function $K=K(\mathbf{A};x,y)=[K^{a,b}(\mathbf{A};x,y)]$ for the operator $L$ in Proposition \[ModGreen\], that is, a distributional solution to the equation $$L^{a,b}(\mathbf{A};x)K^{b,d}(\mathbf{A};x,y)=\delta^{a,d}\delta(x-y),$$ is given by the convergent series in $\mathcal{S}^{\prime}(\mathbf{R}^3,\mathbf{R}^{K\times K})$ $$\label{ser} \begin{split} K^{b,d}(\mathbf{A};x,y)=&\frac{\delta^{b,d}}{4\pi|x-y|}+g\varepsilon^{d,e,b}\int_{\mathbf{R}^3}d^3u_1\frac{1}{4\pi|x-u_1|}A_k^e(u_1)\partial_k\left(\frac{1}{4\pi|u_1-y|}\right)+\\ &+\dots+\\ &+(-1)^{n-1}\frac{n+1}{2}g^n\varepsilon^{b,e_1,s_1}\cdots\varepsilon^{s_{n-1},e_n,d}\int_{\mathbf{R}^3}d^3u_1\frac{1}{4\pi|x-u_1|}A_k^{e_1}(u_1)\cdot\\ &\cdot\partial_k\int_{\mathbf{R}^3}d^3u_2\frac{1}{4\pi|u_1-u_2|}\dots\int_{\mathbf{R}^3}d^3u_n\frac{1}{4\pi|u_{n-1}-u_n|}A_l^{e_n}(u_n)\partial_l\left(\frac{1}{4\pi|u_n-y|}\right)+\\ &+\dots \end{split}$$ The series (\[ser\]) converges because of the integrability of the connection $A$ and the fact that $g<1$. Recall that $\frac{1}{|x|}\in L^1_{loc}(\mathbf{R}^3)\subset\mathcal{S}^{\prime}(\mathbf{R}^3)$. We now check that it represents a Green’s function for $L$: $$L(\mathbf{A};x)^{a,b}K^{b,d}(\mathbf{A};x,y)=\delta^{a,d}\delta(x-y)+\lim_{n\rightarrow+\infty}\text{Rest}_n,$$ where, after having evaluated a “telescopic sum”, the remainder part reads $$\begin{split} \text{Rest}_n=&(-1)^{n-1}\frac{n+1}{2}g^{n+1}\varepsilon^{b,e_1,s_1}\cdots\varepsilon^{s_{n-1},e_n,d}\varepsilon^{a,c,b}A^c_k(x)\int_{\mathbf{R}^3}d^3u_1\frac{-x_k}{4\pi|x-u_1|^3}A_k^{e_1}(u_1)\cdot\\ &\cdot\partial_k\int_{\mathbf{R}^3}d^3u_2\frac{1}{4\pi|u_1-u_2|}\dots\int_{\mathbf{R}^3}d^3u_n\frac{1}{4\pi|u_{n-1}-u_n|}A_l^{e_n}(u_n)\partial_l\left(\frac{1}{4\pi|u_n-y|}\right). \end{split}$$ Because of the integrability of the connection, there exists a constant $C>0$ such that for any $\varphi\in\mathcal{S}(\mathbf{R}^3,\mathbf{R})$ $$|\text{Rest}_n(\varphi)|\le Cg^{n+1}\|\varphi\|_{L^2(\mathbf{R}^3,\mathbf{R})}\rightarrow0\quad(n\rightarrow+\infty),$$ and the proposition follows. For the Minkowskian $\mathbf{R}^4$ the assumption of a (dimensionless) coupling constant $g<1$ is well posed, as the experimental results on running coupling constants show: for the Yang-Mills theory for QCD with SU(3) one is basically allowed to choose any value $g>0$ (see [@SaSc10]). Moreover, we will later have to analyze the case where $g\rightarrow0^+$. \[cor\] Under the same assumptions as Proposition \[prop\] the distribution $$G(\mathbf{A};x,y)= K(\mathbf{A};x,y)+K(\mathbf{A};x,y)^{\dagger}-\sum_{n:\lambda_n=0}\psi_n(\mathbf{A};x)\psi_n^{\dagger}(\mathbf{A};y)$$ is a (symmetric) modified Green’s function for the operator $L$. After this preparation we can turn to the Hamiltonian formulation of Yang-Mills’ equations, following the results in [@Br03] and [@Pe78], which just need to be adapted for the generic case that $L$ has a mixture of discrete and continuous spectral resolution. \[CHam\]. For a simple Lie group as structure group and for canonical variables satisfying the Coulomb gauge condition, the Yang-Mills equations for the Minkowskian $\mathbf{R}^4$ can be written as Hamilton equations $$\left\{ \begin{array}{ll} \frac{d\mathbf{E}}{dt}&=-\frac{\partial H}{\partial \mathbf{A}}(\mathbf{A},\mathbf{E})\\ \\ \frac{d\mathbf{A}}{dt}&=+\frac{\partial H}{\partial \mathbf{E}}(\mathbf{A},\mathbf{E}) \end{array} \right.$$ for the following choices: - **Position variable:** $\mathbf{A}=[A_{i}^a(t,x)]_{\substack{ a=1,\dots,K \\ i=1,2,3}}$ also termed [*potentials*]{} , - **Momentum variable:** $\mathbf{E}=[E_{i}^a(t,x)]_{\substack{ a=1,\dots,K \\ i=1,2,3}}$ termed [*chromoelectric fields*]{}, - **Hamilton Function:** defined as function of $\mathbf{A}$ and $\mathbf{E}$ as $$\label{classicH} H=H(\mathbf{A},\mathbf{E}):=\frac{1}{2}\int_{\mathbf{R}^3}d^3x\left(E_i^a(t,x)^2+B_i^a(t,x)^2+f^a(t,x))\Delta f^a(t,x)+2\rho^c(t,x)A^c_0(t,x)\right),$$ where $\mathbf{B}=[B_i^a]$, termed [*chromomagnetic fields*]{} is the matrix valued function defined as $$\begin{split} B_i^a&:=\frac{1}{4}\varepsilon_{i}^{j,k}\left(\partial_jA_k^a-\partial_kA^a_j+g\varepsilon^a_{b,c}A_j^bA_k^c\right), \end{split}$$ and $\rho=[\rho^a(t,x)]$, termed [*charge density*]{} is the vector valued function defined as $$\rho^a:=g\varepsilon^{a,b,c}E^b_iA^c_i,$$ and where $$\label{deff} \begin{split} f^a(t,x)&:=-g\int_{\mathbf{R}^3}d^3y\,G^{a,b}(\mathbf{A};x,y)\rho^b(t,x)\\ A_0^a(t,x)&:=\int_{\mathbf{R}^3}d^3y\,G^{a,b}(\mathbf{A};x,y)\Delta f^b(t,x), \end{split}$$ for the modified Green’s function $G(\mathbf{A};x,y)$ for the operator $L(\mathbf{A};x)$. As shown in [@Pe78] the ambiguities discussed by Gribov in [@Gr78] concerning the gauge fixing (see also [@Si78] and [@He97]) can be traced precisely to the existence of zero eigenfunctions of the operator $L$. Wightman’s Axioms of Constructive Quantum Field Theory ====================================================== In year 1956 Wightman first stated the axioms needed for CQFT in his seminal work [@Wig56], which remained not very widely spread in the scientific community till 1964, when the first edition of [@SW10] appeared. We will list the axioms in the slight refinement of [@BLOT89] and [@DD10]. A scalar (respectively vectorial-spinorial) quantum field theory consists of a separable Hilbert space $\mathcal{H}$, whose elements are called states, a unitary representation $U$ of the Poincaré group $\mathcal{P}$ in $\mathcal{H}$, an operator valued distributions $\Phi$ (respectively $\Phi_1,\dots,\Phi_d$) on $\mathcal{S}(\mathbf{R}^4)$ with values in the unbounded operators of $\mathcal{H}$, and a dense subspace $\mathcal{D}\subset\mathcal{H}$ such that the following properties hold: (W1) Relativistic invariance of states: : The representation $U:\mathcal{P}\rightarrow\mathcal{U}(\mathcal{H})$ is strongly continuous. (W2) Spectral condition: : Let $P_0,P_1,P_2,P_3$ be the infinitesimal generators of the one-parameter groups $t\mapsto U(te_\mu,I)$ for $\mu=0,1,2,3$. The operators $P_0$ and $P_0^2-P_1^2-P_2^2-P_3^2$ are positive. This is equivalent to the spectral measure $E_{\cdot}$ on $\mathbf{R}^4$ corresponding to the restricted representation $\mathbf{R}^4\ni a\mapsto U(a,I)$ having support in the positive light cone (cf. [@RS75], Chapter IX.8). (W3) Existence and uniqueness of the vacuum: : There exists a unique state $\psi_0\in\mathcal{D\subset\mathcal{H}}$ such that $U(a,I)\psi_0=\psi_0$ for all $a\in\mathbf{R}^4$. (W4) Invariant domains for fields: : The maps $\Phi:\mathcal{S}(\mathbf{R}^4)\rightarrow\mathcal{O}(\mathcal{H})$, and, respectively $\Phi_1,\dots,\Phi_d:\mathcal{S}(\mathbf{R}^4)\rightarrow\mathcal{O}(\mathcal{H})$, from the Schwartz space of test function to (possibly) unbounded selfadjoint operators on the Hilbert space satisfies (satisfy) following properties (a) : For all $\varphi\in\mathcal{S}(\mathbf{R}^4)$ and all field maps the domain of definitions $\mathcal{D}(\Phi(\varphi))$, $\mathcal{D}(\Phi(\varphi)^*)$, and respectively $\mathcal{D}(\Phi_j(\varphi))$, $\mathcal{D}(\Phi_j(\varphi)^*)$, all contain $\mathcal{D}$ and the restrictions of all operators to $\mathcal{D}$ agree. (b) : $\Phi(\varphi)(\mathcal{D})\subset\mathcal{D}$, and, respectively $\Phi_j(\varphi)(\mathcal{D})\subset\mathcal{D}$. (c) : For any $\psi\in\mathcal{D}$ fixed the maps $\varphi\mapsto\Phi(\varphi)\psi$, and, respectively $\varphi\mapsto\Phi_j(\varphi)\psi$, are linear. (W5) Regularity of fields: : For all $\psi_1,\psi_2\in\mathcal{D}$ the map $\varphi\mapsto(\psi_1,\Phi(\varphi)\psi_2)$, and, respectively the maps $\varphi\mapsto(\psi_1,\Phi_j(\varphi)\psi_2)$, are tempered distributions i.e. elements of $\mathcal{S}^{\prime}(\mathbf{R}^4)$. (W6) Poincaré invariance: : For all $(a,\Lambda)\in\mathcal{P}$, $\varphi\in\mathcal{S}(\mathbf{R}^4)$ and $\psi\in\mathcal{D}$ the inclusion $U(a,\Lambda)\mathcal{D}\subset\mathcal{D}$ must hold and (Scalar field case): : The following equation must hold for all $\Lambda\in \text{O}(1,3)$ $$U(a,\Lambda)\Phi(\varphi)U(a,\Lambda)^{-1}\psi=\Phi((a,\Lambda)\varphi)\psi.$$ (Vectorial/Spinorial field case): : There exists a representation of $\text{SL}(2,\mathbf{C})$ on $\mathbf{C}^d$ denoted by $\rho$ and satisfying $\rho(-I)=I$ or $\rho(I)=I$ such that for all $\Lambda\in \text{SO}^+(1,3)$ and $\Phi:=[\Phi_1,\dots,\Phi_d]^{\dagger}$ $$U(a,\Lambda)\Phi(\varphi)U(a,\Lambda)^{-1}\psi=\rho(s^{-1}(\Lambda))\Phi((a,\Lambda)\varphi)\psi,$$ where $s$ denotes the spinor map $s:\text{SL}(2,\mathbf{C})\rightarrow\text{SO}^{+}(1,3)$ defined as following. The vector spaces of Hermitian matrices $\mathfrak{H}$ in $\mathbf{C}^{2\time 2}$ and $\mathbf{R}^4$ are isomorphically mapped by $$\begin{split} X:\mathbf{R}^4&\longrightarrow \mathfrak{H}\\ x=(x^0, x^1, x^2, x^3)&\mapsto X(x):=\left[ \begin{array}{cc} x^0+x^3 & x^1-ix^2 \\ x^1+ix^2 & x^0-x^3 \\ \end{array} \right]. \end{split}$$ The group $\text{SL}(2,\mathbf{C})$ acts on $\mathfrak{H}$ by $$\begin{split} \text{SL}(2,\mathbf{C})\times\mathfrak{H} &\longrightarrow \mathfrak{H}\\ (P,X)&\mapsto P.X:=PXP^*. \end{split}$$ The spinor map is defined as $$\label{spinormap} \begin{split} s: \text{SL}(2,\mathbf{C})&\longrightarrow \text{SO}^+(1,3) \\ P&\mapsto s(P):x\mapsto s(P)x:=X^{-1}(PX(x)P^*). \end{split}$$ (W7) Microscopic casuality or local commutativity: : Let $\varphi,\chi\in\mathcal{S}(\mathbf{R}^4)$, whose supports are spacelike separated, i.e. $\phi(x)\chi(y)=0$ if $x-y$ is not in the positive light cone. Then, (Scalar field case): : The images of the test functions by the map field must commute $$[\Phi(\varphi), \Phi(\chi)]=0.$$ (Vectorial/Spinorial field case): : For any field maps $j,i=1,\dots,d$ either the commutations $$[\Phi_j(\varphi), \Phi_i(\chi)]=0,\quad [\Phi_j^*(\varphi), \Phi_i(\chi)]=0,$$ or the anticommutations $$[\Phi_j(\varphi), \Phi_i(\chi)]_{+}=0,\quad[\Phi_j^*(\varphi), \Phi_i(\chi)]_{+}=0$$ hold. (W8) Cyclicity of the vacuum: : The sets (Scalar field case): : The set $$\mathcal{D}_0:=\{\Phi(\varphi_1)\cdots\Phi(\varphi_n)\psi_0\,|\,\varphi_j\in\mathcal{S}(\mathbf{R}^4), n\in\mathbf{N}_0\}$$is dense in $\mathcal{H}$. (Vectorial/Spinorial field case): : The set $$\mathcal{D}_0:=\{\Psi(\varphi_1)\cdots\Psi(\varphi_n)\psi_0\,|\,\varphi_j\in\mathcal{S}(\mathbf{R}^4), \Psi\in\{\Phi_1,\dots,\Phi_d,\Phi_1^*,\dots,\Phi_d^*\},\,n\in\mathbf{N}_0\}$$is dense in $\mathcal{H}$. We remark that Osterwalder and Schrader (cf. [@OS73], [@OS73Bis]) utilized the Wick rotation technique to pass from the Minkowskian to the Euclidean space and formulate axioms equivalent to Wightman’s ones in terms of Euclidean Green’s functions. In [@OS75] they defined free Bose and Fermi fields and proved a Feynman-Kač formula for boson-fermion models. For a general overview see also [@GJ81]. The dynamics of the quantized system obtained by Wightman’s axioms is given by the Schrödinger equation $$\imath\frac{\partial}{\partial x^0}\psi=H\psi,$$ where $\psi\in\mathcal{H}$ and $H$ is the unbounded, selfadjoint Hamilton operator, obtained by a quantization procedure of the Hamilton function in the classical description of the physical system. Second Quantization ------------------- To extend a quantum mechanical model accounting for a fixed number of particles to one accounting for an arbitrary number, the following procedure is required. Let $\mathcal{H}$ be the Hilbert space whose unit sphere corresponds to the possible pure quantum states of the system with a fixed number of particles. The **Fock space** $\mathcal{F}(\mathcal{H}):= \bigoplus_{n=0}^\infty\mathcal{H}^{(n)}$ where, $\mathcal{H}^{(0)}:=\mathbf{C}$ and $\mathcal{H}^{(n)}:=\mathcal{H}\otimes\cdots\mathcal{H}$ ($n$ times tensor product) is the vector space representing the states of a quantum system with a variable number of particles. The vector $\Omega_0:=(1,0\dots)\in\mathcal{F}(\mathcal{H})$ is called the **vacuum vector**. Given $\psi\in\mathcal{F}(\mathcal{H})$, we write $\psi^{(n)}$ for the orthogonal projection of $\psi$ onto $\mathcal{H}^{(n)}$. The set $F_0$ consisting of those $\psi$ such that $\psi^{(n)}=0$ for all sufficiently large $n$ is a dense subspace of the Fock space, called the **space of finite particles**. The **symmetrization** and [anti-symmetrization]{} operators $$\begin{split} S_n(\psi_1\otimes\dots\otimes\psi_n)&:=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}^n}\psi_{\sigma(1)}\otimes\dots\otimes\sigma_{\sigma(n)}\\ A_n(\psi_1\otimes\dots\otimes\psi_n)&:=\frac{1}{n!}\sum_{\sigma\in\mathfrak{S}^n}(-1)^{\text{sgn}(\sigma)}\psi_{\sigma(1)}\otimes\dots\otimes\sigma_{\sigma(n)} \end{split}$$ extend by linearity to $\mathcal{H}^{(n)}$ and are projections. The state space for $n$ fermions is defined as $\mathcal{H}^{(n)}_a:=A_n(\mathcal{H}^{n})$ and that for $n$ bosons as $\mathcal{H}^{(n)}_s:=S_n(\mathcal{H}^{n})$. The **fermionic Fock space** is defined as $$\mathcal{F}_a(\mathcal{H}):=\bigoplus_{n=0}^{\infty}\mathcal{H}^{(n)}_a,$$ and the **bosonic Fock space** as $$\mathcal{F}_s(\mathcal{H}):=\bigoplus_{n=0}^{\infty}\mathcal{H}^{(n)}_s.$$ A *unitary* operator $U:\mathcal{H}\rightarrow\mathcal{H}$ can be uniquely extended to a *unitary* operator $\Gamma(U):\mathcal{F}(\mathcal{H})\rightarrow\mathcal{F}(\mathcal{H})$ as $$\Gamma(U)|_{\mathcal{H}^{(n)}}:=\bigotimes_{j=1}^n U.$$ A *selfadjoint* operator $A$ on $\mathcal{H}$ with dense subspace $\mathcal{D}(A)\subset\mathcal{H}$ can be uniquely extended to a *selfadjoint* operator $d\Gamma(A)$ on $\mathcal{F}(\mathcal{H})$ as closure of the essentialy selfadjoint operator $$d\Gamma(A)|_{\mathcal{D}(d\Gamma A)\cap\mathcal{H}^{(n)}}:=\bigoplus_{j=1}^n\mathbb{1}\otimes\dots\mathbb{1}\otimes\underbrace{A}_{j}\otimes\mathbb{1}\dots\mathbb{1},$$ where $$\mathcal{D}(d\Gamma(A)):=\left\{\psi\in F_0\left|\,\psi^{(n)}\in\bigotimes_{j=1}^n\mathcal{D}(A)\text{ for each }n\right.\right\}.$$ The operator $d\Gamma(A)$ is called the **second quantization** of $A$. It is easy to prove that the the spectrum of the second quantization can be infered from the spectrum of the first. \[spec\] Let $A$ be a selfadjoint operator with a discrete spectral resolution, i.e. $A\varphi_j=\lambda_j\varphi_j$, where $\{\lambda_j\}_{j\ge0}\subset\mathbb{R}$ and $\{\varphi_j\}_{j\ge0}$ is a o.n.B in $\mathcal{H}$. Then, $d\Gamma(A)|_{\mathcal{H}^{(n)}}$ has a discrete spectral resolution given by $$d\Gamma(A)|_{\mathcal{H}^{(n)}}\varphi_{i_1}\otimes\dots\otimes\varphi_{i_n}=\left(\sum_{j=1}^n\lambda_{i_j}\right)\varphi_{i_1}\otimes\dots\otimes\varphi_{i_n}\quad(i_j\ge0, 1\le j\le n).$$ If $A$ is selfadjoint operator with continuous spectrum $\operatorname{spec}_c(A)$, then $d\Gamma(A)|_{\mathcal{H}^{(n)}}$ has a continuous spectrum given by $$\operatorname{spec}_c\left(d\Gamma(A)|_{\mathcal{H}^{(n)}}\right)=\left\{\left.\sum_{j=1}^n\lambda_j\right|\,\lambda_j\in\operatorname{spec}_c(A)\text{ for }1\le j\le n\right\}.$$ Segal Quantization ------------------ Let $f\in\mathcal{H}$ be fixed. For vectors in $\mathcal{H}^{(n)}$ of the form $\eta=\psi_1\otimes\psi_2\otimes\dots\otimes\psi_n$ we define a map $b^{-}(f):\mathcal{H}^{(n)}\rightarrow \mathcal{H}^{(n-1)}$ by $$b^{-}(f)\eta:=(f,\psi_1)\psi_2\otimes\dots\otimes\psi_n.$$ The expression $b^{-}(f)$ extends by linearity to a bounded operator on $\mathcal{F}(\mathcal{H})$. The operator $N:=d\Gamma(I)$ is termed **number operator** and $$a^{-}(f):=\sqrt{N+1}b^{-}(f)$$ the **annihilation operator** on $\mathcal{F}_s(\mathcal{H})$. Its adjoint, $a^{-}(f)^*$ is called the **creation operator**. Finally, the real linear (but not complex linear) operator $$\Phi_S(f):=\frac{1}{\sqrt{2}}\left(a^{-}(f)+a^{-}(f)^*\right)$$ is termed **Segal field operator**, and the map $$\begin{split} \Phi_S:\mathcal{H}&\rightarrow\mathcal{F}_s(\mathcal{H})\\ f&\mapsto\Phi_S(f) \end{split}$$ the **Segal quantization** over $\mathcal{H}$. \[Segal\] Let $\mathcal{H}$ be a complex Hilbert space and $\Phi_S$ the corresponding Segal quantization. Then: (a) : For each $f\in\mathcal{H}$ the operator $\Phi_S(f)$ is essentially selfadjoint on $F_0$. (b) : The vacuum $\Omega_0$ is in the domain of all finite products $\Phi_S(f_1)\Phi_S(f_2)\dots\Phi_S(f_n)$ and the linear span of $\left\{\Phi_S(f_1)\Phi_S(f_2)\dots\Phi_S(f_n)\Omega_0\,|\,f_i\in\mathcal{H},n\ge0\right\}$ is dense in $\mathcal{F}_s(\mathcal{H})$. (c) : For each $\psi_0\in F_0$ and $f,g\in\mathcal{H}$ $$\begin{split} &\Phi_S(f)\Phi_S(g)\psi-\Phi_S(g)\Phi_S(f)\psi=\imath\text{Im}(f,g)\psi\\ &\exp\left(\imath\Phi_S(f+g)\right)=\exp\left(-\frac{\imath}{2}\text{Im}(f,g)\right)\exp\left(\imath\Phi_S(f)\right)\exp\left(\imath\Phi_S(g)\right). \end{split}$$ (d) : If $f_n\rightarrow f$ in $\mathcal{H}$, then: $$\begin{split} \Phi_S(f_n)\rightarrow \Phi_S(f)\\ \exp\left(\imath\Phi_S(f_n)\right)\rightarrow \exp\left(\imath\Phi_S(f)\right)\\ \end{split}$$ (e) : For every unitary operator $U$ on $\mathcal{H}$, $\Gamma(U):\mathcal{D}(\overline{\Phi_S(f)})\rightarrow\mathcal{D}(\overline{\Phi_S(Uf)})$ and for $\psi\in\mathcal{D}(\overline{\Phi_S(Uf)})$ and for all $f\in\mathcal{H}$ $$\Gamma(U)\overline{\Phi_S(f)}\Gamma(U)^{-1}\psi=\overline{\Phi_S(Uf)}\psi.$$ See Theorem X.41 in [@RS75]. Quantization of Yang-Mills Equations and Positive Mass Gap ========================================================== Functional Derivation, Functional Integration and Hilbert Space Construction ---------------------------------------------------------------------------- In this subsection we follow the work of Kuo ([@Ku96]) which is aligned with the framework for functional integration developed by Cartier/DeWitt-Morette (see [@Ca97], [@La04] and [@CDM10]). We briefly summarize the construction without proofs. Given the space of tempered distributions $\mathcal{S}^{\prime}(\mathbf{R}^N)$, the dual of the space of test functions $\mathcal{S}(\mathbf{R}^N)$ we have the following Gel’fand triple: $$\label{Gelf1} \mathcal{S}(\mathbf{R}^N)\subset L^2(\mathbf{R}^N)\subset\mathcal{S}^{\prime}(\mathbf{R}^N),$$ where $\mathcal{S}(\mathbf{R}^N)$ and $\mathcal{S}^{\prime}(\mathbf{R}^N)$ are Banach spaces and $L^2(\mathbf{R}^N)$ an Hilbert space. This Gel’fand triple is backed by the continuous inclusions $$\mathcal{S}(\mathbf{R}^N)\subset L^{s,2}(\mathbf{R}^N)\subset L^2(\mathbf{R}^N)\subset L^{-s,2}(\mathbf{R}^N))\subset\mathcal{S}^{\prime}(\mathbf{R}^N),$$ where $L^{s,2}(\mathbf{R}^N)$ denotes the Sobolev space of distributions such that all weak derivatives up to order $s$ are square integrable. By Minlos theorem there exist a unique probability measure $\mu$ such that $$\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}d\mu(F)\,e^{\imath F(\varphi)}=e^{-\frac{1}{2}\|\varphi\|^2_{L^2(\mathbf{R}^N)}}\quad\text{ for all } \varphi\in\mathcal{S}(\mathbf{R}^N).$$ The probability space $(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$ is termed *white noise space* and the measure $\mu$ is called the standard *Gaussian measure* on $\mathcal{S}^{\prime}(\mathbf{R}^N)$. The construction of the square integrable functions can be mimicked in infinite dimensions for the integration domain, where the white noise space $(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$ plays the role of its finite dimensional analogue $(\mathbf{R}^N,dx^N)$. We obtain the space of the square integrable functionals over the white noise space $L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$. Furthermore, following the *Kubo-Takenaka construction* explained in Chapter 4.2 of [@Ku96], which utilizes the Wiener-itô theorem and the second quantization of the partial differential operator $\Pi_{i=1}^N[-\partial_i^2+x_i^2+1]$, it is possible to obtain a nuclear space $\mathcal{E}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$, *the space of the Hida test functions* and its dual $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$, *the space of the Hida distributions*, and the Gelf’and triple: $$\mathcal{E}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset \mathcal{E}^{\prime}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu).$$ More exactly, the Kubo-Takenaka construction produces the following continuous inclusions for all $s>0$: $$\mathcal{E}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^{s,2}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset L^{-s,2}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)\subset \mathcal{E}^{\prime}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu),$$ and $L^{s,2}(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$ the Hida-Sobolev space of test functions. We can conclude that the Feynman integral can be properly defined for functionals on the white noise space with a standard Gaussian measure: $$\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}\mathcal{D}(F)\,\varphi(F):=\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}d\mu(F)\,\varphi(F),$$ and that it can be formally defined for functionals in the space of Hida distribution as the value of such distributions on the Hida test function $1$: $$\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}\mathcal{D}(F)\,\Phi(F):=\int_{\mathcal{S}^{\prime}(\mathbf{R}^N)}\mathcal{D}(F)\,\Phi(F)\cdot 1:=\Phi(1).$$ Let $\varphi\in L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$. Its functional derivative is defined as Gâteaux derivative as $$\frac{\delta\varphi}{\delta F}(F).G:=\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}\varphi(F+\epsilon G),$$ for any $G\in\mathcal{S}^{\prime}(\mathbf{R}^N)$. The Gel’fand-Kostyuchenko spectral theorem (Theorem \[GK\]) can be lifted to an analogous version for selfadjoint operators on the Hilbert space $L^2(\mathcal{S}^{\prime}(\mathbf{R}^N),\mu)$. Quantization ------------ To account for functionals on transversal fields as required by the Coulomb gauge, we perform the Kubo-Takenaka construction for $$L^2_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}):=\{\left.\mathbf{A}\in L^2(\mathbf{R}^4,\mathbf{R}^{K \times 3})\right|\,\mathbf{A}^{\parallel}=0\},$$ obtaining the Gel’fand triples $$\begin{split} &\mathcal{S}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3})\subset L^2_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3})\subset\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3})\\ &\mathcal{E}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)\subset L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)\subset \mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu). \end{split}$$ We introduce the Hamilton operator originated by the quantization of the Hamiltonian formulation of Yang-Mills equations, and verify that the Wightman’s axioms are satisfied. Quantizing Thorem \[CHam\] we obtain \[quantization\] Let $H=H(A,E)$ be the Hamilton function of the Hamilton equations equivalent to the Yang-Mills equations as in Theorem \[CHam\] for a simple Lie group as structure group. The quantization of the position variable $A$, of the momentum variable $E$ and of the Hamilton function means the following substitution: $$\begin{split} \mathbf{A} \in C^{\infty}(\mathbf{R}^4,\mathbf{R}^{K\times 3})&\longrightarrow \mathbf{A}\in\mathcal{O}(L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu))\\ \mathbf{E} \in C^{\infty}(\mathbf{R}^4,\mathbf{R}^{K\times 3})&\longrightarrow \frac{1}{\imath}\frac{\delta}{\delta \mathbf{A}}\in\mathcal{O}(L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu))\\ H \in C^{\infty}(\mathbf{R}^{K\times 3}\times\mathbf{R}^{K\times 3},\mathbf{R})&\longrightarrow H\left(\mathbf{A},\frac{1}{\imath}\frac{\delta}{\delta \mathbf{A}}\right)\in\mathcal{O}(L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)). \end{split}$$ The Hamilton operator for the quantized Yang-Mills equations, denoted again by $H$ reads $$H=H_I+H_{II}+H_{III},$$ where $$\begin{split} H_I&=-\frac{1}{2}\int_{\mathbf{R}^3}d^3x\left[\frac{\delta}{\delta A_i^a(t,x)}\frac{\delta}{\delta A_i^a(t,x)}\right]\\ & \\ H_{II}&=\frac{1}{16}\int_{\mathbf{R}^3}d^3x\left[\varepsilon_i^{j,k}\varepsilon_i^{p,q}(\partial_jA^a_k-\partial_kA_j^a+g\varepsilon^{a,b,c}A_j^bA_k^c)(\partial_pA^a_q-\partial_qA_p^a+g\varepsilon^{a,b,c}A_p^bA_q^c)\right]\\ & \\ H_{III}&=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\partial_iG^{a,b}(\mathbf{A};x,y)\varepsilon^{b,c,d}A_k^d(t,y)\frac{\delta}{\delta A_k^c(t,y)}\right]\\ &\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\left[\partial_iG^{a,b}(\mathbf{A};x,\bar{y})\varepsilon^{b,c,d}A_k^d(t,\bar{y})\frac{\delta}{\delta A_k^c(t,\bar{y})}\right]. \end{split}$$ With the domain of definition $$\mathcal{D}(H):=\left\{\Psi\in L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)\left|\,H\Psi\in L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3},\mu)\right.\right\}$$ the operator $H$ is selfadjoint. The operator $H_I$ is the Laplace operator in infinite dimensions for functionals of the potential fields. The operator $H_II$ is a multiplication operator corresponding to the fibrewise multiplication with the square of the connection curvature. Both operators $H_I$ and $H_{II}$ do not vanish if $g=0$ and do not contribute to the existence of a mass gap. The operator $H_{III}$ vanishes if $g=0$ and -as we will see- is responsible for the existence of a mass gap. The expressions for the operators $H_I$ and $H_{II}$ are obtained by a straightforward calculation, after having inserted the quantization in the first two addenda of (\[classicH\]). For the operator $H_{III}$ some more work is needed. Inserting (\[deff\]) in the last addendum of (\[classicH\]) $$\label{classicH3} \begin{split} &\int_{\mathbf{R}^3}d^3x\left(\frac{1}{2}f^a(t,x)\Delta f^a(t,x)+\rho_c(t,x)A_0^c(t,x)\right)=\\ &=\int_{\mathbf{R}^3}d^3x\left[\frac{1}{2}\left(-\int_{\mathbf{R}^3}d^3y\,G^{a,b}(\mathbf{A};x,y)\rho_b(t,y)\right)\left(-\int_{\mathbf{R}^3}d^3\bar{y}\,G^{a,b}(\mathbf{A};x,\bar{y})\rho_b(t,\bar{y})\right)\right.+\\ &\qquad\qquad\quad+\left.\rho_c(t,x)\int_{\mathbf{R}^3}d^3y\,G^{c,b}(\mathbf{A};x,y)\int_{\mathbf{R}^3}d^3\bar{y}\,\Delta G^{b,d}(\mathbf{A};y,\bar{y})\rho_d(t,\bar{y})\right]=\\ &=\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\frac{1}{2}\left(G^{a,b}(\mathbf{A};x,y)\rho_b(t,y)\Delta G^{a,d}(\mathbf{A};x,\bar{y})\rho_d(t,\bar{y})\right)\right.+\\ &\qquad\qquad\quad-\left.\rho_c(t,x)G^{c,b}(\mathbf{A};x,y)\Delta G^{b,d}(\mathbf{A};y,\bar{y})\rho_d(t,\bar{y})\right]=\\ &=\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\frac{1}{2}\left(G^{a,b}(\mathbf{A};x,y)\Delta G^{a,d}(\mathbf{A};x,\bar{y})\rho_b(t,y)\rho_d(t,\bar{y})\right)\right.+\\ &\qquad\qquad\quad-\left.G^{a,b}(\mathbf{A};y,x)\Delta G^{a,d}(\mathbf{A};x,\bar{y})\rho_b(t,y)\rho_d(t,\bar{y})\right]=\\ &=\frac{1}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\partial_iG^{a,b}(\mathbf{A};x,y)\rho_b(t,y)\right]\left[\partial_iG^{a,d}(\mathbf{A};x,\bar{y})\rho_d(t,\bar{y})\right], \end{split}$$ where in the last formula transformation we have utilized integration by parts in the variables $x_1, x_2, x_3$ and the fact that $G(\mathbf{A};x,y)=G(\mathbf{A};y,x)$. Introducing the quantization into the charge we obtain $$\rho_b(t,x)=-\frac{g}{\imath}\varepsilon^{b,c,d}A_k^c(t,x)\frac{\delta}{\delta A_k^c(t,x)},$$ which inserted into \[classicH3\] leads to $$\begin{split} H_{III}&=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\left[\partial_iG^{a,b}(\mathbf{A};x,y)\varepsilon^{b,c,d}A_k^d(t,y)\frac{\delta}{\delta A_k^c(t,y)}\right]\\ &\qquad\qquad\qquad\qquad\qquad\qquad\quad\left[\partial_iG^{a,b}(\mathbf{A};x,\bar{y})\varepsilon^{b,c,d}A_k^d(t,\bar{y})\frac{\delta}{\delta A_k^c(t,\bar{y})}\right]. \end{split}$$ The proof is completed. In the physical literature (see f.i. [@Sch08]) the operator $C=C(\mathbf{A};x,y)$ $$C^{a,b}(\mathbf{A};x,y):=-\int_{\mathbf{R}^3}d^3\bar{y}G^{a,c}(\mathbf{A};x\bar{y})\Delta G^{c,b}(\mathbf{A};y,\bar{y})$$ is termed *Coulomb operator* and $G=G(\mathbf{A};x,y)$ is also called the *Faddeev-Popov operator*. \[corspec\] The spectra of $H_I$, $H_{II}$ and $H_{III}$ are: $$\label{specH} \begin{split} &\operatorname{spec}(H_I)=\operatorname{spec}_c(H_I)=[0,+\infty[\\ &\operatorname{spec}(H_{II})=\operatorname{spec}_c(H_{II})=[0,+\infty[\\ &\operatorname{spec}(H_{III})=\operatorname{spec}_c(H_{III})=\{0\}\cup[\eta,+\infty[, \text{ for a }\eta>0. \end{split}$$ Moreover $\eta=O(g^2)$. We will construct generalized eigenvectors to show that the spectra have only a continuous part depicted as in (\[specH\]). First, we analyze the operator $H_I$, which can be seen as $$H_I=-\frac{1}{2}\int_{\mathbf{R}^3}d^3 x\Delta_{\mathbf{A}(t,x)}.$$ Let $x\in\mathbf{R}^3$ and $t\in\mathbf{R}$ now be fixed. For any $R>0$ the Laplace operator $\Delta_{\mathbf{A}}$ on $[-\frac{R}{2},+\frac{R}{2}]^{3K}$ under Dirichlet boundary conditions has a discrete spectral resolution $(\lambda_k,\psi_k)_{k\ge0}$, where $\lambda_k=-\frac{\pi^2}{R^2}(k+1)$ and $\psi_j=\psi_j(\mathbf{A})\in C^{\infty}_0([-\frac{R}{2},+\frac{R}{2}]^{3K},\mathbf{C})$. We can extend $\psi_j$ outside the cube by setting its value to $0$ obtaining an approximated eigenvector for the approximated eigenvalue $\lambda_j$, which is in line with the fact that the Laplacian on $L^2(\mathbf{R}^{3K},\mathbf{C})$ has solely a continuous spectrum, which is $]-\infty,0]$. The functional $$\Psi^{x_0}_k(\mathbf{A}):=\delta(\cdot-x_0)\psi_k(\mathbf{A}(t,x_0))$$ is a generalized eigenvector in $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the operator $H_I$ on the rigged Hilbert space $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the generalized eigenvalue $\frac{\pi^2}{R^2}(k+1)$, which, by Theorem \[GK\], is an element of the continuous spectrum of the non negative operator $H_I$. By varying the generalized eigenvalue over $j$ and $R$, the claim about the spectrum follows. Next, we analyze the operator $H_{II}$, which can be seen as $$H_{II}=\frac{1}{2}\int_{\mathbf{R}^3}d^3 x|R^{\nabla^\mathbf{A}}(t,x)|^2,$$ where $R^{\nabla^\mathbf{A}}$ is the curvature operator associated to the connection $\mathbf{A}$. Let $x\in\mathbf{R}^3$ now be fixed. Any non zero $\psi\in L^2(\mathbf{R}^{3K},\mathbf{C})$ is eigenvector of the multiplication with the non negative real $|R^{\nabla^\mathbf{A}}(t,x)|^2$. The functional $$\Psi^{x_0}(\mathbf{A}):=\delta(\cdot-x_0)\psi(\mathbf{A}(t,x_0))$$ is a generalized eigenvector in $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the operator $H_{II}$ on the rigged Hilbert space $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the generalized eigenvalue $|R^{\nabla^A}(t,x_0)|^2$, which, by Theorem \[GK\], is an element of the continuous spectrum of the non negative operator $H_{II}$. By varying the generalized eigenvalue over $A$, and taking into account that $$\inf_{\mathbf{A}\in\mathcal{A}} |R^{\nabla^\mathbf{A}}(t,x_0)|^2=0,$$ the claim about the spectrum follows. Finally, we analyze the operator $H_{III}$, which we write as $$H_{III}=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\,D_i^a(\mathbf{A};x,y)D_i^a(\mathbf{A};x,\bar{y}),$$ for the operator $D=D(\mathbf{A;x,y})$ defined as $$D_i^a(\mathbf{A};x,y):=\partial_iG^{a,b}(\mathbf{A};x,y)\varepsilon^{b,c,d}A_k^d(t,y)\frac{\delta}{\delta A_k^c(t,y)}.$$ Let $x$ and $y\in\mathbf{R}^3$ now be fixed. For any $R>0$ the operator $D_i^a(\mathbf{A};x,y)$ on $[-\frac{R}{2},+\frac{R}{2}]^{3K}$ under periodic boundary conditions has a discrete spectral resolution $(\mu_{i,k}^a(x,y),\psi_{i,k}^a(\mathbf{A};x,y))_{k\ge0}$, where $$\begin{split} \mu_{i,k}^a(x,y)&=\frac{2\pi\imath k}{\int_{-\frac{R}{2}}^{+\frac{R}{2}}dA_j^c(t,y)[\partial_iG^{a,b}(A;x,y)\varepsilon^{b,c,d}A_j^d(t,y)]^{-1}}\\ \psi_{i,k}^a(A;x,y)&=C_{i,k}^a\exp\left(\mu_{i,k}^a(x,y)\int_0^{A_j^c(t,y)}d\bar{A}_j^c(t,y)[\partial_iG^{a,b(\bar{\mathbf{A}};x,y)}\varepsilon^{b,c,d}\bar{A}_j^d(t,y)]^{-1}\right), \end{split}$$ where $C_{i,k}^a\in\mathbf{C}$ is a constant. We can extend $\psi_{i,k}^a(\cdot;x,y)$ outside the cube by setting its value to $0$ obtaining an approximated eigenvector for the approximated eigenvalue $\mu_{i,k}^a(x,y)$ for the operator $D_i^a(\mathbf{A};x,y)$ on $L^2(\mathbf{R}^{3K}, \mathbf{C})$, which means that $\mu_{i,k}^a(x,y)\in\operatorname{spec}_c(D_i^a(\mathbf{A};x,y))$. For fixed $i$ and $a$ the functional $$\Psi^{a,x_0,y_0}_{i,k}(\mathbf{A}):=\delta(\cdot-x_0)\delta(y-y_0)\delta(\bar{y}-y_0)\psi_{i,k}^a(\mathbf{A};x_0,y_0)\psi_{i,k}^a(\mathbf{A};x_0,y_0)$$ is a generalized eigenvector in $\mathcal{E}^{\prime}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the operator $$H_{i,III}^a:=-\frac{g^2}{2}\int_{\mathbf{R}^3}d^3x\int_{\mathbf{R}^3}d^3y\int_{\mathbf{R}^3}d^3\bar{y}\,D_i^a(\mathbf{A};x,y)D_i^a(\mathbf{A};x,\bar{y})$$ on the rigged Hilbert space $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ for the generalized eigenvalue $$\lambda_{i,k}^a(x_0,y_0)=\frac{2\pi^2g^2k^2}{\left[\int_{-\frac{R}{2}}^{+\frac{R}{2}}dA_j^c(t,y_0)[\partial_iG^{a,b}(\mathbf{A};x_0,y_0)\varepsilon^{b,c,d}A_j^d(t,y_0)]^{-1}\right]^2},$$which, by Theorem \[GK\] is an element of the continuous spectrum of the operator $H_{i,III}^a$. Since, by Proposition \[prop\] and Corollary \[cor\] $$[\partial_iG^{a,b}(\mathbf{A};x_0,y_0)\varepsilon^{b,c,d}A_j^d(t,y_0)]^{-1}=O((1+|\mathbf{A}(t,y_0)|^2)^{-1})\qquad (|\mathbf{A}(t,y_0)|\rightarrow+\infty),$$ the denominator of the generalized eigenvalue has a uniform bound in the connection $\mathbf{A}$ given by $$\left[\int_{-\frac{R}{2}}^{+\frac{R}{2}}dA_j^c(t,y_0)[\partial_iG^{a,b}(A;x_0,y_0)\varepsilon^{b,c,d}A_j^d(t,y_0)]^{-1}\right]^2\le \left[C\int_{-\infty}^{\infty}d\alpha\,\frac{1}{1+\alpha^2}\right]^2=\pi^2 C^2.$$ By varying the generalized eigenvalue over $k$ and $R$, the claim about the spectrum follows for $\eta:=\frac{6K g^2}{C^2}$. Note that $0$ is an element of the continuous spectrum but not of the discrete spectrum, which is empty. The proof is complete. Main Theorem ------------ We describe the construction of field maps and verify Wightman’s axioms. **Quantum Field Settting for Yang-Mills Theory**\[QFTYM\] Pseudoriemannian manifold: : $M:=\mathbf{R}^{1,3}$ with the Minkowski metric tensor $g:=dx^0\otimes dx^0-dx^1\otimes dx^1-dx^2\otimes dx^2-dx^3\otimes dx^3$. Positive light cone: : $\mathcal{C}_+:=\{x\in M\,|\,g(x,x)\ge0\}$. Bundles over the manifold: : $P$ is a principle fibre bundle over $M$ with simple structure group $G$. Hilbert space: : $\mathcal{H}:=L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4}),\mu)$. Bosonic Fock space: : $\mathcal{F}_s(\mathcal{H})$. Vacuum state: : $\Omega_0:\equiv (1,0,0,\dots)\in\mathcal{F}_s(\mathcal{H})$. Dense subspace: : The space of test functionals $\mathcal{D}:=\mathcal{E}(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4}),\mu)$. Unitary representation of the Poincaré group: : $$U(a,\Lambda)\Psi(A):=\Psi(A(\Lambda\cdot+a)).$$ It is defined on any $\Psi\in\mathcal{H}$ and extended to $\mathcal{F}_s(\mathcal{H})$ by $\Gamma(U)$. Hamilton operator: : $H$ defined in Theorem (\[quantization\]), with domain of definition $\mathcal{D}(H)$. The definition on the Hilbert space $\mathcal{H}$ is extended to $\mathcal{F}_s(\mathcal{H})$ by $d\Gamma(H)$. Note that a functional $\Psi=\Psi(A)=\Psi([A_0;\mathbf{A}])\in L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4}),\mu)$ can be seen as an element of $L^2(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 3}),\mu)$ if we consider only the dependence on the space components $\mathbf{A}$. Field maps: : Let us consider a connection $A$ for the principal fibre bundle $P$. Since the connection $A$ is a Lie algebra $\mathcal{G}$ valued $1$-form on $M$, we can write $$A(x)=\sum_{\mu=0}^3A_\mu(x)dx^\mu$$ for the fields $A_{\mu}(x):=A(y)e_\mu=\sum_{a=1}^KA_\mu^a(x)t_a$, where $\{t_1,\dots,t_K\}$ is a basis of $\mathcal{G}$. We define now following operators for $\mu=0,1,2,3$ and $a=1,\dots, K$: $$\begin{split} E_\mu^a:\mathcal{S}(\mathbb{R}^4)&\rightarrow\mathcal{H}\\ \varphi&\mapsto E_\mu^a(\varphi),\text{ defined as }E_\mu^a(\varphi)(A):=\int_{\mathbf{R}^4}d^4x\,\varphi(x)A_\mu^a(x) \end{split}$$ and for $\varphi\in\mathcal{S}(\mathbb{R}^4)$ $$\Phi_\mu^a(\varphi):=\Phi_S(\text{Re}(E_\mu^a(\varphi)))+\imath\Phi_S(\text{Im}(E_\mu^a(\varphi))),$$ where $\Phi_S$ is the closure of the Segal operator. The field maps are well defined. As a matter of fact, for any indices $\mu=0,1,2,3$, $k=1,\dots,K$ and all $\varphi\in\mathcal{S}(\mathbf{R}^4)$ the functional $E_\mu^a(\varphi)$ is an element of $\mathcal{H}$: $$\begin{split} \int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)|E_\mu^a(\varphi)(A)|^2&=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\left|\int_{\mathbf{R}^4}d^4x\,\varphi(x)A_\mu^a(x)\right|^2=\\ &\le \int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\int_{\mathbf{R}^4}d^4x\,|\varphi(x)|^2|A_\mu^a(x)|^2=\\ &\le \|\varphi\|^2_{L^2(\mathbf{R}^4,\mathbf{C})}\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\|A_\mu^a\|^2_{L^2(\mathbf{R}^4,\mathbf{C})}<+\infty, \end{split}$$ because $\mu$ is a probability measure and, hence, $\mu\left(\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})\right)=1$. \[CMIThm\] The construction of Definition \[QFTYM\] satisfies Wightman’s axioms (W1)-(W8) and the spectra of the Hamilton operator $H$ contains $0$ as simple eigenvalue for the vacuum eigenstate. There exist a constant $\eta>0$ such that $\text{spec}(H)=\{0\}\cup[\eta,+\infty[$. Moreover $\eta=O(g^2)$. First we prove that Wightman’s axioms are satisfied. (W1): With the definition $U(a,\Lambda)\Psi(A):=\Psi(A(\Lambda\cdot+a))$ the strong continuity follows by Lebesgue’s dominated convergence for the Feynman’s integral and the equality $$\|U(a,\Lambda)\Psi-U(a^{\prime},\Lambda^{\prime})\Psi\|_{L^2}^2=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)|\Psi(A(\Lambda\cdot+a))-\Psi(A(\Lambda^{\prime}\cdot+a^{\prime}))|^2,$$ which holds for all $\Lambda, \Lambda^{\prime}\in\text{SO}^+(1,3)$ and all $a, a^{\prime}\in\mathbf{R}_+\times\mathbf{R}^3$. (W2): The computation of the infinitesimal generators shows that $P_\mu=\frac{1}{\imath}\frac{\partial}{\partial x\mu}$: $$P_\mu\Psi(A):=\frac{1}{\imath}\left.\frac{d}{dt}\right|_{t=0}U(te_\mu,I)\Psi(A)=\frac{1}{\imath}\left.\frac{d}{dt}\right|_{t=0}\Psi(A(\cdot+te_\mu))=\frac{1}{\imath}\frac{\partial \Psi\circ A}{\partial x_\mu}=\frac{1}{\imath}\frac{\delta \Psi}{\delta A_k}\frac{\partial A_k}{\partial X_\mu}$$The operator $\Delta:=P_0^2-P_1^2-P_2^2-P_3^2$ is the Laplacian for the Minkowskian metric and is positive in the light cone. $P_0$ is unitary equivalent to the multiplication operator $x_0$ which is positive on $\mathcal{C}_+$. Therefore, the spectral measure in $\mathbf{R}^4$ corresponding to the restricted representation $\mathbf{R}^4\ni a\mapsto U(a,I)$ has support in the positive light cone. (W3): The vacuum state satisfies for all $a,x\in\mathbf{R}^4$ $$\Gamma(U(a,I))\Omega_0(A)=\Omega_0(A\cdot+a)\equiv\Omega_0\equiv(1,0,\dots).$$ (W4): follows from Theorem \[Segal\] (a) (W5): follows from Theorem \[Segal\] (d). (W6): For all $(a,\Lambda)\in\mathcal{P}$ the inclusion $U(a,\Lambda)\mathcal{D}\subset\mathcal{D}$ holds, because the test functional space is invariant under affine transformation of the domain. Since, by Theorem \[Segal\] (e) for any $\mu=0,1,2,3$, $k=1,\dots,K$, $(a,\Lambda)\in\mathcal{P}$, $\varphi\in\mathcal{S}(\mathbf{R}^4)$ and $\Psi\in\mathcal{D}$ $$\label{tr} \Gamma(U(a,\Lambda))\Phi_\mu^k(\varphi)\Gamma(U(a,\Lambda))^{-1}\Psi=\Phi_\mu^k(U(a,\Lambda).\varphi)\Psi,$$ which can be rewritten as $$\label{tr2} \Gamma(U(a,\Lambda))\Phi(\varphi)\Gamma(U(a,\Lambda))^{-1}\Psi=\rho(s^{-1}(\Lambda))\Phi(U(a,\Lambda).\varphi)\Psi,$$ where $s$ denotes the spinor map $s:\text{SL}(2,\mathbf{C})\rightarrow\text{SO}^{+}(1,3)$, $$\Phi:=[\Phi_0^1,\dots,\Phi_0^K,\Phi_1^1,\dots,\Phi_1^K,\Phi_2^1,\dots,\Phi_2^K,\Phi_3^1,\dots,\Phi_3^K]^{\dagger},$$ and $$\rho\equiv\mathbb{1}_{\text{GL}(\mathbf{C}^d)}$$ for $d:=4K$ is a representation of $\text{SL}(2,\mathbf{C})$ on $\mathbf{C}^d$. In particular, it satisfies $\rho(-\mathbb{1})=\mathbb{1}$, which means that representation has integer spin, as expected in the bosonic case. (W7): follows from Theorem \[Segal\] (c). More exactly, for two test functions $\varphi$ and $\chi$ and any indices $a,b\in\{1,\dots,K\}$, $\mu,\nu\in\{0,1,2,3\}$ the commutator of the field maps can be computed as $$\begin{split} &[\Phi_\mu^a(\varphi), \Phi_\nu^b(\chi)]=\\ &=[\Phi_S(\text{Re}(E_\mu^a(\varphi))), \Phi_S(\text{Re}(E_\nu^b(\chi)))]-[\Phi_S(\text{Im}(E_\mu^a(\varphi))), \Phi_S(\text{Im}(E_\nu^b(\chi)))]+\\ &\imath\left\{[\Phi_S(\text{Re}(E_\mu^a(\varphi))), \Phi_S(\text{Im}(E_\nu^b(\chi)))]+[\Phi_S(\text{Im}(E_\mu^a(\varphi))), \Phi_S(\text{Re}(E_\nu^b(\chi)))]\right\}=\\ &=\imath\left\{(\text{Re}(E_\mu^a(\varphi)), \text{Re}(E_\nu^b(\chi)))_{L^2}-(\text{Im}(E_\mu^a(\varphi)), \text{Im}(E_\nu^b(\chi)))_{L^2}\right\}+\\ &-\left\{(\text{Re}(E_\mu^a(\varphi)), \text{Im}(E_\nu^b(\chi)))_{L^2}+(\text{Im}(E_\mu^a(\varphi)), \text{Re}(E_\nu^b(\chi)))_{L^2}\right\}. \end{split}$$ All the scalar products vanish because the supports of the test functions are space-like separated and hence disjoint. For example: $$\begin{split} &(\text{Re}(E_\mu^a(\varphi)), \text{Re}(E_\nu^b(\chi)))_{L^2(\mathcal{A},\mathbf{C})}=\\ &=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\text{Re}\left[\int_{\mathbf{R}^4}d^4x\,\varphi(x)A^a_\mu(x)\right]\text{Re}\left[\int_{\mathbf{R}^4}d^4y\,\chi(y)A^b_\nu(y)\right]=\\ &=\int_{\mathcal{S}^{\prime}_{\bot}(\mathbf{R}^4,\mathbf{R}^{K \times 4})}\mathcal{D}(A)\int_{\mathbf{R}^4}d^4x\int_{\mathbf{R}^4}d^4y\,\text{Re}(\varphi(x))\text{Re}(\chi(y))A^a_\mu(x)A^b_\nu(y)=0. \end{split}$$ By Theorem \[Segal\] (a) the field map $\Phi_\mu^a(\varphi)$ is selfadjoint and thus $$[\Phi_\mu^a(\varphi)^*, \Phi_\nu^b(\chi)]=[\Phi_\mu^a(\varphi), \Phi_\nu^b(\chi)]=0.$$ (W8): follows from Theorem \[Segal\] (b). Finally, we prove the existence of a positive lower spectral bound. It suffices to prove the existence of a bound $\eta>0$ for the spectrum of $H$ on $\mathcal{H}$. From Corollary \[corspec\] we infer the existence of a $\eta=O(g^2)>0$ such that $$\operatorname{spec}(H)=\operatorname{spec}_c(H)=\{0\}\cup[\eta,+\infty[$$ for the Hamilton operator $H$ on the Hilbert space $\mathcal{H}$. For the spectrum of $d\Gamma(H)$ on the Fock space $\mathcal{F}(\mathcal{H})$ we obtain by Proposition \[spec\] $$\text{spec}(d\Gamma(H))= \{0\}\cup[\eta,+\infty[,$$ where we have to consider that $0$ is a proper eigenvalue for the vacuum state $\Omega_0$.\ The proof is completed.\ The representation $\rho$ has integer spin and lead therefore to connection fields $(A_\mu)_{\mu=0,\dots 3}$ satisfying the Bose statistics, in accordance with the standard spin-statistics relation for gluons which are bosons. Conclusion ========== We have quantized Yang-Mills equations for the positive light cone in the Minkowskian $\mathbf{R}^{1,3}$ obtaining field maps satisfying Wightman’s axioms of Constructive Quantum Field Theory. Moreover, the spectrum of the corresponding Hamilton operator is positive and bounded away from zero except for the case of the vacuum state which has vanishing energy level. Acknowledgement {#acknowledgement .unnumbered} =============== We would like to express our gratitude to Gian-Michele Graf and to Lee Smolin for the challenging discussion leading to various important corrections of the first versions of this paper. [refer]{} M. S. AGRANOVICH, [*Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains*]{}, Springer Monographs in Mathematics, 2015. T. BALABAN, [*Propagators and renormalization transformations for lattice gauge theories I*]{}, Comm. Math. Phys., Vol. 95, (17-40), 1984. T. BALABAN, [*Propagators and renormalization transformations for lattice gauge theories. II*]{}, Comm. Math. Phys., Vol. 96, (223-250), 1984. T. BALABAN, [*Averaging operations for lattice gauge theories*]{}, Comm. Math. Phys., Vol. 98, (17-51), 1985. T. BALABAN, [*Various titles*]{}, Comm. Math. Phys. Vol. 99, (75-102), 1985. T. BALABAN, [*Propagators for lattice gauge theories in a background field*]{}, Comm. Math. Phys., Vol 99, (389-434), 1985. T. BALABAN, [*Ultraviolet stability of three-dimensional lattice pure gauge field theories*]{}, Comm. Math. Phys., Vol. 102, (255-275), 1985. T. BALABAN, [*Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions*]{}, Comm. Math. Phys. Vol. 109, (249-301), 1987. T. BALABAN, [*Renormalization group approach to lattice gauge field theories. II. Cluster expansions*]{}, Comm. Math. Phys. Vol. 116, (1-22), 1988. T. BALABAN, [*Convergent renormalization expansions for lattice gauge theories*]{}, Comm. Math. Phys. Vol. 119, (243-285), 1988. T. BALABAN, [*Large field renormalization. I. The basic step of the $\mathbf{R}$ operation*]{}, Comm. Math. Phys. Vol. 122, (175-202), 1989. T. BALABAN, [*Large field renormalization. II. Localization, exponentiation, and bounds for the $\mathbf{R}$ operation*]{}, Comm. Math. Phys. Vol. 122, (355-392), 1989. C. M. BENDER, T. EGUCHI, and H. PAGELS, [*Gauuge-Fixing Degeneracies and Confinement in NOn-Abelian Gauge Theories*]{}, Vol. 17, No. 4, (1086-1096), 1978. N. N. BOGOLUBOV, A. A. LOGUNOV, A. I. OKSAK and I. TODOROV, [*General Principles of Quantum Field Theory*]{}, Mathematical Physics and Applied Mathematics, Kluwer, 1989. P. CARTIER (with C. DEWITT-MORETTE, A. WURM and D. COLLINS), [*A Rigorous Mathematical Foundation of Functional Integration*]{}, Functional Integration: Basics and Applications, NATO ASI Series, Series B: Physics Vol. 361, (1-50), 1997. P. CARTIER, C. DEWITT-MORETTE, [*Functional Integration: Action and Symmetries*]{}, Cambridge Monographs on Mathematical Physics, 2010. P. BRACKEN, [*Hamiltonian Formulation of Yang-Mills Theory and Gauge Ambiguity in Quantization*]{}, Can. J. Phys., Vol. 81, (545-554), 2003. E. DE FARIA and W. DE MELO, [*Mathematical Aspects of Quantum Field Theory*]{}, Cambridge Studies in Advanced Mathematics, 2010. L. I. DAIKHIN and D. A. POPOV, [*Einstein Spaces, and Yang-Mills Fields*]{}, Dokl. Akad. Nauk SSSR , 225:4, (790–793), 1975. M. R. DOUGLAS, [*Report on the Status of the Yang-Mills Millenium Prize Problem*]{}, Clay Mathematics Institute, 2004. K. VAN DEN DUNGEN, M. PASCHKE and A. RENNIE, [*Pseudo-Riemannian Spectral Triples and the Harmonic Oscillator*]{}, Journal of Geometry and Physics, Vol. 73, (37-55), 2013. A. DYNIN, [*Mass gap in Quantum Energy-Mass Spectrum of Relativistic Yang-Mills Fields*]{}, arXiv:1005.3779, 2013. D. M. EARDLEY and V. MONCRIEF, [*The Global Existence of Yang-Mills-Higgs Fields in $4$-dimensional Minkowski Space I. Local Existence and Smoothness Properties*]{}, Comm. Math. Phys., Vol. 83, No. 2, (171-191), 1982. D. M. EARDLEY and V. MONCRIEF, [*The Global Existence of Yang-Mills-Higgs Fields in $4$-dimensional Minkowski Space II. Completion of Proof*]{}, Comm. Math. Phys., Vol. 83, No. 2, (193-212), 1982. Encyclopedia of Mathemathics, [*Yang-Mills Fields*]{}, Kluwer, 2002. L. D. FADDEEV, [*Mass in Quantum Yang-Mills Theory*]{}, Perspective in Analysis, Math. Phys. Studies, Vol. 27, Springer, 2005. M. FRASCA, [*Mass Gap in a Yang-Mills Theory in the Strong Coupling Limit*]{}, arXiv:0511173, 2007. P. B. GILKEY, [*Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem*]{}, Second Edition, Studies in Advanced Mathematics, CRC Press, 1995. M. GEL’FAND and G. E. SHILOV, [*Generalized Functions, Volume 3*]{}, Academic Press, 1964. J. GLIMM and A. JAFFE, [*Quantum Physics: A Functional Integral Point of View*]{}, Springer, 1981. V. N. GRIBOV, [*Quantization of Non-Abelian Gauge Theories*]{}, Nucl. Phys. B, Vol. 139, Issues 1–2, (1-19), 1978. K. HALLER, [*Gauge-Invariant QCD: an Approach to Long-Range Color-Confining Forces*]{}, CERN preprint,\ `https://cds.cern.ch/record/361784/files/9808028.pdf`, 1997. T. HEINZL, [*Hamiltonian Formulations of Yang-Mills Quantum Theory and the Gribov Problem*]{}, 5th Meeting on Light Cone Quantization and Nonperturbative Q Conference: C96-03-02, C95-06-20, Proceedings TPR-95-30, e-Print: hep-th/9604018, June 1995. T. HEINZL, [*Hamiltonian Approach to the Gribov Problem*]{}, Nucl. Phy. B, Proc. Suppl., Vol. 54 A, (194-197), 1997. A. JAFFE and E. WITTEN, [*Quantum Yang-Mills Theory*]{}, Official Yang-Mills and Mass Gap Problem Description, Clay Mathematics Institute, 2004. J. JORMAKKA, [*Solutions to Yang-Mills Equations*]{}, arXiv:1011.3962, 2010. H.-H. KUO, [*White Noise Distribution Theory*]{}, Probability and Stochastics Series, CRC Press, 1996. J. LACHAPPELLE, [*Path Integral Solution of Linear Second Order Partial Differential Equations I. The General Construction*]{}, Ann. Phys., Vol. 314, (362-395), 2004. J. LACHAPPELLE, [*Path Integral Solution of Linear Second Order Partial Differential Equations I. The General Construction*]{}, Ann. Phys., Vol. 314, (362-395), 2004. R. DE LA MADRID, [*The Role of the Rigged Hilbert Space in Quantum Mechanics*]{}, Eur. J. Phys., Vol. 26, No. 2, (287-, 2005. J. MAGNEN, V. RIVASSEAU AND R. SENEOR, [*Construction of YM$4$ with an Infrared Cutoff*]{}, Comm. Math. Phys. Vol. 155, (325-383), 1993. R. L. MILLS and C. N. YANG, [*Conservation of Isotopic Spin and Gauge Invariance*]{}, Phy. Rev. Vol. 96, (191-195), 1954. K. OSTERWALDER and R. SCHRADER, [*Axioms for Euclidean Green’s Functions*]{}, Comm. Math. Phys., Vol. 31, (83-112), 1973. K. OSTERWALDER and R. SCHRADER, [*Euclidean fermi Fields and a Feynman-kac Formula for Boson-Fermion Models*]{}, Helv. Phys. Acta Vol. 46, (277-302), 1973. K. OSTERWALDER and R. SCHRADER, [*Axioms for Euclidean Green’s Functions II*]{}, Comm. Math. Phys., Vol. 42, (281-305), 1975. T. PAUSE, [*Diploma Thesis*]{}, Universität Regensburg, 1995. R. D. PECCEI, [*Resolution of Ambiguities of Non-Abelian Gauge Fied Theories in the Coulomb Gauge*]{}, Phys. Rev. D: Particle and Fields, Vol. 17, No. 4, (1097-1110), 1978. R. D. RICHTMYER, [*Principles of Advanced Mathematical Physics*]{}, Springer Texts and Monographs in Physics, Second Edition, 1985. M. REED and B. SIMON, [*Fourier Analysis, Self-Adjointness*]{}, Methods of Modern Mathematical Physics, Vol. 2, Academic Press, 1975. J. E. ROBERTS, [*Rigged Hilbert Spaces in Quantum Mechanics*]{}, Comm. Math. Phys., Vol. 3, No. 2, (98-119), 1966. F. SANNINO and J. SCHECHTER, [*Nonperturbative Results for Yang-Mills Theories*]{}, Phys. Rev. D Vol. 82, No. 7, 096008, 2010. W. SCHLEIFENBAUM, [*Nonperturbative Aspects of Yang-Mills Theory*]{}, Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften, Universität Tübingen, 2008. G. SCHWARZ, [*Hodge Decomposition — A Method for Solving Boundary Value Problems*]{}, Lecture Notes in Math., 1607, Springer-Verlag, Berlin, 1995. G. SCHWARZ and J. ŚNIATYCKI, [*The Existence and Uniqueness of Solutions of Yang-Mills Equations with Bag Boundary Conditions*]{}, Comm. Math. Phys., Vol. 159, No. 3, (593-604), 1994. G. SCHWARZ and J. ŚNIATYCKI, [*Yang-Mills and Dirac Fields in a Bag, Existence and Uniqueness Theorems*]{}, Comm. Math. Phys., Vol. 168, No. 2, (441-453), 1995. C. SCHULTE and R. TWAROCK, [*The Rigged Hilbert Space Formulation of Quantum Mechanics and its Implications for Irreversibility*]{}, based on lectures by Arno Bohm, ` http://arxiv.org/abs/quant-ph/9505004`, 1998. I. E. SEGAL, [*General Properties of the Yang-Mills Equations in Physical Space*]{}, Proc. Nat. Acad. Sci. USA, Vol. 75, No. 10, Applied Physical and Mathematical Sciences, (4608-4639), 1978. I. E. SEGAL, [*The Cauchy Problem for the Yang-Mills Equations*]{}, J. Funct. Anal, Vol. 33, (175-194), 1979. I. M. SINGER, [*Some Remarks on the Gribov Ambiguity*]{}, Comm. Math. Phys. 60, (7—12), 1978. R. F. STREATER and A. S. WIGHTMAN, [*PCT, Spin and Statistics, and All That*]{}, Princeton Landmarks in Physics, 2010. S. WEINBERG, [*The Quantum Theory of Fields, Volume 2: Modern Applications*]{}, Cambridge University Press, 2005. A. S. WIGHTMAN, [*Quantum Field Theory in terms of its Vacuum Expectation Values*]{}, Phy. Rev. 101, (860-866), 1956. E. WITTEN, [*What We Can Hope To Prove About 3d Yang-Mills Theory*]{}, Presentation held at Stony Brook on 17.01.2012, available on ` http://media.scgp.stonybrook.edu/presentations/` `20120117_3_Witten.pdf`, 2012. E. ZEIDLER, [*Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists*]{}, corrected 2nd printing, Springer 2009, [^1]: Simone Farinelli, Ettenbergstrasse 13, CH-8907 Wettswil, Switzerland, e-mail [email protected]

--- author: - 'S. Freddy,' - 'C.S. Kim,' - 'R.H. Li,' - 'and Z.T. Zou' title: 'Charmless $B_{u,d,s}\to VT$ decays in perturbative QCD approach' --- Introduction ============ Flavor physics has been being thoroughly investigated for many years with the advent of $B$-factories. As more and more experimental data is accumulated, flavor physics plays an important role in the precision test of the standard model (SM) and beyond the SM as well as studying the properties of many light hadrons. A few $B\to VT$ decay channels have been already reported by the BaBar collaboration [@Aubert:2009sx; @Aubert:2008bc; @Aubert:2008zza], which makes the $B$ to tensor meson[^1] decays gain more and more attention. Even before the experimental reports there had been already a couple of works [@LopezCastro:1997im; @Munoz:1998sn], which studied the $B \to VT$ decays involving a charmed tensor meson under the quark model. Here we would like to consider the charmless $B\to VT$ decays instead. As early works, these charmless decays had been also studied in the framework of generalized factorization [@Kim:2001py] and in Isgur-Scora-Grinstein-Wise updated model [@Kim:BtoVT]. Later these decays were again studied in the covariant light-front approach in Ref. [@JHMunoz:2009aba]. Polarizations of $B\to VT$ decays as well are studied in Ref. [@Datta:2007yk]. However, most of the branching ratios in the early works are not predicted precisely, which are usually one or two order smaller than the experimental data. This may indicate that some contributions, such as the nonfactorizable and annihilation contributions, are very important in these decays, which are not included in those early works. It has become very urgent to investigate these contributions by employing proper theoretical models. In Ref. [@Cheng:2010yd] the authors accommodated the experimental data with the QCD factorization (QCDF) approach [@Beneke:QCDF], which deals with these additional contributions in a very subtle and technical way. Here we want to adopt yet another theoretical approach, the perturbative QCD (pQCD) approach [@HNLi:pQCD], which calculate the nonfactorizable and annihilation contributions in a theoretically systematical way. These investigations will offer us more detailed knowledge about the dynamics of $B\to VT$ decays, which is one reason why those decays are worthy to be studied again. Another reason why $B\to VT$ decays are meaningful is the interesting polarization phenomena. In $B \to VV$ decays, the transversely polarized contributions of some penguin dominating channels, such as $B\to (\phi,\rho)K^*$, are nearly the same as the longitudinal ones [@Amsler:2008zzb]. This is quite different from the prediction of the naive factorization, in which the longitudinal polarization always dominates. However, in $B\to VT$ decays, such as $B\to \phi K_2^*$, the experimental data indicate that the longitudinal polarization is much larger, while for $B\to \omega K_2^*$ the longitudinal polarization takes only about a half contribution. Earlier models such as naive factorization cannot give us any satisfied explanation. Therefore, employing theoretically more complete models, such as the QCDF, the pQCD and the soft collinear effective theory, to understand the phenomena becomes very important. In a recent paper [@Cheng:2010hn], the authors studied the light cone distribution amplitudes of the tensor mesons, which make the calculation of $B\to VT$ decays possible for the QCDF and the pQCD approach. In their following paper [@Cheng:2010yd], by extracting inputs from the experimental data they accommodated $B\to VT$ decays in the frame of the QCDF. However, some subtle dynamical phenomena is not yet fully understood, which inspires us to explore these decays under another approach. The pQCD approach based on the $k_T$ factorization has already been used to explore many two body exclusive decays of $B$ meson. The form factors of $B$ to a tensor meson transition has already been calculated under this approach [@Wang:2010ni]. There are already a few investigations on the $B$ to a pseudoscalar and a tensor[@Zou:2012td] as well as a charmed meson and a tensor meson decays [@Zou:2012zk; @Zou:2012sx; @Zou:2012sy]. Though there still exist few controversies [@DescotesGenon:2001hm; @Feng:2008zs] on its feasibility, the predictions based on the pQCD can accommodate many experimental data well, for example, see Ref. [@Li:2004ti]. In this work, we will put the controversies aside and adopt this approach to our analysis. The paper is organized as follows. In Sec. \[section:TheoryFrame\], all the details of the theoretical frameworks are listed, including the notation conventions, the Hamiltonian, the kinematics definitions, the wave functions which are used as the inputs in the pQCD approach, and the analytic formulas for the Feynman diagrams in the pQCD approach. The numerical results and discussions are given in Sec. \[section:Ndata\], and the last section is for the summary. In appendix \[appendix:forHD\] we collect the expressions of common pQCD functions. Theoretical details {#section:TheoryFrame} =================== Hamiltonian and kinematics {#section:Hamiltonian} -------------------------- We start from the common low energy effective hamiltonian used in $B$ physics calculations, which are given [@Buchalla:1995vs] as $$\begin{aligned} {\cal H}_{eff} &=& \frac{G_{F}}{\sqrt{2}} \bigg\{ \sum\limits_{q=u,c} V_{qb} V_{qD}^{*} \big[ C_{1}({\mu}) O^{q}_{1}({\mu}) + C_{2}({\mu}) O^{q}_{2}({\mu})\Big]\nonumber\\ &&-V_{tb} V_{tD}^{*} \Big[{\sum\limits_{i=3}^{10}} C_{i}({\mu}) O_{i}({\mu}) \big ] \bigg\} + \mbox{H.c.} , \label{eq:hamiltonian}\end{aligned}$$ where $D=s,d$ stands for a down type light quark, $V_{qb(D)}$ and $V_{tb(D)}$ are Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Functions $O_{i}$ ($i=1,...,10$) are local four-quark operators or the moment type operators: - current–current (tree) operators $$\begin{aligned} O^{q}_{1}=({\bar{q}}_{\alpha}b_{\beta} )_{V-A} ({\bar{D}}_{\beta} q_{\alpha})_{V-A}, \ \ \ \ \ \ \ \ \ O^{q}_{2}=({\bar{q}}_{\alpha}b_{\alpha})_{V-A} ({\bar{D}}_{\beta} q_{\beta} )_{V-A}, \label{eq:operator12} \end{aligned}$$ - QCD penguin operators $$\begin{aligned} O_{3}=({\bar{D}}_{\alpha}b_{\alpha})_{V-A}\sum\limits_{q^{\prime}} ({\bar{q}}^{\prime}_{\beta} q^{\prime}_{\beta} )_{V-A}, \ \ \ \ \ \ \ \ \ O_{4}=({\bar{D}}_{\beta} b_{\alpha})_{V-A}\sum\limits_{q^{\prime}} ({\bar{q}}^{\prime}_{\alpha}q^{\prime}_{\beta} )_{V-A}, \label{eq:operator34} \\ \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! \!\!\!\! O_{5}=({\bar{D}}_{\alpha}b_{\alpha})_{V-A}\sum\limits_{q^{\prime}} ({\bar{q}}^{\prime}_{\beta} q^{\prime}_{\beta} )_{V+A}, \ \ \ \ \ \ \ \ \ O_{6}=({\bar{D}}_{\beta} b_{\alpha})_{V-A}\sum\limits_{q^{\prime}} ({\bar{q}}^{\prime}_{\alpha}q^{\prime}_{\beta} )_{V+A}, \label{eq:operator56} \end{aligned}$$ - electro-weak penguin operators $$\begin{aligned} O_{7}=\frac{3}{2}({\bar{D}}_{\alpha}b_{\alpha})_{V-A} \sum\limits_{q^{\prime}}e_{q^{\prime}} ({\bar{q}}^{\prime}_{\beta} q^{\prime}_{\beta} )_{V+A}, \ \ \ \ O_{8}=\frac{3}{2}({\bar{D}}_{\beta} b_{\alpha})_{V-A} \sum\limits_{q^{\prime}}e_{q^{\prime}} ({\bar{q}}^{\prime}_{\alpha}q^{\prime}_{\beta} )_{V+A}, \label{eq:operator78} \\ O_{9}=\frac{3}{2}({\bar{D}}_{\alpha}b_{\alpha})_{V-A} \sum\limits_{q^{\prime}}e_{q^{\prime}} ({\bar{q}}^{\prime}_{\beta} q^{\prime}_{\beta} )_{V-A}, \ \ \ \ O_{10}=\frac{3}{2}({\bar{D}}_{\beta} b_{\alpha})_{V-A} \sum\limits_{q^{\prime}}e_{q^{\prime}} ({\bar{q}}^{\prime}_{\alpha}q^{\prime}_{\beta} )_{V-A}, \label{eq:operator9x} \end{aligned}$$ where $\alpha$ and $\beta$ are color indices and $q^\prime$ are the active quarks at the scale $m_b$, i.e. $q^\prime=(u,d,s,c,b)$. At the tree level, the operators $O_{7\gamma}$ and $O_{8g}$ do not contribute, thus they are not listed here. The left handed current is defined as $({\bar{q}}^{\prime}_{\alpha} q^{\prime}_{\beta} )_{V-A}= {\bar{q}}^{\prime}_{\alpha} \gamma_\nu (1-\gamma_5) q^{\prime}_{\beta} $ and the right handed current $({\bar{q}}^{\prime}_{\alpha} q^{\prime}_{\beta} )_{V+A}= {\bar{q}}^{\prime}_{\alpha} \gamma_\nu (1+\gamma_5) q^{\prime}_{\beta}$. The projection operators are defined as $P_{L}=(1-\gamma_5)/2$ and $P_{R}=(1+\gamma_5)/2$. The combinations $a_i$ of Wilson coefficients are defined as usual [@Ali:1998eb]: $$\begin{aligned} a_1= C_2+C_1/3, &~a_2= C_1+C_2/3, &~ a_3= C_3+C_4/3, ~a_4= C_4+C_3/3,~a_5= C_5+C_6/3,\nonumber \\ a_6= C_6+C_5/3, &~a_7= C_7+C_8/3, &~a_8= C_8+C_7/3,~a_9= C_9+C_{10}/3, ~a_{10}= C_{10}+C_{9}/3.\end{aligned}$$ The calculation is carried out in the rest frame of $B$ meson, the momenta of $B$ meson ($p_B$), tensor meson ($p_2$) and vector meson ($p_3$) are defined in the light cone coordinates as $$\begin{aligned} p_B=\frac{m_B}{\sqrt{2}}(1,1,{\bf 0_T})\;,\; p_2=\frac{m_B}{\sqrt{2}}(1,r_2^2,{\bf 0_T})\;,\; p_3=\frac{m_B}{\sqrt{2}}(r_3^2,1,{\bf 0_T})\;\label{eq:momenta}\end{aligned}$$ with $r_2=m_T/m_B$ and $r_3=m_V/m_B$. In the calculation of the pQCD, the momenta of the quarks are also related, and they are defined as follows: $$\begin{aligned} k_1=(0,x_1\frac{m_B}{\sqrt{2}},{\bf k_{1T}})\;,\; k_2=(x_2\frac{m_B}{\sqrt{2}},0,{\bf k_{2T}})\;,\; k_3=(0,x_3\frac{m_B}{\sqrt{2}},{\bf k_{3T}})\;,\;\end{aligned}$$ where $k_{1,2,3}$ are the momenta of the light anti-quark in $B$ meson, quarks in tensor and vector mesons, respectively. Wave functions {#section:WF} -------------- ### $B$ meson The $B_{(s)}$ meson wave functions are decomposed into the following Lorentz structures: $$\begin{aligned} &&\int\frac{d^4z}{(2\pi)^4}e^{ik_1\cdot z}\langle0|\bar b_{\alpha}(0)d_{\beta}(z)|B_{(s)}(P_1)\rangle\nonumber\\ &=&\frac{i}{\sqrt{2N_c}}\left\{(\not P_1+m_{B_{(s)}})\gamma_5[\phi_{B_{(s)}}(k_1)-\frac{\not n-\not v}{\sqrt{2}} \bar\phi_{B_{(s)}}(k_1)]\right\}_{\beta\alpha}, \end{aligned}$$ where $\phi_{B_{(s)}}(k_1)$ and $\bar\phi_{B_{(s)}}(k_1)$ are the leading twist distribution amplitudes. After neglecting the numerically small contribution term $\bar\phi_{B_{(s)}}(k_1)$  [@Lu:2002ny], the expression for $\Phi_{B_{(s)}}$ in the momentum space becomes $$\begin{aligned} \Phi_{B_{(s)}}=\frac{i}{\sqrt{2N_c}}{(\not{P_1}+m_{B_{(s)}})\gamma_5\phi_{B_{(s)}}(k_1)}.\end{aligned}$$ The calculation of the pQCD is always carried out in the $b$-space, in which we adopt the following model function $$\begin{aligned} \phi_{B_{(s)}}(x,b)&=&N_{B_{(s)}}x^2(1-x)^2\exp\left[-\frac{1}{2}(\frac{xm_{B_{(s)}}} {\omega_b})^2-\frac{\omega_b^2b^2}{2}\right],\label{eq:Bwave}\end{aligned}$$ where $b$ is the conjugate space coordinate of $\textbf{k}_{1 T}$. $N_{B_{(s)}}$ is the normalization constant, which is determined by the normalization condition $$\begin{aligned} \int^1_0 dx\phi_{B_{(s)}}(x,b=0)=\frac{f_{B_{(s)}}}{2\sqrt{2N_c}}.\end{aligned}$$ For $B^{\pm}$ and $B_d^0$ decays, we adopt the value $\omega_b=0.40~\rm{GeV}$ [@Bauer:1988fx], which is supported by intensive pQCD studies [@pqcd]. For $B_s$ meson, we will follow the authors in Ref. [@Ali:2007ff] and adopt the value $\omega_{b_s}=(0.50\pm0.05)~\rm{GeV}$. ### Vector meson    $f_{\rho}$ $f_{K^*}$ $f_{\omega}$ $f_{\phi}$ $f_{\rho}^T$ $f_{K^*}^T$ $f_{\omega}^T$ $f_{\phi}^T$ --------------- ----------- -------------- ------------ -------------- ------------- ---------------- --------------    $209\pm2$ $217\pm5$ $195\pm3$ $231\pm4$ $165\pm9$ $185\pm10$ $151\pm9$ $186\pm9$ : The decay constants of vector mesons (in MeV) \[vector\_decay\_constants\] The decay constants of the vector mesons are defined by $$\langle 0|\bar q_1\gamma_\mu q_2|V(p_3,\epsilon)\rangle=f_Vm_V\epsilon_\mu,\;\;\; \langle 0|\bar q_1\sigma_{\mu\nu}q_2|V(p_3,\epsilon)\rangle =if^T_V(\epsilon_\mu P_{3\nu}-\epsilon_\nu P_{3\mu}).$$ The longitudinal decay constants of the charged mesons can be extracted experimentally from $\tau^-$ decays and those of the neutral ones can be extracted from their $e^+e^-$ decays [@Ball:2006eu], whereas, the transverse decay constants can be calculated by the QCD sum rules [@Ball:2005vx]. All the constants for the vector mesons in this paper are collected in Table \[vector\_decay\_constants\]. Up to twist-3 the distribution amplitudes of the light vector mesons are summarized as $$\begin{aligned} \langle V(p_3,\epsilon^*_L)|q_{1\alpha}(0)\bar q_{2\beta}(z)|0\rangle&=&-\frac{1}{\sqrt{2N_C}} \int_0^1dxe^{ixp_3\cdot z}\left[m_V\not\epsilon^*_L\phi_V(x)+\not\epsilon^*_L\not p_3\phi_V^t(x)+m_V\phi_V^s(x)\right]_{\alpha\beta},\nonumber\\ % \langle V(p_3,\epsilon^*_T)|q_{1\alpha}(0)\bar q_{2\beta}(z)|0\rangle&=&-\frac{1}{\sqrt{2N_C}} \int_0^1dxe^{ixp_3\cdot z}\left[m_V\not\epsilon^*_T\phi_V^v(x)+\not\epsilon^*_T\not p_3\phi_V^T(x)\right.\nonumber\\ &&\left.+m_Vi\epsilon_{\mu\nu\rho\sigma}\gamma_5\gamma^\mu\epsilon^{*\nu}_Tn^\rho v^\sigma \phi_V^a(x)\right]_{\alpha\beta}, \end{aligned}$$ where $x$ is the momentum fraction of the $q_2$ quark. Here $n$ is the light cone direction along which the meson moves and $v$ is the opposite direction. With $t=2x-1$ the expression for the twist-2 distribution amplitudes are given by $$\begin{aligned} \phi_V(x)=\frac{3f_V}{\sqrt{2N_C}}x(1-x)\left[1+a^\parallel_1C_1^{3/2}(t) +a^\parallel_2C_2^{3/2}(t)\right],\nonumber\\ \phi_V^T(x)=\frac{3f_V}{\sqrt{2N_C}}x(1-x)\left[1+a^\perp_1C_1^{3/2}(t) +a^\perp_2C_2^{3/2}(t)\right].\label{vwavef1}\end{aligned}$$ and the corresponding values of the Gegenbauer moments are [@vdas]: $$\begin{aligned} a_{2\rho}^\parallel=a_{2\omega}^\parallel=0.15\pm0.07\;,\;a_{1K^*}^\parallel=0.03\pm0.02\;,\; a_{2K^*}^\parallel=0.11\pm0.09\;,\;a_{2\phi}^\parallel=0.18\pm0.08\;,\;\nonumber\\ a_{2\rho}^\perp=a_{2\omega}^\perp=0.14\pm0.06\;,\;a_{1K^*}^\perp=0.04\pm0.03\;,\; a_{2K^*}^\perp=0.10\pm0.08\;,\;a_{2\phi}^\perp=0.14\pm0.07\;.\; \end{aligned}$$ We adopt the asymptotic form for the twist-3 distribution amplitudes: $$\begin{aligned} &\phi_V^t(x) = \frac{3f_V^T}{2\sqrt{6}} t^2\;,\;&\phi_V^s(x)=\frac{3f_V^T}{2\sqrt{6}}(-t)\;,\nonumber\\ &\phi_V^v(x) = \frac{3f_V}{8\sqrt{6}} (1+t^2)\;,\;&\phi_V^a(x)=\frac{3f_V}{4\sqrt{6}}(-t)\;.\;\label{vwavef2} \end{aligned}$$ ### Tensor meson In the quark model, the tensor meson with $J^{PC}=2^{++}$ has the angular momentum $L=1$ and spin $S=1$. The ground $SU(3)$ nonet states are consist of $a_2(1320)$, $f_2(1270)$, $f_2^{\prime}(1525)$, and $K_2^*(1430)$. Mixing exists for the $f_2(1270)$ and $f_2^{\prime}(1525)$, just as the $\eta$ and $\eta^{\prime}$ mixing, and their wave functions can be expressed as $$\begin{aligned} f_2&=&f^q \cos\theta_{f_2}+f^s\sin\theta_{f_2},\nonumber\\ f_2^{\prime}&=&f^q \cos\theta_{f_2}-f^s\sin\theta_{f_2},\end{aligned}$$ where $f^q=\frac{1}{\sqrt{2}}(u\bar u + d\bar d)$ and $f^s=s\bar s$. The mixing angle $\theta_{f_2}$ is found to be very small, $\theta_{f_2}=7.8^{\circ}$ [@Amsler:2008zzb] and $\theta_{f_2}=(9\pm1)^{\circ}$ [@mixing2]. Therefore, $f_2$ is nearly an $f^q$ state and $f_2^{\prime}$ is mainly $f^s$. The spin-2 polarization tensor, which is symmetric and traceless, satisfies $\epsilon^{\mu\nu}p_{2\nu}=0$ and can be constructed by spin-1 polarization vectors $\epsilon$ by $$\begin{aligned} \epsilon_{\mu\nu}(\pm2)&=& \epsilon_\mu(\pm)\epsilon_\nu(\pm),\;\;\;\;\nonumber\\ \epsilon_{\mu\nu}(\pm1)&=&\frac{1}{\sqrt2} [\epsilon_{\mu}(\pm)\epsilon_\nu(0)+\epsilon_{\nu}(\pm)\epsilon_\mu(0)],\nonumber\\ \epsilon_{\mu\nu}(0)&=&\frac{1}{\sqrt6} [\epsilon_{\mu}(+)\epsilon_\nu(-)+\epsilon_{\nu}(+)\epsilon_\mu(-)] +\sqrt{\frac{2}{3}}\epsilon_{\mu}(0)\epsilon_\nu(0).\label{eq:tesorPL}\end{aligned}$$ In the case that the tensor meson is moving along the $z$-axis, the polarizations $\epsilon$ can be defined as $$\begin{aligned} \epsilon(0)=(|p_2|,0,0,E_2)/m_T\;\;,\;\;\epsilon(\pm1)=(0,\mp1,i,0)/\sqrt{2},\end{aligned}$$ with $E_2$ as the energy of the tensor meson. Associating with the tensor momentum defined in Eq. (\[eq:momenta\]), the polarization vectors are given in the light cone coordinates by $$\begin{aligned} \epsilon(0)=(1,-r_2^2,{\bf{0_T}})/(\sqrt{2}r_2)\;\;,\;\;\epsilon(\pm1)=(0,0,\mp1,i,0)/\sqrt{2}.\label{eq:PLexpression}\end{aligned}$$ The decay constants of the tensor mesons are defined as $$\begin{aligned} \langle T(p_2)|j_{\mu\nu}(0)|0\rangle&=&f_T m_T^2 \epsilon^*_{\mu\nu},\nonumber\\ \langle T(p_2)|j_{\mu\nu\rho}(0)|0\rangle&=&-if_T^T m_T\left(\epsilon^*_{\mu\delta}p_{2\nu}-\epsilon^*_{\nu\delta}p_{2\mu}\right),\end{aligned}$$ where the currents are defined as $$\begin{aligned} j_{\mu\nu}(0)&=&\frac{1}{2}[\bar q_1(0)\gamma_{\mu}i{\buildrel\leftrightarrow\over D}_{\nu}q_2(0)+\bar q_1(0)\gamma_{\nu}i{\buildrel\leftrightarrow\over D}_{\mu}q_2(0)],\nonumber\\ j_{\mu\nu\rho}^{\dagger}(0)&=&\bar q_2(0)\sigma_{\mu\nu}i{\buildrel\leftrightarrow\over D}_{\rho}q_1(0)\end{aligned}$$ with ${\buildrel\leftrightarrow\over D}_{\mu}={\buildrel\rightarrow\over D}_{\mu} - {\buildrel\leftarrow\over D}_{\mu}$, ${\buildrel\rightarrow\over D}_{\mu}={\buildrel\rightarrow\over\partial}_{\mu}+ig_s A_{\mu}^a\lambda^a/2$ and ${\buildrel\leftarrow\over D}_{\mu}={\buildrel\leftarrow\over\partial}_{\mu}-ig_s A_{\mu}^a\lambda^a/2$, respectively. These decay constants have already been studied [@Aliev:1981ju; @Aliev:1982ab; @Aliev:2009nn] and we use the recently updated ones with the QCD sum rules [@Cheng:2010hn], which are summarized in Table \[Table:Tdecayconstant\]. -------------- --------------- --------------- ----------------- ------------------ ------------------- ------------------- -------------------- -- -- $f_{a_2} $ $ f_{a_2}^T $ $ f_{K_2^*} $ $ f_{K_2^*}^T $ $f_{f_2(1270)} $ $f_{f_2(1270)}^T$ $f_{f_2'(1525)} $ $f_{f_2'(1525)}^T$    $107\pm6$ $105\pm 21$ $ 118\pm 5$ $77\pm 14$    $102\pm 6$ $117\pm25$ $126\pm 4$ $65\pm 12$ -------------- --------------- --------------- ----------------- ------------------ ------------------- ------------------- -------------------- -- -- : Decay constants (in unit of MeV) of tensor mesons from Ref. [@Cheng:2010hn]. \[Table:Tdecayconstant\] The light cone distribution amplitudes (LCDAs) of the tensor mesons are also recently studied by Ref. [@Cheng:2010hn] and we follow the notations in Ref. [@Wang:2010ni] to summarize them up to twist-3 as $$\begin{aligned} \langle T(p_2,\epsilon)|q_{1\alpha} (0)\bar q_{2\beta}(z) |0\rangle &=&\frac{1}{\sqrt{2N_c}}\int_0^1 dx e^{ixp_2\cdot z} \left[m_T\not\! \epsilon^*_{\bullet L} \phi_T(x) +\not\! \epsilon^*_{\bullet L}\not\! p_2 \phi_{T}^{t}(x) +m_T^2\frac{\epsilon_{\bullet} \cdot v}{p_2\cdot v} \phi_T^s(x)\right]_{\alpha\beta}, \label{eq:lpwf}\\ \langle T(p_2,\epsilon)|q_{1\alpha} (0) \bar q_{2\beta}(z) |0\rangle &=&\frac{1}{\sqrt{2N_c}}\int_0^1 dx e^{ixp_2\cdot z} \left[m_T\not\! \epsilon^*_{\bullet T}\phi_T^v(x)+ \not\!\epsilon^*_{\bullet T}\not\! p_2\phi_T^T(x)+m_T i\epsilon_{\mu\nu\rho\sigma}\gamma_5\gamma^\mu\right.\nonumber\\ &&\left.\times\epsilon_{\bullet T}^{*\nu} n^\rho v^\sigma \phi_T^a(x)\right]_{\alpha\beta}\;, \label{eq:tpwf}\end{aligned}$$ with $\epsilon^{0123}=1$ adopted. Eq. (\[eq:lpwf\]) is for the longitudinal polarized mesons ($h=0$) and Eq. (\[eq:tpwf\]) for the transverse polarized ones ($h=\pm 1$). $x$ is the momentum fraction associated with the $q_2$ quark. $n$ is the light cone direction along with tensor meson moves and $v$ is the opposite direction. $\epsilon_{\bullet}$ is defined by $$\epsilon_{\bullet\mu}\equiv\frac{\epsilon_{\mu\nu} v^\nu}{p_2\cdot v}m_T.\label{eq:epsilondot}$$ With the momenta and polarizations defined in the above paragraphs, Eq. (\[eq:epsilondot\]) can be reexpressed by $$\epsilon_{\bullet\mu}=\frac{2m_T}{m_B^2}\epsilon_{\mu\nu}p_B^{\nu}$$ up to the leading power of $r_2$. We follow the symbols in Ref. [@Wang:2010ni], and list the expressions of LCDAs as $$\begin{aligned} &&\phi_{T}(x)=\frac{f_{T}}{2\sqrt{2N_c}}\phi_{||}(x),\;\;\; \phi_{T}^t(x)=\frac{f_{T}^T}{2\sqrt{2N_c}}h_{||}^{(t)}(x),\nonumber\\ &&\phi_{T}^s(x)=\frac{f_{T}^T}{4\sqrt{2N_c}} \frac{d}{dx}h_{||}^{(s)}(x),\hspace{3mm} \phi_{T}^T(x)=\frac{f_{T}^T}{2\sqrt{2N_c}}\phi_{\perp}(x) ,\nonumber\\ &&\phi_{T}^v(x)=\frac{f_{T}}{2\sqrt{2N_c}}g_{\perp}^{(v)}(x), \hspace{3mm}\phi_{T}^a(x)=\frac{f_{T}}{8\sqrt{2N_c}} \frac{d}{dx}g_{\perp}^{(a)}(x).\end{aligned}$$ The twist-2 LCDAs can be expanded in terms of the Gegenbauer polynomials, and their asymptotic form are given by $$\begin{aligned} \phi_{\parallel,\perp}(x)=30x(1-x)(2x-1)\end{aligned}$$ with the normalization conditions $$\int_0^1dx(2x-1)\phi_{\parallel,\perp}(x)=1.$$ By using the QCD equations of motion, the twist-3 two partons distribution amplitudes (DAs) can be related to the twist-2 ones and the tree partons DAs [@Ball:1998ff; @Ball:1998sk]. Their expressions for the asymptotic forms are given by [@Cheng:2010hn] $$\begin{aligned} h_\parallel^{(t)}(x) & = & \frac{15}{2}(2x-1)(1-6x+6x^2) ,\;\;\; h_{||}^{(s)}(x) = 15x(1-x)(2x-1),\\ %\frac{dh_{||}^{(s)}(x)}{dx} & = & 6 (1-2x),\\ g_\perp^{(a)}(x) & = & 20x(1-x)(2x-1) ,\;\;\; g_\perp^{(v)}(x) =5(2x-1)^3. %\frac{dg_\perp^{(a)}(x)}{dx}& = & 6\{ (1-2x) , \\ ,\end{aligned}$$ Analytic formulae {#section:Aformula} ----------------- In this subsection, we list the pQCD formulas for all the possible Feynman diagrams. In the diagrams we use $M_{2,3}$ to denote the tensor and vector mesons, respectively. At the tree level, the Feynman diagrams in the pQCD can be divided into two types according to their typological structures: the emission diagrams, in which the light quark in $B$ meson enter one of the light mesons as a spectator, and the annihilation diagrams, in which both of the two quarks in $B$ mesons are absorbed by the electro-weak operator. According to the polarizations, we can list the formulas in two parts, the longitudinal polarizations and the transverse ones. For simplicity we only list the amplitude functions for the longitudinal ones. The transverse polarized ones can be calculated in the same way with the corresponding wave functions. ![The emission diagrams with a vector meson emitted.[]{data-label="fig:m3e"}](m3ef.eps "fig:") ![The emission diagrams with a vector meson emitted.[]{data-label="fig:m3e"}](m3en.eps "fig:") The factorizable emission diagrams are shown as the first two diagrams in Fig. \[fig:m3e\]. Since the tensor meson can not be generated from the vector or axial vector current, only the vector meson can be emitted. The expressions for all possible Lorentz structures are given as follows. - (V-A)(V-A) factorizable emission diagrams: $$\begin{aligned} F_{vef}^{LL}(a_i)&=&8\sqrt{\frac{2}{3}}\pi m_B^4 f_V C_F\int_0^1 dx_1 dx_2 \int_0^{1/\Lambda_{QCD}}b_1 db_1 b_2 db_2 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2(2x_2-1)(\phi_T^t(x_2)-\phi_T^s(x_2))+(2-x_2)\phi_T(x_2)\right)a_i(t_{vef}^1)E_e(t_{vef}^1) \right.\nonumber\\ &&\left. \times h_e(\sqrt{|\alpha_{ef1}^2|},\sqrt{|\beta_{ef1}^2|}, b_2,b_1)S_t(x_2)-2r_2\phi_T^s(x_2)a_i(t_{vef}^2)E_e(t_{vef}^2)\right.\nonumber\\ &&\left.\times h_e(\sqrt{|\alpha_{ef2}^2|},\sqrt{|\beta_{ef2}^2|},b_1,b_2) S_t(x_1)\right\}\;,\end{aligned}$$ where the explicit expressions of scales $t_{vef}^{1,2}$, the production of coupling $\alpha_s$ and Sudakov factor $E_e(t_{vef}^{1,2})$, the function of hard kernel $h_e$, the parameters $\alpha_{(ef1,ef2)}^2$ and $\beta_{(ef1,ef2)}^2$, and the jet function $S_t(x)$ are all collected in Appendix \[appendix:forHD\]. In the following analytic formulas, the expressions of all the additional functions can also be found in the same appendix. - (V-A)(V+A) factorizable emission diagrams: $$F_{vef}^{LR}(a_i)=F_{vef}^{LL}(a_i)\;,$$ - (S-P)(S+P) factorizable emission diagrams: $$F_{vef}^{SP}(a_i)=0\;.$$ ![The nonfactorizable emission diagrams with a tensor meson emitted.[]{data-label="fig:m2e"}](m2en.eps) There are two possible types of nonfactorizable emission diagrams, one has the vector meson emitted and the other has the tensor meson emitted. They are depicted by the last two diagrams of Fig. \[fig:m3e\] and Fig. \[fig:m2e\] respectively. We use the index $ten$ to represent the tensor meson emission and $ven$ for vector meson emission. The expressions are given by: - (V-A)(V-A) nonfactorizable emission diagrams with vector meson emission: $$\begin{aligned} F_{ven}^{LL}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\phi_V(x_3)\nonumber\\ &&\left\{\left(r_2(1-x_2)(\phi_T^s(x_2)+\phi_T^t(x_2))+x_3\phi_T(x_2)\right)a_i(t_{ven}^1)E_{en}(t_{ven}^1,1,2)\right.\nonumber\\ &&\left. \times h_{en}(\sqrt{|\alpha_{en1}^2|},\sqrt{|\beta_{en1}^2|}, b_1,b_3)+\left(r_2(x_2-1)(\phi_T^s(x_2)-\phi_T^t(x_2))+(x_2+x_3-2)\phi_T(x_2))\right)\right.\nonumber\\ &&\left. \times a_i(t_{ven}^2)E_{en}(t_{ven}^2,1,2) h_{en}(\sqrt{|\alpha_{en2}^2|},\sqrt{|\beta_{en2}^2|},b_1,b_3)\right\}\;,\end{aligned}$$ - (V-A)(V+A) nonfactorizable emission diagrams with vector meson emission: $$\begin{aligned} F_{ven}^{LR}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F r_3 \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2((x_2-1)(\phi_T^s(x_2)-\phi_T^t(x_2))(\phi_V^s(x_3)+\phi_V^t(x_3))-x_3(\phi_T^s(x_2)+\phi_T^t(x_2))(\phi_V^s(x_3)-\phi_V^t(x_3)))\right.\right.\nonumber\\ &&\left.\left.+x_3\phi_T(x_2)(\phi_V^s(x_3)-\phi_V^t(x_3))\right)a_i(t_{ven}^1)E_{en}(t_{ven}^1,1,2)h_{en}(\sqrt{|\alpha_{en1}^2|},\sqrt{|\beta_{en1}^2|},b_1,b_3)\right.\nonumber\\ &&\left.-\left(r_2(x_2(\phi_V^s(x_3)-\phi_V^t(x_3))(\phi_T^s(x_2)-\phi_T^t(x_2))+x_3(\phi_V^s(x_3)+\phi_V^t(x_3))(\phi_T^s(x_2)+\phi_T^t(x_2))\right.\right.\nonumber\\ &&\left.\left.-2(\phi_V^s(x_3)\phi_T^s(x_2)+\phi_V^t(x_3)\phi_T^t(x_2)))-(x_3-1)\phi_T(x_2)(\phi_V^s(x_3)+\phi_V^t(x_3))\right)\right.\nonumber\\ &&\left.\times a_i(t_{ven}^2)E_{en}(t_{ven}^2,1,2)h_{en}(\sqrt{|\alpha_{en2}^2|},\sqrt{|\beta_{en2}^2|},b_1,b_3)\right\}\;,\end{aligned}$$ - (S-P)(S+P) nonfactorizable emission diagrams with vector meson emission: $$\begin{aligned} F_{ven}^{SP}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\phi_V(x_3)\nonumber\\ &&\left\{\left(r_2(1-x_2)(\phi_T^s(x_2)-\phi_T^t(x_2))+\phi_T(x_2)(-x_2+x_3+1)\right) a_i(t_{ven}^1)E_{en}(t_{ven}^1,1,2)\right.\nonumber\\ &&\left. \times h_{en}(\sqrt{|\alpha_{en1}^2|}, \sqrt{|\beta_{en1}^2|},b_1,b_3) +\left(r_2(x_2-1)(\phi_T^t(x_2)+\phi_T^s(x_2))+\phi_T(x_2)(x_3-1)\right)a_i(t_{ven}^2)\right.\nonumber\\ &&\left.\times E_{en}(t_{ven}^2,1,2) h_{en}(\sqrt{|\alpha_{en2}^2|},\sqrt{|\beta_{en2}^2|},b_1,b_3)\right\}\;.\end{aligned}$$ - (V-A)(V-A) nonfactorizable emission diagrams with tensor meson emission: $$\begin{aligned} F_{ten}^{LL}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_2 db_2 \phi_B(x_1)\phi_T(x_2)\nonumber\\ &&\left\{\left(x_2\phi_V(x_3)-r_3(x_3-1)(\phi_V^s(x_3)+\phi_V^t(x_3))\right)a_i(t_{ten}^1)E_{en}(t_{ten}^1,1,3)h_{en}(\sqrt{|\alpha_{en1}^{\prime 2}|},\sqrt{|\beta_{en1}^{\prime 2}|},b_1,b_2)\right.\nonumber\\ &&\left.+\left(r_3(x_3-1)(\phi_V^s(x_3)-\phi_V^t(x_3))+\phi_V(x_3)(x_2+x_3-2)\right)a_i(t_{ten}^2)E_{en}(t_{ten}^2,1,3)\right.\nonumber\\ &&\left. \times h_{en}(\sqrt{|\alpha_{en2}^{\prime 2}|},\sqrt{|\beta_{en2}^{\prime 2}|},b_1,b_2)\right\}\;,\end{aligned}$$ - (V-A)(V+A) nonfactorizable emission diagrams with tensor meson emission: $$\begin{aligned} F_{ten}^{LR}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F r_2 \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_2 db_2 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_3(x_2(\phi_V^s(x_3)+\phi_V^t(x_3))(\phi_T^t(x_2)-\phi_T^s(x_2))+(1-x_3)(\phi_T^s(x_2)+\phi_T^t(x_2))(\phi_V^t(x_3)-\phi_V^s(x_3)))\right.\right.\nonumber\\ &&\left.\left.+x_2\phi_V(x_3)(\phi_T^s(x_2)-\phi_T^t(x_2))\right)a_i(t_{ten}^1)E_{en}(t_{ten}^1,1,3)h_{en}(\sqrt{|\alpha_{en1}^{\prime 2}|},\sqrt{|\beta_{en1}^{\prime 2}|},b_1,b_2)\right.\nonumber\\ &&\left.+\left(r_3(-x_2(\phi_V^s(x_3)+\phi_V^t(x_3))(\phi_T^s(x_2)+\phi_T^t(x_2))+x_3(\phi_V^s(x_3)-\phi_V^t(x_3))(\phi_T^t(x_2)-\phi_T^s(x_2))\right.\right.\nonumber\\ &&\left.\left.+2(\phi_V^t(x_3)\phi_T^t(x_2)+\phi_V^s(x_3)\phi_T^s(x_2)))+(x_2-1)\phi_V(x_3)(\phi_T^s(x_2)+\phi_T^t(x_2))\right)\right.\nonumber\\ &&\left.\times a_i(t_{ten}^2)E_{en}(t_{ten}^2,1,3)h_{en}(\sqrt{|\alpha_{en2}^{\prime 2}|},\sqrt{|\beta_{en2}^{\prime 2}|},b_1,b_2)\right\}\;,\end{aligned}$$ - (S-P)(S+P) nonfactorizable emission diagrams with tensor meson emission: $$\begin{aligned} F_{ten}^{SP}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_2 db_2 \phi_B(x_1)\phi_T(x_2)\nonumber\\ &&\left\{\left(\phi_V(x_3)(x_2-x_3+1)-r_3(x_3-1)(\phi_V^s(x_3)-\phi_V^t(x_3))\right)a_i(t_{ten}^1)E_{en}(t_{ten}^1,1,3)\right.\nonumber\\ &&\left. \times h_{en}(\sqrt{|\alpha_{ef1}^{\prime 2}|},\sqrt{|\beta_{ef1}^{\prime 2}|},b_1,b_2)+\left(r_3(x_3-1)(\phi_V^s(x_3)+\phi_V^t(x_3))+(x_2-1)\phi_V(x_3)\right)a_i(t_{ten}^2)\right.\nonumber\\ &&\left.\times E_{en}(t_{ten}^2,1,3)h_{en}(\sqrt{|\alpha_{en2}^{\prime 2}|},\sqrt{|\beta_{en2}^{\prime 2}|},b_1,b_2)\right\}\;.\end{aligned}$$ According to which meson has the anti-quark generated from the weak vertex, the annihilation diagrams are also divided into two types, as depicted in Figs. \[fig:m2a\] and \[fig:m3a\]. We use the first letter of the index “$v$" to denote the case that the quark enters the vector meson and “$t$" to denote that the quark enters the tensor meson. ![The annihilation diagrams with the electro-weak generated anti-quark entering the tensor meson.[]{data-label="fig:m2a"}](m3af.eps "fig:") ![The annihilation diagrams with the electro-weak generated anti-quark entering the tensor meson.[]{data-label="fig:m2a"}](m3an.eps "fig:") ![The annihilation diagrams with the electro-weak generated anti-quark entering the vector meson.[]{data-label="fig:m3a"}](m2af.eps "fig:") ![The annihilation diagrams with the electro-weak generated anti-quark entering the vector meson.[]{data-label="fig:m3a"}](m2an.eps "fig:") For the factorizable annihilation diagrams, which are the first two diagrams in Figs. \[fig:m2a\] and \[fig:m3a\], the corresponding functions are given as follows. - (V-A)(V-A) factorizable annihilation diagrams with the quark entering the vector meson: $$\begin{aligned} F_{vaf}^{LL}(a_i)&=&8\sqrt{\frac{2}{3}}\pi m_B^4 f_B C_F\int_0^1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_2 db_2 b_3 db_3\nonumber\\ &&\left\{\left(-2r_2r_3\phi_V^s(x_3)(-\phi_T^s(x_2)(x_2+1)+\phi_T^t(x_2)(1-x_2))-x_2\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{vaf}^1)\right.\nonumber\\ &&\left.\times E_a(t_{vaf}^1)h_a(\sqrt{|\alpha_{af1}^{2}|},\sqrt{|\beta_{af1}^{2}|},b_2,b_3)S_t(x_2)\right.\nonumber\\ &&\left.+\left(-2r_2r_3\phi_T^s(x_2)((2-x_3)\phi_V^s(x_3)+x_3\phi_V^t(x_3))+(1-x_3)\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{vaf}^2)\right.\nonumber\\ &&\left.\times E_a(t_{vaf}^2)h_a(\sqrt{|\alpha_{af2}^{2}|},\sqrt{|\beta_{af2}^{2}|},b_3,b_2)S_t(x_3)\right\}\;,\end{aligned}$$ - (V-A)(V+A) factorizable annihilation diagrams with the quark entering the vector meson: $$\begin{aligned} F_{vaf}^{LR}(a_i)=F_{vaf}^{LL}(a_i)\;,\end{aligned}$$ - (S-P)(S+P) factorizable annihilation diagrams with the quark entering the vector meson: $$\begin{aligned} F_{vaf}^{SP}(a_i)&=&16\sqrt{\frac{2}{3}}\pi m_B^4 f_B C_F\int_0^1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_2 db_2 b_3 db_3\nonumber\\ &&\left\{\left(r_2x_2\phi_V(x_3)(\phi_T^t(x_2)-\phi_T^s(x_2))+2r_3\phi_T(x_2)\phi_V^s(x_3)\right)a_i(t_{vaf}^1)E_a(t_{vaf}^1) \right.\nonumber\\ &&\left.\times h_a(\sqrt{|\alpha_{af1}^{2}|},\sqrt{|\beta_{af1}^{2}|},b_2,b_3)S_t(x_2)+\left(r_3(1-x_3)\phi_T(x_2)(\phi_V^s(x_3)+\phi_V^t(x_3))-2r_2\phi_T^s(x_2) \phi_V(x_3)\right)\right.\nonumber\\ &&\left.\times a_i(t_{vaf}^2)E_a(t_{vaf}^2)h_a(\sqrt{|\alpha_{af2}^{2}|},\sqrt{|\beta_{af2}^{2}|},b_3,b_2)S_t(x_3)\right\}\;.\end{aligned}$$ - (V-A)(V-A) factorizable annihilation diagrams with the quark entering the tensor meson: $$\begin{aligned} F_{taf}^{LL}(a_i)&=&8\sqrt{\frac{2}{3}}\pi m_B^4 f_B C_F\int_0^1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_2 db_2 b_3 db_3\nonumber\\ &&\left\{\left(2r_2r_3\phi_T^s(x_2)((x_3+1)\phi_V^s(x_3)+(x_3-1)\phi_V^t(x_3))-x_3\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{taf}^1)\right.\nonumber\\ &&\left.\times E_a(t_{taf}^1)h_a(\sqrt{|\alpha_{af1}^{\prime 2}|},\sqrt{|\beta_{af1}^{\prime 2}|},b_3,b_2)S_t(x_3)\right.\nonumber\\ &&\left.+\left(2r_2r_3\phi_V^s(x_3)((x_2-2)\phi_T^s(x_2)-x_2\phi_T^t(x_2))+(1-x_2)\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{taf}^2)\right.\nonumber\\ &&\left.\times E_a(t_{taf}^2)h_a(\sqrt{|\alpha_{af2}^{\prime 2}|},\sqrt{|\beta_{af2}^{\prime 2}|},b_2,b_3)S_t(x_2)\right\}\;,\end{aligned}$$ - (V-A)(V+A) factorizable annihilation diagrams with the quark entering the tensor meson: $$\begin{aligned} F_{taf}^{LR}(a_i)=F_{taf}^{LL}(a_i)\;,\end{aligned}$$ - (S-P)(S+P) factorizable annihilation diagrams with the quark entering the tensor meson: $$\begin{aligned} F_{taf}^{SP}(a_i)&=&16\sqrt{\frac{2}{3}}\pi m_B^4 f_B C_F\int_0^1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_2 db_2 b_3 db_3\nonumber\\ &&\left\{\left(r_3x_3\phi_T(x_2)(\phi_V^t(x_3)-\phi_V^s(x_3))+2r_2\phi_T^s(x_2)\phi_V(x_3)\right)a_i(t_{taf}^1)E_a(t_{taf}^1)\right.\nonumber\\ &&\left.\times h_a(\sqrt{|\alpha_{af1}^{\prime 2}|},\sqrt{|\beta_{af1}^{\prime 2}|},b_3,b_2)S_t(x_3)+\left(r_2\phi_V(x_3)(1-x_2)(\phi_T^s(x_2)+\phi_T^t(x_2)) -2r_3\phi_T(x_2)\phi_V^s(x_3)\right)\right.\nonumber\\ &&\left.\times a_i(t_{taf}^2)E_a(t_{taf}^2)h_a(\sqrt{|\alpha_{af2}^{\prime 2}|},\sqrt{|\beta_{af2}^{\prime 2}|},b_2,b_3)S_t(x_2)\right\}\;.\end{aligned}$$ The nonfactorizable annihilation diagrams are depicted by the last two diagrams in Figs. \[fig:m2a\] and \[fig:m3a\], and their corresponding functions are given in the following. - (V-A)(V-A) nonfactorizable annihilation diagrams with the quark entering the vector meson: $$\begin{aligned} F_{van}^{LL}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(x_2-x_3+3)+\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-r_2r_3\phi_T^t(x_2)(\phi_V^s(x_3)(x_2+x_3-1)+\phi_V^t(x_3)(x_2-x_3-1))\right.\right.\nonumber\\ &&\left.\left.+(x_3-1)\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^1)E_{an}(t_{van}^1)h_{an}(\sqrt{|\alpha_{an1}^{2}|},\sqrt{|\beta_{an1}^{2}|},b_2,b_1)\right.\nonumber\\ &&\left.+\left(r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(-x_2+x_3-1)+\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-r_2r_3\phi_T^t(x_2)(\phi_V^s(x_3)(x_2+x_3-1)+\phi_V^t(x_3)(-x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.+x_2\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^2)E_{an}(t_{van}^2)h_{an}(\sqrt{|\alpha_{an2}^{2}|},\sqrt{|\beta_{an2}^{2}|},b_2,b_1)\right\}\;,\end{aligned}$$ - (V-A)(V+A) nonfactorizable annihilation diagrams with the quark entering the vector meson: $$\begin{aligned} F_{van}^{LR}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2\phi_V(x_3)(2-x_2)(\phi_T^s(x_2)+\phi_T^t(x_2))+r_3(x_3+1)\phi_T(x_2)(\phi_V^s(x_3)-\phi_V^t(x_3))\right)a_i(t_{van}^1)\right.\nonumber\\ &&\left.\times E_{an}(t_{van}^1)h_{an}(\sqrt{|\alpha_{an1}^{2}|},\sqrt{|\beta_{an1}^{2}|},b_2,b_1)\right.\nonumber\\ &&\left.+\left(r_2x_2\phi_V(x_3)(\phi_T^s(x_2)+\phi_T^t(x_2))+r_3(1-x_3)\phi_T(x_2)(\phi_V^s(x_3)-\phi_V^t(x_3))\right)a_i(t_{van}^2) \right.\nonumber\\ &&\left.\times E_{an}(t_{van}^2)h_{an}(\sqrt{|\alpha_{an2}^{2}|},\sqrt{|\beta_{an2}^{2}|},b_2,b_1)\right\}\;,\end{aligned}$$ - (S-P)(S+P) nonfactorizable annihilation diagrams with the quark entering the vector meson: $$\begin{aligned} F_{van}^{SP}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(x_2-x_3+3)-\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-r_2r_3\phi_T^t(x_2)(\phi_V^t(x_3)(x_2-x_3-1)-\phi_V^s(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-x_2\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^1)E_{an}(t_{van}^1)h_{an}(\sqrt{|\alpha_{an1}^{2}|},\sqrt{|\beta_{an1}^{2}|},b_2,b_1)\right.\nonumber\\ &&\left.+\left(r_2r_3(-\phi_T^s(x_2)(\phi_V^s(x_3)(x_2-x_3+1)+\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.+r_2r_3\phi_T^t(x_2)(\phi_V^t(x_3)(x_2-x_3+1)+\phi_V^s(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-(x_3-1)\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^2)E_{an}(t_{van}^2)h_{an}(\sqrt{|\alpha_{an2}^{2}|},\sqrt{|\beta_{an2}^{2}|},b_2,b_1)\right\}\;.\end{aligned}$$ - (V-A)(V-A) nonfactorizable annihilation diagrams with the quark entering the tensor meson: $$\begin{aligned} F_{tan}^{LL}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(-x_2+x_3+3)-\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.+r_2r_3\phi_T^t(x_2)(\phi_V^s(x_3)(x_2+x_3-1)-\phi_V^t(x_3)(-x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-(1-x_2)\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^1)E_{an}(t_{tan}^1)h_{an}(\sqrt{|\alpha_{an1}^{\prime 2}|},\sqrt{|\beta_{an1}^{\prime 2}|},b_2,b_1)\right.\nonumber\\ &&\left.+\left(-r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(-x_2+x_3+1)+\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.+r_2r_3\phi_T^t(x_2)(\phi_V^s(x_3)(x_2+x_3-1)+\phi_V^t(x_3)(-x_2+x_3+1))\right.\right.\nonumber\\ &&\left.\left.+x_3\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^2)E_{an}(t_{tan}^2)h_{an}(\sqrt{|\alpha_{an2}^{\prime 2}|},\sqrt{|\beta_{an2}^{\prime 2}|},b_2,b_1)\right\}\;,\end{aligned}$$ - (V-A)(V+A) nonfactorizable annihilation diagrams with the quark entering the tensor meson: $$\begin{aligned} F_{tan}^{LR}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{-\left(r_2\phi_V(x_3)(x_2+1)(\phi_T^t(x_2)-\phi_T^s(x_2))+r_3\phi_T(x_2)(x_3-2)(\phi_V^s(x_3)+\phi_V^t(x_3))\right)a_i(t_{van}^1)\right.\nonumber\\ &&\left.\times E_{an}(t_{tan}^1)h_{an}(\sqrt{|\alpha_{an1}^{\prime 2}|},\sqrt{|\beta_{an1}^{\prime 2}|},b_2,b_1)\right.\nonumber\\ &&\left.+\left(r_3x_3\phi_T(x_2)(\phi_V^s(x_3)+\phi_V^t(x_3))-r_2(1-x_2)\phi_V(x_3)(\phi_T^t(x_2)-\phi_T^s(x_2))\right)a_i(t_{van}^2)\right.\nonumber\\ &&\left.\times E_{an}(t_{tan}^2)h_{an}(\sqrt{|\alpha_{an2}^{\prime 2}|},\sqrt{|\beta_{an2}^{\prime 2}|},b_2,b_1)\right\}\;,\end{aligned}$$ - (S-P)(S+P) nonfactorizable annihilation diagrams with the quark entering the tensor meson: $$\begin{aligned} F_{tan}^{SP}(a_i)&=&\frac{32}{3}\pi m_B^4 C_F \int_0^1 dx_1 dx_2 dx_3 \int_0^{1/\Lambda_{QCD}}b_1db_1 b_3 db_3 \phi_B(x_1)\nonumber\\ &&\left\{\left(r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(-x_2+x_3+3)+\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-r_2r_3\phi_T^t(x_2)(\phi_V^s(x_3)(x_2+x_3-1)+\phi_V^t(x_3)(-x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-x_3\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^1)E_{an}(t_{tan}^1)h_{an}(\sqrt{|\alpha_{an1}^{\prime 2}|},\sqrt{|\beta_{an1}^{\prime 2}|},b_2,b_1)\right.\nonumber\\ &&\left.+\left(-r_2r_3\phi_T^s(x_2)(\phi_V^s(x_3)(-x_2+x_3+1)-\phi_V^t(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.+r_2r_3\phi_T^t(x_2)(\phi_V^t(x_3)(-x_2+x_3+1)-\phi_V^s(x_3)(x_2+x_3-1))\right.\right.\nonumber\\ &&\left.\left.-(x_2-1)\phi_T(x_2)\phi_V(x_3)\right)a_i(t_{van}^2)E_{an}(t_{tan}^2)h_{an}(\sqrt{|\alpha_{an2}^{\prime 2}|},\sqrt{|\beta_{an2}^{\prime 2}|},b_2,b_1)\right\}\;.\end{aligned}$$ Similar to the $B\to VV$ decays, the amplitude of $B\to VT$ can be decomposed as $$\begin{aligned} {\cal A}(\epsilon_{2},\epsilon_{3})&=&i{\cal A}^N + i(\epsilon^*_{T} \cdot \epsilon^*_{\bullet T}){\cal A}^s + (\epsilon_{\mu \nu \alpha \beta}n^{\mu} \bar n^{\nu} \epsilon^{*\alpha}_{T} \epsilon^{*\beta}_{\bullet T}) {\cal A}^p,\end{aligned}$$ where ${\cal A}^N$ contains the contribution from the longitudinal polarizations, ${\cal A}^s$ and ${\cal A}^p$ represent the transversely polarized contributions. With the amplitude functions obtained in this section, the amplitude for the decay channels can be expressed. Considering the length of the paper, we will not list all the expressions of the amplitudes, but give one of the $B$ decay amplitude as an example: [ $$\begin{aligned} {\cal M}(B^-\to \rho^- a_2^0)=\frac{G_F}{\sqrt{2}}&\bigg\{& V_{ub}V^*_{ud}[F^{LL}_{vef}(\frac{1}{\sqrt{2}}a_{ 1})+F^{LL}_{ven}(\frac{1}{\sqrt{2}}C_{ 1})+F^{LL}_{ten}(\frac{1}{\sqrt{2}}C_{ 2})+F^{LL}_{vaf}(\frac{1}{\sqrt{2}}a_{ 1}) \nonumber\\ && +F^{LL}_{taf}(-\frac{1}{\sqrt{2}}a_{ 1})+F^{LL}_{van}(\frac{1}{\sqrt{2}}C_{ 1})+F^{LL}_{tan}(-\frac{1}{\sqrt{2}}C_{ 1})]-V_{tb}V^*_{td}[F^{LL}_{vef}(\frac{1}{\sqrt{2}}a_{ 4}+\frac{1}{\sqrt{2}}a_{10}) \nonumber\\ && +F^{LL}_{ven}(\frac{1}{\sqrt{2}}C_{ 3}+\frac{1}{\sqrt{2}}C_{ 9})+F^{LL}_{ten}(-\frac{1}{\sqrt{2}}C_{ 3}+\frac{1}{2\sqrt{2}}C_{ 9}+\frac{3}{2\sqrt{2}}C_{10}) \nonumber\\ && +F^{LR}_{ven}(\frac{1}{\sqrt{2}}C_{ 5}+\frac{1}{\sqrt{2}}C_{ 7})+F^{LR}_{ten}(-\frac{1}{\sqrt{2}}C_{ 5}+\frac{1}{2\sqrt{2}}C_{ 7})+F^{SP}_{vef}(\frac{1}{\sqrt{2}}a_{ 6}+\frac{1}{\sqrt{2}}a_{ 8}) \nonumber\\ && +F^{SP}_{ten}(\frac{3}{2\sqrt{2}}C_{ 8})+F^{LL}_{vaf}(\frac{1}{\sqrt{2}}a_{ 4}+\frac{1}{\sqrt{2}}a_{10})+F^{LL}_{taf}(-\frac{1}{\sqrt{2}}a_{ 4}-\frac{1}{\sqrt{2}}a_{10}) \nonumber\\ && +F^{LL}_{van}(\frac{1}{\sqrt{2}}C_{ 3}+\frac{1}{\sqrt{2}}C_{ 9})+F^{LL}_{tan}(-\frac{1}{\sqrt{2}}C_{ 3}-\frac{1}{\sqrt{2}}C_{ 9})+F^{LR}_{van}(\frac{1}{\sqrt{2}}C_{ 5}+\frac{1}{\sqrt{2}}C_{ 7}) \nonumber\\ && +F^{LR}_{tan}(-\frac{1}{\sqrt{2}}C_{ 5}-\frac{1}{\sqrt{2}}C_{ 7})+F^{SP}_{vaf}(\frac{1}{\sqrt{2}}a_{ 6}+\frac{1}{\sqrt{2}}a_{ 8})+F^{SP}_{taf}(-\frac{1}{\sqrt{2}}a_{ 6}-\frac{1}{\sqrt{2}}a_{ 8})] \bigg\}. \end{aligned}$$ ]{} Numerical results and discussions {#section:Ndata} ================================= With the amplitudes calculated in Sec. \[section:Aformula\], the decay width is given as $$\begin{aligned} \Gamma&=&\frac{[(1-(r_2+r_3)^2)(1-(r_2-r_3)^2)]^{1/2}}{16\pi m_B}\sum_{i}|A_i|^2,\end{aligned}$$ where $i$ represents all the polarization states, and the branching ratio is obtained through ${\cal BR}=\Gamma\tau_B$. The key observables of the decays related in this paper are the CP averaged branching ratios, polarization fractions, as well as direct CP asymmetries ($A_{\rm{CP}}^{\rm{dir}}$). Readers are referred to Ref. [@forcp] for reviews on CP violation. First, we define four amplitudes as follows: $$\begin{aligned} A_f&=&\langle f|{\cal H}|B\rangle,\;\;\; \bar A_f=\langle f|{\cal H}|\bar B\rangle,\nonumber\\ A_{\bar f}&=&\langle\bar f|{\cal H}|B\rangle,\;\;\; \bar A_{\bar f}=\langle\bar f|{\cal H}|\bar B\rangle,\end{aligned}$$ where $\bar B$ meson has a $b$ quark in it and $\bar f$ is the CP conjugate state of $f$. The direct CP asymmetry $A_{\rm{CP}}^{\rm{dir}}$ is defined by $$\begin{aligned} A_{\rm{CP}}^{\rm{dir}}&=&\frac{|\bar A_{\bar f}|^2-|A_f|^2} {|\bar A_{\bar f}|^2+|A_f|^2}\;.\label{eq:Dcp}\end{aligned}$$ Our results for CP averaged branching ratios and CP asymmetries are listed in Tables \[tab:BmtoVT\], \[tab:BdtoVT\] and \[tab:BstoVT\]. In these tables, we also list the results of the longitudinal polarization fractions ${\cal R}_L$, which is defined by $$\begin{aligned} {\cal R}_L&=&\frac{|A_0|^2}{\sum_i |A_i|^2}\;,\end{aligned}$$ where $A_0$ is the amplitude of the longitudinal polarization. The first error entries of our results are from the parameters in the wave functions, the decay constant $f_B$ and the shape parameter $\omega_b$. The second ones are from $\Lambda_{\rm{QCD}}$, which varies $20\%$ for error estimates, and from the scale $t$, which are listed in appendix \[appendix:forHD\]. ------------------------------------------- ---------------------------------- --------------- ------------------------------ ----------------------     $\cal BR$ $$ ${\cal R}_L$ $$    Decay This Work Experiments This Work Experiments    $B^-\to K_2^*(1430)^-\omega$ $0.81^{+0.62+1.10}_{-0.54-0.62}$ $21.5\pm 4.3$ $47.0^{+0.8+0.3}_{-4.2-5.2}$ $56\pm 11$    $\bar B^0\to \bar K_2^*(1430)^0\omega$ $0.93^{+0.71+1.04}_{-0.51-0.73}$ $10.1\pm 2.3$ $55.6^{+3.1+3.0}_{-1.5-3.2}$ $45\pm12$    $B^-\to K_2^*(1430)^-\phi$ $9.1^{+3.4+2.9}_{-2.6-2.0}$ $8.4\pm2.1$ $82.1^{+6.2+8.7}_{-6.6-9.2}$ $80\pm10$    $\bar B^0\to \bar K_2^*(1430)^0\phi$ $8.7^{+3.1+2.7}_{-2.5-1.9}$ $7.5\pm1.0$ $82.0^{+6.5+8.1}_{-6.2-9.7}$ $90.1^{+5.9}_{-6.9}$ ------------------------------------------- ---------------------------------- --------------- ------------------------------ ---------------------- : The pQCD results for $B\to VT$ decays which have experimental data, where the experimental data is from the BaBar collaboration [@Amsler:2008zzb; @Aubert:2009sx; @Aubert:2008bc]. Unit $10^{-6}$ for branching ratios, and $\%$ for the ${\cal R}_L$.[]{data-label="tab:4channels"}     Decays $\cal BR$ Decays $\cal BR$ ------------------------------------------- --------------- --------------------------------- -------------------    $B^-\to K_2^*(1430)^-\omega$ $21.5\pm 4.3$ $B^-\to K^{*-}\omega$ $2.4\pm1.0\pm0.2$    $\bar B^0\to \bar K_2^*(1430)^0\omega$ $10.1\pm 2.3$ $\bar B^0\to \bar K^{*0}\omega$ $2.0\pm0.5$    $B^-\to K_2^*(1430)^-\phi$ $8.4\pm2.1$ $B^-\to K^{*-}\phi$ $10.0\pm2.0$    $\bar B^0\to \bar K_2^*(1430)^0\phi$ $7.5\pm1.0$ $\bar B^0\to \bar K^{*0}\phi$ $9.8\pm0.6$ : The experimental branching ratios of $B\to K_2^*(1430)(\omega,\phi)$ decays [@Amsler:2008zzb; @Aubert:2009sx; @Aubert:2008bc] and their corresponding $B\to VV$ decays [@Aubert:2008zza]. The unit is $10^{-6}$.[]{data-label="tab:compare"} Before we go to the numerical discussions of Tables \[tab:BmtoVT\], \[tab:BdtoVT\] and \[tab:BstoVT\], we note a few comments on the present experimental status. Only four channels, $B^-\to K_2^{*-}(\phi,\omega)$ and $\bar B^0 \to K_2^{*0}(\phi,\omega)$, are reported by BaBar [@Amsler:2008zzb; @Aubert:2009sx; @Aubert:2008bc], which are shown in Tables \[tab:4channels\] and \[tab:compare\]. We also collect the corresponding decays in $B\to VV$ mode [@Aubert:2008zza] for comparison. For the helicity structures of $B\to VT$ decays are very similar to the $B\to VV$ ones, a comparison between $B\to VT$ and $B\to VV$ would be very enlightening. Comparing with the experimental data, one can see that the pQCD can give good predictions for the $B \to \phi (K_2^{*-},\bar K_2^{*0})$ decays. For the $B \to \omega (K_2^{*-},\bar K_2^{*0})$ decays, only the polarization fractions can be accommodated well, and large deviations exist in the branching ratios. Comparing our predictions with the experimental data, here we would like to make a few comments: 1. ${\cal BR}(B \to \phi (K_2^{*-},\bar K_2^{*0}))$ is very similar to ${\cal BR}(B \to \phi (K^{*-},\bar K^{*0}))$, but a little smaller, which might be understood easily by the effect of a heavier tensor mass on the phase space. It also indicates that only small effects are brought in the branching ratios when $K^*$ is substituted for $K_2^*$. 2. However, the experimental data shows totally opposite behavior for the $B \to \omega (K_2^{*-},\bar K_2^{*0}, K^{*-},\bar K^{*0})$ decays, where ${\cal BR}(B \to \omega (K_2^{*-},\bar K_2^{*0}))$ is much larger than ${\cal BR}(B \to \omega (K^{*-},\bar K^{*0}))$. In the $B\to VV$ case, ${\cal BR}(B \to \phi (K^{*-},\bar K^{*0}))$ is about five times larger than ${\cal BR}(B \to \omega (K^{*-},\bar K^{*0}))$, while in the $B\to VT$ case ${\cal BR}(B \to \phi (K_2^{*-},\bar K_2^{*0}))$ is even smaller than ${\cal BR}(B \to \omega (K^{*-},\bar K^{*0}))$. 3. The pQCD predictions for the branching ratios of the $B\to VT$ decays are very similar to but a little smaller than the experimental data of $B\to VV$. Taking the errors into consideration, the similar numerical relationship between ${\cal BR}(B \to \phi (K^{*-},\bar K^{*0}))$ and ${\cal BR}(B \to \omega (K^{*-},\bar K^{*0}))$ mentioned above can also be accommodated in $B\to VT$ decays. As is well known, the $B\to VT$ decay is very similar to the $B\to VV$ decay mode theoretically, therefore the branching ratios in these two decay modes are expected to have the similar behavior. Based on such prejudice, the present experimental data is a little difficult to be understood. 4. However, only BaBar collaboration reported the results for $B \to \omega (K_2^{*-},\bar K_2^{*0})$ up to now, thus the experimental data need to be confirmed later. On the theoretical side, the tensor meson may bring forth new mechanism, which needs further investigations. In Ref. [@Cheng:2010yd] the authors approached those channels in a different way. They used the experimental data of those channels to extract the penguin-annihilation parameters of the QCDF and predicted the other channels. By adopting the way, the experimental data could also be accommodated. However, we note more investigations are in need to understand the underlying dynamics totally. ---------------------------------- ---------------------------------- ---------------------------- -------------------------- ------------------------------ ---------------------------------- -----------------------------------     ${\cal{R}}_L$ $A_{\rm{CP}}^{\rm{dir}}$     This Work QCDF[@Cheng:2010yd] ISGW2[@Kim:BtoVT] CLF[@JHMunoz:2009aba] This Work This Work    $B^-\to \omega K_2^{*-}$ $0.81^{+0.62+1.10}_{-0.54-0.62}$ $7.5^{+19.7}_{-7.0}$ $0.112$ $0.06$ $47.0^{+0.8+0.3}_{-4.2-5.2}$ $-4.4_{-0.0-2.2}^{+0.4+3.7}$    $B^-\to \phi K_2^{*-}$ $9.1^{+3.4+2.9}_{-2.6-2.0}$ $7.4^{+25.8}_{-5.2}$ $2.180$ $9.24$ $82.1^{+6.2+8.7}_{-6.6-9.2}$ $1.5_{-0.1-0.3}^{+0.2+0.1}$    $B^-\to \rho^- a_2^0$ $12.8^{+7.1+2.4}_{-5.1-2.4}$ $8.4^{+4.7}_{-2.9}$ $7.432$ $19.34$ $93.4^{+0.8+1.2}_{-0.9-1.5}$ $6.5^{+1.4+1.6}_{-1.6-1.5}$    $B^-\to \rho^- \bar K_2^{*0}$ $3.9^{+1.2+1.8}_{-0.9-1.0}$ $18.6^{+50.1}_{-17.2}$ $$&$$ $57.6^{+1.6+1.8}_{-0.8-1.8}$ $0.43^{+0.50+0.56}_{-0.39-0.07}$    $B^-\to \rho^0 a_2^-$ $0.67^{+0.30+0.37}_{-0.20-0.20}$ $0.82^{+2.30}_{-0.95}$ $0.007$ $0.071$ $50.8^{+5.1+9.5}_{-2.4-8.1}$ $-6.5^{+0.8+11.2}_{-0.4-7.9}$    $B^-\to \rho^0 K_2^{*-}$ $2.3^{+0.6+0.8}_{-0.6-0.5}$ $10.4^{+18.8}_{-9.2}$ $0.253$ $0.74$ $67.6^{+2.2+1.9}_{-2.9-4.0}$ $-4.8^{+1.2+1.0}_{-1.8-0.9}$    $B^-\to \omega a_2^-$ $0.41^{+0.14+0.07}_{-0.14-0.06}$ $0.38^{+1.84}_{-0.36}$ $0.010$ $0.14$ $64.5^{+0.5+2.4}_{-2.8-5.1}$ $5.91^{+2.4+4.2}_{-6.9-7.0}$    $B^-\to \phi a_2^-$ $0.01^{+0.01+0.01}_{-0.00-0.00}$ $0.0003^{+0.013}_{-0.001}$ $0.004$ $0.019$ $67.4^{+0.8+4.3}_{-0.2-0.4}$ $--$    $B^-\to K^{*-} a_2^0$ $3.2^{+1.4+1.1}_{-0.9-0.6}$ $2.9^{+11.7}_{-2.5}$ $1.852$ $2.80$ $59.4^{+8.1+9.9}_{-7.9-10.7}$ $-4.1^{+2.9+5.8}_{-3.7-5.2}$    $B^-\to K^{*-} K_2^{*0}$ $0.39^{+0.09+0.15}_{-0.10-0.07}$ $2.1^{+4.2}_{-1.8}$ $$&$$ $68.1^{+0.0+5.3}_{-2.1-2.9}$ $-3.4^{+0.9+0.9}_{-1.6-5.1}$    $B^-\to K^{*0} K_2^{*-}$ $0.19^{+0.08+0.07}_{-0.06-0.05}$ $0.56^{+1.09}_{-0.38}$ $0.014$ $0.59$ $60.7^{+8.2+6.9}_{-8.8-9.8}$ $22.1^{+7.5+1.4}_{-7.7-5.2}$    $B^-\to \bar K^{*0} a_2^-$ $7.6^{+3.4+2.3}_{-2.7-1.9}$ $6.1^{+23.8}_{-5.4}$ $4.495$ $8.62$ $61.8^{7.3+8.9}_{-8.1-10.7}$ $-0.82^{+0.01+0.32}_{-0.25-0.27}$    $B^-\to \rho^- f_2$ $15.6^{+8.2+1.8}_{-6.1-2.2}$ $7.7^{+4.8}_{-2.9}$ $8.061$ $$ $96.9^{+0.0+0.0}_{-0.0-0.0}$ $7.2^{+0.3+1.2}_{-0.6-1.3}$    $B^-\to \rho^- f_2^{\prime}$ $0.11^{+0.04+0.02}_{-0.03-0.02}$ $0.07^{+0.11}_{-0.04}$ $0.103$ $$ $99.3^{+0.1+0.5}_{-0.0-0.8}$ $--$    $B^-\to K^{*-} f_2$ $7.3^{+2.8+2.4}_{-2.2-1.5}$ $8.3^{+17.3}_{-6.7}$ $2.032$ $$ $76.3^{+4.1+1.2}_{-3.6-1.5}$ $-38.6^{+1.7+3.5}_{-0.7-2.7}$    $B^-\to K^{*-} f_2^{\prime}$ $1.7^{+0.5+1.0}_{-0.3-0.5}$ $12.6^{+24.0}_{-11.1}$ $0.025$ $$ $15.1^{+4.2+5.1}_{-3.6-5.6}$ $-1.6^{+0.6+0.8}_{-1.0-1.2}$ ---------------------------------- ---------------------------------- ---------------------------- -------------------------- ------------------------------ ---------------------------------- ----------------------------------- : The branching ratios ($\cal{BR}$ in unit of $10^{-6}$), polarization fractions (${\cal R}_L$ in unit of $\%$) and direct CP violation ($A_{\rm{CP}}^{\rm{dir}}$ in unit of $\%$) of $B^-\to VT$ decays.[]{data-label="tab:BmtoVT"} --------------------------------------------- ---------------------------------- --------------------------- -------------------------- ------------------------------ ---------------------------------- ---------------------------------     ${\cal{R}}_L$ $A_{\rm{CP}}^{\rm{dir}}$     This Work QCDF[@Cheng:2010yd] ISGW2[@Kim:BtoVT] CLF[@JHMunoz:2009aba] This Work This Work    $\bar B_d^0\to \omega \bar K_2^{*0}$ $0.93^{+0.71+1.04}_{-0.51-0.73}$ $8.1^{+21.7}_{-7.6}$ $0.104$ $0.053$ $55.6^{+3.1+3.0}_{-1.5-3.2}$ $5.4_{-1.7-3.3}^{+1.9+3.5}$    $\bar B_d^0\to \phi \bar K_2^{*0}$ $8.7^{+3.1+2.7}_{-2.5-1.9}$ $7.7^{+26.9}_{-5.5}$ $2.024$ $8.51$ $82.0^{+6.5+8.1}_{-6.2-9.7}$ $--$    $\bar B_d^0\to \rho^- a_2^+$ $26.7^{+13.6+3.5}_{-10.2-3.9}$ $11.3^{+5.3}_{-4.6}$ $14.686$ $36.18$ $94.8^{+0.3+0.5}_{-0.3-0.6}$ $1.3^{+0.5+1.0}_{-0.4-0.9}$    $\bar B_d^0\to \rho^+ a_2^-$ $0.74^{+0.19+0.16}_{-0.20-0.13}$ $1.2^{+2.6}_{-1.0}$ $$&$$ $88.7^{+0.8+1.3}_{-1.6-2.6}$ $7.7^{+3.2+0.6}_{-4.3-4.4}$    $\bar B_d^0\to \rho^+ K_2^{*-}$ $3.4^{+1.1+1.6}_{-0.8-0.9}$ $19.8^{+52.0}_{-18.2}$ $$&$$ $53.9^{+1.0+1.3}_{-0.0-1.4}$ $-2.2^{+0.7+0.8}_{-0.9-0.8}$    $\bar B_d^0\to \rho^0 a_2^0$ $0.68^{+0.30+0.22}_{-0.21-0.12}$ $0.39^{+1.35}_{-0.20}$ $0.003$ $0.03$ $90.7^{+1.0+1.5}_{-0.0-0.6}$ $13.6^{+2.1+5.3}_{-0.4-3.1}$    $\bar B_d^0\to \rho^0 \bar K_2^{*0}$ $1.7^{+0.5+0.7}_{-0.4-0.4}$ $9.5^{+33.4}_{-9.5}$ $0.235$ $0.68$ $47.0^{+0.5+2.7}_{-0.8-1.7}$ $3.3^{+1.6+1.3}_{-1.1-1.9}$    $\bar B_d^0\to \omega a_2^0$ $0.37^{+0.11+0.0}_{-0.12-0.03}$ $0.25^{+1.14}_{-0.19}$ $0.005$ $0.07$ $89.5^{+0.3+1.2}_{-1.6-2.5}$ $5.7^{+8.9+11.1}_{-7.0-9.9}$    $\bar B_d^0\to \phi a_2^0$ $\sim 10^{-3}$ $0.001^{+0.006}_{-0.001}$ $0.002$ $0.009$ $45.0^{+1.5+4.8}_{-1.1-3.5}$ $--$    $\bar B_d^0\to K^{*-} a_2^+$ $6.1^{+2.8+2.1}_{-1.7-1.1}$ $6.1^{+24.3}_{-5.3}$ $3.477$ $7.25$ $59.3^{+8.3+9.1}_{-5.8-9.6}$ $-17.9^{+1.6+2.6}_{-4.4-4.9}$    $\bar B_d^0\to K^{*-} K_2^{*+}$ $3.0^{+0.7+1.4}_{-0.7-0.8}$ $0.43^{+0.54}_{-0.31}$ $$&$$ $49.8^{+0.3+0.3}_{-1.0-0.8}$ $5.9^{+3.7+4.3}_{-1.0-0.3}$    $\bar B_d^0\to K^{*+} K_2^{*-}$ $3.5^{+0.9+1.7}_{-0.8-0.9}$ $0.06^{+0.09}_{-0.03}$ $$&$$ $49.4^{+0.3+1.2}_{-0.6-0.4}$ $-1.2^{+0.4+1.6}_{-0.6-1.2}$    $\bar B_d^0\to K^{*0}\bar K_2^{*0}$ $4.5^{+1.3+2.2}_{-1.1-1.2}$ $0.44^{+0.88}_{-0.30}$ $0.026$ $0.55$ $40.6^{+1.9+2.7}_{-1.5-2.1}$ $6.1^{+1.4+1.3}_{-0.9-1.9}$    $\bar B_d^0\to \bar K^{*0} a_2^0$ $3.5^{+1.7+1.4}_{-1.2-0.8}$ $3.4^{+12.4}_{-2.8}$ $2.109$ $4.03$ $60.3^{+7.9+9.7}_{-7.7-9.9}$ $-11.1^{+0.2+2.2}_{-0.1-2.0}$    $\bar B_d^0\to \bar K^{*0} K_2^{*0}$ $5.1^{+1.4+2.3}_{-1.2-1.5}$ $1.1^{+2.9}_{-1.0}$ $$&$$ $55.1^{+0.7+0.0}_{-0.2-1.9}$ $0.41^{+0.22+0.31}_{-0.19-0.93}$    $\bar B_d^0\to \rho^0 f_2$ $0.41^{+0.26+0.15}_{-0.16-0.07}$ $0.42^{+1.90}_{-0.44}$ $0.004$ $0.019$ $85.3^{+1.7+3.1}_{-1.1-2.0}$ $-1.9^{+0.0+5.2}_{-2.4-9.4}$    $\bar B_d^0\to \rho^0 f_2^{\prime}$ $0.05^{+0.02+0.01}_{-0.01-0.01}$ $0.03^{+0.06}_{-0.02}$ $5\times 10^{-5}$ $$ $99.3^{+0.0+0.4}_{-0.0-0.8}$ $--$    $\bar B_d^0\to \omega f_2$ $0.56^{+0.19+0.12}_{-0.16-0.11}$ $0.69^{+0.97}_{-0.36}$ $0.005$ $$ $95.5^{+0.5+0.6}_{-0.5-0.9}$ $-5.7^{+5.9+7.5}_{-8.3-7.3}$    $\bar B_d^0\to \omega f_2^{\prime}$ $0.04^{+0.01+0.01}_{-0.01-0.01}$ $0.03^{+0.04}_{-0.01}$ $6\times10^{-5}$ $$ $99.2^{+0.0+0.0}_{-0.0-0.0}$ $--$    $\bar B_d^0\to \phi f_2$ $\sim 10^{-3}$ $0.001^{+0.007}_{-0.000}$ $0.002$ $$ $73.0^{+1.12+5.53}_{-0.00-0.00}$ $--$    $\bar B_d^0\to \phi f_2^{\prime}$ $0.06^{+0.03+0.02}_{-0.01-0.05}$ $0.006^{+0.034}_{-0.005}$ $2\times10^{-5}$ $$ $10.0^{+33.1+24.6}_{-0.0-0.3}$ $0.67^{+0.0+0.59}_{-4.98-6.03}$    $\bar B_d^0\to \bar K^{*0} f_2$ $7.1^{+3.2+2.5}_{-2.1-1.3}$ $9.1^{+8.8}_{-7.3}$ $2.314$ $$ $73.8^{+5.4+4.3}_{-3.3-1.4}$ $6.1^{+0.1+1.1}_{-0.4-1.2}$    $\bar B_d^0\to \bar K^{*0} f_2^{\prime}$ $1.8^{+0.6+1.1}_{-0.4-0.6}$ 13.5$^{+25.4}_{-11.9}$ $0.029$ $$ $17.4^{+6.7+6.0}_{-1.5-2.5}$ $--$ --------------------------------------------- ---------------------------------- --------------------------- -------------------------- ------------------------------ ---------------------------------- --------------------------------- : The branching ratios ($\cal{BR}$ in unit of $10^{-6}$), polarization fractions (${\cal R}_L$ in unit of $\%$) and direct CP violation ($A_{\rm{CP}}^{\rm{dir}}$ in unit of $\%$) of $\bar B_d^0 \to VT$ decays.[]{data-label="tab:BdtoVT"} We collect our results for $B_{u,d}\to VT$ decays in the Table \[tab:BmtoVT\] and \[tab:BdtoVT\], as well as the results of branching ratios under the QCDF and from the other two models. Most results of the pQCD and the QCDF agree with each other very well. For those channels, where the pQCD and the QCDF have obviously different central values, such as $B^-\to \rho^-\bar K_2^{*0}$ and $\bar B_d^0\to \rho^+\bar K_2^{*-}$, the penguin-annihilation parameters of the QCDF contribute the differences. The penguin-annihilation parameters are the key point of the QCDF to enhance the branching ratios of $B^-\to \omega\bar K_2^{*-}$ and $\bar B_d^0\to \omega\bar K_2^{*0}$ to accommodate the experimental data. We presume those parameters are the main factors for the large differences between the central values of these channels. However, taking the errors into consideration, the two different theoretical approaches can still agree with each other. On the other hand, future experimental observation of these channels may offer an opportunity to test the dynamics of the pQCD and the QCDF. For the channels dominated by the $W$-emission diagrams, especially with a vector meson emitted, such as $\bar B_d^0\to a_2^+\rho^-$, the longitudinal contribution is dominating, and the polarization fraction ${\cal R}_L$ is around $90\%$. The polarization fractions of those decays dominated by the penguin diagrams are very profound. The polarization fractions of some penguin-dominating decays of $B\to VV$ decay mode like $B\to \phi K^{*0}$ are reported to be around $50\%$ [@Amsler:2008zzb], which are out of expectation of the SM. This is so-called the polarization puzzle in $B$ physics. However, one can find that the polarization fraction of $B\to\phi K_2^{*0}$ in the $B\to VT$ behaves as the SM expectation, while the $B\to \omega K_2^{*0}$ gives about $90\%$. In our calculation, we find that the polarization of $B\to VT$ for these channels are near to the $B\to VV$ ones. However, after we consider $r^2=(m_T/m_B)^2$ contributions carefully, the polarizations can be accomodated to the experimental data, although the branching fractions cannot be accomodated. From Eq. (\[eq:Dcp\]) one can see that the generation of the direct CP violation requires that the amplitude $A_f$ consists of at least two parts with different weak phases. Usually they are the tree contribution and penguin contributions in the SM. Readers are referred to Ref. [@Amsler:2008zzb] for the related formulas and reviews. The interference of these parts will bring the direct CP violation. The magnitude of the direct CP violation is proportional to the ratio of the penguin and tree contributions. Therefore, the direct CP violation in the SM is very small, since the penguin contribution is almost always sub-dominating, which can be seen in previous section. However, there are a few very special channels in which the penguin contributions may be comparable to the tree one, as a result, sizeable direct CP violation appears. Take $B^-\to K^{*-} K_2^{*0}$ as an example. In this channel, the CKM matrix elements for the tree ($V_{ub}V^*_{ud}$) and penguin contributions ($V_{tb}V^*_{td}$) are at the same order. Although the Wilson coefficients for the tree contributions are much larger, the tree operator only appears in the annihilation diagrams, not in the emission ones. Therefore the tree contributions are suppressed, and the penguin ones become comparable, which brings to a relatively large direct CP violation for this channel.     $\cal{BR}$ ${\cal{R}}_L$ $A_{\rm{CP}}^{\rm{dir}}$ ----------------------------------------- -------------------------------------- --------------------------------- ----------------------------------    $\bar B_s^0\to \rho^- a_2^+$ $0.35^{+0.08+0.12}_{-0.11-0.08}$ $75.6^{+0.1+2.9}_{-1.5-3.4}$ $-11.0^{+3.7+4.9}_{-2.2-2.9}$    $\bar B_s^0\to \rho^- K_2^{*+}$ $17.1^{+7.7+1.2}_{-6.0-1.2}$ $93.5^{+0.2+0.5}_{-0.3-0.6}$ $4.2^{+0.7+0.8}_{-0.6-1.2}$    $\bar B_s^0\to \rho^+ a_2^-$ $0.15^{+0.04+0.08}_{-0.04-0.05}$ $35.7^{+0.4+6.0}_{-1.2-3.3}$ $9.2^{+1.1+8.2}_{-1.3-7.6}$    $\bar B_s^0\to \rho^0 a_2^0$ $0.03^{+0.01+0.02}_{-0.01-0.01}$ $99.1^{+0.3+0.6}_{-0.1-1.2}$ $-18.1^{+3.8+2.5}_{-1.3-5.2}$    $\bar B_s^0\to \rho^0 K_2^{*0}$ $0.30^{+0.13+0.14}_{-0.10-0.08}$ $42.6^{+6.2+8.0}_{-5.9-8.5}$ $7.0^{+1.4+3.3}_{-2.0-4.7}$    $\bar B_s^0\to \omega a_2^0$ $\sim 10^{-3}$ $59.8^{+12.0+16.7}_{-5.6-13.3}$ $-18.8^{+1.7+12.7}_{-5.3-3.3}$    $\bar B_s^0\to \omega K_2^{*0}$ $0.13^{+0.05+0.07}_{-0.04-0.03}$ $49.6^{+2.6+6.3}_{-2.1-3.8}$ $-19.2^{+4.2+6.2}_{-4.8-4.2}$    $\bar B_s^0\to \phi a_2^0$ $0.04^{+0.01+0.005}_{-0.01-0.006}$ $99.2^{+0.1+0.3}_{-0.1-0.6}$ $-1.1^{+0.1+0.4}_{-1.1-2.2}$    $\bar B_s^0\to \phi K_2^{*0}$ $0.36^{+0.10+0.18}_{-0.09-0.10}$ $62.2^{+3.1+2.3}_{-0.9-0.9}$ $--$    $\bar B_s^0\to K^{*-}K_2^{*+}$ $4.5^{+1.6+1.7}_{-1.1-0.8}$ $39.9^{+7.8+11.3}_{-4.5-8.3}$ $-12.5^{+1.2+6.0}_{-0.6-4.3}$    $\bar B_s^0\to K^{*+}a_2^-$ $0.66^{+0.18+0.24}_{-0.17-0.15}$ $77.8^{+1.6+2.6}_{-0.7-3.2}$ $7.1^{+3.3+3.1}_{-3.0-3.0}$    $\bar B_s^0\to K^{*+}K_2^{*-}$ $6.1^{+1.5+2.5}_{-1.6-1.6}$ $59.9^{+0.3+0.7}_{-1.4-1.9}$ $-0.9^{+0.7+0.5}_{-0.6-0.4}$    $\bar B_s^0\to K^{*0}a_2^0$ $0.88^{+0.25+0.15}_{-0.23-0.13}$ $90.5^{+0.6+2.3}_{-0.6-3.1}$ $5.1^{+1.6+2.5}_{-2.6-3.5}$    $\bar B_s^0\to K^{*0}\bar K_2^{*0}$ $8.9^{+2.6+3.7}_{-2.2-2.1}$ $62.9^{+0.4+1.4}_{-1.6-2.9}$ $--$    $\bar B_s^0\to \bar K^{*0} K_2^{*0}$ $6.2^{+1.9+2.2}_{-1.7-1.5}$ $34.1^{+6.3+11.8}_{-5.2-12.2}$ $-4.2^{+0.2+0.8}_{-0.3-0.6}$    $\bar B_s^0\to \rho^0 f_2$ $\sim 10^{-3}$ $69.5^{+1.8+8.1}_{-7.1-9.9}$ $-23.8^{+3.1+5.6}_{-1.6-3.8}$    $\bar B_s^0\to \rho^0 f_2^{\prime}$ $0.12^{+0.05+0.01}_{-0.04-0.01}$ $89.8^{+0.0+0.4}_{-0.0-0.4}$ $14.2^{+0.8+2.1}_{-0.7-1.9}$    $\bar B_s^0\to \omega f_2$ $0.02^{+0.004+0.008}_{-0.003-0.009}$ $99.2^{+0.1+0.2}_{-0.3-1.0}$ $-13.9^{+2.6+2.4}_{-6.4-8.1}$    $\bar B_s^0\to \omega f_2^{\prime}$ $0.28^{+0.13+0.09}_{-0.10-0.06}$ $26.4^{+1.7+10.3}_{-0.4-7.4}$ $-1.3^{+0.5+1.6}_{-0.0-0.5}$    $\bar B_s^0\to \phi f_2$ $2.9^{+1.0+0.7}_{-0.9-0.7}$ $98.7^{+0.1+0.6}_{-0.0-1.1}$ $0.84^{+0.07+0.19}_{-0.35-0.41}$    $\bar B_s^0\to \phi f_2^{\prime}$ $3.1^{+1.8+0.6}_{-1.4-0.6}$ $75.3^{+3.0+3.5}_{-3.2-1.7}$ $--$    $\bar B_s^0\to K^{*0}f_2$ $0.51^{+0.17+0.11}_{-0.16-0.13}$ $92.2^{+1.6+2.4}_{-2.7-5.1}$ $-11.9^{+4.3+5.9}_{-2.5-2.4}$    $\bar B_s^0\to K^{*0}f_2^{\prime}$ $0.39^{+0.13+0.14}_{-0.09-0.08}$ $59.7^{+3.2+3.3}_{-2.2-3.1}$ $--$ : The branching ratios ($\cal{BR}$ in unit of $10^{-6}$), polarization fractions (${\cal R}_L$ in unit of $\%$) and direct CP violation ($A_{\rm{CP}}^{\rm{dir}}$ in unit of $\%$) of $\bar B_s^0 \to VT$ decays.[]{data-label="tab:BstoVT"} For most of the $B\to VT$ decays, the pQCD predicts the branching ratios at the order of $10^{-6}$, which would be easy for the experimental observation. We also calculate the branching ratios, polarization fractions and the direct CP violations of $\bar B_s\to VT$ decays, which are collected in Table \[tab:BstoVT\]. Most of the $\bar B_s$ decays are penguin-dominated, whose branching ratios are mainly at the order of $10^{-7}$, therefore, whose observation requires more accumulation of experimental data. However, it would be easy for the forth-coming future flavor physics experiments. If a vector meson, generated by the tree operator whose decay constant is nonzero, is emitted in a $\bar B_s$ decay, then such channels have a large possibility to gain a relatively large branching ratios with the order of $10^{-6}$. Summary ======= One of the valuable topics in flavor physics is studying the hadrons in the $B$ meson decays. In recent years, inspired by the interesting experimental data, more and more studies on the $B$ to tensor meson decays are carried on. The pQCD approach, which has been being developed for years and predicts many $B$ meson decays successfully, is a powerful tool in the study of two body non-leptonic $B$ meson decays. In this paper, we investigated the $B\to VT$ decays under the frame of the pQCD. We calculated all the tree level diagrams in the approach and collected all the necessary expressions in our paper, with which we can study the $39$ $B\to VT$ and $23$ $B_s\to VT$ decays. The branching ratios, polarization fractions, and direct CP violations are predicted. Four channels in $B\to VT$ are reported by the experiments: $B\to \phi(K_2^{*-}, \bar K_2^{*0})$ and $B\to \omega(K_2^{*-}, \bar K_2^{*0})$. Comparing with their similar decays in the $B \to VV$ mode, these four channels have very interesting phenomena. On the experimental side, unlike the polarization puzzle in $B\to VV$ decays, the longitudinal polarization fractions of $B\to \phi(K_2^{*-}, \bar K_2^{*0})$ decays are around $90\%$, while those of the $B\to \omega(K_2^{*-}, \bar K_2^{*0})$ decays are around $50\%$. The branching ratios of $B\to \omega(K_2^{*-}, \bar K_2^{*0})$ are much larger than those of $B\to \phi(K_2^{*-}, \bar K_2^{*0})$. This is quite different from the $B\to VV$ case, where the branching ratio of $B\to \omega(K^{*-}, \bar K^{*0})$ are about $5$ times larger than the ones of $B\to \phi(K^{*-}, \bar K^{*0})$. By considering the $r^2=(m_T/m_B)^2$ corrections, although the polarization fractions can be accommodated, the branching ratios are not predicted well. This may need further experimental confirmation and theoretical investigation. Most of the branching ratios for $B^-$ and $\bar B^0$ decays are predicted to be at the order of $10^{-6}$. Most of our results agree with the the ones of the QCDF. Some channels do not agree so well by the central values, which may be caused by the different dynamics, since the QCDF introduce the penguin-annihilation parameters to accommodate the experimental data and their behavior seems different from the pQCD approach. However, taking the errors into consideration, they can still agree. For the decays which contributed by the $W$-emission diagram, especially when the vector meson is emitted, the polarization fraction is about $90\%$, which is just as the expectation of SM. The polarization fractions for the penguin dominated decays are complicated. Some are around $90\%$ and some are $50\%$, just like the cases of the four channels observed by the experiments. Fortunately, the main order of the $B^-$ and $\bar B^0$ decays is $10^{-6}$, which would be easy for the experimental observation. In the $\bar B_s^0$ decays, the branching ratios are smaller. Such tree-dominated decays as $\bar B^0_s \to \rho^- K_2^{*+}$, when a vector meson is emitted, have the mechanism to gain a relatively large branching ratio at the order of $10^{-6}$. Most of the others are at the order of $10^{-7}$, whose observation need more accumulation of experimental data. Acknowledgement =============== The work was supported by the National Research Foundation of Korea (NRF) grant funded by Korea government of the Ministry of Education, Science and Technology (MEST) (Grant No. 2011-0017430) and (Grant No. 2011-0020333). The work of Z.T.Z. was supported by the National Science Foundation of China under the Grant No.11075168, 11228512, and 11235005. Functions for hard kernel, Sudakov factors and scales {#appendix:forHD} ===================================================== The parameters in the hard part is given as follows. $$\begin{aligned} \beta^2_{ef1} &=& x_1(1-x_2)m_B^2\, ,\, \beta^2_{ef2}=\beta^2_{ef1}\, ,\nonumber\\ \alpha^2_{ef1} &=& (1-x_2)m_B^2\, ,\, \alpha_{ef2}^2=x_1m_B^2\, ,\nonumber\\ \beta^2_{en1} &=& (1-x_2)(x_1-x_3)m_B^2\, ,\,\beta_{en2}^2=(1-x_2)(x_3+x_1-1)m_B^2\, ,\nonumber\\ \alpha^2_{en1} &=& (1-x_2)x_1 m_B^2\, ,\, \alpha_{en2}^2=\alpha^2_{en1}\, ,\nonumber\\ \beta^2_{af1} &=& -x_2(1-x_3)m_B^2\, ,\, \beta^2_{af2}=\beta^2_{af1}\, ,\nonumber\\ \alpha^2_{af1}&=& -x_2 m_B^2\, ,\, \alpha^2_{af2}=(x_3-1)m_B^2\, ,\nonumber\\ \beta^2_{an1} &=& [1-(x_3-x_1)(1-x_2)]m_B^2\, ,\,\beta^2_{an2}=(x_3+x_1-1)x_2 m_B^2\, ,\nonumber\\ \alpha^2_{an1} &=& -x_2(1-x_3)m_B^2\, ,\,\alpha^2_{an2}=\alpha^2_{an1}\, ,\nonumber\\ \beta^{\prime 2}_{i} &=& \beta^2_{i}(x_2\leftrightarrow x_3)\,,\,\alpha^{\prime 2}_{i}=\alpha^2_{i}(x_2\leftrightarrow x_3)\, ,\end{aligned}$$ where $i$ represents any indices. The functions for the hard parts are given by $$\begin{aligned} h_e(\alpha,\beta,b_1,b_2)&=&K_0(\beta b_2)[\theta(b_2-b_1)I_0(b_1\alpha)K_0(b_2\alpha) +\theta(b_1-b_2) I_0(b_2\alpha) K_0(b_1\alpha)],\nonumber\\ %---------------------------------------------------------------------------- h_{en}(\alpha,\beta,b_1,b_2)&=&[\theta(b_2-b_1)I_0(b_1\alpha)K_0(b_2\alpha) +\theta(b_1-b_2) I_0(b_2\alpha) K_0(b_1\alpha)]\times \left\{ \begin{array}{l} K_0(\beta b_2) \,\,\,\,\,\,\mbox{for }\beta^2>0\\ \frac{i\pi}{2}H^{(1)}_0(\beta b_2)\,\,\,\mbox{for }\beta^2<0 \end{array}\right\},\nonumber\\ h_a(\alpha,\beta,b_1,b_2)&=&\left(\frac{i\pi}{2}\right)^2 H_0^{(1)}(\beta b_2)[\theta(b_2-b_1)J_0(b_1\alpha)H^{(1)}_0(b_2\alpha) +\theta(b_1-b_2) J_0(b_2\alpha) H^{(1)}_0(b_1\alpha)],\nonumber\\ % h_{an}(\alpha,\beta,b_1,b_2)&=&\frac{i\pi}{2}[\theta(b_2-b_1)J_0(b_1\alpha)H^{(1)}_0(b_2\alpha)\nonumber\\ &&+\theta(b_1-b_2) J_0(b_2\alpha) H^{(1)}_0(b_1\alpha)]\times\left\{ \begin{array}{l} K_0(\beta b_2) \,\,\,\,\,\,\mbox{for }\beta^2>0\\ \frac{i\pi}{2}H^{(1)}_0(\beta b_2)\,\,\,\mbox{for }\beta^2<0 \end{array}\right\},\end{aligned}$$ where the $K_0$, $I_0$, $J_0$ and $H_0^{(1)}$ are all Bessel functions, and $H_0^{(1)}(z)=J_0(z)+i Y_0(z)$. The scales are defined as $$\begin{aligned} t_{v..}^l=\max\left[c\sqrt{|\alpha_{..l}^2|},c\sqrt{|\beta_{..l}^2|},1/b_k,1/b_l\right],\; t_{t..}^l=\max\left[c\sqrt{|\alpha_{..l}^{\prime 2}|},c\sqrt{|\beta_{..l}^{\prime 2}|},1/b_k,1/b_l\right],\end{aligned}$$ where $l=1,2$, the “$..$” represents $ef$, $en$, $af$ or $an$, and $b_{k,l}$ represent the two corresponding $b$ coordinates in the measurement of the integration. The parameter $c=1$, and in our error estimation, we choose $c=0.75$ and $1.25$ for a rough estimation. The expressions for the Sudakov factors and coupling constants are given as $$\begin{aligned} E_e(t)&=&\alpha_s(t)\exp[-S_B(t)-S_V(t)]\;,\nonumber \\ E_a(t)&=&\alpha_s(t)\exp[-S_T(t)-S_V(t)]\;,\nonumber\\ E_{en}(t)&=&\left\{ \begin{array}{l} \alpha_s(t)\exp[-S_B(t)-S_T(t)-S_V(t)|_{b_2=b_1}],\,\,\mbox{if vector meson emits} \\ \alpha_s(t)\exp[-S_B(t)-S_T(t)-S_V(t)|_{b_3=b_1}],\,\,\mbox{if tensor meson emits} \end{array} \right\} \;,\nonumber \\ E_{an}(t)&=&\alpha_s(t)\exp[-S_B(t)-S_T(t)-S_V(t)|_{b_3=b_2}]\;, \end{aligned}$$ where $$\begin{aligned} S_B(t)&=&s(x_1\frac{m_{B}}{\sqrt{2}},b_1)+\frac{5}{3}\int^t_{1/b_1}\frac{d\bar \mu}{\bar \mu}\gamma_q(\alpha_s(\bar \mu)),\nonumber\\ S_T(t)&=&s(x_2\frac{m_{B}}{\sqrt{2}},b_2)+s((1-x_2)\frac{m_{B}}{\sqrt{2}},b_2)+2\int^t_{1/b_2}\frac{d\bar \mu}{\bar \mu}\gamma_q(\alpha_s(\bar \mu)),\nonumber\\ S_V(t)&=&s(x_3\frac{m_{B}}{\sqrt{2}},b_3)+s((1-x_3)\frac{m_{B}}{\sqrt{2}},b_3)+2\int^t_{1/b_3}\frac{d\bar \mu}{\bar \mu}\gamma_q(\alpha_s(\bar \mu)), \end{aligned}$$ with the quark anomalous dimension $\gamma_q=-\alpha_s/\pi$. The explicit form for the function $s(Q,b)$ is: $$\begin{aligned} s(Q,b)&=&~~\frac{A^{(1)}}{2\beta_{1}}\hat{q}\ln\left(\frac{\hat{q}} {\hat{b}}\right)- \frac{A^{(1)}}{2\beta_{1}}\left(\hat{q}-\hat{b}\right)+ \frac{A^{(2)}}{4\beta_{1}^{2}}\left(\frac{\hat{q}}{\hat{b}}-1\right) %\nonumber \\ -\left[\frac{A^{(2)}}{4\beta_{1}^{2}}-\frac{A^{(1)}}{4\beta_{1}} \ln\left(\frac{e^{2\gamma_E-1}}{2}\right)\right] \ln\left(\frac{\hat{q}}{\hat{b}}\right) \nonumber \\ &&+\frac{A^{(1)}\beta_{2}}{4\beta_{1}^{3}}\hat{q}\left[ \frac{\ln(2\hat{q})+1}{\hat{q}}-\frac{\ln(2\hat{b})+1}{\hat{b}}\right] +\frac{A^{(1)}\beta_{2}}{8\beta_{1}^{3}}\left[ \ln^{2}(2\hat{q})-\ln^{2}(2\hat{b})\right],\end{aligned}$$ where the variables are defined by $$\begin{aligned} \hat q\equiv \mbox{ln}[Q/(\sqrt 2\Lambda)],~~~ \hat b\equiv \mbox{ln}[1/(b\Lambda)], \end{aligned}$$ and the coefficients $A^{(i)}$ and $\beta_i$ are $$\begin{aligned} \beta_1=\frac{33-2n_f}{12},~~\beta_2=\frac{153-19n_f}{24},\nonumber\\ A^{(1)}=\frac{4}{3},~~A^{(2)}=\frac{67}{9} -\frac{\pi^2}{3}-\frac{10}{27}n_f+\frac{8}{3}\beta_1\mbox{ln}(\frac{1}{2}e^{\gamma_E}),\end{aligned}$$ $n_f$ is the number of the quark flavors and $\gamma_E$ is the Euler constant. We will use the one-loop running coupling constant, i.e. we pick up the four terms in the first line of the expression for the function $s(Q,b)$. [1]{} B. Aubert [*et al.*]{} \[BABAR Collaboration\], Phys. Rev. D [**79**]{}, 052005 (2009) \[arXiv:0901.3703 \[hep-ex\]\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], Phys. Rev. Lett.  [**101**]{}, 161801 (2008) \[arXiv:0806.4419 \[hep-ex\]\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], Phys. Rev. D [**78**]{}, 092008 (2008) \[arXiv:0808.3586 \[hep-ex\]\]. G. Lopez Castro and J. H. Munoz, Phys. Rev. D [**55**]{}, 5581 (1997) \[hep-ph/9702238\]. J. H. Munoz, A. A. Rojas and G. Lopez Castro, Phys. Rev. D [**59**]{}, 077504 (1999) \[hep-ph/9812274\]. C. S. Kim, B. H. Lim and S. Oh, Eur. Phys. J. C [**22**]{}, 695 (2002) \[Erratum-ibid. C [**24**]{}, 665 (2002)\] \[hep-ph/0108054\]. C.S. Kim, J.P. Lee, and S. Oh, Phys. Rev. D [**67**]{},014002(2003). J H Munoz and N Quintero 2009 J. Phys. G: Nucl. Part. Phys. 36 095004. A. Datta, Y. Gao, A. V. Gritsan, D. London, M. Nagashima and A. Szynkman, Phys. Rev. D [**77**]{}, 114025 (2008) \[arXiv:0711.2107 \[hep-ph\]\]. H. -Y. Cheng and K. -C. Yang, Phys. Rev. D [**83**]{}, 034001 (2011) \[arXiv:1010.3309 \[hep-ph\]\]. M. Beneke, G. Buchallla, M. Neubert, and C.T. Sachrajda, Phys. Rev. Lett. [**83**]{},1914 (1999); Nucl. Phys. [**B 591**]{}, 313 (2000); [**B 606**]{}, 245 (2001). Y.Y. Keum, H.-n. Li, and A.I. Sanda, Phys. Rev. D [**63**]{}, 054008 (2001). C. Amsler [*et al.*]{} \[Particle Data Group Collaboration\], Phys. Lett. B [**667**]{}, 1 (2008). D.M. Li, H. Yu, and Q.X. Shen, J. Phys. G [**27**]{},807(2001). H. -Y. Cheng, Y. Koike and K. -C. Yang, Phys. Rev. D [**82**]{}, 054019 (2010) \[arXiv:1007.3541 \[hep-ph\]\]. W. Wang, Phys. Rev. D [**83**]{}, 014008 (2011) \[arXiv:1008.5326 \[hep-ph\]\]. Z. -T. Zou, X. Yu and C. -D. Lu, arXiv:1203.4120 \[hep-ph\]. Z. -T. Zou, Z. Rui and C. -D. Lu, arXiv:1204.3144 \[hep-ph\]. Z. -T. Zou, X. Yu and C. -D. Lu, arXiv:1205.2971 \[hep-ph\]. Z. -T. Zou, X. Yu and C. -D. Lu, arXiv:1208.4252 \[hep-ph\]. S. Descotes-Genon and C. T. Sachrajda, Nucl. Phys.  B [**625**]{}, 239 (2002) \[arXiv:hep-ph/0109260\]. F. Feng, J. P. Ma and Q. Wang, Phys. Lett.  B [**674**]{}, 176 (2009) \[arXiv:0807.0296 \[hep-ph\]\];\ H. n. Li and S. Mishima, Phys. Lett.  B [**674**]{}, 182 (2009) \[arXiv:0808.1526 \[hep-ph\]\];\ F. Feng, J. P. Ma and Q. Wang, Phys. Lett.  B [**677**]{}, 121 (2009) \[arXiv:0808.4017 \[hep-ph\]\]. H. n. Li and S. Mishima, Phys. Rev.  D [**71**]{}, 054025 (2005) \[arXiv:hep-ph/0411146\];\ H. n. Li, S. Mishima and A. I. Sanda, Phys. Rev.  D [**72**]{}, 114005 (2005) \[arXiv:hep-ph/0508041\];\ H. n. Li and S. Mishima, Phys. Rev.  D [**74**]{}, 094020 (2006) \[arXiv:hep-ph/0608277\]. G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys.  [**68**]{}, 1125 (1996) \[arXiv:hep-ph/9512380\]. A. Ali, G. Kramer and C. D. Lu, Phys. Rev.  D [**58**]{}, 094009 (1998) \[arXiv:hep-ph/9804363\]. C. D. Lu and M. Z. Yang, Eur. Phys. J.  C [**28**]{}, 515 (2003) \[arXiv:hep-ph/0212373\]. M. Bauer and M. Wirbel, Z. Phys.  C [**42**]{}, 671 (1989). Y.-Y. Keum, H.-n. Li and A. I. Sanda, Phys. Lett. B504, 6 (2001); Phys. Rev. D63, 054008 (2001); C.-D. Lü, K. Ukai and M.-Z. Yang, Phys. Rev. D63, 074009 (2001); C.-D. Lü and M.-Z. Yang, Eur. Phys. J. C23, 275-287 (2002);R. H. Li, C. D. Lu and H. Zou, Phys. Rev.  D [**78**]{}, 014018 (2008) \[arXiv:0803.1073 \[hep-ph\]\]; R. H. Li, X. X. Wang, A. I. Sanda and C. D. Lu, Phys. Rev.  D [**81**]{}, 034006 (2010) \[arXiv:0910.1424 \[hep-ph\]\]. A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, Phys. Rev.  D [**76**]{}, 074018 (2007) \[arXiv:hep-ph/0703162\]. P. Ball, G. W. Jones and R. Zwicky, Phys. Rev.  D [**75**]{} (2007) 054004 \[arXiv:hep-ph/0612081\]. P. Ball and R. Zwicky, Phys. Lett.  B [**633**]{}, 289 (2006) \[arXiv:hep-ph/0510338\]. P. Ball and R. Zwicky, Phys. Rev.  D [**71**]{}, 014029 (2005); P. Ball and R. Zwicky, JHEP [**0604**]{}, 046 (2006); P. Ball and G. W. Jones, JHEP [**0703**]{}, 069 (2007). T. M. Aliev and M. A. Shifman, Phys. Lett.  B [**112**]{}, 401 (1982). T. M. Aliev and M. A. Shifman, Sov. J. Nucl. Phys.  [**36**]{} (1982) 891 \[Yad. Fiz.  [**36**]{} (1982) 1532\]. T. M. Aliev, K. Azizi and V. Bashiry, J. Phys. G [**37**]{}, 025001 (2010) \[arXiv:0909.2412 \[hep-ph\]\]. P. Ball and V. M. Braun, Nucl. Phys.  B [**543**]{}, 201 (1999) \[arXiv:hep-ph/9810475\]. P. Ball, V. M. Braun, Y. Koike and K. Tanaka, Nucl. Phys.  B [**529**]{}, 323 (1998) \[arXiv:hep-ph/9802299\]. I.I.Y. Bigi and A.I. Sanda, Cambridge Monogr. Part. Phys., Nucl. Phys., Cosmol. 9, 1(2000); G.C. Branco, L. Lavoura, and J.P. Silva, *CP Violation* (Oxford University Press, Oxford, 1999). [^1]: Tensor mesons with $J^P=2^+$ have recently become one of many hot topics.

--- abstract: 'Superbursts are very energetic Type I X-ray bursts discovered in recent years by long term monitoring of X-ray bursters, and believed to be due to unstable ignition of carbon in the deep ocean of the neutron star. In this Letter, we follow the thermal evolution of the surface layers as they cool following the burst. The resulting light curves agree very well with observations for layer masses in the range $10^{25}$–$10^{26}\ {\rm g}$ expected from ignition calculations, and for an energy release $\gtrsim 10^{17}$ erg per gram during the flash. We show that at late times the cooling flux from the layer decays as a power law $F\propto t^{-4/3}$, giving timescales for quenching of normal Type I bursting of weeks, in good agreement with observational limits. We show that simultaneous modelling of superburst lightcurves and quenching times promises to constrain both the thickness of the fuel layer and the energy deposited.' author: - Andrew Cumming and Jared Macbeth title: The Thermal Evolution following a Superburst on an Accreting Neutron Star --- Introduction ============ Type I X-ray bursts from accreting neutron stars in low mass X-ray binaries involve unstable thermonuclear burning of accreted hydrogen (H) and helium (He) (Lewin, van Paradijs, & Taam 1995). In the last few years, long term monitoring of X-ray bursters by BeppoSAX and the Rossi X-Ray Timing Explorer (RXTE) has revealed a new class of very energetic Type I X-ray bursts, now known as “superbursts” (see Strohmayer & Bildsten 2003; Kuulkers 2003 for reviews). The $10^{42}\ {\rm erg}$ energies and several hour durations of superbursts are $100$–$1000$ times greater than usual Type I bursts. In addition, they are rare: so far 8 have been seen from 7 sources, with recurrence times not well-constrained, but estimated as $\sim 1$ year (Kuulkers 2002; in ’t Zand et al. 2003; Wijnands 2001), instead of hours to days for usual Type I bursts. The current picture is that superbursts are due to unstable ignition of carbon at densities $\rho\sim 10^8$–$10^9\ {\rm g\ cm^{-3}}$. Hydrogen and helium burn at $\rho\sim 10^5$–$10^6\ {\rm g\ cm^{-3}}$ via the rp-process (Wallace & Woosley 1981), producing chiefly heavy elements beyond the iron group (including nuclei as massive as $A = 104$; Schatz et al. 2001), but with some residual carbon (mass fraction $X_C\sim 0.01$–$0.1$) (Schatz et al. 2003). Cumming & Bildsten (2001) (hereafter CB01) showed that this small amount of carbon can ignite unstably once the mass of the ash layer reaches $\sim 10^{25}\ {\rm g}$ (see also Strohmayer & Brown 2002). This fits well with observed superburst energies for $X_C\approx 0.1$ and an energy release from the nuclear burning of 1 MeV per nucleon. The low thermal conductivity of the rp-process ashes gives a large temperature gradient and ignition at the required mass (CB01). The heavy nuclei may also photodisintegrate to iron group during the flash, enhancing the nuclear energy release (Schatz, Bildsten, & Cumming 2003). Therefore superbursts offer an opportunity to study the rp-process ashes. Previous authors used one-zone models to estimate the time-dependence of the flash (CB01; Strohmayer & Brown 2002). In this paper, we present the first multi-zone models of the cooling phase of superbursts. Unlike normal Type I bursts, the time to burn the fuel is much less than the convective turnover time. We therefore assume that the fuel burns locally and instantaneously in place, without significant vertical mixing. We do not calculate ignition conditions, but rather treat the amount of energy deposited and the thickness of the fuel layer as free parameters[^1]. In §2, we describe our calculations of the subsequent thermal evolution of the layer, and present a simple analytic model which helps to understand the numerical results. At late times, the flux evolves as a power law in time rather than the exponential decay found by CB01 for a one-zone model. In §3, we use the long term flux evolution of the layer to predict the timescale of quenching of Type I bursts after the superburst, and compare to observations. Time Evolution of the Superburst ================================ After the fuel burns, the cooling of the layer is described by the entropy equation $$c_P {\partial T\over\partial t} = -\epsilon_\nu -{1\over\rho}{\partial F\over \partial r}$$ where the heat flux is $F = -K(\partial T/\partial r)$, and $\epsilon_\nu$ is the neutrino energy loss rate. The layer remains in hydrostatic balance, in which case a useful independent coordinate is the column depth $y$ into the star (units: ${\rm g\ cm^{-2}}$), where $dy = - \rho dr$, giving a pressure $P =gy$. The surface gravity is $g = (GM/R^2)(1 + z)$, where $1 + z = (1 - 2GM/Rc^2)^{-1/2}$ is the gravitational redshift factor. In this paper, we assume $M = 1.4 M_\odot$ and $R = 10 {\rm km}$, giving $z = 0.31$ and $g_{14} = g/10^{14}\ {\rm cm\ s^{-2}} = 2.45$. To find the temperature profile just after the fuel burns, we deposit an energy $E_\mathrm{nuc}=E_{17} 10^{17}\ {\rm erg\ g^{-1}}$ throughout the layer. Since carbon burning to iron gives $\approx 10^{18}\ {\rm erg\ g^{-1}}$, we expect $E_{17}=1$ to correspond to $X_C\approx 0.05$–$0.1$, depending on how much energy is contributed by photodisintegration (Schatz et al. 2003). At each depth, we calculate the temperature of the layer, $T_f$, from $\int^{T_f}_{T_i} c_P dT = E_{\rm nuc}$, where $T_i$ the initial temperature. Following CB01, a simple analytic estimate of $T_f$ is as follows. The electrons are degenerate and relativistic for $\rho\gtrsim 10^7\ {\rm g\ cm^{-3}}$, giving $\rho Y_e = 1.5\times 10^8\ {\rm g\ cm^{-3}}\ P_{26}^{3/4}$, Fermi energy $E_F=2.7\ {\rm MeV}\ P_{26}^{1/4}$, and pressure scale height $H=y/\rho=67\ {\rm m}\ P^{1/4}_{26}Y_e/g_{14}$, where $P=P_{26}\ 10^{26}\ {\rm erg\ cm^{-3}}$. The heat capacity $c_P$ is determined mainly by the electrons, $c_P\approx \pi^2(Y_ek_B/m_p)(k_BT/E_F)= 2.6\times 10^7\ {\rm erg\ g^{-1}\ K^{-1}}\ T_9Y_eP^{-1/4}_{26}$. Integrating and assuming $T_i\ll T_f$, we find the convectively stable temperature profile $T_f = 3.6\times 10^9\ {\rm K}\ E^{1/2}_{17}P^{1/8}_{26}Y_e^{-1/8}$, insensitive to depth and depending mainly on $E_{17}$. Our thermal evolution code uses the method of lines, in which the right hand side of equation (1) is differenced over a spatial grid, and the resulting set of ordinary differential equations integrated using a stiff integrator. We choose a uniform grid in $\sinh^{-1}(\log y/y_b)$, which concentrates grid points around the base of the layer $y_b$, resolving the initial temperature discontinuity. We place the outer boundary at $y = 10^8\ {\rm g\ cm^{-2}}$, and set flux $\propto T^4$ there; at the inner boundary, typically $y\approx 10^{14}\ {\rm g\ cm^{-2}}$, we assume vanishing flux. We assume the layer is heated before the flash by a $10^{21}\ {\rm erg\ cm^{-2}\ s^{-1}}$ flux from the crust, and that all the fuel burns to $^{56}$Fe. Our results are not sensitive to the details of the grid, or boundary conditions (for times longer than the thermal time at the top zone). We calculate the equation of state, opacity, neutrino emissivity, and heat capacity as described by Schatz et al. (2003). Figure 1 shows temperature and flux profiles 10 minutes, 1 hour, 1 day, and 10 days after ignition for a model with $E_{17}=1$ and $y_b = 10^{12}\ {\rm g\ cm^{-2}}$. Figure 2 shows a series of lightcurves for different $y_b$ and $E_{17}$. At early times, as the outer parts of the layer thermally adjust, the radiative flux depends mostly on $E_{17}$. At late times, after the cooling wave reaches the base of the layer, the flux depends mostly on $y_b$, and falls off as a power law $F\propto t^{-4/3}$. Figure 3 shows the cumulative energy release for $y_b = 10^{12}$ and $10^{13}\ {\rm g\ cm^{-2}}$ and $E_{17}=1$–$3$. In the first few hours, the energy released from the surface is $\approx 10^{42}\ {\rm ergs}$, the exact value being mainly sensitive to $E_{17}$, rather than depth. A significant fraction of the heat is initially conducted inwards and released on a longer timescale, as pointed out by Strohmayer & Brown (2002). The physical reason for the late-time power law flux decay is that as time evolves, the peak of the temperature profile moves to greater depths where the thermal timescale to the surface is longer[^2] (see Fig. 1). The simplest analytic model is a slab with constant thermal diffusivity $D$, whose temperature is perturbed close to the surface, for example at a depth $x=a$ (where $x=0$ is the surface). For a delta-function perturbation initially, the temperature evolution is given by the Green’s function[^3] $$T(x,t) = {1\over \sqrt{\pi Dt}} \sinh\left({ax\over 2Dt}\right) \exp\left(-{x^2+a^2\over 4Dt}\right)$$ and the surface flux is $F\propto (\partial T/\partial x)_{x=0}\propto t^{-3/2}\exp(-\tau/t)$, where $\tau$ is the thermal time at the initial heating depth $\tau = 4a^2/D$. For an initial “top hat” temperature profile $T (x < a) = 1$, $T (x > a) = 0$, the surface flux is $F\propto (\tau/t)^{1/2}\left[1-\exp(-\tau /t)\right]$. For $t <\tau$, before the cooling wave reaches the base of the layer, $F\propto t^{-1/2}$; for $t >\tau$, the solution is independent of the initial temperature profile, and $F\propto t^{-3/2}$. The numerical results show a similar behavior, although with different power law indices. The relevant timescale in this case is $t_{\rm cool} = H^2/D$, where $D = K/\rho c_P$. The electron conductivity is $K=\pi^2n_ek^2_BT/3m_\star\nu_c$, where $m_\star = E_F/c^2$, and $\nu_c$ is the electron collision frequency. When electrons dominate the heat capacity, the thermal diffusivity takes the particularly simple temperature-independent form $D = c^2/3\nu_c$. For electron-ion collisions, $\nu_c = 9.3\times 10^{16}\ {\rm s^{-1}}\ P_{26}^{1/4}\langle Z^2/A\rangle\Lambda_{ei}/Y_e$ (e.g. see Appendix of Schatz et al. 1999), giving $$t_{\rm cool} = 3.8\ {\rm hrs}\ y^{3/4}_{12} \left({Y_e\langle Z^2/A\rangle\Lambda_{ei}\over 6}\right) \left({g_{14}\over 2.45}\right)^{-5/4}$$ (see also eq. \[10\] of CB01), where we insert the appropriate numbers for $^{56}$Fe composition. The simple “top hat” solution for constant conductivity motivates a fit to the numerical solutions, $$\label{eq:fit} F_{25} = 0.2\ t_{\rm hr}^{-0.2} \ E_{17}^{7/4} \left[1-\exp\left(-0.63\ t_{\rm cool}^{4/3}E_{17}^{-5/4}t_{\rm hr}^{-1.13}\right)\right],$$ where $t_{\rm hr} = t/1\ {\rm hour}$. For $t > t_{\rm cool}$, $F_{25}=0.13\ E^{1/2}_{17}(t/t_{\rm cool})^{-4/3}$. The transition from $F\propto t^{-0.2}$ to $F\propto t^{-4/3}$ occurs when $t/t_{\rm cool}\approx E_{17}^{-1.1}$. Equation (\[eq:fit\]) fits the numerical results to better than a factor of two for models without substantial neutrino emission. As emphasised by Strohmayer & Brown (2002), neutrino cooling is important for large carbon fractions: it depresses the flux at $t\approx 5$–$10$ hours for the models with $E_{17}=2$ and $3$, $y =10^{13}\ {\rm g\ cm^{-2}}$ in Figure 2. Whenever neutrinos dominate the cooling, the peak temperature is large enough that emission is by pair annhilation. A good fit to the neutrino energy loss rate is $\epsilon_\nu\approx 10^4\ {\rm erg\ g^{-1}\ s^{-1}} T^{12}_9 y^{-3/2}_{12}$, giving a cooling time $t_\nu=c_PT/\epsilon_\nu=2.5\times 10^{12}\ {\rm s}\ y^{5/4}_{12}T^{-10}_9$. Inserting the peak temperature from equation (2) gives $t_\nu\approx 300\ {\rm hrs}\ E^{-5}_{17}$. Neutrinos dominate when $t_\nu<t_{\rm cool}$, or when $E_{17}>2.3\ y^{-3/20}_{12}$. Comparison to Observations ========================== The cooling curves in Figure \[fig:gop3\] compare well with observed lightcurves, including a rapid initial decay on hour timescales, followed by an extended tail of emission (as observed following some superbursts, e.g. KS 1731-260, Kuulkers et al. 2002; Ser X-1, Cornelisse et al. 2002). We will present a detailed comparison with the observed superburst lightcurves in a future paper. The initial decay from the peak depends mostly on $E_{17}$, and so it should be possible to constrain the amount of fuel consumed in the superburst. Our models do not resolve the peak itself, since this depends on the details of how the burning propagates out to the surface; however, for $E_{17}\gtrsim 2$–$3$, the flux exceeds the Eddington flux, $F_{\rm Edd}=3\times 10^{25}\ \mathrm{erg\ cm^{-2}\ s^{-1}}/(1+X)$, where $X$ is the H fraction, for timescales of minutes. Superburst peak luminosities are generally less than the Eddington luminosity (Kuulkers 2003), implying $E_{17}\lesssim 2$. The one exception is the superburst from 4U 1820-30, which showed dramatic photospheric radius expansion lasting for several minutes (Strohmayer & Brown 2002). This is consistent with the proposal that this source, which accretes and burns He rich material, produces large quantities of carbon (Strohmayer & Brown 2002; Cumming 2003a). The transition to the late-time power law occurs after $t\approx 4\ {\rm h}\ E_{17}^{-1.11}y_{12}^{3/4}$ (eq. \[5\]), which corresponds to $F_{25}\approx 0.13 E_{17}^2$. It may therefore be possible to measure the power law decay using superburst tails, although this depends upon being able to subtract out the underlying accretion luminosity, $F_{\rm accr,25}\approx 0.1\ (\dot M/0.1 \dot M_{\rm Edd})$, in a reliable way. Another way to probe the late-time cooling is to use the remarkable observation that Type I bursts disappear (are “quenched”) for $t_{\rm quench}\approx$ weeks following the superburst (e.g. Kuulkers 2003). CB01 proposed that the cooling flux from the superburst temporarily stabilizes the H/He burning. An estimate of the critical stabilizing flux, $F_{\rm crit}$, is as follows. The condition for temperature fluctuations to grow, and unstable He ignition to occur, is $\nu\epsilon_{3\alpha} = \eta\epsilon_{\rm cool}$ (Fushiki & Lamb 1987), where $\epsilon_{3\alpha}$ is the triple alpha ($3\alpha$) energy production rate, $\epsilon_{\rm cool}$ is a local approximation to the cooling rate, and $\nu$ and $\eta$ are the respective temperature sensitivities. For a large flux from below, the He burns stably before reaching this ignition condition, at a depth where the time to accumulate the layer equals the He burning time, $y/\dot m = YQ_{3\alpha}/\epsilon_{3\alpha}$, where $Y$ is the He mass fraction, $\dot m$ is the local accretion rate per unit area, and $Q_{3\alpha} = 5.84\times 10^{17}\ {\rm erg\ g^{-1}} = 0.606$ MeV per nucleon is the $3\alpha$ energy release. At the transition from unstable to stable burning, both criteria are satisfied at the base of the H/He layer. Using the first condition to eliminate $\epsilon_{3\alpha}$ from the second, and writing $\epsilon_{\rm cool}\approx F/y$, gives $F=\nu\dot m Q_{3\alpha}Y/\eta=6.2\times 10^{22}\ {\rm erg\ cm^{-2}\ s^{-1}} (\dot m/\dot m_{\rm Edd})(Y/0.3)(\nu/4\eta)$. Some of this flux is provided by hot CNO burning of accreted H, $F_H\approx\epsilon_Hy=5.8\times 10^{21}\ {\rm erg\ cm^{-2}\ s^{-1}} y_8 (Z/0.01)$ (Cumming & Bildsten 2000; $Z$ is the metallicity); the remainder is $F_{\rm crit}=F-F_H$. This estimate agrees well with a more detailed calculation using the ignition models of Cumming & Bildsten (2000), in which we find $F_{\rm crit}\approx\dot m Q_{3\alpha} \approx 0.7\ {\rm MeV}$ per accreted nucleon, almost independent of $\dot M$. Therefore, $$\label{eq:fcrit} F_{{\rm crit},22}\approx 6\ (\dot m/ \dot m_{\rm Edd})$$ (see also Paczynski 1983a; Bildsten 1995). Equation (5) in the limit $t\gg t_{\rm cool}$ gives $$t_{\rm quench} = 38\ t_{\rm cool}\ F_{{\rm crit},22}^{-3/4} E_{17}^{3/8} = 6\ {\rm days}\ y^{3/4}_{12} F^{-3/4}_{{\rm crit},22} E^{3/8}_{17},$$ which gives $t_\mathrm{quench}$ in terms of the thickness of the layer and energy release. Figure 5 compares the predicted and observed quenching times. The observations of $t_{\rm quench}$ and accretion rates (used to find $F_{\rm crit}$ from eq. \[\[eq:fcrit\]\]) are taken from Kuulkers (2003) (except for 4U 1636-53, which has a revised upper limit of $23$ days, Kuulkers private communication). The observations are upper or lower limits only: nonetheless, the general agreement is very good and supports the quenching picture suggested by CB01. There is much to learn from a careful comparison of superburst lightcurves and the corresponding quenching times, separately constraining both $E_{17}$ and $y_b$. Summary and Conclusions ======================= We have presented the first multi-zone models of the cooling phase of superbursts. The flux decay is not exponential, but power-law (eq. \[5\]). For $t<t_{\rm cool}$, where $t_{\rm cool}$ is the cooling time at the base of the layer, the flux depends mostly on the energy release $E_{17}$, and is insensitive to depth: the inwards travelling cooling wave does not yet know that the layer has a finite thickness. For $t>t_{\rm cool}$, the flux decays as a power law $F\propto t^{-4/3}$, independent of the initial temperature profile. The power law decay at late times gives predicted Type I burst quenching times of weeks (eq. \[6\]), consistent with observational limits. Future comparisons of both superburst lightcurves and quenching times with observations will constrain both the thickness of the fuel layer and the energy deposited, particularly when combined with models of normal Type I bursts from the same source (Cumming 2003a,b). There is still much to be done in terms of theory. Perhaps the most important issues are the physics of the rise (which sets the initial condition for our simulations), and production of the fuel. Important clues to the first are the observed precursors to superbursts, which may be normal Type I bursts ignited by the superburst. Recent progress has been made on the second, with indications from both theory (Schatz et al. 2003; Woosley et al. 2003) and observations (in ’t Zand et al. 2003) that stable burning may be required to produce enough carbon to power superbursts. A self-consistent model of H/He burning, followed by accumulation and ignition of the ashes may require a better understanding of the transition from unstable to stable burning observed in normal Type I bursting (e.g. Cornelisse et al. 2003). We thank P. Arras, E. Brown, R. Cornelisse, E. Kuulkers, G. Ushomirsky, S. Woosley, and J. in’t Zand for useful comments and discussions. AC is supported by NASA through Hubble Fellowship grant HF-01138 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. JM acknowledges support from DOE grant No. DE-FC02-01ER41176 to the Supernova Science Center/UCSC. Bildsten, L. 1995, , 438, 852 Brown, E. F., & Bildsten, L. 1998, , 496, 915 Cornelisse, R., Kuulkers, E., in’t Zand, J. J. M., Verbunt, F., & Heise, J. 2002, , 382, 174 Cornelisse, R. et al. 2003, , 405, 1033 Cumming, A. 2003a, , 595, 1077 Cumming, A. 2003b, in “The Restless High-Energy Universe” (Amsterdam, May 5-8, 2003), ed. E.P.J. van den Heuvel, J.J.M. in ’t Zand, and R.A.M.J. Wijers (astro-ph/0309626) Cumming, A., & Bildsten, L. 2001, , 559, L127 (CB01) Eichler, D., & Cheng, A. F. 1989, , 336, 360 Fushiki, I., & Lamb, D. Q. 1987, , 323, L55 in’t Zand, J. J. M., Kuulkers, E., Verbunt, F., Heise, J., & Cornelisse, R. 2003, , 411, L487 Itoh, N., Hayashi, H., Nishikawa, A., & Kohyama, Y. 1996, , 102, 411 Kuulkers, E., 2002, , 383, L5 Kuulkers, E., 2003, in “The Restless High-Energy Universe” (Amsterdam, May 5-8, 2003), ed. E.P.J. van den Heuvel, J.J.M. in ’t Zand, and R.A.M.J. Wijers (astro-ph/0310402) Kuulkers, E. et al. 2002, , 382, 503 Lewin, W. H. G., van Paradijs, J., & Taam, R. E. 1995, in X-Ray Binaries, ed. W. H. G. Lewin, J. van Paradijs, & E. P. J. van den Heuvel (Cambridge: CUP), 175 Lyubarsky, Y., Eichler, D., & Thompson, C. 2002, , 580, L69 Paczynski, B., 1983, , 264, 282 Paczynski, B., 1983, , 267, 315 Potekhin, A. Y. & Chabrier, G. 2000, , 62, 8554 Sang, Y. & Chanmugam, G. 1987, , 323, L61 Schatz, H., et al. 2001, , 86, 3471 Schatz, H., Bildsten, L., & Cumming, A. 2003, , 583, L87 Schatz, H., Bildsten, L., Cumming, A., & Wiescher, M. 1999, , 524, 1014 Schatz, H., Bildsten, L., Cumming, A., & Ouellette, M. 2003, Nucl. Phys. A, 718, 247 Strohmayer, T. E., & Bildsten, L. 2003, in Compact Stellar X-Ray Sources, eds. W.H.G. Lewin and M. van der Klis (Cambridge: Cambridge University Press) (astro-ph/0301544) Strohmayer, T. E., & Brown, E. F., 2002, , 566, 1045 Urpin, V. A., Chanmugam, G., & Sang, Y. 1994, , 433, 780 Wallace, R. K. & Woosley, S. E. 1981, , 45, 389 Wijnands, R. 2001, , 554, L59 Woosley, S. E. et al. 2003, , in press (astro-ph/0307425) [^1]: This is a similar approach to Eichler & Cheng (1989) who studied the thermal response of a neutron star to energy deposition at different depths. However, the transient events they consider are less energetic than superbursts. [^2]: A similar problem is ohmic decay of crustal magnetic fields, where power law decay is also expected (Sang & Chanmugam 1987; Urpin, Chanmugam, & Sang 1994) [^3]: A simple way to obtain this result is to apply the method of images to the Green’s function for an unbounded domain $T(x,t)\propto t^{-1/2} \exp\left(-x^2/4Dt\right)$. Eichler & Cheng (1989) derive a similar result for a power law dependence of conductivity on depth, which also shows self-similar behavior at late times (see Lyubarsky, Eichler, & Thompson 2002 for a recent application to cooling of SGR 1900+14 after an outburst).

--- abstract: '[The detection of high-energy (HE) $\gamma$-ray emission up to $\sim 3$ GeV from the giant lobes of the radio galaxy Centaurus A has been recently reported by the Fermi-LAT Collaboration based on ten months of all-sky survey observations. A data set more than three times larger is used here to study the morphology and photon spectrum of the lobes with higher statistics. The larger data set results in the detection of HE $\gamma$-ray emission (up to $\sim 6$ GeV) from the lobes with a significance of more than $10$ and $20~\sigma$ for the north and the south lobe, respectively. Based on a detailed spatial analysis and comparison with the associated radio lobes, we report evidence for a substantial extension of the HE $\gamma$-ray emission beyond the WMAP radio image for the northern lobe of Cen A. We reconstructed the spectral energy distribution (SED) of the lobes using radio (WMAP) and Fermi-LAT data from the same integration region. The implications are discussed in the context of hadronic and time-dependent leptonic scenarios.]{}' author: - 'Rui-zhi Yang' - Narek Sahakyan - Emma de Ona Wilhelmi - Felix Aharonian - Frank Rieger title: Deep observation of the giant radio lobes of Centaurus A with the Fermi Large Area Telescope --- Introduction ============ The bright, nearby radio galaxy Centaurus A (Cen A; NGC 5128) has been extensively studied from radio to very-high-energy (VHE) $\gamma$-rays (e.g., see [@israel98; @steinle10] for reviews). Its unique proximity (d$\sim$3.7 Mpc; [@ferrarese07]) and peculiar morphology allow a detailed investigation of the non-thermal acceleration and radiation processes occurring in its active nucleus and its relativistic outflows. At radio frequencies Cen A reveals giant structures, the so-called “lobes”, with a total angular size of $\sim 10^{\circ}$ ([@shain58; @burns83]), corresponding to a physical extension of $\sim 600~{\rm kpc}~ (\rm d/3.7 \rm~ Mpc)$.\ At high-energy (HE; 200 MeV$<\rm E<100$ GeV) Fermi-LAT has recently detected $\gamma$-ray emission from the core (i.e., within ) and the giant radio lobes of Cen A ([@abdo10a; @abdo10b]): An analysis of the available ten-month data set reveals a point-like emission region coincident with the position of the radio core of Cen A, and two large extended emission regions detected with a significance of $5$ and $8\sigma$ for the northern and the southern lobe, respectively. The HE emission from the core extends up to $\sim 10$ GeV and is well described by a power-law function with photon index . It can be successfully interpreted as originating from synchrotron self-Compton (SSC) processes in the innermost part of the relativistic jet. However, a simple extrapolation of the HE core spectrum to the TeV regime tends to under-predict the TeV flux observed by H.E.S.S. ([@aharonian09]), which may indicate an additional contribution related to, e.g. non-thermal magnetospheric processes emerging at the highest energies (see [@rieger11] for review). The extended HE emission regions, on the other hand, seem to be morphologically correlated with the giant radio lobes and contribute more than one-half to the total HE source emission. These regions are again spectrally well described by a power-law function extending up to 2 or 3 GeV with photon indices of $\Gamma\sim 2.6$.\ If the extended HE emission is generated by indeed inverse-Compton up-scattering of CMB and EBL (extragalactic background light) photons, this could offer a unique possibility to spatially map the underlying relativistic electron distribution in this source. The detection of GeV $\gamma$-rays from the radio lobes implies magnetic field strengths ${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}1~\mu$G (e.g., [@abdo10a]). This estimate can be obtained quite straightforwardly from the comparison of radio and $\gamma$-rays, assuming that these radiation components are produced in the same region by the same population of electrons through synchrotron and inverse-Compton processes. In general, however, the radio and the $\gamma$-ray region do not need to coincide. While the radio luminosity depends on the product of the relativistic electron density $\rm N_e$ and the magnetic-field square $\rm B^2$, the inverse-Compton $\gamma$-ray luminosity only depends on $\rm N_e$. This implies that $\gamma$-rays can give us model-independent information about both the energy and the spatial distribution of electrons, while the radio image of synchrotron radiation strongly depends on the magnetic field. As a consequence, the $\gamma$-ray image can be larger than the radio image if the magnetic field drops at the periphery of the region occupied by electrons. This provides one of the motivations for a deeper study of the extended HE (lobe) emission region in Cen A. In principle, X-ray observations could also offer valueable insights into the electron (albeit in a different energy band) and the magnetic field distributions in the lobes (e.g., [@hardcastle05]). Gamma-ray observations, however, bear a certain advantage in that the characteristic cooling timescales for electrons that upscatter CMB photons into the GeV domain is much shorter ($t_c \sim 10^6 [10^6/ \gamma]$ yr), thereby enabling one to trace local conditions much more closely. Since the TeV electrons responsible for GeV $\gamma$-rays cannot move much beyond (${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}$1 kpc) their acceleration points, the very fact of the existence of extended $\gamma$-ray lobes in Cen A implies that we deal with a huge (distributed) 200 kpc-size TeV electron accelerator. Moreover, for a hard uncooled spectrum of low-energy electrons, the inverse Compton emission in the X-ray band will be suppressed. Therefore, an X-ray detection of the very extended lobes is complicated (cf. [@isobe01]). In the present paper we analyze 3 yr of Fermi LAT data, increasing the available observation time by more than a factor of three with respect to the previously reported results. The larger data set allows a detailed investigation of the spectrum and morphology of the lobes with better statistics, especially above $1$ GeV, where the spectral shape may reflect cooling effects and/or maximum energy constraints on the parent population of particles generating the HE $\gamma$-ray emission. We also re-analyze radio data from WMAP ([@page03]) for the same region from where the HE emission is evaluated from, and discuss the implications of the resulting spectral energy distribution (SED) for different emission scenarios.\ The paper is structured as follows. In Sec. 2 the spectral and spatial HE analysis results are described, while the analysis of the WMAP results for the lobes is presented in Sec. 3. Implications for leptonic and hadronic emission models are discussed in Sec. 4 and conclusions are presented in Sec. 5. Fermi-LAT data analysis ======================= The Large Area Telescope (LAT) on board the Fermi $\gamma$-Ray Space Telescope, operating since August 4, 2008, can detect $\gamma-$ray photons with energies in the range between $100$ MeV and a few $100$ GeV. Details about the LAT instrument can be found in Atwood et al. (2009). Here we analyze the field of view (FoV) of Cen A, which includes the bright core and the giant radio lobes. We selected data obtained from the beginning of the operation until November 14, 2011, amounting to $\sim 3$ yr of data (MET 239557417– 342956687). We used the standard LAT analysis software (v9r23p1)[^1]. To avoid systematic errors due to poor determination of the effective area at low energies, we selected only events with energies above 200 MeV. The region-of-interest (ROI) was selected to be a rectangular region of size $14^ \circ \times 14^ \circ$ centered on the position of Cen A ($\rm RA=201^{\circ}21^{\prime}54^{\prime\prime}, \rm DEC=-43^{\circ}1^{\prime}9^{\prime\prime}$). To reduce the effect of Earth albedo backgrounds, time intervals when the Earth was appreciably in the FoV (specifically, when the center of the FoV was more than $52^ \circ$ from zenith) as well as time intervals when parts of the ROI were observed at zenith angles $> 105^\circ$ were also excluded from the analysis. The spectral analysis was performed based on the P7v6 version of the post-launch instrument response functions (IRFs). We modeled the Galactic background component using the LAT standard diffuse background model [*gal\_2yearp7v6\_v0*]{} and we left the overall normalization and index as free parameters. We also used [*iso\_p7v6source*]{} as the isotropic $\gamma$-ray background.\ The resulting Fermi-LAT counts map for the 3 yr data set is shown in Fig. 1(a). The (green) crosses show the position of the point-like sources from the 2FGL catalog ([@abdo11]) within the ROI. The core of Cen A is clearly visible with a test statistic of TS $>$ 800, corresponding to a detection significance of 28 $\sigma$. Extended emission to the north and south of Cen A is detected with significances of TS $>100$ (10 $\sigma$) and TS $>400$ (20 $\sigma$), respectively. Spatial analysis ---------------- Events with energies between $200$ MeV and $30$ GeV were selected. The residual image after subtracting the diffuse background and point-like sources including the core of Cen A is shown in Fig. 1(b). The fluxes and spectral indices of 11 other point-like sources generated from the 2FGL catalog within the ROI are also left as free parameters in the analysis. The 2FGL catalog source positions are shown in Fig. 1(a), where 2FGL J1324.0-4330e accounts for the lobes (both north and south). A new point-like source (2FGL J1335.3-4058), located at $\rm RA=203^{\circ}49^{\prime}30^{\prime\prime}, \rm DEC=-40^{\circ}34^{\prime} 48^{\prime\prime}$ accounts for some residual emission from the north lobe, although no known source at other wavelengths is found to be associated. We treat it as part of the north lobe here. The core of Cen A is modeled as a point-like source. Then the following steps were performed:\ (1) To evaluate the total (extended) HE $\gamma$-ray emission we first used a template based on the residual map (T1; corresponding to the blue contours in Fig. 2). The TS values for the south and the north lobe in this template are 411 and 155, respectively. The residual map was also compared with radio (WMAP, 22 GHz) lobe contours (green contours overlaid on Fig. 1(b)). While lower-frequency radio maps exist, we expect the higher-frequency 22 GHz map to better represent the GeV-emitting particles. We find that the south lobe of the HE $\gamma$-ray image is similar to the south lobe of the radio one, whereas the HE emission in the north extends beyond the radio lobe emission region.\ (2) To understand this feature better, we re-fitted the excess using an additional template (T2; red contours in Fig. 3) generated from the radio (WMAP) image. The two templates are shown in Fig. 2, and the corresponding residual maps are shown in Fig. 1. While there is some residual emission to the north of Cen A for template T2, this residual emission is obviously absent from template T1. The qualitative features of the different residual maps are confirmed by the corresponding TS values, which are listed in Table 1. Accordingly, the HE south lobe seems to agree reasonably well with the radio south lobe, whereas for the north lobe, the template generated from the radio lobe (T2) fits the HE excess substantially worse than T1 (110 vs 155). Model north Lobe south Lobe ------- ------------ ------------ -- T1 155 411 T2 110 406 : TS value for the two templates used.[]{data-label="tab:1"} \(3) To further investigate a possible extension (or contribution of a background source) of the north lobe, we evaluated the projection of a rectangular region on the excess image (in white in Fig. 1(b)). Fig. 3 shows the projection for the north and south regions for Fermi-LAT image (in black) and the radio one (in red). The south projection for the radio map is well-fitted by a single Gaussian centered at $\sim 0.05$ (0 is defined as the center of the rectangle on $RA=201^{\circ}21^{\prime}54^{\prime\prime}, DEC=-43^{\circ}1^{\prime} 9^{\prime\prime}$) with an extension of $\sigma = 0.99^{\circ}$. For the Fermi-LAT map the Gaussian is centered at $\sim 0.5$ and has $\sigma = 1.01^{\circ}$, compatible with the radio map projection. In contrast, the north projection for the Fermi-LAT map has a Gaussian profile with $\sigma = 1.68^{\circ}$, while for the radio map $\sigma$ is $0.97^{\circ}$. The extension in the north projections for the Fermi-LAT map indicates that the $\gamma$-ray north lobe is more extended than the radio one or that an (otherwise unknown) source in the background may be contributing to the total emission.\ Spectral analysis ----------------- Our morphological analysis indicates some incongruity between the morphology of the radio lobe and $\gamma$-ray lobe in the north. Hence, to model the $\gamma$-ray lobe as self-consistently as possible, we used the template generated with the residual map (T1). Integrating the whole $\gamma$-ray emission observed, we then derived the total flux and index in the $100$ MeV to $30$ GeV energy range. For the north lobe the integral HE flux is $(0.93\pm 0.09) \times 10^{-7} \rm ph~cm^{-2}s^{-1}$ and the photon index is $2.24\pm 0.08$, while for the south lobe we find $(1.4\pm 0.2) \times 10^{-7} \rm ph~cm^{-2}s^{-1}$ and $2.57\pm 0.07$, respectively. The core region of Cen A has a flux $(1.4\pm 0.2) \times 10^{-7} \rm ph~cm^{-2}s^{-1}$ while the photon index is $2.7 \pm 0.1$. The results are summarized in Table 2, where the subscripts 3a and 10m refer to the three-year data (analyzed here) and the ten-month data ( reported in Abdo et al. 2010a), respectively. We find that the flux and photon indices in the T2 templates are similar to the ten-month data. On the other hand, the analysis using the T1 template results in a harder spectrum for the north lobe.\ To derive the spectral energy distribution (SED) we divided the energy range into logarithmically spaced bands and applied [*gtlike*]{} in each of these bands. Only the energy bins for which a signal was detected with a significance of at least $2 \sigma$ wereconsidered, while an upper limit was calculated for those below. As a result, there are seven bins in the SED for the south lobe. The SED is shown in Fig. 4.\ To clarify the origin of the $\gamma$-ray emission, we evaluated the spectrum in different parts of each lobe. To this end, we divided each lobe into two parts (see Fig. 5) and used [*gtlike*]{} to evaluate the spectrum. In the south lobe the resulting photon index is $2.8\pm 0.2$ near the Cen A core and $2.3\pm 0.1$ far away from the core. Unfortunately, the statistics are still not high enough to claim a clear hardening of the spectrum. For the northern lobe, both parts appear to be consistent with values of $2.2\pm 0.2$. The spectra for the four regions are shown in Fig. 6 and Fig. 7. Source Name $\Phi_{3a}(\rm T1)$ $\Gamma_{3a}(\rm T1)$ $\Phi_{10m}$ $\Gamma_{10m}$ $\Phi_{3a}(\rm T2)$ $\Gamma_{3a}(\rm T2)$ ------------- --------------------- ----------------------- ---------------- ---------------- --------------------- ----------------------- South Lobe $1.43\pm 0.15$ $2.57\pm 0.07$ $1.09\pm 0.24$ $2.60\pm 0.15$ $1.40\pm 0.15$ $2.56\pm 0.08$ North Lobe $0.93\pm 0.09$ $2.24\pm 0.08$ $0.77\pm 0.20$ $2.52\pm 0.16$ $0.64\pm 0.15$ $2.56\pm 0.08$ : $\Phi$ is the integral flux (100 MeV to 30 GeV) in units of $10^{-7} \rm ph \cdot cm^{-2}s^{-1}$ and $\Gamma$ is the photon index. The subscripts “3a” and “10m” refer to the three-year data analyzed here and to the ten-month results (based on a WMAP template) reported in Abdo et al. (2010a), respectively. \ WMAP data ========= In Sec. 2 we found evidence that the HE $\gamma$-ray emission regions do not coincide well with the radio lobes, especially for the north lobe. To correctly compare the $\gamma$-ray emission with the radio emission, an analysis of the radio data for the same region is required, rather than simply for the north radio lobe itself. We performed this analysis following the method described in Hardcastle et al. (2009). Seven years of WMAP data were analyzed ([@komatsu11]). The WMAP maps in all five bands are convolved with a $0.83^{\circ}$ Gaussian to obtain a similar resolution for all bands. The internal linear combination (ILC) cosmic microwave background (CMB) map was treated as background and subtracted from all maps. The intensity maps (in $\rm mK$) were converted to flux maps (in $\rm Jy/beam$) ([@page03]) and integrated in the region defined by the north $\gamma$-ray lobe to obtain the total flux. Flux values almost twice as high were obtained when the $\gamma$-ray region was used. To cross-check our analysis method, we also derived the flux of the north radio lobe for the same region as used in Hardcastle et al. (2009). All results are summarized in Tab. 4. Our results for the north radio lobe are compatible within errors with those obtained by Hardcastle et al. (2009) (shown in parenthesis in Tab. 3). band north $\gamma$-ray Lobe north radio Lobe ------------ ------------------------- --------------------------------- K(22.5GHz) $5.34\pm0.37$ $2.71\pm0.19$ ($2.61\pm 0.25$) Ka(33GHz) $5.24\pm 0.41$ $2.55\pm 0.22$ ($2.50\pm 0.24$) Q(41GHz) $5.53\pm 0.56$ $2.94\pm0.29$ ($2.58\pm 0.25$) V(61GHz) $7.65\pm 1.19$ $<4.13$ W(94GHz) $<25.6$ $<9.43$ : Radio flux for the north lobe (in $10^{11}\rm Jy\cdot Hz$) from WMAP data.[]{data-label="tab:4"} Note that in the V and the W band the signal is detected at less than the 5 $\sigma$ level, so we only give upper limits here. The origin of the non-thermal lobe emission =========================================== Using the WMAP and Fermi-LAT results reported here, we can characterize the spectral energy distributions for the north and the south lobe. While the radio emission is usually taken to be caused by electron synchrotron emission, the origin of the HE $\gamma$-ray emission could in principle be related to both leptonic (inverse-Compton scattering) and hadronic (e.g., pp-interaction) processes. In the following we discuss possible constraints for the underlying radiation mechanism as imposed by the observed SEDs. Inverse-Compton origin of $\gamma$-rays --------------------------------------- Both the HE $\gamma$-ray and the radio emission could be accounted for in a leptonic scenario. In the simplest version, a single population of electrons $N(\gamma,t)$ is used to model the SED through synchrotron and inverse-Compton emission, with particle acceleration being implicitly treated by an effective injection term $Q=Q(\gamma,t)$. The latter allows us to distinguish acceleration caused by, e.g., multiple shocks or stochastic processes (e.g., [@osullivan09]) from emission, and enables a straightforward interpretation. The kinetic equation describing the energetic and temporal evolution of the radiating electrons can then be written as $$\frac{\partial N}{\partial t}=\frac{\partial}{\partial \gamma}\left(P\:N \right)-\frac{N}{\tau_{esc}}+Q\:,$$ where $P=P(\gamma)=-\frac{d\gamma}{dt}$ is the (time-independent) energy loss rate and $\tau_{\rm esc}$ is the characteristic escape time. For negligible escape (as appropriate here, given the large size of the $\gamma$-ray emitting region) and quasi continuous injection (considered as a suitable first-order approximation given the short lifetime of TeV electrons and the scales of the lobes) $Q(\gamma,t)=Q(\gamma)$, the solution of the kinetic equation becomes $$\label{Nsolution} N(\gamma,t)=\frac{1}{P(\gamma)} \int^{\gamma_{0}}_{\gamma}Q(\gamma)d\gamma\;,$$ where $\gamma_{0}$ is found by solving the characteristic equation for a given epoch $t$, $t=\int^{\gamma_{0}}_{\gamma} \frac{d\gamma}{P(\gamma)}$ (e.g., [@atoyan99]). If synchrotron or inverse-Compton (Thomson) losses ($P(\gamma)=a\: \gamma^{2}$) provide the dominant loss channel, then $\gamma_0=\gamma/(1-a\gamma t)$, so that at the energy $\gamma_{\rm br} =\frac{1}{a\: t}$ the stationary power-law electron injection spectrum steepens by a factor of $1$ (cooling break) due to radiative losses, naturally generating a broken power-law.\ We used the above particle distribution described in eq. (\[Nsolution\]) for a representation of the observed lobe SEDs. The magnetic field strength B, the maximum electron energy $\gamma_{\rm max}$ and the epoch time t were left as free parameters to model the data. Klein-Nishina (KN) effects on the inverse-Compton-scattered HE spectrum were taken into account (following [@aharonian81]).\ Fig. 8 shows the SED results obtained for the north and south lobes. The HE part of both spectra can be described by a power-law with photon index $\Gamma_{\gamma} \simeq 2.2$ and $2.6$ for the north and the south lobe, respectively. At low energies, the south lobe spectrum shows a synchrotron peak at about $5$ GHz, while the north lobe is well described by a power-law with an index $>2$. Note that if one would use a simple power-law electron injection spectrum $Q(\gamma) \propto \gamma^{-\alpha}$, evolving in time with a cooling break, to describe the HE $\gamma$-ray spectrum, a power index $\alpha=3.2$ would be required for the south lobe. Yet, assuming that the same electron population is responsible for both the radio-synchrotron and HE inverse-Compton emission, such a value would be in conflict with the results obtained from the WMAP data analysis, indicating an electron population with power-law index $\alpha\simeq 2$ based on the detected synchrotron emission. As it turns out, however, this problem could be accommodated by considering a more natural spectral input shape, e.g., an electron injection spectrum with an exponential cut-off $$Q(\gamma)=Q_{0}\:\gamma^{-\alpha}\exp\left(-\frac{\gamma}{\gamma_{\rm max}}\right)\:,$$ where the constant $Q_{0}$ can be obtained from the normalization to the injection power $L=m_ec^{2} \int Q(\gamma)\; \gamma\;d\gamma$.\ The age of the giant lobe emission, and associated with this, the duration of particle acceleration activity, is somewhat uncertain. Dynamical arguments suggest a lower limit $>10^7$ yr for the giant radio lobes, while synchrotron spectral aging arguments indicate an age ${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}3 \times 10^7$ yr (e.g., [@israel98; @alvarez00; @hardcastle09]). The observed GeV extension in itself would already imply an extreme lower limit of $R/c >10^6$ yr. In the following we therefore discuss the SED implications for an epoch time $t$ between $10^{7}$ yr and $10^{8}$ yr. As it turns out, the modeling of the GeV data provides support for a maximum lobe age of $\sim 8 \times 10^7$ yr.\ For the south lobe, the radio data suggest a break frequency $\nu_{\rm br}=5$ GHz above which the spectrum drops abruptly. The break in the synchrotron spectrum is related to the break in the electron spectrum via $\nu_{ \rm br}=1.3\:\gamma_{\rm br}^{2}\: B_{1\mu G}$ Hz. In principle, a change in the spectral shape of the electron population might be caused by cooling effects or/and the existence of a maximum energy for the electron population. For a minimum epoch time $t_{\rm min}=10^{7}$ yr, cooling would affect the synchrotron spectrum at frequency $\approx 80 B_{1\mu G}$ GHz, much higher than inferred from the radio data. Therefore, to obtain a break at $5$ GHz in the south lobe, a maximum energy in the electron population ($\gamma_{\rm max}$), lower than $\gamma_{\rm br}$ defined by $t=t_{\rm min}$ would be needed. On the other hand, for a maximum epoch time $t_{\rm max}=8\times 10^{7}$ yr, the power-law spectral index changes at frequency $\simeq 1 B_{1\mu G}$ GHz, providing a satisfactory agreement with the radio data. In this case the maximum electron energy is obtained from the radio data above the break frequency 5 GHz. Results for the considered minimum and maximum epoch time, and for a fixed power-law electron index $\alpha=2$ are illustrated in Figs. 8 and 9. Note that for $B\leq 3 \mu$G, the energy loss rate $P(\gamma)$ is dominated by the IC channel only, so that the results of the calculations are quite robust.\ Figure 8 shows a representation of the SED for the north and the south lobe, respectively, using the parameters $t_{\rm min} =10^{7}$ yr and $\gamma_{\rm max}=1.5\times 10^5$. The dashed line shows the HE contribution produced by inverse-Compton scattering of cosmic microwave background photons by relativistic electrons within the lobes. In this case the resulting $\gamma$-ray flux can only describe the first two data points and then drops rapidly. Consequently, to be able to account for the observed HE spectrum, extragalactic background light photons need to be included in addition to CMB photons (see dot-dashed line in Figure 8 ). Upscattering of infrared-to-optical EBL photons was already required in the stationary leptonic model reported in the original Fermi paper ([@abdo10a]). In our approach we adopt the model by [@franceschini08] to evaluate this EBL contribution. The solid line in Fig. 8 represents the total (CMB+EBL) inverse-Compton contribution. The maximum total energy of electrons in both lobes is found to be $\sim 2\times10^{58}$ erg and the energy in the magnetic fields is roughly $25\%$ of this. Dividing the total energy by the epoch time $10^7$ yr would imply a mean kinetic power of the jets inflating the lobes of $\simeq 7\times 10^{43}$ erg/s, roughly two orders of magnitude lower than the Eddington power inferred for the black hole mass in Cen A, yet somewhat above the estimated power of the kpc-scale jet in the current epoch of jet activity ([@croston09]). This could indicate that the jet was more powerful in the past. Obviously, the requirement on the mean jet power can be significantly reduced if one assumes an older age of the lobes.\ Figure 9 shows a representation of the SED for an epoch time $t_{\rm max}=8 \times 10^{7}$ yr, with a maximum electron Lorentz factor $\gamma_{\rm max}= 2.5 \times 10^{6}$ and $1.5 \times 10^{6}$ for the north lobe and the south lobe, respectively. Note that in this case the contribution by inverse-Compton scattering of CMB photons alone is sufficient to account for the observed HE spectrum (see the solid line in Figure 9). The inverse-Compton contribution of EBL photons only becomes important at higher energies (see the dot-dashed line in Fig. 9). On the other hand, for an epoch time $t$ exceeding $t_{\rm max}=8\times 10^{7}$ yr, the high-energy part of the SED would no longer be consistent with the data (see the dashed line in Fig. 9 for $t=10^{8}$ yr). This could be interpreted as additional evidence for a finite age $< 10^8$ yr of the lobes. The maximum total energy of electrons in both lobes is found to be $\approx6\times10^{57}$ erg, with the total energy in particles and fields comparable to the $10^7$yr-case, thus requiring only a relatively modest mean kinetic jet power of $\sim 10^{43}$ erg/s. Hadronic $\gamma$-rays? ----------------------- Once protons are efficiently injected, they are likely to remain energetic since the cooling time for pp-interactions is $t_{\rm pp} \approx 10^{15}(n/1\;cm^{-3})^{-1}$ s. High-energy protons interacting with the ambient low-density plasma can then produce daughter mesons and the $\pi^{0}$ component decays into two $\gamma$-rays. The data reported here allow us to derive an upper limit on the energetic protons contained in the lobes of Cen A. As before, we use a power-law proton distribution with an exponential cut-off, i.e., $$N(\gamma_p)=N_{0}\;\gamma_p^{-\alpha } \exp\left(-\frac{\gamma_p}{\gamma_{\rm max}}\right)$$ where the constant $N_{0}$ can be expressed in terms of the total proton energy $W_{p}=m_p c^2 \int \gamma_p\; N(\gamma_p)\:d\gamma_p$. Current estimates for the thermal plasma density in the giant radio lobes of Cen A suggest a value in the range $n \simeq (10^{-5}-10^{-4})$ cm$^{-3}$ (e.g., [@isobe01; @feain09]). We used $n=10^{-4}$ cm$^{-3}$ for the model representation shown in dotted line in Fig. 9. In both lobes, the power-law index of the proton population is $\alpha=2.1$, and the high-energy cut-off is $E_{\rm max} \simeq 55$ GeV. The maximum total energy $W_{p}$ is proportional to the gas number density $n$, so that $W_{p} \simeq 10^{61}(n/10^{-4}\;{\rm cm}^{-3})^{-1}$ erg, obtained here, should be considered as an upper limit. In principle, protons could be accumulated over the whole evolutionary timescale of the lobes. For a long timescale of $\geq 10^9$ yr, an average injection power $\leq 3 \times 10^{44}$ erg/s and a mean cosmic-ray diffusion coefficient of $D \sim R^2/t {\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}3 \times 10^{30} (R/100~\rm{kpc})^2$ cm$^2$/s would be needed. Discussion and conclusion ========================= Results based on an detailed analysis of 3 yr of Fermi-LAT data on the giant radio lobes of Cen A are described in this paper. We have shown that the detection of the HE lobes with a significance more than twice as high as reported before (i.e., with more than $10$ and $20 \sigma$ for the northern and the southern lobe, respectively) allows a better determination of their spectral features and morphology. A comparison of the Fermi-LAT data with WMAP data indicates that the HE $\gamma$-ray emission regions do not fully coincide with the radio lobes. There is of course no a priori reason for them to coincide. The results reported here particularly support a substantial HE $\gamma$-ray extension beyond the WMAP radio image for the northern lobe of Cen A. We have reconstructed the SED based on data from the same emission region. A satisfactory representation is possible in a time-dependent leptonic scenario with radiative cooling taken into account self-consistently and injection described by a single power-law with exponential cut-off. The results imply a finite age $<10^8$ yr of the lobes and a mean magnetic field strength $B{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}1 \mu$G. While for lobe lifetimes on the order of $8\times 10^7$ yr, inverse-Compton up-scattering of CMB photons alone would be sufficient to account for the observed HE spectrum, up-scattering of EBL photons is needed for shorter lobe lifetimes. In a leptonic framework, the HE emission directly traces (via inverse-Compton scattering) the underlying relativistic electron distribution and thereby provides a spatial diagnostic tool. The radio emission, arising from synchrotron radiation, on the other hand also traces the magnetic field distribution. Together, the HE $\gamma$-ray and the radio emission thus offer important insights into the physical conditions of the source. That the HE emission seems extended beyond the radio image could then be interpreted as caused by a change in the magnetic field characterizing the region. This would imply that our quasi-homogeneous SED model for the HE lobes can only serve as a first-order approximation and that more detailed scenarios need to be constructed to fully describe the data. This also applies to the need of incorporating electon re-acceleration self-consistently. Extended HE emission could in principle also be related to a contribution from hadronic processes. The cooling timescales for protons appear much more favorable. On the other hand, both the spectral shape of the lobes and the required energetics seem to disfavor pp-interaction processes as sole contributor. One of the insights emerging from the present paper is the need for a more detailed theoretical SED approach to be able to take full advantage of the current observational capabilities. Constructive comments by the anonymous referee are gratefully acknowledged. {width="120mm"} {width="120mm"} {width="120mm"} \[h!!\] [{width="85mm"}]{} [{width="85mm"}]{}\ \[h!!\] [{width="85mm"}]{} [{width="85mm"}]{} [99]{} Abdo, A.A. et al. (Fermi-LAT) 2010a, Science, 328, 725 Abdo, A.A. et al. (Fermi-LAT) 2010b, ApJ, 719, 1433 Abdo, A.A. et al. (Fermi-LAT) 2011, arXiv:1108.1435 \[astro-ph.HE\]. Aharonian, F.A., and Atoyan, A.M. 1981, Ap & SS, 79, 321 Aharonian, F. et al. (H.E.S.S. Collaboration) 2009, ApJ, 695, L40 Alvarez, H. et al. 2000, A&A, 355, 863 Atoyan, A.M., and Aharonian, F.A. 1999 MNRAS, 302, 253 Atwood, W.B. et al. (Fermi-LAT Collaboration) 2009, ApJ, 697, 1071 Burns, J.O. et al. 1983, ApJ, 273, 128 Croston, J.H. et al. 2009, MNRAS, 395, 1999 Feain, I.J. et al. 2009, ApJ, 707, 114 Ferrarese, L. et al. 2007, ApJ, 654, 186 Franceschini A., Rodighiero G., Vaccari M. 2008, Astron. Astrophys. 487, 837 Hardcastle, M.J., and Croston, J.H. 2005, MNRAS, 363, 649 Hardcastle, M.J., Cheung, C.C., Feain, I.J., Stawarz, L. 2009, MNRAS, 393, 104 Isobe, N. et al. 2001, ASPC, 250, 394 Israel, F.P. 1998, A&ARv 8, 237 O’Sullivan, S., Reville, B., Taylor, A. 2009, MNRAS, 400, 248 Page, L. et al. 2003, ApJS, 148, 39 Komatsu, E. et al. (WMAP Collaboration) 2011 ApJS, 192, 18 Rieger, F.M. 2011, IJMPD 20, 1547 Shain, C.A. 1958, Aust. J. Phys. 11, 517 Steinle, H. 2010, PASA, 27, 431 http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/ http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Pass7\_usage.html [^1]: http://fermi.gsfc.nasa.gov/ssc

--- abstract: 'The extra space paradigm plays a significant role in modern physics and in cosmology as the specific case. In this review, the relation between the main cosmological parameters - Planck mass and Cosmological constants - and a metric of extra space is discussed. Matter distribution inside extra space and its effect on the 4-dimensional observational parameters is of particular interest. The ways to solve the Fine-tuning problem and the Hierarchy problem are analyzed.' author: -   bibliography: - 'Ru-Article.bib' date: June 2019 title: Cosmology and matter induced branes --- Introduction ============ The Cosmology encompasses physical laws at all distances. The impressive interpenetration of microphysics and physics at extremely large distances has been noticed some time ago [@1981RvMP...53....1D] and is discussed up to now [@KhlopovRubin]. The origin of the physical parameters like masses and coupling constants is the matter of future theory, but there are two general facts that are worth discussing. The first one relates to the smallness of all observable parameters as compared to the Planck mass (Hierarchy problem). The second one is known as the Fine-tuning problem [@2007unmu.book..231D; @Page:2003zm]. The cosmological constant smallness is its most amazing illustration. Even more serious problem concerns the Anthropic observational fact - the increasing of the cosmological constant value in several times leads to a serious variation of our Universe structure so that we, observers would not exist. This is the particular case the general fact of the Fine-tuning of our Universe. The Planck mass and the cosmological constant seem to be “more fundamental” and important for different cosmological research. These two parameters and their dependence on an extra space structure are used throughout the text as the illustration of the discussion. **The Planck mass** is one of the natural units introduced by Max Planck. It is connected to the Newton constant $G_N$ as $M_P=\sqrt{8\pi/G_N}, \, (h=c=1)$. Its value is known with poor accuracy that is the reason for a variety of speculation on its origin and time variation. For example, the authors of the paper [@vandeBruck:2015gsa] consider the Planck mass depending on a scalar field that tends to constant shortly after the inflation. The Planck mass is in many orders of magnitude greater the electroweak scale. This puzzle is known as the Hierarchy problem is not clarified yet. The standard Einstein-Hilbert gravity with the Lambda term $\Lambda$ is described by the action $$\label{EH} S=\int d^4 x \left[\frac{1}{16\pi G_N}R-\Lambda\right], M_P^2 = \frac{1}{8\pi G_N}.$$ Here $R$ is the Ricci scalar of our 4-dim space and $\Lambda$ is the cosmological constant. The Newton constant does not vary with time by definition. There are research studied its possible slow time variation [@Bronnikov:2003rf], but we do not discuss this direction here. Action is firmly confirmed at the energies lower than $\sim 10$TeV. A variety of modified gravitational actions are studied at higher scales, [@Gogberashvili:1998iu; @Lyakhova:2018zsr]. The necessary condition for all of them is the reduction to the standard form at the low energies. We will see how does it work for multidimensional models. The vacuum energy is connected to **the cosmological constant** (the $\Lambda$ term) which is included now into the Standard Cosmological Model. The substantial review may be found in [@Sahni:1999gb]. Common opinion nowadays is that this energy is positive and extremely small which leads to several consequences. First, our Universe is expanding with acceleration; second, the modern horizon is shrinking with time and third, the large scale structure was formed under the strong influence of the positive energy density uniformly distributed in the space. There are two riddles related to the $\Lambda$ term. One of them is its smallness which can not be explained by now. Another one is the coincidence problem: the energy density of usual matter distributed at large scales is very close to the vacuum energy density nowadays. A substantial amount of attempts to solve the problems mentioned above is based on the idea of the extra dimensions [@1999NuPhB.537...47D]. The extra dimensions ==================== In modern physics, the idea of extra dimensions is used for explanation of a variety of phenomena. It is applied for the elaboration of physics beyond the Standard Model, cosmological scenarios including inflationary models and the origin of the dark component of the Universe (dark matter and energy), the number of fermion generations and so on. Gradually, this direction becomes the main element for a future theory. Sometimes extra dimensions are endowed by scalar fields and form fields to stabilize their metric. There are models where the Cazimir effect is attracted for the same reason [@Fischbach:2009ud], [@2018GrCo...24..154B]. Our experience indicates that we live in the 4-dimensional world so that a mechanism to hide the extra dimensions is a necessary element of each model. Let us describe some models focusing on the Planck mass, the Lambda term and the two related problems mentioned at the beginning of the Introduction. From D dimensions to 4 dimensions. General remark ------------------------------------------------- Necessary element of all multidimensional models is a reduction of a D-dimensional action to an effective 4-dimensional form: $$\label{reduction} \int d^DX\sqrt{|g_D|} L_D (\alpha_D,g_D)\rightarrow \int d^4x\sqrt{|g_4|} L_4 (\alpha_4,g_4)$$Here $\alpha_D$ is a set of $D-$dim parameters and $\alpha_4$ is general notation for the observable parameters like masses, coupling constant. The Planck mass $M_P$ and the cosmological constant $\Lambda$ are of particular interest. Fields dependence is assumed but not shown explicitly in . An extra dimensional part $g_{extra}$ of the $D-$dimensional metric $g_D$ is hidden in the 4-dimensional parameters $\alpha_4$ so that $$\label{extra} \alpha_4 =\alpha_4(\alpha_D, g_{extra}).$$ A variety of observational parameters can be obtained by a variation of the extra space metrics. This remark is important for further discussion. Evidently, formula relates to the Planck mass and the cosmological constant as well. The Planck mass and the extra space structure --------------------------------------------- Kaluza-Klein model {#kaluza-klein-model .unnumbered} ------------------ The action for this model has the form $$\label{KK} S_g=\int d^4 xd^ny\sqrt{|g|} \left[\frac{m_D^{D-2}}{2}R-\Lambda\right].$$ After integration out the extra dimensional coordinates $y$ we obtain effective action with the Planck mass related to the $D$-dimensional Planck mass $m_D$. The relation $$\label{MPKK} M_P^2 = m_D^{D-2}v_n$$ is the simplest realization of formula . Here $v_n$ stands for an extra space volume, $n=D-4$. Classical behavior of the system is possible if the inequality $v_n^{1/n}>1/m_D$ and hence $m_D<M_P$ take place. The latter is usually assumed, but it is optional, as we will see later. It is supposed that the fluctuations of known fields within the extra dimensions are very massive and can not be excited in the course of low energy processes. Hyperbolic extra dimensions {#hyperbolic-extra-dimensions .unnumbered} --------------------------- The conclusion on the size of the extra dimensions made above assumed the constant positive curvature of this extra space. The more encouraging result can be obtained if we attract a constant negative metric. In this case, there is no rigid connection between the Ricci scalar and a characteristic size $L$ of a compact hyperbolic space which is the significant feature of such spaces [@2002PhRvD..66d5029N]. The volume of such manifold is $$v_n=r_c^n e^{\alpha}, \quad \alpha \simeq (n-1)L/r_c$$ where $r_c$ is the curvature radius and $L$ is the size of extra space which is not a Lagrangian parameter but an accidental value. The Planck mass exponentially depends on the independent linear size $$\label{MP2} M^2 _{P}=m_D ^{n+2}v_n\simeq m_D ^{n+2}r_c^ne^{(n-1)L/r_c}$$ and hence can be sufficiently large even if the Lagrangian parameters are fixed. f(R) theories {#fR .unnumbered} ------------- Nowadays, the $f(R)$ theories of gravity or more generally the theories with higher derivatives are widely used as the tool for the theoretical research. The interest in $f(R)$ theories is motivated by inflationary scenarios starting with the pioneering work of Starobinsky [@Starobinsky:1980te]. A number of viable $f(R)$ models in 4-dim space that satisfies the observable constraints are discussed in Refs. [@2014JCAP...01..008B; @2007PhLB..651..224N; @Sokolowski:2007rd] Consider the gravity with higher order derivatives and the action in the form, $$\begin{aligned} \label{act1} && S=\frac{m_D ^{D-2}}{2}\int d^{D}Z \sqrt{|g_D|}f(R) \end{aligned}$$ The metric is assumed to be the direct product $M_4\times V_n$ of the 4-dim space $M_4$ and $n$-dim compact space $V_n$ $$\label{metric} ds^2 =g_{6,AB}dz^A dz^B = \eta_{4,\mu\nu}dx^{\mu}dx^{\mu} + g_{n,ab}(y)dy^a dy^b.$$ Here $\eta_{4,\mu\nu}$ is the Minkowski metric of the manifolds $M_4$ and $g_{n,ab}(y)$ is metric of the manifolds $V_n$. $x$ and $y$ are the coordinates of the subspaces $M_4$ and $V_n$. We will refer to 4-dim space $M_4$ and $n$-dim compact space $V_n$ as the main space and an extra space respectively. The metric has the signature (+ - - - ...), the Greek indexes $\mu, \nu =0,1,2,3$ refer to 4-dimensional coordinates. Latin indexes run over $a,b = 4, 5,...$. According to , the Ricci scalar represents a simple sum of the Ricci scalar of the main space and the Ricci scalar of extra space $$R=R_4 + R_n .$$ In this subsection, the extra space is assumed to be maximally symmetric so that its Ricci scalar $R_n=const$. In the following, natural inequality $$\label{ll} R_4 \ll R_n$$ is assumed. This suggestion looks natural for the extra space size $L_n < 10^{-18}$ cm if one compares it to the Schwarzschild radius $L_n \ll r_g \sim 10^6$cm of stellar mass black hole where the largest curvature in the modern Universe exists. Below we follow the method developed in [@Bronnikov:2005iz] Using inequality the Taylor expansion of $f(R)$ in Eq. \[act1\] gives $$\begin{aligned} \label{act2} && S= \frac{m_D ^{D-2}}{2}\int d^4 x d^n y \sqrt{|g_4(x)|} \sqrt{|g_n(y)|} f(R_4 + R_n )\\ && \simeq \frac{m_D ^{D-2}}{2}\int d^4 x d^n y\sqrt{|g_4(x)|} \sqrt{|g_n(y)|}[ R_4(x) f' (R_n) + f(R_n)] \nonumber %&& = \int d^4x \sqrt{g(x)}\left[\frac{M^2 _{Pl}}{2}R_4 + \frac{m_D ^{D-2}}{2}\int d^n y \sqrt{G(y)} \left( f(R_n)+L_m \right)\right] . \end{aligned}$$ The prime denotes the derivation of function on its argument. Thus, $f'(R)$ stands for $df/dR$ in the formula written above. In this paper a stationary and uniform distribution of the matter fields in the 4-dimensional part of our Universe is assumed. Comparison of the second line in expression with the Einstein-Hilbert action $$S_{EH}=\frac{M^2_{P}}{2} \int d^4x \sqrt{|g(x)|}(R-2\Lambda)$$ gives the expression $$\label{MPl} M^2 _{P}=m_{D} ^{D-2}v_n f' (R_n)$$ for the Planck mass. Here $v_n$ is the volume of the extra space. The term $$\label{Lambda0} \Lambda \equiv -\frac{m_D ^{D-2}}{2M^2 _{Pl}}v_n f(R_n)$$ represents the cosmological $\Lambda$ term. Both the Planck mass and the $\Lambda$ term depends on a function $f(R)$. According to , the Planck mass could be smaller than $D$-dim Planck mass, $M_P<m_D$ for specific functions $f$. This leads to nontrivial consequences. For example, the classical 4-dim observer is limited by the smallest scale $l_{quantum}\sim 1/M_P$. At high energy scales, an observer “feel” extra dimensions and hence the classical behaviour starts at the scale $l_{D,quantum}\sim 1/m_D$ which can be much smaller then $10^{-43}$cm, the standard Planck scale. This question deserves a separate discussion in future. Brane models {#brane-models .unnumbered} ------------ The first brane models have appeared two decades ago [@ArkaniHamed:1998rs; @Gogberashvili:1998iu; @1999PhRvL..83.3370R; @2012arXiv1201.0614C], see also nice review [@Shifman:2009df], though the very first idea was declared in 1983 [@Rubakov:1983bb] where it was proposed that we are living in the 4-dimensional manifold that is immersed in a manifold of larger dimensions. A large but compact extra dimension was invented by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali [@ArkaniHamed:1998rs] (the ADD model). In this approach, the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions. The Planck mass relates to the extra space radius as $$M_P=(2\pi R)^{n/2}m_D^{\frac{n+2}{2}}$$ The Randall-Sundrum model [@1999PhRvL..83.3370R] is based on the 1-dim extra space representing $S^1/Z_2$ orbifold. Here $S^1$ is the circle and $Z_2$ is the multiplicative group $\{ -1, 1\}$. Two 3-dim branes are attached to two fixed points with coordinates $y=0$ and $y=L$. The 5-dim action is described by the expression $$S_{RS}=S_g - \int d^4x\int dy\sqrt{|g_5|}\sigma_1 \delta(y) - \int d^4x\int dy\sqrt{|g_5|}\sigma_2 \delta(L-y).$$ The first term is represented in . The second and the third terms describe the branes with the constant tensions $\sigma_1$ and $\sigma_2$. The metric of the model describes the warped space with interval $$ds^2 = e^{-2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^2$$ This model solves the Hierarchy problem, the price of which is the connection of the Lagrangian parameters $$\label{RSconnec} \sigma_1 = -\sigma_2 = \sqrt{-6m_D^3\Lambda}. (\Lambda<0)$$ The 4-dim Planck mass is expressed in terms the model parameters as $$\label{MPRS} M_P^2=\frac{1}{2k} (1-e^{-2kL})m_D^3, \quad k=\sqrt{\frac{-\Lambda}{6m_D^3}}.$$ Evidentely, the 4-dim Planck mass $M_P$ and its 5-dim analog $m_D$ are of the same order of the magnitude for not very small value of the parameter $k$ and $L$. One can conclude that a solution to the Hierarchy problem looks solvable and the extra-dimensional paradigm is the important idea allowing the progress in this direction. In general, the brane idea represents a powerful tool to solve deep questions of modern physics. For example, the large value of the Planck mass as compared to the electro-weak scale can be justified. The Fine-tuning problem remains unsolved yet. The fact of the fine-tuning is supported by a lot of examples [@2007unmu.book..231D; @Bauer:2010wj]. There are many attempts to solve each problem separately. In the paper [@Krause:2000uj] warped geometry is used to the solution of the small cosmological constant problem. The hybrid inflation [@2002PhRvD..65j5022G] was developed to avoid the smallness of the inflaton mass. The paper [@1999PhRvL..83.3370R] describes the way to solve the smallness of the Gravitational constant. Nevertheless, all of them suffer the fine-tuning of Lagrangian parameters. We devote the following discussion to this subject. Brane as a clump of matter? --------------------------- The first brane models postulated the extra space metric and 3-dim spaces (branes) that are attached to their critical points. The modern trend consists of involving thick branes into consideration which are soliton-like solutions extended in extra coordinates. To build such solutions, the scalar field potential with several vacua states [@Peyravi:2015bra] is usually proposed. The one-dimensional kinks are studied for a long time and represent a substantial ground for the branes construction. The serious shortage of the approach mentioned above consists of a firm connection of model parameters and the effective low energy parameters . Even if a model including extra dimensions is able to solve the Hierarchy Problem, the Fine-tuning enigma is still far from resolution. The problem is simply translated from the observable parameters to parameters of a specific model. An important feature of branes is their ability to concentrate the matter nearby. But what is the effect of matter on the very structure of brane? This subject is studied below. The encouraging analogy is that a gravitating substance can experience the Jeans instability, as we know from four-dimensional physics. One may expect the same effect in the extra space which should lead to the brane formation. Here we discuss the new mechanism of the branes construction which was revealed in [@Rubin:2015pqa]. A complicated form of the scalar field potential is not necessary for it is known that the scalar field with the potential $V(\phi)\propto \phi^2$ experiences the gravitational instability [@Khlopov:1985jw]. In analogy with the 4-dim case, the scalar field could form stable clumps within the extra dimensions due to the gravitational interaction. This subject has been also studied in [@Gani:2014lka], [@Rubin:2014ffa], [@2017JCAP...10..001B] and we shortly discuss it in next Section. The solution describing the brane depends on an initial amount of matter and hence such solutions form a continuous set. This property is extremely important for the discussion of the Fine-tuning problem and the Lambda term problem as a particular case. Matter induced branes ===================== Matter distribution within extra space -------------------------------------- This section discusses extra dimensions filled with matter. An ordinary scalar field is accepted as representative of matter. It is assumed that its potential has a single minimum. Solutions of the system of equations indicate that the distribution of the scalar field has a critical point. The back reaction of the scalar field significantly affects the extra metric, forming non-trivial static configurations. Let us come back to the action with a scalar field $\phi$ $$\begin{aligned} \label{actfL} S=\frac{m_D^{D-2}}{2}\int d^{D}z \sqrt{|g_D|}\left[f(R)+L_m\right]; \quad f(R)=aR^2 + bR +c \end{aligned}$$ and the corresponding equations of motion $$\label{eqn} R_{AB} f' -\frac{1}{2}f(R)g_{AB} - \nabla_A\nabla_B f_R + g_{AB} \square f' = \frac{1}{m_D^{D-2}}T_{AB}.$$ Here $\square$ stands by the d’Alembert operator $$\square =\square_D=\frac{1}{\sqrt{|g_D|}}\partial_A ( {g_D}^{AB}\sqrt{|g_D|}\partial_B),\quad A,B=1,2,..,D$$ Evidently, there is a continuum set of solutions to system depending on additional conditions. Maximally symmetric extra spaces represent a small subset of this continuum set. One of the reasons to choose this particular case has been discussed in [@Kirillov:2012gy]. As was shown there, the entropy outflow from the extra space into the large 3-dimensional space of our Universe leads to the maximally symmetric extra space at the final state. Essentially new element changing the situation is the matter (the scalar field) inclusion into the consideration. System contains equation for the scalar field $$\label{scalareq} \square_D \phi =- V'(\phi)$$ Let us consider the class of the homogeneous in 4-dim space solutions $\phi(x,y)=\Phi(y)$ and suppose that the potential possesses unique minimum at $\Phi = 0$, i.e. $V'(0)=0$. In this case, the solution $\Phi = const = 0$ looks natural. If $\Phi>0$, the system radiate waves of different kinds thus increasing the entropy of a thermostat. This process lasts until the energy is settled in a minimum, which is zero in our case. Such a picture is true if the gravity is absent. The latter leads to the gravitational instability like Jeans instability that is the reason for the large scale structure formation in our Universe. Scalar field instability in the framework of the Einstein gravity was discussed in [@Khlopov:1985jw] where the instability in the wavenumber range $$0< k^2< k_J^2=4\sqrt{\pi G_N}m^2a_0,$$\[Jeans\] were found. Here $a_0$ is an initial amplitude of the field and $m$ is its mass. The final state could be compact objects made from the scalar field [@Carneiro:2018url]. Suppose that such compact object has been created within the extra dimensions provided that its density distribution along the $x$ coordinate is the uniform (i.e. the scalar field depends only on the extra coordinates $y$). Its stability may be supported by general arguments. Indeed, if such configuration decays, a 4-dimensional observer must detect a final state consisting of point-like defect of the scalar field distribution and massive scalar particles that have been instantly nucleated from the homogeneous state. Such a process is forbidden due to energy conservation. This argument for stability is not absolutely strict but reliable and we will keep it in mind postponing thorough study for the future. Numerical solutions of differential equations depending on additional conditions and the scalar field acting in the extra space were studied in [@Gani:2014lka; @Rubin:2015pqa]. It was assumed that the metric of our 4-dim space is the Minkowski metric, $g_{4} = diag (1,-1,-1,-1)$. The compact 2-dim manifold is supposed to be parameterized by the two spherical angles $y_1=\theta$ and $y_2=\phi$ $(0 \leq\theta \leq \pi, 0 \leq \phi < 2\pi)$. The choice of the extra space metric is as follows $$\label{metric2} g_{2,\theta\theta} = -r(\theta)^2;\quad g_{2,\phi \phi}= -r(\theta)^2 \sin^2(\theta).$$ In the absence of matter, the extra metric is supposed to be the maximally symmetric, i.e. $R=const,\quad r=\sqrt{2/R}=const$. Let us fix the metric at the point $\theta =\pi$ $$\label{bound2} R(\pi)=const, \quad R'(\pi)=0, \quad r(\pi)=\sqrt{2/R}, \quad r'(\pi)=0.$$ The system of equations together with these additional conditions completely determine the form of extra space metric. The horizontal line in Fig.\[MetricVsMatter\] (the scalar field is absent, $r(\theta)=const$) is the solution to the system which coincides with our intuition and hence validates the applied method. Nontrivial results can be obtained for the nonzero value of the scalar field density within the extra space. The solutions to the extra metric are presented in Fig. \[MetricVsMatter\]. The more scalar field is inserted into the extra dimensions, the deeper the well is formed. The matter density relates to the additional conditions at point $\theta =\pi$ which represent a set of the cardinality of the continuum. We conclude that the extra space metric continuously depends on the scalar field distribution in the extra space. {width="10cm"} . \[MetricVsMatter\] Matter induced branes and variation of 4-dim physical parameters ---------------------------------------------------------------- In this subsection, the way to obtain effective 4-dim action for matter fields is discussed. To facilitate analysis, a scalar field is used as an example. Its action is written in the standard form $$\begin{aligned} \label{act0} &&S_{\chi}=\int d^D z \sqrt{|g_D|}\left[\frac12 \partial_{A} \chi g_{D}^{AB}\partial_{B} \chi - \frac{m_{\chi}^2}{2}\chi^2 \right]. \end{aligned}$$ Let us decompose the field around its classical part $$\label{xY} \chi(x,y) =\chi_{cl}(y)+\delta\chi;\quad \delta\chi\equiv\sum_{k=1}^{\infty}\chi_k (x) Y_k(y)\ll \chi_{cl}(y),$$ where $Y_k(y)$ are the orthonormal eigenfunctions of the d’Alembert operator acting in the inhomogeneous extra space $$\label{eqY1} \square_n Y_k(\theta)=l_k Y_k(y).$$ The term $\chi_{cl}(y)$ is the solution to classical equation $$\square_D \chi_{cl}(y)=\square_n \chi_{cl}(y)=-U'(\chi_{cl}(y))$$ If we take into account the form of metric in the numerical example discussed above, the trial scalar field distribution can be written in the form $$\label{cut} \chi_{cl}(\theta)= C\exp\{-m_{\chi}\int^{\theta} _0 d\theta' r (\theta')\}$$ valid for not very small coordinate $\theta$, see discussion in [@Rubin:2015pqa]. The normalization multiplier $C$ defines the density of the scalar field distributed over the extra dimensions relates to an amount of the field stored in the extra dimensions from the beginning. Below, we limit ourselves by only first term in the sum so that $$\delta\chi=\chi_0(x)Y_0(y), \quad \square_nY_0=0.$$ After substitution into expression we get the following form of the effective 4-dim action for the gravity with the scalar field $$\begin{aligned} S_{\chi}=\frac{1}{2}\int d^4 x \sqrt{|g_4|}\left[ \frac{1}{2}\partial_{\mu}\chi_0(x)g^{\mu\nu}\partial_{\nu}\chi_0(x) - \frac{m^2}{2}\chi_0 (y)^2 - ... -\Lambda_{\chi} \right] , \label{Ssc} \end{aligned}$$ where $$\begin{aligned} &&m^2=\int d^n y \sqrt{|g_n(y)|} \left[m^2_{\chi}Y_0(y)^2-\partial_aY_0 (y)g_n^{ab}(y)\partial_b Y_0(y)\right] , \label{m2}\\ &&\Lambda_{\chi}= \int d^n y \sqrt{|g_n(y)|} \left[\frac12 m^2_{\chi}\chi_{cl}^2 -\frac12 \partial_a\chi_{cl} (y)g_n^{ab}(y)\partial_b \chi_{cl}(y)\right] \label{La} \end{aligned}$$ The effective mass and $\Lambda_{\chi}$ term are the functions of the classical field distribution $\chi_{cl}(y)$ in the extra dimensions. The latter depends on an accidental conditions just after the D-dim manifold was formed. Therefore, the mass of the scalar field $\chi_0$ (and the Lambda term) varies depending on initial conditions that have been realized at the early Universe. The Higgs field is responsible for nonzero masses of the fermions and gauge bosons of the Standard Model. Hence, it is worth discussing the parameters of the Higgs potential and their possible variation. The simplest way is to introduce an interaction of the Higgs field and the field $\chi$ in the spirit of the moduli field approach, see [@Trigiante:2016mnt] and references therein. To this end, consider $D$-dim action as an example: $$\begin{aligned} &&S_H=\int d^4xd^ny\sqrt{g_4g_n}[\partial H^+\partial H +\mu^2(\chi)H^+H- \lambda(\chi)(H^+H)^2]. \end{aligned}$$ Here $\lambda(\chi)$ and $\mu^2(\chi)$ are arbitrary functions of the field $\chi(y)\simeq \chi_{cl}(y)$. Integration out the extra coordinates $y$ leads to the standard form of the action for the Higgs field with the parameters $$\mu_{eff}^2=\int d^n y\mu^2(\chi_{cl}),\quad \lambda_{eff}=\int d^n y\lambda(\chi_{cl})$$ where only zero mode of the Higgs field $H=H(x)$ is taken into account. We have got the effective action $$\begin{aligned} &&S_{H,eff}=\int d^4xd^ny\sqrt{g_4g_n}[\partial H^+\partial H +\mu_{eff}^2H^+H- \lambda_{eff}(H^+H)^2]. \end{aligned}$$ with 4-dim parameters depending on the matter distribution in the extra dimensions. The same can be said about the Planck mass and the Cosmological constant. Indeed, formulae and are easily converted to the expressions $$\label{MPl2} M^2 _{P}=m_{D} ^{D-2}\int d^n y \sqrt{|g_n(y)|}f' (R_n (y) )$$ for the Planck mass and $$\label{Lambda2} \Lambda \equiv -\frac{m_D ^{D-2}}{2M^2 _{P}}\int d^n y \sqrt{|g_n(y)|} [f(R_n(y)) + L_m (y)]$$ for the cosmological $\Lambda$ term. Both the Planck mass and the $\Lambda$ term depend on a stationary geometry $g_{n,ab} (y)$ which are now functions of not only the Lagrangian parameters, but also on the accidental value of the initial scalar field density. The preliminary conclusion is as follows. The matter uniformly distributed in our 3-dim space can be a reason for the branes formation due to a nontrivial distribution of the matter within the extra dimensions. The brane properties depend not only on the Lagrangian parameters but also on the density of the matter. The latter is a random value that is formed in the early Universe when the quantum fluctuations were important. One can conclude that a variety of branes with different properties can be formed in different spatial regions which can be a basis for the idea of the Multiverse. Therefore, this property could lead to the solution of the Fine-tuning enigma. Below we discuss this topic, bearing in mind the problem of the cosmological constant. Fine-tuning of the Lambda term and matter induced branes ========================================================= The situation with the Lambda term remains intriguing [@Sahni:1999gb] despite two decades of discussion. Cosmological observations indicate that the current acceleration is described by the general relativity with the extremely small Cosmological Constant (CC). At the same time, the quantum fluctuations lead to the vacuum energy density which is in many orders of magnitude higher than the observed value of the CC. There are many models elaborated to explain the smallness of the $\Lambda$ term, see e.g. [@Yurov:2005zn; @Garriga:2000cv]. General discussion on the subject can be found in [@Weinberg:1987dv; @Sahni:1999gb; @Loeb:2006en; @Wetterich:2008sx; @Bousso:2000xa; @Brown:2013fba; @Burgess:2013ara]. The role of quantum corrections is not clarified up to now. The quantum corrections are of the order of cutting parameter which is compatible with a highest energy scale of a specific model. Hence natural values of physical parameters defined at this energy scale are of the same order of magnitude as this highest energy scale and it is not clear how to neutralize them except by a strong parameter selection. We have to admit that the observed Lambda term value is hardly be explained in terms of the physical parameters determined at low energies. The problem is deepened because if this value were several times larger, intelligent life would be absent. This represents the particular case of the fine-tuning problem. The question “How the physical parameters acquire the observable values?” divides the physical community into two groups. The first one does not bother with questions of such kind. They are interested in the study of physical laws that explain experimental facts. This point of view is quite firm but slightly inconsistent. Indeed, there is the experimental fact of fine-tuning of the physical parameters necessary for the existence of intelligent life. The range of the parameter values is very narrow and like any observed phenomena, it must be explained. This is the reason for the second group of physicists to make efforts in answering this question. The first step has been done decades ago when the Anthropic principle was proclaimed: “there are a lot of different patches (or universes) with different properties and the life originates in universes with appropriate conditions”. The immediate question is formulated as follows: What is the mechanism for the creation of a variety of universes (Multiverse) with different properties? As we will see, an attempt to answer this question consists of several ideas that deserve further development. The Anthropic principle is not the solution to the fine-tuning problem but the small step forward. The string theory is the well-known idea supplying us with the multiverse - the landscape in its terms [@Susskind:2003kw]. Unfortunately, this approach has a weakness. Indeed, even if a number of final states is as huge as $10^{500}$ in the string theory, they could be distributed non uniformly in parameter space and there is no certainty that the necessary physical parameters can be realized. This shortcoming can be eliminated if the set of low energy parameters has the cardinality of the continuum. This relates to the discussion made above. The branes induced by matter depend on accidental values of the initial energy density of matter produced by the quantum fluctuations. Therefore a set of such branes has the cardinality of the continuum. Evident logical chain is: *continuum set of initial metrics $\rightarrow$ continuum set of final metrics $\rightarrow$ continuum set of the $\Lambda$-terms.* The picture looks as follows. In the spirit of the inflationary scenario, quantum fluctuations at high energies produce a huge variety of space volumes characterized by different energy density and hence by branes with different properties. This is the reason for the formation of different cosmological constants within such volumes. According to formula , the Lambda term varies with the matter variation. One can see from Fig.\[Lambda\] that the cosmological constant varies from negative to positive values due to variation of the matter distribution. The latter relates to the additional condition that fixes the scalar field at the boundary $\phi(\theta=\pi)$. The universes differ from each other due to the initial distribution of the matter along the extra dimensions. The problem of the $\Lambda$-term smallness is reduced now to the question “does this set contain the term $\Lambda =0$?”. Fig. \[Lambda\] gives the positive answer to the question if one keeps in mind continuous kind of the curve. In particular, there exists a set of universes with such initial matter distributions that gives the cosmological $\Lambda$ terms be arbitrarily close to zero. ![Solution to equation - $\Lambda$-term (arbitrary units) depending on the accidentally chosen additional conditions $\phi_i(\theta=\pi),\quad i=1,...,5$. []{data-label="Lambda"}](LambdaVar.pdf){width="10cm"} Conclusion ========== There are at least two general problems of the cosmology that worth discussing - the Hierarchy problem and the Fine-tuning one. It seems that a multidimensional paradigm allows us to solve the first puzzle, while the exact adjustment of the physical parameters remains unresolved. The perspective way to solve it is an elaboration of mechanism of the Multiverse formation containing a continuum set of different Universes. The mechanism of such sort is discussed in this article. The main point is the use of matter to obtain a non-trivial metric of extra space. The 4-dim analogy can be useful. Indeed, the formation of compact dense objects due to the Jeans instability leads to the formation of a variety of objects, the mass of which depends on an initial matter distribution. The same process could take place in the extra dimensions where compact objects - branes - are formed under the influence of matter. Each universe belonging to the Multiverse is described by the specific distribution of matter and hence by specific extra space metric. This leads to the formation of causally disconnected regions endowed by branes that differ with each other. Therefore the physical parameters in such volumes are also different as was discussed above. Initial conditions form a continuous set. Hence, the extra space metrics also form a set of the cardinality of the continuum. The low energy physical parameters depend on the extra space metrics and hence represent a continuous set as well. This means that those space areas where the Lambda term has the observable value do exist thus providing the basis for the Anthropic argument. The quantum fluctuations seem to destroy the analysis made on the classical level. This problem is discussed in [@Rubin:2016ude] where it was shown that the situation looks solvable in the framework of the effective field theory. The discussion in this article shows that the matter distribution within the extra space is a promising way to describe the fine-tuning of the physical parameters. Acknowledgement =============== The work was supported by the Ministry of Education and Science of the Russian Federation, MEPhI Academic Excellence Project (contract N 02.a03.21.0005, 27.08.2013). The work was also supported by the Ministry of Education and Science of the Russian Federation, Project N 3.4970.2017/BY and is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.

--- abstract: '\[abstract\] At zero temperature magnetic phases of the quantum spin-$1/2$ Heisenberg antiferromagnet on a simple cubic lattice with competing first and second neighbor exchanges ($J_1$ and $J_2$) is investigated using the non-linear spin wave theory. We find existence of two phases: a two sublattice Néel phase for small $J_2$ (AF), and a collinear antiferromagnetic phase at large $J_2$ (CAF). We obtain the sublattice magnetizations and ground state energies for the two phases and find that there exists a first order phase transition from the AF-phase to the CAF-phase at the critical transition point, $p_c=0.28$. Our results for the value of $p_c$ are in excellent agreement with results from Monte-Carlo simulations and variational spin wave theory. We also show that the quartic $1/S$ corrections due spin-wave interactions enhance the sublattice magnetization in both the phases which causes the intermediate paramagnetic phase predicted from linear spin wave theory to disappear.' author: - Kingshuk Majumdar - Trinanjan Datta bibliography: - 'SC.bib' title: 'Zero temperature phases of the frustrated J$_1$-J$_2$ antiferromagnetic spin-1/2 Heisenberg model on a simple cubic lattice' --- Introduction ============ Frustrated quantum Heisenberg magnets with competing nearest neighbor (NN) and next-nearest-neighbor (NNN) antiferromagnet (AF) exchange interactions, $J_1$ and $J_2$ respectively, have been under intense investigation both theoretically and experimentally in condensed matter physics for more than a decade. [@diep1] At low temperatures these systems exhibit new types of magnetic order and novel quantum phases. [@diep1; @subir1; @subir2] A well-known example is the quantum spin-$1/2$ antiferromagnetic $J_1-J_2$ model on a square lattice, which has been studied extensively by various analytical and numerical methods. [@tassi1; @tassi2; @oleg; @subir3; @dot; @chubu1; @gelfand; @sushkov; @weihong; @singh; @irkhin] For this two-dimensional square lattice system with $J_2=0$ the ground state is antiferromagnetically ordered at zero temperature. Addition of next nearest neighbor interactions induces a strong frustration and break the antiferromagnetic (AF) order. The competition between the NN and NNN interactions for the square lattice is characterized by the frustration parameter $p= J_2/J_1$. It has been found that a disordered quantum spin liquid phase exists between $p_{1c} \approx 0.38$ and $p_{2c} \approx 0.60$. For $p<p_{1c}$ the square lattice is AF-ordered whereas for $p>p_{2c}$ a collinear phase emerges. In the collinear state the NN spins have a parallel orientation in the vertical direction and antiparallel orientation in the horizontal direction or vice versa. The nature of phase transition from AF-ordered state to disordered state at $p_{1c}$ is of second order and from the disordered state to the collinear state at $p_{2c}$ is of first order. The properties of quantum magnets depend strongly on the lattice dimensionality since the tendency to order is more pronounced in three dimensional (3D) systems than in the lower dimensional systems. Furthermore, in 3D the available phase space is more and we expect quantum fluctuations to play a lesser role as compared to 1D and 2D. In 1D and 2D the available phase space is limited and quantum fluctuations play a dominant role in determining the quantum critical points. Despite this fact a magnetically disordered phase has been observed in frustrated 3D systems such as the Heisenberg AF on the pyrochlore lattice [@canals] or on the stacked kagome lattice [@Fak08SL; @subir92; @chubu92Kagome; @harris92Kagome]. Studies on the Heisenberg AF on the pyrochlore lattice (a geometrically frustrated system) have revealed the existence of a spin liquid state. [@canals] On the other hand, for the 3D J1-J2 model on the body-centered cubic (BCC) lattice there are no signs of an intermediate quantum paramagnetic phase at zero temperature. [@oitmaa; @Schmidt; @kingBCC] For the BCC lattice competing interactions and not lattice geometry generates the frustration. This comparison illustrates how the magnetic phase diagram may dramatically change based on whether the frustration is generated by *competing interactions* or *by geometry*. Most of the efforts on quantum 3D magnets have primarily focused on geometrically frustrated lattices. [@diep1; @subir1] There exists some computational[@azaria; @derrida1; @derrida2; @lallemand; @karchev1; @karchev2; @banavar; @oitmaa; @Schmidt; @oguchi; @katanin; @diep4; @ader] and very few analytical studies[@villain1; @villain2; @banavar; @kingBCC] of the magnetic phase diagrams and magnetic order of spin-$1/2$ Heisenberg AF on 3D lattices where on the study of magnetic phase diagrams and magnetic order of spin-$1/2$ Heisenberg AF on 3D lattices where *competing interactions induce frustration*. [@villain1; @villain2; @azaria; @derrida1; @derrida2; @lallemand; @karchev1; @karchev2; @banavar; @oitmaa; @Schmidt; @kingBCC; @oguchi; @katanin; @diep4; @ader] Very few analytical and numerical results exist for the the frustrated $J_1 - J_2$ isotropic Heisenberg model on a simple cubic lattice. [@diep2; @diep3; @irkhin; @kishi] This model has been studied previously using Monte Carlo simulation [@diep2], variational spin wave theory [@kishi], and modified spin wave theory [@irkhin]. In a recent work the critical properties of the 3D anisotropic quantum spin-$1/2$ model on a simple cubic (SC) lattice has been investigated within the framework of the differential operator technique and by using an effective field theory in a two-spin cluster. [@viana] The study revealed that at zero temperature there is a AF-lamellar (first order) phase transition. The motivation for the present work is to investigate the zero temperature phases of this model in the framework of non-linear spin wave theory (NLSWT) and to obtain the critical transition points of this model. Also we will compare our results from NLSWT with the prediction from the linear spin wave theory (LSWT). The paper is organized as follows. In Section \[sec:model\] we set-up the Hamiltonian for the spin-$1/2$ Heisenberg AF on the SC lattice. The classical ground state configurations of the model and the different phases are then discussed. In Section \[sec:SCNLSWT\] we map the spin Hamiltonian to the Hamiltonian of interacting bosons and develop the NLSWT sublattice magnetization and energy expressions. The sublattice magnetizations and the ground state energies for the two phases are numerically calculated and the results are plotted and discussed in Section \[sec:results\]. Finally we summarize our findings in Section \[sec:conclusion\]. Classical Ground State Configurations \[sec:model\] =================================================== The Hamiltonian for a spin-$1/2$ Heisenberg antiferromagnet with first and second neighbor interactions on a simple cubic (SC) lattice is H = J\_1 \_[ij ]{}[**S**]{}\_i \_j + J\_2 \_[\[ij\]]{}[**S**]{}\_i \_j. \[ham\] $J_1$ is the NN and $J_2$ is the frustrating NNN (which are along the face diagonals of the cube) exchange constants. Both couplings are considered antiferromagnetic, i.e. $J_1,J_2>0$. For the SC lattice the number of nearest and next-nearest neighbors are $z_1=6$ and $z_2=12$. The limit of infinite spin, $S \rightarrow \infty$, corresponds to the classical Heisenberg model. We assume that classically the spin configurations of the system are described by $S_i=S{\bf u}e^{i{\bf q}\cdot{\bf r_i}}$, where ${\bf u}$ is a vector expressed in terms of an arbitrary orthonormal basis and ${\bf q}$ defines the relative orientation of the spins on the lattice. The classical ground state energy of the lattice in terms of the frustration parameter, p, is given by E\_[**k**]{}/NJ\_1 = S\^2z\_1\[\_[1 [**k**]{}]{}+p\_[2 [**k**]{}]{}\], with the structure factors $$\begin{aligned} \gamma_{1{\bf k}} &=& \frac 1{3}\Big[ \cos(k_x)+\cos(k_y)+\cos(k_z)\Big], \\ \gamma_{2{\bf k}} &=& \frac 1{3}\Big[\cos(k_x)\cos(k_y)+\cos(k_y)\cos(k_z) +\cos(k_z)\cos(k_x)\Big],\end{aligned}$$ where we define the parameter of frustration as $p=z_2J_2/z_1J_1$. [@note1] The wave-vectors along the $x$, $y$, and $z$ directions are denoted by $k_x,k_y$, and $k_z$. The number of lattice sites are given by $N$ and we have set the lattice spacing $a=1$. At zero temperature the classical ground state (GS) for the SC lattice can be characterized by the values of $p$. $p=0$ corresponds to the unfrustrated case (only AF interactions between NN). For $p<1/2$ or $J_2/J_1 <1/4$, there is a single minimum in energy $E_0/NJ_1=-3S^2(1-p)$ for the wave-vector $(\pm \pi,\pm \pi,\pm \pi)$. They correspond to the classical two sublattice Néel state (AF phase) where spins in A and B-sublattices point in opposite directions \[Fig. \[fig:phase1\]\]. For $p>1/2$ apart from the global rotation the classical ground state has an infinite degeneracy – the frustration is uniformly distributed on all the spins, causing a non-collinear GS with very large degeneracy. In general, the GS for $p>1/2$ can be decomposed into two NNN tetrahedra. The spin configurations in each of these two NNN tetrahedra can be characterized by two angles $\theta$ and $\phi$. This results in a four sublattice $A,B,C,D$ \[Fig. \[fig:phase2\]\] antiferromagnetic structure. Out of these infinite possibilities there are three collinear configurations (one line up, one line down). The wave-vectors corresponding to these collinear states are $(\pi,\pi,0),\;(\pi,0,\pi),\;(0,\pi,\pi)$. The classical GS energy for these states is $E_0/NJ_1=-S^2(1+p)$. Thermal or quantum fluctuations lift these degeneracies and select specific discrete states and it has been conjectured that thermal or quantum disorder favors collinear states ([*order by disorder*]{}). [@shender; @henley] The four fold rotational symmetry of the lattice is spontaneously broken in this state. By employing a spin wave theory based on the general four sublattice mean field ground state it has been shown that the quantum fluctuations stabilize a collinear spin ordering. [@kubo] Quantum Monte Carlo simulations on the frustrated SC lattice for $p>1/2$ also confirm this conjecture. [@diep2] In the present article, for $p>1/2$, we consider the system to be in one of these three collinear configurations (collinear antiferromagnet or CAF). $p=1/2$ corresponds to the case where both $J_1$ and $J_2$ compete – causing frustration in the system. This critical value $p_{\rm class}=0.5$ is the classical phase transition point where a phase transition from AF to CAF phase occurs. In this work we will investigate the role of quantum fluctuations in the two different phases (AF and CAF) of the model and how these fluctuations shift the critical transition point. \[sec:SCNLSWT\] Self-consistent non-linear spin wave theory =========================================================== The Hamiltonian in Eq. \[ham\] can be mapped into an equivalent Hamiltonian of interacting bosons by transforming the spin operators to bosonic operators $a, a^\dag$ and $b, b^\dag$ using the well-known Holstein-Primakoff transformations. For the AF-phase ($J_2<J_1$) the operators $a, a^\dag$ and $b, b^\dag$ are for the $A$ and $B$ sublattices. On the other hand for the CAF phase ($J_2>J_1$) we have used the operators $a, a^\dag$ and $b, b^\dag$ for the up and down spin configurations. $$\begin{aligned} S_{Ai}^+ &\approx& \sqrt{2S}\Big(1- \frac {a_i^\dag a_i}{4S} \Big)a_i,\;\; S_{Ai}^- \approx\sqrt{2S}a_i^\dag \Big(1-\frac {a_i^\dag a_i}{4S} \Big), \non \\ S_{Ai}^z &=& S-a^\dag_ia_i, \non \\ S_{Bj}^+ &\approx&\sqrt{2S}b_j^\dag \Big(1-\frac {b_j^\dag b_j}{4S}\Big),\;\; S_{Bj}^- \approx\sqrt{2S}\Big(1-\frac {b_j^\dag b_j}{4S} \Big)b_j, \non \\ S_{Bj}^z &=& -S+b^\dag_jb_j,\label{spinmap}\end{aligned}$$ In these transformations we only kept terms up to the order of $1/S$. Next using the Fourier transforms $$a_i = \sqrt{\frac 2{N}}\sum_{\bf k} e^{-i{\bf k \cdot R_i}}a_{\bf k},\;\;\; b_j = \sqrt{\frac 2{N}}\sum_{\bf k} e^{-i{\bf k \cdot R_j}}b_{\bf k},$$ the real space Hamiltonian is transformed to the ${\bf k}$-space Hamiltonian. In the following two sections we study the cases $J_2<J_1$ and $J_2>J_1$ separately. ### \[sec: smallp\]$J_2<J_1$: AF phase In this phase the classical ground state is the two-sublattice Néel state \[Fig. \[fig:phase1\]\]. For the NN interaction, spins in $A$ sublattice interacts with spins in $B$ sublattice and vice versa. On the other hand the NNN exchange $J_2$ connects spins on the same sublattice, $A$ with $A$ and $B$ with $B$. Substituting equations (\[spinmap\]) into (\[ham\]), the ${\bf k}$-space Hamiltonian takes the form: H=H\^[(0)]{}+H\^[(2)]{}+H\^[(4)]{}. The classical ground state energy $H^{(0)}$ and the quadratic terms $H^{(2)}$ are H\^[(0)]{} &=& -NJ\_1S\^2 z\_1 (1-p), \[cgs\]\ H\^[(2)]{} &=& J\_1S z\_1 \_[**k**]{}. \[smallp\] with the coefficients $A^{(1)}_{0{\bf k}}$ and $B^{(1)}_{0{\bf k}}$ defined as $$\begin{aligned} A^{(1)}_{0{\bf k}}&=&1-p(1-\gamma_{2{\bf k}}), \label{Ak0smallp}\\ B^{(1)}_{0{\bf k}}&=& \gamma_{1{\bf k}}. \label{Bk0smallp}\end{aligned}$$ The quartic terms in the Hamiltonian $H^{(4)}$ involve interactions between $A-B$ (for NN terms) and $A-A,\; B-B$ (for NNN terms) sublattices. The Hamiltonian for these interaction are stated in Appendix \[Smallp\], Eq. \[quartic\]. These terms are evaluated by applying the Hartree-Fock decoupling process. The contributions of the decoupled quartic terms to the harmonic Hamiltonian in Eq. \[smallp\] are to redefine the values of $A^{(1)}_{0{\bf k}}$ and $B^{(1)}_{0{\bf k}}$ which are now A\^[(1)]{}\_[**k**]{} &=& ( 1-) - p\[1-\_[2 [**k**]{}]{}\]( 1- ), \[Acoeff\]\ B\^[(1)]{}\_[**k**]{} &=& \_[1[**k**]{}]{}( 1- ), \[Bcoeff\]\ \^[(1)]{}\_[**k**]{}&=& . \[omegak\] The coefficients $u_1,v_1$ and $w_1$ are in Appendix \[Smallp\]. They are evaluated self-consistently from equations (\[Acoeff\]), (\[Bcoeff\]), (\[u1\])–(\[w1\]). The quartic corrections to the ground state energy is calculated from the four-boson averages. In the leading order they are decoupled into the bilinear combinations (equations (\[u1\]) – (\[w1\])) using Wick’s theorem. The corresponding four boson terms are, a\^\_i a\_i b\^\_j b\_j &=& u\_1\^2+v\_1\^2, a\^\_i b\^\_j b\_jb\_j = 2u\_1v\_1,\ a\^\_i a\_i a\_i b\_j &=& 2u\_1v\_1, a\^\_i a\_i a\_j\^a\_j = u\_1\^2+w\_1\^2,\ a\_i a\_j\^a\_j\^a\_j &=& 2u\_1w\_1, a\^\_i a\_i a\_i a\_j\^= 2u\_1w\_1. This yields the ground state energy correction from the quartic terms: E\^[(4)]{}= -NJ\_1z\_1. \[EquarticsmallJ\] Adding all the corrections together the ground state energy takes the form E/NJ\_1 &=&-z\_1S(S+1)(1-p)+ z\_1S\ &+& z\_1\[Esmallp\] and the average sublattice magnetization $\langle S_\alpha \rangle$ is given by S\_= S. \[magsmallp\] Using equations (\[Acoeff\])–(\[omegak\]), we numerically evaluate $E/NJ_1$ and $\langle S_\alpha \rangle$. ### \[sec: largep\]$J_2>J_1$: CAF phase The classical ground state for $J_2>J_1$ is considered to be in one of the three collinear states \[Fig. \[fig:phase2\]\]. For NN and NNN exchanges there are $A-B$, $A-C$, $A-D$, $B-C$, $B-D$, and $C-D$ interactions between the four sublattices \[See Fig. \[fig:phase2\]\]. Considering all their contributions together up to the quadratic terms the harmonic Hamiltonian takes the same form as before with H\^[(0)]{} &=& - NJ\_1S\^2z\_1(1+p),\ A\^[(2)]{}\_[0[**k**]{}]{}&=& (1+k\_z)+p(1+k\_x k\_y),\[Ak0largep\]\ B\^[(2)]{}\_[0[**k**]{}]{}&=& (k\_x + k\_y)(1+pk\_z). \[Bk0largep\] The quartic terms in the Hamiltonian for this case are shown in Appendix \[Bigp\]. These terms are decoupled and evaluated in the same way as before. The renormalized values of the coefficients $A^{(2)}_{\bf k}$ and $B^{(2)}_{\bf k}$ are A\^[(2)]{}\_[**k**]{} &=& A\^[(2)]{}\_[0[**k**]{}]{} + 1[3S]{}, \[largeA\]\ B\^[(2)]{}\_[**k**]{} &=& B\^[(2)]{}\_[0[**k**]{}]{} - 1[3S]{}{([u]{}+[v\_1]{}) + p([u]{}+[v\_2]{})k\_z}(k\_x+k\_y), \[largeB\]\ \^[(2)]{}\_[**k**]{}&=&. \[omega2k\] The coefficients ${\overline u},{\overline v_{1}},{\overline v_2},{\overline w_1}, {\overline w_z}$ are in Appendix \[Bigp\]. As before these coefficients are calculated self-consistently from equations (\[largeA\])–(\[omega2k\]) and (\[u2\])–(\[w2\]). The quartic correction to the ground state energy (following the same Hartree-Fock decoupling process as done in the AF-case) is E\^[(4)]{}= -NJ\_1z\_1. \[EquarticbigJ\] Combining all these corrections, the ground state energy takes the following form: E/NJ\_1 &=& - z\_1S(S+1)(1+p) +z\_1S\ &-&z\_1\ &-&z\_1. \[Elargep\] The sublattice magnetization and the ground state energy are then obtained numerically using equations (\[magsmallp\]) and (\[Elargep\]). Results\[sec:results\] ====================== In Fig. \[fig:param\] we show the self-consistent values of the different parameters $u_1,v_1,w_1$ (AF phase) and ${\overline u_{1}},{\overline v_{1}}, {\overline v_{2}},{\overline w_{1}},{\overline w_{z}}$ (CAF phase) of our model. These parameters which provide the quartic corrections to our model do not appear in the LSWT calculations for the sublattice magnetization, $\langle S_\alpha \rangle$ and the ground state energy, $E$. We see from Fig. \[fig:param\] that most of these coefficients vary significantly with $p$ especially as $p$ approaches $0.5$ from both ends. This demonstrates that non-linear corrections due to the spin-wave interactions play a significant role in determining the different phases of our model. Figure \[fig:MagSC\] shows the result for the average sublattice magnetization for the SC lattice for both AF and CAF phase without (dashed line) and with (solid line) quartic corrections. In the AF ordered phase or the two sublattice Neél phase where A and B sublattice spins point in the opposite directions, sublattice magnetization decreases monotonically with increase in $p$ until $p \approx 0.49$. This gradual decrease in $\langle S_\alpha \rangle$ with increase in $p$ is expected as increasing strength of NNN interaction $J_2$ disorders the antiferromagnetic spin alignments. With only quadratic terms in the Hamiltonian (linear spin wave theory) we find that $\langle S_\alpha \rangle$ approaches zero as $p \rightarrow p_{c1}$ where $p_{c1} \approx 0.48$ indicating a order-disorder phase transition to a disordered paramagnetic (PM) state at this point. In the CAF phase with lines of spins up and down, LSWT calculations show that that $\langle S_\alpha \rangle$ decreases as $p$ approaches 0.5 from above and at $p=p_{c2}=0.50$ there is an another phase transition from the CAF state to the disordered PM state. This is similar to the two dimensional AF-square lattice with Heisenberg spins where we have a line of quantum critical points between $0.38<p<0.60$. However, self-consistent calculations with quartic $1/S$ corrections drastically alter the zero temperature phase diagram. We find that in the AF-phase with increase in $p$ the system aligns the spins antiferromagnetically along the horizontal and vertical directions – thus decreasing the sublattice magnetization from $\approx 0.42$ for $p=0$ to $\approx 0.30$ for $p=0.49$. In the CAF phase $\langle S_\alpha \rangle$ steadily decreases from $\approx 0.41$ for $p=1$ to $\approx 0.27$ for $p=0.52$. There is no existence of any disordered state as predicted by the linear spin-wave theory (quadratic corrections). The disordered PM region disappears completely and we only obtain two phases: AF and CAF. This is one of our main findings in the present work. This significant change due to the quartic corrections is due to the enhancement of order by quantum fluctuations. At $p=0$ (no frustration) there is no quartic corrections to $\langle S_\alpha \rangle$. This can be observed from equations (\[Acoeff\]) – (\[omegak\]) as the correction factor $(1-(u_1+v_1)/S)$ cancels out in equation \[magsmallp\]. Our non-linear spin wave theory calculations become unstable close to the classical transition point $p_{\rm class}=0.5$ since the coefficient $A_{\bf k}^{1}$ becomes equal to $B_{\bf k}^{1}$. We have also applied the NLSWT technique to compute the quartic corrections in the spin-$1/2$ Heisenberg AF on a body-centered lattice. [@kingBCC] The LSWT calculation for the BCC lattice does not predict any intermediate disordered state and the quartic corrections play a role in stabilizing the sublattice magnetization (see Fig. 2 of Ref. ). However, the effect of quartic corrections is more pronounced in the SC lattice where the intermediate disordered phase disappears. In Fig. \[fig:GSE\] we show the ground state energy per site, $E/NJ_1$, for the AF and the CAF phases with (solid line) and without (dashed line) quartic corrections as a function of the frustration parameter $p$. Classically $p_{\rm class}=0.5$ or $J_2/J_1=0.25$ is the critical point where a phase transition from the AF phase to one of the three CAF phases occur. With increase in frustration (as $p$ approaches 0.5 from both sides) we expect the GS energy to increase as the system goes from an energetically favored ordered state to a more disordered state. However, linear spin-wave theory calculation fails to capture this. Especially when $p$ is close to 0.5 we find a slight downward turn in energy. This has been reported in Ref. . On the other hand, NLSWT calculation correctly produces the expected energy increase. At $p=0$ the calculated energy with the quartic corrections is slightly lower than the energy obtained without the quartic corrections. This small decrease from the LSWT calculation is due to the ground state energy correction, which is negative (as seen in Eq. \[EquarticsmallJ\] – these terms originate from the self-energy Hartree diagrams). As our spin-wave theory calculation becomes unstable in the regime $0.49<p<0.52$ we used a spline fit for the AF-phase energy data points and then extrapolated the line so that it intersects the CAF-phase energy line. The extrapolated curve is shown by dotted lines (color online) in the figure. After extrapolation, we find that the two energies meet at $p_c \approx 0.56$ or $J_2/J_1 \approx 0.28$. The symmetries of the two phases are different: SO(3)/SO(2) for the AF phase and Z$_3 \times$SO(3)/SO(2) for the CAF phase. [@diep1] Due to the different symmetries of the two phases the transition is of first order. This is confirmed by the kink in the energy at $p_c \approx 0.28$. Our obtained value from our self-consistent NLSWT calculations for the quantum critical point is $J_2/J_1 \approx 0.28$. This is another major finding of our work. Using the variational spin-wave theory the authors in Ref.  obtained an upper bound of 0.27 for the ground state energy. However, their variational calculation slightly overestimates the value of GS energy. This is quite noticeable at $p=0$, where their energy value is higher than the LSWT prediction. We have found the $p=0$ energy value to be slightly less than the LSWT prediction. This is due to the quartic corrections explained earlier. The authors in Ref.  used a modified spin-wave theory based on Dyson-Maleev representation of the spins to study this model. They numerically obtained the value of critical transition point to be $p_c=0.30$. The other known existing numerical work is by Diep et. al. [@diep2] By extensive standard and histogram Monte-Carlo simulations, they obtained the transition point to be around 0.26. Our result is in excellent agreement with the results obtained from Monte-Carlo and variational spin-wave theory calculations (our $p_c$ differs by less than 3.5% from the variational spin-wave theory prediction). Conclusion\[sec:conclusion\] ============================ In this work we have investigated the zero temperature phases of a spin-$1/2$ Heisenberg frustrated AF on a SC lattice by considering the quartic $1/S$ corrections due to the spin-wave interactions. We have compared our results obtained from NLSWT calculations with the predictions from LSWT. It is known that LSWT predicts the existence of three phases: a two sublattice Néel phase for smaller values of the NNN exchange $J_2$, an intermediate paramagnetic phase, and a collinear phase for larger values of $J_2$. At zero temperature there are two quantum phase transitions - one from the AF-state to the disordered paramagnetic state and the other from the disordered state to one of the three collinear states. Both these transitions occur at the quantum critical point $p_c \approx 0.5$ or $J_2/J_1 \approx 0.25$. We have found that quartic corrections significantly alter this phase diagram as intermediate paramagnetic phase disappears. We find the existence of two phases at zero temperature: a two sublattice AF phase for small $J_2$ and a collinear phase phase for large $J_2$. With the inclusion of quartic interactions the intermediate paramagnetic phase completely disappears. Due to the different symmetries of the two phases (AF and CAF) the transition between the two phases is of first order and we find that the critical point for transition to be 0.28. Our obtained result is in excellent agreement with existing numerical results from Monte Carlo simulations. One of us (KM) thanks O. Starykh and S. D. Mahanti for helpful discussions. \[Smallp\] Quartic terms for the AF phase ========================================= The quartic terms for the AF phase from the NN interactions involve $A-B$ interactions and for the NNN interactions involve $A-A$ and $B-B$ type interactions. Considering all these interactions the quartic Hamiltonian takes the form: H\^[(4)]{} &=& -J\_1 \_[ij ]{}\ &+& J\_2\_[\[ij\]]{}. \[quartic\] In the harmonic approximation the following Hartree-Fock averages are non-zero for the SC-lattice Heisenberg antiferromagnet: u\_1 &=& a\_i\^a\_i = b\_i\^b\_i = , \[u1\]\ v\_1 &=& a\_i b\_j = a\_i\^b\_j\^=-,\[v1\]\ w\_1 &=& a\_i\^a\_j = b\_i\^b\_j = , \[w1\] where $\omega^{(1)}_{{\bf k}}=\sqrt{\Big(A^{(1)}_{{\bf k}}\Big)^2 - \Big(B^{(1)}_{{\bf k}}\Big)^2}.$ \[Bigp\] Quartic terms for the CAF phase ======================================== The quartic terms for the collinear phase from the NN interactions involve interactions between the sublattices $B-C$ and $A-D$ along the $x$ axis, $A-C$ and $B-D$ along the $y$ axis, and $A-B$ and $C-D$ along the $z$ axis. For the NNN interactions spin-spin interactions are between sublattices $A-B$ and $C-D$ (in the $xy$-plane), $A-C$ and $B-D$ (in the $xz$-plane), and $A-D$ and $B-C$ (in the $yz$-plane) as shown in Fig. \[fig:phase2\]. Adding all the contributions together yield H\^[(4)]{} &=&-J\_1 \_[ij\_x ]{}\ &-& J\_1 \_[ij\_y ]{}\ &+& J\_1\_[ij\_z ]{}\ &+&J\_2\_[ij\_[xy]{} ]{}\ &-& J\_2 \_[ij\_[yz]{} ]{}\ &-& J\_2 \_[ij\_[xz]{} ]{}. \[quartlargeJ\] Above $j_x,j_y,j_z$ are NN lattice sites along $x,y,z$ axes and $j_{xy},j_{yz},j_{xz}$ connects one lattice site with a NNN corner lattice sites on the $xy,yz,xz$ planes. The different coefficients that originate from Hartree-Fock decoupling process are &=& a\_i\^a\_i = b\_i\^b\_i = , \[u2\]\ [v\_[1x]{}]{} &=& a\_i b\_[j\_x]{} = a\_i\^b\_[j\_x]{}\^=-,\ [v\_[1y]{}]{} &=& a\_i b\_[j\_y]{} = a\_i\^b\_[j\_y]{}\^=-,\ [v\_[2yz]{}]{} &=& a\_i b\_[j\_[yz]{}]{} = a\_i\^b\_[j\_[yz]{}]{}\^=-,\ [v\_[2xz]{}]{} &=& a\_i b\_[j\_[xz]{}]{} = a\_i\^b\_[j\_[xz]{}]{}\^=-,\ [w\_z]{} &=& a\_i\^a\_[j\_z]{} = b\_i\^b\_[j\_z]{} = ,\ [w\_1]{} &=& a\_i\^a\_[j\_[xy]{}]{} = b\_i\^b\_[j\_[xy]{}]{} = , \[w2\] where $\omega^{(2)}_{{\bf k}}=\sqrt{\Big(A^{(2)}_{{\bf k}}\Big)^2 -\Big(B^{(2)}_{{\bf k}}\Big)^2}.$ By symmetry ${\overline v_{1x}}={\overline v_{1y}}={\overline v_1}$ and ${\overline v_{2yz}}={\overline v_{2xz}}={\overline v_2}$. ![\[fig:param\] Self-consistent results for the different parameters, $u_1,v_1,w_1$ (for the AF-phase) and ${\overline u_{1}},{\overline v_{1}}, {\overline v_{2}},{\overline w_{1}},{\overline w_{z}}$ (for the CAF phase) are plotted with the frustration parameter $p=z_2J_2/z_1J_1$ (for the SC lattice $z_1=6$ and $z_2=12$). These coefficients vary significantly with $p$, which shows that the quartic interaction terms play a significant role in determining the different phases of our model.](Parameters_SC.eps){width="5in"} ![\[fig:MagSC\] Average sublattice magnetization, $\langle S_\alpha \rangle $, is plotted with the frustration parameter $p$ for AF and one of the three CAF phases with (solid lines) and without (dashed lines) quartic corrections. At zero temperature without the quartic $1/S$ corrections (linear spin-wave theory) $\langle S_\alpha \rangle \rightarrow 0$ at $p_{c1} \approx 0.48$ indicating a phase transition from the AF-ordered state to the disordered paramagnetic state. At $p_{c2}=0.50$ there is a second phase transition from the collinear state to the disordered state for $T=0$. Non-linear spin wave theory provides significant corrections to this phase diagram. With the quartic $1/S$ corrections the disordered PM region disappears completely and we only obtain two phases: AF and CAF. There is no existence of any disordered state as predicted by the linear spin-wave theory (quadratic corrections). For both the phases the quartic corrections to the Hamiltonian enhance the magnetic order.](Mag_SC.eps){width="5in"} ![\[fig:GSE\] Zero temperature ground state energy per site, $E/NJ_1$, is plotted versus $p$ without (dashed lines) and with (solid lines) quartic corrections for both AF ($p<0.5$) and CAF ($p>0.5$) ordered phases. Spin wave theory becomes unstable close to the classical transition point ($p_{\rm class} \approx 0.5$) between the two phases. After extrapolation (shown by the dotted line), we find that the two energies meet at the quantum transition point, $p_c \approx 0.56$ or $J_2/J_1 \approx 0.28$. This kink in the energy indicates a first-order quantum phase transition from the AF to CAF phase. Compared to the results without quartic corrections (long dashed lines) we find that the quartic corrections provide significant corrections to the ground state energy especially near the AF-CAF phase transition point.](Energy_SC.eps){width="5in"}

--- abstract: | A disk wind can cause perturbations that propagate throughout the disk via diffusive processes. On reaching the inner disk, these perturbations can change the disk luminosity, which in turn, can change the wind mass loss rate, $\dot{M}_w$. It has been argued that this so-called “wind driven relaxation cycle" might explain the observed variability in some disk accreting objects. Here, we study the response of the innermost mass accretion rate $\dot{M}_a$ to the loss of matter at different rates and radii. We allow the wind launching radius, $R_L$, to scale with $\dot{M}_a$. We computed a grid of time-dependent models for various $\dot{M}_w$-$\dot{M}_a$ and $R_{L}$-$\dot{M}_a$ dependencies. We find that the disk behaviour significantly differs for the ‘variable $R_L$’ case compared to the ‘fixed $R_L$’ case. In particular, much stronger winds are required to destabilize the disk in the former than the latter case. However, the $\dot{M}_a$ amplitude does not grow significantly even for unstable cases because the oscillations saturate at a low level either due to disk depletion or due to the wind being launched at very small radii, or both. This result implies that disk winds are unlikely to be responsible for state transitions as those require large changes in the inner disk. Despite modest changes at the inner disk regions, the disk surface density at large radii can vary with a large amplitude, i.e., from 0 to a few factors of the steady state value. This dramatic variation of the outer disk could have observable consequences.\ author: - Shalini - Daniel title: 'On the wind-driven relaxation cycle in accretion disks' --- Introduction {#sec:1} ============ Various astrophysical systems including X-ray binaries, young stellar objects (YSOs), cataclysmic variables (CVs) and active galactic nuclei (AGNs) are powered by accretion disk processes. The relatively high luminosity of these objects is due to the efficient conversion of accretion power into radiation. The luminosity generated from such accretion processes tend to vary with time which provides us with important clues as to the nature of the accretion disk, the accretor, and also the object or source supplying matter to the disk. For a constant rate of mass supply, time variability could be attributed to spatial and temporal variations in the disk structure or the strength and configuration of the disk magnetic field. Such variations can arise due to the extended nature of the disks, with the inner and outer radii differing by orders of magnitude (ranging between two orders in CVs, to about seven orders in AGNs). The surface properties at small radii differ from those at large radii and there is a host of different physical processes that may cause the accreting material to undergo time-dependent evolution. They include a variety of local instabilities such as convective, thermal or magneto-rotational instability [@BH98; @FL19]. Yet there are several non-local processes that could also lead to time variability. One of the consequences of the large radial extent is the huge variation in escape velocity throughout the disk. Thus, as long as a disk has a slightly concave surface, the high-energy radiation that is emitted by the inner disk can irradiate the outer disk leading to the formation of a high-temperature surface layer in which thermal speeds can exceed the escape velocity. This can drive a strong wind from outer radii and cause a disruption in the accretion flow. When information about this disruption reaches the inner disk, it changes the local emission which, in turn, affects the disk irradiation. Thus, the radiation from the inner disk acts as a coupling between the inner and outer disks. The self-irradiated disk is an example of a “self-regulated accretion" process [@Shields S86 hereafter] with feedback. The Compton-heated corona and eventual disk wind [@Shields1] have widespread applications in understanding the absorption lines observed in AGNs [@W96] and X-ray binaries [@PK02; @2010Lu; @M15; @WP18]. However, it is unclear as to what degree a disk wind can destabilize the accretion disk and be responsible for the observed variability in the luminosity and spectral energy distribution (SED). To assess this role of the disk winds, we may define a variable $\eta_{ w} \equiv \dot{M}_{ w}/\dot{M}_{ a}$, where $\dot{M}_{ w}$ and $\dot{M}_{ a}$ are the wind mass loss rate (at the outer disk region) and mass accretion rate onto an accretor, respectively. The ratio measures the efficiency of wind driving due to the accretion power and indicates how strongly the wind is coupled to the latter. The model of instantaneous response of wind-to-accretion and vice versa showed that a wind with $\eta_{ w}$ as low as one, destabilizes the disk [@Shields1]. However, taking the effect of viscosity into account, found that the accretion at the inner disk edge responded much slower to the change in the disk surface density, $\Sigma$, at large radii. Viscosity stabilizes the disk by producing a “delay" or “relaxation time" to the propagation of perturbations throughout the disk. Therefore, a much higher $\eta_{ w}$ was needed to generate variability in the disk. For systems with relatively high luminosities, the escape velocity from a disk at a given radius could be reduced by the radiation pressure on free electrons and due to opacity from spectral lines and bound-free processes. In X-ray binaries, the latter two are negligible because the gas is highly ionized and few lines as well as few bound transitions are present. Yet, as expected, and shown both numerically and theoretically, the radiation pressure on free electrons introduces a linear scaling between the launching radius of the thermal wind and the luminosity, $L$ [@PK02]. In AGNs and CVs, the radiation pressure on lines (line driving) can produce a wind with $\dot{M}_{ w} \propto L^{1/\alpha}$, where $\alpha$ is the force multiplier parameter. The value of $\alpha$ depends on the SED, the chemical composition, and the physical conditions in the gas but it generally ranges from  0.2 to 0.8 [e.g., @CAK]. The above scaling is quite universal as it holds for 1-D stellar wind [@CAK] as well as for 2-D [e.g., @PSD98; @P99] and even 3-D disk winds [@DP18]. In this paper, we study a more generalized model of disk wind coupling to accretion power. We verify our results against the classic case of linear dependence of wind on accretion rate and then extend our analysis to non-linear dependencies of the wind. This allows us to test our model against different possibilities. Our main focus is to study the dynamic variability of the launching radius of the wind and explore our model for different free parameters. The outline of the paper is as follows: In section \[sec:2\], we describe in brief the mathematical and computational techniques used in our analysis. In section \[sec:3\], we introduce our calculation and verification of the result (with increased numerical resolutions). Our generalized approach towards the problem allows us to look at the results of two different models holistically and identify several previously unexplored cases. Finally, in section \[sec:4\], we summarize the applications of disk oscillations that have been studied in the past and what future prospects it might hold. ![Schematic of the model described as “*$\delta$-function wind at $R_d$ with $\nu\propto R$*" studied in S86. Matter enters the disk at a constant rate at $R_d$ while the X-ray source accretes matter from the innermost disk radius. The accretion leads to X-ray irradiation of outer disk region which drives the wind. The inset shows the outermost radial grid, where diffusion of matter at a rate $\dot{M}$ and removal of matter by a wind at rate $\dot{M}_w$ are compensated by mass injection ($\dot{M}_{in}$). Diffusive process disperses matter throughout the disk.[]{data-label="fig:scheme"}](schematic.pdf) Methods {#sec:2} ======= Equations and analytical results {#sec:2.a} -------------------------------- We assume azimuthal symmetry and perform 1D simulations on a geometrically thin and optically thick disk along the radial direction. We have adopted similar formulations and notations as used in . Their approach involves solving the diffusion equation that describes the disk evolution [@P81; @1974L]. As in S86, we assume a constant rate of mass injection at the outermost disk radius, $R_d$, from an external source. This is true for all the cases studied henceforth. The loss of mass in the form of wind takes place at the very same radius for the first model, which corresponds to the model described by eq.(3.5) in S86 (see $\S$III.(b) in S86). Fig. \[fig:scheme\] shows a schematic of this model. We also assume a simple radius-dependent viscosity $\nu \propto R$, similar to that used in [@1974L], for all our models. The mass continuity and angular momentum conservation gives us the following diffusion equation, $$\begin{aligned} &\frac{\partial \Sigma}{\partial t} = \frac{1}{2\pi R} \frac{\partial \dot{M}}{\partial R} + S(R,t) \textrm{,}\label{eq4}\end{aligned}$$ where, $$\begin{aligned} \dot{M} =& 6\pi R^{1/2} \frac{\partial}{\partial R}(\nu \Sigma R^{1/2}) \label{eq5} \qquad \textrm{and,}\\ S(R,t) =& S_{in}(R) - S_w(R,t)\textrm{.}\end{aligned}$$ Here $\dot{M}$ is the mass accretion rate at a given radius, $\nu$ is the kinematic viscosity and $\Sigma$ is the local surface mass density. The net source term, $S$, accounts for both a steady mass input to the disk ($S_{in}$), as well as a mass loss due to wind ejection from the disk ($S_w$). The general mass input and output rates respectively, are defined by $$\begin{aligned} \dot{M}_{in} = 2\pi \int_{R_i}^{R_d} S_{in}(R) R\quad dR \qquad \textrm{and,}\end{aligned}$$ $$\begin{aligned} \dot{M}_w(<R) = 2\pi \int_{R_i}^{R} S_w(R^\prime) R^\prime\quad dR^\prime \textrm{,}\end{aligned}$$ where $R_i$ and $R_d$ are the innermost and outermost disk radii, respectively, and the mass loss is calculated up to an arbitrary radius R within which mass is being lost. The above equations are recast using dimensionless variables as in eqs. (2.5)-(2.10) in S86. We present these equations below for clarity of our method description, $$\begin{aligned} R_* \equiv & \quad R/R_d\textrm{,}\\ \nu_* \equiv & \quad \nu/\nu_0\textrm{,}\\ \dot{M}_* \equiv & \quad \dot{M}/\dot{M}_{in}\textrm{,}\\ \Sigma_* \equiv & \quad \Sigma/\Sigma_0\textrm{,} \\ t_* \equiv & \quad t/t_0\textrm{,}\\ S_* \equiv & \quad SR_d^2/\dot{M}_{in}\textrm{,}\end{aligned}$$ where $\nu_0$ is the characteristic viscosity, $\Sigma_0 \equiv \dot{M}_{in}/\nu_0$ is the characteristic surface density and $t_0 \equiv R_d^2/\nu_0$ is the characteristic viscous time scale. Using the coordinate transformation, $x = R_*^{1/2}$, equations \[eq4\] and \[eq5\] can be rewritten as, $$\begin{aligned} \frac{\partial \Sigma_*}{\partial t_*} =& \frac{1}{4\pi x^3} \frac{\partial \dot{M}_*}{\partial x} + S_*(x,t) \textrm{,} \label{eq:diff}\\ \dot{M}_* =& 3\pi \frac{\partial}{\partial x}(x\nu_* \Sigma_*) \textrm{.}\end{aligned}$$ As in the wind model studied by , we treat mass loss using a delta function (launching of wind at a given radius), and defined the main model parameter as $$\begin{aligned} C \equiv& \dot{M}_w/\dot{M}_a \textrm{.}\label{simrel}\end{aligned}$$ where $\dot{M}_a \equiv \dot{M}(R_{in})$. In this paper, we alternatively refer to this ratio as the wind efficiency, $\eta_w$. While discussing or referring to the classic case studied in , we use the ‘$C$’ notation for comparison. As demonstrated by , once the disk has attained a steady state, perturbations in disk surface density would either persist, grow, or decay with time. This forms the basis of the “self-regulated accretion" and $\eta_w$ (or $C$) determines how strongly coupled the wind is to the central accretion rate and hence, to the luminosity of the central source. At steady state, the mass conservation relation applied to the disk, gives $$\begin{aligned} \dot{M}_{in} =& \dot{M}_a^{(s)} + \dot{M}_w^{(s)} \textrm{,}\end{aligned}$$ where the superscript $(s)$ denotes the steady state value. The expression for steady state mass accretion rate normalized to the mass input rate $\dot{M}_{in}$ reads in the following way, $$\begin{aligned} \dot{M}_{a*}^{(s)} =& \frac{1}{1+C} \textrm{.}\end{aligned}$$ Using a straightforward radius-dependent viscosity law, performed an analytical calculation to find the critical value of $C$ that would lead to a perpetual oscillation in disk density about its steady state value. The amount of matter depleted through wind and accretion is continually replenished by the constant supply of matter. This results in a variability of mass accretion rate and luminosity at the inner edge of the disk. For the parameter $C$, obtained the value required for sustained stable (critical) oscillations analytically as, $$\begin{aligned} C_{crit} = \cosh \pi \approx 11.6 \label{eq3} \textrm{,}\end{aligned}$$ such that when $C < C_{crit}$, the oscillations decay, whereas for $C > C_{crit}$, they grow. The wind mass loss rate might not be a linear function of mass accretion rate. considers such a possibility and they show based on their analytical treatment that for a power-law dependence, $C \propto \dot{M}_a^{k_l}$, where $k_l$ is an arbitrary constant, the $C_{crit}$ would be reduced by a factor of $(1+k_l)^{-1}$ (see analogous equation 4.5 in ). The case with $k_l=0$ corresponds to the case discussed above, where equation \[eq3\] gives the value of $C_{crit}$. We formally approach this possibility by generalizing our equation \[simrel\] and writing it in the following way: $$\begin{aligned} \dot{M}_{w*} \equiv C^\prime \dot{M}_{a*}^p \textrm{.}\label{eq7}\end{aligned}$$ where the subscript $*$ stands for mass loss rates normalized to the input mass rate $\dot{M}_{in}$, where any remaining constant of proportionality has been absorbed into $C^\prime$, and $p$ is an arbitrary constant exponent. Note that equation \[eq7\] reduces to equation \[simrel\] when $p=1$. The requirement for the steady state condition becomes $$\begin{aligned} \dot{M}_{a*}^{(s)} +& C^\prime (\dot{M}_{a*}^{(s)})^p - 1 = 0 \textrm{,}\label{eq1}\end{aligned}$$ but this equation needs to be solved numerically. In the above analyses, a fixed wind launching radius has been assumed. However, in general, this may not be an ideal condition. Our main focus here is to examine the effects of relaxing this assumption by allowing the radiation from the inner disk to irradiate the entire disk and causing reduced local escape velocity. We can express luminosity $L$ in units of the Eddington factor $\Gamma$, such that $L = \Gamma L_{Edd}$, where $L_{Edd}$ is the Eddington luminosity. Here we assume that the irradiation luminosity $L$ of the disk equals to the total accretion luminosity. We express the coupling between the launching radius and $\dot{M}_{a}$ using the following expression: $$\begin{aligned} R_{L*} = 1 - \Gamma \frac{\dot{M}_{a*}}{\dot{M}_{a*}^{(s)}} \textrm{,}\label{eq2}\end{aligned}$$ where $R_{L*}$ is the launching radius normalised to $R_d$. Our equation \[eq2\] is similar to equation (22) in @PK02 that was derived for the launching radius of a Compton-heated wind, corrected for radiation driving. When $\Gamma=0$, $R_L=R_d$ and we have the case studied by . The only numerical constraint is that $R_L\geq0$. Using their analytical method for the simplest case ($\Gamma=0$, $p=1$), derived an expression for the period of oscillations as $$\begin{aligned} P_* \propto \frac{R_L}{R_d}\end{aligned}$$ We expect that in our variable $R_L$ model, the disk stability condition and the variability period will be sensitive to $\Gamma$. Numerical methods {#sec:2.b} ----------------- We have used numerical methods similar to that used in [@BP]. We have developed a Python code to study the effects of self-regulated accretion as discussed in the previous section. The wind launching zone is a delta function, with mass loss taking place from a single radial grid zone. $$\begin{aligned} S_{w*} = \frac{C\dot{M}_{a*}}{\pi (R_{j*}^{\prime 2}-R_{j-1*}^{\prime 2})} \label{eq:wind}\end{aligned}$$ where $R_{j*}^\prime=x_j^{\prime 2}$ and $x_j^\prime = (x_j + x_{j+1})/2$. Here $R_j$ refers to the wind launching radius and $R_{j*}^\prime$ is the averaged launching radius value used to calculate $S_w$. The disk surface density $\Sigma_*$ is assigned an initial value of 0. We take $\dot{M}_{in}=1$, since all mass rates are normalised to $\dot{M}_{in}$ and together with eq.\[eq:wind\], calculate the source term $S_*$. The diffusion equation \[eq:diff\] is then solved using forward difference method while updating $\Sigma_*$. The boundary condition is obtained by imposing the condition that mass flux is finite and conserved at the disk edges, i.e. $\partial(\nu\Sigma)/\partial R = 0$. Our resolution study shows that the calculated value of the accretion rate depends on the width of each radial zone. In particular, the variable $\Gamma$ model is sensitive to smaller resolution in $x$, $N_x$ ($N_x<$ 200). Hence, for most of our models presented below, we use $N_x=200$ (see $\S$\[sec:3\] for more discussion). To ensure that the value of $\Sigma$ remains realistic at all times, it is quite common to impose the condition $\Sigma_*=0$, whenever the numerical solution leads to $\Sigma_*<0$. This condition is especially important when we deal with growing oscillations. Instead of allowing the disk to deplete completely, we introduce a floor value of 0.001 for $\Sigma$. Three time scales that enter this problem are: 1) The mass outflow time scale, $t_{\textrm{out}}$, 2) mass inflow time scale, $t_{\textrm{in}}$ and, 3) diffusion time scales, $t_{\textrm{diff}}$. These scales can be defined as $$\begin{aligned} t_{\textrm{out}} = \dot{\Sigma}(R_L)/\dot{\Sigma}_w(R_L)\textrm{,}\\ t_{\textrm{in}} = \dot{\Sigma}(R_{\textrm{in}})/\dot{\Sigma}_{\textrm{in}}(R_L)\textrm{ and,}\\ t_{\textrm{diff}} = 0.25 \frac{4\Delta x^2}{3}\label{eq:t} \textrm{.}\end{aligned}$$ where eq. \[eq:t\] expresses the stability criterion of eq. \[eq:diff\] which is a diffusion equation (see e.g., Press et al. 2007). In our numerical calculations, we choose a time step to be 20$\%$ of the shortest of the above three time scales. The initial condition is to set the disk surface density to be 0 and let matter diffuse from the surrounding source until the accretion disk reaches a steady state. We then perturb it by switching on the wind. In practice, we allow the system to reach $(1-\epsilon)\dot{M}_{a*}^{(s)}$, where $\epsilon$ is a very small number of the order of $10^{-3}$. It takes roughly 3 viscous time scales to reach this near steady state. We use the *fsolve* method in SciPy’s [^1] optimization library to find the root of equation \[eq1\]. Results {#sec:3} ======= First we checked the results from our simulations against the results presented in . Our resolution is higher than theirs by a factor of 10. This increased resolution does not considerably affect the key result although it does substantially reduce the amplitude of oscillation. This can be attributed to the fact that we have limited our analysis to a single zone wind regardless of the radial resolution. Namely, we still inject and eject the same amount of matter but now from a radial ring with smaller area. We find that $C_{crit} \approx 11.3$ as opposed to 11.4 obtained by . Fig.  \[fig:shields result\] illustrates our results in a similar manner to the fig.$\sim$ 1 by . As expected, we observe damped oscillation for $C<11.3$ and growing oscillations for $C>11.3$. We note that we consider only those oscillations to be stable whose amplitudes vary by less than $10^{-4}$. This condition is consistently followed throughout our analysis, and is used to classify the oscillations into categories. ![Central mass accretion rate (normalized to the steady state mass accretion rate) evolution of the disk for different wind strength parameter $C$. The top panel shows decayed oscillations in the disk, the middle panel shows the critical $C$ case where the oscillation persists with constant amplitude. The bottom panel shows a growing oscillation phase, which saturates after some time due to local disk depletion.[]{data-label="fig:shields result"}](shields) In the growing oscillation cases, the mass accretion rate grows until it saturates after some time. This saturation of oscillation is caused by the local surface density reaching negligible values or, in other words, complete depletion of matter from that region of the disk. We expect that if S86 continued their calculations to longer times, they would likely find the same behaviour. For $C=14.4$ shown above, our simulations showed that during this phase, $\Sigma_{*,min}$ approaches 0, whereas $\Sigma_{*,max}$ approaches 2.5 times its steady state value. The outcome of this instability is a small amplitude $\dot{M}_{a*}$ oscillation but a large modification of the disk solution. Table  \[tab:1\] summarizes the combination of $p$ and $C^\prime$ (for $\Gamma=0$) values used in our simulations. The wind to accretion ratio sets the stability of the disk. The efficiency factor $\eta_w$ is a function of time and hence, we consider the value of $\eta_w$ at the beginning of the perturbation. The value of $\eta_w$ decreases with increasing $p$ with a slope of about -1 on a log-log plot (Fig. \[fig:sl\]). This confirms the inverse relationship between $\eta_{w,\textrm{crit}}$ and $p$, that we discussed in $\S$\[sec:2.a\] and demonstrates that the disk is easily destabilized if the wind is more strongly coupled to accretion (i.e. higher $p$). ![The critical wind efficiency $\eta_{w,crit}$ as a function of the power-law index, $p$, in accordance with eq. \[eq7\].[]{data-label="fig:sl"}](pcons) p $C^\prime_{d}$ $C^\prime_{crit}$ $C^\prime_{g}$ ----- ---------------- ------------------- ---------------- 0.5 4.5 4.95 5 2/3 6 7.05 7.5 3/4 8 8.6 9 4/5 9 9.5 9.9 1 11 11.3 14.4 5/4 18.5 20 24 4/3 21 22.8 24 3/2 20.5 26 30.5 7/4 37 39 40.5 2 30.5 48.5 50.5 : Summary of all parameters ($C^\prime$ and p) used in simulations for the model described by eqn \[eq7\]. The subscripts d, crit and g denote values of $C^\prime$ for which the oscillations decay, persist and grow, respectively. The value of $C_{crit}^\prime$ increases with increasing $p$. In Fig. \[fig:sl\], we use the corresponding $C^\prime_{crit}$ to plot the wind efficiency $\eta_{w,crit}$, which decreases with increasing $p$. \[tab:1\] Table \[tab:2\] contains our parameter survey for the variable $R_L$ model. The $\Gamma$ factor strongly controls and alters the outcome of disk evolution. In particular, we identified new cases for high $\Gamma$ ($\Gamma\geq 0.7$). In low $\Gamma$ cases ($\Gamma\sim 0.2$), we observe some deviation from the classical cases of stable oscillations. ![Examples of the time evolution of the accretion rate for $\Gamma=0.2, p=0.5$, and various $C^\prime$. The top panel shows an example of an initial decay that is followed by a constant amplitude oscillation. The second, third and fourth panels depict the range of $C_{crit}^\prime$ that lead to stable oscillations.[]{data-label="fig:g2"}](consolidated1) We limit our presentation to 3 cases, $p = 0.5, 1$ and $2$. The case $p = 1$ corresponds to the special case studied in . The other two cases are representative of the lowest and highest $p$ values considered (see Table \[tab:1\]). For $0.001\leq \Gamma \leq 0.1$, the disk behaviour does not deviate much from the classical behaviour depicted in Fig. \[fig:shields result\]. However, for $\Gamma\geq0.2$, we find some new results. We describe these cases in detail below. ![As Fig. \[fig:g2\] but for $\Gamma=0.7, p=1$. The top three panels indicate a damped high amplitude oscillation superposed on the fundamental mode oscillating around the steady state value of 1. The lowest panel shows growing oscillations that saturate but has a very high frequency of oscillation, which increases with time.[]{data-label="fig:g4"}](consolidated2) - $\Gamma$=0.2 and 0.3: We observe more than one $C^\prime_{crit}$ value resulting in stable oscillations. For example, constant amplitude oscillations occur when $C_{crit}^\prime$ is between 6.2 and 12. The lowermost panel of Fig. \[fig:g2\] shows how the sinusoidal nature of the stable oscillations starts to change for $C^\prime\gtrsim 13$. We also find that for increasing $C^\prime$, the frequency of oscillation starts to increase. For $\Gamma=0.3$, the results are similar to those of $\Gamma=0.2$. - $\Gamma=0.4$: For $p=0.5$ and $p=1$, we find a single value of $C^\prime_{crit}$. For $p=2$, we see that $C_{crit}^\prime$ lies between 130 and 300. The range of $C_{crit}^\prime$ and the critical value of $C^\prime$ rise considerably. - $\Gamma = 0.5$: For all $p$ values, we obtain single-valued $C^\prime_{crit}$. We note that $C^\prime_{crit}$ for $\Gamma=0.5$ is less than that for $\Gamma=0.4$. This result holds true for all three p-values investigated. - $\Gamma = 0.7$: Fig. \[fig:g4\] summarizes the distinct cases obtained for $p=1$. We see that up to a certain value of $C^\prime$, the accretion rate initially increases and at the same time oscillates with relatively high frequency and small amplitude. In the case of $C^\prime=6.5$, we see that eventually the fundamental oscillation dominates and steadies around 1. As $C^\prime$ increases, we see a growing oscillation that eventually saturates. - **$\Gamma = 0.9$:** The behaviour is similar to the $\Gamma=0.7$ case. For higher $C^\prime$, the frequency increases rapidly with time. The amplitude of oscillations in $\dot{M}_{a*}$ remain constrained to $\sim 50\%$ of its steady state value in all our simulations. This can be mostly attributed to the amount of matter available to be launched as wind and also the constraint on the lowest possible launching radius. In all of the cases studied, the amplitude of oscillation increases for increasing $\eta_w$. As stated in $\S$\[sec:2.b\], we have chosen a spatial resolution of $N_x = 200$ for the above cases. Our resolution study showed that there are quantitative changes in most cases as well as qualitative changes in some of the extreme cases. For example, for $p=1, C^\prime=6, \Gamma=0.7$, the oscillations cease earlier for $N_x=400$, while for $N_x=800$, they persist for a longer number of time steps. In addition, the amplitude of oscillations also decrease with increasing resolution. For higher $N_x$, there is a clear tendency towards convergence. This resolution study was conducted for several other cases in our parameter survey. Convergence was evident for most cases with $\Gamma\leq 0.7$. For higher $\Gamma$, the behaviour was more erratic and unpredictable for different $N_x$. We restricted our $N_x$ to 200 despite this fact, since the nature of oscillations and the feedback on $\dot{M}_w$ remained unaffected from a qualitative perspective. ![The steady state critical wind efficiency $\eta_{w}$ vs $p$ for different $\Gamma$ values. For some $\Gamma$, stable oscillations occur not just for a single value of $\eta_w$, but for a range of $\eta_w$ (see Fig. \[fig:g2\] for some examples, e.g. second, third and fourth panels there). We shaded the regions for $\Gamma$s where this happens.[]{data-label="fig:g1"}](gcons.png) In Fig. \[fig:g1\], we show the steady state critical $\eta_w$-$p$ relation for various $\Gamma$. We find that the slope of this relation is nearly constant for $\Gamma\leq 0.2$. For higher $\Gamma$, the slope changes and more than one $\eta_{w,crit}$ exists. For such cases we have shaded the region between all the possible straight line fits. For $\Gamma>0.5$, oscillations are distinctly different from that of the classical cases and we cannot group them under simple categories (see above and Fig. \[fig:g4\]). We do not plot these points in Fig. \[fig:g1\] or list them in our table of classification of disk oscillations. To visualize our results for the wind efficiency in a different way, we also plot steady state critical $\eta_w$ as a function of $\Gamma$ for different $p$ values (the upper panel of Fig. \[fig:g3\]). The curves for different $p$ values generally resemble each other. For $\Gamma\leq 0.2$, $\eta_{w,crit}$ increases with increasing $\Gamma$. For higher $\Gamma$, $\eta_w$ decreases with increasing $\Gamma$, in all three $p$ cases. To more directly compare the results for various $p$, in the bottom panel of Fig. \[fig:g3\], we plot $p\eta_{w,crit}$ vs $\Gamma$. For $\Gamma<0.1$, we see that $p\eta_{w,crit}$ is nearly constant, which is what we concluded from Fig. \[fig:sl\]. ![The steady state critical wind efficiency $\eta_{w,crit}$ vs $\Gamma$ for different $p$ values (top panel). In the bottom panel, we plot $p\eta_{w,crit}$ vs $\Gamma$ for different $p$ values. Once again, the shaded region highlights the possible $\eta_{w,crit}$ values for a particular combination of $p$ and $\Gamma$.[]{data-label="fig:g3"}](gcons2.png) Concluding Remarks {#sec:4} ================== The study of disk winds in the context of state transition has evoked a lot of interest over the past decades [e.g. @RF05; @RF06]. The wind launching mechanism and location may lead to time variability, the effects of which could be coupled to the signatures of mass accretion by the accretor. We explore this aspect through a self-regulated accretion disk. We found that our model is unlikely to explain state transitions in accretion disk spectra. However, it could be responsible for persistent small amplitude regular single-mode oscillations in the mass accretion rate. Admittedly, our treatment is very simple. We use one viscosity law in all our analyses, $\nu_* = R_* = x^2$. We did not consider thermodynamic effects. We also did not incorporate magnetic fields in our analysis. Moreover, we assumed a $\delta$-function model for the wind, which exaggerates the role of mass removal. Despite the simplifications, this analysis might be a stepping stone towards developing models that can account for the existence of state transition signatures in accretion disks. Time variability in accretion rates may be explained by physical processes considered here. They are rich in features and may hold the key to understanding the coupling forces operating in an accretion disk. For example, the study of GRS 1915+105 by [@N11] is a very detailed analysis in that direction. They nicknamed the oscillations in X-ray spectra as the ‘heartbeat’ state and conducted a study of the geometry of the accretion disk using the X-ray continuum and emission lines in the optical spectrum. They demonstrate a strong correlation between mass loss in the form of wind and oscillations in the accretion rate that would explain the long-term effects in the disk. A study of the same source, GRS 1915+105, by @Z16 considered in detail the reflection spectrum during the oscillatory phase of the source. Their calculations indicate winds being launched from very small disk radius which remains unchanged during this phase. Our variable $R_L$ model allows the launching of wind from as close as the innermost disk region for high $\Gamma$ cases. The time evolution of the accretion rate from our simulation does not indicate any distinct spike or sharp flare, which leads us to conclude that the system does not produce outbursts. [@HP17] used their photoionization modelling of the SEDs to argue that thermally driven winds may hold the key to explaining the state changes in such systems. On the other hand, @N13 goes on to demonstrate how heavy outflows not only quench the disk, thus affecting the formation of jets and causing state transition, but also influence the further production of winds. Disk variability and the state changes could also be caused or affected by instabilities of the disk itself. For instance, [@J02] studies the effect of radiation instabilities leading to limit-cycle behaviour and modulations in accretion rates. As opposed to our model, this radiation-driven instability may result in sharp spikes indicating a high outflow from the system. There are a number of studies related to fast outflows with high mass loss rates leading to state transitions and subsequent detection of jets in the system [@NL09; @K13; @G19]. A similar situation has been discussed in @RF19, where a particular case of V404 Cyg indicates the presence of massive outflows, almost 2 orders of magnitude higher than central mass accretion rate $\dot{M}_a$. They speculated that these outflows are produced by radiation-driven winds coupled with classical thermal winds. This work stands out as providing direct observational evidence of powerful outflows leading to a quenching of accretion. The optical $H_\alpha$ line profile clearly indicates disk contraction following the massive outflow phase, consistent with what we would expect happens when irradiation is reduced. Our analysis of the $\Sigma$ radial profile showed that $\Sigma$ approaching 0 is responsible for saturating the oscillations. Recently, @T19 showed that the detection of a blue-shifted line from a black hole binary source H1743-322 strongly suggests a thermal-radiative disk wind. Their work indicates a disappearing wind in the hard state which could be attributed to the shadowing of outer disk region by the inner corona. They also went on to state that the absorption features in other black hole sources such as GRS 1915+105 and GRO J1655-40 are most likely due to thermal-radiative winds as opposed to previously speculated magnetic effects. Another recent paper @D19 studies the effect of thermal-viscous instability on the light curves and stability diagrams associated with black hole binary systems. Additionally, they consider a fraction of the X-ray irradiation to be scattered by the wind and partially impinge on the outer disk regions. This has a stabilizing effect and can explain the shortened outbursts but cannot explain the rapid decay of outbursts. They studied a particular BHXB, GRO J1655-40, for which their model was able to reproduce the observed features of the light curve. They speculate that magnetic fields would need to be considered for a more promising explanation for the outbursts. These studies indicate several possible aspects of our simplistic approach towards the study of wind-accretion coupling. The criteria for instability derived by is often invoked when discussing consequences of observed or model disk winds [e.g., @2010Lu]. The main conclusion of our work is that upon satisfying this criteria, a disk wind might not be responsible for large scale variations in luminosity because instability saturates at a relatively low level in terms of $\dot{M}_a$. However, it could result in a harder-to-detect significant reduction of $\Sigma$ at large radii. This work was supported by NASA under ATP grant 80NSSC18K1011. We thank Drs. Tim Waters and Rebecca Martin for their valuable comments and discussions. natexlab\#1[\#1]{}\[1\][[\#1](#1)]{} \[1\][doi: [](http://doi.org/#1)]{} \[1\][[](http://ascl.net/#1)]{} \[1\][[](https://arxiv.org/abs/#1)]{} , S. A., & [Hawley]{}, J. F. 1998, Reviews of Modern Physics, 70, 1, , G. T., & [Pringle]{}, J. E. 1981, , 194, 967, , M. C., [McKee]{}, C. F., & [Shields]{}, G. A. 1983, , 271, 70, , J., [Mu[ñ]{}oz-Darias]{}, T., [Mata S[á]{}nchez]{}, D., [et al.]{} 2019, , 488, 1356, , J. I., [Abbott]{}, D. C., & [Klein]{}, R. I. 1975, , 195, 157, , G., [Done]{}, C., [Tetarenko]{}, B. E., & [Hameury]{}, J.-M. 2019, arXiv e-prints. , S., & [Proga]{}, D. 2018, , 481, 5263, , R., [Belloni]{}, T., & [Gallo]{}, E. 2005, , 300, 1, , S., & [Lesur]{}, G. 2019, in EAS Publications Series, Vol. 82, EAS Publications Series, 391–413 , E., [D[í]{}az Trigo]{}, M., [Miller-Jones]{}, J. C. A., & [Migliari]{}, S. 2019, , 482, 2597, , N., [Proga]{}, D., [Knigge]{}, C., & [Long]{}, K. S. 2017, , 836, 42, , A., [Czerny]{}, B., & [Siemiginowska]{}, A. 2002, , 576, 908, , A. L., [Miller]{}, J. M., [Raymond]{}, J., [et al.]{} 2013, , 762, 103, , E. G., [Jester]{}, S., & [Fender]{}, R. 2006, , 372, 1366, , S., [Proga]{}, D., [Kallman]{}, T. R., [Raymond]{}, J. C., & [Miller]{}, J. M. 2010, , 719, 515, , D., & [Pringle]{}, J. E. 1974, , 168, 603, , J. M., [Fabian]{}, A. C., [Kaastra]{}, J., [et al.]{} 2015, , 814, 87, , J. 2013, Advances in Space Research, 52, 732, , J., & [Lee]{}, J. C. 2009, , 458, 481, , J., [Remillard]{}, R. A., & [Lee]{}, J. C. 2011, , 737, 69, , J. E. 1981, , 19, 137, , D. 1999, , 304, 938, , D., & [Kallman]{}, T. R. 2002, , 565, 455, , D., [Stone]{}, J. M., & [Drew]{}, J. E. 1998, , 295, 595, , G. A., [McKee]{}, C. F., [Lin]{}, D. N. C., & [Begelman]{}, M. C. 1986, , 306, 90, , R., [Done]{}, C., [Ohsuga]{}, K., [Nomura]{}, M., & [Takahashi]{}, T. 2019, arXiv e-prints. , T., & [Proga]{}, D. 2018, , 481, 2628, , D. T., [Klein]{}, R. I., [Castor]{}, J. I., [McKee]{}, C. F., & [Bell]{}, J. B. 1996, , 461, 767, Zoghbi, A., Miller, J. M., King, A. L., [et al.]{} 2016, arXiv e-prints, [| c | &gt;m[2cm]{} | &gt;m[2.5cm]{} | &gt;m[2.5cm]{} | &gt;m[2.5cm]{} | c|]{}\ **$\Gamma$**& **$C^\prime_{sd}$** & **$C^\prime_d$** & **$C^\prime_{crit}$** & **$C^\prime_g$** & **$C^\prime_{sg}$**\ 0.001 & - & 3.5 $\longrightarrow$ 4.7$^*$ & 4.71 & 4.715 $\longrightarrow$ 4.75 & 4.8\ 0.01 & - & 4 $\longrightarrow$ 4.71 & 4.72 & 4.8 $\longrightarrow$ 5 & -\ 0.05 & - & 4.7 $\longrightarrow$ 4.75 & 4.772 & 4.8 $\longrightarrow$ 5 & -\ 0.1 & - & 5 $\longrightarrow$ 5.05 & 5.067 & 5.1 $\longrightarrow$ 5.2 & -\ 0.2 & 5.2 $\longrightarrow$ 5.5 & 5.8 $\longrightarrow$ 6 & 6.2 $\longrightarrow$ 12 & - &\ 0.3 & - & 4 $\longrightarrow$ 5.45 & 5.48 $\longrightarrow$ 6.2 & 6.5 $\longrightarrow$ 9 & 10 $\longrightarrow$ 15\ 0.4 & 4 $\longrightarrow$ 4.417 & - & 4.418 & - & 4.419 $\longrightarrow$ 5\ 0.5 & 3.4 $\longrightarrow$ 3.6 & -$^{**}$ & 3.63 & - & 3.7 $\longrightarrow$ 4\ \ **$\Gamma$** & **$C^\prime_{sd}$** & **$C^\prime_d$** & **$C^\prime_{crit}$** & **$C^\prime_g$** & **$C^\prime_{sg}$**\ 0.001 & - & 11.4 & 11.6 & - & 11.65\ 0.01 & - & 11.6 & 11.65 & 11.7 & -\ 0.05 & - & 11.8 & 11.87 $\longrightarrow$ 11.9 & 11.95 & -\ 0.1 & - & 12.6 & 12.65 & 12.7 & -\ 0.2 & - & 16.5 & 16.55 & 16.6 & -\ 0.3 & 18 & - & 20 $\longrightarrow$ 32 & - & 45\ 0.4 & 15 & - & 15.93 & - & 20 $\longrightarrow$ 25\ 0.5 & - & 11.8 $\longrightarrow$ 11.88 & 11.89 & - & 11.9 $\longrightarrow$ 11.95\ \ **$\Gamma$**& **$C^\prime_{sd}$** & **$C^\prime_d$** & **$C^\prime_{crit}$** & **$C^\prime_g$** & **$C^\prime_{sg}$**\ 0.001 & - & 39.4 & 35 $\longrightarrow$ 39.3 & 39.5 $\longrightarrow$ 40 & 50\ 0.01 & - & 11 $\longrightarrow$ 39.5 & 39.7 & 40 & -\ 0.05 & - & 39.7 $\longrightarrow$ 41 & 41.3 & 41.5 $\longrightarrow$ 42 & -\ 0.1 & - & 42 $\longrightarrow$ 43.5 & 43.7 & 44 & -\ 0.2 & - & 43 $\longrightarrow$ 55 & 60.35 & 62 $\longrightarrow$ 65 & -\ 0.3 & - & 60 $\longrightarrow$ 100 & 107.8 & 110 $\longrightarrow$ 120 & -\ 0.4 & - & 110 & 130 $\longrightarrow$ 300 & 500 & -\ 0.5 & 78 & 80 $\longrightarrow$ 83 & 83.7 & 85 & 100\ \[tab:2\] [^1]: Python 3.6.7

--- abstract: 'We simulate the dynamics of a quantum dot coupled to the single resonating mode of a metal nano-particle. Systems like this are known as metamolecules. In this study, we consider a time-dependent driving field acting onto the metamolecule. We use the Heisenberg equations of motion for the entire system, while representing the resonating mode in Wigner phase space. A time-dependent basis is adopted for the quantum dot. We integrate the dynamics of the metamolecule for a range of coupling strengths between the quantum dot and the driving field, while restricting the coupling between the quantum dot and the resonant mode to weak values. By monitoring the average of the time variation of the energy of the metamolecule model, as well as the coherence and the population difference of the quantum dot, we observe distinct non-linear behavior in the case of strong coupling to the driving field.' author: - 'Daniel A. Uken' - Alessandro Sergi title: 'Quantum dynamics of a plasmonic metamolecule with a time-dependent driving' --- Introduction ============ Quantum plasmonics is a relatively new area of research that studies the interaction of surface plasmons with quantum emitters [@plasmonics-review-tame; @plasmonics-chen; @plasmonics-zuloaga; @plasmonics-marinica; @plasmonics-bouillard]. The surface plasmon and the quantum emitter constitute what is known as a metamaterial or metamolecule. Such plasmonic metamaterials, at variance from those that operate in the microwave regime [@metamaterials-valentine; @metamaterials-pratibha; @metamaterials-leonhardt], require the downscaling to nanometers. Metal nano-particles (MNP’s) seem to be an ideal candidate in order to build metamaterials at optical frequencies [@mnp-lance-kelly; @mnp-lal; @mnp-nehl; @mnp-pelton]. They also have the convenient property that their resonant frequency can be tuned by changes in their geometries [@mnp-lance-kelly]. A typical example of such systems is provided by metamolecules comprising MNP’s coupled to quantum dots (QD’s) [@savasta; @plasmonics-tame]. These advances strengthen the need for techniques capable of simulating the dynamics of MNP’s coupled to QD’s. In the past two decades, much progress has been made in the development of methods for the simulation of quantum dynamics, involving path integral formulations [@path], mean field approximations [@meanfield], semiclassical approximations and surface-hopping schemes [@tully; @miller; @pechukas1; @pechukas2; @heller; @shenvi]. An alternative approach is provided by the partial Wigner representation of quantum mechanics [@wigner1; @wigner2; @wigner3; @wigner4; @wigner5; @wigner6]. In such a representation, exact algorithms for subsystems embedded in harmonic environments can be developed (when the coupling to the environment is bilinear) [@num-app-quant]. In this paper, we present a method for simulating, within the partial Wigner representation of quantum mechanics, a MNP-QD metamolecule subject to an external driving field. To this end, we use a piece-wise deterministic algorithm which utilizes a time-dependent basis. We study the dynamics of the population difference of the QD, in the case of weak coupling to the MNP, when the QD is subjected to an external driving field of varying strengths. By monitoring the average of the time variation of the energy of the metamolecule model, as well as the coherence and the population difference of the quantum dot, we observe distinct non-linear behavior in the case of strong coupling to the driving field. For validating our approach, we perform a comparison with the results obtained by employing a brute-force numerical approach, which uses a discretized phase-space grid and deals with the partial differential equation system associated with the evolution of the density matrix in the partial Wigner representation. The structure of the paper is as follows. In Sec. \[sec:sec2\], the formalism of quantum dynamics in the partial Wigner representation is presented. Section \[sec:sec3\] outlines the generalization of the formalism to a time-dependent Hamiltonian. Section \[sec:sec4\] introduces the model for the QD coupled to the MNP. In the same section, a brief outline of the propagation algorithm is discussed. In Sec. \[sec:sec5\], the results of the numerical simulations are presented. Finally in Sec. \[sec:sec6\], we give our conclusions. Appendix \[app:app1\] details the adimensional coordinates used in our study while App. \[app:app2\] outlines the phase-space-grid algorithm used to verify the results. Quantum dynamics in the partial Wigner representation {#sec:sec2} ===================================================== Consider a system defined by the following Hamiltonian operator: $$\begin{aligned} \hat{H} = \hat{H}_{\rm S}(\hat{r},\hat{p}) + \hat{H}_{\rm B}(\hat{R},\hat{P}) + \hat{H}_{\rm C}(\hat{r},\hat{R}) + \hat{H}_{\rm E}(\hat{r},\hat{p},\hat{R},\hat{P},t) \,,\end{aligned}$$ where $\rm S$, $\rm B$ and $\rm C$ are subscripts denoting the subsystem, bath and the coupling respectively. The Hamiltonian $H_{\rm E}(t)$ describes an external time-dependent field which may interact with both the subsystem and bath. The lower case coordinates describe the subsystem degrees of freedom, while the upper case coordinates describe the bath. The equation of motion for an arbitrary operator $\hat{\chi}(t)$ in the Heisenberg picture is written in symplectic form as $$\begin{aligned} \frac{\partial}{\partial t}\hat{\chi}(t)=\frac{i}{\hbar} \left[\begin{array}{cc}\hat{H} & \hat{\chi}(t)\end{array}\right] {\mathcal{B}}^c \left[\begin{array}{c}\hat{H}\\ \hat{\chi}(t)\end{array}\right]\,,\end{aligned}$$ where the matrix elements of the symplectic matrix [@goldstein] are defined as $B_{ij}^c=\epsilon_{ij}$, with $\epsilon_{ij}$ being the complete antisymmetric tensor. It is assumed that the Hamiltonian of the bath depends on a pair of canonically conjugate operators, $\hat{X} = (\hat{R}, \hat{P})$, and that the coupling Hamiltonian $\hat{H}_{\rm C}$ depends only on the position coordinates and not on the momenta. The partial Wigner transform for the operator $\hat{\chi}$ is defined as $$\begin{aligned} \hat{\chi}_{\rm W}(X) = \int dz\, e^{i Pz/\hbar}\Big\langle R - \frac{z}{2}\Big|\hat{\chi}\Big| R + \frac{z}{2}\Big\rangle\,,\end{aligned}$$ where $X= (R,P)$ is the phase space point, defined in terms of the canonically conjugate positions and momenta. Upon taking the partial Wigner transform of the Heisenberg equation, one obtains the Wigner-Heisenberg equation of motion: $$\begin{aligned} \frac{\partial}{\partial t}\hat{\chi}(X,t) = \frac{i}{\hbar}\left[\begin{array}{cc} \hat{H}_{\rm W}(X) & \hat{\chi}_{\rm W}(X,t)\end{array}\right]{\mathcal{D}} \left[\begin{array}{c}\hat{H}_{\rm W}(X) \\ \hat{\chi}_{\rm W}(X,t)\end{array}\right]\,, \label{eq:WH-equation-of-motion}\end{aligned}$$ where $$\begin{aligned} {\mathcal{D}} = \left[\begin{array}{cc} 0 & e^{\frac{i\hbar}{2}\stackrel{\leftarrow} \partial_{k}{\mathcal{B}}^{c}_{kj}\stackrel{\rightarrow}\partial_{j}} \\ -e^{\frac{i\hbar}{2}\stackrel{\leftarrow}\partial_{k}{\mathcal{B}}^{c}_{kj} \stackrel{\rightarrow}\partial_{j}} & 0 \end{array}\right]\,.\end{aligned}$$ The symbols $\stackrel{\leftarrow}\partial_{k} = \stackrel{\leftarrow}\partial/\partial X_{k}$ and $\stackrel{\rightarrow}\partial_{k} = \stackrel{\rightarrow}\partial/\partial X_{k}$ denote the operators of derivation with respect to the phase-space coordinates acting to the left and right respectively. Summation over repeated indices is implied. The partial Wigner-transformed Hamiltonian takes the form: $$\begin{aligned} \hat{H}_{\rm W}(X,t)=\hat{H}_{\rm S}+H_{\rm B,W}(X)+\hat{H}_{\rm C,W}(R) +\hat{H}_{\rm E,W}(X,t)\,. \label{eq:H_W-total}\end{aligned}$$ When the Hamiltonians $\hat{H}_{\rm B,W}$, $\hat{H}_{\rm C,W}$ and $\hat{H}_{\rm E,W}(X,t)$ are at most quadratic in $R$ and $P$, the action of the terms in the matrix $\mathcal{D}$ are equivalent to their linear order Taylor series expansion, since any higher order terms acting upon the Hamiltonian will yield zero. Under these conditions, algorithms for simulating exact quantum dynamics in the partial Wigner representation can be devised. The above theory is also applicable in the case when the external field Hamiltonian, $\hat{H}_{\rm E}$, depends upon the bath coordinates, however, in the following we restrict our study to situations where it depends only upon the subsystem coordinates. Representation in a time-dependent basis {#sec:sec3} ======================================== When considering Eq. (\[eq:WH-equation-of-motion\]), one can define the following time-dependent Hamiltonian $$\begin{aligned} \hat{h}_{\rm W}(R,t) &=&\hat{H}_{\rm S}+V_{\rm B,W}(R)+\hat{H}_{\rm C,W}(R)+\hat{H}_{\rm E,W}(t)\;, \label{eq:h_W}\end{aligned}$$ where $V_{\rm B,W}(R)$ is the potential energy of the bath. A time-dependent basis can be defined in terms of the eigenstates of $\hat{h}_{\rm W}(R,t)$: $\hat{h}_{\rm W}(R,t)|\alpha; R,t\rangle = E_{\alpha}(R,t)|\alpha; R,t\rangle$. In this basis the quantum evolution takes the form $$\begin{aligned} \chi_{\rm W}^{\alpha\alpha'}(X,t) = {\cal T}\left\{ \sum_{\beta\beta'}\left(e^{i\int_{t_0}^t d\tau \mathcal{L}^{t}(\tau)}\right)_{\alpha\alpha',\beta\beta'}\right\} \chi_{\rm W}^{\beta\beta'}(X,t_0)\,, \label{eq:quant-evolution}\end{aligned}$$ where $t_0$ is the initial time, the symbol $\cal T$ denotes time-ordering, and $$\begin{aligned} i{\mathcal{L}}_{\alpha\alpha'\beta\beta'}^{t}(t) =i\tilde{\mathcal{L}}_{\alpha\alpha'\beta\beta'}(t) +J_{\alpha\alpha'\beta\beta'}^{t}(t)\;. \label{eq:calL^t}\end{aligned}$$ In Eq. (\[eq:calL\^t\]), we have introduced the transition operator $$\begin{aligned} J^{t}_{{\alpha\alpha',\beta\beta'}} = \langle\dot\alpha|\beta\rangle\delta_{\alpha'\beta'} + \langle\beta|\dot\alpha'\rangle\delta_{{\alpha\beta}}\;,\end{aligned}$$ which arises explicitly from the time dependence of the basis. The operator $J_{{\alpha\alpha',\beta\beta'}}^t(t)$ determines the quantum transitions of the subsystem caused by the interaction with the external field. The time-dependent Liouville operator $i\tilde{\mathcal{L}}_{\alpha\alpha'\beta\beta'}$ is similar in form to those first obtained in Refs. [@sergi-theor-chem; @mqc] $$\begin{aligned} i\tilde{\mathcal{L}}_{\alpha\alpha',\beta\beta'} &=& i\tilde{\mathcal{L}}^{0}_{\alpha\alpha'} \delta_{\alpha\beta}\delta_{\alpha'\beta'} + \tilde{J}_{\alpha\alpha',\beta\beta'}(t)\,,\end{aligned}$$ with $i\tilde{\mathcal{L}}_{\alpha\alpha'}^0= i\tilde{\omega}_{\alpha\alpha'}(t) +i\tilde{L}_{\alpha\alpha'}(t)$, and the Bohr frequency given by $\tilde\omega_{\alpha\alpha'}(R,t)=(E_{\alpha}(R,t)-E_{\alpha'}(R,t))/\hbar$. The Liouville operator for the bath degrees of freedom is given by $i\tilde{L}_{\alpha\alpha'}=(P/M)\cdot(\partial/\partial R) +(1/2)(\tilde{F}^{\alpha}_{\rm W}(t)+\tilde{F}^{\alpha'}_{\rm W}) \cdot\partial/\partial P$, where $\tilde{F}^{\alpha}_{\rm W}(R,t)$ is a time-dependent Hellman-Feynman force for the energy surface $E_{\alpha}(R,t)$. The quantum transition operator is also similar in form to that given in Ref. [@mqc]: $$\begin{aligned} \tilde{J}_{{\alpha\alpha',\beta\beta'}}(t)=\tilde{\mathcal{T}}_{\alpha\rightarrow\beta}(t) \delta_{\alpha'\beta'} +\tilde{\mathcal{T}}_{\alpha'\rightarrow\beta'}^*(t) \delta_{\alpha\beta}\,,\end{aligned}$$ with $$\begin{aligned} \tilde{\mathcal{T}}_{\alpha\rightarrow\beta}(t)&=& \frac{P}{M}\cdot d_{\alpha\beta}(R,t)\left( 1+{\frac{1}{2}}\frac{\Delta E_{\alpha\beta}(t)d_{\alpha\beta}(R,t)}{\frac{P}{M} \cdot d_{\alpha\beta}(R,t)} {\frac{\partial}{\partial P}}\right)\,, \\ \tilde{\mathcal{T}}^{*}_{\alpha'\rightarrow\beta'} &=&\frac{P}{M}\cdot d^{*}_{\alpha'\beta'}(R,t)\left( 1+{\frac{1}{2}}\frac{\Delta E_{\alpha'\beta'}(t)d_{\alpha'\beta'}^*(R,t)} {\frac{P}{M}\cdot d^{*}_{\alpha\beta}(R,t)} {\frac{\partial}{\partial P}}\right)\,,\end{aligned}$$ and $\Delta E_{\alpha\beta}(t) = E_{\alpha}(R,t) - E_{\beta}(R,t)$. In the above, the coupling vector for the time-dependent states has been introduced as $d_{\alpha\beta}(R,t)=\langle\alpha;R,t| \overrightarrow{\partial}/\partial R|\beta;R,t\rangle$. The operator $\tilde{J}_{{\alpha\alpha',\beta\beta'}}(t)$ describes the quantum transitions arising from the interaction between the system and the bath. If such an interaction is weak, the effect of $\tilde{J}_{{\alpha\alpha',\beta\beta'}}(t)$ is negligible. Metamolecule model {#sec:sec4} ================== The metamolecule model considered in this work comprises a two-level system (the QD) coupled to a single resonating mode (RM), and subjected to a time-dependent external field. At this point, a single harmonic mode was considered, as the computational resources required by the phase-space grid algorithm, with which results were being compared, rises exponentially with the dimension of the bath. In the following, we will use adimensional coordinates and parameters; they are expounded in detail in App. \[app:app1\]. The QD Hamiltonian is $$\hat{H}_{\rm S} = -\frac{\Omega}{2}\hat\sigma_{z}\;. \label{eq:H_S}$$ The resonant single-mode Hamiltonian, describing the MNP, is defined as $$H_{\rm B,W}=\frac{P^2}{2}+ {\frac{1}{2}}\omega^2 R^2\;,\label{eq:H_B}$$ while the coupling to the RM is $$\hat{H}_{\rm C,W} = -c R\hat\sigma_{x}\;, \label{eq:H_C}$$ where $c$ is a coupling constant. The external driving field is represented through the Hamiltonian $$\hat{H}_{\rm E}(t)= g\cos(\omega_{d}t)\hat\sigma_{x}\,, \label{eq:H_E}$$ where $g$ denotes the driving strength of the external field, and $\omega_{d}$ is the driving frequency. The symbols $\hat\sigma_x$ and $\hat\sigma_z$ denote the Pauli matrices. The total Hamiltonian is given by Eq. (\[eq:H\_W-total\]). The energy eigenvalues of the Hamiltonian $\hat{h}_{\rm W}(R,t)$ in Eq. (\[eq:h\_W\]) are $$\begin{aligned} E_{1,2}(R,t) &=& {V_{b}}\pm \sqrt{{\frac{\Omega^2}{4}}+ \gamma^2 + {g^2\cos^2(\omega t)}+ 2\gamma{g{\cos{(\omega t)}}}}\,,\end{aligned}$$ where $\gamma = -cR$. In the basis of $\hat{h}_{\rm W}(R,t)$, the dynamics of arbitrary quantum operators is defined by Eq. (\[eq:quant-evolution\]). One can discretize time and obtain the evolution equation $$\begin{aligned} \chi_{\rm W}^{{\alpha\alpha'}}(t) = \sum_{\beta\beta'}{\mathcal{T}} \bigg\{\exp\left[i\sum_{n}\tau_{n} {\mathcal{L}}^{t}(\tau_{n})\right]\bigg\}_{{\alpha\alpha',\beta\beta'}}\chi_{\rm W}^{\beta\beta'}(t_{0})\,,\end{aligned}$$ where $\sum_{n}\tau_{n} = t - t_0$. Using very small time steps $\tau_n$ and the Dyson identity, one obtains $$\begin{aligned} \chi_{\rm W}^{{\alpha\alpha'}}(t)&=&\sum_{\beta\beta'}{\mathcal{T}}\prod_{n} \Big\{\exp\left[i\tau_{n} \tilde{\mathcal{L}}^{0}_{{\alpha\alpha'}}(\tau_{n})\right] \nonumber\\ &\times& \left(1+\tau_{n}\tilde{J}_{{\alpha\alpha',\beta\beta'}} +\tau_{n}J^{t}_{{\alpha\alpha',\beta\beta'}}\right)\Big\} \chi_{\rm W}^{\beta\beta'}(t_0)\,.\end{aligned}$$ In the case of weak coupling to the bath, the action of $\tilde{J}_{{\alpha\alpha',\beta\beta'}}$ can be disregarded. For $\tau_n = \tau$ for every $n$, one then obtains: $$\begin{aligned} \chi_{\rm W}^{{\alpha\alpha'}}(R,P,t) &=& \sum_{\beta\beta'}{\mathcal{T}}\prod_{n}\Big\{\exp\left[i\tau \tilde{\mathcal{L}}^{0}_{{\alpha\alpha'}}(\tau)\right]\left(1+\tau J^{t}_{{\alpha\alpha',\beta\beta'}}\right)\Big\} \chi_{\rm W}^{\beta\beta'}(t_0)\;. \label{eq:chi-discretised-weak-coupling}\end{aligned}$$ Equation (\[eq:chi-discretised-weak-coupling\]) can be implemented by means of a stochastic algorithm. One can sample with probability $1/2$ one of the two terms in $J^{t}_{{\alpha\alpha',\beta\beta'}}$ acting at each time step. For each phase space point $(R,P)$, one propagates a single deterministic step, dictated by $i\tilde{\cal L}_{{\alpha\alpha',\beta\beta'}}^0(t)$. At the end of such a step, the quantum transition, due to the external field, is sampled. Transition probabilities can be defined as: $$\begin{aligned} {\mathcal{P}}_{\beta\rightarrow\alpha} = \frac{\tau| \langle\dot{\alpha}|\beta\rangle |} {1 + \tau|\langle\dot{\alpha}|\beta\rangle|}\,. \label{eq:transition}\end{aligned}$$ The probability of rejecting the transition will then be given by $$\begin{aligned} {\mathcal{Q}}_{\beta\rightarrow\alpha} = \frac{1}{1+\tau|\langle\dot{\alpha}|\beta\rangle|}\,. \label{eq:no-transition}\end{aligned}$$ For the two-level model that we are studying, the eigenstates can be calculated exactly and therefore, so can the transition probabilities in Eqs. (\[eq:transition\]) and (\[eq:no-transition\]). Results {#sec:sec5} ======= The initial state of the system is defined as $$\begin{aligned} \hat{\rho}_{\rm W}(R,P) = \left(\begin{array}{cc} 0\,\,\,\, & 0 \\ 0\,\,\,\, & 1\end{array}\right)\times \rho_{\rm B,W}(R,P)\,, \label{eq:inirho}\end{aligned}$$ where the $\rho_{\rm B,W}(R,P)$ is the Wigner function for the bath, given by $$\begin{aligned} \rho_{\rm B,W}(R,P) = \frac{\tanh(\beta\omega/2)}{\pi}\exp\left[-\frac{2\tanh(\beta\omega/2)} {\omega}\left(\frac{P^{2}}{2}+\frac{\omega^{2}R^{2}}{2}\right)\right]\,,\end{aligned}$$ and the matrix on the right hand side of Eq. (\[eq:inirho\]) is given in the subsystem basis. The average values of an arbitrary phase-space dependent operator, $\hat{\chi}_{\rm W}(R,P,t)$, are calculated as $$\langle\hat{\chi}_{\rm W}(R,P,t)\rangle ={\rm Tr}'\int dRdP \hat{\rho}_{\rm W}(R,P) \hat{\chi}_{\rm W}(R,P,t) \;. \label{eq:averages}$$ The partial trace and the evolution of $\hat{\chi}_{\rm W}(R,P,t)$ are calculated in the time-dependent basis and with the algorithm sketched in Secs. \[sec:sec3\] and \[sec:sec4\]. In order to determine the effect of the external field upon the metamolecule, we kept the values of the system parameters unchanged while varying the coupling strength of the external field. The values of the system parameters are $\beta = 12.5$, $c = 0.01$, $\Omega = 0.8$, $\omega = 0.5$, and $\omega_{d} = 0.05$. These values lead to a weak coupling between the QD and the RM, so that the quantum transitions caused by the interaction with the resonant mode can be neglected. The values for the coupling strength of the external field used in this study were $g = 0.1, 0.3, 0.5, 0.7, 0.9, 1.5$. However, in order to demonstrate the differences between weak and strong coupling, only the results for $g = 0.1$ and $g = 1.5$ are shown in the figures. In the case of the piece-wise deterministic algorithm, described in Secs. \[sec:sec3\] and \[sec:sec4\], a time step of $\tau = 0.1$ was employed, and a total of $10^5$ trajectories were propagated in each calculation. Instead, the phase-space grid algorithm (described in App. \[app:app1\]) requires a smaller time step of $\tau = 0.001$. The phase-space grid spacing was $\Delta R = \Delta P = 0.1$. In Fig. \[fig:fig1\], a comparison of the results obtained with the two different algorithms, for the average coherence of the quantum dot as a function of time, is shown. With a value of $g = 0.1$, this calculation corresponds to a weak driving field. The two results agree almost exactly, so that they cannot be distinguished by the human eye. Moreover, the error bars are negligible. In this weak coupling case, the oscillations remain relatively small, with the values of $\langle\sigma_{x}\rangle$ ranging between $-0.5$ and $0.5$. A very slow mode of oscillation, with angular frequency $\approx 0.05$ appears to be superimposed to a fast mode with angular frequency $\approx 0.81$. The slow frequency is basically that of the driving field while the slow one corresponds to the tunnel splitting, shifted by a very small amount because of the weak coupling to the field. As expected, in the case of weak coupling to the external field it is the tunnel splitting that dominates the evolution in time of $\langle\sigma_x\rangle$. Figure \[fig:fig2\] shows the results of the calculations with $g = 1.5$, corresponding to a strong driving field. The results produced by the two algorithms are indistinguishable also in this case, with error bars remaining smaller than the points for the entire simulation time. In this case, the evolution of the coherence in time displays a non-linear pattern: it starts with fast oscillations around $\langle\sigma_x\rangle=-0.25$ from $t=0$ to $t\approx20$; within $20<t<40$ it undergoes large oscillations, and then it switches to oscillating fast around the value of $\langle\sigma_x\rangle=0.25$; it does so until $t=80$, when the large oscillations start again. We can therefore conclude that the QD switches between two different dynamical regimes, one where the coherence is positive, and the other where it is negative. We can associate a period to such a switching dynamics, whose numerical value matches that of the driving field. Because of the strong coupling, the frequency of the fast oscillations is about 200% greater than that obtained in the case of weak coupling to the external field. The inspection of Fig. \[fig:fig2\] reveals that the dynamics of $\langle\sigma_x\rangle$ is now dominated by the strong coupling to the driving field. In Fig. \[fig:fig3\], we show the evolution in time of the average of the population difference in the strong driving regime. Such a quantity also shows a non-linear pattern, with regions of large and fast oscillations separated by regions of slow and small oscillations. At weak driving strengths, this behavior is not noticeable, and the oscillations are much smaller. This corresponds to a lower percentage of the ensemble of trajectories of the piece-wise deterministic algorithm being driven into the excited state by the external field. In Fig. \[fig:fig4\], we plot the rate of change of the expectation value of the energy of the QD-RM metamolecule in two different cases: $g = 0.1$ and $g = 1.5$. For $g=1.5$, such a quantity shows a non-linear pattern, with regions of large and fast oscillations separated by regions of slow and small oscillations, in the same time intervals where $\langle\sigma_x\rangle$ and $\langle\sigma_z\rangle$ do. Instead, for $g = 0.1$ the rate of change of the average energy of the metamolecule has significantly smaller oscillations around the mean and the structured pattern is almost absent. The calculation of the rate of change of the energy of RM was also performed, and it was found that the RM reacts slowly to the change in energy of the QD. This justifies the neglect of the quantum transitions due to the coupling to the RM. Conclusions {#sec:sec6} =========== Employing the partial Wigner representation of quantum mechanics, we have studied the dynamics of a model for a quantum dot coupled to a single resonating mode of a metal nano-particle. We have treated the case in which the Hamiltonian is explicitly time-dependent, due to the presence of a driving field directly coupled to the quantum dot. An explicitly time-dependent basis has been used for the representation of the equations of motion and generalized propagation schemes have been devised both in terms of a piece-wise deterministic algorithm and of a grid-based numerical integration. The results obtained by using these two algorithms were compared. We have shown that both schemes of integration produce numerically indistinguishable results. However, the piece-wise deterministic algorithm has the definite advantage of being able to treat systems with a higher number of discrete energy levels (such as three- or four-level quantum dots) and many more (hundreds or thousands) of resonating modes in an affordable computational time (see, for example, Ref. [@num-app-quant]). We have studied the effect of the driving strength of the external field upon the quantum dot. By monitoring the average of the time variation of the energy of the metamolecule model, as well as the coherence and the population difference of the quantum dot, we observe distinct non-linear behavior in the case of strong coupling to the driving field. Both the algorithms presented in this work and the results obtained can be considered as a first step toward the development of an effective approach (alternative to the use of master equations) for studying plasmonic metamolecules. Adimensional coordinates and parameters {#app:app1} ======================================= Upon indicating the dimensional coordinates and parameters with a prime, and introducing the energy scale $\hbar\omega_{\rm a}'$, we have the following definitions: $$\begin{aligned} \Omega&=&\frac{\Omega'}{\omega_{\rm a}'} \;, \\ P&=&\frac{P'}{\sqrt{M'\hbar\omega_{\rm a}'} } \;, \\ R&=&\sqrt{\frac{\hbar \omega_{\rm a}'}{\hbar}} R' \;, \\ \omega&=&\frac{\omega'}{\omega_{\rm a}'} \;,\\ c&=& \frac{c'}{\sqrt{ \hbar\omega_{\rm a}^{\prime 3}M'}} \;,\\ g&=&\frac{g'}{\hbar\omega_{\rm a}'} \;,\\ \beta&=&\hbar\omega_{\rm a}'\beta'\;.\end{aligned}$$ The symbol $M$ is the inertial parameter of the oscillator, which has been set to unity. Upon choosing $\omega_{\rm a}'$ in such a way that the the frequency $\omega=0.5$ of the oscillator in Eq. (\[eq:H\_B\]) corresponds to the value $\omega'=8.9 \times 10^{12}$ Hz, which is typical for the dynamics of metal nano-particles [@plasmonics-tame], we obtain a spanned time-scale in our simulations of $5.62 \times 10^{-12}$ s and an energy variation for the quantum dot, as shown in Fig. \[fig:fig4\], of $24.6$ meV. Phase-space Grid Algorithm {#app:app2} ========================== The equation of the density matrix in the partial Wigner representation is analogous to that Given in Eq. (\[eq:WH-equation-of-motion\]): $$\begin{aligned} \frac{\partial}{\partial t}\hat{\rho}(X,t)=-\frac{i}{\hbar}\left[\begin{array}{cc} \hat{H}_{\rm W}(X) & \hat{\rho}_{\rm W}(X,t)\end{array}\right]{\mathcal{D}} \left[\begin{array}{c}\hat{H}_{\rm W}(X) \\ \hat{\rho}_{\rm W}(X,t)\end{array}\right]\;. \label{eq:WH-rho-equation-of-motion}\end{aligned}$$ Using the basis defined by $\hat{H}_{\rm S}|\alpha\rangle = \epsilon_{\alpha} |\alpha\rangle$ ($\alpha=1,2$), Eq. (\[eq:WH-rho-equation-of-motion\]) can be written as $$\begin{aligned} {\frac{\partial}{\partial t}}\rho_{\rm W}^{{\alpha\alpha'}}(R,P,t) &=& -i{\tilde\omega}_{{\alpha\alpha'}}{\rho_{\rm W}^{{\alpha\alpha'}}}-L{\rho_{\rm W}^{{\alpha\alpha'}}}- {\frac{i}{\hbar}}\left(H_{\rm C,W}^{\alpha\beta}\rho_{\rm W}^{\beta\alpha'} - \rho_{\rm W}^{\alpha\beta'}H_{\rm C,W}^{\beta'\alpha'}\right) \nonumber \\ &-& {\frac{i}{\hbar}}\left(H_{E}^{\alpha\beta}\rho_{\rm W}^{\beta\alpha'} - \rho_{\rm W}^{\alpha\beta'}H_{E}^{\beta'\alpha'}\right) \nonumber \\ &+& {\frac{1}{2}}\left(\frac{\partial H_{\rm C,W}^{\alpha\beta}}{\partial R}\frac{\partial\rho_{\rm W}^{\beta\alpha'}} {\partial P} + \frac{\partial\rho_{\rm W}^{\alpha\beta'}}{\partial P}\frac{\partial H_{\rm C,W}^{\beta'\alpha'}} {\partial R}\right)\;, \label{eq:qcle-eq-sub}\end{aligned}$$ where the frequency ${\tilde\omega}_{{\alpha\alpha'}} = \left(\epsilon_{\alpha} - \epsilon_{\alpha'}\right)/\hbar$ has been defined. The Liouville operator, $L$, is given by $$\begin{aligned} L = P{\frac{\partial}{\partial R}}- \frac{\partial H_{B,W}}{\partial R}{\frac{\partial}{\partial P}}\;.\end{aligned}$$ One can introduce the following frequencies $\tilde{\omega}^{{\alpha\alpha'}}$: $$\begin{aligned} \label{eq:frequencies} &&\tilde{\omega}^{11} = 0, \hspace{1.3cm} \tilde{\omega}^{22} = 0, \nonumber \\ &&\tilde\omega^{12} = \Omega, \hspace{1cm} \tilde\omega^{21} = -\hbar\Omega.\end{aligned}$$ The equations of motion for matrix elements of the density operator can be written explicitly as: $$\begin{aligned} {\frac{\partial}{\partial t}}{\rho_{\rm W}^{11}}(R,P,t) &=& {\frac{i}{\hbar}}cR\left(2i\text{Im}\left[{\rho_{\rm W}^{21}}\right]\right) -{\frac{i}{\hbar}}{g\cos({\omega_{d}}t)}\left(2i\text{Im}\left[{\rho_{\rm W}^{21}}\right]\right) \nonumber \\ &-& L{\rho_{\rm W}^{11}}- \frac{c}{2}{\frac{\partial}{\partial P}}\left(2\text{Re}\left[{\rho_{\rm W}^{21}}\right]\right)\;, \\ {\frac{\partial}{\partial t}}{\rho_{\rm W}^{21}}(R,P,t) &=& -i{\tilde\omega}_{21}{\rho_{\rm W}^{21}}+ {\frac{i}{\hbar}}cR\left(\rho_{\rm W}^{11} - \rho_{\rm W}^{22}\right) -{\frac{i}{\hbar}}{g\cos({\omega_{d}}t)}\left({\rho_{\rm W}^{11}}- {\rho_{\rm W}^{22}}\right) \nonumber \\ &-& L{\rho_{\rm W}^{21}}- \frac{c}{2}{\frac{\partial}{\partial P}}\left({\rho_{\rm W}^{11}}+{\rho_{\rm W}^{22}}\right)\;, \\ {\frac{\partial}{\partial t}}{\rho_{\rm W}^{22}}(R,P,t) &=& -{\frac{i}{\hbar}}cR\left(2i\text{Im}\left[{\rho_{\rm W}^{21}}\right]\right) +{\frac{i}{\hbar}}{g\cos({\omega_{d}}t)}\left(2i\text{Im}\left[{\rho_{\rm W}^{21}}\right]\right) \nonumber \\ &-& L{\rho_{\rm W}^{22}}- \frac{c}{2}{\frac{\partial}{\partial P}}\left(2\text{Re}\left[{\rho_{\rm W}^{21}}\right]\right)\,.\end{aligned}$$ In order to simplify the integration, one can use the definition $$\rho_{\rm W}^{{\alpha\alpha'}}(X,t)=\eta_{\rm W}^{{\alpha\alpha'}}(X,t) e^{-i\tilde\omega^{{\alpha\alpha'}}t}\;.$$ Hence, using dimensionless coordinates, the equations of motion become $$\begin{aligned} {\frac{\partial}{\partial t}}{\eta_{\rm W}^{11}}&=& -2cR\left(-\text{Re}\left[{\eta_{\rm W}^{21}}\right]\sin\left({\tilde\omega}_{21}t\right) + \text{Im}\left[{\eta_{\rm W}^{21}}\right]\cos\left({\tilde\omega}_{21}t\right)\right) \nonumber \\ &+& 2g{\cos{(\omega t)}}\left(-\text{Re}\left[{\eta_{\rm W}^{21}}\right]\sin\left({\tilde\omega}_{21}t\right) + \text{Im}\left[{\eta_{\rm W}^{21}}\right]\cos\left({\tilde\omega}_{21}t\right)\right) \nonumber \\ &-& L{\eta_{\rm W}^{11}}- c{\frac{\partial}{\partial P}}\left(\text{Re}\left[{\eta_{\rm W}^{21}}\right]\cos\left({\tilde\omega}_{21}t\right) + \text{Im}\left[{\eta_{\rm W}^{21}}\right]\sin\left({\tilde\omega}_{21}t\right)\right)\;, \label{eq:pde-11} \\ {\frac{\partial}{\partial t}}{\eta_{\rm W}^{22}}&=& 2cR\left(-\text{Re}\left[{\eta_{\rm W}^{21}}\right]\sin\left({\tilde\omega}_{21}t\right) + \text{Im}\left[{\eta_{\rm W}^{21}}\right]\cos\left({\tilde\omega}_{21}t\right)\right) \nonumber \\ &-& 2g{\cos{(\omega t)}}\left(-\text{Re}\left[{\eta_{\rm W}^{21}}\right]\sin\left({\tilde\omega}_{21}t\right) + \text{Im}\left[{\eta_{\rm W}^{21}}\right]\cos\left({\tilde\omega}_{21}t\right)\right) \nonumber \\ &-& L{\eta_{\rm W}^{22}}- c{\frac{\partial}{\partial P}}\left(\text{Re}\left[{\eta_{\rm W}^{21}}\right]\cos\left({\tilde\omega}_{21}t\right) + \text{Im}\left[{\eta_{\rm W}^{21}}\right]\sin\left({\tilde\omega}_{21}t\right)\right)\;, \label{eq:pde-22} \\ {\frac{\partial}{\partial t}}\left(\text{Re}\left[{\eta_{\rm W}^{21}}\right]\right) &=& -cR\left({\eta_{\rm W}^{11}}- {\eta_{\rm W}^{22}}\right)\sin({\tilde\omega}_{21}t)+{g\cos({\omega_{d}}t)}\left({\eta_{\rm W}^{11}}- {\eta_{\rm W}^{22}}\right)\sin({\tilde\omega}_{21}t) \nonumber \\ &-& L\left(\text{Re}\left[{\eta_{\rm W}^{21}}\right]\right) - \frac{c}{2}{\frac{\partial}{\partial P}}\left({\eta_{\rm W}^{11}}+ {\eta_{\rm W}^{22}}\right)\cos({\tilde\omega}_{21}t)\;, \label{eq:pde-21re} \\ {\frac{\partial}{\partial t}}\left(\text{Im}\left[{\eta_{\rm W}^{21}}\right]\right) &=& cR\left({\eta_{\rm W}^{11}}- {\eta_{\rm W}^{22}}\right)\cos({\tilde\omega}_{21}t) -{g\cos({\omega_{d}}t)}\left({\eta_{\rm W}^{11}}- {\eta_{\rm W}^{22}}\right)\cos({\tilde\omega}_{21}t) \nonumber \\ &-& L\left(\text{Im}\left[{\eta_{\rm W}^{21}}\right]\right) - \frac{c}{2}{\frac{\partial}{\partial P}}\left({\eta_{\rm W}^{11}}+ {\eta_{\rm W}^{22}}\right)\sin({\tilde\omega}_{21}t)\;. \label{eq:pde-21im}\end{aligned}$$ Equations (\[eq:pde-11\]), (\[eq:pde-22\]), (\[eq:pde-21re\]) and (\[eq:pde-21im\]) describe a set of coupled partial differential equations (PDE’s) which can be solved to obtain the elements of the density matrix as functions of time. In order to solve these coupled PDE’s, the numerical integration approach known as the method of lines can be employed. The method of lines transforms the PDE’s into a set of ordinary differential equations (ODE’s) by using finite difference approximations for all but one of the integration variables. In this case, the phase-space derivatives are approximated, while the time variable is left un-approximated. For the types of potentials studied in this work, a fourth-order finite difference approximation for the phase-space derivatives proves to be more than sufficient: $$\begin{aligned} \frac{\textrm{d}f}{\textrm{d} X} = \left(\frac{f(X - 2{\textrm{d}}X) - 8f(X - {\textrm{d}}X) + 8f(X + {\textrm{d}}X) - f(X + 2{\textrm{d}}X)}{12{\textrm{d}}X}\right)\,.\end{aligned}$$ Once the conversion from PDE’s to ODE’s has been performed, a numerical integration method can be utilized. In this work, a Runge-Kutta 5 Cash-Karp method was found to be suitable for stable numerical results. In such an approach, the phase-space of the system is essentially discretized into a numerical grid, upon which the coupled ODE’s are solved. We are grateful to Prof. Mark Tame for useful discussions and for having stimulated our interest in quantum plasmonics. This work is based upon research supported by the National Research Foundation of South Africa. Tame MF, McEnery KR, Özdemir SK, Lee J, Maier SA, Kim MS (2013) Nature Phys 9: 329 Chen X, Li S, Xue C, Banholzer MJ, Schatz GC, Mirkin CA, (2009) ACS Nano 3: 87 Zuloaga J, Prodan E, Nordlander P (2010) ACS Nano 4: 5269 Marinica DC, Kazansky AK, Nordlander P, Aizpurua J, Borisov AG (2012) Nano Lett 12: 1333 Bouillard JG, Dickson W, O’Connor DP, Wurtz GA, Zayats AV (2012) Nano Lett 12: 1561 Valentine J, Zhang S, Zentgraf T, Ulin-Avila E, Genov DA, Bartal G, Zhang X (2008) Nature 455: 376 Pratibha R, Park K, Smalyukh II, Park W, (2009) Opt Express 17: 19459 Leonhardt U (2007) Nature Photon 1: 207 Kelly KL, Coronado E, Zhao LL, Schatz GC (2003) J Phys Chem B 107: 668 Lal S, Link S, Halas NJ (2007) Nature Photon 1: 641 Nehl CL, Hafner JH (2008) J Mater Chem 18: 2415 Pelton M, Aizpurua J, Bryant G (2008) Laser Photon Rev 2: 136 Ridolfo A, Di Stefano O, Fina N, Saija R, Savasta S (2010) Phys Rev Lett 105: 263601 McEnery KR, Tame MS, Maier SA, Kim MS (2014) Phys Rev A 89: 013822 Makarov DE, Makri N (1994) Chem Phys Lett 221: 482 McLachlan AD (1964) Mol Phys 8: 39 Tully JC, Preston RK (1971) J Chem Phys 55: 562 Miller WH, George FF (1972) J Chem Phys 56: 5637 Pechukas P (1969) Phys Rev 181: 166 Pechukas P (1969) Phys Rev 174: 166 Heller EJ, Segev B, Sergeev AV (2002) J Phys Chem B 106: 8471 Shenvi N (2009) J Chem Phys 130: 124177 Aleksandrov IV (1981) Z Nat Forsch A 36: 902 Gerasimenko VI (1982) Teor Mat Fiz 150: 7 Boucher W and Traschen J (1988) Phys Rev D 37: 3522 Zhang WY and Balescu R (1988) J Plasma Phys 40: 199 Balescu R and Zhang WY (1988) J Plasma Phys 40: 215 Osborn TA, Kondrat’eva MF, Tabisz GC, McQuarrie BR (1999) J Phys A: Math Gen 32: 4149 Sergi A, Sinayskiy I, Petruccione F (2009) Phys Rev A 80: 012108 Goldstein H (1980) [*Classical Mechanics*]{}, Addison-Wesley, New York Sergi A, MacKernan D, Ciccotti G, Kapral R (2003) Theor Chem Acc 110: 49 Kapral R, Ciccotti G (1999) J Chem Phys 110: 8919

--- abstract: 'We solve the problem of counting elliptic curves with fixed j-invariant in projective space with tangency conditions. This is equivalent to couting rational nodal curves with condition on the node of the image. The solution is given in the form of effective recursions. We give explicit formulas when the dimension of the ambient projective space is at most $5$. Many numerical examples are provided. A C++ program implementing all of the recursions is available upon request.' author: - Dung Nguyen title: 'Characteristic numbers of elliptic curves with fixed j-invariant' --- introduction ============ Charateristic numbers of curves in projective spaces is a classical problem in algebraic geometry: how many curves in $\proj$ of given degree and genus that pass through a general set of linear subspaces, and are tangent to a general set of hyperplanes? Presented in this form, the problem seems almost unattackable, as not much is known even in the case of genus two space curves. However, the cases of genus zero and genus one space curves are well understood. Incidence-only (meaning no tangency condition is considered) characteristic numbers of rational plane curves were first computed by Kontsevich, see [@fp]. The method was to pull back the WDVV equation on $\mbar_{0,4}$ onto the moduli space of stable maps $\mbar_{0,n}(2,d)$ to obtain a recursion counting rational plane curves. The same method works equally well for rational space curves. In [@idq], Lemma $2.3.1$, it was shown that the tangency divisor is numerically equivalent to a linear combination of the incident divisor and boundary divisors on $\mbar_{0,n}(r,d).$ Hence one can write down a recursion computing full characteristic numbers of rational space curves. In genus one, there are at least two counting problems. One could try to obtain enumeration of genus one curves with generic $j-$ invariant, or of genus one curves with fixed $j-$ invariant. This note will deal with the latter. Incidence-only characteristic numbers for genus one space curves with fixed-j invariant have been computed in [@eln] and [@zin]. In this note, recursions computing all characteristic numbers will be provided. In case of incidence-only numbers, we obtain an algebraic solution that works over any closed field of zero charactersistic, in contrast to the analytic method in [@eln] and [@zin]. The results in this note will also be used to compute characteristic numbers of elliptic space curves in an upcoming paper by the author. All the recursions are based on our algorithm counting rational two nodal reducible curves. These are projective curves having two rational smooth component intersecting at two points (or with a choice of two intersection points in the case of plane curves). Counting these curves is in turn based on an algorithm counting rational curves, now with an additional type of conditions: special tangent conditions. This will be defined in Section $2$. We work out in detail the algorithm counting rational curves with special tangent conditions in ambient space of dimension at most $5$. For dimension $6$ or higher, the numbers could in theory be expressed as intersections of tautological classes on a blowup of $\mbar_{0,1}(r,d)$, but this is much less implementable. We use the following results to obtain our recursions. We use the WDVV equation on $\mbar_{0,n}(r,d)$. We use the divisor theory on $\mbar_{0,n}(r,d)$ as developed in [@idq]. We do not use any outside input, and our method for incidence-only characteristic numbers is different from those in [@eln], [@zin]. The author is very grateful to R.Vakil, his advisor, for numerous helpful conversations and ideas, and for introducing him to the beautiful subject of enumerative geometry.\ \ Definitions and Notations ========================= The moduli space of stable maps of genus $0$ in $\proj$. -------------------------------------------------------- As usual, $\mbar_{0,n}(r,d)$ will denote the Kontsevich compactification of the moduli space of genus zero curves with $n$ marked points of degree $d$ in $\proj$. We will also be using the notation $\mbar_{0,S}(r,d)$ where the markings are indexed by a set $S$. The following are Weil divisors on $\mbar_{0,S}(r,d)$: - The divisor $(U \sep V)$ of $\mbar_{0,S}(r,d)$ is the closure in $\mbar_{0,S}(r,d)$ of the locus of curves with two components such that $U \cup V = S$ is a partition of the marked points over the two components. - The divisor $(d_1,d_2)$ is the closure in $\mbar_{0,S}(r,d)$ of the locus of curves with two components, sucht that $d_1 + d_2 = d$ is the degree partition over the two components. - The divisor $(U,d_1 \sep V,d_2)$ is the closure in $\mbar_{0,S}(r,d)$ of the locus of curves with two components, where $U \cup V = S$ and $d_1 +d_2 = d$ are the partitions of markings and degree over the two components respectively. The constraints and the ordering of constraints. ------------------------------------------------ We will be concerned with the number of curves passing through a constraint. Each constraint is denoted by a $(r+1)-$tuple $\d$ as follows :\ [**(i)**]{} $\d(0)$ is the number of hyperplanes that the curves need to be tangent to.\ [**(ii)**]{} For $0<i \leq r$, $\d(i)$ is the number of subspaces of codimension $i$ that the curves need to pass through.\ [**(iii)**]{} If the curves in consideration have a node and we place a condition on the node, that is the node has to belong to a general codimension $k$ linear subspace, then $\d$ has $r+2$ elements and the last element, $\d(r+1)$, is $k$. Note that because in general a curve of degree $d$ will always intersect a hyperplane at $d$ points, introducing an incident condition with a hyperplane essentially means multiplying the cycle class cut out by other conditions by $d$. For example, if we ask how many genus zero curves of degree $4$ in $\mathbb P^3$ that pass through the constraint $\d = (1,2,3,4,0) (\d(1) = 2)$, that means we ask how many genus zero curves of degree $4$ pass through three lines, four points, are tangent to one hyperplane, and then multiply that answer by $4^2$. We will also refer to $\d$ as a set of linear spaces, hence we can say, pick a space $p$ in $\d$. We consider the following ordering on the set of constraints, in order to prove that our algorithm will terminate later on. Let $r(\d) = -\sum_{i >1}^{i\leq r} \d[i]\cdot i^2$, and this will be our rank function. We compare two constraints $\d,\d'$ using the following criteria, whose priority are in the following order : - If $\d(0) = \d'(0)$ and $\d$ has fewer non-hyperplane elements than $\d'$ does, then $\d<\d'$. - If $\d(0) > \d'(0)$ then $\d < \d'$. - If $r(\d) < r(\d')$ then $\d < \d'$. Informally speaking, characteristic numbers where the constraints are more spread out at two ends are computed first in the recursion. We write $\d = \d_1\d_2$ if $\d = \d_1 + \d_2$ as a parition of the set of linear spaces in $\d.$ The stacks ${\mathbbmss{R}},\n, {\mathbbmss{R}}{\mathbbmss{R}}, {\mathbbmss{R}}{\mathbbmss{R}}_2$ . --------------------------------------------------------------------------------------------------- We list the following definitions of stacks of stable maps that will occur in our recursions.\ \ [**1)**]{} Let ${\mathbbmss{R}}(r,d)$ be the usual moduli space of genus zero stable maps $\mbar_{0,0}(r,d)$.\ \ [**2)**]{} Let $\n(r,d)$ be the closure in $\mbar_{0,\{A,B\} }(r,d)$ of the locus of maps of smooth rational curves $\gamma$ such that $\gamma(A) = \gamma(B)$. Informally, $\n(r,d)$ parametrizes degree $d$ rational nodal curves in $\proj$.\ \ [**3)**]{} For $d_1,d_2>0,$ let ${\mathbbmss{R}}{\mathbbmss{R}}(r,d_1,d_2)$ be $\mbar_{0,\{C\}}(r,d_1) \times \mbar_{0,\{C\}}(r,d_2)$ where the fibre product is taken over evaluation maps $ev_{C}$ to $\proj.$\ \ [**4)**]{} Similarly we can define $\n{\mathbbmss{R}}(r,d_1,d_2)$ (see figure 1).\ \ [**5)**]{} For $d_1,d_2>0$, let ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ be the closure in $\mbar_{0,\{A,C\}}(r,d_1) \times_{\proj} \mbar_{0,\{B,C\}}(r,d_2)$ (the projections are evaluation maps $e_C$) of the locus of maps $\gamma$ such that $\gamma(A) = \gamma(B)$. We call maps in this family rational two-nodal reducible curves.\ \ $$\includegraphics[width = 150mm]{StacksRNRRNRRR2.pdf}$$\ $$\text{Fig 1. Pictorial description of a general curve in the stacks ${\mathbbmss{R}},\n, {\mathbbmss{R}}{\mathbbmss{R}}, \n{\mathbbmss{R}}, {\mathbbmss{R}}{\mathbbmss{R}}_2$}$$ Special Tangent Condition ------------------------- It is necessary to understand the enumerative geometry of rational curves, now considering extra conditions of the form: there is a fixed marked point $A$ on the curve, and the projective tangent line at $A$ passes through a given codimension $2$ linear subspace $M$. The corresponding (Weil) divisor is denoted by $\W_A^M$. When there is no need to consider any particular codimension $2$ subspace $M$, we will only write $\W_A$. We would also need to consider the case where there is a condition on $A$, which means it could be specified to lie on a certain linear subspace. By characteristic numbers of rational space curves with special tangent conditions, we mean the numbers of rational space curves having a marked point $A$ that satisfy the following conditions : - Pass through various linear spaces and are tangent to various hyperplanes. - The tangent line at $A$ to the curve passes through various codimension $2$ linear spaces. - The point $A$ may or may not lie on a given linear space. $$\includegraphics[width = 60mm]{SpecialTangent.pdf}$$ $$\text{Fig 2. A curve with a special tangent condition}$$ Stacks of stable maps with constraints. --------------------------------------- Let $\F$ be a maps of curves into $\proj$. For a constraint $\d$, we define $(\F, \d)$ be the closure in $\F $ of the locus of maps that satisfy the constraint $\d$. If the stack of maps $\F$ has two marked points $A$ and $B$, we define $(\F, \L_A^u\L_B^v)$ to be the closure in $\F$ of the locus of maps $\gamma$ such that $\gamma(A)$ lies on $u$ general hyperplanes, and that $\gamma(B)$ lies on $v$ general hyperplanes. If $\F$ has one marked point $A$ then we define $(\F,\L_A^u\W^v_A)$ to be the closure of maps $\gamma$ such that $\gamma(A)$ lies on $u$ general hyperplanes, and that the image of $\gamma$ is smooth at $\gamma(A)$ and the tangent line to the image of $\gamma$ at $\gamma(A)$ passes through $v$ general codimension $2$ subspaces ($v$ special tangent conditions) If a stack $\F$ is supported on a finite number of points then we denote $\# \F$ to be the stack-theoretic length of $\F$. If $\F$ is a closed substack of the stacks $\n{\mathbbmss{R}}, {\mathbbmss{R}}{\mathbbmss{R}}$ then we denote $(\F,\gam_1,\gam_2,k)$ to be the closure in $\F$ of the locus of maps $\gamma$ such that the restriction of $\gamma$ on the $i-$th component satisfies constraint $\gam_i$ and that $\gamma(C)$ lies on $k$ general hyperplanes. We use the notation $(\F,\d,k)$ if we don’t want to distinguish the conditions on each compo.nent. If $\F$ is a closed substack of ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ then we denote $(\F,\gam_1,\gam_2,k,l)$ to be the closure in $\F$ of the locus of maps $\gamma$ such that the restriction of $\gamma$ on the $i-$th component satisfies constraint $\gam_i$ and that $\gamma(C)$ lies on $l$ general hyperplanes, and that $\gamma(A) = \gamma(B)$ lies on $k$ general hyperplanes. Similary, we use the notation $(\F,\d,k,l)$ if we don’t want to distinguish the conditions on each component. Note that for maps of reducible source curves, tangency condition include the case where the image of the node lies on the tangency hyperplane, as the intersection multiplicity is $2$ in this case. Counting one-nodal reducible curves in $\proj$ =============================================== In this section we discuss how to count maps with reducible source curves. Let $\F_1$ and $\F_2$ be two families of stable maps with marked point $C$. Let $\gam_1$ and $\gam_2$ be two constraints. Then we have $$\#(\F_1 \times_{ev_{C}} \F_2,\gam_1,\gam_2,k) = \#( \F_1 , \d_1') \cdot \# (\F_2, \d_2')$$ where $\d_i'$ are determined as follows. Let $e_1$ be the dimension of the pushforward under $ev_C$ of $(\F_1,\gam_1)$ into $\proj.$ Let $e_2$ be the dimension of the pushforward under $ev_C$ of $(\F_2, \gam_2)$ into $\proj$. Then $\d_i'$ is obtained from $\gam_i$ by adding a subspace of codimension $e_i$. [ [*Proof.* ]{}]{}Let $\alpha_i$ be the class of ${ev_C}_*(\F_i,\gam_i)$ in the Chow ring of $\proj$. Let $h$ be the class of a subspace of codimension $k$. Then $\#(\F_1 \times_{ev_{C}} \F_2,\gam_1,\gam_2,k)$ is equal to the intersection product $\alpha_1 \cdot \alpha_2 \cdot h$ which is $\deg(\alpha_1) \cdot \deg(\alpha_2).$ To compute $\deg(\alpha_i)$, we intersect $\alpha_i$ with a subspace of codimension $e_i$, thus $$\deg(\alpha_i) = \#( \F_i , \d_i')$$ which proves the proposition.\ \ The following lemma is useful because it allow us to express the tangency condition on maps of reducible curves in terms of tangency conditions on maps of each component and condition on the node. Let $\X_1,\X_2$ be stacks of stable maps into $\proj$. Assume each map in each family carries at least one marked point $C$. Let $\X = \X_1 \times_{ev_C} \X_2$ . Let $\T$ be the tangency divisortangenttangent on $\X$, and $\T_i$ be the pull-back of the tangency divisor on the $i-$th component. Then on $\X$ we have this divisorial equation: $\T = \T_1 + \T_2 + 2\L_C.$ [ [*Proof.* ]{}]{}Let $\C$ be a general curve in $\X$. $\C$ has the following description. There is a family of nodal curves over $\C,$ $\pi : S \to \C$ such that $S$ is the union of two families of nodal curves $X_1,X_2$ along a section $s : \C \to S$. The section $s$ represents the marked point $C$ of each family. There is also a map $\mu : S \to \proj$ such that the restriction of $\mu$ on each fiber is an element (a map) of $\X_1 \times_{ev_C} \X_2$. Now choose a general hyperplane $H$ in $\proj.$ Then the restriction of the tangency divisor $\T$ on $\C$ is the branched divisor of the map $\pi : \mu^{-1}(H) = \D \to C$. This map is a $d_1 + d_2$ sheet covering of $\C$. The ramification points of this map come from three sources : - The ramification points on $\mu^{-1}(H)_{|X_1}.$ - The ramification points on $\mu^{-1}(H)_{|X_2}.$ - The intersections $\mu^{-1}(H) \cap s.$ The first two sources contribute to the pull backs $\T_1 \cdot \C$ and $\T_2 \cdot \C$ respectively. The intersections points $\mu^{-1}(H) \cap s$ correspond precisely to the maps $\gamma$ with $\gamma(C) \in H.$ These points are the nodes of the curve $\D$, because through each of them, there are two branches : one from $\mu^{-1}(H)_{|X_1}$, one from $\mu^{-1}(H)_{|X_2}.$ If $P \in \D $ is one of such points, then the branched divisor of $\pi$ contains $\pi(P)$ with multiplicity $2$. Thus we have $\T \cdot \C = \T_1 \cdot \C + \T_2 \cdot \C + 2\L_C\cdot \C$. Using the lemma, we can “expand" the tangency conditions on $\F_1\times_{ev_C}\F_2$ until we have tangency conditions only on each individual component. Let $\d$ be a constraint and let $\d_l$ be the constraint obtained from $\d$ by removing $l$ tangency conditions. Then we have the following equality : $$\begin{aligned} \# (\F_1 \times_{ev_C} \F_2, \d,k) &=& \sum_{l=0}^{\d(0)} 2^l \binom{\d(0)}{l} \sum_{\gam_1\gam_2 = \d_l} \# (\F_1\times_{ev_C} \F_2,\gam_1,\gam_2,k+l). \\\end{aligned}$$ [ [*Proof.* ]{}]{}There are $(^n_l)$ ways to remove $l$ tangency conditions. Doing this results in a codimension $k+l$ condition on the node (the image of $C$) , and the multiplicity is $2^l.$ Applying the proposition to the family ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ we have : $$\#( {\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\d,k,k') = \sum_{l=0}^{\d(0)} 2^l\binom{\d(0)}{l}\sum_{\gam_1 \gam_2 = \d_l}\#({\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\gam_1,\gam_2,k,k'+l).$$ Counting Rational Space Curves With Special Tangent Conditions ============================================================== In this section, we will describe the algorithm counting rational space curves with special tangent conditions in $\proj.$ Let $\X = \mbar_{0,\setA}(r,d)$ throughout this section. Following the notation in [@idq] let $\H$ be the incident divisor (incident to a codimension $2$ subspace), and let $\K^{A,j}$ be the boundary divisor of $\mbar_{0,\{A\}}(r,d)$ whose points represent reducible curves in which the component containing $A$ is mapped with degree $j$. The main difficulty when we have multiple special tangent conditions is excess intersection: any special tangent divisor $\W_A^M$ passes through the locus of maps $\gamma$ where $\gamma(A)$ is not a smooth point of the image. However, we have the following result that helps us reduce the number of special tangent divisors in our computation. Any characteristic number of rational curves with $l \geq r-1$ special tangent conditions is expressible in terms of characteristic numbers of rational curves with at most $r-2$ special tangent conditions. Proof of this statement will be given in section 5. Thus, we only need to care about excess intersection locus in codimension at most $r-2$. The following proposition lists all components of this locus. Let $S_n$ be the closure of locus of maps $\gamma$ in $\X$ such that the source curve has $n+1$ components, and the component containing $A$, called the principal component, is incident with $n$ other components. Moreover, $\gamma$ contracts the principal component. Then $S_2,\ldots,S_{r-2}$ are the components of codimension at most $r-2$ of the excess intersection locus of the special tangent divisors. Furthermore, $S_n$ contributes to the excess intersection only if there are at least $2n-2$ special tangent conditions. In particular, only $S_n$’s with $2n \leq r$ are relevant in counting curves with special tangent conditions. [ [*Proof.* ]{}]{} Let $\gamma$ be a map in $\X$ such that $\gamma(A)$ is not a smooth point of its image. If $\gamma$ does not contract the component of the source curve containing $A$ then $\gamma(A)$ is at least a nodal singularity. Maps of this type vary in a family of codimension at least $r-1$. Thus if $\gamma$ belongs to a component (of the excess locus) of codimension at most $r-2$, $\gamma$ must contract the component of the source curve that containts $A$. For a multi-index $I(d,n) = (d_1, \ldots,d_n)$ with $\sum_id_i =d$, let $\V_{I(d,n)}$ be $\prod_i \mbar_{0,\setA}(r,d_i)$ where the product is taken over the evaluation maps $ev_A$. It is easy to see that each component of $S_n$ is a finite quotient of a $\mbar_{0,n+1}\times \V_{I(d,n)}$, where $\mbar_{0,n+1}$ is the moduli space of genus zero stable curves with $n+1$ marked points. Now $\mbar_{0,n+1}$ is of dimension $n-2$, which means the “enumerative codimension” of $S_n$ is $n-2$ less than its codimension, hence is $2n-2$. Since we will only need to count rational curves with at most $r-2$ special tangent conditions, only $S_n$ in which $2n-2 \leq r-2$, or equivalently $2n \leq r$, is relevant.\ \ $$\includegraphics[width = 40mm]{S_4.pdf}$$\ $$\text{Fig 3. ${S_4}$ }$$ We will blow up $S_n$’s in order to discount the excess contribution. The above proposition provides us with an useful guideline. In $\mathbb {P}^3$, no blowup is needed. One blowup of $S_2$ is needed for $\mathbb P^4$ and $\mathbb P^5$. More generally, we need one more blowup for each increase by two in the dimension of the ambient space. In the rest of this section, we provide explicit formula for the cases $\mathbb P^3, \mathbb P^4, \mathbb P^5$, which only requires at most one blowup as expect.\ \ [**Case 1:**]{} Counting rational curves with one special tangent condition in $\proj, r\geq 3.$\ \ We can express the special tangent divisor as linear combinations of boundary divisors and incident divisors, as shown in the following lemma. The following equality holds in the group $A^1(\X) \otimes \rationals$, for $r > 2$ : $${\mathcal W}_A = 2\L_A + \psi_A$$ where $\psi_A$ is the psi-class. In particular, we have $$\W_A = \left(2-\frac{2}{d} \right)\L_A + \frac{1}{d^2}\H + \sum_{j=1}^{j<d}\frac{(d-j)^2}{d^2}\K^{A,j}$$ [ [*Proof.* ]{}]{}We use the method as described in [@idq], intersecting the two sides of the equations with a general curve $\C$ in $\X$. Let $\gamma$ denote the image of $\C$ under the evaluation map $ev_A$. Let $M$ be the codimension $2$ subspace in $\proj$ corresponding to the special tangent condition $\W_A$. Beccause $\C$ is a general curve, we can assume $\gamma$ is smooth. Let $L$ be a general line in $\proj$, and let $\pi_M : \proj - M \to L$ be the projection onto $L$ from $M$. Let $\phi_A$ be the line bundle on $\gamma $ described as follows. For each point $p \in \gamma$, $ev_A^{-1}(p)$ is a map $\alpha \in \C$. The fibre of $\phi_A$ over $p$ is then the tangent vector to the image of $\alpha$ at $\alpha(A)$. Let $R$ be the zero scheme of the bundle map $\phi_A \to {\pi_M}^*(T_L)$, with $T_L$ being the tangent bundle of $L$. Geometrically, $R$ represents the locus pf points $p \in \gamma$, such that the map $ev_A^{-1}(p)$ satisfies special tangent condition with respect to the subspace $M$. Thus $$\deg R = R \cap [\gamma] = \C \cap \W_A.$$ We have $$\deg R = -c_1(\phi_A) + \deg({\pi_M}_{|\gamma})c_1(T_L).$$ Now $c_1(T_L) = 2$\[class of a point\], and $\deg({\pi_M}_{|\gamma}) = \deg \gamma = \L_A \cap \C$. The pullback of $\phi_A$ by $ev_A$ is isomorphic to the line bundle on $C$ obtained by attaching to each map the tangent vector at $A$ to the source curve. Hence $-c_1(\phi_A)\cap \gamma = -c_1(ev_A^*(\phi_A)) \cap \C = \psi_A \cap \C$ is the usual psi-class. In short, we have $${\mathcal W}_A = 2\L_A + \psi_A.$$ The second equality follows from the fact that $\psi_A = - \pi_*(s_A^2)$ on $\mbar_{0,\seta}(r,d)$ and Lemma 2.2.2 in [@idq].\ \ [**Case 2:**]{} Counting rational curves with two special tangent conditions in $\proj, r\geq 4.$\ \ Let $\pi: \wt{\X} \to \X$ the blowup of $\X$ along $S_2$. Let $S_2^j$ be the component of $S_2$ with degree partition $(j,0,d-j)$, and let $E_2^j$ be the corresponding exceptional divisor. We have that $S_2^j$ is a $\mathbb{Z}_2$-quotient of ${\mathbbmss{R}}{\mathbbmss{R}}(j,d-j)$. A general element $E_2^j$ has following geometric interpretation: it is a pair $(\gamma,l)$ where $\gamma$ is a map in ${\mathbbmss{R}}{\mathbbmss{R}}(j,d-j)$ , and $l$ is a line in $\proj$. $l$ must lie on the plane $(l_1,l_2)$ where $l_i$ is the projective tangent line to the image (under $\gamma$) of the $i$-th component at the image (under $\gamma$) of $A$ (here we use $A$ to denote the node of the family ${\mathbbmss{R}}{\mathbbmss{R}}(j,d-j)$, instead of using $C$ as in the definition in Section 2.2, but this does not change anything). For each divisor $\D$ of $\X$, let $\wt{\D}$ be its proper transformation. The next lemma allows us to compute the class $\pi_*(\wt{\W}_A^2)$ The following equality holds in $A^2(\X) \otimes \mathbb Q$: $$\begin{aligned} \pi_* (\wt{\W}_A^2) &=& \left( 2 - \frac{2}{d} \right)\W_A\L_A + \frac{1}{d^2}\W_A\H + \sum_{j=1}^{j<d}\frac{(j-d)^2}{d^2}\pi_*(\wt{\W}_A \wt{\K}^{A,j}) + \sum_{j=1}^{j\leq d/2} \frac{2j^2 - 2jd}{d^2}S_2^j\end{aligned}$$ The class $\pi_*(\wt{\W}_A \wt{\K}^{A,j})$ is the class of the closure of the locus of maps with reducible source curves, where the restriction onto the component containing $A$ satisfies one special tangent condition. Counting maps in $\pi_*(\wt{\W}_A \wt{\K}^{A,j})$ is doable by Lemma $4.1$ and results in section $3$. Counting maps in $S_2^j$ is equivalent to counting maps in ${\mathbbmss{R}}{\mathbbmss{R}}(j,d-j)$ which is also doable by results in section $3$.\ \ [ [*Proof.* ]{}]{}We pull back the main equation of Lemma 4.3: $$\pi^*\W_A = \left(2-\frac{2}{d} \right)\wt{\L}_A + \frac{1}{d^2}\wt{\H} + \sum_{j=1}^{j<d}\frac{(d-j)^2}{d^2}\pi^*\K^{A,j}$$ $\pi^*\W_A = \wt{\W}_A + \sum_j E_2^j$ and $\pi^*\K^{A,j} = \wt{\K}^{A,j} + m_jE_2^j$ where $m_j$ is $1$ if $j \neq d-j$ and $2$ if $j=d-j$. Rearranging the terms, we have $$\wt{\W}_A= \left(2-\frac{2}{d} \right)\wt{\L}_A + \frac{1}{d^2}\wt{\H} + \sum_{j=1}^{j<d}\frac{(d-j)^2}{d^2}\wt{\K}^{A,j} + \sum_{j=1}^{j\leq d/2} \frac{2j^2 - 2jd}{d^2}E_2^j$$ Now it is obvious that $\pi_*(\wt{\W}_AE_2^j)= S_2^j$. Multiply the above equation with $\wt{\W}_A$ and pushforward yields the desired equation.\ \ Using Lemma $4.4$, we can reduce a counting problem involving two special tangent conditions into various counting problems involving at most one special tangent condition.\ \ [**Case 3:**]{} Counting rational curves with three special tangent conditions in $\proj, r\geq 5$.\ \ View ${\mathbbmss{R}}{\mathbbmss{R}}(j,d-j)$ as $\mbar_{0,\setA}(r,j) \times_{ev_A} \mbar_{0,\setA}(r,d-j)$. Let $\W^{(i)}$ be the pullback of the special tangent divisor of the $i$-th factor. Let $p:{\mathbbmss{R}}{\mathbbmss{R}}(j,d-j) \to S_2^j$ be the natural projection. We have the following lemma. The following equality holds in $A^3(\X)\otimes \rationals$: $$\begin{aligned} \pi_* (\wt{\W}_A^3) &=& \left( 2 - \frac{2}{d} \right)\pi_*(\wt{\W}^2_A)\L_A + \frac{1}{d^2}\pi_*(\wt{\W}_A)^2\H + \sum_{j=1}^{j<d}\frac{(j-d)^2}{d^2}\pi_*(\wt{\W}^2_A \wt{\K}^{A,j}) \\ &+& \sum_{j=1}^{j\leq d/2} \frac{2j^2 - 2jd}{d^2}\pi_*(\wt{\W}_A^2E_2^j)\end{aligned}$$ $\pi_*(\wt{\W}^2_A \wt{\K}^{A,j})$ is the closure in $\X$ of the locus of maps with reducible source curves, where the restriction of the map on the component containing $A$ satisfies two special tangent conditions. Counting maps in this locus is doable by Lemma $4.4$ and results in section $3$. Furthermore, for any constraint $\d$ we have $$(\pi_*(\wt{\W}_A^2E_2^j),\d) = (\W^{(1)} + \W^{(2)},\d)$$ if both sides are finite. [ [*Proof.* ]{}]{}Only the last equality needs proving. Because the constraint $\d$ cuts out a one-dimensional family on ${\mathbbmss{R}}{\mathbbmss{R}}(j,d-j)$, proving the equality is an intersection theory problem on a $\mathbb P^1$-bunlde over a curve. We reformulate the problem as follows. Let $\F_1$ be a one-dimensional family of projective rational curves of degree $j$ with a marked point $A$. We associated with $\F_1$ the line bundle $l_1$ which is the line bundle of the projective tangent lines at $A.$ Similarly, we have $\F_2$ and $l_2$, where curves in $\F_2$ have degree $d-j.$ Let $\C = \F_1 \times_{ev_A} \F_2$, which is a curve ($\F_i$’s are choosen so that $\C$ is not empty). Let $\P$ be the projectivization of $l_1 \oplus l_2$. Thus $\pi: \P \to \C $ is a rank-one projective bundle. A general element of $\P$ is a pair of curve-line $(\gamma,l)$ with $\gamma \in \C$ and $l \subset (l_1,l_2).$ Let $\W$ be the divisor on $\P$ define as follows. For a general codimension $2$ subspace $M\in \proj$, a pair $(\gamma,l) \in \P$ is in $\W$ if and only if $l \subset M$. We have a natural inclusion $\F_i = P(l_i) \subset \P$, with $P(l_i)$ being the projectivization of the line bundle $l_i$. Let $\D$ be the canonical line bundle on $\P$, and let $\G$ be the pullback of a point $\pi^{(-1)}(p)$ for any $p\in \C$. With this reformulation, the equality that we need to prove becomes $$\W^2 = \W\F_1 +\W\F_2$$ Let $a_i = - c_1(l_i) \cdot \C$. We have $$\F_1 = \D + \pi^*(c_1(\phi_2) \cap C) = \D - a_2\G$$ hence $$\deg(\F_1^2) = \deg(\pi_*(\D^2 - 2a_2\D\G + a_2^2\G^2)) = \deg (s_1(F) \cap C) - 2a_2 = a_1 + a_2 - 2a_2 = a_1 - a_2$$ which means that $\F_1^2 = a_1 - a_2$ as $F_1^2$ is of dimension $0$ in the Chow ring of $\P$. Similarly $\F_2^2 = a_2 - a_1$, thus $F_1^2 + F_2^2 = 0$. Now let $\W =a\F_1 + b\G$. Then we have $\W\G = 1 = a(\F\G) \Rightarrow a = 1$. Now we have $\W\F_1 = \F_1^2 + b \Rightarrow b = \W\F_1 - \F_1^2$. That leads to $\W^2 = 2\W\F_1 - \F_1^2$. Similarly $\W^2 = 2\W\F_2 - \F_2^2$. Add the two equalities together we have $$\W^2= \frac{1}{2}(2\W\F_1 + 2\W\F_2 -\F_1^2 - \F_2^2) = \W\F_1 + \W\F_2.$$\ \ Using Lemma $4.5$, we can reduce a counting problem involving three special tangent conditions into various counting problem involving at most $2$ special tangent conditions.\ \ We end this section with some examples. How many conics in $\mathbb P^3$ passing through $3$ points, that have a marked point $A$ which must lie on a fixed line $M$, and that the tangent line at $A$ to the curve passes through a fixed line $L$? The answer is $1$. [ [*Proof.* ]{}]{}Because the three points that the conic passes through determine its plane $H$, this problem reduces to an enumerative problem in $\mathbb{P}^2$ : how many conics in $\mathbb{P}^2 $ that pass through $3$ points and is tangent to a line at a fixed point? The answer is therefore $1$. Now we will compute this number in a different way, using Lemma $4.3.$ Let $\d = (0,0,0,3)$, and $\d'=(0,0,1,3)$. We need to compute $\# ((\mbar_{0, \{A\}}(3,2), \d),\L^2_AW_A)$. On $\mbar_{0,\{A\}}(3,2)$, there is one boundary divisor, $\K= (\emptyset, 1 \sep \{A \}, 1)$, which parametrize pair of lines intersecting at one point, and the marked point $A$ is on one of them. Using lemma $4.3$ we have $$\W_A = \L_A + \frac{\H}{4} + \frac{\K}{4}$$ Thus $$\begin{aligned} \# ((\mbar_{0, \{A\}}(3,2), \d),\L^2_AW_A) &=& \# ((\mbar_{0, \{A\}}(3,2),\d),\L^3_A) + \frac{1}{4}\# ((\mbar_{0, \{A\}}(3,2), \d'),\L^2_A) \\ && + \frac{1}{4} \# ((\K,\d),\L^2_A) \\ &=& 0 + \frac{1}{4} + \frac{1}{4}3 = 1.\end{aligned}$$ The first “$\#$” term of the right hand side is the number of conics in $\proj$ passing through $4$ points. The second “$\#$” term is the number of conics in $\proj$ passing through $3$ points and $2$ lines. The last “$\#$” term is the number of pair of lines in $\proj$ with one common point, that pass through $3$ points, and that the component with the marked point $A$ intersect a line at $A$. There are $2$ conics in $\mathbb{P}^4$ satisfying the following conditions. The conics pass through $3$ points and a plane, and there is a marked point $A$ on the curve, the projective tangent line at which passes through $2$ other planes. [ [*Proof.* ]{}]{}Again, the three point conditions determine the plane $H$ for the conics. Thus in fact we have a plane curve counting problem. The conics must pass through $4$ points (the plane condition now become point condition), and the tangent line at $A$ must pass through $2$ other points on the plane $H$. Thus the problem is equivalent to counting plane conics through $4$ points and tangent to $1$ line, thus the answer is two. We must show that $$\#((\mbar_{0,{A}}(4,2),\d),\W_A^2) = 2$$ with $\d = (0,1,0,0,3)$. From the proof of Lemma $4.4$ we have $$\wt{\W}_A= \wt{\L}_A + \frac{\wt{\H}}{4} + \frac{\wt{\K}^{A,1}}{4} - \frac{E_2^j}{2}$$ Multiply the equation with $\wt{\W}_A$, pushforward and integrate against $(\mbar_{0,\{A\}}(4,2),\d)$ we have $$\begin{aligned} \#((\mbar_{0,{A}}(4,2),\d),\W_A^2) &=& \#((\mbar_{0,{A}}(4,2),\d),\W_A\L_A) + \frac{1}{4}((\mbar_{0,{A}}(4,2),\d'),\W_A) \\ &+& \frac{1}{4}\#((\K^{A,1},\d),\W_A) - \frac{1}{2} \#(E^j_2,\d) \\ &=& = 3 + \frac{2}{4} + 0 - \frac{3}{2} = 2 \end{aligned}$$ where $\d' = (0,2,0,0,3)$. We list below several numbers of curves with special tangent conditions in $\mathbb{P}^3, \mathbb{P}^4, \mathbb{P}^5$. The special class $(a,b)$ means the marked point as a codimension $a$ condition and there are $b$ special tangent conditions.\ \ [ | l | c | c | l |]{} Degree & Condition & Special Classes & Numbers\ Cubic & $(1,2,3)$ & $(3,1)$ & 34\ Cubic & $(4,2,2)$ & $(2,1)$ & 4736\ Quartic & $(7,2,3)$ & $(1,1)$ & 35131904\ Quintic & $(4,4,6)$ & $(0,1)$ & 280111872\ Quintic & $(2,2,7)$ & $(2,1)$ & 352176\ Sextic & $(3,4,7)$ & $(3,1)$ & 340403776\ $$\text{Table 1. Some enumerative numbers with special class in $\mathbb{P}^3$}$$ [ | l | c | c | l | ]{} Degree & Condition & Special Classes & Numbers\ Conic & $(1,1,2,1)$ & $ (1,2)$ & 38\ Cubic & $(2,1,1,3)$ & $(1,2)$ & $980$\ Quartic & $(2,2,1,4)$ & $(2,2)$ & $37792$\ Quintic & $(3,3,1,5)$ & $(2,2)$ & $31565232$\ Sextic &$(3,3,4,5)$ &$(1,2)$ & $49679646304$\ $$\text{Table 2. Some enumerative numbers with special classes in $\mathbb{P}^4$}$$ [ | l | c | c | l | ]{} Degree & Condition & Special Classes & Numbers\ Conic & $(1,1,1,0,2)$ & $(0,3)$ & $20$\ Cubic & $(1,1,1,2,2)$ & $(0,3)$ & $1240 $\ Quartic &$(2,3,1,2,2)$ & $(3,3)$ & $1181400$\ Quintic & $(2,2,3,4,2)$ & $(0,3)$ & $ 1654232816 $\ $$\text{Table 3. Some enumerative numbers with special classes in $\mathbb{P}^5$}$$ Counting curves in ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ ================================================================ First we need a result about the Chow ring of $Bl_{\D}(\proj \times \proj)$, which is the blowup of $\proj \times \proj$ along the diagonal. For details of the derivation, we refer the readers to [@dn2]. The Chow ring of $Bl_{\D}(\proj \times \proj)$ is generated by $h,k$, the hyperplane class of the first and second factor, and the exceptional divisor $e$ with the following relations : $$\begin{aligned} h^{r+1} &=& k^{r+1} =0, \\ he&=& ke, \\ e^r &=& \sum_{i>0}^{i < r}(-1)^{i-1}(^{r+1}_{ \ i})h^ie^{r-i} + \sum_{i\geq 0}^{i\leq r}h^ik^{r-i}.\end{aligned}$$ [**Example.**]{} The following are the third relation in the case $r=1,2,3,4$: $$\begin{aligned} e &=& h + k. \\ e^2 &=& 3he - (h^2+hk+k^2). \\ e^3 &=& 4he^2 - 6h^2e + (h^3 +h^2k + hk^2 + k^3). \\ e^4 &=& 5he^3 - 10h^2e^2 + 5h^3e - (h^4 + h^3k + h^2k^2 + hk^3 + k^4).\end{aligned}$$ Recall that ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ is a substack of $\mbar_{0, \{A,C\}}(r,d_1) \times_{ev_C} \mbar_{0, \{B,C\}}(r,d_2)$ of maps $\gamma $ such that $\gamma(A) = \gamma(B)$. We rephrase the problem of counting maps in ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ as follows : *Given two families $\F_1$ and $\F_2$ of maps of rational curves with two marked points $A,C$. How many times a map $\gamma_1$ from $\F_1$ and a map $\gamma_2 $ from $\F_2$ intersect in such a way that :* - $\gamma_1(A) = \gamma_2(A)$ and $\gamma_1(C) = \gamma_2(C).$ - $\gamma_i(A)$ lies on a fixed linear space of codimension $p$. - $\gamma_i(C)$ lies on a fixed linear space of codimension $q$. We consider the evaluation map $$ev_{AC}: \F_i \longrightarrow(\proj\times\proj)$$ Let $T_i$ be the closure in $Bl_{\D}(\proj \times \proj)$ of $ev_{AC}(\F_i)$. Let $h,k$ be the hyperplane classes of the first and second factor in $Bl_{\D}(\proj \times \proj).$ Then the answer to our enumerative problem above is the intersection number $$T_1T_2h^pk^q$$ where the product is evaluated in the Chow ring of $Bl_{\D}(\proj \times \proj)$. ($T_i$ parametrizes ordered pair of points on the curves in $\F_i$. The blowup is to prevent us from counting in the case where two points run into each other). To count maps in ${\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2)$ satisfying the constraint $(\d,p,q),$ we first consider all the partitions $\d = \gam_1\gam_2$, and for each such partition, assign constraint $\gam_i$ to the $i$-th component. If $\d(0) \neq 0$, meaning if there are tangency conditions, we also have to distribute the tangency conditions over each component first, in the sense of Proposition $3.3$. Then the constraint $\gam_1$ cuts out a family $\F_1$ on $\mbar_{0,\{A,C\}}(r,d_1)$. Similarly, $\gamma_2$ cuts out a family $\F_2$ on $\mbar_{0,\{A,C\}}(r,d_2)$. Let $T_i$ be the closure of $ev_{AC}(\F_i)$ in $Bl_{\D}(\proj \times \proj)$ . We then calculate the product $$T_1T_2h^pk^q$$ in the Chow ring $A^*(Bl_{\D}(\proj \times \proj))$. Then we take the sum over all partitions $\d = \gam_1\gam_2$ to get the number of maps $\# ({\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2), \d ,p,q).$ We need a result to calculate the classes of $T_i$ in $A^*(Bl_{\D}(\proj \times \proj)).$ The following lemma is useful: Let $\F$ be a family of stable maps in $\mbar_{0,\{A,C\}}(r,d)$ such that $A,C$ moves freely, that is, the forgetful map $\mbar_{0,\{A,C\}}(r,d) \to \mbar_{0,0}(r,d)$ has fibre dimension $2$. Let $T$ be the closure in $Bl_{\D}(\proj \times \proj)$ of the image of $\F$ under the evaluation map $ev_{AC}: \F \to \proj \times \proj$. Let $\G$ be the family of stable maps in $\mbar_{0,\{A\}}(r,d)$ that is the image of $\F$ under the forgetful morphism $\mbar_{0,\{A,C\}}(r,d) \to \mbar_{0,\{A\}}(r,d)$. Assume $\dim T \leq 2r$. Then we have - For $m,n$ such that $m+n = \dim T$ : $$Th^mk^n= \# (\F,\L_A^m\L_C^n).$$ - For $m$ such that $m + 1 = \dim T$ : $$Th^me = \# (\G,\L_A^m).$$ - For $m,n$ such that $m+n = \dim T$ , we have $$Th^me(h+k-e)^{n-1}= \#(\G,\L_A^m\W_A^{(n-1)}).$$ [ [*Proof.* ]{}]{}The first equality is trivial. The number $Th^mk^n$ is the number of maps $ \gamma \in \F$ such that $\gamma(A)$ belongs to $h$ hyperplanes, and that $\gamma(C)$ belongs to $k$ hyperplanes. That is precisely the number $\#(\F,\L_A^m\L_C^n).$ The second equality follows from the fact that multiplying with $e$ is the same as replacing the family $\F$ by the family $G$. Now we prove the third equality. Let $$[x_0:x_1:\cdots:x_n] \times [y_0:y_1:\cdots:y_n]$$ be a homogeneous coordinate system of $\proj \times \proj$. Let $H$ be the hypersurface $$x_0y_n = x_ny_0$$ in $\proj \times \proj.$ $H$ contains $\D$ with multiplicity one and $T = h +k$ in $A^*(\proj \times \proj)$, hence the proper transformation $\widetilde{H}$ of $H$ in $Bl_{\D}(\proj \times \proj)$ satisfies $$\widetilde{H} = h + k -e.$$ Let us examine what it means to intersect ${T}$ with $e$ and $\widetilde{H}.$ Let $\pi : Bl_{\D}(\proj \times \proj) \to \proj \times \proj$ be the blow up, and let $S = \pi(T)$. We have a map $\gamma: S \to S \cap \D$ defined as folows. For each point $x \in S$, let $P_x$ be the subspace $ \{p\} \times \proj \subset \proj \times \proj$, where $\{p\} \in \proj$ is chosen so that $x \in P_x$. The intersection $S \cap P_x$ is a genus zero curve $f_x$ in $P_x,$ and $\gamma$ maps the entire curve $f_x$ onto $x$. The intersection $H \cap P_x$ is a hyperplane in $P_x$ which is the span of $x$ and the codimension $2$ subspace $x_0=y_0 = 0.$ Then for a point $y \in {T}$ with $\pi(y) = x$, we have $ y \in {T} \cap e \cap \wt{H}$ iff $f_x$, as a curve in the projective space $P_x$ is tangent to $H_x$ at $x.$ Thus intersecting with $\widetilde{H}$ (after intersecting with $e$) has the effect of imposing one special tangent condition on the family $\G.$ It follows that intersecting with $n-1$ instances of $\wt{T}$ has the effect of imposing $n-1$ special tangent conditions. $$\includegraphics[width = 70mm]{SpecialProof.pdf}$$ $$\text{Fig 4.}$$\ \ Now we have enough to be able compute the class of $T = {ev_{AC}}_*(\F)$ in $A^*(Bl_{\D}(\proj \times \proj)).$ The formal statement of that fact is the following proposition, whose proof is trivial. Let $T \in A^*(Bl_{\D}(\proj \times \proj))$ be a class of codimension $d, 0 \leq d \leq 2r$ . Then the following intersection products determine $T$ : - $Th^mk^n$ with $0 \leq m \leq r, 0\leq n \leq r.$ - $Th^{m}e(h+k-e)^n$ with $0\leq m \leq r, 0\leq n \leq r-2$. - $Th^{d-1}e$. with $m,n$ appropriately choosen so that the intersection number is well-defined. The reason the power $n$ of $h+k-e$ is at most $r-2$ is because $e^r$ is expressible as polynomials in $h$ and $k$, so we never need to multiply $T$ with a power of $e$ that is more than $r-1$, in order to determine $T$. In particular, if we know all characteristic numbers of rational curves with at most $r-2$ special tangent conditions, then that is enough to count maps in ${\mathbbmss{R}}{\mathbbmss{R}}_2(d_1,d_2).$ [*Proof of Proposition 4.1.*]{} If the number of special tangent conditions $l$ is greater than $2r-2$, then the number is $0$ because the tangent line at $\gamma(A)$ can pass through at most $2r-2$ general codimension $2$ subspaces. Now assume $l \leq 2r-2$. Let $\d$ be the constraint (beside the special tangent conditions). Let $\F$ be $(\mbar_{0,\{A,C\}}(r,d),\d)$ and $T$ be the closure in $Bl_{\D}(\proj \times \proj)$ of the image of $\F$ under $ev_{AC}$. We have $\dim T < 2r.$ If we know all the characteristic numbers with at most $r-2$ special tangent conditions, then Proposition $5.3$ shows that we can determine $T$. Then the characteristic number with constraint $\d$ (and $\L_A^m$) and $l$ special tangent conditions is the intersection number $Th^me(h+k-e)^l$.\ \ We end the section with some examples. How many pair of lines $(L_1,L_2)$ in $\mathbb P^3$ such that they intersect twice, and that each of them passes through $3$ lines? The answer is $0.$ The answer is obvious because two distinct lines can never intersect twice. But our algorithm does not know that. Let $\d = (0,0,3,0)$. We need to compute $$\frac{1}{2}\# ({\mathbbmss{R}}{\mathbbmss{R}}_2(3,1,1), \d,\d).$$ The factor $1/2$ accounts for the fact that the statement of the problem does not distinguish the two intersection points. Let $\F_i$ be the family of the lines $L_i$ with a choice of two marked points $A,C$ on them. Let $T_i$ be the pushforward of $\F_i$ under the evaluation maps $ev_{AC} : \F_i \to Bl_D(\proj \times \proj).$ $T_1$ is three dimensional, so we can assume $$T_1 = \alpha(h^3+k^3) + \beta(h^2k +hk^2) + \gamma eh^2 + \mu e^2h.$$ The coefficients of $h^3$ and $k^3$ must be the same due to symmetry. Similarly the coefficients of $h^2k$ and $hk^2$ must be the same. $$\begin{aligned} \alpha &=& \alpha h^3k^3 = T_1k^3 = \# ((\mbar_{0,\{A,C\}}(3,1),\d),\L_A^3) = 0 \\ \beta &=& \beta h^3k^3 = T_1kh^2 = \# ((\mbar_{0,\{A,C\}}(3,1),\d),\L_A^2\L_C) = 2 \\ \mu &=& \mu h^3e^3 = T_1h^2e = \# ((\mbar_{0,\{A\}}(3,1),\d),\L_A^2) = 2 \end{aligned}$$ Computation of $\gamma$ is a little bit lengthier. First we have $$\begin{aligned} \gamma &=& \gamma h^3k^3 = T_1 he^2 - \mu e^4h^2 = \left (2T_1 h^2e - T_1he(k+k-e) \right) - 4\mu \\ &=& -2\mu - T_1he(h+k-e). \end{aligned}$$ Now $T_1he(h+k-e) = \# ((\mbar_{0, \{A\}}(3,1),\d),\L_A\W_A)$ is the number of lines with a marked point $A$ in $\mathbb P^3$ that pass through $3$ lines, such that $A$ lies on a fixed plane, and such that the tangent line at $A$ passes through a general line. This number is the same as the number of lines passing through $4$ general lines in $\proj$, which is $2$. Thus $\gamma = -2\mu - T_1hk(h+k-e) = -4 - 2 = -6$. Therefore $$T_1 = 2(h^2k+hk^2) - 6h^2e + 2he^2$$ Obviously $T_1 = T_2$, so after a bit of algebra we have $$T_1T_2 = \left(2(h^2k+hk^2) - 6h^2e + 2he^2 \right)^2= 0.$$ How many pair of conics-twisted cubics in $\mathbb{P}^5$ intersecting at two nodes, with the first node being on a fixed hyperplane and the second node being on a fixed $3-$space, such that the conic passes through one $3-$space, one general plane, one general line, one general point, and the cubic passes through two general $3-$spaces, one general plane, one general line, two general points? The answer is $956$. Let $\gam_1 = (0,0,1,1,1,1,0)$ and $\gam_2 = (0,0,2,1,1,2,0)$. We need to compute $$\#({\mathbbmss{R}}{\mathbbmss{R}}_2(5,2,3), \gam_1,\gam_2,1,2).$$ Let $\F_1$ be a family of lines conics in $\mathbb P^5$ with a choice of two marked points $A,C$ on them, such that the conics satisfy $\gam_1.$ Let $\F_2$ be the a family of twisted cubics in $\mathbb P^5$ with a choice of two marked points $A,C$ on them, such that the cubics satisfy $\gam_2.$ Let $T_i$ be the pushforward of $\F_i$ under ${ev_{AC}}_*$ onto the Chow ring $A^*(Bl_{\D}(\mathbb P^5 \times \mathbb P^5)).$ The we need to compute the intersection product $hk^2T_1T_2.$ Using Lemma $5.2$ and Proposition $5.3$, we can find the classes of $T_i$ to be : $$\begin{aligned} T_1&=& 2h^4 + 6h^3k + 8h^2k^2 + 6hk^3 + 2k^4 - 42h^3e + 29h^2e^2 - 9he^3 + e^4 \\ T_2&=& 45h^3 + 88h^2k + 88hk^2 + 45k^3 - 308h^2e+ 140he^2 -23e^3\end{aligned}$$ Using proposition $5.1$, we can calculate the product: $$\begin{aligned} &&(2h^4 + 6h^3k + 8h^2k^2 + 6hk^3 + 2k^4 - 42h^3e + 29h^2e^2 - 9he^3 + e^4) \\ &\times& ( 45h^3 + 88h^2k + 88hk^2 + 45k^3 - 308h^2e+ 140he^2 -23e^3)hk^2 = 956.\end{aligned}$$ Some numbers; [| l | l | c | c | c | l |]{} & [**Degree**]{} & [**Constraint**]{} & [**Constraint**]{} & [**Nodes**]{} & [**Number**]{}\ Conic & Conic & $(2,3,1)$ & $(2,3,1)$ & $(0,0)$ & $ 3360 $\ Conic & Cubic & $(2,3,1)$ & $(3,4,1)$ & $(1,1)$ & $614656 $\ Line & Quartic & $(0,1,0)$ &$(3,4,3)$ & $(2,2)$ & $570752$\ Cubic & Cubic & $(3,3,2) $ &$(1,4,2)$ &$(0,3)$ & $963360$\ Conic & Quartic & $(3,3,1) $ & $(0,6,4)$ & $(0,0)$ & $2253312$\ $$\text{Table 4. Some enumerative numbers of pair of rational curves in $\mathbb{P}^3$}$$ [| l | l | c | c | c | l |]{} & [**Degree**]{} & [**Constraint**]{} & [**Constraint**]{} & [**Nodes**]{} & [**Number**]{}\ Conic & Conic & $(1,1,2,1)$ & $(0,0,0,3)$ & $(0,0)$ & $4$\ Conic & Cubic & $(1,2,1,1)$ & $(1,1,2,2)$ & $(1,2)$ & $4816$\ Line & Conic & $(0,1,1,0)$ & $(1,1,1,1)$ & $(1,2)$ & $18$\ Cubic & Cubic & $(3,1,0,3)$ & $(3,1,0,3)$ & $(1,1)$ & $2297664$\ $$\text{ Table 5. Some enumerative numbers of pair of rational curves in $\mathbb{P}^4$}$$ [| l | l | c | c | c | l |]{} & [**Degree**]{} & [**Constraint**]{} & [**Constraint**]{} & [**Nodes**]{} & [**Number**]{}\ Conic & Conic & $(0,0,0,2,1)$ & $(0,0,0,2,1)$ & $(1,1)$ & $2$\ Conic & Cubic & $(1,0,1,0,2)$ & $(1,1,0,1,3)$ & $(0,0)$ & $144$\ Line & Quartic & $(0,0,0,0,1)$ & $(2,0,0,2,3)$ & $(1,3)$ & $844$\ Cubic & Cubic & $(3,4,1,1,1)$ & $(2,1,1,2,1)$ & $(1,2)$ & $1027324928$\ $$\text{ Table 6. Some enumerative numbers of pair of rational curves in $\mathbb{P}^5$.}$$ Counting rational nodal curves in $\proj$ ========================================= First we gave a recursion counting incidence-only characteristic numbers of rational nodal curves (with condition on the node) in $\proj$. Let $\d$ be a constraint that $\d(0)= 0.$ Let $k = \d(r+1).$ Choose a subspace $u$ in $\d$ which is not a hyperplane, such that the dimension of $u$ is largest possible. Then choose any two other subspaces $s,t$ in $\d$. The following constraints are derived from $\d$ :\ 0) $\wt{\d}$ by removing $u,s,t$ from $\d.$\ 1) $\d_0$ by replacing $u$ with two subspaces : a hyperplane $p$ and a subspace $q$ such that $p \cap q = u.$\ 2) $\d_1$ is derived from $\d_0$, by replacing $p$ and $s$ with $p\cap s.$\ 3) $\d_2$ is derived from $\d_0$, by replacing $q$ and $t$ with $q\cap t.$\ 4) $\d_3$ is derived from $\d_0$, by replacing $s$ and $t$ with $s\cap t.$\ \ If $\gam$ is a set of linear spaces, and $a$ and $b$ are two linear spaces, denote $\gam^{(a,b)}$ the set obtained from $\gam$ by adding $a $ and $b.$ Then the following formula holds : $$\begin{aligned} \#(\n(r,d),\d) &=& - \sum_{d_1+d_2 = d}^{\gam_1\gam_2 = \wt{\d}} \binom{\wt{\d}}{\gam_1}\#(\n{\mathbbmss{R}}(r,d_1,d_2), \gam_1^{(s,t)}, \gam_2^{(p,q)},0) \\ &-& \sum_{d_1+d_2=0}^{\gam_1\gam_2 = \wt{\d}} \binom{\wt{\d}}{\gam_1} \#(\n{\mathbbmss{R}}(r,d_1,d_2), \gam_1^{(p,q)}, \gam_2^{(s,t)},0) \\ &-& 2\sum_{d_1+d_2 =d}^{\gam_1\gam_2 = \wt{\d}} \binom{\wt{\d}}{\gam_1}\#({\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\gam_1^{(p,q)}, \gam_2^{(s,t)},k,0) \\ &+& \sum_{d_1+d_2 =d}^{\gam_1\gam_2 = \wt{\d}}\binom{\wt{\d}}{\gam_1} \#(\n{\mathbbmss{R}}(r,d_1,d_2),\gam_1^{(q,t)}, \gam_2^{(p,s)},0) \\ &+& \sum_{d_1+d_2 =d}^{\gam_1\gam_2 = \wt{\d}} \binom{\wt{\d}}{\gam_1}\#(\n{\mathbbmss{R}}(r,d_1,d_2),\gam_1^{(p,s)}, \gam_2^{(q,t)},0) \\ &+& 2\sum_{d_1+d_2 =d}^{\gam_1\gam_2 = \wt{\d}} \binom{\wt{\d}}{\gam_1}\#({\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\gam_1^{(p,s)}, \gam_2^{(q,t)},k,0 ) \\ &-& \# (\n(r,d),\d_3) + \# (\n(r,d),\d_1) + \# (\n(r,d),\d_2).\end{aligned}$$ Furthermore, $\d_1,\d_2,\d_3$ are all of lower rank than that of $\d.$ Here $\binom{\alpha}{\beta} = \prod \binom{\alpha(i)}{\beta(i)}$ for any two tuples $\alpha,\beta$ having the same length. [ [*Proof.* ]{}]{} Let $S$ be a set of markings that is in one-to-one correspondence $\mu : \d_0 \to S$ with the linear spaces in $\d_0$. Let $\X$ be the moduli space $\mbar_{0, \{ A,B\} \cup S}(r,d)$, and let $\n^{(S)}(r,d)$ be the closure in $\X$ of the locus of maps $\gamma$ such that $\gamma(A) = \gamma(B).$ Let $\Y$ be the closure in $\n^{(S)}$ of the locus of maps $\gamma$ such that $\gamma(\mu(m)) \in m$ for all $m \in \d_0.$ Because $\# (\n(r,d),\d)$ is finite, $\Y$ is one-dimensional. We consider two equivalent divisors on $\X$ : $$(\{\mu(p),\mu(q) \} \sep \{ \mu(s), \mu(t)\})= (\{ \mu(p), \mu(s) \}\sep \{\mu(q), \mu(t)\}).$$ Let $\K_1 = (\{\mu(p),\mu(q)\} \sep \{\mu(s), \mu(t)\}), $ and let $ \K_2 = (\{\mu(p),\mu(s)\} \sep \{\mu(q), \mu(t)\})$. Then we have $$\# \left(\Y \cap \K_1\right) = \# \left( \Y \cap \K_2 \right).$$ Let us analyze the left-hand side of the equation. Let $\gamma$ be a general point of $\Y \cap \K_1$. Then $\gamma$ is a stable map whose source curve has two components $C_1,C_2$ joined at a node, such that $\mu(p), \mu(q) \in C_1$ and $\mu(s), \mu(t) \in C_2.$ There are several cases to consider: - $\deg \gamma_{|C_1} =0.$ If only $A$ or $C$ is on $C_1$ then by dimension couting we have that this case has no contribution. If both $A,C$ are on $C_1$ then the image curve has a cusp, on which we impose condition like those we impose on $p,q$. By dimension count again, we also have that the case has no contribution. The quick reason is that if a map contracted a component containing at least $4$ special points (marked or nodes), then the dimension of the family of image curves is less than the dimension of the family of maps, therefore is enumeratively irrelevant. Now if $A,B \in C_2$, $\gamma_{|C_2}$ is a rational nodal curve and satisfies the constraint $\d$ (but these conditions are marked). The contribution to $\# (\Y \cap \K_1)$ in this case is $\#(\n(r,d),\d).$ - $\deg \gamma_{|C_2} = 0.$ Arguing similarly, we have that the contribution to $\# (\Y \cap \K_1)$ is $\# (\n(r,d), \d_3)$ - $\gamma$ has positive degree $d_i$ component $C_i.$ There are three subcases : - $A,B \in C_1:$ In this case, $\gamma_{|C_1}$ is a rational nodal curve and $\gamma_{|C_2}$ is a rational curve. The contribution in this case is $$\sum_{d_1+d_2=0}^{\gam_1\gam_2 = \wt{\d}}\#(\n{\mathbbmss{R}}(r,d_1,d_2), \gam_1^{(p,q)}, \gam_2^{(s,t)},0).$$ - $A,B \in C_2$ : The contribution is $$\sum_{d_1+d_2 = d}^{\gam_1\gam_2 = \wt{\d}} \#(\n{\mathbbmss{R}}(r,d_1,d_2), \gam_1^{(s,t)}, \gam_2^{(p,q)},0).$$ - $A \in C_1, B\in C_2$ or vice versa. In this case the image of $\gamma$ is a curve having two components that intersect twice at distinguished points. The contribution is therefore $$2\sum_{d_1+d_2 =d}^{\gam_1\gam_2 = \wt{\d}} \#({\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\gam_1^{(p,q)}, \gam_2^{(s,t)},k,0 ).$$ We can analyze $\Y \cap \K_2$ in the same way and after rearranging the terms, we derive the equation in the statement of the theorem. It is now possible to use the results so far to compute the characteristic number of rational nodal curves. Let $\d$ be a constraint such that $\d(0) >0$. Let $\d(r+1) = k$ Let $\d''$ be the constraint obtained from $\d$ by removing a tangency hyperplane. Let $\d'$ be the constraint obtained from $\d''$ by adding an incident codimension $2$ subspace. Then we have the following equality, provided that the left hand side is finite. $$\begin{aligned} \#(\n(r,d), \d) &=& \frac{d-1}{d} \# (\n(r,d), \d') \\ &+& \sum_{d_1+d_2 = d} \big( \# (\n{\mathbbmss{R}}(r,d_1,d_2),\d'') + \#( {\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\d'',k,0) \big).\end{aligned}$$ [**Warning :**]{} if $\d(0) \neq 0$ then those summands above involving reducible curves contain (twice) the case where the node is mapped to a tangency hyperplane. Also, in computing those summands, one needs to consider all possible splitting of constraints over two components (see Proposition $3,3$ and Corollary $3,4$). [ [*Proof.* ]{}]{}We have the following equality of divisors on $\mbar_{0,\{A.B\}}(r,d)$ $$\T = \frac{d-1}{d}\H + \sum_{d>0}^{j \leq d/2}\frac{j(d-j)}{d}(j,d-j).$$ For a proof of this see [@idq], Lemma $2.3.1$. Thus $$\begin{aligned} \# (\n(r,d), \d) &=& \# \left ((\n(r,d),\d''),\T \right ) \\ &=& \frac{d-1}{d} \# \left ( (\n(r,d),\d''),\H \right) + \sum_{j>0}^{j \leq d/2} \# \left( \n(r,d) \cap (j,d-j), \d'' \right).\end{aligned}$$ Now we will analyze $\#( \n(r,d) \cap (j,d-j), \d'')$. A general map $\gamma \in \n(r,d) \cap (j,d-j)$ has two-component source curve. There are two cases: - $A,B$ belong to a same component. The contribution is $\#(\n{\mathbbmss{R}}(j,d-j),\d'') + \#(\n{\mathbbmss{R}}(d-j,j),\d'')$ if $j < d-j$ depending on whether $A,B$ are in the component of lower or higher degree. If $j = d-j,$ the contribution is just $\#(\n{\mathbbmss{R}}(j,d-j),\d'').$ - $A,B$ belong to different components. The contribution is $2\#( {\mathbbmss{R}}{\mathbbmss{R}}_2(j,d-j),\d'',k,0)$ if $j < d-j$ and is $\#( {\mathbbmss{R}}{\mathbbmss{R}}_2(j,d-j), \d'',k,0)$ if $j =d-j.$ Sum up all possibilities, we derive the formula in Theorem $6.2$.\ \ Calculation of $\#({\mathbbmss{R}}{\mathbbmss{R}}_2(r,d_1,d_2),\d'',k,0)$ should make use of Corollary $3.4.$ One point worth mentioning when counting rational nodal curves with tangency conditions and with condition on the node is that maps with degree $2$ do contribute enumeratively. Rational nodal curves with degree two are rational degree two covers of $\mathbb P^1$ with a marked point specified as the node. For these maps, having a hyperplane passing through the branched points count as tangency. From characteristic number of rational nodal curves, it is easy to get characteristic number of rational nodal curves. Let $m = \d(0)$, and $\d_i$ be the constraint received by removing $i$ tangency conditions and replace them by a codimension $i$ on the node. Then we have the number of elliptic curves with fixed $j-$ invariant, with $j$ generic, of degree $d$ in $\proj$ satisfying constraint $\d$ denoted $\#(\J(r,d),\d)$, is : $$\#(\J(r,d),\d) = \sum_{i=0}^m 2^i \binom{n}{i}\#(\n(r,d),\d_i).$$ Now we give several numerical examples. We recover all previously known numbers in literature. The characteristic numbers of plane nodal cubics were computed in [@luf]. The charactersitic numbers of elliptic plane curves with fixed $j-$ invariant were computed in [@char]. Charactersitic numbers of rational plane cubics in $\mathbb{P}^3$ were computed in [@hmx]. Let $N,N_l,N_p$ be the family of rational nodal curves, rational nodal curves with the node on a fixed line, rational nodal curves with the node on a fixed point. Similarly, we denote $N_{s},N_{b}, N_{f}$ for the same family with the node on a fixed plane, a fixed $3-$space, or a fixed $4-$space. The following tables list the characteristic numbers of such families and of elliptic curves with fixed $j-$ invariant (denoted by $\J$). Below are tables of characteristic numbers of such families of low degree ($2,3,4,5$). In some tables, we put some point conditions so that the numbers are small enouch to fit in the table. The only other conditions are tangency, and top incident condition. For example, in the table for quartics in $\mathbb{P}^4$, the curves must pass through $2$ points, the other conditions are combination of tangency and incident to planes. [ | c | l | l | l | l |]{} $\#$ tang & $N$ & $N_l$ & $N_p$ & $\J$\ $0$& $0$ & $0$ & $0$ & $0$\ $1$ & $0$ & $0$ & $0$ & $0$\ $2$ & $0$ & $2$ & $1$ & $0$\ $3$ & $0$ & $3$ & $3/2$ & $12$\ $4$ & $0$ & $3/2$ & & $48$\ $5$ & $0$ & & & $75$\ $$\text{Table 7. Plane conics.}$$ [| c | l | l| l| l |]{} $\#$ tang & $N$ & $N_l$ & $N_p$ & $\J$\ $0$ & $12$ & $6 $ & $1$ & $12$\ $1$ & $36$ & $22$ & $4$ & $48$\ $2$ & $100$ & $80$ & $16$ & $192$\ $3$ & $240$ & $240$ & $52$ & $768$\ $4$ & $480$ & $604$ & $142$ & $2784$\ $5$ & $712$ & $1046$ & $256$ & $8832$\ $6$ & $756$ & $1212$ & $304$ & $21828$\ $7$ & $600$ & $1000$ & & $39072$\ $8$ & $400$ & & & $50448$\ $$\text{Table 8. Plane cubics.}$$ [| c | l | l| l| l |]{} $\#$ tang & $N$ & $N_l$ & $N_p$ & $\J$\ $ 0 $ & $ 1860 $ & $ 768 $ & $ 96 $ & $ 1860 $\ $ 1 $ & $ 6552 $ & $ 2952 $ & $ 384 $ & $ 8088 $\ $ 2 $ & $ 21600 $ & $ 10712 $ & $ 1448 $ & $ 33792 $\ $ 3 $ & $ 65328 $ & $ 35616 $ & $ 4992 $ & $ 134208 $\ $ 4 $ & $ 178272 $ & $ 106752 $ & $ 15516 $ & $ 497952 $\ $ 5 $ & $ 429120 $ & $ 281348 $ & $ 42416 $ & $ 1696320 $\ $ 6 $ & $ 886632 $ & $ 633972 $ & $ 99024 $ & $ 5193768 $\ $ 7 $ & $ 1515960 $ & $ 1166352 $ & $ 187248 $ & $ 13954512 $\ $ 8 $ & $ 2097648 $ & $ 1705856 $ & $ 279152 $ & $ 31849968 $\ $ 9 $ & $ 2350752 $ & $ 1986672 $ & $ 329496 $ & $ 60019872 $\ $ 10 $ & $ 2184480 $ & $ 1893528 $ & & $ 92165280 $\ $ 11 $ & $ 1745712 $ & & &$ 115892448 $\ $$\text{Table 9. Plane quartics.}$$ [| c | l | l| l| l | l |]{} $\#$ tang & $N$ & $N_s$ & $N_l$ & $N_p$ & $\J$\ $ 0 $ & $ 0 $ & $ 0 $ & $ 0 $ & $ 0$ & $ 0 $\ $ 1 $ & $ 0 $ & $ 0 $ & $ 0 $ & $ 0 $ & $ 0 $\ $ 2 $ & $ 0 $ & $ 16 $ & $ 8 $ & $ 2 $ & $ 0 $\ $ 3 $ & $ 0 $ & $ 24 $ & $ 12 $ & $ 3 $ & $ 96 $\ $ 4 $ & $ 0 $ & $ 20 $ & $ 10 $ & $ 7/2 $ & $ 384 $\ $ 5 $ & $ 0 $ & $ 10 $ & $ 5 $ & & $ 840 $\ $ 6 $ & $ 0 $ & $ 5 $ & & & $ 1200 $\ $ 7 $ & $ 0 $ & & & & $ 1470 $\ $$\text{Table 10. Conics in $\mathbb{P}^3$.}$$ [| c | l | l| l| l | l |]{} $\#$ tang & $N$ & $N_s$ & $N_l$ & $N_p$ & $\J$\ $ 0 $ & $ 12960 $ & $ 5040 $ & $ 904 $ & $ 72 $ & $ 12960 $\ $ 1 $ & $ 29520 $ & $ 13120 $ & $ 2512 $ & $ 216 $ & $ 39600 $\ $ 2 $ & $ 61120 $ & $ 32048 $ & $ 6568 $ & $ 612 $ & $ 117216 $\ $ 3 $ & $ 109632 $ & $ 64608 $ & $ 13904 $ & $ 1384 $ & $ 332640 $\ $ 4 $ & $ 167616 $ & $ 107072 $ & $ 23904 $ & $ 2524 $ & $ 849024 $\ $ 5 $ & $ 214400 $ & $ 144960 $ & $ 33304 $ & $ 3732 $ & $ 1890240 $\ $ 6 $ & $ 230240 $ & $ 162760 $ & $ 38432 $ & $ 4656 $ & $ 3625440 $\ $ 7 $ & $ 211200 $ & $ 155288 $ & $ 37808 $ & $ 5112 $ & $ 5994096 $\ $ 8 $ & $ 170192 $ & $ 130048 $ & $ 32864 $ & $ 5424 $ & $ 8631120 $\ $ 9 $ & $ 124176 $ & $ 98352 $ & $ 25664 $ & & $ 11038224 $\ $ 10 $ & $ 85440 $ & $ 70880 $ & & & $ 12875520 $\ $ 11 $ & $ 56960 $ & & & & $ 14422080 $\ $$\text{Table 11. Cubics in $\mathbb{P}^3$.}$$ [| c | l | l| l| l | l |]{} $\#$ tang & $N$ & $N_s$ & $N_l$ & $N_p$ & $\J$\ $ 0 $ & $ 247191840 $ & $ 61582704 $ & $ 7487280 $ & $ 402216 $ & $ 247191840 $\ $ 1 $ & $ 519424512 $ & $ 138566640 $ & $ 17469840 $ & $ 975192 $ & $ 642589920 $\ $ 2 $ & $ 1034619648 $ & $ 295896480 $ & $ 38636160 $ & $ 2242512 $ & $ 1618835328 $\ $ 3 $ & $ 1932171072 $ & $ 588656160 $ & $ 79348512 $ & $ 4785408 $ & $ 3920405760 $\ $ 4 $ & $ 3353134848 $ & $ 1079389056 $ & $ 149728320 $ & $ 9378160 $ & $ 9020858112 $\ $ 5 $ & $ 5361957120 $ & $ 1808973504 $ & $ 257515200 $ & $ 16752296 $ & $ 19509189120 $\ $ 6 $ & $ 7841572992 $ & $ 2752793920 $ & $ 401264800 $ & $ 27140752 $ & $ 39298619520 $\ $ 7 $ & $ 10431095808 $ & $ 3788712880 $ & $ 564734880 $ & $ 39830752 $ & $ 73227372288 $\ $ 8 $ & $ 12599060192 $ & $ 4716456320 $ & $ 718744512 $ & $ 53161088 $ & $ 125665152480 $\ $ 9 $ & $ 13851211968 $ & $ 5333385216 $ & $ 831757440 $ & $ 65099040 $ & $ 198307833792 $\ $ 10 $ & $ 13948252800 $ & $ 5522229504 $ & $ 883153920 $ & $ 74131776 $ & $ 288227491200 $\ $ 11 $ & $ 12986719872 $ & $ 5292561600 $ & $ 870495360 $ & $ 79929312 $ & $ 387635041920 $\ $ 12 $ & $ 11309818368 $ & $ 4757882880 $ & $ 807883200 $ & $ 84550992 $ & $ 486058242048 $\ $ 13 $ & $ 9330496512 $ & $ 4070594880 $ & $ 715629312 $ & & $ 574243507200 $\ $ 14 $ & $ 7394421888 $ & $ 3381893376 $ & & & $ 648194719872 $\ $ 15 $ & $ 5703866880 $ & & & & $ 715490590080 $\ $$\text{Table 12. Quartics in $\mathbb{P}^3$.}$$ $\#$ tang $N$ $N_s$ $N_l$ $N_p$ $\J$ ----------- ------------------- ------------------ ----------------- ---------------- -------------------- $ 0 $ $ 2987074368 $ $ 597069288 $ $ 59293632 $ $ 2757288 $ $ 2987074368 $ $ 1 $ $ 6654861504 $ $ 1393675584 $ $ 142403568 $ $ 6890568 $ $ 7849000080 $ $ 2 $ $ 14302171008 $ $ 3141287760 $ $ 330349200 $ $ 16691344 $ $ 20114047872 $ $ 3 $ $ 29534616768 $ $ 6800411520 $ $ 736077600 $ $ 38978688 $ $ 50113244448 $ $ 4 $ $ 58394890752 $ $ 14081928256 $ $ 1569037056 $ $ 87466348 $ $ 120947061888 $ $ 5 $ $ 110164217088 $ $ 27795971008 $ $ 3189343752 $ $ 188200508 $ $ 281761911168 $ $ 6 $ $ 197654921184 $ $ 52144209544 $ $ 6165495488 $ $ 387843208 $ $ 631585386720 $ $ 7 $ $ 336286484448 $ $ 92755042440 $ $ 11312688400 $ $ 765476504 $ $ 1358700870672 $ $ 8 $ $ 541376364848 $ $ 156271230640 $ $ 19684719200 $ $ 1449944208 $ $ 2800306366128 $ $ 9 $ $ 823917940992 $ $ 249556959696 $ $ 32520764016 $ $ 2653490208 $ $ 5526457857888 $ $ 10 $ $ 1186459103808 $ $ 379132252128 $ $ 51221741472 $ $ 4769939328 $ $ 10455705197568 $ $ 11 $ $ 1621483284864 $ $ 552185368704 $ $ 77488852608 $ $ 19030887269760 $ $ 12 $ $ 2114474172288 $ $ 783085854720 $ $ 33559605535872 $ $ 13 $ $ 2648546358528 $ $ 58098921777408 $ $$\text{ Table 13. Quintics in $\mathbb{P}^3$, passing through $3$ points.}$$ [| c | l | l | l | l| l | l |]{} $\#$ tang & $N$ & $N_b$ & $N_s$ & $N_l$ & $N_p$ & $\J$\ $ 0 $ & $ 7833840 $ & $ 2565720 $ & $ 468935 $ & $ 52140 $ & $ 2865 $ & $ 7833840 $\ $ 1 $ & $ 14708400 $ & $ 5294270 $ & $ 1017980 $ & $ 119400 $ & $ 6984 $ & $ 19839840 $\ $ 2 $ & $ 25085900 $ & $ 10073080 $ & $ 2038520 $ & $ 252192 $ & $ 15720 $ & $ 48138720 $\ $ 3 $ & $ 37705920 $ & $ 16296840 $ & $ 3416336 $ & $ 440272 $ & $ 28924 $ & $ 110777280 $\ $ 4 $ & $ 49732080 $ & $ 22491008 $ & $ 4833312 $ & $ 644504 $ & $ 44470 $ & $ 232897920 $\ $ 5 $ & $ 57643520 $ & $ 26854560 $ & $ 5889580 $ & $ 812540 $ & $ 59250 $ & $ 439941120 $\ $ 6 $ & $ 59232320 $ & $ 28240140 $ & $ 6319450 $ & $ 906690 $ & $ 70854 $ & $ 745702080 $\ $ 7 $ & $ 54660200 $ & $ 26636130 $ & $ 6095150 $ & $ 916962 $ & $ 78360 $ & $ 1141405440 $\ $ 8 $ & $ 45993500 $ & $ 22938610 $ & $ 5383586 $ & $ 858012 $ & $ 82584 $ & $ 1593774300 $\ $ 9 $ & $ 35861700 $ & $ 18337518 $ & $ 4423952 $ & $ 755184 $ & $ 85440 $ & $ 2055201960 $\ $ 10 $ & $ 26323500 $ & $ 13808900 $ & $ 3420200 $ & $ 626640 $ & $ 87360 $ & $ 2480472300 $\ $ 11 $ & $ 18497240 $ & $ 9949360 $ & $ 2513120 $ & $ 480480 $ & & $ 2841879120 $\ $ 12 $ & $ 12649200 $ & $ 6978480 $ & $ 1786880 $ & & & $ 3137555760 $\ $ 13 $ & $ 8510880 $ & $ 4808480 $ & & & & $ 3385230720 $\ $ 14 $ & $ 5673920 $ & & & & & $ 3589051200 $\ $$\text{Table 14. Cubics in $\mathbb{P}^4$.}$$ [| c | l | l | l | l| l | l |]{} $\#$ tang & $N$ & $N_b$ & $N_s$ & $N_l$ & $N_p$ & $\J$\ $ 0 $ & $ 264271032 $ & $ 61079694 $ & $ 8388348 $ & $ 749421 $ & $ 34860 $ & $ 264271032 $\ $ 1 $ & $ 493716948 $ & $ 120918936 $ & $ 17290038 $ & $ 1630488 $ & $ 81252 $ & $ 615876336 $\ $ 2 $ & $ 878434848 $ & $ 228232116 $ & $ 33980664 $ & $ 3390452 $ & $ 181836 $ & $ 1395663984 $\ $ 3 $ & $ 1479817080 $ & $ 405964896 $ & $ 62797160 $ & $ 6629800 $ & $ 383672 $ & $ 3062685600 $\ $ 4 $ & $ 2353692768 $ & $ 678089744 $ & $ 108738088 $ & $ 12151512 $ & $ 761888 $ & $ 6469681248 $\ $ 5 $ & $ 3530480992 $ & $ 1063566824 $ & $ 176508768 $ & $ 20905076 $ & $ 1429930 $ & $ 13101001152 $\ $ 6 $ & $ 4995675728 $ & $ 1569827616 $ & $ 269290448 $ & $ 33879818 $ & $ 2556172 $ & $ 25387171536 $\ $ 7 $ & $ 6680908448 $ & $ 2189197336 $ & $ 387775734 $ & $ 51989792 $ & $ 4399696 $ & $ 47102511264 $\ $ 8 $ & $ 8472417440 $ & $ 2900923506 $ & $ 529920660 $ & $ 75922720 $ & $ 7378752 $ & $ 83878893600 $\ $ 9 $ & $ 10234272948 $ & $ 3679075344 $ & $ 691414728 $ & $ 105627552 $ & $ 12126048 $ & $ 143940578328 $\ $ 10 $ & $ 11836475952 $ & $ 4504817304 $ & $ 867212688 $ & $ 138946656 $ & & $ 239302639872 $\ $ 11 $ & $ 13167563808 $ & $ 5374257696 $ & $ 1054871808 $ & & & $ 387833169936 $\ $ 12 $ & $ 14112721248 $ & $ 6278297856 $ & & & & $ 616383262944 $\ $ 13 $ & $ 14531107200 $ & & & & & $ 963518793600 $\ $$\text{Table 15. Quartics in $\mathbb{P}^4$ passing through $2$ points.}$$ $\#$ tang $N$ $N_b$ $N_s$ $N_l$ $N_p$ $\J$ ----------- ------------------- ------------------ ----------------- ---------------- --------------- -------------------- $ 0 $ $ 5264130996 $ $ 960390870 $ $ 105886953 $ $ 7801695 $ $ 311311 $ $ 5264130996 $ $ 1 $ $ 10335707556 $ $ 1973618742 $ $ 224710598 $ $ 17371678 $ $ 742316 $ $ 12256489296 $ $ 2 $ $ 19791788388 $ $ 3960252460 $ $ 465840460 $ $ 37911496 $ $ 1746624 $ $ 28109811168 $ $ 3 $ $ 36896035320 $ $ 7737537944 $ $ 940326944 $ $ 80796848 $ $ 4041128 $ $ 63416490816 $ $ 4 $ $ 66880583024 $ $ 14699954352 $ $ 1845469104 $ $ 167905648 $ $ 9189708 $ $ 140521932288 $ $ 5 $ $ 117792292576 $ $ 27145486560 $ $ 3519654728 $ $ 340028520 $ $ 20558296 $ $ 305497218816 $ $ 6 $ $ 201506364736 $ $ 48745168872 $ $ 6523861268 $ $ 670681448 $ $ 45308086 $ $ 651327035136 $ $ 7 $ $ 334871977648 $ $ 85223104580 $ $ 11759484440 $ $ 1287078386 $ $ 98524384 $ $ 1362231952128 $ $ 8 $ $ 540951986840 $ $ 145379939744 $ $ 20637848154 $ $ 2397410108 $ $ 211715288 $ $ 2797819372056 $ $ 9 $ $ 850242885024 $ $ 242702404542 $ $ 35332114224 $ $ 4312424928 $ $ 5652591017568 $ $ 10 $ $ 1301286873156 $ $ 397849014300 $ $ 59181220928 $ $ 11257978051236 $ $ 11 $ $ 1938666465816 $ $ 641728301752 $ $ 22149199999776 $ $ 12 $ $ 2804649121008 $ $ 43096623642288 $ $$\text{Table 16. Quintics in $\mathbb{P}^4$ passing through $4$ points.}$$ $\#$ tang $N$ $N_f$ $N_b$ $N_s$ $N_l$ $N_p$ $\J$ ----------- ----------------- ---------------- ---------------- --------------- -------------- ------------- ------------------ $ 0 $ $ 3580435656 $ $ 1034759292 $ $ 189136374 $ $ 24039939 $ $ 2009982 $ $ 85745 $ $ 3580435656 $ $ 1 $ $ 5820250128 $ $ 1803057816 $ $ 343203840 $ $ 45424176 $ $ 3974516 $ $ 178640 $ $ 7889768712 $ $ 2 $ $ 8641680264 $ $ 2888520852 $ $ 572163144 $ $ 78755588 $ $ 7205344 $ $ 341240 $ $ 16610457024 $ $ 3 $ $ 11507535984 $ $ 4048138080 $ $ 824350976 $ $ 116897472 $ $ 11089152 $ $ 549128 $ $ 33149426688 $ $ 4 $ $ 13759570272 $ $ 4992894416 $ $ 1036797728 $ $ 150683904 $ $ 14773856 $ $ 764324 $ $ 61362323712 $ $ 5 $ $ 14867247680 $ $ 5502189760 $ $ 1161050240 $ $ 172833416 $ $ 17554792 $ $ 954832 $ $ 104391383040 $ $ 6 $ $ 14650427520 $ $ 5502894720 $ $ 1179603568 $ $ 180279708 $ $ 19079772 $ $ 1102606 $ $ 163351745280 $ $ 7 $ $ 13303631040 $ $ 5066847184 $ $ 1104900496 $ $ 174051444 $ $ 19343536 $ $ 1204100 $ $ 236503108800 $ $ 8 $ $ 11252393152 $ $ 4350397184 $ $ 967029476 $ $ 157723006 $ $ 18576208 $ $ 1267280 $ $ 319397674176 $ $ 9 $ $ 8959119120 $ $ 3522421644 $ $ 799569876 $ $ 135605388 $ $ 17095224 $ $ 1305896 $ $ 405992118672 $ $ 10 $ $ 6782773704 $ $ 2715749316 $ $ 629998440 $ $ 111418656 $ $ 15173120 $ $ 1331840 $ $ 490193697672 $ $ 11 $ $ 4929887760 $ $ 2011043040 $ $ 476256768 $ $ 87775688 $ $ 12973792 $ $ 1349216 $ $ 567210910536 $ $ 12 $ $ 3472645440 $ $ 1442366496 $ $ 347592224 $ $ 66354624 $ $ 10586880 $ $ 1360832 $ $ 634363027200 $ $ 13 $ $ 2392303152 $ $ 1010425424 $ $ 246674816 $ $ 48224736 $ $ 8073728 $ $ 691172850672 $ $ 14 $ $ 1624181888 $ $ 696607744 $ $ 171675392 $ $ 34118336 $ $ 738716078016 $ $ 15 $ $ 1092498624 $ $ 474968256 $ $ 117859840 $ $ 778457098944 $ $ 16 $ $ 730705920 $ $ 321392512 $ $ 811258656768 $ $ 17 $ $ 487137280 $ $ 838048055040 $ $$\text {Table 17. Cubics in $\mathbb{P}^5$}$$ $\#$ tang $N$ $N_f$ $N_b$ $N_s$ $N_l$ $N_p$ $\J$ ----------- ---------------- --------------- --------------- -------------- ------------- ------------ ----------------- $ 0 $ $ 17793468 $ $ 4315338 $ $ 675729 $ $ 82815 $ $ 7629 $ $ 408 $ $ 17793468 $ $ 1 $ $ 33892524 $ $ 8728578 $ $ 1428506 $ $ 187086 $ $ 18804 $ $ 1122 $ $ 42523200 $ $ 2 $ $ 61915284 $ $ 16962956 $ $ 2898296 $ $ 406116 $ $ 44736 $ $ 3012 $ $ 99532512 $ $ 3 $ $ 108109320 $ $ 31398264 $ $ 5580216 $ $ 834384 $ $ 100788 $ $ 7728 $ $ 227691648 $ $ 4 $ $ 180450912 $ $ 55359984 $ $ 10188624 $ $ 1618620 $ $ 214248 $ $ 18948 $ $ 507304944 $ $ 5 $ $ 288477120 $ $ 93327232 $ $ 17697268 $ $ 2968056 $ $ 429304 $ $ 44638 $ $ 1099292256 $ $ 6 $ $ 442955328 $ $ 151262244 $ $ 29385528 $ $ 5155156 $ $ 807974 $ $ 101692 $ $ 2318653056 $ $ 7 $ $ 655304328 $ $ 237174048 $ $ 46930448 $ $ 8512992 $ $ 1413096 $ $ 4771225200 $ $ 8 $ $ 936129552 $ $ 361876128 $ $ 72589134 $ $ 13497600 $ $ 9605588880 $ $ 9 $ $ 1291589856 $ $ 539604810 $ $ 109323720 $ $ 18969484704 $ $ 10 $ $ 1716845652 $ $ 788940756 $ $ 36822211764 $ $ 11 $ $ 2184938712 $ $ 70374247152 $ $$\text{Table 18. Quartics in $\mathbb{P}^5$ passing through $3$ points.}$$ [\[1\]]{} P. Aluffi [*The enumerative geometry of plane cubics II: nodal and cuspidal cubics,*]{} Math. Annalen 289 (1991), 543-572. W.Fulton [*Intersection Theory,*]{} Second Edition, Springer 1996. W.Fulton and R. Pandharipande, [*Notes on stable maps and quantum cohomology*]{}, preprint 1996, alg-geom/9608011. X. Hernandez, J. M. Miret, [*The characteristic numbers of cuspidal plane cubics in $\mathbb P^3$*]{} , Bull. Belg. Math. Soc. Simon Stevin, [**10**]{} (2003) No. 1, 115–124. X. Hernandez, J. M. Miret and S. Xambo-Descamps, [*Computing the characteristic numbers of the variety of nodal plane cubics in $\mathbb P^3$*]{} , J. Symb. Comp. [**42**]{} (2007) 192–202. E. Ionel, [*Genus-one enumerative invariants in $\mathbb{P}^n$ with fixed j-invariant*]{}, Duke Math. J. [**94 (2)**]{} (1998) 279–324. E. Getzler, [*Intersection theory on $\mbar_{1,4}$ and elliptic Gromov-Witten invariants*]{}, J. Amer. Math. Soc. [**10**]{} No. 4 (1997) 973–998. [www.stanford.edu/ dhnguyen/Code/ell.](https://www.stanford.edu/~dhnguyen/Code/ell) D. Nguyen, [*Doctoral thesis at Stanford University*]{}, in preparation. R. Pandharipande, [*Intersection of $\Q$-divisors on Kontsevich’s moduli space $\mbar_{0,n}(\mathbb{P}^r,d)$ and enumerative geometry*]{}, Trans. Amer. Math. Soc, [**351**]{} (1999), 1481-1505. R. Pandharipande, [*A note on elliptic plane curves with fixed $j$-invariant*]{}, Proc. Amer. Math. Soc., [**125**]{}, No. 12, 3471–3479. R. Vakil, [*The enumerative geometry of rational and elliptic plane curves in projective space*]{}, J. Reine Angew. Math. (Crelle’s Journal), [**529**]{} (2000), 101–153. R. Vakil, [*Recursions for characteristic numbers of genus one plane curves*]{}, Arkiv for Matematik, [**39**]{} (2001), no. 1, 157–180. R. Vakil, A. Zinger, [*A desingularization of the main component of the moduli space of genus-one stable maps to projective space*]{}, Geom. Topol. [**12**]{} (2008), no. 1, 1-95. A. Zinger, [*Enumeration of one-nodal rational curves in projective spaces* ]{}, Topology [**43**]{} (2004) 793–829.

--- abstract: 'This paper presents a secure and private implementation of linear time-invariant dynamic controllers using Paillier’s encryption, a semi-homomorphic encryption method. To avoid overflow or underflow within the encryption domain, the state of the controller is reset periodically. A control design approach is presented to ensure stability and optimize performance of the closed-loop system with encrypted controller.' author: - 'Carlos Murguia[^1], Farhad Farokhi, [^2]and Iman Shames' bibliography: - 'ref.bib' title: 'Secure and Private Implementation of Dynamic Controllers Using Semi-Homomorphic Encryption' --- Introduction ============ Internet of Things (IoT) has brought opportunities for flexibility of deployment and efficiency improvements. However, it threatens security and privacy of individuals and businesses as IoT devices, by design, share their information for processing over the cloud. This information can be secured from adversaries over the network by using encrypted communication channels [@Patel2009ICS15387881538820]. This approach, although effective and necessary, does not address vulnerability of the data on servers running cloud-computing services. These services themselves can use the data for targeted advertisement or can be hacked for malicious purposes. Therefore, there is a need for a more secure methodology that addresses the security and privacy of data while being processed. Thankfully secure cloud computing is possible with the use of homomorphic encryption methods – encryption methods that allow computation over plain data by performing appropriate computations on the encrypted data [@Gentry2009; @Paillier1999; @1057074]. The use of homomorphic encryption allows a controller to be remotely realised without needing to openly sharing private and sensitive data (and consenting to its use in an unencrypted manner). This paper specifically discusses secure and private implementation of linear time-invariant dynamic controllers with the aid of the Paillier’s encryption [@Paillier1999], a semi-homomorphic encryption method. The use of homomorphic encryption for secure control has been studied previously [@kogiso2015cyber; @farokhi2017secure; @farokhi2016secure; @darup2018towards; @kim2016encrypting; @darup2018encrypted; @lin2018secure; @alexandru2018cloud]. However, all these studies consider static controllers. This is because, when dealing with dynamical control laws (with an encrypted memory/state that must be maintained remotely), the number of bits required for representing the state of the controller grows linearly with the number of iterations. This renders the memory useless after a few iterations due to an overflow or an underflow (i.e., number of fractional bits required for representing a number becomes larger than the number of fractional bits in the fixed-point number basis). In fact, using rough calculations, it can be seen that for a system with sampling time of 10 milliseconds, 16 bits quantized controller parameters and measurements, and within an encryption space of [[2048]{}]{} bits[^3], the state of the controller becomes incorrect after roughly [[1.2]{}]{} seconds due to an overflow or underflow. [[The unstabilizing effect of restricting the memory of controllers to finite rings is illustrated in Section \[sec:example\] for the key length of 2048 bits using a controller that can easily stabilize a batch chemical reactor in the absence of encryption.]{}]{} There are multiple ways to deal with this issue: 1. We should decrypt the state of the encrypted controller, project it into the desired set of fixed-point rational numbers, and encrypt it again. To avoid this issue, the encrypted state can be sent to a trust third-party (e.g., an IoT device) to be decrypted, rounded, encrypted, and transmitted back. This adds unnecessary communication overhead and overburdens the computational units of the IoT device. Furthermore, by decrypting the state, the risk of a privacy or security breach increases. 2. We should restrict the controller parameters so that the state of the dynamic controller remains within the set of fixed-point rational numbers. This approach was pursued in [@8619600]. This makes the problem of designing the controller into a mixed-integer optimization problem, which can be computationally exhaustive. However, a robust control approach can be taken to ensure that converting non-integer controllers to integer ones does not ruin stability [@8619600]. 3. We should reset the controller, i.e., the state of the controller is set to a publicly known number (e.g., zero) periodically. In this case, the controller must be redesigned to ensure stability/performance, which this paper shows to remain a tractable optimisation problem. This is the approach chosen by the current paper. [[In this paper, we only focus on encrypting the outputs of the system and the state of the controller. This is because the parameters of the controller are often not sensitive in practice. For instance, in autonomous vehicles, the location and velocity are sensitive as they reveal private information about the user, e.g., home/work address and travel habits, while the controller parameters are implicitly related to the dynamics of the vehicle.]{}]{} Resetting controllers have been previously studied in [@1100479; @bakkeheim2008lyapunov; @clegg1958nonlinear; @krishnan1974synthesis; @beker2004fundamental; @prieur2018analysis; @guo2012stability]. However, the synthesis approach in this paper is more general than those studies and further it is designed to accommodate challenges associated with the implementation of dynamical controllers over the cipher space. [[Particularly, majority of existing work on reset controllers focus on state dependent triggers. Due to the nature of our problem, where the controller cannot access to the unencrypted state, those results are not applicable. Along the same lines, since we always have to reset the controller to the same state regardless of the state of the plant, the existing results for switched systems seem to be not applicable. ]{}]{} The rest of the paper is organized as follows. Preliminary materials on homomorphic encryption are presented in Section \[sec:prelim\]. The design and implementation of the controller is discussed in Section \[sec:implementation\]. Finally, numerical results are presented in Section \[sec:example\] and the paper is concluded in Section \[sec:conclusions\]. All proofs are presented in the appendices to improve the overall presentation of the paper. Preliminary Material {#sec:prelim} ==================== In this paper, a tuple $(\mathbb{P},\mathbb{C},\mathbb{K},\mathfrak{E},\mathfrak{D})$ denotes a public key encryption scheme, where $\mathbb{P}$ is the set of plaintexts, $\mathbb{C}$ is the set of ciphertexts, $\mathbb{K}$ is the set of keys, $\mathfrak{E}$ is the encryption algorithm, and $\mathfrak{D}$ is the decryption algorithm. Each $\kappa=(\kappa_p,\kappa_s)\in\mathbb{K}$ is composed of a public key $\kappa_p$ (which is shared with and used by everyone for encrypting plaintexts) and a private key $\kappa_s$ (which is maintained only by the trusted parties for decryption). Algorithms $\mathfrak{E}$ and $\mathfrak{D}$ are publicly known while the keys, which set the parameters of these algorithms, are generated and used in each case. The use of the term “algorithm”, instead of mapping or function, is due to the presence of random[^4] elements in the encryption procedure possibly resulting in one plaintext being mapped to multiple ciphertexts. A necessary requirement for the encryption scheme is to be invertible, i.e., $\mathfrak{D}(\mathfrak{E}(x,\kappa_p),\kappa_p,\kappa_s)=x$ for all $x\in\mathbb{P}$ given $\kappa=(\kappa_p,\kappa_s)\in\mathbb{K}$. \[def:homomorphic\] Assume there exist operators $\circ$ and $\diamond$ such that $(\mathbb{P},\circ)$ and $(\mathbb{C},\diamond)$ form groups. A public key encryption $(\mathbb{P},\mathbb{C},\mathbb{K},\mathfrak{E},\mathfrak{D})$ is called called homomorphic if ${{\color{black}\mathfrak{D}(}}\mathfrak{E}(x_1,\kappa_p)\diamond\mathfrak{E}(x_1,\kappa_p){{\color{black},\kappa_p,\kappa_s)}}=x_1\circ x_2$ for all $x_1,x_2\in\mathbb{P}$ and $\kappa\in\mathbb{K}$. [[Throughout this paper, $|\mathbb{A}|$ denotes the cardinality of any set $\mathbb{A}$. Further, we define the notation $\mathbb{Z}_q:=\{0,\dots,q-1\}=\{n\operatorname{mod}q:\forall n\in\mathbb{Z}\}$]{}]{} for all positive integers $q\in\mathbb{N}$. [[In this paper, we assume that $\mathbb{P}=\mathbb{Z}_{n_p}$ and $\mathbb{C}=\mathbb{Z}_{n_c}$]{}]{} with $n_p=|\mathbb{P}|$ and $n_c=|\mathbb{C}|$. A public key encryption $(\mathbb{P},\mathbb{C},\mathbb{K},\mathfrak{E},\mathfrak{D})$ is additively homomorphic if there exists an operator $\diamond$ such that Definition \[def:homomorphic\] is satisfied when the operator $\circ$ is defined as $x_1\circ x_2:=(x_1+x_2)\,\operatorname{mod}\,n_p$ for all $x_1,x_2\in\mathbb{P}$. For additively homomorphic schemes, in this paper, the notation $\oplus$ is used to denote the equivalent operator in the ciphertext domain ($\diamond$ in the definition above). Similarly, a public key encryption is multiplicatively homomorphic if there exists an operator $\diamond$ such that Definition \[def:homomorphic\] is satisfied with $\circ$ defined as $x_1\circ x_2:=(x_1x_2)\,\operatorname{mod}\,n_p$ for all $x_1,x_2\in\mathbb{P}$. If a public key encryption is both additively and multiplicatively homomorphic, it is fully homomorphic but, if only one of these conditions is satisfied, it is semi-homomorphic. Homomorphism shows there exist operations over ciphertexts that can generate encrypted versions of sumed or multiplied plaintexts without the need of decrypting their corresponding cuphertexts. An example of additively homomorphic encryption scheme is the Paillier’s encryption method [@Paillier1999]. ElGamal is an example of multiplicatively homomorphic encryption schemes [@1057074]. Recently, several fully homomorphic encryption methods have been also developed, see, e.g., [@Gentry2009]. Now, we define semantic security[[, borrowed from [@katz2014introduction]]{}]{}. A key $\kappa=(\kappa_p,\kappa_s)\in\mathbb{K}$ is randomly generated. A probabilistic polynomial time-bounded adversary proposes $x_1,x_2\in\mathbb{P}$. The agent chooses [[$x$]{}]{} at random from $\{x_1,x_2\}$ with equal probability, encrypts $x$ according to $y=\mathfrak{E}(x,\kappa_p)$, and sends $y$ to the adversary (along with the public key $\kappa_p$). The adversary produces $x'$, which is an estimate of $x$ based on all the avialable information (everything except $\kappa_s$, i.e., $x_1$, $x_2$, $y$, $\mathfrak{E}$, $\mathfrak{D}$, $\kappa_p$). The adversary’s advantage (in comparison to that of a pure random number generator) is given by $\mathrm{Adv}(|\mathbb{K}|):=|\mathbb{P}\{x=x'\}-1/2|$. The public key encryption $(\mathbb{P},\mathbb{C},\mathbb{K},\mathfrak{E},\mathfrak{D})$ is semantically secure (alternatively known as indistinguishability under chosen plaintext attack) if $\mathrm{Adv}$ is negligible[^5]. In this paper, the results are presented for the Paillier’s encryption method. It is [[noteworthy]{}]{} that the Paillier’s encryption method is semantically secure under the *Decisional Composite Residuosity Assumption*, i.e., it is “hard” to decide whether there exists $y\in\mathbb{Z}_{N^2}$ such that $x= y^N \operatorname{mod}N$ for $N\in\mathbb{Z}$ and $x\in\mathbb{Z}_{N^2}$. More information regarding the assumption can be found in [@Paillier1999; @yi2014homomorphic]. This can be used to establish the security of the proposed framework. The Paillier’s encryption scheme is as follows. First the public and private keys are generated. To do so, large prime numbers $p$ and $q$ are selected randomly and independently of each other such that $\gcd(pq,(1-p)(1-q))=1$, where $\gcd(a,b)$ refers to the greatest common divisor of integers $a$ and $b$. The public key (which is shared with all the parties and is used for encryption) is $\kappa_p=pq$. The private key (which is only available to the entity that needs to decrypt the data) is $\kappa_s=(\lambda,\mu)$ with $\lambda=\operatorname{lcm}(p-1,q-1)$ and $\mu=\lambda^{-1}\operatorname{mod}\kappa_p$, where $\operatorname{lcm}(a,b)$ is the least common multiple of integers $a$ and $b$. The ciphertext of plain message $x\in\mathbb{P}=\mathbb{Z}_{\kappa_p}$ is $\mathfrak{E}(x,\kappa_p)=(\kappa_p+1)^xr^{\kappa_p}\operatorname{mod}\kappa_p^2,$ where $r$ is randomly selected with uniform probability from $\mathbb{Z}^*_{\kappa_p}:=\{x\in\mathbb{Z}_{\kappa_p}|\gcd(x,\kappa_p)=1\}$. Finally, to decrypt any ciphertext $c\in\mathbb{C}=\mathbb{Z}_{\kappa_p^2}$, $\mathfrak{D}(c,\kappa_p,\kappa_s)=(L(c^\lambda \operatorname{mod}\kappa_p^2)\mu)\operatorname{mod}\kappa_p,$ where $L(z)=(z-1)/\kappa_p$. *[@Paillier1999]* 1) For $r,r'\in\mathbb{Z}^*_{\kappa_p}$ and $t,t' \in\mathbb{P}$ such that $t+t'\in\mathbb{P}$, $\mathfrak{E}(t,\kappa_p)\mathfrak{E}(t',\kappa_p)\operatorname{mod}\kappa_p^2=\mathfrak{E}(t+t',\kappa_p)$; 2) For $r\in\mathbb{Z}^*_{\kappa_p}$ and $t,t' \in\mathbb{P}$ such that $tt'\in\mathbb{P}$, $\mathfrak{E}(t,\kappa_p)^{t'}\operatorname{mod}\kappa_p^2=\mathfrak{E}(t't,\kappa_p)$. \[prop:cipher\] Proposition \[prop:cipher\] shows that the Paillier’s encryption is a semi-homomorphic encryption scheme, i.e., algebraic manipulation of the plain data is possible without decryption using appropriate operations over the encrypted data. The Paillier’s encryption is additively homomorphic with operator $\oplus$ being defined as $x_1\oplus x_2=(x_1x_2)\operatorname{mod}\kappa_p^2$ for all $x_1,x_2\in\mathbb{C}$. Note that the Paillier’s method is not multiplicatively homomorphic as $t'$ in the identity $\mathfrak{E}(t,\kappa_p)^{t'}\operatorname{mod}\kappa_p^2=\mathfrak{E}(t't,\kappa_p)$ in Proposition \[prop:cipher\] is not encrypted. Define $\triangle$ such that $x_1\triangle x_2=x_1^{x_2}\operatorname{mod}N^2$ for all $x_1\in\mathbb{C}$ and $x_2\in\mathbb{P}$. Note that $\triangle$ is not an operator (in the mathematical sense) as its operands belong to two difference sets; it is just a mapping. Dynamic Controller Implementation {#sec:implementation} ================================= Consider the discrete-time linear time invariant system $$\begin{aligned} \label{eqn:system} \mathcal{P}: \left\{ \begin{array}{rlc} x[k+1]\hspace*{-.1in}&=Ax[k]+Bu[k],& x[0]=x_0,\\ y[k]\hspace*{-.1in}&=Cx[k], \end{array} \right.\end{aligned}$$ with $k \in \mathbb{N}$, state $x[k]\in\mathbb{R}^{n_x}$, control input $u[k]\in\mathbb{R}^{n_u}$, and output $y[k]\in\mathbb{R}^{n_y}$. [[Many linear time-invariant systems cannot be stabilized by static output feedback controllers [@SYRMOS1997125; @CAO19981641; @Bara]. Therefore, dynamic output feedback controllers have been used for decades to stabilize system using only output measurements, e.g., standard Kalman-filter (or Luenberger observer) based linear regulators [@Astrom], and general dynamic output feedback controllers for quadratic performance [@Scherer_IEEE].]{}]{} System is controlled by a dynamic output feedback controller of the form $$\begin{aligned} \label{eqn:controller} \hspace{-.04in}\mathcal{C}\hspace{-.04in}:\hspace{-.04in} \left\{\hspace*{-.07in} \begin{array}{rl} \hspace{-.03in}x_c[k+1]\hspace*{-.14in}&=\hspace{-.04in} \begin{cases} \hspace{-.04in}A_cx_c[k]+B_cy[k], & \hspace{-.1in}(k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in}T\hspace{-.04in}\neq\hspace{-.02in} 0, \\ \hspace{-.03in}0, & \hspace{-.1in}(k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in}T\hspace{-.04in}=\hspace{-.02in} 0, \end{cases} \\[1.5em] u[k]\hspace*{-.14in}&=\hspace{-.04in}C_cx_c[k]+D_cy[k], \end{array} \right.\end{aligned}$$ with controller state $x_c[k]\in\mathbb{R}^{n_c}$. It is assumed that the state of the controller resets every $T$ time steps, i.e., $x_c[\ell T]=0$ for all $\ell\in\mathbb{N}$. *This is because implementing encrypted controllers over an infinite horizon is impossible due to memory issues (by multiplication of fractional numbers, the number of bits required for representing fractional and integer parts grow)*. Combining the dynamics in  and  results in the augmented system: $$\begin{aligned} z[k+1]\hspace{-.04in}=\hspace{-.04in} \begin{cases} F(\mathcal{P},\mathcal{C})z[k], & (k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in}T\hspace{-.02in}\neq\hspace{-.02in} 0, \\ \begin{bmatrix} I & 0\\ 0 & 0 \end{bmatrix} F(\mathcal{P},\mathcal{C})z[k], & (k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in}T\hspace{-.02in}=\hspace{-.02in} 0, \end{cases}\end{aligned}$$ where $z[k]:=\begin{bmatrix} x[k]^\top & x_c[k]^\top \end{bmatrix}^\top$ and $$\begin{aligned} F(\mathcal{P},\mathcal{C}):= \begin{bmatrix} A+BD_cC & BC_c \\ B_cC & A_c \end{bmatrix}. \label{Synthesis5b}\end{aligned}$$ The following theorem provides a sufficient condition for the asymptotic stability of the origin of in feedback with the resetting controller . \[tho:1\] The closed-loop dynamics - is globally asymptotically stable if there exist [[$P\in\mathbb{R}^{(n_c+n_x)\times (n_c+n_x)}$]{}]{}, $\varepsilon\in(0,1)$, $\mu\in [-1,0)$, $\delta\in[1,\infty)$, and $\epsilon\in(0,\infty)$ satisfying: \[eqn:tho:1:cond\] $$\begin{aligned} &P\succ \epsilon I,\label{eqn:tho:1:cond1}\\ &F(\mathcal{P},\mathcal{C})^\top P F(\mathcal{P},\mathcal{C})\preceq (1+\mu) P,\label{eqn:tho:1:cond2}\\ & F(\mathcal{P},\mathcal{C})^\top \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} P\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} F(\mathcal{P},\mathcal{C}) \preceq \delta P,\label{eqn:tho:1:cond3}\\ &\delta(1+\mu)^{T-1}<\varepsilon.\label{eqn:tho:1:cond4}\end{aligned}$$ See Appendix \[proof:tho:1\]. The following result provides a sufficient condition for the stabilizability of the system using the resetting controller. \[prop:stars\] If $n_c\geq n_x$, $(A,B)$ is stabilizable, and $(A,C)$ is detectable, there exist $\mu = \mu^*\in [-1,0)$ and $\epsilon = \epsilon^*\in(0,\infty)$ such that  and  are satisfied. See Appendix \[proof:prop:stars\]. For $\mu^*\in [-1,0)$ and $\epsilon^*\in(0,\infty)$ in Proposition \[prop:stars\], the following problem can be solved to find the smallest resetting horizon $T$ for the dynamical controller: \[eqn:optimization\] $$\begin{aligned} \min_{T\in\mathbb{N}}\hspace{-.1in} \min_{\scriptsize\begin{array}{c} \varepsilon\in(0,1) \\ \delta\in[1,\infty) \end{array}} & \hspace{-.1in}T,\\ \mathrm{s.t.}\hspace{.20in}&\hspace{-.1in}P\succ \epsilon^* I,\quad \delta(1+\mu^*)^{T-1}<\varepsilon,\\ &\hspace{-.1in}F(\mathcal{P},\mathcal{C})^\top PF(\mathcal{P},\mathcal{C})\hspace{-.04in}\preceq\hspace{-.04in} (1+\mu^*) P,\\ & \hspace{-.1in}F(\mathcal{P},\mathcal{C})^\top \hspace{-.04in}\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}\hspace{-.04in} P\hspace{-.04in}\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}\hspace{-.04in} F(\mathcal{P},\mathcal{C}) \hspace{-.04in}\preceq \delta P.\end{aligned}$$ [[Note that the conditions in Theorem \[tho:1\], or optimization problem , are sufficient but not necessary. This is always the case when working with Lyapunov-based techniques for stability of dynamical systems [@Scherer_IEEE; @Kha02]. In the next subsection, we provide change of variables to cast these conditions as linear matrix inequalities that can be solved off-line only once and passed to the cloud for real-time control.]{}]{} Synthesis of Resetting Controllers ---------------------------------- [[In this subsection, we use appropriate change of variables to linearize the matrix inequalities in Theorem \[tho:1\] without generating conservatism.]{}]{} We provide tools for designing full order ($n_c=n_x$) resetting controllers of the form satisfying . That is, we look for matrices $(A_c,B_c,C_c,D_c)$ satisfying the inequalities in for some positive definite $P \in {{\mathbb R}}^{2n_x \times 2n_x}$, $\mu \in [-1,0)$, $\delta \in (0,\infty)$, $\varepsilon \in (0,1)$, and $T \in \mathbb{N}$. Let $n_c = n_x$ and $P$ be positive definite. Consider $F(\mathcal{P},\mathcal{C})$ in and define: $$\begin{aligned} &\begingroup \renewcommand*{\arraycolsep}{3pt} \tilde{F}(\mathcal{P},\mathcal{C}) := \begin{bmatrix} I & 0\\ 0 & 0 \end{bmatrix} F(\mathcal{P},\mathcal{C}) = \begin{bmatrix} A+BD_cC & BC_c \\ 0 & 0 \end{bmatrix}.\label{Synthesis5bb}\endgroup\end{aligned}$$ For simplicity of notation, in this subsection, $F(\mathcal{P},\mathcal{C})$ and $\tilde{F}(\mathcal{P},\mathcal{C})$ are denoted by $F$ and $\tilde{F}$, respectively. Then, and can be written as $$\begin{aligned} \label{Synthesis5c} &F^\top P F - (1+\mu)P \preceq 0,\quad \tilde{F}^\top P \tilde{F} - \delta P \preceq 0,\end{aligned}$$ where $0$ denotes the zero matrix of appropriate dimensions. Using properties of the Schur complement, inequalities are fulfilled if and only if the following is satisfied: $$\begin{aligned} \label{Synthesis5e} &\mathcal{L} := \begin{bmatrix} (1+\mu)P & F^\top P \\ PF & P \end{bmatrix} \succeq 0,\tilde{\mathcal{L}} := \begin{bmatrix} \delta P & \tilde{F}^\top P \\ P\tilde{F} & P \end{bmatrix} \succeq 0.\end{aligned}$$ Note that the blocks $PF$ and $P\tilde{F}$ are nonlinear functions of $(P,A_c,B_c,C_c,D_c)$. [[ In what follows, we propose a change of variables: $\left(P,A_c,B_c,C_c,D_c \right) \rightarrow \nu,$ so that, in the new variables $\nu$, we can obtain *affine* matrix inequalities equivalent to .]{}]{} In particular, for positive definite $P$ and *nonlinear* matrix inequalities $\mathcal{L} \geq0$ and $\tilde{\mathcal{L}}\geq 0$, we aim at finding two invertible matrices $\mathcal{Y}$ and $\mathcal{T}$, and variables $\nu$ such that the congruence transformations $P \rightarrow \mathcal{Y}^\top P \mathcal{Y}$, $\mathcal{L} \rightarrow \mathcal{T}^\top \mathcal{L} \mathcal{T}$, and $\tilde{\mathcal{L}} \rightarrow \mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T}$ lead to new Linear Matrix Inequalities (LMIs) $\mathcal{Y}^\top P \mathcal{Y} >0$, $\mathcal{T}^\top \mathcal{L} \mathcal{T} \geq 0$, and $\mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T} \geq 0$ in the variables $\nu$. Let $P$ be positive definite and partitioned as follows: $$\label{Synthesis1} \begingroup \renewcommand*{\arraycolsep}{1.5pt} P := \begin{bmatrix} X \hspace{2mm} & U \\ U^\top & \tilde{X} \end{bmatrix},\endgroup$$ with $X,U,\tilde{X} \in {{\mathbb R}}^{n_x \times n_x}$ and positive definite $X,\tilde{X}$. Define $$\label{Synthesis2} \begingroup \renewcommand*{\arraycolsep}{1.5pt} {{\color{black}P}}^{-1} =: \begin{bmatrix} Y \hspace{2mm} & V \\ V^\top & \tilde{Y} \end{bmatrix}, \hspace{1mm} \mathcal{Y} := \begin{bmatrix} Y \hspace{2mm} & I \\ V^\top & 0 \end{bmatrix}, \hspace{1mm} \mathcal{Z} := \begin{bmatrix} I & 0 \\ X & U \end{bmatrix}.\endgroup$$ Using block matrix inversion formulas, it can be verified that $YX+VU^\top =I$ and $YU+V\tilde{X}=0$, which yields $\mathcal{Y}^\top P = \mathcal{Z}$. Then, $P \rightarrow \mathcal{Y}^\top P \mathcal{Y}$ takes the form: $$\begin{aligned} \label{Synthesis5} &\begingroup \renewcommand*{\arraycolsep}{3pt} \mathcal{Y}^\top P \mathcal{Y} = \mathcal{Z} \mathcal{Y} = \begin{bmatrix} Y & I \\ I & X \end{bmatrix} =: \mathbf{P}(\nu).\endgroup\end{aligned}$$ Define $\mathcal{T}:= \text{diag}[\mathcal{Y},\mathcal{Y}]$ with $\mathcal{Y}$ as introduced in . Then, the transformations $\mathcal{L} \rightarrow \mathcal{T}^\top \mathcal{L} \mathcal{T}$ and $\tilde{\mathcal{L}} \rightarrow \mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T}$ can be written as $$\begin{aligned} \label{Synthesis5fpp} &\mathcal{T}^\top \mathcal{L} \mathcal{T} = \begin{bmatrix} (1+\mu)\mathbf{P}(\nu) & \mathcal{Y}^\top F^\top \mathcal{Z}^\top \\ \mathcal{Z}F\mathcal{Y} & \mathbf{P}(\nu) \end{bmatrix},\\&\mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T} = \begin{bmatrix} \delta \mathbf{P}(\nu) & \mathcal{Y}^\top \tilde{F}^\top \mathcal{Z}^\top \\ \mathcal{Z}\tilde{F}\mathcal{Y} & \mathbf{P}(\nu) \end{bmatrix}. \label{Synthesis5fppp}\end{aligned}$$ Using the structure of $F$ and $\tilde{F}$ and the change of variables: $$\begin{aligned} \label{change_of_coordinates} &\begingroup \hspace{-.1in} \renewcommand*{\arraycolsep}{2pt} \begin{pmatrix} K_1-XAY & K_2 \\ K_3 & K_4 \end{pmatrix} \hspace{-.05in}:=\hspace{-.05in} \begin{pmatrix} U & XB \\ 0 & I_{n_u} \end{pmatrix} \hspace{-.05in}\begin{pmatrix} A_c & B_c\\ C_c & D_c \end{pmatrix} \hspace{-.05in}\begin{pmatrix} V^\top & 0 \\ CY & I_{n_y} \end{pmatrix} \hspace{-.05in}\endgroup,\end{aligned}$$ the blocks $\mathcal{Z}F\mathcal{Y}$ and $\mathcal{Z}\tilde{F}\mathcal{Y}$ can be written as $$\begin{aligned} &\begingroup \renewcommand*{\arraycolsep}{3pt} \mathcal{Z}F\mathcal{Y} = \begin{bmatrix} AY + BK_3 & A+BK_4C \\ K_1 & XA + K_2C \end{bmatrix} =: \mathbf{F}(\nu),\endgroup \label{Synthesis7}\\ &\begingroup \renewcommand*{\arraycolsep}{3pt} \mathcal{Z}\tilde{F}\mathcal{Y} = \begin{bmatrix} AY + BK_3 & A+BK_4C \\ XBK_3 + XAY & XA + XBK_4C \end{bmatrix} =: \tilde{\mathbf{F}}(\nu).\endgroup \label{Synthesis7B}\end{aligned}$$ Therefore, under $\mathcal{T}$ and the change of variables in , we can write $\mathcal{T}^\top \mathcal{L}\mathcal{T}$ and $\mathcal{T}^\top \tilde{\mathcal{L}}\mathcal{T}$ as follows: $$\begin{aligned} \label{Synthesis5fpppp} &\begingroup \renewcommand*{\arraycolsep}{0pt} \mathcal{T}^\top \mathcal{L} \mathcal{T} = \begin{bmatrix} (1+\mu)\mathbf{P}(\nu) & \hspace{2mm} \mathbf{F}(\nu)^\top \\ \mathbf{F}(\nu) & \mathbf{P}(\nu) \end{bmatrix} =: \mathbf{L}(\nu), \endgroup \\[2mm] &\begingroup \renewcommand*{\arraycolsep}{0pt} \mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T} = \begin{bmatrix} \delta \mathbf{P}(\nu) & \hspace{2mm} \tilde{\mathbf{F}}(\nu)^\top \\ \tilde{\mathbf{F}}(\nu) & \mathbf{P}(\nu) \end{bmatrix} =: \mathbf{S}(\nu), \endgroup \label{Synthesis5fpppppp}\end{aligned}$$ with $\mathbf{P}(\nu),\mathbf{F}(\nu)$, and $\tilde{\mathbf{F}}(\nu)$ as defined in , , and , respectively. Therefore, the original matrix inequality, $\mathcal{L} \succeq 0$ defined in , that depends non-linearly on the decision variables $(P,A_c,B_c,C_c,D_c)$ is transformed into a new inequality, $\mathbf{L}(\nu) \succeq 0$, that is an affine function of the variables $\nu$. Note, however, that $\mathbf{S}(\nu) \succeq 0$ (the block $\tilde{\mathbf{F}}(\nu)$) is still nonlinear in the new variables $\nu$. In the following lemma, we give a sufficient condition, in terms of an affine inequality $\tilde{\mathbf{L}}(\nu) \succeq 0$, for $\mathbf{S}(\nu)$ to be positive semidefinite. \[lemma:bound\] Consider $\mathbf{P}(\nu)$ and $\mathbf{S}(\nu)$ defined in and , respectively. Define the matrices: $$\begin{aligned} \mathbf{R}(\nu) &:= \begin{pmatrix} AY + BK_3 & A + BK_4C \end{pmatrix}, \label{Synthesis31}\\[1mm] \tilde{\mathbf{L}}(\nu) &:= \begin{bmatrix} \delta \mathbf{P}(\nu) & \hspace{2mm} \mathbf{R}(\nu)^\top \\ \mathbf{R}(\nu) & 2I_n - X \end{bmatrix}. \label{Synthesis34}\end{aligned}$$ Then, $\tilde{\mathbf{L}}(\nu) \succeq 0 \Rightarrow \mathbf{S}(\nu) \succeq 0$. See Appendix \[proof:lemma:bound\]. Lemma \[lemma:bound\] provides a sufficient condition, $\tilde{\mathbf{L}}(\nu) \succeq 0$, for the nonlinear matrix $\mathbf{S}(\nu)$ to be positive semidefinite. This $\tilde{\mathbf{L}}(\nu)$ is an affine function of $\nu$. Note, however, that finding $\nu$ satisfying $(\mathbf{P}(\nu) \succ 0,\hspace{.5mm} \mathbf{L}(\nu) \succeq 0,\hspace{.5mm} \tilde{\mathbf{L}}(\nu) \succeq 0)$ might not be sufficient to guarantee the existence of $(P,A_c,B_c,C_c,D_c)$ satisfying $(P \succ 0,\hspace{.5mm} \mathcal{L} \succeq 0,\hspace{.5mm} \tilde{\mathcal{L}} \succeq 0)$. For this to be true, matrices $\mathcal{Y}$ and $\mathcal{T}$ must be invertible so that the transformations $P \rightarrow \mathcal{Y}^\top P \mathcal{Y} = \mathbf{P}(\nu)$, $\mathcal{L} \rightarrow \mathcal{T}^\top \mathcal{L} \mathcal{T} = \mathbf{L}(\nu)$, and $\tilde{\mathcal{L}} \rightarrow \mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T} = \mathbf{S}(\nu)$ are congruence transformations; and $\nu$ must render the change of variables in invertible. \[lemma:congruence\] Consider matrices $\mathcal{Y}$ and $\mathbf{P}(\nu)$ defined in and , respectively. Let $(X,Y)$ be such that $\mathbf{P}(\nu)\succ 0$. Then, $\mathcal{Y}$ and $\mathcal{T} = \text{diag}(\mathcal{Y},\mathcal{Y})$ are nonsingular and the change of variables in is invertible. See Appendix \[proof:lemma:congruence\]. Therefore, by Lemma \[lemma:congruence\], if $\mathbf{P}(\nu) \succ 0$, the transformations $P \rightarrow \mathbf{P}(\nu)$, $\mathcal{L} \rightarrow \mathbf{L}(\nu)$, and $\tilde{\mathcal{L}} \rightarrow \mathbf{S}(\nu)$ are congruence transformations. The latter and the fact that (by Lemma \[lemma:bound\]) $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0 \Rightarrow \mathbf{S}(\nu) {{\color{black}\,\succeq\,}} 0$ imply that $$\label{Synthesis11} \left\{ \begin{array}{c} ( \mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0, \text{ }\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0, \text{ and } \tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0 )\\ \Downarrow \\ ( P {{\color{black}\,\succ\,}} 0, \text{ }\mathcal{L} {{\color{black}\,\succeq\,}} 0, \text{ and } \tilde{\mathcal{L}} {{\color{black}\,\succeq\,}} 0 ), \end{array} \right.$$ for $P = \mathcal{Y}^{-\top}\mathbf{P}(\nu)\mathcal{Y}^{-1}$ and the controller matrices in obtained by inverting . In the following lemma, we summarize the discussion presented above. \[synthesis\_lemma1\] For given system matrices $(A,B,C)$. If there exist matrices $\nu = (X,Y,K_1,K_2,K_3,K_4)$, $K_2 \in {{\mathbb R}}^{n_x \times n_u}$, $K_3 \in {{\mathbb R}}^{n_y \times n_x}$, $K_4 \in {{\mathbb R}}^{n_u \times n_y}$, $X,Y,K_1 \in {{\mathbb R}}^{n_x \times n_x}$ satisfying $\mathbf{P}(\nu) {{\color{black}\,\succ\,}}0$, $\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$ with $\mathbf{P}(\nu)$, $\mathbf{L}(\nu)$, and $\tilde{\mathbf{L}}(\nu)$ as defined in , , and , respectively; then, there exist $(P,A_c,B_c,C_c,D_c)$ satisfying $P {{\color{black}\,\succ\,}}0$, $\mathcal{L} {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathcal{L}} {{\color{black}\,\succeq\,}} 0$ with $P$, $\mathcal{L}$, and $\tilde{\mathcal{L}}$ as defined in and , respectively. Moreover, for every $\nu$ such that $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0$, $\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$, the change of variables in and matrix $\mathcal{Y}$ in are invertible and the $(P,A_c,B_c,C_c,D_c)$ obtained by inverting and are unique and satisfy the analysis inequalities . See Appendix \[proof:synthesis\_lemma1\]. By Lemma \[synthesis\_lemma1\], the matrices $(P,A_c,B_c,C_c,D_c)$ obtained by inverting and satisfy inequalities (and thus also and ). Moreover, because the reconstructed $P$ is positive definite, inequality is satisfied with $\epsilon = \lambda_{\min}(P)$, where $\lambda_{\min}(P) \in {{\mathbb R}}_{>0}$ denotes the smallest eigenvalue of $P$. Next, we give the synthesis result corresponding to Theorem \[tho:1\]. \[synthesis\_theorem\] For given system matrices $(A,B,C)$ and constants $\varepsilon\in(0,1)$, $\mu\in [-1,0)$, $\delta\in[1,\infty)$, and $T \in {{\mathbb N}}$ satisfying , if there exist matrices $\nu = (X,Y,K_1,K_2,K_3,K_4)$ satisfying $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0$, $\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$; then, $P = \mathcal{Y}^{-\top}\mathbf{P(\nu)}\mathcal{Y}$ and the controller matrices in satisfy the analysis inequalities - and thus render the closed-loop dynamics - asymptotically stable. See Appendix \[proof:synthesis\_theorem\]. **Controller Reconstruction.** For given $\nu$ satisfying the synthesis inequalities ($\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0, \mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$): 1. For given $X$ and $Y$, compute via singular value decomposition a full rank factorization $VU^\top = I-YX$ with square and nonsingular $V$ and $U$. 2. For given $\nu$ and invertible $V$ and $U$, solve the system of equations $\mathcal{Y}^\top P \mathcal{T} = \mathbf{P}(\nu)$ and to obtain the matrices $(P,A_c,B_c,C_c,D_c)$. Note that $\varepsilon,\mu,\delta$, and $T$, in Theorem \[synthesis\_theorem\] must be fixed before looking for feasible solutions $\nu$ satisfying the synthesis LMIs: $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0$, $\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$. However, for any $\mu \in [-1,0)$ and $\delta\in[1,\infty)$, there always exist $\varepsilon \in (0,1)$ and $T \in {{\mathbb N}}$ satisfying . Moreover, the synthesis LMIs depend on $\nu$, $\delta$, and $\mu$ but not on $\varepsilon$ and $T$. Therefore, to find feasible controllers, we only have to fix $(\mu,\delta)$ and look for $\nu$ satisfying the synthesis LMIs. The constants $(\mu,\delta)$ are, in fact, variables of the synthesis problem; however, to linearize some of the constraints, we fix their value and search over $\mu \in [-1,0)$ and $\delta\in[1,\infty)$ to find feasible $\nu$. The latter increases the computations needed to find controllers; however, we can perform a bisection search over $\delta\in[1,\infty)$ and, because $\mu \in [-1,0)$ (a bounded set), a grid search over $\mu$ to decrease the required computations. [[Finally, note that the characteristics (e.g., unstable poles) of the system in  make the feasibility of the design LMIs $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0$, $\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$ “easier or harder” for fixed resetting horizon $T$ and constants $\varepsilon,\mu,\delta$. Exploring this dependence, in general, is an avenue for future research.]{}]{} Dynamic Controller Implementation {#dynamic-controller-implementation} --------------------------------- [[In this subsection, we present the necessary transformations required for implementing encrypted dynamic control laws. ]{}]{} Before stating the next result, we introduce some notation. Define $\|A\|_{\max}:=\max_{i,j}|a_{ij}|$, where $a_{ij}$ denotes the entry in $i$-th row and $j$-th column of matrix $A$, and $\mathbb{Q}(n,m):=\{b\,|\,b=-b_n2^{n-m-1}+\sum_{i=1}^{n-1}2^{i-m-1}b_i,b_i\in\{0,1\}\,\forall i\in\{1,\dots,n\}\}.$ For any $x\in\mathbb{R}^q$ and $\mathbb{A}\subseteq\mathbb{R}^q$, let $ \operatorname{proj}(x,\mathbb{A})\in{\mathop{\text{arg\,min}}}_{x'\in\mathbb{A}}\|x'-x\|_\infty$ and $\operatorname{dist}(x,\mathbb{A}):=\min_{x'\in\mathbb{A}}\|x'-x\|_\infty$. The quantization of $x\in\mathbb{R}^q$ is $\operatorname{proj}(x,\mathbb{Q}(n,m)^q)$ and the quantization error is $\|\operatorname{proj}(x,\mathbb{Q}(n,m)^q)-x\|_\infty=\operatorname{dist}(x,\mathbb{Q}(n,m)^q)$. The quantization of $X\in\mathbb{R}^{p\times q}$ is defined as $\operatorname{proj}(x,\mathbb{Q}(n,m)^{p\times q})\in{\mathop{\text{arg\,min}}}_{x'\in\mathbb{A}}\|x'-x\|_{\max}$ and the quantization error as $\|\operatorname{proj}(x,\mathbb{Q}(n,m)^{p\times q})-x\|_{\max}$. Please refer to [@farokhi2017secure] for details about the quantization scheme. \[tho:2\] Let \[eqn:quantized\] $$\begin{aligned} \bar{A}_c&=\operatorname{proj}(A_c,\mathbb{Q}(n,m)^{n_c\times n_c}),\\ \bar{B}_c&=\operatorname{proj}(B_c,\mathbb{Q}(n,m)^{n_c\times n_y}),\\ \bar{C}_c&=\operatorname{proj}(C_c,\mathbb{Q}(n,m)^{n_u\times n_x}),\\ \bar{D}_c&=\operatorname{proj}(D_c,\mathbb{Q}(n,m)^{n_u\times n_y}).\end{aligned}$$ Then, there exists $\bar{n}\geq \bar{m}>0$ such that $F(\mathcal{P},\mathcal{C})$ satisfies  if and only if $F(\mathcal{P},\bar{\mathcal{C}})$, where $\bar{\mathcal{C}}$ denotes the controller in  with quantized parameters $\bar{A}_c$, $\bar{B}_c$, $\bar{C}_c$, and $\bar{D}_c$ in , satisfies  with the same $P$ for all $n\geq \bar{n}$ and $m\geq \bar{m}$. The proof follows from continuity of the eigenvalues. Note that, by increasing $n$ and $m$, the quantization error decreases (actually, it tends to zero). In what follows, we discuss the implementation of quantized resetting controllers using homomorphic encryption schemes and quantized sensor measurements. The controller, in this case, is given by $$\begin{aligned} \label{eqn:controller_quantized} \bar{\bar{\mathcal{C}}}\hspace{-.04in}:\hspace{-.04in} \left\{\hspace*{-.07in} \begin{array}{rl}\hspace{-.03in} x_c[k+1]\hspace*{-.14in}&=\hspace{-.04in} \begin{cases} \hspace{-.04in}\bar{A}_cx_c[k]\hspace{-.02in}+\hspace{-.02in}\bar{B}_c\bar{y}[k], & \hspace{-.1in}(k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in}T\hspace{-.03in}\neq\hspace{-.02in} 0, \\ \hspace{-.03in}0, & \hspace{-.1in}(k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in} T\hspace{-.03in}=\hspace{-.02in} 0, \end{cases} \\[1.5em] u[k]\hspace*{-.14in}&=\hspace{-.04in}\bar{C}_cx_c[k]+\bar{D}_c\bar{y}[k], \end{array} \right.\end{aligned}$$ where $\bar{A}_c$, $\bar{B}_c$, $\bar{C}_c$, and $\bar{D}_c$ are defined in  and $$\begin{aligned} \label{eqn:output_quantized} \bar{y}[k]&\in{\mathop{\text{arg\,min}}}_{y\in \mathbb{Q}(n,m)^{n_y}} \|y-y[k]\|_{\infty}.\end{aligned}$$ The difference between $\bar{\bar{\mathcal{C}}}$ in  and $\bar{\mathcal{C}}$ in Theorem \[tho:2\] is the quantization of the output measurements $y[k]$. The following standing assumption is made in this paper to ensure the stability of the closed-loop system. \[assum:1\] $n\geq \bar{n}$ and $m\geq \bar{m}$ where $\bar{n}$ and $\bar{m}$ are given in Theorem \[tho:2\]. The following theorem proves the stability of the system $\mathcal{P}$ with the quantized resetting controller $\bar{\bar{\mathcal{C}}}$. Note that, for any $r\in\mathbb{R}_{\geq 0}$, $\mathbb{B}(r):=\{x\,|\,\|x\|_2^2\leq r\}$. \[tho:3\] Under Assumption \[assum:1\], if there exist $\varepsilon\in(0,1)$, $\mu\in [-1,0)$, $\delta\in[1,\infty)$, $T \in {{\mathbb N}}$, and $\epsilon\in(0,\infty)$ such that inequalities in  are satisfied and $$\begin{aligned} \label{bound_on_n} n>\log_2\bigg(\dfrac{\lambda_{\max}(C^\top C)}{\epsilon}x_0^\top \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}P\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}x_0\bigg)+1;\end{aligned}$$ then, the system dynamics with the quantized resetting controller in  is stable and, for some constant[^6] $\varrho>0$, $\lim_{k\rightarrow\infty} \operatorname{dist}(x[k],\mathbb{B}(\varrho2^{-m}))=0$. See Appendix \[proof:tho:3\] Theorem \[tho:3\] implies that the state of the system converges to a vicinity of the origin (instead of the origin itself) due to quantization effects. The volume of the this area can be arbitrarily reduced by increasing $m$ and thus the performance of the system can be arbitrarily improved. \[lemma:2\] For the resetting quantized controller in , $x_c[k]\in\mathbb{Q}((n_c+1)(k\operatorname{mod}T-1)+n_y+n(k\operatorname{mod}T+1),m(k\operatorname{mod}T+1))^{n_c}$ and $u_c[k]\in\mathbb{Q}((n_c+1)(k\operatorname{mod}T)+n_y+n(k\operatorname{mod}T+2),m(k\operatorname{mod}T+2))^{n_u}$. See Appendix \[proof:lemma:2\]. Using the change of variables: \[eqn:transformtointeger\] $$\begin{aligned} \tilde{A}_c&=(2^m\bar{A}_c)\operatorname{mod}2^{\tilde{n}},\\ \tilde{B}_c{{\color{black}[k]}}&=({{\color{black}2^{m(k\operatorname{mod}T+1)}}}\bar{B}_c)\operatorname{mod}2^{\tilde{n}},\\ \tilde{C}_c&=(2^m\bar{C}_c)\operatorname{mod}2^{\tilde{n}},\\ \tilde{D}_c{{\color{black}[k]}}&=({{\color{black}2^{m(k\operatorname{mod}T+1)}}}\bar{D}_c)\operatorname{mod}2^{\tilde{n}},\\ \tilde{x}_c[k]&=(2^{m(k\operatorname{mod}T+1)}\bar{x}_c[k])\operatorname{mod}2^{\tilde{n}},\\ \tilde{y}[k]&=(2^{m(k\operatorname{mod}T+1)}\bar{y}[k])\operatorname{mod}2^{\tilde{n}},\\ \tilde{u}[k]&=(2^{m(k\operatorname{mod}T+2)}\bar{u}[k])\operatorname{mod}2^{\tilde{n}},\end{aligned}$$ with $\tilde{n}>(n_c+1)T+n_u+n(T+2)$, the resetting quantized controller in  can be rewritten as $$\begin{aligned} \label{eqn:controller_positiveint} \hspace{-.1in}\tilde{\mathcal{C}}\hspace{-.04in}:\hspace{-.04in} \left\{\hspace*{-.07in} \begin{array}{rl}\hspace{-.04in} \tilde{x}_c[k+1]\hspace*{-.14in}&=\hspace{-.04in} \begin{cases} \hspace{-.04in}\tilde{A}_c\tilde{x}_c[k]\hspace{-.02in}+\hspace{-.02in}\tilde{B}_c{{\color{black}[k]}}\tilde{y}[k], & \hspace{-.1in}(k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in}T\hspace{-.03in}\neq\hspace{-.02in} 0, \\ \hspace{-.03in}0, & \hspace{-.1in}(k\hspace{-.02in}+\hspace{-.02in}1)\hspace{-.02in}\operatorname{mod}\hspace{-.02in} T\hspace{-.03in}=\hspace{-.02in} 0, \end{cases} \\[1.5em] \tilde{u}[k]\hspace*{-.14in}&=\hspace{-.04in}\tilde{C}_c\tilde{x}_c[k]+\tilde{D}_c{{\color{black}[k]}}\tilde{y}[k]. \end{array} \right.\end{aligned}$$ Note that, by Lemma \[lemma:2\], $\tilde{A}_c,\tilde{B}_c,\tilde{C}_c,\tilde{D}_c,\tilde{x}_c,\tilde{y},\tilde{u}$ are *positive integers*. This is useful because the Paillier’s scheme can only work with finite ring of positive integers. Therefore, the update equation can now be implemented using Paillier’s encryption scheme. The correctness of this implementation follows from the results of [@farokhi2017secure] on fixed-point rational numbers. ![Norm of the state of the closed-loop system $\|x[k]\|_2$ with quantized controller and quantizer resolution $(n,m) = (24,14)$.[]{data-label="Fig1"}](Fig1d.eps) ![Norm of the state of the quantized controller $\|x_c[k]\|_2$ in with quantizer resolution $(n,m) = (24,14)$.[]{data-label="Fig2"}](Fig2d.eps) First, the public and private keys must be generated such that $\kappa_p\geq 2^{\tilde{n}+1}$ to ensure that no unintended overflow occurs when using the encrypted numbers. The sensors measure, quantize, and encrypt the output to obtain $$\begin{aligned} \label{encrypted_y} \check{y}_i[k]:=\mathfrak{E}(\tilde{y}_i[k],\kappa_p).\end{aligned}$$ The controller follows the encrypted version of  to update its state and compute the actuation signal as $$\begin{aligned} \label{encrypted_xc} (\check{x}_c)_i[k+1]=& \begin{cases} \bigg(\oplus_{j=1}^{n_x} (\check{x}_c)_j[k]\triangle(\tilde{A}_c)_{ij}\bigg)\\ \hspace{.2in}\oplus \bigg(\oplus_{j=1}^{n_y} \check{y}_j[k]\triangle(\tilde{B}_c)_{ij}\bigg),\\ &\hspace{-1in} (k+1)\operatorname{mod}T\neq 0,\\ \mathfrak{E}(0,\kappa_p), &\hspace{-1in} (k+1)\operatorname{mod}T=0, \end{cases}\end{aligned}$$ $$\begin{aligned} \label{encrypted_u} \check{u}_i[k]=&\bigg(\oplus_{j=1}^{n_c} (\check{x}_c)_j[k]\triangle(\tilde{C}_c)_{ij}\bigg)\nonumber\\ &\oplus \bigg(\oplus_{j=1}^{n_y} (\check{y})_j[k]\triangle(\tilde{D}_c)_{ij}\bigg).\end{aligned}$$ Finally, the actuator extract the control signal by $\tilde{u}_i[k]=\mathfrak{D}(\check{u}_i[k],\kappa_p,\kappa_s)\operatorname{mod}2^{\tilde{n}},$ and implements $u_i[k]=2^{-m(k\operatorname{mod}T+2)}(\tilde{u}_i[k]-2^{\tilde{n}}\mathds{1}_{\tilde{u}_i[k]\geq 2^{\tilde{n}-1}}).$ National Institute of Standards and Technology (NIST) recommends the use of key length of 2048 bits for factoring-based asymmetric encryption to guarantee that brute-force attacks are not physically possible during the life-time of the services based on projections of computing technologies. This high standard might not be necessary for some applications, such as remote control of autonomous vehicles. To demonstrate this, consider RSA, which is a similar encryption methodology and also a semi-homomorphic encryption relying on hardness of prime number factorization. RSA encryption has been attacked repeatedly using a brute-force methodology; see RSA Challenge[^7]. Factorization of 663 bit numbers has been shown to take approximately 55 CPU-Years[^8] [@elbirt2009understanding]. Using IBM Watson (used recently for natural language processing to win quiz show Jeopardy), factorization of 663 bit numbers takes approximately 2.5 years. These numbers are certainly not safe for use in finance or military. However, for remote control of autonomous vehicles, these keys may provide strong-enough guarantees as, by the time that an adversary breaks the code, the autonomous vehicle is in an entirely different location. ![Norm of the state of the closed-loop system $\|x[k]\|_2$ with quantized controller and quantizer resolution $(n,m) = (24,14)$.[]{data-label="Fig3"}](Fig3.eps) ![Norm of the state of the quantized controller $\|x_c[k]\|_2$ in with quantizer resolution $(n,m) = (24,14)$.[]{data-label="Fig4"}](Fig4.eps) Case Study of a Chemical Batch Reactor {#sec:example} ====================================== We illustrate the performance of our results through a case study of a batch chemical reactor. This case study has been developed over the years as a benchmark example for networked control systems, see, e.g., [@NCS1; @NCS3; @NCS2]. The reactor considered here is open-loop unstable, has one input, and two outputs (please refer to [@NCS1; @NCS3; @NCS2] for details about the system dynamics). We exactly discretize the reactor dynamics introduced in [@NCS3] with sampling period $h=0.1$. The resulting discrete-time linear system is of the form with matrices $A,B,C$ as follows: $$\begin{aligned} \label{Simul1b} &\hspace{-.1in}\begin{bmatrix}[c|c|c] A & B & C^\top \end{bmatrix}\nonumber\\ &= \hspace{-.05in} \begin{bmatrix}[cccc|c|cc] \hspace{2.5mm}1.18 & \hspace{2.5mm}0 & \hspace{2.5mm}0.51 & -0.40 & 0 & \hspace{2.5mm}1 & \hspace{.5mm}0\\ -0.05 & \hspace{2.5mm}0.66 & -0.01 & \hspace{2.5mm}0.06 & 0.47 & \hspace{2.5mm}0 & \hspace{.5mm}1\\ \hspace{2.5mm}0.08 & \hspace{2.5mm}0.34 & \hspace{2.5mm}0.56 & \hspace{2.5mm}0.38 & 0.21 & \hspace{2.5mm}1 & \hspace{.5mm}0\\ \hspace{2.5mm}0 & \hspace{2.5mm}0.34 & \hspace{2.5mm}0.09 & \hspace{2.5mm}0.85 & 0.21 & -1 & \hspace{.5mm}0 \end{bmatrix} \hspace{-.03in}.\end{aligned}$$ Note that $\text{ eig}[A] = \{1.22,1.01,.60,.42 \}$; thus the system is open-loop unstable. Moreover, it can be verified (e.g., using the tools in [@Bara Theorem 3.3]) that there does not exist a static output feedback controllers of the form $u[k] = Ly[k]$, $L \in \mathbb{R}^{1 \times 2}$, stabilizing system with matrices $(A,B,C)$ as in . First, using the synthesis results in Section \[sec:implementation\], we design switching dynamic output feedback controllers of the form . Using Theorem \[synthesis\_theorem\], and conducting a bisection in $\delta \in [1,\infty)$, and a line search in $\mu \in [-1,0)$, we look for the smallest $\delta$ for which there exist $\mu \in [-1,0)$ and $\nu$ satisfying the synthesis LMIs in Theorem \[synthesis\_theorem\]. The obtained $\delta$ is given by $\delta= \delta^* = 55.0$, the corresponding $\mu$ is $\mu = \mu^* = -0.15$, the resetting horizon is $T^* = \text{argmin}_{T \in {{\mathbb N}}} \hspace{1mm} \delta^*(1+\mu^*)^T = 25$, and the reconstructed $A_c,B_c,C_c,D_c$ (see Section \[sec:implementation\]) are given in $$\begin{aligned} &\begin{bmatrix}[c|c] A_c & B_c\\ \cmidrule(lr){1-2} C_c & D_c \end{bmatrix}\nonumber \\ &\hspace{-.05in}=\hspace{-.05in} \begin{bmatrix}[cccc|cc] \hspace{2.5mm}0.26 & -0.03 & -0.29 & \hspace{2.5mm}0.31 & -0.52 & -0.03\\ -0.32 & \hspace{2.5mm}1.24 & \hspace{2.5mm}1.40 & -3.05 & \hspace{2.5mm}5.46 & \hspace{2.5mm}1.25\\ -0.45 & \hspace{2.5mm}0.02 & \hspace{2.5mm}0.87 & -0.75 & \hspace{2.5mm}2.32 & -0.01\\ -0.05 & -0.04 & \hspace{2.5mm}0.72 & -0.51 & \hspace{2.5mm}2.28 & -0.08\\ \cmidrule(lr){1-6} \hspace{2.5mm}1.02 & -2.65 & -2.65 & \hspace{2.5mm}6.28 & -11.3 & -4.09 \end{bmatrix}\hspace{-.05in}. \label{Simul2b}\end{aligned}$$ This controller satisfies the original inequalities in with $\epsilon = 0.0026$ and any $\varepsilon \in (0.9459,1)$. For comparison, let $\mu = \mu^* = -0.65$ and search for the smallest $\delta$ for which there exists $\nu$ satisfying the synthesis LMIs in Theorem \[synthesis\_theorem\]. In this case, $\delta^* = 3000$, the smallest resetting horizon is $T^* = \text{argmin}_{T \in {{\mathbb N}}} \hspace{1mm} \delta^*(1+\mu^*)^T = 8$, the reconstructed $A_c,B_c,C_c,D_c$ are in $$\begin{aligned} &\begin{bmatrix}[c|c] A_c & B_c\\ \cmidrule(lr){1-2} C_c & D_c \end{bmatrix}\nonumber\\ & \hspace{-.05in}=\hspace{-.05in} \begin{bmatrix}[cccc|cc] -0.18 & -0.01 & -0.77 & \hspace{2.5mm}0.84 & -1.11 & -0.01\\ \hspace{2.5mm}9.17 & \hspace{2.5mm}0.43 & \hspace{2.5mm}13.4 & -16.2 & \hspace{2.5mm}22.8 & \hspace{2.5mm}0.42\\ \hspace{2.5mm}1.24 & \hspace{2.5mm}0.10 & \hspace{2.5mm}3.82 & -4.22 & \hspace{2.5mm}7.81 & \hspace{2.5mm}0.06\\ \hspace{2.5mm}1.32 & \hspace{2.5mm}0.10 & \hspace{2.5mm}3.47 & -3.87 & \hspace{2.5mm}7.89 & \hspace{2.5mm}0.06\\ \cmidrule(lr){1-6} -19.6 & -0.93 & -28.8 & \hspace{2.5mm}34.9 & -49.0 & -2.33 \end{bmatrix}\hspace{-.05in}.\label{Simul3b}\end{aligned}$$ This controller satisfies the original inequalities in with $\epsilon =2.8 \times 10^{-5}$, and $\varepsilon \in (0.6756,1)$. Next, we quantize the controller matrices according to to obtain $(\bar{A}_c,\bar{B}_c,\bar{C}_c,\bar{D}_c)$ with quantizer resolution $(n,m)=(24,14)$. It can be verified that for $A_c,B_c,C_c,D_c$ in and , the corresponding $\bar{A}_c,\bar{B}_c,\bar{C}_c,\bar{D}_c$ satisfy the conditions of Theorem \[tho:3\] with $(n,m)=(24,14)$. We quantize sensor measurements $y[k]$ according to with the same resolution $(n,m)=(24,14)$, and close the system dynamics with the quantized controller in . By Theorem \[tho:3\], the quantizer resolution must satisfy inequality to ensure practical stability of in feedback with in the sense of Theorem \[tho:3\]. Inequality , with initial condition $[x(0)^T,x_c(0)^T] = [-6.83,-5.18,-4.05,-3.12,0,0,0,0]$, amounts to $n>17$ for the controller in and to $n>23$ for the controller in . Therefore, $(n,m)=(24,14)$ is enough for practical stabilization using the controllers in and . Figures \[Fig1\] and \[Fig2\] show $\|x(k)\|_2$ and $\|x_c(k)\|_2$ of the closed-loop dynamics for quantized controllers corresponding to the controllers in and with $(n,m)=(24,14)$. To illustrate the need for the proposed resetting controller, we naively implement a standard quantized dynamic controller of the form: $$\begin{aligned} \label{eqn:controller_quantized2} \left\{ \begin{array}{l} \bar{x}_c[k+1] = \bar{A}_c\bar{x}_c[k] + \bar{B}_c\bar{y}[k],\\[1mm] \hspace{7.5mm}\bar{u}[k] = \bar{C}_c\bar{x}_c[k]+\bar{D}_c\bar{y}[k]. \end{array} \right.\end{aligned}$$ We use the same quantizer resolution $(n,m)=(24,14)$, and compute the matrices $(\bar{A}_c,\bar{B}_c,\bar{C}_c,\bar{D}_c)$ using with $(A_c,B_c,C_c,D_c)$ from . This controller is stabilizing even without resets. Note that the Paillier’s encryption only works over the ring of positive integers $\mathbb{Z}_{\kappa_p}$, and thus the controller needs to be transformed so that its states and parameters always belong to this ring. Therefore, as also required for the resetting controller, we must transform $\bar{y}[k]$ and $(\bar{A}_c,\bar{B}_c,\bar{C}_c,\bar{D}_c)$ into positive integers, which can be done using the change of variables in  replacing $k\operatorname{mod}T$ with $k$ (as there is no resetting after $T$ steps in this case). Let the integer representations of $\bar{y}[k]$ and $(\bar{A}_c,\bar{B}_c,\bar{C}_c,\bar{D}_c)$ be similarly denoted by $\tilde{y}[k]$ and $(\tilde{A}_c,\tilde{B}_c,\tilde{C}_c,\tilde{D}_c)$. The equivalent controller in the integer domain is then given by $$\begin{aligned} \label{eqn:controller_integer2} \left\{ \begin{array}{l} \tilde{x}_c[k+1] = \big( \tilde{A}_c\tilde{x}_c[k] + \tilde{B}_c\tilde{y}[k] \big)\operatorname{mod}2^{\tilde{n}},\\[1mm] \hspace{7.5mm}\tilde{u}[k] = \big( \tilde{C}_c\tilde{x}_c[k]+\tilde{D}_c\tilde{y}[k] \big)\operatorname{mod}2^{\tilde{n}}. \end{array} \right.\end{aligned}$$ Finally, given $\tilde{u}[k]$, the actuators implement the control action: $$\begin{aligned} \label{eqn:controller_integer2_extractred} u_i[k]= 2^{-m(k+2)}(\tilde{u}_i[k]-2^{\tilde{n}}\mathds{1}_{\tilde{u}_i[k]\geq 2^{\tilde{n}-1}}).\end{aligned}$$ Because we must ensure that $2^{\tilde{n}}\leq \kappa_p$, we need to select a large, yet finite $\tilde{n}$. Here, for illustration purposes, we selected a key length of 2048bits and $\tilde{n} = 2014$. Figure \[Fig3\] and \[Fig4\] illustrate the norm of the state of the closed-loop system, $\|x[k]\|_2$, with controller -, and the state of the controller in the quantized domain, $\|x_c[k]\|$, respectively. Note that, even though - is a stabilizing controller (if no under/over flow occur), when implementing -, the closed-loop system is unstable due to under/over flows. Conclusions and Future Work {#sec:conclusions} =========================== A secure and private implementation of linear time-invariant dynamic controllers using the Paillier’s encryption was presented. The state is reset to zero periodically to avoid data overflow or underflow within the encryption space. A control design approach was presented to ensure the stability and performance of the closed-loop system with encrypted controller. Future work can focus on nonlinear dynamical systems and controllers. Proof of Theorem \[tho:1\] {#proof:tho:1} ========================== For the sake brevity $F(\mathcal{P},\mathcal{C})$ is denoted by $F$. Consider the quadratic Lyapunov function $V(z[k])=z[k]^\top Pz[k]$, where $P$ is a positive semi-definite matrix. For all $k\in\{\ell T,\dots,(\ell+1)T-2\}$ with $\ell\in\mathbb{N}$ the Lyapunov function evolves as $$\begin{aligned} V(z[k+1])-V(z[k])&=z[k]^\top (F^\top PF-P)z[k]\nonumber\\ &\leq \mu z[k]^\top P z[k]\nonumber\\ &= \mu V(z[k]). \label{eqn:proof:0}\end{aligned}$$ Therefore, $$\begin{aligned} \label{eqn:proof:1} V(z[(\ell+1)T-1])\leq (1+\mu)^{T-1} V(z[\ell T]).\end{aligned}$$ Now, note that $$\begin{aligned} V(z[(\ell+1)T]) &= \begin{bmatrix} x[(\ell+1)T] \nonumber \\ 0 \end{bmatrix}^\top P\begin{bmatrix} x[(\ell+1)T] \\ 0 \end{bmatrix}\nonumber\\ &= z[(\ell+1)T-1]^\top F^\top\nonumber \\ &\hspace{.2in}\times\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} P\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix} F z[(\ell+1)T-1]\nonumber\\ &\leq \delta z[(\ell+1)T-1]^\top P z[(\ell+1)T-1]\nonumber\\ &=\delta V(z[(\ell+1)T-1]).\label{eqn:proof:2}\end{aligned}$$ Combining  and  results in $ V(z[(\ell+1)T])\leq \delta(1+\mu)^{T-1} V(z[\ell T]). $ This shows that $\lim_{\ell\rightarrow\infty} V(z[\ell T])=0$. Following , it can be seen that $\lim_{k\rightarrow\infty} V(z[k])=0$. This proves the stability of the system. Proof of Proposition \[prop:stars\] {#proof:prop:stars} =================================== [[We state the proof for two cases where $n_c=n_x$ and $n_c>n_x$. ]{}]{}[[Case I ($n_c=n_x$): ]{}]{} [[Since]{}]{} $(A,B)$ is stabilizable and $(A,C)$ is detectable, a Luenberger observer [[exists such that]{}]{} conditions  and  [[are satisfied]{}]{}. [[Case II ($n_c>n_x$):]{}]{} Unobservable and uncontrollable states can be [[incorporated into]{}]{} the controller in addition to the Luenberger observer [[so that and  are satisfied]{}]{}. Proof of Lemma \[lemma:bound\] {#proof:lemma:bound} ============================== The block $\tilde{\mathbf{F}}(\nu)$ can be factored as $\tilde{\mathbf{F}}(\nu) = \mathbf{P}(\nu) \begin{pmatrix} 0 & I_n \end{pmatrix}^\top \mathbf{R}(\nu).$ Then, by properties of the Schur complement, $\mathbf{S}(\nu) \succeq 0$ if and only if $$\label{Synthesis32} \delta \mathbf{P}(\nu) - \tilde{\mathbf{F}}(\nu)^\top \mathbf{P}(\nu)^{-1} \tilde{\mathbf{F}}(\nu) = \delta \mathbf{P}(\nu) - \mathbf{R}(\nu)^\top X \mathbf{R}(\nu) \succeq 0;$$ therefore, using the Schur complement again, $\mathbf{S}(\nu) \succeq 0$ if and only if $$\label{Synthesis33} \begin{bmatrix} \delta \mathbf{P}(\nu) & \hspace{2mm} \mathbf{R}(\nu)^\top \\ \mathbf{R}(\nu) & X^{-1} \end{bmatrix} \succeq 0.$$ Because $(X^{-1/2} - X^{1/2})^\top (X^{-1/2} - X^{1/2}) \succeq 0$ for any $X$, we have that $X^{-1} \succeq 2I_n - X$. It follows that $$\begin{bmatrix} \delta \mathbf{P}(\nu) & \hspace{2mm} \mathbf{R}(\nu)^\top \\ \mathbf{R}(\nu) & X^{-1} \end{bmatrix} \succeq \tilde{\mathbf{L}}(\nu) = \begin{bmatrix} \delta \mathbf{P}(\nu) & \hspace{2mm} \mathbf{R}(\nu)^\top \\ \mathbf{R}(\nu) & 2I_n - X \end{bmatrix};$$ therefore, $\tilde{\mathbf{L}}(\nu) \succeq 0 \Rightarrow \mathbf{S}(\nu) \succeq 0$. Proof of Lemma \[lemma:congruence\] {#proof:lemma:congruence} =================================== Let $\mathbf{P}(\nu)$ be positive definite; then, by properties of the Schur complement, $Y {{\color{black}\,\succ\,}}0$ and $X - Y^{-1} {{\color{black}\,\succ\,}}0$, and because $YX + VU^\top = I$ by construction (see ), $VU^\top = I - YX {{\color{black}\,\prec\,}} 0$, i.e., the matrix $VU^\top $ is nonsingular. Therefore, if $\mathbf{P}(\nu) {{\color{black}\,\succ\,}}0$, it is always possible to find nonsingular $U$ and $V$ satisfying $YX + VU^\top = I$. [[The existence of a nonsingular]{}]{} $V$ implies that $\mathcal{Y}$[[, introduced in ,]{}]{} and $\mathcal{T}{{\color{black}:=\text{diag}[\mathcal{Y},\mathcal{Y}]}}$ are invertible. Moreover, nonsingular $U$ and $V$ imply that the matrices: $$\begin{pmatrix} U & XB \\ 0 & I_{n_u} \end{pmatrix} \text{ and } \begin{pmatrix} V^\top & 0 \\ CY & I_{n_y} \end{pmatrix},$$ are invertible. Therefore, the controller matrices: $$\begin{aligned} \label{CONTROLLER} \begin{pmatrix} A_c & B_c\\ C_c & D_c \end{pmatrix} = &\begin{pmatrix} U & XB \\ 0 & I_{n_u} \end{pmatrix}^{-1} \begin{pmatrix} K_1- XAY & K_2 \\ K_3 & K_4 \end{pmatrix}\\ \notag &\times \begin{pmatrix} V^\top & 0 \\ CY & I_{n_y} \end{pmatrix}^{-1},\end{aligned}$$ are the unique solution of the matrix equation . Proof of Lemma \[synthesis\_lemma1\] {#proof:synthesis_lemma1} ==================================== Assume that $\nu$ is such that $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0, \text{ }\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$. Then, by Lemma \[lemma:congruence\], $\mathcal{Y}$ and $\mathcal{T}$ are square and nonsingular and thus the transformations $P \rightarrow \mathcal{Y}^\top P\mathcal{Y} = \mathbf{P}(\nu)$, $\mathcal{L} \rightarrow \mathcal{T}^\top \mathcal{L} \mathcal{T} = \mathbf{L}(\nu)$, and $\tilde{\mathcal{L}} \rightarrow \mathcal{T}^\top \tilde{\mathcal{L}} \mathcal{T} = \mathbf{S}(\nu)$ are congruent. By Lemma \[lemma:bound\], $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0 \Rightarrow \mathbf{S}(\nu) {{\color{black}\,\succeq\,}}0$. It follows that ($\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0, \mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$) [[implies]{}]{} ($P {{\color{black}\,\succ\,}}0,\mathcal{L} {{\color{black}\,\succeq\,}} 0,\tilde{\mathcal{L}} {{\color{black}\,\succeq\,}} 0$) because $(\mathbf{P}(\nu),\mathbf{L}(\nu),\mathbf{S}(\nu))$ have the same signature as ($P,\mathcal{L},\tilde{\mathcal{L}}$), respectively, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0 \Rightarrow \mathbf{S}(\nu) {{\color{black}\,\succeq\,}} 0$. Because $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0$, the matrices $U$ and $V$ are nonsingular. This implies that the change of variables in and $\mathcal{T}$ are invertible and lead to unique $(P,A_c,B_c,C_c,D_c)$ by inverting and . Proof of Theorem \[synthesis\_theorem\] {#proof:synthesis_theorem} ======================================= If $\nu$ satisfies $\mathbf{P}(\nu) {{\color{black}\,\succ\,}} 0$, $\mathbf{L}(\nu) {{\color{black}\,\succeq\,}} 0$, and $\tilde{\mathbf{L}}(\nu) {{\color{black}\,\succeq\,}} 0$, by Lemma \[synthesis\_lemma1\], the change of variables in and matrix $\mathcal{Y}$ in are invertible and the controller matrices and $P$ obtained by inverting and are unique and satisfy inequalities - with $\epsilon = \lambda_{\min}(P)$. Hence, because is satisfied by assumption, by Theorem \[tho:1\], the controller matrices in render the closed-loop dynamics - asymptotically stable. Proof of Theorem \[tho:3\] {#proof:tho:3} ========================== For the sake [[of]{}]{} brevity $F(\mathcal{P},\bar{\mathcal{C}})$ is denoted by $\bar{F}$. First, note that $$\begin{aligned} z[\ell T+i]=\bar{F}^i z[\ell T]+\sum_{j=0}^{i-1}\bar{F}^{i-j-1}G\theta_{\ell T+j},\end{aligned}$$ for all $i\in\{1,\dots,T-1\}$, where $\theta_k{{\color{black}:=}}\bar{y}[k]-y[k]$ and $$\begin{aligned} G= \begin{bmatrix} A+B\bar{D}_c \\ \bar{B}_c \end{bmatrix}.\end{aligned}$$ For $i=T$, it can be seen that $$\begin{aligned} z[\ell T+i]=\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}\bigg(\bar{F}z[\ell T+i-1]+G\theta_{\ell T+i-1}\bigg).\end{aligned}$$ Combining these update rules shows that $$\begin{aligned} \label{eqn:update_lifted_z} z[(\ell+1) T]=H\bar{F}^T z[\ell T]+w_\ell,\end{aligned}$$ where $$\begin{aligned} H:=\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix},\quad w_\ell:=\sum_{j=0}^{T-1} \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}\bar{F}^{T-j-1}G\theta_{\ell T+j}.\end{aligned}$$ Note that $$\begin{aligned} z[(\ell+1) T]^\top Pz[(\ell+1) T] =&z[\ell T]^\top \bar{F}^{\top T} H^\top P H \bar{F}^T z[\ell T]\\ &+2w_\ell^\top P H \bar{F}^Tz[\ell T]+w_\ell^\top P w_\ell\\ \leq &\delta z[\ell T]^\top \bar{F}^{\top T-1} P \bar{F}^{T-1} z[\ell T]\\ &+2w_\ell^\top P H \bar{F}^Tz[\ell T]+w_\ell^\top P w_\ell\\ \leq &\delta(1+\mu)^{T-1} z[\ell T]^\top P z[\ell T]\\ &+2w_\ell^\top P H \bar{F}^Tz[\ell T]+w_\ell^\top P w_\ell,\end{aligned}$$ where the inequalities follow from $\bar{F}^{\top } H^\top P H \bar{F}\leq \delta P$ and $\bar{F}^{\top }P \bar{F}\leq (1+\mu) P$. Thus, $$\begin{aligned} z[(\ell+1) T]^\top &Pz[(\ell+1) T] -z[\ell T]^\top P z[\ell T]\\ \leq &(\delta(1+\mu)^{T-1}-1) z[\ell T]^\top P z[\ell T]\\ &+2w_\ell^\top P H \bar{F}^Tz[\ell T]+w_\ell^\top P w_\ell\\ \leq &(\delta(1+\mu)^{T-1}-1) \|P^{1/2}z[\ell T]\|_2^2\\ &+2\|P^{-1/2}\bar{F}^{\top T} H^\top P\|_2 W\|P^{1/2}z[\ell T]\|_2\\ &+\lambda_{\max}(P) W^2,\end{aligned}$$ where $$\begin{aligned} W :=&\sup_{\ell} \|w_\ell\|_2\\ =&\sup_{\ell} \bigg\|\sum_{j=0}^{T-1} \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}\bar{F}^{T-j-1}G\theta_{\ell T+j}\bigg\|_2\\ \leq & \sum_{j=0}^{T-1} \bigg\|\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}\bar{F}^{T-j-1}G\theta_{\ell T+j}\bigg\|_2\\ \leq &\sum_{j=0}^{T-1} \|\bar{F}\|^{T-j-1}\|G\|\sup_{\ell}\|\theta_{\ell T+j}\|_2\\ \leq &\bigg(\sum_{j=0}^{T-1} \|\bar{F}\|^{T-j-1}\|G\|\bigg){{\color{black}n_y}}2^{-m},\end{aligned}$$ where the last equality follows from that the quantization error is bounded by $\|\theta_{\ell T+j}\|_2\leq {{\color{black}n_y}} 2^{-m}$ if $n$ is selected large enough[[, i.e., $n$ is selected such that $\|y[k]\|_\infty< 2^{n-1}$ (see the last paragraph of this proof for ensuring this)]{}]{}. Let $s_1$ denote the largest real root of the polynomial equation $s\mapsto [(\delta(1+\mu)^{T-1}-1)\lambda_{\max}(P)] s^2+[2\|P^{-1/2}\bar{F}^{\top T} H^\top P\|_2 W]s+ [\lambda_{\max}(P) W^2]$. Define $\mathbb{M}_1:=\{z\,|\,z^\top P z\leq s_1^2\}$. Notice that if $z[\ell T]\notin \mathbb{M}_1$, $z[(\ell+1) T]^\top Pz[(\ell+1) T]-z[\ell T]^\top P z[\ell T]<0$ (and thus $z[(\ell+1) T]\in \mathbb{M}_1$) since $(\delta(1+\mu)^{T-1}-1{{\color{black})}}<0$. However, if $z[\ell T]\in \mathbb{M}_1$, it can be deduce that $z[(\ell+1) T]^\top Pz[(\ell+1) T] \leq s_2 :=\delta(1+\mu)^{T-1} s_1^2+\lambda_{\max}(P) W^2 +2\|P^{-1/2}\bar{F}^{\top T} H^\top P\|_2 Ws_1.$ Define $\mathbb{M}_2:=\{z\,|\,z^\top P z\leq s_2^2\}$. Evidently, if $z[\ell T]\in \mathbb{M}_1$, then $z[(\ell+1) T]\in \mathbb{M}_2$. Combining these results, it can be seen that $\mathbb{M}_1\cup \mathbb{M}_2$ is an invariant set for . This is because two distinct cases can happen if $z[\ell T]{{\color{black}\,\in\,}}\mathbb{M}_1\cup \mathbb{M}_2$; either $z[\ell T]\in \mathbb{M}_2\setminus \mathbb{M}_1$ or $z[\ell T]\in \mathbb{M}_1$ must happen. If $z[\ell T]\in \mathbb{M}_2\setminus \mathbb{M}_1$, it means that $z[\ell T]\notin \mathbb{M}_1$, thus $z[(\ell+1) T]\in \mathbb{M}_1\subseteq \mathbb{M}_1\cup \mathbb{M}_2$. On the other hand, if $z[\ell T]\in \mathbb{M}_1$, then $z[(\ell+1) T]\in \mathbb{M}_2\subseteq \mathbb{M}_1\cup \mathbb{M}_2$. Furthermore, $\mathbb{M}_1\cup \mathbb{M}_2$ is an invariant set for the dynamical system in feedback loop with . This is because, for all $k\in\{\ell T,\dots,(\ell+1)T-2\}$, $z[k+1]^\top P z[k+1]-z[k]^\top P z[k]=V(z[k+1])-V(z[k])\leq \mu z[k]^\top P z[k]<0$. It can be seen that the set $\mathbb{M}_1\cup \mathbb{M}_2$ is attractive for . This because if $z[\ell T]\notin \mathbb{M}_1\cup \mathbb{M}_2$, it must also $z[\ell T]\notin \mathbb{M}_1$. Therefore, $z[(\ell+1) T]^\top Pz[(\ell+1) T]-z[\ell T]^\top P z[\ell T]<0$. All these results in the fact that $\lim_{k\rightarrow\infty} \operatorname{dist}(z[k],\mathbb{M}_1\cup \mathbb{M}_2)=0$. Now, note that $\max(s_1,s_2)=\mathcal{O}(2^{-m})$. Therefore, there exists $\varrho>0$ such that $\lim_{k\rightarrow\infty} \operatorname{dist}(z[k],\mathcal{B}(\varrho 2^{-m}))=0$. It only remains to find the bound on $n$. Note that the largest value of the Lyapunov function is $$\begin{aligned} z[0]^\top Pz[0]=c:=x_0^\top \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}P\begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}x_0.\end{aligned}$$ This implies that $z[k]^\top Pz[k]\leq c$. Therefore, $x[k]^\top x[k]+x_c[k]^\top x_c[k]=z[k]^\top z[k]\leq c/\lambda_{\min}(P):=c/\epsilon$. As a result, $x[k]^\top x[k]\leq c/\epsilon$. Noting that $y[k]=Cx[k]$, it can be deduced that $y[k]^\top y[k]\leq \lambda_{\max}(C^\top C)c/\epsilon$. Finally, because of the relationship between norms, it can be seen that $\|y[k]\|_{\infty}\leq \lambda_{\max}(C^\top C)c/\epsilon$. This implies that $2^{n-1}> \lambda_{\max}(C^\top C)c/\epsilon$ and thus $n>\log_2(\lambda_{\max}(C^\top C)c/\epsilon)+1$. Proof of Lemma \[lemma:2\] {#proof:lemma:2} ========================== Noting that the controller resets every $T$ steps, we only need to prove this result for $k\in\{0,\dots,T-1\}$. At $k=0$, $x_c[k]=0$ and thus $x_c[1]\in\mathbb{Q}(n_y+2n,2m)$ because the entries of $\bar{B}_c\bar{y}[k]$ belong to $\mathbb{Q}(n_y+2n,2m)$. For all $k\in\{1,\dots,T-1\}$, if the entries of $x_c[k]$ belong to $\mathbb{Q}(n',m')$, the entries of $\bar{A}_cx_c[k]$ (at worst case) belong to $\mathbb{Q}(n_c+n+n',m+m')$ and the entries of $\bar{B}_c\bar{y}[k]$ belong to $\mathbb{Q}(n_y+2n,2m)$, therefore the entries of $x[k+1]=\bar{A}_cx_c[k]+\bar{B}_c\bar{y}[k]$ must belong to $\mathbb{Q}(n_c+n+n'+1,m+m')$ because $n_y\leq n_c$, $n\leq n'$, and $m\leq m'$. Furthermore, $u[k]$ must belong to $\mathbb{Q}(n_c+n+n'+1,m+m')$. This proves that the entries of $x_c[k]$ and $u[k]$ must, respectively, belong to $\mathbb{Q}((n_c+n+1)(k\operatorname{mod}T-1)+n_y+2n,m(k\operatorname{mod}T-1)+2m)$ and $\mathbb{Q}((n_c+n+1)(k\operatorname{mod}T)+n_y+2n,m(k\operatorname{mod}T)+2m)$. [^1]: C. Murguia and I. Shames are with the Department of Electrical and Electronic Engineering at the University of Melbourne. [^2]: F. Farokhi is with the CSIRO’s Data61 and the Department of Electrical and Electronic Engineering at the University of Melbourne. [^3]: Encryption keys with the length of [[2048]{}]{} bits is recommended by National Institute of Standards and Technology (NIST) for data [[over 2016-2030]{}]{}; see [[<https://www.keylength.com/en/4/>.]{}]{} [^4]: These random elements are replaced with pseudo-random ones when implementing encryption and decryption algorithms. [^5]: A function $f:\mathbb{N}\rightarrow\mathbb{R}_{\geq 0}$ is called negligible if, for any $c\in\mathbb{N}$, there exists $n_c\in\mathbb{N}$ such that $f(n)\leq 1/n^c$ for all $n\geq n_c$. [^6]: See Appendix \[proof:tho:3\] for a description of this constant. [^7]: <https://en.wikipedia.org/wiki/RSA_Factoring_Challenge> [^8]: A CPU-Year is the amuont of computing work done by a 1 Giga Floating Point Operations Per Second (FLOP) reference machine in a year of dedicated service (8760 hours).

--- abstract: 'We demonstrate how dipolar interactions can have pronounced effects on the structure of vortices in atomic spinor Bose-Einstein condensates and illustrate generic physical principles that apply across dipolar spinor systems. We then find and analyze the cores of singular vortices with non-Abelian charges in the point-group symmetry of a spin-3 $^{52}$Cr condensate. Using a simpler model system, we analyze the underlying dipolar physics and show how a characteristic length scale arising from the magnetic dipolar coupling interacts with the hierarchy of healing lengths of the $s$-wave scattering, and leads to simple criteria for the core structure: When the interactions both energetically favor the ground-state spin condition, such as in the spin-1 ferromagnetic phase, the size of singular vortices is restricted to the shorter spin-dependent healing length. Conversely, when the interactions compete (e.g., in the spin-1 polar phase), we find that the core of a singular vortex is enlarged by increasing dipolar coupling. We further demonstrate how the spin-alignment arising from the interaction anisotropy is manifest in the appearance of a ground-state spin-vortex line that is oriented perpendicularly to the condensate axis of rotation, as well as in potentially observable internal core spin textures. We also explain how it leads to interaction-dependent angular momentum in nonsingular vortices as a result of competition with rotation-induced spin ordering. When the anisotropy is modified by a strong magnetic field, we show how it gives rise to a symmetry-breaking deformation of a vortex core into a spin-domain wall.' author: - 'Magnus O. Borgh' - Justin Lovegrove - Janne Ruostekoski title: 'Internal structure and stability of vortices in a dipolar spinor Bose-Einstein condensate' --- Introduction {#sec:introduction} ============ The achievement of Bose-Einstein condensation using atoms with large magnetic dipole moments, such as $^{52}$Cr [@griesmaier_prl_2005; @pasquiou_prl_2011; @de-paz_pra_2013], $^{168}$Er [@aikawa_prl_2012], and several Dy isotopes [@lu_prl_2011; @tang_njp_2015] as well as creation of a degenerate dipolar Fermi gas [@aikawa_prl_2014] have opened up a new avenue for studying the effects of long-range and anisotropic interactions in ultracold atomic gases [@lahaye_rpp_2009]. In such systems, long-range magnetic order can coexist with superfluidity, making possible, e.g., ferro-superfluids [@lahaye_nature_2007]. The interaction can then lead to novel instabilities, e.g., toward formation of droplet crystals [@kadau_nature_2016; @bisset_pra_2015; @xi_pra_2016], and formation of a condensate may be strongly influenced by the spin dynamics [@naylor_prl_2016]. The interaction can also profoundly affect the stability and structure of vortices [@yi_pra_2006b; @o'dell_pra_2007; @wilson_prl_2008; @klawunn_prl_2008; @abad_pra_2009], e.g., inducing a phase transition from straight to twisted vortex lines [@klawunn_njp_2009]. Simultaneously, the structure of topological defects and textures is a central topic in the study of spinor Bose-Einstein condensates (BECs), where the atomic spin degree of freedom is not frozen out by strong magnetic fields [@kawaguchi_physrep_2012]. This gives rise to a rich phenomenology of the internal structure of vortices [@yip_prl_1999; @mizushima_pra_2002; @mizushima_prl_2002; @martikainen_pra_2002; @saito_prl_2006; @ji_prl_2008; @lovegrove_pra_2012; @kobayashi_pra_2012; @borgh_prl_2012; @lovegrove_prl_2014; @lovegrove_pra_2016; @borgh_prl_2016]. Recent experiments have demonstrated controlled preparation of nonsingular vortices [@leanhardt_prl_2003; @leslie_prl_2009; @choi_prl_2012; @choi_njp_2012] as well as point defects [@ray_nature_2014; @ray_science_2015] and particle-like solitons [@hall_nphys_2016]. The *in situ* observation of splitting of singly quantized vortices into pairs of half-quantum vortices [@seo_prl_2015], theoretically predicted in Ref. [@lovegrove_pra_2012], marks an increasing experimental interest in the internal core structure. The spin degree of freedom in spinor BECs also implies that dipolar interactions (DIs) arising from the magnetic dipole moment of the atoms can have a strong impact on the spin texture [@kawaguchi_prl_2007; @kawaguchi_pra_2010; @huhtamaki_pra_2010a]. Nontrivial textures arising spontaneously due to DI have been observed in experiment [@vengalattore_prl_2008; @eto_prl_2014]. However, the potentially large impact on the internal structure of vortices has so far been little studied. Even a very weak DI can influence the relaxation of vortices by changing the longitudinal magnetization, whose conservation in $s$-wave scattering can be important for stability and structure, e.g., of a coreless vortex in a polar condensate [@lovegrove_prl_2014; @lovegrove_pra_2016]. Theoretical works on vortices in dipolar spinor BECs have predicted a superfluid Einstein–de Haas effect, where magnetic relaxation induces vortex formation [@santos_prl_2006; @kawaguchi_prl_2006a; @gawryluk_prl_2011; @swislocki_pra_2011], as well as a stable spin vortex in a nonrotating system [@yi_prl_2006; @kawaguchi_prl_2006b; @takahashi_prl_2007]. In a rotating highly oblate condensate, complex multivortex states and stable higher-order defects have been described [@simula_jpsj_2011]. Here we demonstrate how dipolar interactions can have pronounced effects on the internal structure of vortices in atomic spinor BECs. To clearly illustrate the underlying physical principles, we employ the comparative simplicity of a spin-1 model system. The dipolar effects arise from generic properties of the DI and spinor systems and the corresponding principles may therefore be applied more broadly to understand properties of vortices in dipolar spinor BECs. Dipolar spin-1 BECs could potentially also be realized using alkali-metal atoms by suppressing the $s$-wave scattering lengths via optical or microwave Feshbach resonances [@fatemi_prl_2000; @papoular_pra_2010]. As an example of experimentally realized dipolar BEC, we numerically find and analyze the stable core structure of a singular vortex in a spin-3 condensate of $^{52}$Cr. The $^{52}$Cr atom possesses a relatively large magnetic dipole moment [@santos_prl_2006] and is predicted to exhibit a dihedral-6 point-group order-parameter symmetry in the ground state, supporting non-Abelian vortices. The multicomponent condensate wave function of a spinor BEC allows the condensate to maintain the superfluid density in the core of singular vortices. In addition to depleting the condensate density, the wave function can also be excited out of the ground-state manifold to accommodate the order-parameter singularity and form a filled defect core. Here we numerically find the superfluid cores of singular vortices when the atoms exhibit a long-range and anisotropic magnetic DI. The DI gives rise to a new spin-dependent healing length, adding to the hierarchy of characteristic length scales arising form the contact interaction to determine the structure of singular-vortex cores [@ruostekoski_prl_2003; @lovegrove_pra_2012; @borgh_prl_2016]. We analyze the interplay of these length scales and demonstrate how the size of a singular-vortex core is determined by the shorter of the spin-dependent healing lengths when the DI and contact interaction both restrict breaking of the ground-state spin condition (the ground-state phase of the bulk superfluid), e.g., in the ferromagnetic (FM) spin-1 BEC. On the other hand, when the contact interaction and DI compete, such as in the spin-1 polar phase, we explain how a singular-vortex core expands with increasing DI, beyond the size in its absence. In addition, the anisotropy of the interaction leads to an internal spin texture that is potentially observable in a spin-3 $^{52}$Cr condensate. We further analyze manifestations of the interaction anisotropy in the spin-1 model system and and show it leads to a ground-state spin vortex that appears perpendicularly to the axis of a slow rotation. The structure is then the result of interplay between dipolar spin alignment and rotation as the vortex line bends to adapt to the latter. At more rapid rotation, we demonstrate a nontrivial interaction dependence of the angular momentum carried by a ground-state coreless vortex. We show how it may be understood from a competition between dipolar spin alignment and the adaptation of the spin texture to rotation. Drastically different spin-ordering effects can appear in the presence of a sufficiently strong external magnetic field, such that the DI may be averaged over the rapid spin precession [@kawaguchi_prl_2007]. We show how the resulting modified interaction anisotropy leads to a symmetry-breaking core deformation with increasing DI in a stable singular vortex. At sufficiently strong DI, the vortex core deforms into a domain wall separating regions with opposite spin polarization. Mean-field theory of the dipolar BEC {#sec:mft} ==================================== We treat the spinor BEC in the classical Gross-Pitaevskii mean-field theory, which can be straightforwardly extended to include DI between the atoms. Here we first give a brief overview of the salient points (for full details see, e.g., Ref. [@kawaguchi_physrep_2012]) in the spin-1 case, and then show how the theory is modified for spin-3 atoms. Spin-1 ------ The spin-1 condensate wave function $\Psi$ may be expressed in terms of the atomic density and a normalized three-component spinor as $$\Psi({\ensuremath{\mathbf{r}}}) = \sqrt{n({\ensuremath{\mathbf{r}}})}\zeta({\ensuremath{\mathbf{r}}}) = \sqrt{n({\ensuremath{\mathbf{r}}})}{\left(\begin{array}{c}\zeta_+\\#2\\#3\end{array}\right)}, \quad \zeta^\dagger\zeta = 1.$$ The expectation value ${\langle\mathbf{\hat{F}}\rangle}= \zeta^\dagger_{\alpha}\mathbf{\hat{F}}_{\alpha\beta}\zeta_{\beta}$ of the spin operator, defined as the vector of spin-1 Pauli matrices, gives the condensate spin. This relates to the magnetic dipole moment arising from the intrinsic angular momentum of the atom as ${\mathbf{m}}= -g_F\mu_{\mathrm{B}}{\langle\mathbf{\hat{F}}\rangle}$ [@yi_prl_2004], where $g_F$ is the Landé factor and $\mu_{\mathrm{B}}$ is the Bohr magneton. The Hamiltonian density including the DI is then given by $$\label{eq:hamiltonian-density} {\cal H} = h_0 + \frac{c_0}{2}n^2 + \frac{c_2}{2}n^2|{\langle\mathbf{\hat{F}}\rangle}|^2 + \frac{{\ensuremath{c_\mathrm{d}}}}{2} \int D({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}^\prime}})\,d^3r^\prime,$$ where $$\label{eq:dipolar-energy-density} D({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}^\prime}}) = \frac{{\mathbf{F}}({\ensuremath{\mathbf{r}}})\cdot{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}}) - 3[{\mathbf{F}}({\ensuremath{\mathbf{r}}})\cdot{\ensuremath{\mathbf{\hat{n}}}}][{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})\cdot{\ensuremath{\mathbf{\hat{n}}}}]} {|{\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}}|^3}$$ describes the interaction of dipoles at ${\ensuremath{\mathbf{r}}}$ and ${\ensuremath{\mathbf{r}^\prime}}$ given by the local condensate spin with the coupling constant ${\ensuremath{c_\mathrm{d}}}= \mu_0\mu_{\mathrm{B}}^2g_F^2/(4\pi)$ [@yi_prl_2004], where $\mu_0$ is the vacuum permeability. We define ${\mathbf{F}}=n{\langle\mathbf{\hat{F}}\rangle}$ and denote the unit vector along ${\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}}$ by ${\ensuremath{\mathbf{\hat{n}}}}$. The single-particle Hamiltonian density $$\label{eq:h0} h_0=\frac{\hbar^2}{2M}{\ensuremath{\left| \nabla\Psi \right|}}^2 + \frac{1}{2}M\omega^2r^2n,$$ where $M$ is the atomic mass, includes the external trapping potential, which we take to be an isotropic harmonic oscillator with frequency $\omega$. The contact-interaction strengths are given by the scattering lengths $a_f$ in the spin-$f$ channel of colliding spin-1 atoms as $c_0=4\pi\hbar^2(2a_2+a_0)/(3M)+c_0^{\mathrm{d}}$ and $c_2=4\pi\hbar^2(a_2-a_0)/(3M)+c_2^{\mathrm{d}}$. Here we have made it explicit that the coupling constants may be modified by contributions $c_{0,2}^{\mathrm{d}}$ arising from an absorbed contact-interaction part of the DI (see below and Appendix \[app:ft\]). The spin-1 BEC exhibits two ground state phases: a FM phase that maximizes the condensate spin ${|\langle\mathbf{\hat{F}}\rangle|}=1$ and a polar phase where ${|\langle\mathbf{\hat{F}}\rangle|}=0$ in a uniform system. Without DI, the ground-state phase is determined by the sign of $c_2$ with a negative value favoring the FM phase. When magnetic DI, where the dipole moment is proportional to the condensate spin, is present, the ground state depends also on ${\ensuremath{c_\mathrm{d}}}$. In particular, from Eq.  we can see that the DI is minimized when spins ${|\langle\mathbf{\hat{F}}\rangle|}=1$ are aligned head-to-tail. The DI will therefore also favor formation of a FM phase, and sufficiently large ${\ensuremath{c_\mathrm{d}}}$ may overcome also a positive $c_2$ [@yi_prl_2006]. From Eq.  the familiar coupled Gross-Pitaevskii equations (GPEs) describing the condensate dynamics may be derived. Following Ref. [@kawaguchi_physrep_2012], we write the contribution from the DI term in the equation for the $\psi_m=\sqrt{n}\zeta_m$ spinor component as $$\label{eq:dipolar-gpe} i\hbar{\frac{\partial\psi_m({\ensuremath{\mathbf{r}}})}{\partialt}} = \ldots + {\ensuremath{c_\mathrm{d}}}\sum_j\hat{\mathbf{F}}_{mj}\psi_j({\ensuremath{\mathbf{r}}})\cdot{\ensuremath{\boldsymbol{\mathcal{B}}}}({\ensuremath{\mathbf{r}}}).$$ The vector ${\ensuremath{\boldsymbol{\mathcal{B}}}}$ is given by $$\label{eq:b-vector-def} {\ensuremath{\boldsymbol{\mathcal{B}}}}({\ensuremath{\mathbf{r}}}) = \int \frac{{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})-3{\ensuremath{\mathbf{\hat{n}}}}[{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})\cdot{\ensuremath{\mathbf{\hat{n}}}}]}{|{\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}}|^3}\, d^3r^\prime\\$$ and is related to the magnetic field $$\label{eq:dipole-field-condensate} \begin{split} &{\mathbf{B}}({\ensuremath{\mathbf{r}}}) = \frac{\mu_0\{3{\ensuremath{\mathbf{\hat{n}}}}[{\mathbf{m}}({\ensuremath{\mathbf{r}^\prime}})\cdot{\ensuremath{\mathbf{\hat{n}}}}]-{\mathbf{m}}({\ensuremath{\mathbf{r}^\prime}})\}} {4\pi |{\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}}|^3} + \frac{2\mu_0}{3}{\mathbf{m}}({\ensuremath{\mathbf{r}^\prime}})\delta({\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}})\\ &= -g_F{\ensuremath{\mu_\mathrm{B}}}\mu_0\left\{\frac{3{\ensuremath{\mathbf{\hat{n}}}}[{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})\cdot{\ensuremath{\mathbf{\hat{n}}}}]-{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})} {|{\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}}|^3} + \frac{2}{3}{\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})\delta({\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}})\right\} \end{split}$$ at ${\ensuremath{\mathbf{r}}}$ arising from the condensate dipole moment at ${\ensuremath{\mathbf{r}^\prime}}$. The factor $g_F{\ensuremath{\mu_\mathrm{B}}}\mu_0/(4\pi)$ enters the coupling constant ${\ensuremath{c_\mathrm{d}}}$ and the $\delta$-function term yields a contact-interaction contribution that is absorbed in $c_{0,2}$ as above (see also Appendix \[app:ft\]). Integrating the remaining term over ${\ensuremath{\mathbf{r}^\prime}}$ yields ${\ensuremath{\boldsymbol{\mathcal{B}}}}$. The DI term in Eq.  is nonlocal, and its evaluation involves finding the integral over ${\ensuremath{\mathbf{r}^\prime}}$, which is computationally expensive. However, since the integral in Eq.  has the form of a convolution, it can be computed efficiently in Fourier space, where the convolution of two functions becomes a multiplication of their Fourier transforms. For our computations we follow the formalism of Ref. [@kawaguchi_physrep_2012], rewriting ${\ensuremath{\boldsymbol{\mathcal{B}}}}$ as $$\label{eq:b-vector} {\ensuremath{\mathcal{B}}}_\alpha = -\sum_\beta \int {\ensuremath{\mathsf{Q}}}_{\alpha\beta}({\ensuremath{\mathbf{r}}}-{\ensuremath{\mathbf{r}^\prime}}) F_\beta({\ensuremath{\mathbf{r}^\prime}})\, d^3r^\prime,$$ where the tensor ${\ensuremath{\mathsf{Q}}}$ is defined as $$\label{eq:q-tensor} {\ensuremath{\mathsf{Q}}}_{\alpha\beta}({\ensuremath{\mathbf{r}}}) = \frac{3\hat{r}_\alpha\hat{r}_\beta-\delta_{\alpha\beta}}{r^3},$$ for ${\ensuremath{\mathbf{\hat{r}}}}= {\ensuremath{\mathbf{r}}}/r$. In Eq.  the convolution is explicit and Fourier transformation immediately gives $$\label{eq:b-transf} \tilde{{\ensuremath{\mathcal{B}}}}_\alpha({\ensuremath{\mathbf{k}}}) = - \sum_\beta {\ensuremath{\mathsf{\tilde{Q}}}}_{\alpha\beta}({\ensuremath{\mathbf{k}}})\tilde{F}_\beta({\ensuremath{\mathbf{k}}}).$$ To compute ${\ensuremath{\boldsymbol{\mathcal{B}}}}$ we then need the Fourier transforms on the right-hand side, where $\tilde{{\mathbf{F}}}({\ensuremath{\mathbf{k}}})$ must be found numerically, while ${\ensuremath{\mathsf{\tilde{Q}}}}({\ensuremath{\mathbf{k}}})$ can be found analytically as (see Appendix \[app:ft\] and Ref. [@kawaguchi_physrep_2012]) $$\label{eq:q-transf} {\ensuremath{\mathsf{\tilde{Q}}}}_{\alpha\beta}({\ensuremath{\mathbf{k}}}) = -\frac{4\pi}{3}(3\hat{k}_\alpha\hat{k}_\beta-\delta_{\alpha\beta}),$$ where ${\ensuremath{\mathbf{\hat{k}}}}={\ensuremath{\mathbf{k}}}/k$. Note that the derivation of this Fourier transform rests on nontrivial assumptions. We provide the details in the Appendix \[app:ft\]. In practical numerical calculations using Fast Fourier Transforms, the long-range nature of the DI can lead to aliasing problems that yield erroneous results, and accuracy may more generally be reduced. These problems can be avoided or mitigated by truncating the dipolar interaction [@ronen_pra_2006; @blakie_pre_2009]. In a spherical or nearly spherical system, where computations are performed on a grid with all sides equal, the simplest solution is to truncate the dipolar interaction at a radius $R$, such that ${\ensuremath{\mathsf{Q}}}({\ensuremath{\mathbf{r}}})=0$ for $r>R$. The Fourier transform of the truncated interaction is then $$\label{eq:q-transf-trunc} \begin{split} &{\ensuremath{\mathsf{\tilde{Q}}}}^{r<R}_{\alpha\beta}({\ensuremath{\mathbf{k}}}) = \int_{r<R} {\ensuremath{\mathsf{Q}}}_{\alpha\beta}({\ensuremath{\mathbf{r}}})e^{-i{\ensuremath{\mathbf{k}}}\cdot{\ensuremath{\mathbf{r}}}}\,d^3r \\ &= -4\pi(3\hat{k}_\alpha\hat{k}_\beta-\delta_{\alpha\beta}) \left(\frac{1}{3} + \frac{kR\cos(kR) - \sin(kR)}{(kR)^3}\right), \end{split}$$ which is the spherical cut-off found in Ref. [@ronen_pra_2006], straightforwardly generalized to the spinor case (see Appendix \[app:ft\]). In the presence of an external magnetic field $\mathbf{B}_{\mathrm{ext}}=B_{\mathrm {ext}}{\ensuremath{\mathbf{\hat{z}}}}$, the condensate spin precesses with the Larmor frequency $\omega_{\mathrm{L}}=g_F\mu_{\mathrm{B}}B_{\mathrm {ext}}/\hbar$. In a sufficiently strong field, the precession is rapid compared with the DI-induced spin dynamics. It is then convenient to describe the condensate in the spin-space frame rotating at the Larmor frequency through the transformation $\zeta_m({\ensuremath{\mathbf{r}}},t) \to \zeta_m({\ensuremath{\mathbf{r}}},t)e^{-im\omega_{\mathrm{L}}t}$ [@kawaguchi_prl_2007; @kawaguchi_pra_2010]. This leaves all terms of Eq.  invariant, except the dipolar interaction. (Also the linear Zeeman term that would arise from the magnetic field is canceled and we assume any quadratic Zeeman energy to be small.) The modified dipolar interaction is found as a time average over the period of the Larmor precession [@kawaguchi_pra_2010]: $$\label{eq:q-tensor-l} {\ensuremath{\mathsf{Q}}}_{\alpha\beta}^\mathrm{L}({\ensuremath{\mathbf{r}}}) = \frac{3\hat{r}_z^2-1}{r^3} \frac{3\delta_{z\alpha}\delta_{z\beta}-\delta_{\alpha\beta}}{2}$$ Also in this case we truncate the dipolar interaction at a radius $R$ for computational purposes. Its Fourier transform then becomes (see Appendix \[app:ft\]) $$\label{eq:q-transf-l-trunc} \begin{split} {\ensuremath{\mathsf{\tilde{Q}}}}^{\mathrm{L},r<R}_{\alpha\beta}({\ensuremath{\mathbf{k}}}) = & -2\pi(\hat{k}_z^2-1) (3\delta_{z\alpha}\delta_{z\beta}-\delta_{\alpha\beta})\times\\ &\left[\frac{1}{3} + \frac{kR\cos(kR) - \sin(kR)}{(kR)^3}\right]. \end{split}$$ Spin-3 ------ The spin-1 condensate provides a useful system where the physical principles underlying the dipolar effects in a spinor BEC can be illustrated. Dipolar spin-1 BECs could potentially be realized using Na or Rb atoms by suppressing the $s$-wave scattering lengths via optical or microwave Feshbach resonances. However, large magnetic dipole moments are exhibited, e.g., by $^{52}$Cr, which is a spin-3 atom. In this case, the condensate wave function becomes a seven-component spinor with $\zeta = (\zeta_{+3},\dots,\zeta_{-3})^T$ and a Hamiltonian density given by [@kawaguchi_physrep_2012; @diener_prl_2006; @santos_prl_2006] $$\label{eq:spin-3-hamiltonian-density} \begin{split} {\cal H} = &h_0 + \frac{c_0}{2}n^2 + \frac{c_2}{2}n^2|{\langle\mathbf{\hat{F}}\rangle}|^2 + \frac{c_4}{2}n^2|A_{00}|^2\\ &+ \frac{c_6}{2}n^2\sum_{j=-2}^{+2}|A_{2j}|^2 + \frac{{\ensuremath{c_\mathrm{d}}}}{2} \int D({\ensuremath{\mathbf{r}}},{\ensuremath{\mathbf{r}^\prime}})\,d^3r^\prime, \end{split}$$ where two additional interaction terms, compared with Eq. , appear as a result of the $s$-wave scattering of spin-3 atoms. These depend on the amplitudes $$\label{eq:A00} A_{00} = \frac{1}{\sqrt{7}} \left(2\zeta_{+3}\zeta_{-3} - 2\zeta_{+2}\zeta_{-2} + 2\zeta_{+1}\zeta_{-1} - \zeta_0^2\right)$$ and $$\label{eq:A2j} \begin{split} &A_{20} = \frac{1}{\sqrt{7}} \left(\frac{5}{\sqrt{3}}\zeta_{+3}\zeta_{-3} - \sqrt{3}\zeta_{+1}\zeta_{-1} + \sqrt{\frac{2}{3}}\zeta_0^2\right),\\ &A_{2\pm1} = \frac{1}{\sqrt{7}} \left(\frac{5}{\sqrt{3}}\zeta_{\pm3}\zeta_{\mp2} - \sqrt{5}\zeta_{\pm2}\zeta_{\mp1} + \sqrt{\frac{2}{3}}\zeta_{\pm1}\zeta_0\right),\\ &A_{2\pm2} = \frac{1}{\sqrt{7}} \left(\frac{10}{\sqrt{3}}\zeta_{\pm3}\zeta_{\mp1} - \sqrt{\frac{20}{3}}\zeta_{\pm2}\zeta_{0} + \sqrt{2}\zeta_{\pm1}^2\right), \end{split}$$ respectively. The interaction strengths $c_{0,2,4,6}$ are found from the scattering lengths of the four spin channels of colliding spin-3 atoms as $c_0=4\pi\hbar^2(9a_4+2a_6)/(11M)+c_0^{\mathrm{d}}$, $c_2=4\pi\hbar^2(a_6-a_4)/(11M)+c_2^{\mathrm{d}}$, $c_4=4\pi\hbar^2(11a_0-21a_4+10a_6)/(11M)+c_4^{\mathrm{d}}$, and $c_6=4\pi\hbar^2(11a_2-18a_4+7a_6)/(11M)+c_6^{\mathrm{d}}$, where the coupling constants may again be modified by contact part of the DI. The dipolar interaction is again given by Eq. , where, the spin operator $\mathbf{\hat{F}}$ is now the vector of $7\times7$ spin-3 Pauli matrices. From Eq.  seven coupled GPEs for the components of the spinor wave function may be derived. Using the spin-3 $\mathbf{\hat{F}}$ operator in Eqs.  and yields the DI contribution, which can then be treated analogously to the spin-1 case. The spin-3 BEC exhibits a complex family of phases exhibiting different symmetries [@kawaguchi_pra_2011]. Here we concentrate on $^{52}$Cr, where current measurements of the scattering lengths [@werner_prl_2005; @pasquiou_pra_2010; @de-paz_pra_2014] predict an $A$-phase ground state with ${|\langle\mathbf{\hat{F}}\rangle|}=0$ in a uniform system. Nevertheless, the DI may influence the structure of singular vortices as they develop superfluid cores with nonzero spin. Results {#sec:results} ======= We now employ the mean-field theory outlined in Section \[sec:mft\] to study the internal core structure of vortices. We find the vortex solutions by solving the coupled GPEs derived from Eq. \[eq:hamiltonian-density\] in the frame rotating with frequency $\Omega$ around the $z$ axis: $\mathcal{H} \to \mathcal{H} - \Omega\langle\hat{L}_z\rangle$, where $\hat{L}_z$ is the $z$ component of the angular-momentum operator. This is done using a successive overrelaxation method [@numerical-recipes] to find stationary solutions in the spin-1 model, while we have used imaginary-time propagation for particular results and to solve the spin-3 GPEs for $^{52}$Cr. We first find our main results considering a spin-1 BEC. While higher-spin atoms are necessary to reach large magnetic dipole moment, the spin-1 system provides a useful model where the physics arising from the DI can be illustrated and compared with known results in a non-dipolar condensate. Since the DI is given by Eq.  regardless of the atomic spin and its effects arise from generic properties of the interaction and the spinor condensates, the physical principles illustrated by the results can be expected to apply more broadly also in higher-spin systems. Dipolar BECs can also be realized with weak dipolar interactions, provided that the other nonlinearities are even weaker (see aslo Appendix \[app:interaction-strengths\]). We keep the spin-independent interaction strength fixed at $Nc_0=10^4\hbar\omega\ell^3$, where $\ell$ is the oscillator length $\ell=\sqrt{\hbar/(M\omega)}$ of the harmonic trap and $N$ is the number of atoms in the condensate. We allow $c_2$ to vary around $c_0/c_2\simeq-216$, which corresponds to $^{87}$Rb, the most commonly used atom with FM interactions in spin-1 experiments. We then study how the vortex structure varies with ${\ensuremath{c_\mathrm{d}}}$. We further briefly consider a polar BEC with $c_0/c_2\simeq28$, corresponding to $^{23}$Na. We then find the stable core structure of singular half-quantum vortices in a spin-3 $^{52}$Cr BEC with with and without the corresponding DI, and analyze these in light of the spin-1 model. Weak magnetic field ------------------- ### Spin vortex For our spin-1 model, we first consider a condensate in the FM interaction regime, $c_2<0$, such that in a uniform system ${|\langle\mathbf{\hat{F}}\rangle|}$ is maximized. In this case, different FM spinors are related by three-dimensional rotations of the orthonormal triad formed by ${\langle\mathbf{\hat{F}}\rangle}$ and two vectors perpendicular to it. The order-parameter space is therefore ${\mathrm{SO}}(3)$, which supports only two topologically distinct classes of vortices [@kawaguchi_physrep_2012]: nonsingular coreless vortices and singly quantized singular vortices. Here we first consider the singular vortex, whose core structure in the absence of DI we studied in detail in Refs. [@lovegrove_pra_2012; @lovegrove_pra_2016]. In a nondipolar condensate, the stabilization of a vortex line usually requires a sufficiently rapid external rotation. While the ground state in a rotating FM spin-1 condensate is generally predicted to be made up of coreless vortices [@mizushima_prl_2002; @martikainen_pra_2002], a singular FM vortex can also be energetically (meta-)stable for a range of trap-rotation frequencies [@lovegrove_pra_2012], and is predicted to form the ground state when the coreless vortex is destabilized by conservation of a weak magnetization [@lovegrove_prl_2014; @lovegrove_pra_2016]. Even though spin vortices that carry no mass circulation can form in a spinor BEC, one would not generally expect them to be energetically stable. In a FM spin-1 BEC, a spin vortex can be stabilized by magnetic fields, e.g., in a Ioffe-Pritchard trap [@bulgakov_prl_2003], and a BEC with FM interactions initially in a polar state can be dynamically unstable towards spin-vortex formation [@saito_prl_2006]. In the FM phase, mass circulation alone is not quantized and a spin vortex can continuously pick up angular momentum through local spin rotations to stabilize it in a rotating trap [@mizushima_pra_2002; @lovegrove_pra_2012]. When a magnetic DI is present, however, the situation changes due to the anisotropy of the interaction, which strives to arrange the dipoles in a head-to-tail configuration that minimizes the interaction energy. Beyond a critical ${\ensuremath{c_\mathrm{d}}}$, it then becomes possible for a singular spin vortex carrying no circulation to form in the ground state even in a nonrotating condensate [@yi_prl_2006; @kawaguchi_prl_2006b; @takahashi_prl_2007]. The structure of the stable spin vortex is shown in Fig. \[fig:spin-vortex-core\]. ![Top left: Spin structure of a spin vortex in a nonrotating condensate. Surface plot indicates ${|\langle\mathbf{\hat{F}}\rangle|}$, showing the vortex core. Cones indicate ${\langle\mathbf{\hat{F}}\rangle}$, exhibiting a tangential disgyration that minimizes the DI energy. Bottom left: Size of the vortex core ($\bullet$) compared with $\xi_F$ ($+$) and ${\ensuremath{\xi_\mathrm{d}}}^\prime$ ($\times$) as functions of ${\ensuremath{c_\mathrm{d}}}$ (see also Appendix \[app:interaction-strengths\]). Subpanels from top to bottom: $Nc_2 = -463\hbar\omega\ell^3$, $-46.3\hbar\omega\ell^3$, and $-4.63\hbar\omega\ell^3$ ($Nc_0 = 10^4\hbar\omega\ell^3$). The spin vortex is stable above a critical ${\ensuremath{c_\mathrm{d}}}$ that depends on $c_2$. The core size is then well predicted by the smaller of the two healing lengths. Right: Surface plot of $F={|\langle\mathbf{\hat{F}}\rangle|}$ (color scale) showing the different sizes of the vortex core for $N{\ensuremath{c_\mathrm{d}}}= 16\hbar\omega\ell^3$ (top) and $N{\ensuremath{c_\mathrm{d}}}= 100\hbar\omega\ell^3$ (bottom) for $Nc_2=-46.3\hbar\omega\ell^3$. []{data-label="fig:spin-vortex-core"}](singular-core-size-cb-a.eps){width="\columnwidth"} The tangential disgyration exhibited by the condensate spin is a consequence of the DI. The singular FM vortex in the spin-1 BEC can exhibit a wide range of associated spin textures that can be transformed into each other through local and continuous transformations. In addition to the tangential disgyration shown in Fig. \[fig:spin-vortex-core\], radial and cross disgyrations are possible, as well as asymptotically uniform spin textures. In the absence of DI, these different spin structures are energetically (near) degenerate [@lovegrove_pra_2012]. Here, this degeneracy is broken by the directional dependence of the DI. The tangential disgyration corresponds to the greatest head-to-tail alignment of the spins, and therefore minimizes the DI energy, energetically locking in the spin texture. The vortex can then be described by the spinor wave function $$\label{eq:spin-vortex} \zeta = \frac{i}{\sqrt{2}} {\left(\begin{array}{c}-\sqrt{2}e^{-i\phi}\cos^2\frac{\beta}{2}\\#2\\#3\end{array}\right)},$$ where $\phi$ is the azimuthal angel and $F_z=\cos\beta$. For $\beta=\pi/2$, such that ${\langle\mathbf{\hat{F}}\rangle}$ lies in the $xy$ plane as in Fig. \[fig:spin-vortex-core\], the vortex is a pure spin vortex that carries no angular momentum. In a scalar BEC, the superfluid density vanishes on the line singularity of the order parameter that constitutes a vortex line. In the multicomponent order parameter of a spinor BEC, by contrast, a vortex-line singularity can also be accommodated by exciting the wave function out of its ground-state manifold. In a spin-1 BEC, the resulting filled vortex core becomes energetically favorable when $c_2$ is small compared with $c_0$, which is usually the case. This can be understood from the healing lengths arising from the contact-interaction terms. These are the density and spin healing lengths $\xi_n=\hbar/(2Mc_0n)^{1/2}$ and $\xi_F=\hbar/(2M|c_2|n)^{1/2}$ that describe the characteristic length scales of deviations from the corresponding ground-state condition. By breaking the spin condition instead of depleting the density, the defect core can expand to the larger healing length and lower its energy [@ruostekoski_prl_2003; @lovegrove_pra_2012]. A singular FM vortex then develops a superfluid core exhibiting the polar phase on the line singularity. When the vortex is represented by Eq. , this corresponds to the $\zeta_0$ component occupying the singular lines in $\zeta_\pm$. In the dipolar spinor BEC considered here, an additional interaction term appears in the Hamiltonian density, Eq. . Unlike the interaction terms arising from the $s$-wave scattering, the DI term is nonlocal. However, it is still possible to associate with it a length scale $$\label{eq:xidd} {\ensuremath{\xi_\mathrm{d}}}= \frac{\hbar}{\sqrt{2M|{\ensuremath{c_\mathrm{d}}}|n}}.$$ It was shown in Ref. [@kawaguchi_prl_2006b] that dipole-induced spin textures such as the ground-state spin vortex can form when the extent of the condensate exceeds ${\ensuremath{\xi_\mathrm{d}}}$. Here we show that the dipolar healing length also interacts nontrivially with the other characteristic length scales to affect the vortex-core structure in the dipolar spinor condensate. Specifically, we find that the dipolar healing length becomes important for the structure of a singular-vortex core when it is *shorter* than the spin healing length: ${\ensuremath{\xi_\mathrm{d}}}\lesssim\xi_F$. This is contrary to the nondipolar case, where the core structure of a singular vortex is determined by the largest healing length [@lovegrove_pra_2012]. The healing lengths in that case are associated with different and independent ground state conditions (superfluid density and spin magnitude). In the case of magnetic DI, however, $\xi_F$ and ${\ensuremath{\xi_\mathrm{d}}}$ both relate to the condensate spin. In particular, when $c_2<0$ the contact interaction and the DI both energetically favor ${|\langle\mathbf{\hat{F}}\rangle|}=1$. Any perturbation of ${|\langle\mathbf{\hat{F}}\rangle|}$ must then heal back to the bulk value over the shortest of the spin-dependent healing lengths. Consequently, the size of the core becomes dependent on ${\ensuremath{c_\mathrm{d}}}$ when ${\ensuremath{\xi_\mathrm{d}}}$ becomes comparable to $\xi_F$ as illustrated in Fig. \[fig:spin-vortex-core\]. Unlike the contact interaction, the effective strength of the DI depends on the relative orientation of the dipoles. In a head-to-tail arrangement, the effective strength of the interaction is $-2{\ensuremath{c_\mathrm{d}}}$ \[cf.Eq.  for ${\mathbf{F}}({\ensuremath{\mathbf{r}^\prime}})={\mathbf{F}}({\ensuremath{\mathbf{r}}})$\]. In the context of the spin vortex shown in Fig. \[fig:spin-vortex-core\], $\xi_F$ should therefore be compared with ${\ensuremath{\xi_\mathrm{d}}}^\prime \equiv {\ensuremath{\xi_\mathrm{d}}}/\sqrt{2}$. In the bottom left panels of Fig. \[fig:spin-vortex-core\], we plot both $\xi_F$ and ${\ensuremath{\xi_\mathrm{d}}}^\prime$, together with the vortex core size (defined as the diameter of the core at ${|\langle\mathbf{\hat{F}}\rangle|}=1-e^{-1}$), as functions of ${\ensuremath{c_\mathrm{d}}}$. \[For simplicity we here treat ${\ensuremath{c_\mathrm{d}}}$ as a free parameter within the spin-1 model system. Appendix \[app:interaction-strengths\] outlines how the dimensionless nonlinearity $N{\ensuremath{c_\mathrm{d}}}/(\hbar\omega\ell^3)$ can be varied also for fixed ${\ensuremath{c_\mathrm{d}}}$, corresponding to some particular magnetic dipole moment.\] The middle subpanel corresponds to $Nc_2=-46.3\hbar\omega\ell^3$ (corresponding to $^{87}$Rb, whose physical dipole moment also gives $N{\ensuremath{c_\mathrm{d}}}\simeq4.2\hbar\omega\ell^3$, for comparison), while in the top and bottom panels, $c_2$ is one order of magnitude stronger and weaker, respectively. For the strong $c_2$, $\xi_F<{\ensuremath{\xi_\mathrm{d}}}^\prime$ over the range of the plot, and the core size remains nearly unaffected by ${\ensuremath{c_\mathrm{d}}}$ and is well predicted by $\xi_F$ (except at the very onset of stability). Conversely, for the weak $c_2$, $\xi_F>{\ensuremath{\xi_\mathrm{d}}}^\prime$, with the latter quantity corresponding well to the core size. In the middle panel, $\xi_F \simeq {\ensuremath{\xi_\mathrm{d}}}^\prime$ and the two cross as ${\ensuremath{c_\mathrm{d}}}$ increases. For large ${\ensuremath{c_\mathrm{d}}}$ the core size follows ${\ensuremath{\xi_\mathrm{d}}}^\prime$, while at small ${\ensuremath{c_\mathrm{d}}}$ the influence of the now smaller $\xi_F$ becomes evident. These fairly simple principles then characterize the behavior of a singular-defect core as DI is varied. For a nonrotating cloud in a 3D isotropic trap, there is no preferred direction for the spin-vortex line (in the absence of Zeeman shifts). In Fig. \[fig:spin-vortex-core\], the vortex line coincides with the $z$ axis, while the left panel of Fig. \[fig:perp-spin-vortex\] shows a spin vortex line in the $xy$ plane. Considering now a slowly rotating trap, the axis of rotation represents a preferred spatial direction. Vortices stabilized by rotation would then form parallel to the rotation axis, as in the nondipolar case [@lovegrove_pra_2012]. In a highly oblate dipolar spinor condensate, the spin vortex with axial symmetry around the $z$ direction persists up to a critical rotation frequency [@simula_jpsj_2011]. For our isotropic trap, however, we find that the ground state in a slowly rotating condensate exhibits a spin vortex forming perpendicularly to the rotation axis when ${\ensuremath{c_\mathrm{d}}}$ is sufficiently large (Fig. \[fig:perp-spin-vortex\]). While a solution with a spin vortex parallel to the rotation also exists, this has a higher energy. The orientation of the vortex line shows that similarly to the nonrotating case, the vortex forms due to minimization of the DI energy, rather than because of the rotation. The effect of the rotation is instead to increasingly bend the vortex line, as illustrated in Fig. \[fig:perp-spin-vortex\]. ![Spin vortex in nonrotating (left) and slowly rotating, $\Omega=0.10\omega$ (right), trap. Color map shows $F={|\langle\mathbf{\hat{F}}\rangle|}$ highlighting the polar vortex core, while cones show the spin vector in the $y=0$ plane. In the rotating system, the vortex line forms perpendicular to the rotation axis. $Nc_0=10^4\hbar\omega\ell^3$, $Nc_2=-46.3\hbar\omega\ell^3$, $N{\ensuremath{c_\mathrm{d}}}=50\hbar\omega\ell^3$. []{data-label="fig:perp-spin-vortex"}](horizontal-spinvortex-ax-a.eps){width="\columnwidth"} ### Coreless vortex When the condensate rotates sufficiently rapidly, we find that the ground state is a coreless vortex along the rotation axis. This was also found to be the case in the highly oblate trap in Ref. [@simula_jpsj_2011]. Coreless vortices are also predicted to make up the ground state in a rotating nondipolar FM condensate [@mizushima_prl_2002; @martikainen_pra_2002], unless destabilized through conservation of a sufficiently weak magnetization [@lovegrove_prl_2014]. The coreless vortex is characterized by a nonsingular spin texture in which the superfluid circulation varies continuously as the spin bends from ${\langle\mathbf{\hat{F}}\rangle}={\ensuremath{\mathbf{\hat{z}}}}$ on the vortex line toward its asymptotic direction. The boundary condition on the spin texture away from the vortex line is however not fixed, allowing it to adapt to the imposed rotation, bending more sharply towards the $-{\ensuremath{\mathbf{\hat{z}}}}$ direction as rotation increases. The DI introduces a competing mechanism that strives to align the spins in the head-to-tail configuration of a tangential disgyration similar to the spin vortex. This leads to the formation of a coreless vortex with the spin texture shown in Fig. \[fig:cl\], where the spin vector bends gradually into the tangential disgyration. Far from the vortex line, the DI thus determines the spin texture by the same mechanism as for the spin vortex. Note, however, that while the two vortices appear superficially similar, both exhibiting tangential disgyrations of the spin vector at large length scales, their topology is distinctly different. This is easily established from the complex phases of the individual spinor components, which exhibit $(2\pi,0,-2\pi)$ winding in the singular spin vortex, but wind by $(0,2\pi,4\pi)$ in the coreless vortex. In the latter case, the vortex can be removed through purely local spin rotations, provided that the value of the spin is free to rotate at the edge of the cloud. The structure of the coreless vortex can be understood as the result of competition between rotation and DI. While the spin texture strives to adapt to the imposed rotation asymptotically forming an angle with the $z$ axis that depends on the rotation frequency, the DI strives to align the asymptotic texture in the $xy$ plane. This competition is reflected in the total angular momentum $L$ carried by the vortex, which becomes dependent on ${\ensuremath{c_\mathrm{d}}}$ at fixed trap rotation. When the trap rotates slowly, the rotation alone is not enough to bring the asymptotic texture into the $xy$ plane. Increasing ${\ensuremath{c_\mathrm{d}}}$ will then result in a more sharply bending texture that carries additional angular momentum, such that $L$ increases as a function of ${\ensuremath{c_\mathrm{d}}}$. On the other hand, a rapid rotation causes the asymptotic spin texture to acquire a negative $F_z$ component in order to provide sufficient circulation. Increasing ${\ensuremath{c_\mathrm{d}}}$ then has the opposite effect, causing the spin to bend less sharply in order to bring it more in line with the $xy$ plane. This causes $L$ to decrease with with DI strength. Consequently, as the DI becomes more dominant with increasing ${\ensuremath{c_\mathrm{d}}}$, $L$ becomes less sensitive to the trap rotation frequency. ![Top left: Spin texture of the coreless vortex for $\Omega=0.175\omega$ and $N{\ensuremath{c_\mathrm{d}}}=10\hbar\omega\ell^3$. Cones show the spin vector in the $x,y$ plane, perpendicular to the vortex line, with the color scale indicating the $z$ component. The DI causes the condensate spin to bend toward a tangential disgyration away from the vortex line. Top right: Total angular momentum carried by the coreless vortex as a function of ${\ensuremath{c_\mathrm{d}}}$. The lines from bottom to top correspond to trap rotation $\Omega=0.14\omega$ through $\Omega=0.20\omega$ in steps of $0.01\omega$. The varying dependence on ${\ensuremath{c_\mathrm{d}}}$ and the convergent behavior at large values is the result of the spin texture simultaneously adapting to DI and imposed rotation. Bottom: Angular-momentum density distribution for $\Omega=0.18\omega$ and $N{\ensuremath{c_\mathrm{d}}}=10\hbar\omega\ell^3$ (left) and $100\hbar\omega\ell^3$ (right). The panels show the same $13\ell$ by $13\ell$ cut-out and use the same color scale. In all panels $Nc_0=10^4\hbar\omega\ell^3$, $Nc_2=-46.3\hbar\omega\ell^3$[]{data-label="fig:cl"}](coreless-a.eps){width="\columnwidth"} ### Polar condensate {#sec:polar} The DI couples to the condensate spin through Eq. . So far we have explored how this leads to consequences for the formation, stability and structure of in the FM phase, where ${|\langle\mathbf{\hat{F}}\rangle|}$ is maximized in the bulk condensate. In the polar phase of the spin-1 BEC, ${|\langle\mathbf{\hat{F}}\rangle|}=0$ in a uniform system. However, when a singular vortex is present, nonzero ${|\langle\mathbf{\hat{F}}\rangle|}$ can appear in the defect core, and DI can still affect its structure. The physics, however, exhibits differences from the FM case, where the contact and dipolar interactions both strive to maximize the condensate spin. Here, by contrast, the interactions compete, with the contact interaction favoring ${|\langle\mathbf{\hat{F}}\rangle|}=0$. The size of a superfluid core with nonzero spin is then not limited by the dipolar interaction, which now favors breaking of the ground-state spin condition. One may then expect the presence of the DI to lead to an enlarged core, and our numerical simulations confirm these simple principles. We explore this by considering a stable singular half-quantum vortex in a rotating system. We keep $c_0$ the same as in the FM examples, but now take $c_0/c_2\simeq28$ corresponding to $^{23}$Na. Energy relaxation causes the vortex to develop a superfluid core that breaks the ground-state spin condition, reaching the FM phase on the singular line [@lovegrove_pra_2016]. In the absence of DI the size of the vortex core is determined by the spin healing length $\xi_F$. We find that as the strength of the DI increases from ${\ensuremath{c_\mathrm{d}}}= 0$ to $c_0/{\ensuremath{c_\mathrm{d}}}= 200$, the size of the vortex core increases by $\sim35\%$ for $Nc_0=10^4\hbar\omega\ell^3$ and rotation frequency in the range $0.12\omega \leq \Omega \leq 0.17\omega$. Precession-averaged dipolar interaction in a magnetic field ----------------------------------------------------------- In experiment, the condensate may be subject to residual or deliberately imposed external magnetic fields. In the presence of the magnetic field, the condensate spin exhibits precession around the field direction at the Larmor frequency $\omega_{\mathrm{L}}$. If the field is sufficiently strong, the Larmor precession will be rapid compared with the spin dynamics resulting from the DI. On the latter time scale, the condensate then experiences an effective DI that corresponds to the averaging of the bare DI over the period of the Larmor precession. The resulting reduced DI is given by Eq. . This removes some of the anisotropy of the bare DI and therefore leads to degeneracy between some spin configurations that would otherwise have different energies. Here we show that this can have a profound effect on the spin structure of singular vortices. Figure \[fig:larmor-singular\] shows the vortex core and spin texture of a stable singular vortex in a rotating system. For sufficiently small ${\ensuremath{c_\mathrm{d}}}$, the vortex is axially symmetric, exhibiting a radial disgyration in the $xy$ components of the spin vector. Increasing ${\ensuremath{c_\mathrm{d}}}$ results in a deformation of the vortex core, breaking the axial symmetry. The spins rotate toward a configuration where the asymptotic spin texture exhibits a nearly uniform projection onto the $xy$ plane, while $F_z$ bends across the condensate. At the same time, the polar core of the vortex deforms to exhibit an elliptical cross section whose eccentricity increases with ${\ensuremath{c_\mathrm{d}}}$. Eventually the deformation of the vortex core becomes large enough that its extent along the major axis of the ellipse reaches the condensate size. As shown in Fig. \[fig:larmor-singular\] the condensate the exhibits a polar domain wall separating two halves of the cloud with oppositely aligned spin ${\langle\mathbf{\hat{F}}\rangle}=\pm{\ensuremath{\mathbf{\hat{z}}}}$. In the isotropic spinor condensate, there is no analog of this deformation when the DI cannot be averaged over the Larmor precession period. However, a similar anisotropic deformation has been predicted in a two-dimensional, scalar BEC with fixed dipole moments [@mulkerin_prl_2013] ![Structure of the stable singular-vortex core ($z=0$ cross section, perpendicular to the vortex line) as the effective DI averaged over the Larmor precession increases. Color map shows $F={|\langle\mathbf{\hat{F}}\rangle|}$, while cones show the spin vector ${\langle\mathbf{\hat{F}}\rangle}$ (color indicates $F_z$). From top left to bottom right $N{\ensuremath{c_\mathrm{d}}}= 0, 6, 8, 10, 50$ and $100\hbar\omega\ell^3$. The core deforms by breaking axial symmetry to form an ellipsoidal cross section. Eventually the core covers the diameter of the condensate and forms a domain wall between regions of $F_z=\pm1$. In all panels $Nc_0=10^4\hbar\omega\ell^3$, $Nc_2=-46.3\hbar\omega\ell^3$, and $\Omega=0.13\omega$. []{data-label="fig:larmor-singular"}](larmor-singular-a.eps){width="\columnwidth"} Singular vortex in a spin-3 $^{52}$Cr BEC {#sec:spin-3-results} ----------------------------------------- The spin-1 BEC provides a good model system for theoretically exploring the physical principles of DI effects on vortices in spinor BECs. However, a stronger dipole moment of $6{\ensuremath{\mu_\mathrm{B}}}$ is found in $^{52}$Cr, which can be used to create a spin-3 condensate [@diener_prl_2006; @santos_prl_2006]. Measurements of the $s$-wave scattering lengths [@werner_prl_2005; @pasquiou_pra_2010; @de-paz_pra_2014] yield $c_0/c_2 \simeq 20$, $c_0/c_4 \simeq -4.6$, $c_0/c_6 \simeq -1.5$ for the contact-interaction strengths in Eq. , while $c_0/{\ensuremath{c_\mathrm{d}}}\simeq 177$. The ground-state determined by the $s$-wave interaction is then the so-called $A$-phase in a uniform system [@kawaguchi_pra_2011], with a representative order-parameter $\zeta=(1/\sqrt{2},0,0,0,0,0,1/\sqrt{2})^T.$ This exhibits ${|\langle\mathbf{\hat{F}}\rangle|}=0$ and the order parameter has the discrete hexagonal symmetry of the dihedral-6 group $D_6$ [@yip_pra_2007; @barnett_pra_2007; @kawaguchi_pra_2011], illustrated in Fig. \[fig:spin3\] using the spherical-harmonics representation $$\label{eq:spherical-harmonics} Z(\theta,\phi)=\sum_{m=-3}^{+3}Y_{3,m}(\theta,\phi)\zeta_m.$$ In a biaxial-nematic spin-2 BEC, the related but simpler dihedral-4 point-group symmetry already leads to highly complex core structures of a half-quantum vortex [@borgh_prl_2016]. As a result of the $D_6$ symmetry, the spin-3 $A$-phase vortices are also non-Abelian (i.e., the different topological charges do not all commute), leading to the restricted reconnection dynamics of vortices also predicted in the cyclic and biaxial-nematic spin-2 phases [@kobayashi_prl_2009; @borgh_prl_2016]. The $A$-phase $D_6$ order parameter supports a half-quantum vortex where the $\pi$ winding of the condensate phase is compensated by a $\pi/3$ spin rotation. This is the simplest vortex that carries angular momentum and could therefore be stabilized by rotation. Figure \[fig:spin3\] shows the relaxed core structure of one out of a pair of such half-quantum vortices in a condensate with and without the DI corresponding to $^{52}$Cr, for the case where any external magnetic field is negligible (i.e., the spin precession is not assumed to be rapid). We find that energy relaxation leads to the condensate approaching the $H$-phase [@kawaguchi_pra_2011] on the singular line. A representative $H$-phase order parameter can be written $\zeta=(\sqrt{(2+F)/5},0,0,0,0,\sqrt{(3-F)/5},0)^T$, where $F={|\langle\mathbf{\hat{F}}\rangle|}$. This phase exhibits a five-fold rotational symmetry, shown in Fig. \[fig:spin3\]. The bottom left panel also uses the spherical-harmonics representation, Eq. , to illustrate the change of the order parameter form the bulk $A$-phase to the vortex core. The $H$-phase further exhibits a parameter-dependent condensate spin magnitude that is determined by energy relaxation. Here we find ${|\langle\mathbf{\hat{F}}\rangle|}>0$ in the vortex core, which therefore breaks the ground-state spin condition. The effects of DI are then similar to the polar half-quantum vortex with FM core in the spin-1 model (section \[sec:polar\]), where increasing DI leads to an increase in core size since the DI favors the nonzero spin. In the spin-3 vortex, however, ${|\langle\mathbf{\hat{F}}\rangle|}$ is not restricted to a particular value on the vortex line, but is determined by energy relaxation and depends on the interaction parameters. Comparing the stable vortex cores, we find a slightly increased spin magnitude in the presence of DI, illustrating the general principle that was also demonstrated for the spin-1 dipolar BEC. ![Top: Core structure of a singular half-quantum vortex in the $A$-phase of a spin-3 BEC in the absence (left) and presence (right) of DI corresponding to $^{52}$Cr. Color map shows $F={|\langle\mathbf{\hat{F}}\rangle|}$, while cones show the spin vector ${\langle\mathbf{\hat{F}}\rangle}$ (color indicates $F_z$). The panels show a $2.4\ell\times2.4\ell$ region around the vortex line, which is stable together with a second half-quantum vortex (not shown) in a system rotating with frequency $\Omega=0.43\omega$. The contact interaction corresponds to $^{52}$Cr with $Nc_0=10^3\hbar\omega\ell^3$. Bottom left: Change of the order parameter symmetry showing the transition from the $A$-phase bulk towards the $H$-phase in the core when DI is present (background color map shows $F$ for reference). Bottom right: Spherical-harmonics representations of the $A$- and $H$-phase order parameters for reference. []{data-label="fig:spin3"}](spin3-4p-a.eps){width="\columnwidth"} However, the presence of the DI is also reflected in the internal spin texture of the vortex core, which is reminiscent of the coreless vortex in Fig. \[fig:cl\]. By the same mechanism, the anisotropy of the DI here leads to the formation of a spin texture across the vortex core that approaches a tangential disgyration as shown in Fig. \[fig:spin3\]. In a condensate where $\xi_F$, and therefore the vortex core, is not small on the scale of experimental resolution, this effect could be observed in experiment. Conclusions {#sec:conlusions} =========== We have demonstrated how DI can have several pronounced effects on the internal structure of vortices in spinor BECs, and determined and analyzed the stable core structure of singular vortices in a spin-3 $^{52}$Cr condensate. In addition to exhibiting relatively strong dipolar interactions, the $^{52}$Cr interaction parameters predict a ground-state order parameter exhibiting a hexagonal point-group symmetry that makes it a candidate for experimental observation of non-Abelian vortices. We used a spin-1 model system to analyze the underlying physical principles and established simple criteria that determine the defect core structure in the presence of DI. We have shown how a new characteristic length scale arising from the DI adds to the hierarchy of healing lengths to restrict the size of singular-vortex cores when DI and $s$-wave scattering both favor the ground-state spin condition (e.g., in the spin-1 FM phase), but can lead to core enlargement when they compete, as in the spin-1 polar phase. These dipolar effects arise from generic properties of the interaction and the spinor system and our results can therefore be expected to apply generally across condensates of atoms with different atomic spin. We have also shown how the spin ordering induced by the anisotropy of the DI has several different manifestations in both singular and nonsingular vortices. These include a nontrivial interaction dependence of the angular momentum carried by a coreless vortex, arising as a result of competition between dipolar spin ordering and rotation, as well as the deformation of a singular vortex when the DI is modified by a sufficiently strong magnetic field. The spin ordering can also give rise to internal spin textures in the superfluid vortex cores with nonzero spin, which are potentially observable in non-Abelian vortices in $^{52}$Cr condensates. Similar studies of the effects of DI could be extended beyond vortices to other more complex defects and textures [@ruostekoski_prl_2003; @tiurev_pra_2016; @savage_prl_2003] in which case their symmetries and stability properties could be altered. We acknowledge financial support from the EPSRC. The numerical results were obtained using the Iridis 4 high-performance computing facility at the University of Southampton. We acknowledge discussions with T. P. Simula. Explicit derivation of Fourier Transforms {#app:ft} ========================================= In numerical computations, it is convenient to calculate DI contributions in Fourier space, where the convolution integrals arising from the long-range nature of the DI become a simple matter of multiplication of Fourier transforms. However, the $\delta$-function contribution to the magnetic dipole field and its absorption into the $s$-wave interaction introduces subtleties into the derivation of these Fourier transforms. Here we first carefully derive the Fourier transform of the magnetic dipole field, keeping track of all contributions, and show how it yields Eq.  after explicitly subtracting the contact-interaction part. We then show how the Fourier transform is modified by the introduction of a spherical long-range cut-off and also indicate how the corresponding derivation is modified when the DI is averaged over a rapid Larmor precession. The magnetic field ${\mathbf{B}}({\ensuremath{\mathbf{r}}})$ from a magnetic point dipole ${\mathbf{m}}$ at the origin is given by [@jackson] $$\label{eq:dipole-field} {\mathbf{B}}({\ensuremath{\mathbf{r}}}) = \frac{\mu_0}{4\pi r^3}[3{\ensuremath{\mathbf{\hat{r}}}}({\mathbf{m}}\cdot{\ensuremath{\mathbf{\hat{r}}}})-{\mathbf{m}}] + \frac{2\mu_0}{3}{\mathbf{m}}\delta(r),$$ where ${\ensuremath{\mathbf{\hat{r}}}}={\ensuremath{\mathbf{r}}}/r$. We can write Eq.  on tensor form as $$\label{eq:dipole-field-tensor} B_\alpha = \frac{\mu_0}{4\pi} \sum_\beta{\ensuremath{\mathsf{B}}}_{\alpha\beta}m_\beta,$$ where $$\label{eq:b-tensor} {\ensuremath{\mathsf{B}}}_{\alpha\beta}({\ensuremath{\mathbf{r}}}) = \frac{3\hat{r}_\alpha\hat{r}_\beta-\delta_{\alpha\beta}}{r^3} + \frac{8}{3}\delta_{\alpha\beta}\delta(r).$$ The $\delta$-function contribution to the field follows from $$\label{eq:dipole-integral} \int_\delta{\mathbf{B}}\,d^3r = \frac{2\mu_0}{3}{\mathbf{m}},$$ with the convention that the integral of the first term in Eq.  vanishes on any infinitesimal sphere $\delta$ surrounding the point dipole (integrating over angles first to make the integral converge). In writing the dipolar Gross-Pitaevskii Hamiltonian, Eq. , however, this contact-interaction contribution is absorbed by the $s$-wave interaction, yielding an effective field $$\label{eq:effective-field} {\mathbf{B}}^\prime({\ensuremath{\mathbf{r}}}) \equiv {\mathbf{B}}({\ensuremath{\mathbf{r}}})-\frac{2\mu_0}{3}{\mathbf{m}}\delta({\ensuremath{\mathbf{r}}})$$ that corresponds to the tensor $${\ensuremath{\mathsf{Q}}}_{\alpha\beta} \equiv {\ensuremath{\mathsf{B}}}_{\alpha\beta} - \frac{8}{3}\delta_{\alpha\beta}\delta(r) = \frac{3\hat{r}_\alpha\hat{r}_\beta-\delta_{\alpha\beta}}{r^3},$$ appearing in Eqs.  and and whose Fourier transform is needed in Eq. . In finding the Fourier transforms of ${\ensuremath{\mathsf{B}}}$ and ${\ensuremath{\mathsf{Q}}}$ we need to ensure that the $\delta$-function contribution and its subtraction are correctly accounted for. In order to compute the Fourier transform, it is convenient to rewrite the dipole field as $$\label{eq:dipole-rewritten} {\mathbf{B}}= \frac{\mu_0}{4\pi}({\mathbf{m}}\times{\boldsymbol{\nabla}})\times{\boldsymbol{\nabla}}\frac{1}{r}.$$ In the tensor notation, this corresponds to rewriting ${\ensuremath{\mathsf{B}}}$ as $$\label{eq:dipole-rewritten-tensor} {\ensuremath{\mathsf{B}}}_{\alpha\beta} = \sum_{\gamma\mu\nu} \epsilon_{\alpha\gamma\mu}\epsilon_{\gamma\beta\nu}\partial_\nu\partial_\mu \frac{1}{r},$$ where $\epsilon_{\alpha\beta\gamma}$ is the fully antisymmetric Levi-Civita tensor. It is straightforward to check that Eq.  gives the correct field away from the origin \[corresponding to the first term of Eq. \]. However, we now need to check that the condition  is satisfied. To do this, we first rewrite the integral of ${\mathbf{B}}$ as a surface integral by rearranging the cross products and using the divergence theorem: $$\begin{split} I &\equiv \int_\delta \frac{\mu_0}{4\pi}({\mathbf{m}}\times{\boldsymbol{\nabla}})\times{\boldsymbol{\nabla}}\frac{1}{r}\,d^3r\\ &= -\int_{\partial\delta} \frac{\mu_0}{4\pi}{\ensuremath{\mathbf{\hat{r}}}}\times({\mathbf{m}}\times{\boldsymbol{\nabla}})\frac{1}{r}\,dS. \end{split}$$ Then using ${\boldsymbol{\nabla}}(1/r)={\ensuremath{\mathbf{r}}}/r^3$ and writing $dS = r^2 d(\cos\theta)d\phi$ in spherical coordinates, the integral becomes $$\label{eq:dipole-rewritten-integral} \begin{split} I &= \frac{\mu_0}{4\pi} \int_{\partial\delta} {\ensuremath{\mathbf{\hat{r}}}}\times({\mathbf{m}}\times{\ensuremath{\mathbf{\hat{r}}}})\,d(\cos\theta)d\phi\\ &= \frac{\mu_0}{4\pi} \int_{\partial\delta} [-{\ensuremath{\mathbf{\hat{r}}}}({\mathbf{m}}\cdot{\ensuremath{\mathbf{\hat{r}}}})+{\mathbf{m}}]\,d(\cos\theta)d\phi,\\ &= \frac{2\mu_0}{3}{\mathbf{m}}, \end{split}$$ in agreement with Eq. . We have thus verified that Eqs.  and correctly yield the magnetic dipole field, Eq. , including the $\delta$-function contribution [^1] We can now proceed to find the Fourier transform of the full magnetic dipole field as $$\label{eq:dipole-transf-1} \begin{split} \tilde{{\mathbf{B}}}({\ensuremath{\mathbf{k}}}) &= \int e^{-i{\ensuremath{\mathbf{k}}}\cdot{\ensuremath{\mathbf{r}}}} \frac{\mu_0}{4\pi}({\mathbf{m}}\times{\boldsymbol{\nabla}})\times{\boldsymbol{\nabla}}\frac{1}{r} \, d^3r \\ &= -\frac{\mu_0}{k^2}({\mathbf{m}}\times{\ensuremath{\mathbf{k}}})\times{\ensuremath{\mathbf{k}}}, \end{split}$$ where we have first used the vector identity $$\label{eq:integral-identity} \int_V f(\mathbf{v}\times{\boldsymbol{\nabla}})\times{\boldsymbol{\nabla}}g \, d^3r = \int_V g(\mathbf{v}\times{\boldsymbol{\nabla}})\times{\boldsymbol{\nabla}}f \, d^3r,$$ and then find the remaining integral as $$\begin{split} \int \frac{e^{-i{\ensuremath{\mathbf{k}}}\cdot{\ensuremath{\mathbf{r}}}-\mu r}}{r}\,d^3r &= \frac{2\pi}{ik}\left(\frac{1}{-ik-\mu}-\frac{1}{ik-\mu}\right)\\ &\stackrel{\mu\to0}{\longrightarrow} \frac{4\pi}{k^2}, \end{split}$$ using the convergence factor $\mu$, to yield the right-hand side of Eq. . Rewriting Eq.  using vector identities we arrive at $$\label{eq:dipole-tranfs-2} \tilde{{\mathbf{B}}}({\ensuremath{\mathbf{k}}}) = -\mu_0\left[{\ensuremath{\mathbf{\hat{k}}}}({\mathbf{m}}\cdot{\ensuremath{\mathbf{\hat{k}}}})-{\mathbf{m}}\right].$$ From the tensor notation $\tilde{B}_\alpha({\ensuremath{\mathbf{k}}}) = [\mu_0/(4\pi)]\sum_{\beta}{\ensuremath{\mathsf{\tilde{B}}}}_{\alpha\beta}m_\beta$, it follows immediately that $$\label{eq:btens-transf} {\ensuremath{\mathsf{\tilde{B}}}}_{\alpha\beta}({\ensuremath{\mathbf{k}}}) = -4\pi(\hat{k}_\alpha\hat{k}_\beta-\delta_{\alpha\beta}).$$ When the contact part of the interaction is absorbed by the $s$-wave interaction, however, we need to consider instead the Fourier transform of Eq. , which is immediately found from linearity as $$\label{eq:dipole-tranfs-3} \tilde{{\mathbf{B}}}^\prime({\ensuremath{\mathbf{k}}}) = -\frac{\mu_0}{3}\left[3{\ensuremath{\mathbf{\hat{k}}}}({\mathbf{m}}\cdot{\ensuremath{\mathbf{\hat{k}}}}) - {\mathbf{m}}\right].$$ Writing this in tensor notation such that $\tilde{B}^\prime_\alpha({\ensuremath{\mathbf{k}}}) = [\mu_0/(4\pi)]\sum_{\beta}{\ensuremath{\mathsf{\tilde{Q}}}}_{\alpha\beta}m_\beta$ immediately yields Eq. . It is common in the literature (see, e.g., Refs. [@kawaguchi_physrep_2012; @ronen_pra_2006]) to arrive at Eq. , or its special case for aligned dipoles, by ignoring the $\delta$-function contribution in Eq. , considering only ${\mathbf{B}}^\prime$ and expressing ${\ensuremath{\mathsf{Q}}}$ in terms of spherical harmonics as $$\label{eq:q-tensor-ylm} {\ensuremath{\mathsf{Q}}}({\ensuremath{\mathbf{r}}}) = -\sqrt{\frac{6\pi}{5}}\frac{1}{r^3} {\left(\begin{array}{ccc}\sqrt{\frac{2}{3}}Y_{2,0}({\ensuremath{\mathbf{\hat{r}}}})-Y_{2,2}({\ensuremath{\mathbf{\hat{r}}}})-Y_{2,-2}({\ensuremath{\mathbf{\hat{r}}}})&iY_{2,2}({\ensuremath{\mathbf{\hat{r}}}})-iY_{2,-2}({\ensuremath{\mathbf{\hat{r}}}})&Y_{2,1}({\ensuremath{\mathbf{\hat{r}}}})-Y_{2,-1}({\ensuremath{\mathbf{\hat{r}}}})\\#4&\sqrt{\frac{2}{3}}Y_{2,0}({\ensuremath{\mathbf{\hat{r}}}})+Y_{2,2}({\ensuremath{\mathbf{\hat{r}}}})+Y_{2,-2}({\ensuremath{\mathbf{\hat{r}}}})&-iY_{2,1}({\ensuremath{\mathbf{\hat{r}}}})-iY_{2,-1}({\ensuremath{\mathbf{\hat{r}}}})\\#7&-iY_{2,1}({\ensuremath{\mathbf{\hat{r}}}})-iY_{2,-1}({\ensuremath{\mathbf{\hat{r}}}})&-2\sqrt{\frac{2}{3}}Y_{2,0}({\ensuremath{\mathbf{\hat{r}}}})\end{array}\right)}.$$ One then makes use of the expansion of a plane wave in terms of spherical harmonics to find $$\label{eq:ylm-transf} \int e^{-i{\ensuremath{\mathbf{k}}}\cdot{\ensuremath{\mathbf{r}}}}Y_{l,m}({\ensuremath{\mathbf{\hat{r}}}})\,d\Omega = 4\pi(-i)^lj_l(kr)Y_{l,m}({\ensuremath{\mathbf{\hat{k}}}}),$$ where $j_l$ is the spherical Bessel function of order $l$. The radial integral $$\label{eq:bessel-int} \int_0^\infty \frac{j_2(kr)}{r^3}\,r^2dr = \int_0^\infty u^2\left(\frac{1}{u}\frac{d}{du}\right)^2\frac{\sin u}{u} \frac{du}{u} = \frac{1}{3},$$ where $u=kr$, can then be combined with Eqs.  and to find Eq. . Note, however, that this derivation drops the contact term from the outset and the Fourier integral is made to converge only by integrating over angles first in Eq. . Nevertheless, having now established that the use of Eqs.  and does in fact give the correct result when the contact part of the DI is absorbed in the $s$-wave interaction, this provides a convenient way to include the long-range cut-off that is necessary in numerical computations, as this affects the Fourier integral only away from the origin. Then truncating the DI at a radius $R$, Eq.  becomes $$\label{eq:bessel-int-trunc} \int_0^R \frac{j_2(kr)}{r^3}\,r^2dr = \frac{1}{3} + \frac{kR\cos(kR) - \sin(kR)}{(kR)^3},$$ from which Eq.  follows immediately. This is the immediate generalization of the spherical cut-off found by Ronen et al [@ronen_pra_2006] for aligned dipoles in a scalar BEC to free dipoles. Finally we consider the effective DI arising when the interaction is averaged over the Larmor precession period in the presence of a sufficiently strong magnetic field. We can rewrite the corresponding tensor ${\ensuremath{\mathsf{Q}}}^{\mathrm{L}}$ given by Eq.  as $$\label{eq:q-tensor-l-y20} {\ensuremath{\mathsf{Q}}}_{\alpha\beta}^\mathrm{L}({\ensuremath{\mathbf{r}}}) = \frac{Y_{2,0}({\ensuremath{\mathbf{\hat{r}}}})}{r^3} \frac{3\delta_{z\alpha}\delta_{z\beta}-\delta_{\alpha\beta}}{2},$$ and proceed as above. Then, from Eqs.  and , Eq.  follows immediately. Relative strength of DI {#app:interaction-strengths} ======================= For our studies, we have employed a suitably simple spin-1 model system to illustrate the physics arising from DI. The DI coupling constant ${\ensuremath{c_\mathrm{d}}}$ is then regarded as a freely variable parameter. For physical atoms, however, the atomic dipole moment, and therefore ${\ensuremath{c_\mathrm{d}}}$, is a fixed quantity. In this Appendix we briefly outline how the dipolar nonlinearity (as given in Fig. \[fig:spin-vortex-core\]) can also be varied within the same spin-1 model by adjusting trap parameters and scattering lengths. The effective nonlinearities in the GPEs, expressed in dimensionless units as in Fig. \[fig:spin-vortex-core\], scale with the number of atoms $N$ in the condensate and the trap frequency $\omega$ as $\sim N\omega^{1/2}$. Therfore the DI nonlinearity $N{\ensuremath{c_\mathrm{d}}}/(\hbar\omega\ell^3)$, used in the figure, can be varied also for constant ${\ensuremath{c_\mathrm{d}}}$ by adjusting $N$ and/or $\omega$. We can illustrate this principle using $^{87}$Rb as a particular example: $Nc_0=10^4\hbar\omega\ell^3$, in Fig. \[fig:spin-vortex-core\] then corresponds to $N \simeq 5\times10^5$ atoms in an $\omega \simeq 2\pi \times 10$Hz trap, and the physical magnetic dipole moment gives $N{\ensuremath{c_\mathrm{d}}}\simeq 4.2\hbar\omega\ell^3$. By doubling the atom number to $N=10^6$ and increasing the trap frequency to $\omega \simeq 2\pi \times 60$Hz, we can reach $N{\ensuremath{c_\mathrm{d}}}\simeq 20\hbar\omega\ell^3$. However, adjusting the trap parameters also scales the contact-interaction nonlinearity $Nc_0/(\hbar\omega\ell^3)$. This can be prevented by simultaneously suppressing the contact-interaction coupling constant $c_0$. The suppression may be achieved using ac Stark shifts to access Feshbach resonances for the $s$-wave scattering lengths without freezing out the atomic spin [@papoular_pra_2010; @borgh_prl_2012]. In the $^{87}$Rb example, the required suppression is on the order of a factor $\sim 5$. Using these techniques, a strongly dipolar BEC could also be achieved using $^{85}$Rb where scattering lengths are tunable across orders of magnitude [@cornish_prl_2000]. [75]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/PhysRevLett.94.160401) [****,  ()](\doibase 10.1103/PhysRevLett.106.255303) [****,  ()](\doibase 10.1103/PhysRevA.87.051609) [****,  ()](\doibase 10.1103/PhysRevLett.108.210401) [****,  ()](\doibase 10.1103/PhysRevLett.107.190401) [****, ()](http://stacks.iop.org/1367-2630/17/i=4/a=045006) [****,  ()](\doibase 10.1103/PhysRevLett.112.010404) [****, ()](http://stacks.iop.org/0034-4885/72/i=12/a=126401) [****, ()](\doibase 10.1038/nature06036) [****, ()](\doibase 10.1038/nature16485) [****,  ()](\doibase 10.1103/PhysRevA.92.061603) [****,  ()](\doibase 10.1103/PhysRevA.93.011604) [****,  ()](\doibase 10.1103/PhysRevLett.117.185302) [****,  ()](\doibase 10.1103/PhysRevA.73.061602) [****,  ()](\doibase 10.1103/PhysRevA.75.013604) [****,  ()](\doibase 10.1103/PhysRevLett.100.245302) [****,  ()](\doibase 10.1103/PhysRevLett.100.240403) [****,  ()](\doibase 10.1103/PhysRevA.79.063622) [****,  ()](http://stacks.iop.org/1367-2630/11/i=5/a=055012) [****,  ()](\doibase 10.1016/j.physrep.2012.07.005) [****, ()](\doibase 10.1103/PhysRevLett.83.4677) [****,  ()](\doibase 10.1103/PhysRevA.66.053610) [****,  ()](\doibase 10.1103/PhysRevLett.89.030401) [****,  ()](\doibase 10.1103/PhysRevA.66.053604) [****,  ()](\doibase 10.1103/PhysRevLett.96.065302) [****,  ()](\doibase 10.1103/PhysRevLett.101.010402) [****,  ()](\doibase 10.1103/PhysRevA.86.013613) [****,  ()](\doibase 10.1103/PhysRevA.86.023612) [****,  ()](\doibase 10.1103/PhysRevLett.109.015302) [****,  ()](\doibase 10.1103/PhysRevLett.112.075301) [****,  ()](\doibase 10.1103/PhysRevA.93.033633) [****,  ()](\doibase 10.1103/PhysRevLett.117.275302) [****,  ()](\doibase 10.1103/PhysRevLett.90.140403) [****,  ()](\doibase 10.1103/PhysRevLett.103.250401) [****,  ()](\doibase 10.1103/PhysRevLett.108.035301) [****, ()](http://stacks.iop.org/1367-2630/14/i=5/a=053013) [****,  ()](\doibase 10.1038/nature12954) [****,  ()](\doibase 10.1126/science.1258289) [****,  ()](\doibase 10.1038/nphys3624) [****,  ()](\doibase 10.1103/PhysRevLett.115.015301) [****,  ()](\doibase 10.1103/PhysRevLett.98.110406) [****,  ()](\doibase 10.1103/PhysRevA.82.043627) [****,  ()](\doibase 10.1103/PhysRevA.82.053616) [****, ()](\doibase 10.1103/PhysRevLett.100.170403) [****,  ()](\doibase 10.1103/PhysRevLett.112.185301) [****, ()](\doibase 10.1103/PhysRevLett.96.190404) [****,  ()](\doibase 10.1103/PhysRevLett.96.080405) [****,  ()](\doibase 10.1103/PhysRevLett.106.140403) [****, ()](\doibase 10.1103/PhysRevA.83.063617) [****,  ()](\doibase 10.1103/PhysRevLett.97.020401) [****,  ()](\doibase 10.1103/PhysRevLett.97.130404) [****,  ()](\doibase 10.1103/PhysRevLett.98.260403) [****, ()](\doibase 10.1143/JPSJ.80.013001) [****,  ()](\doibase 10.1103/PhysRevLett.85.4462) [****,  ()](\doibase 10.1103/PhysRevA.81.041603) [****,  ()](\doibase 10.1103/PhysRevLett.91.190402) [****,  ()](\doibase 10.1103/PhysRevLett.93.040403) [****, ()](\doibase 10.1103/PhysRevA.74.013623) [****,  ()](\doibase 10.1103/PhysRevE.80.016703) [****, ()](\doibase 10.1103/PhysRevLett.96.190405) [****, ()](\doibase 10.1103/PhysRevA.84.053616) [****,  ()](\doibase 10.1103/PhysRevLett.94.183201) [****,  ()](\doibase 10.1103/PhysRevA.81.042716) [****,  ()](\doibase 10.1103/PhysRevA.90.043607) @noop [**]{},  ed. (, ) [****,  ()](\doibase 10.1103/PhysRevLett.90.200401) [****, ()](\doibase 10.1103/PhysRevLett.111.170402) [****, ()](\doibase 10.1103/PhysRevA.75.023625) [****,  ()](\doibase 10.1103/PhysRevA.76.013605) [****,  ()](\doibase 10.1103/PhysRevLett.103.115301) [****,  ()](\doibase 10.1103/PhysRevA.93.033638) [****,  ()](\doibase 10.1103/PhysRevLett.91.010403) @noop [**]{},  ed. (, ) [****,  ()](\doibase 10.1103/PhysRevLett.85.1795) [^1]: Note that while Eq.  is here automatically fulfilled when the magnetic dipole field is written on the form of Eq. , the same is not true for the corresponding equations in the case of an electric dipole. In that case, a $\delta$-function correction to the electric-field analogs of Eqs.  and is necessary.

--- bibliography: - 'FPCA\_Reference.bib' --- [Causal Mediation Analysis for Sparse and Irregular Longitudinal Data\ ]{} ABSTRACT Causal mediation analysis aims to investigate how the treatment effect of an exposure on outcomes is mediated through intermediate variables. Although many applications involve longitudinal data, the existing methods are not directly applicable to the settings where the mediator and outcome are measured on sparse and irregular time grids. We extend the existing causal mediation framework from a functional data analysis perspective, viewing the sparse and irregular longitudinal data as realizations of underlying smooth stochastic processes. We define causal estimands of direct and indirect effects accordingly and provide corresponding identification assumptions. For estimation and inference, we employ a functional principal component analysis approach for dimension reduction and use the first few functional principal components instead of the whole trajectories in the structural equation models. We adopt the Bayesian paradigm to accurately quantify the uncertainties. The operating characteristics of the proposed methods are examined via simulations. We apply the proposed methods to a longitudinal data set from a wild baboon population in Kenya to estimate the causal effects between early adversity, the strength of social bonds, and adult glucocorticoid hormone concentrations. We find that early adversity has a significant direct effect (a 9-14% increase) on females’ glucocorticoid concentrations across adulthood, but find little evidence that these effects were mediated by weak social bonds. [Key words]{}: Causal inference, functional principal component analysis, mediation, longitudinal data, sparse and irregular data Introduction\[Sec\_Intro\] ========================== Mediation analysis seeks to understand the role of an intermediate variable (i.e. mediator) $M$ that lies on the causal path between an exposure or treatment $Z$ and an outcome $Y$. The most widely used mediation analysis method, proposed by [@baron1986moderator], fits two linear structural equation models (SEMs) between the three variables and interprets the model coefficients as causal effects. There is a vast literature on the Baron-Kenny framework across a variety of disciplines, including psychology, sociology, and epidemiology ([see @mackinnon2012introduction]). A major advancement in recent years is the incorporation of the potential-outcome-based causal inference approach [@Neyman1923; @Rubin1974]. This led to formal definition of relevant causal estimands, clarification of identification assumptions and new estimation strategies beyond linear SEMs [@robins1992identifiability; @pearl2001direct; @sobel2008identification; @daniels2012bayesian; @tchetgen2012semiparametric; @vanderweele2016mediation]. In particular, [@imai2010identification] proved that the Baron-Kenny estimator can be interpreted as a special case of a causal mediation estimator given additional model assumptions. These methodological advancements also opened up new application areas including imaging, neuroscience and environmental health [@lindquist2011graphical; @lindquist2012functional; @zigler2012estimating; @kim2019bayesian]. Comprehensive reviews on causal mediation analysis are given in [@vanderweele2015explanation; @nguyen2019clarifying]. In traditional settings of mediation analysis, exposure $Z$, mediation $M$ and outcome $Y$ are all univariate variables at a single time point. Recent work has extended to time-varying cases, where at least one of the triplet $(Z, M, Y)$ is longitudinal. This line of research has primarily focused on the case with time varying mediators or outcomes that are observed on sparse and regular time grids [@van2008direct; @roth2012mediation; @lin2017parametric]. For example, [@vanderweele2017timevarying] developed a method for identifying and estimating causal mediation effects with time-varying exposures and mediators based on marginal structural model [@robins2000msm]. Another stream of research, motivated from applications in neuroimaging, focuses on the cases where mediators or outcomes are densely recorded continuous functions, e.g. the blood-oxygen-level-dependent (BOLD) signal collected in a functional magnetic resonance imaging (fMRI) session. In particular, [@lindquist2012functional] introduced the concept of *functional mediation* in the presence of a functional mediator and extended causal SEMs to functional data analysis [@ramsay2005functional]. [@zhao2018functional] further extended this approach to functional exposure, mediator and outcome. Sparse and irregularly-spaced longitudinal data are increasingly available for causal studies. For example, in electronic health records (EHR) data, the number of observations usually varies between patients and the time grids are uneven. The same situation applies in animal behavior studies due to the inherent difficulties in observing wild animals. Such data structure pose challenges to the existing causal mediation methods. First, one cannot simply treat the trajectories of mediators and outcomes as functions as in [@lindquist2012functional] because the sparse observations render the trajectories volatile and non-smooth. Second, with irregular time grids the dependence between consecutive observations changes over time, making the methods based on sparse and regular longitudinal data such as [@vanderweele2017timevarying] not applicable. A further complication arises when the mediator and outcome are measured with different frequencies even within the same individual. In this paper, we propose a framework for causal mediation analysis with sparse and irregular longitudinal data that address the aforementioned challenges. Similar as [@lindquist2012functional] and [@zhao2018functional], we adopt a functional data analysis perspective [@ramsay2005functional], viewing the sparse and irregular longitudinal data as realizations of underlying smooth stochastic processes. We define causal estimands of direct and indirect effects accordingly and provide assumptions for nonparametric identification (Section \[Sec\_Framework\]). For estimation and inference, we proceed under the classical two-SEM mediation framework [@imai2010identification] but diverge from the existing methods in modeling (Section \[Sec\_Modeling\]). Specifically, we employ the functional principal component analysis (FPCA) approach [@yao2005functional; @jiang2010covariatefpca; @jiang2011functional] to project the mediator and outcome trajectories to a low-dimensional representation. We then use the first few functional principal components instead of the whole trajectories as predictors in the structural equation models. To accurately quantifying the uncertainties, we employ a Bayesian FPCA model [@kowal2020bayesian] to simultaneously estimate the functional principal components and the structural equation models. Our motivating application is the evaluation of the causal effects and mechanism between early adversity, social bonds, and stress in wild baboons (Section \[sec:background\]). Here the exposure is early adversity (e.g. drought, maternal death), the mediators are the adult social bonds, and the outcomes are the adult glucocorticoid hormone concentrations, which is a measure of the stress level. The exposure is a binary variable measured at one time point, whereas both the mediators and outcomes are sparse and irregular longitudinal variables. We apply the proposed method to a prospective and longitudinal observational data set from the Amboseli Baboon Research Project [@alberts2012amboseli] (Section \[Sec\_Application\]). We find that experiencing one or more sources of early adversity leads to significant direct effects (a 9-14% increase) on females’ glucocorticoid concentrations across adulthood, but find little evidence that these effects were mediated by weak social bonds. Though motivated from a specific application, the proposed method is readily applicable to other causal mediation studies with similar data structure, including the EHR and ecology studies. Note that our method is also applicable to regular longitudinal observations. Motivating Application: Early Adversity, Social Bond and Stress {#sec:background} =============================================================== Biological Background --------------------- Conditions in early life can have profound consequences for individual development, behavior, and physiology across the life course [@lindstrom1999early; @gluckman2008effect; @bateson2004developmental]. These early life effects are important, in part, because they have major implications for human health. One leading explanation for how early life environments affect adult health is provided by the biological embedding hypothesis, which posits that early life stress causes developmental changes that create a “pro-inflammatory” phenotype and elevated risk for several diseases of aging [@miller2011psychological]. The biological embedding hypothesis proposes at least two, non-exclusive causal pathways that connect early adversity to poor health in adulthood. In the first pathway, early adversity leads to altered hormonal profiles that contribute to inflammation and disease. Under this scenario, stress in early life leads to dysregulation of hormonal signals in the body’s main stress response system, leading to the release of glucocorticoid hormones (GCs), which engage the body’s fight-or-flight response. In turn, such activations is associated with inflammation and elevated disease risk [@mcewen1998stress; @miller2002chronic; @mcewen2008central]. In the second causal pathway, early adversity hampers an individual’s ability to form strong interpersonal relationships. Under this scenario, the social isolation contributes to both altered GCs profiles and inflammation. Hence, the biological embedding hypothesis posits that early life adversity affects both GCs profiles and social conditions in adulthood, and that poor social relationships partly mediate the relationship between early adversity and GCs. Importantly, the second causal pathway—mediated through adult social conditions—suggests an opportunity to mitigate the negative health effect of early adversity. Specifically, strong and supportive social relationships may dampen the stress response or reduce individual exposure to stressful events, which in turn reduces GCs and inflammation. In support, strong and supportive social relationships have repeatedly been linked to reduced morbidity and mortality in humans and other social animals [@holt2010social; @silk2007adaptive]. In addition to the biological embedding hypothesis, this idea of social mitigation is central to several hypotheses that propose causal connections between adult social conditions and adult health, even independent of early life adversity; these hypotheses include the stress buffering and stress prevention hypotheses [@cohen1985stress; @landerman1989alternative; @thorsteinsson1999meta] and the social causation hypothesis [@marmot1991health; @anderson2011effects]. Despite the aforementioned research, the causal relationships between early adversity, adult social conditions, and HPA dysregulation remain to be subject to considerable debate. While social relationships might exert direct effects on stress and health, it is also possible that poor health and high stress limit an individual’s ability to form strong and supportive relationships. As such the causal arrow flows backwards, from stress to social relationships [@case2011long]. In another causal scenario, early adversity exerts independent effects on social conditions and the HPA axis, and correlations between social relationships and GCs are spurious, arising solely as a result of their independent links to early adversity. Data {#sec:data_general} ---- In this paper, we test whether the links between early adversity, the strength of adult social bonds, and adult HPA axis activity are consistent with predictions of the biological embedding hypothesis in a wild primate population. Specifically, we use data from a well-studied population of savannah baboons in the Amboseli ecosystem in Kenya [@alberts2012amboseli]. Founded in 1971, the Amboseli Baboon Research Project [@alberts2012amboseli] has prospective longitudinal data on early life experiences, and fine-grained longitudinal data on adult social bonds and GC hormones levels, which is a measure of HPA axis activation and the “stress response.” Our study sample includes 192 female baboons. Each baboon was at least four years old, and we had complete information on her experience of six well-characterized sources of early adversity (i.e., exposure) [@tung2016cumulative; @zipple2019intergenerational], as well as information on her adult social bonds (i.e. mediators) and fecal GC hormones concentrations (i.e. outcomes). Social bonds and GC hormones levels are measured repeatedly throughout the subjects’ lives on the same grid. For wild baboons, the observations are not on regular basis as the social bonds and GCs levels can be missing or measured multiple times within a year. We have 51.4 observations for each baboon on average for both social bonds and GCs levels, but the number of observations of a single baboon ranges from 3 to 113. Figure \[fig:trajectories\] shows the mediator and outcome trajectories of two randomly selected baboons in the sample. We can see that the frequency of the observations and time grids of the mediator or outcome trajectories vary significantly between baboons. More detailed information about the data, including covariates, is discussed in Section \[sec:data\_detail\], and in [@rosenbaum2020pnas]. ![\[fig:trajectories\] Observed trajectories of social bond and GC hormones level of two randomly selected female baboons in the study sample](Sparse_illu.pdf){width="80.00000%"} Causal Mediation Framework {#Sec_Framework} ========================== The Setup and Estimands {#Sec_Setup} ----------------------- Suppose we have a sample of $N$ units; each unit $i\ (i=1,2,\cdots,N)$ is assigned to a treatment ($Z_{i}=1$) or a control ($Z_{i}=0$) group. For each unit $i$, we make observations at $T_{i}$ different time points $\{t_{ij}\in [0,T], j=1,2,\cdots,T_{i}\}$, and $T_i$ can vary between units. At each time point $t_{ij}$, we measure an outcome $Y_{ij}$ and a mediator $M_{ij}$ prior to the outcome, and a vector of $p$ time-varying covariates $\mathbf{X}_{ij}=(X_{ij,1},\cdots,X_{ij,p})'$. For each unit, the time points are sparse along the time span and irregularly spaced. Also, the observed time grids for the outcome and the mediator are not necessarily the same between units. A key to our framework is to view the observed mediator and outcome values drawn from a smooth underlying process $M_{i}(t)$ and $Y_{i}(t)$, $t\in[0,T]$, with Normal measurement errors, respectively: $$\begin{aligned} M_{ij}&=&M_{i}(t_{ij})+\varepsilon_{ij}, \quad \varepsilon_{ij}\sim \mathcal{N}(0,\sigma_{m}^{2}),\\ Y_{ij}&=&Y_{i}(t_{ij})+\nu_{ij},\quad \nu_{ij}\sim\mathcal{N}(0,\sigma_{y}^{2}).\end{aligned}$$ Hence, instead of directly exploring the relationship between the treatment $Z_{i}$, mediators $M_{ij}$ and outcomes $Y_{ij}$, we investigate the relationship between $Z_{i}$ and the stochastic processes $M_{i}(t_{ij})$ and $Y_{i}(t_{ij})$. In particular, we wish to answer two questions: (a) how much is the causal impact of the treatment on the outcome process, and (b) how much of that impact is mediated through the mediator process. To be consistent with the standard notation of potential outcomes in causal inference, from now on we move the time index of the mediator and outcome process to the superscript: $M_{i}(t)=M_{i}^{t},Y_{i}(t)=Y_{i}^{t}$. Also, we use the following bold font notation to represent a process until time $t$: $\mathbf{M}_{i}^{t}\equiv \{M_{i}^{s},s\leq t\}\in \mathcal{R}^{[0,t]}$, and $\mathbf{Y}_{i}^{t}\equiv\{Y_{i}^{s},s\leq t\} \in \mathcal{R}^{[0,t]}$. Similarly, we denote the covariates up until the time between the $j$th and $j+1$th time point for unit $i$ as $\mathbf{X}_{i}^{t}=\{X_{i1},X_{i2},\cdots,X_{ij}\}$ for $t_{ij}\leq t<t_{ij+1}$. We extend the definition of potential outcomes to define the causal estimands. Specifically, let $\mathbf{M}_{i}^{t}(z)\in \mathcal{R}^{[0,t]}$ for $z=0,1,t\in[0,T]$, denote the potential values of the underlying mediator process for unit $i$ until time $t$ under the treatment status $z$; let $\mathbf{Y}_{i}^{t}(z,\mathbf{m})\in\mathcal{R}^{[0,t]}$ be the potential outcome for unit $i$ until time $t$ under the treatment status $z$ and the mediator process taking value of $\mathbf{M}_{i}^{t}=\mathbf{m}$ with $\mathbf{m}\in \mathcal{R}^{[0,t]}$. The above notation implicitly makes a standard assumption that the potential outcomes are determined solely by the treatment status $z$ and the mediator values $\mathbf{m}$ before time $t$, but not after $t$. For each unit, we can only observe one realization from the potential mediator or outcome process: $$\begin{aligned} &&\mathbf{M}_{i}^{t}=\mathbf{M}_{i}^{t}(Z_{i})=Z_{i}{\mathbf{M}}_{i}^{t}(1)+(1-Z_{i}){\mathbf{M}}_{i}^{t}(0),\\ &&\mathbf{Y}_{i}^{t}=\mathbf{Y}_{i}^{t}(z,\mathbf{M}_{i}^{t}(Z_{i}))=Z_{i}{\mathbf{Y}}_{i}^{t}(1,\mathbf{M}_{i}^{t}(1))+(1-Z_{i}){\mathbf{Y}}_{i}^{t}(0,\mathbf{M}_{i}^{t}(0)).\end{aligned}$$ We define the total effect (TE) of the treatment $Z_{i}$ on the outcome process at time $t$ as: $$\begin{aligned} \label{Definition_ATE_Process} \tau_{{\mbox{\tiny{TE}}}}^{t}&=&E\{Y_{i}^{t}(1,\mathbf{M}_{i}^{t}(1))-Y_{i}^{t}(0,\mathbf{M}_{i}^{t}(0))\}.\end{aligned}$$ In the presence of a mediator, the total effect can be decomposed into direct and indirect effects. Below we extend the framework of [@imai2010identification] to formally define these effects. First, we define the average causal mediation (or indirect) effect (ACME) under treatment $z$ at time $t$ by fixing the treatment status while altering the mediator process: $$\begin{aligned} \label{Definition_ACME_Process} \tau_{{\mbox{\tiny{ACME}}}}^{t}(z)&\equiv&E\{ Y_{i}^{t}(z,\mathbf{M}_{i}^{t}(1))- Y_{i}^{t}(z,\mathbf{M}_{i}^{t}(0))\},\quad z=0,1.\end{aligned}$$ The ACME quantifies the difference between the potential outcomes, given a fixed treatment status $z$, corresponding to the potential mediator process under treatment $\mathbf{M}_{i}^{t}(1)$ and that under control $\mathbf{M}_{i}^{t}(0)$. In the previous literature, variants of the ACME are also called the *natural indirect effect* [@pearl2001direct], or the *pure indirect effect* for $\tau_{{\mbox{\tiny{ACME}}}}^{t}(0)$ and *total indirect effect* for $\tau_{{\mbox{\tiny{ACME}}}}^{t}(1)$ [@robins1992identifiability] Second, we define the average natural direct effect (ANDE) [@pearl2001direct; @imai2010identification] of treatment on the outcome at time $t$ by fixing the mediator process while altering the treatment status: $$\begin{aligned} \label{Definition_ANDE_Process} \tau_{{\mbox{\tiny{ANDE}}}}^{t}(z)&\equiv&E\{ Y_{i}^{t}(1,\mathbf{M}_{i}^{t}(z))-Y_{i}^{t}(0,\mathbf{M}_{i}^{t}(z))\},\end{aligned}$$ The ANDE quantifies the portion in the total effects that does not pass through the mediators. It is easy to verify that the total effect is the sum of ACME and ANDE: $$\begin{aligned} \label{TE_decomposition} \tau_{{\mbox{\tiny{TE}}}}^{t}=\tau_{{\mbox{\tiny{ACME}}}}^{t}(z)+\tau_{{\mbox{\tiny{ANDE}}}}^{t}(1-z), \quad z=0,1.\end{aligned}$$ This implies we only need to identify two of the three quantities $\tau_{{\mbox{\tiny{TE}}}}$, $\tau_{{\mbox{\tiny{ACME}}}}^{t}(z)$, $\tau_{{\mbox{\tiny{ANDE}}}}^{t}(z)$. In this paper, we will focus on the estimation of $\tau_{{\mbox{\tiny{TE}}}}$ and $\tau_{{\mbox{\tiny{ACME}}}}^{t}(z)$. Also we make a common assumption that the ACME and ANDE are the same in the treatment and control groups: $\tau_{{\mbox{\tiny{ACME}}}}^{t}(0)=\tau_{{\mbox{\tiny{ACME}}}}^{t}(1),\tau_{{\mbox{\tiny{ANDE}}}}^{t}(0)=\tau_{{\mbox{\tiny{ANDE}}}}^{t}(1)$. Because we only observe a portion of all the potential outcomes, we cannot directly identify these estimands from the observed data, which would require additional assumptions. Identification assumptions {#Sec_Assumption} -------------------------- In this subsection, we list the causal assumptions necessary for identifying the ACME and ANDEs with sparse and irregular longitudinal data. The first assumption extends the standard ignorability (or unconfoundedness) assumption and rules out the unmeasured treatment-outcome confounding. \[A.1\] Conditional on the observed covariates, the treatment is unconfounded with respect to the potential mediator process and the potential outcomes process: $$\begin{aligned} \{\mathbf{Y}_{i}^{t}(1,\mathbf{m}),\mathbf{Y}_{i}^{t}(0,\mathbf{m}),\mathbf{M}_{i}^{t}(1),\mathbf{M}_{i}^{t}(0) \}{\protect\mathpalette{\protect\independenT}{\perp}}Z_{i}\mid \mathbf{X}_{i}^{t} \end{aligned}$$ for any $t$ and $\mathbf{m}\in \mathcal{R}^{[0,t]}$. The second assumption extends the sequential ignorability assumption in [@imai2010identification] to the functional data setting. \[A.2\]There exists $\varepsilon>0$, such that for any $0<\Delta<\varepsilon$,the increment of mediator process is independent of the increment of potential outcomes process from time $t$ to $t+\Delta$, conditional on the observed treatment status, covariates and the mediator process up to time $t$: $$\begin{aligned} \{Y_{i}^{t+\Delta}(z,\mathbf{m})-Y_{i}^{t}(z,\mathbf{m})\} {\protect\mathpalette{\protect\independenT}{\perp}}(M_{i}^{t+\Delta}-M_{i}^{t})\mid \{Z_{i},\mathbf{X}_{i}^{t},\mathbf{M}_{i}^{t}\} \end{aligned}$$ for any $z,0<\Delta<\varepsilon,t,t+\Delta\in [0,T],\mathbf{m}\in \mathcal{R}^{[0,T]}$. Assumption \[A.2\] implies that conditioning on the observed treatment status, covariates, and the mediator process up to a given time point, the change in the mediator values within a sufficiently small time interval is randomized with respect to the change in the potential outcomes. Namely, there are no unobserved mediator-outcomes confounders in a sufficiently small time interval. Though differs in the specific form, Assumption \[A.2\] shares the essence with the previous sequential ignorability assumptions for the regularly spaced observations in [@bind2015longitudinals] and [@vanderweele2017timevarying]. This is a crucial assumption in mediation analysis, but is strong and generally untestable in practice because it is usually impossible to manipulate the mediator values, even in randomized trials. Assumption \[A.1\] and \[A.2\] are illustrated by the directed acyclic graphs (DAG) in Figure \[fig:DAG\_A1A2\], which condition on the covariates $\mathbf{X}_{i}^{t}$ and a window between two sufficiently closed time points $t$ and $t+\Delta$. The arrows between $Z_{i}$, $M_{i}^{t}$, $Y_{i}^{t}$ represent a causal relationship (i.e., nonparametric structural equation model), with solid and dashed lines representing measured and unmeasured relationships, respectively. Figure \[fig:DAG\_A1\_violation\] and \[fig:DAG\_A2\_violation\] depicts two possible scenarios where Assumption \[A.1\] and \[A.2\] is violated, respectively, where $U_i$ represents an unmeasured confounder. \[ &gt; = stealth, shorten &gt; = 0.5pt, auto, node distance = 2cm, semithick \] =\[ draw = white, thick, fill = white, minimum size = 3mm, \] \(A) (A)[$Z_{i}$]{}; (B)[...$M_{i}^{t}$]{}; (C) [$M_{i}^{t+\Delta}$]{};(D) [...$Y_{i}^{t}$]{};(E) [$Y_{i}^{t+\Delta}$]{}; (A) edge node (B); (B) edge node (C); (B) edge node (D); (D) edge node (E); (C) edge node (E); (B) edge node (E); (A) edge node (D); (A) edge node (E); (A) edge \[bend left\] node (C); \[ &gt; = stealth, shorten &gt; = 0.5pt, auto, node distance = 2cm, semithick \] =\[ draw = white, thick, fill = white, minimum size = 3mm, \] \(A) (A)[$Z_{i}$]{}; (B)[...$M_{i}(t)$]{}; (C) [$M_{i}(t+\Delta)$]{}; (D) [...$Y_{i}(t)$]{}; (E) [$Y_{i}(t+\Delta)$]{}; (U) [$U_{i}$]{}; (A) edge node (B); (B) edge node (C); (B) edge node (D); (D) edge node (E); (C) edge node (E); (B) edge node (E); (A) edge node (D); (A) edge node (E); (A) edge \[bend left\] node (C); (U) edge node (A); (U) edge node (D); \[ &gt; = stealth, shorten &gt; = 0.5pt, auto, node distance = 2cm, semithick \] =\[ draw = white, thick, fill = white, minimum size = 3mm, \] \(A) (A)[$Z_{i}$]{}; (B)[...$M_{i}(t)$]{}; (C) [$M_{i}(t+\Delta)$]{}; (D) [...$Y_{i}(t)$]{}; (E) [$Y_{i}(t+\Delta)$]{}; (U) [$U_{i}$]{}; (A) edge node (B); (B) edge node (C); (B) edge node (D); (D) edge node (E); (C) edge node (E); (B) edge node (E); (A) edge node (D); (A) edge node (E); (A) edge \[bend left\] node (C); (U) edge node (A); (U) edge node (B); \[ &gt; = stealth, shorten &gt; = 0.5pt, auto, node distance = 2cm, semithick \] =\[ draw = white, thick, fill = white, minimum size = 3mm, \] \(A) (A)[$Z_{i}$]{}; (B)[...$M_{i}(t)$]{}; (C) [$M_{i}(t+\Delta)$]{}; (D) [...$Y_{i}(t)$]{}; (E) [$Y_{i}(t+\Delta)$]{}; (L) [$U_{i}$]{}; (A) edge node (B); (B) edge node (C); (B) edge node (D); (D) edge node (E); (C) edge node (E); (B) edge node (E); (A) edge node (D); (A) edge node (E); (A) edge \[bend left\] node (C); (A) edge node (L); (B) edge node (L); (L) edge node (C); (L) edge node (E); \[ &gt; = stealth, shorten &gt; = 0.5pt, auto, node distance = 2cm, semithick \] =\[ draw = white, thick, fill = white, minimum size = 3mm, \] \(A) (A)[$Z_{i}$]{}; (B)[...$M_{i}(t)$]{}; (C) [$M_{i}(t+\Delta)$]{}; (D) [...$Y_{i}(t)$]{}; (E) [$Y_{i}(t+\Delta)$]{}; (A) edge node (B); (B) edge node (C); (B) edge node (D); (D) edge node (E); (C) edge node (E); (B) edge node (E); (A) edge node (D); (A) edge node (E); (A) edge \[bend left\] node (C); (D) edge node (C); Assumptions \[A.1\] and \[A.2\] allow nonparametric identification of the total effect and ACME from the observed data, as summarized in the following theorem. \[T.1\] Under Assumption \[A.1\],\[A.2\], and some regularity conditions (specified in the Appendix), the total effect, ACME and ANDE can be identified nonparametrically from the observed data: for $z=0,1$, we have $$\begin{aligned} \tau_{{\mbox{\tiny{TE}}}}&=&\int_{X} \{E(Y_{i}^{t}|Z_{i}=1,\mathbf{X}_{i}^{t}=\mathbf{x}^{t})-E(Y_{i}^{t}|Z_{i}=0,\mathbf{X}_{i}^{t}=\mathbf{x}^{t})\}\textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t}),\\ \tau_{{\mbox{\tiny{ACME}}}}^{t}(z)&=&\int_{\mathcal{X}} \int_{R^{[0,t]}} E(Y_{i}^{t}|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m}) \textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t})\times\\ && \quad\quad \quad \textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=1,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m}) -\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=0,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\}, \end{aligned}$$ where $F_{W}(\cdot)$ and $F_{W|V}(\cdot)$ denotes the cumulative distribution of a random variable or vector $W$ and the conditional distribution given another random variable or vector $V$, respectively. The proof of Theorem \[T.1\] is provided in the Supplementary A. Theorem \[T.1\] implies that estimating the causal effects requires modeling two components: (a) the distribution of observed outcome process given the treatment, covariates, and the observed mediator process, $E(Y_{i}^{t}|Z_i, \mathbf{X}_{i}^{t}, \mathbf{M}_{i}^{t})$, (b) the distribution of the observed mediator process given the treatment and the covariates, $E(\mathbf{M}_{i}^{t}|Z_i, \mathbf{X}_{i}^{t})$. These two components correspond to the two linear structural equations in the classic mediation framework of [@baron1986moderator]. In the setting of functional data, we can employ more flexible models instead of linear regression models, and express the total effect and ACME as functions of the model parameters. Theorem \[T.1\] can be readily extended to more general scenarios such as discrete mediators and time-to-event outcomes. Modeling mediator and outcome via functional principal component analysis {#Sec_Modeling} ========================================================================= In this section, we propose to employ the functional principal component analysis (FPCA) approach to infer the mediator and outcome processes from sparse and irregular observations [@yao2005functional; @jiang2010covariatefpca; @jiang2011functional]. In order to take into account the uncertainty due to estimating the functional principal components [@goldsmith2013corrected], we adopt a Bayesian model to jointly estimate the principal components and the structural equation models. Specifically, we impose a Bayesian FPCA model similar to that in [@kowal2020bayesian] to project the observed mediator and outcome processes into lower-dimensional representations and then take the first few dominant principal components as the predictors in the structural equation models. We assume the potential processes for mediators $\mathbf{M}_{i}^{t}(z)$ and outcomes $\mathbf{Y}_{i}^{t}(z,\mathbf{m})$ have the following Karhunen-Loeve decomposition, $$\begin{gathered} \label{Mediator_Process} M_{i}^{t}(z)=\mu_{M}(\mathbf{X}_{i}^{t})+\sum_{r=1}^{\infty}\zeta_{i,z}^{r}\psi_{r}(t),\\ \label{Outcome_Process} Y_{i}^{t}(z,\mathbf{m})=\mu_{Y}(\mathbf{X}_{i}^{t})+\int_{0}^{t}\gamma(s,t)\mathbf{m}(s)ds+\sum_{s=1}^{\infty}\theta_{i,z}^{s}\eta_{s}(t).\end{gathered}$$ where $\mu_{M}(\cdot)$ and $\mu_{Y}(\cdot)$ are the mean functions of the mediator process $\mathbf{M}_{i}^{t}$ and outcome process $\mathbf{Y}_{i}^{t}$, respectively; $\mathbf{\psi}_{r}(t)$ and $\mathbf{\eta}_{s}(t)$ are the Normal orthogonal eigenfunctions for $\mathbf{M}_{i}^{t}$ and $\mathbf{Y}_{i}^{t}$, respectively, and $\zeta_{i,z}^{r}$ and $\theta_{i,z}^{s}$ are the corresponding principal scores of unit $i$. The above model assumes that the treatment affects the mediation and the outcome processes only through the principal scores. We represent the mediator and outcome process of each unit with its principal score $\zeta_{i,z}^{r}$ and $\theta_{i,z}^{s}$. Given the principal scores , we can transform back to the smooth process with linear combination. As such, if we are interested in the differences on the process, it is equivalent to investigate the difference on the principal scores. Also, as we usually require only 3 or 4 components to explain the most variation, we reduce the dimensions of trajectories effectively by projecting the difference to principal scores. The underlying processes $\mathbf{M}_{i}^{t}$ and $\mathbf{Y}_{i}^{t}$ are not directly observed. Instead, we assume the observations $M_{ij}$’s and $Y_{ij}$’s are randomly sampled from the respective underlying processes with errors. For the observed mediator trajectories, we posit the following model that truncates to the first $R$ principal components of the mediator process: $$\begin{gathered} \label{eq:mediator_model} M_{ij}=X_{ij}'\beta_{M}+\sum_{r=1}^{R}\zeta_{i}^{r}\psi_{r}(t_{ij})+\varepsilon_{ij},\quad \varepsilon_{ij}\sim \mathcal{N}(0,\sigma_{m}^{2}),\end{gathered}$$ where $\psi_{r}(t)$ ($r=1,..., R$) are the orthogonormal principal components, $\zeta_{i}^{r}$ ($r=1,..., R$) are the corresponding principal scores, and $\varepsilon_{ij}$ is the measurement error. With similar parametrization in [@kowal2020bayesian], we express the principal components as a linear combination of the spline basis $\mathbf{b}(t)=(1,t,b_{1}(t),\cdots,b_{L}(t))'$ in $L+2$ dimensions and choose the coefficients $\mathbf{p}_{r}\in\mathcal{R}^{L+2}$ to meet the normal orthogonality constraints of the $r$th principal component: $$\begin{gathered} \label{eq:mediator_pc} \psi_{r}(t)=\mathbf{b}(t)'\mathbf{p}_{r}, \text{ subject to} \int_{0}^{T}\psi_{r}^{2}(t)dt=1,\int_{0}^{T}\psi_{r'}(t)\psi_{r''}(t)dt=0.\end{gathered}$$ We assume the principal scores $\zeta_{i}^{r}$ are randomly drawn from normal distributions with different means in the treated and control groups, $\chi_{1}^{r}$ and $\chi_{0}^{r}$, and diminishing variance as $r$ increases: $$\begin{gathered} \label{eq:mediator_pc_distribution} \zeta_{i}^{r}\sim \mathcal{N}(\chi_{Z_{i}}^{r}, \lambda_{r}^{2}), \quad \lambda_{1}^{2}\geq\lambda_{2}^{2}\geq\cdots\lambda_{R}^{2}\geq 0.\end{gathered}$$ We select the truncation term $R$ based on the fraction of explained variance (FEV), $\sum_{r=1}^{R}\lambda_{r}^{2}/\sum_{r=1}^{\infty}\lambda_{r}^{2}$ being greater than $90\%$. For the observed outcome trajectories, we posit a similar model that truncates to the first $S$ principal components of the outcome process: $$\begin{gathered} \label{eq:outcome_model} Y_{ij}=X_{ij}^{T}\beta_{Y}+\int_{0}^{t_{ij}} \gamma(u,t) M_{i}^{u}\textup{d}u+\sum_{s=1}^{S}\eta_{s}(t)\theta_{i}^{s}+\nu_{ij}, \quad \nu_{ij}\sim N(0,\sigma_{y}^{2}).\end{gathered}$$ We express the principal components $\eta_{s}$ as a linear combination of the spline basis $\mathbf{b}(t)$, with the normal orthogonality constraints: $$\begin{gathered} \label{eq:outcome_pc} \eta_{s}(t)=\mathbf{b}(t)'\mathbf{q}_{s}, \text{ subject to } \int_{0}^{T}\eta_{s}(t)^{2}dt=1,\int_{0}^{T}\eta_{s'}(t)\eta_{s''}(t)dt=0.\end{gathered}$$ Similarly, we assume that the principal scores of outcome process for each unit come from two different normal distributions in the treated and control group with means $\xi_{1}^{s}$ and $\xi_{0}^{s}$ respectively, and a shrinking variance $\rho_{s}^{2}$: $$\begin{gathered} \label{eq:outcome_pc_distribution} \theta_{i}^{s}\sim\mathcal{N}(\xi_{Z_{i}}^{s},\rho_{s}^{2}), \quad \rho_{1}^{2}\geq \rho_{2}^{2}\geq \cdots\rho_{S}^{2}\geq 0.\end{gathered}$$ We select the truncation term $S$ based on the FEV being greater than $90\%$, namely $\sum_{s=1}^{S}\rho_{s}^{2}/\sum_{s=1}^{\infty}\rho_{s}^{2}\geq 90\%$. We assume the effect of the mediation process on the outcome is concurrent, namely the outcome process at time $t$ does not depend on the past value of the mediation process. As such, $\gamma(u,t)$ can be shrunk to $\gamma$ instead of the integral in Model , $$\begin{gathered} \label{eq:outcome_model_concurrent} Y_{ij}=X_{ij}^{T}\beta_{Y}+\gamma M_{ij}+\sum_{s=1}^{S}\eta_{s}(t)\theta_{i}^{s}+\nu_{ij},\quad \nu_{ij}\sim N(0,\sigma_{y}^{2}).\end{gathered}$$ The causal estimands, the total effect and ACME, can be expressed as functions of the parameters in the above mediator and outcome models: $$\begin{aligned} \label{TT_EXPRESS} \tau_{{\mbox{\tiny{TE}}}}^t&=&\sum_{s=1}^{S}(\zeta_{1}^{s}-\zeta_{0}^{s})\eta_{s}(t)+\gamma\sum_{r=1}^{R} (\chi_{1}^{r}-\chi_{0}^{r})\psi_{r}(t),\\ \label{IT_EXPRESS} \tau_{{\mbox{\tiny{ACME}}}}^{t}&=&\gamma (\chi_{1}^{r}-\chi_{0}^{r})\psi_{r}(t).\end{aligned}$$ To account for the uncertainty in estimating the above models, we adopt the Bayesian paradigm and impose prior distributions for the parameters [@kowal2020bayesian]. For the basis function $\mathbf{b}(t)$ to construct principal components, we choose the thin-plate spline which takes the form $\mathbf{b}(t)=(1,t,(|t-k_{1}|)^{3},\cdots,|t-k_{L}|^{3})'\in \mathcal{R}^{L+2}$, where the $k_{l} \ (l=1,2,\cdots,L)$ is the pre-defined knots on the time span. We set the values of knots $k_{l}$ with the quantiles of observation time grids. For the parameters of principal components, taking the mediator model for an example, we impose the following priors on parameters in : $$\begin{gathered} \phi_{r}\sim N(0,h_{r}^{-1}\Omega^{-1}),h_{r}\sim \textup{Uniform}(\lambda^{2}_{r},10^{4}),\end{gathered}$$ where $\Omega\in \mathcal{R}^{(L+2)\times(L+2)}$ is the roughness penalty matrix and $h_{r}>0$ is the smooth parameter. The implies a Gaussian Process prior on $\psi_{r}(t)$ with mean function zero and covariance function $\textup{Cov}(\psi_{r}(t),\break \psi_{r}(s)) =h_{r}\mathbf{b}'(s)\Omega\mathbf{b}(t)$. We choose the $\Omega$ such that $[\Omega_{r}]_{l,l'}=(k_{l}-k_{l})^{2}$,when $l,l'>2,$ and $[\Omega_{r}]_{l,l'}=0$ when $l,l'\leq 2$. For the distribution of principal scores in , we specify a multiplicative Gamma prior [@bhattacharya2011sparse; @montagna2012bayesian] on the variance to encourage shrinkage as $r$ increasing, $$\begin{gathered} \chi_{0}^{r},\chi_{1}^{r}\sim N(0,\sigma^{2}_{\chi_{r}}),\quad \sigma^{-2}_{\chi_{r}}=\prod_{l\leq r}\delta_{\chi_{l}}, \delta_{\chi_{1}}\sim \textup{Ga}(a_{\chi_{1}},1 ), \delta_{\chi_{l}}\sim \textup{Ga}(a_{\chi_{2}},1),l\geq 2,\\ \lambda_{r}^{-2}=\prod _{l\leq r}\delta_{l},\quad \delta_{1}\sim \textup{Ga}(a_{1},1), \quad \delta_{l}\sim \textup{Ga}(a_{2},1), l\geq 2,\\ a_{1},a_{\chi_{1}}\sim \textup{Ga}(2,1), \quad a_{2},a_{\chi_{2}}\sim \textup{Ga}(3,1).\end{gathered}$$ Further details on the hyperparameters of the priors can be found in [@bhattacharya2011sparse] and [@durante2017note]. For the coefficients of covariates $\beta_{M}$, we specify a diffused normal prior $\beta_{M}\sim \mathcal{N}(0,100^{2}*\textup{I}_{\textup{dim}(X)})$. We impose similar prior distributions for the parameters in the outcome model. Posterior inference can be obtained by Gibbs sampling. The credible intervals of the causal effects $\tau_{{\mbox{\tiny{TE}}}}^t$ and $\tau_{{\mbox{\tiny{ACME}}}}^{t}$ can be easily constructed using the posterior sample of the parameters in the model. Details of the Gibbs sampler are provided in the Supplementary B. Simulations\[Sec\_Simulation\] ============================== In this section, we conduct simulations to evaluate the operating characteristics of the proposed method and compare with two standard methods. Simulation design {#sec:simu_design} ----------------- We generate 200 units to approximate the sample size in our application. For each unit, we make $T_{i}$ observations at the time grid $\{t_{ij}\in [0,1], j=1,2,\cdots,T_{i}\}$. We draw $T_{i}$ from a Poisson distribution with mean $T$ and randomly pick $t_{ij}$ uniformly: $$\begin{gathered} T_{i}\sim \textup{Poisson}(T), \quad t_{ij}\sim \textup{Uniform}(0,1), \quad j=1,2,\cdots,T_{i}.\end{gathered}$$ For each unit $i$ and time $j$, we generate three covariates from a tri-variate Normal distribution, $\mathbf{X}_{ij}=(X_{ij1},X_{ij2}, X_{ij3}) \sim \mathcal{N}([0,0,0]^{T},\sigma_{X}^{2}\textup{I}_{3})$. We simulate the binary treatment indicator from $Z_i={\mathbf{1}}\{c_{i1}> 0\}$, where $c_{i1}\sim \mathcal{N}(0,1)$. To simulate the sparse and irregular mediator trajectories, we first simulate a smooth underlying process $M_{i}^{t}(z)$ for the mediators: $$\begin{gathered} M^{t}_{i}(z)=0.2+\{0.2+2t+\textup{sin}(2\pi t)\})(z+1)-X_{ij1}+0.5X_{ij2}+ \varepsilon_{i}^{m}(t)+c_{i2},\end{gathered}$$ where the error term $\varepsilon_{i}^{m}(t)\sim \textup{GP}(0,\sigma_{m}^{2}\textup{exp}\{-8(s-t)^{2}\})$ is drawn from a Gaussian Process (GP) with an exponential kernel and $\sigma_{m}^{2}$ controlling the volatility of the realized curves, and $c_{i2}\sim\mathcal{N}(0,\sigma_{m}^{2})$ to represent the individual random intercepts. The mean value of the mediator process depends on the covariates and time. The polynomial term and the trigonometric function of $t$ introduce the long term growth trend and periodic fluctuations, respectively. Also, the coefficient of $z$ evolves as the time changes, implying a time varying treatment effect on the mediator. Similarly, we specify a GP model for the outcome process, $$\begin{aligned} Y_{i}^{t}(z,\mathbf{m})&=&\mathbf{m}^{t}+\textup{cos}(2\pi t)+0.1t^{2}+2t+\{\textup{cos}(2\pi t)+0.2t^{2}+3t\}z-\\ && \quad 0.5 X_{ij2}+X_{ij3}+ \varepsilon_{i}^{y}(t)+c_{i3}, \end{aligned}$$ where the error term $\varepsilon_{i}^{y}(t)\sim \textup{GP}(0,\sigma_{y}^{2}\textup{exp}\{-8(s-t)^{2}\})$ is drawn from a GP, and $c_{i3}\sim\mathcal{N}(0,\sigma_{y}^{2})$ controls the individual random effect for the outcome process. The above settings imply non-linear true causal effects ($\tau_{{\mbox{\tiny{TE}}}}^t$ and $\tau_{{\mbox{\tiny{ACME}}}}^{t}$) in time, which are shown as the dashed lines in Figure \[fig:MFPCA\]. Upon simulating the processes, we evaluate the potential values of the mediators and outcomes at the sampled time point $t_{ij}$ to obtain the observed trajectories with measurement error: $$\begin{gathered} M_{ij}\sim\mathcal{N}(\mathbf{M}_{i}^{t_{ij}}(Z_{i}),1), \quad Y_{ij}\sim\mathcal{N}(\mathbf{Y}_{i}^{t_{ij}}(Z_{i},\mathbf{M}_{i}^{t_{ij}}(Z_{i})),1).\end{gathered}$$ We control the sparsity of the mediator and outcome trajectories by varying the value of $T$ in the grid of $(15,30,50,100)$, namely the average number of observations for each individual. We compare the proposed method in Section \[Sec\_Modeling\] (referred to as MFPCA) with two standard methods in longitudinal data analysis: the random effects model [@laird1982random] and the generalized estimating equations (GEE) [@liang1986longitudinal]. To facilitate the comparisons, we aggregate the time-varying mediation effects into the following scalar values: $$\begin{aligned} \tau_{{\mbox{\tiny{ACME}}}} = \int_{0}^{T}\tau_{{\mbox{\tiny{ACME}}}}^{t}\textup{d}t, \quad \tau_{{\mbox{\tiny{TE}}}} = \int_{0}^{T}\tau_{{\mbox{\tiny{TE}}}}^{t}\textup{d}t.\end{aligned}$$ The true values for $\tau_{{\mbox{\tiny{ACME}}}}$ and $\tau_{{\mbox{\tiny{TE}}}}$ in the simulations are $1.20$ and $2.77$ respectively. For the random effects approach, we fit the following two models: $$\begin{aligned} M_{ij} = X_{ij}^{T}\beta_{M}+s_{m}(T_{ij})+\tau_{m}Z_{i}+r^{m}_{ij}+\varepsilon^{m}_{ij},\\ Y_{ij} = X_{ij}^{T}\beta_{Y}+s_{y}(T_{ij})+\tau_{y}Z_{i}+\gamma M_{ij}+r^{y}_{ij}+\varepsilon^{y}_{ij},\end{aligned}$$ where $r^{m}_{ij}$ and $r^{y}_{im}$ are normally distributed random effect with zero means, $s_{m}(T_{ij})$ and $s_{y}(T_{ij})$ are thin plate splines to capture the nonlinear effect of time. To model the time dependency, we specify an AR(1) correlation structure for the random effects, thus $\textup{Corr}(r^{m}_{ij},r^{m}_{ij+1})=p_{1},\textup{Corr}(r^{y}_{ij},r^{y}_{ij+1})=p_{2}$, namely the correlation decay exponentially within the observations of a given unit. Given the above random effect model, the mediation effect and total effect can be calculated as: $\hat{\tau}_{{\mbox{\tiny{ACME}}}}^{{\mbox{\tiny{RD}}}}=\hat{\gamma}\hat{\tau}_{m},\hat{\tau}_{{\mbox{\tiny{TE}}}}^{{\mbox{\tiny{RD}}}}=\hat{\gamma}\hat{\tau}_{m}+\hat{\tau}_{y}$. For the GEE approach, we specify the following estimation equations: $$\begin{aligned} E(M_{ij}|X_{ij},Z_{i})=X_{ij}^{T}\beta_{M}+\tau_{m}Z_{i},\\ E(Y_{ij}|M_{ij},X_{ij},Z_{i})=X_{ij}^{T}\beta_{M}+\tau_{y}Z_{i}+\gamma M_{ij}.\end{aligned}$$ For the working correlation structure, we consider the AR(1) correlation for both the mediators and outcomes. Similarly, we obtain the estimations through $\hat{\tau}_{{\mbox{\tiny{ACME}}}}^{{\mbox{\tiny{GEE}}}}=\hat{\gamma}\hat{\tau}_{m},\hat{\tau}_{{\mbox{\tiny{TE}}}}^{{\mbox{\tiny{GEE}}}}=\hat{\gamma}\hat{\tau}_{m}+\hat{\tau}_{y}$ with two different correlation structures. It is worth noting that both the random effects model and the GEE model generally lack the flexibility to accommodate irregularly-spaced longitudinal data, which renders specifying the correlation between consecutive observations difficult. For example, though the AR(1) correlation takes into account the temporal structure of the data, it still imposes the correlation between any two consecutive observations to be constant, which is unlikely the case in the cases with irregularly-spaced data. Nonetheless, we compare the proposed method with these two models as they are the standard methods in longitudinal data analysis. Simulation results ------------------ We apply the proposed MFPCA method, the random effects model and the GEE model in Section \[sec:simu\_design\] to the simulated data $\{Z_{i}, \mathbf{X}_{ij}, M_{ij}, Y_{ij}\}$, to estimate the causal effects $\tau_{{\mbox{\tiny{TE}}}}$ and $\tau_{{\mbox{\tiny{ACME}}}}$. Figure \[fig:MFPCA\] shows the causal effects and associated 95% credible interval estimated from MFPCA in one randomly selected simulated dataset under each of the four levels of sparsity $T$. Regardless of $T$, MFPCA appears to estimate the time-varying causal effects satisfactorily, with the 95% credible interval covering the true effects at any time. As expected, the accuracy of the estimation increases as the frequency of the observations increases. ![Posterior mean of $\tau_{{\mbox{\tiny{TE}}}}^{t}$,$\tau_{{\mbox{\tiny{ACME}}}}^{t}$ and 95% credible intervals in one simulated dataset under each level of sparsity. The solid line represents the true surface for $\tau_{{\mbox{\tiny{TE}}}}^{t}$ and $\tau_{{\mbox{\tiny{ACME}}}}^{t}$ []{data-label="fig:MFPCA"}](Effect_estimation.pdf){width="95.00000%"} Table \[tab:simu\_comparison\] presents the absolute bias, root mean squared error (RMSE) and coverage rate of the 95% confidence interval of $\tau_{{\mbox{\tiny{TE}}}}$ and $\tau_{{\mbox{\tiny{ACME}}}}$ under the MFPCA, the random effect model and the GEE model based on 200 simulated datasets for each level of sparsity $T$ in $[15,25,50,100]$. The performance of all three methods improve as the frequency of observations increases. With low frequency ($T<100$), i.e. sparse observations, MFPCA consistently outperforms random effect model, which in turn outperforms GEE in all measures. The advantage of MFPCA over the other two methods diminishes as the frequency increases. In particular, with dense observations ($T=100$), MFPCA leads to similar results as random effects, both outperforming GEE. The simulation results validate the use of our method in sparse case. --------------- ------- ------- ---------- ------- ------- ---------- Method Bias RMSE Coverage Bias RMSE Coverage MFPCA 0.103 0.154 88.4% 0.134 0.273 86.4% Random effect 0.165 0.208 78.2% 0.883 1.673 69.5% GEE 0.183 0.304 77.6% 0.987 2.051 61.8% MFPCA 0.092 0.123 92.3% 0.102 0.246 90.6% Random effect 0.124 0.165 81.2% 0.679 1.263 72.3% GEE 0.152 0.273 80.3% 0.860 1.753 64.4% MFPCA 0.087 0.112 93.5% 0.094 0.195 92.3% Random effect 0.109 0.134 90.3% 0.228 0.497 88.8% GEE 0.121 0.175 83.5% 0.236 0.493 80.8% MFPCA 0.053 0.089 94.3% 0.064 0.163 93.1% Random effect 0.046 0.093 93.1% 0.053 0.154 92.8% GEE 0.093 0.124 90.5% 0.098 0.161 90.3% --------------- ------- ------- ---------- ------- ------- ---------- : Absolute bias, RMSE and coverage rate of the 95% confidence interval of MFPCA, the random effect model and the generalized estimating equation (GEE) model under different frequency of observations in the simulations. []{data-label="tab:simu_comparison"} Empirical Application {#Sec_Application} ===================== The data {#sec:data_detail} -------- As discussed in Section \[sec:background\], the goal of this application is to investigate the causal mediation mechanisms between early adversity, adult social bonds, and adult GC hormones levels. We apply the method and models proposed in Section \[Sec\_Framework\] and \[Sec\_Modeling\] to a longitudinal dataset on wild baboons collected in the Amboseli Baboon Research Project. Here we first provide more information about the data. Our sample includes 192 female baboons and 11658 observations in total. We retain 10626 observations after removing the observations with missing social bonds or GCs levels information. The longitudinal observations on the strength of social bonds (mediators) and GC hormones concentrations. The social bonds range from $-1.47$ to $3.31$ with mean value at $1.04$ and standard deviation $0.51$. The fecal GCs levels range from $7.51$ to $982.87$ with mean $74.13$ and standard deviation $38.25$. Age is used to index within-individual observations on both social bond and GCs levels. All the baboons enter into the study after becoming mature at age 5. However, data on both GCs levels and social bond strength for females older than 18 years are extremely sparse and volatile (only about 20% baboons survive until age 18). Therefore, we truncated all trajectories at age 18, resulting in a final sample with 192 female baboons and 9878 observations. The six adversity sources (exposure) are drought, maternal death, competing sibling, high group density, low maternal rank, maternal social isolation. Table \[tab:adversity\] presents the number of baboons experienced early early adversity. Overall, while only a small proportion of subjects experienced any given source of early adversity, most subjects had experienced at least one source of early adversity. Therefore, in our analysis we also create a cumulative exposure variable that summarizes the total number of adversities. --------------------------- --------------------------------- ----------------------------- early adversity no. subjects did not experience no. subjects did experience (control) (exposure) Drought 164 28 Competing Sibling 153 39 High group density 161 31 Maternal death 157 35 Low maternal rank 152 40 Maternal Social isolation 140 52 At least one 48 144 --------------------------- --------------------------------- ----------------------------- : Sources of early adversity and the number of baboons experienced each type of early adversity. The last row summarizes the number of baboons had at least one of six individual adversity sources.[]{data-label="tab:adversity"} The time-varying covariates include reproductive state, density in the social group, max temperature in last 30 days, whether the sample is collected in wet or dry season, the amount of rainfall. More information can be found in [@rosenbaum2020pnas]. Results of FPCA --------------- We first summarize the results of FPCA of the observed trajectories. We posit model for the social bonds and Model for the GCs levels, with some modifications. First, we added two random effects, one for social group and one for hydrological year, in both models. Second, in the outcome model, we use the log transformed GCs level instead of the original scale as the outcome, which allows us to interpret the coefficient of early adversity as the effect on the percentage change of the GCs level. For both the mediator and outcome processes, the first three functional principal components explain more than 90% of the total variation and we will use them in the structural equation model for mediation analysis. Figure \[fig:pc\] shows the first two principal components extracted from the mediator (left panel) and outcome (right panel) processes. For the social bonds process, the first two principal components explain 53% and 31% of the total variation. The first component depicts a drastic change in the early stage of a baboon’s life and stabilizes afterwards. For the GCs process, the first two functional principal components explain 54% and 34% of the total variation. The first component depicts a stable trend throughout the life span. ![The first two functional principal components of the process of the mediator, i.e. social bonds (left panel) and the outcome, i.e., GCs level (right panel).[]{data-label="fig:pc"}](DSI_F_eigen.pdf "fig:"){width="40.00000%"} ![The first two functional principal components of the process of the mediator, i.e. social bonds (left panel) and the outcome, i.e., GCs level (right panel).[]{data-label="fig:pc"}](gc_eigen.pdf "fig:"){width="40.00000%"} The left panel of Figure \[fig:ordination\] displays the observed trajectory of GCs versus the posterior mean of its imputed smooth process of three baboons who experienced zero (EAG), one (OCT) and two (GUI) sources of early adversity, respectively. We can see that the imputed smooth process generally captures the overall time trend of each subject while reduce the noise in the observations. The pattern is similar for the social bonds, which is shown in Supplementary C with a few more randomly selected subjects. Recall that each subject’s observed trajectory is fully captured by its vector of principal scores, and thus the principal scores of the first few dominant principal components adequately summarize the whole trajectory. The right panel of Figure \[fig:ordination\] shows the principal scores of the first (X-axis) versus second (Y-axis) principal component for the GCs process of all subjects in the sample, color-coded based on the number of early adversities experienced. We can see that significant difference exists in the distributions of the first two principal scores between the group who experienced no early adversity and the groups who experienced at least one source of adversity. ![Left panel: Observed trajectory of GCs versus the posterior mean of its imputed smooth process of three baboons who experienced zero (EAG), one (OCT) and two (GUI) sources of early adversity, respectively. Right panel: Principal scores of the first (X-axis) versus second (Y-axis) principal component for the GCs process of all subjects in the sample; color-coded based on the number of early adversities experienced. []{data-label="fig:ordination"}](gc_trajectory.pdf "fig:"){width="40.00000%"} ![Left panel: Observed trajectory of GCs versus the posterior mean of its imputed smooth process of three baboons who experienced zero (EAG), one (OCT) and two (GUI) sources of early adversity, respectively. Right panel: Principal scores of the first (X-axis) versus second (Y-axis) principal component for the GCs process of all subjects in the sample; color-coded based on the number of early adversities experienced. []{data-label="fig:ordination"}](gc_ordination.pdf "fig:"){width="40.00000%"} Results of causal mediation analysis ------------------------------------ Before proceeding to the causal mediation analysis, we first assess the plausibility of the key causal assumptions in the application. The ignorability assumption states that there is no unmeasured confounding, besides the observed covariates, between early adversity and the social bond and GCs processes. This is plausible in our application because the six sources of early adversity (e.g. maternal death, drought, low maternal rank) are all largely randomized by the nature. The sequential ignorability assumption states that there is no unmeasured confounding, besides the observed covariates and the history of social bond strength, between the social bond process and the GCs process. One possible violation can be due to the ‘feedback’ between the social bond and GCs processes. We have performed a sensitivity analysis by adding (a) the most recent prior observed GCs value, or (b) the average of all past observed GCs values, as a predictor in the mediation model, which lead to little difference in the results and thus bolsters sequential ignorability. Though we are not aware of the existence of other sequential confounders, we also cannot rule them out. We perform a separate causal mediation analysis for each source of early adversity. Table \[tab:results\] presents the posterior mean and 95% credible interval of the total effect (TE), direct effect (ANDE) and indirect effect mediated through social bonds (ACME) of each source of early adversity on adult GCs level, as well as the effects of early adversity on the mediator (social bonds). First, from the first column of Table \[tab:results\] we can see that experiencing any source of early adversity would reduce a baboon’s strength of social bond with other baboons in the adulthood. The negative effect is particularly severe for those experiencing drought, high group density or maternal death in early life. For example, compared with the baboons who did not experience any early adversity, the baboons who experienced maternal death have a $0.221$ unit decrease in social bonds, translating to approximately $0.4$ standard deviation of social bonds in the population. Overall, experiencing at least one source of adversity have a decrease of $0.2$ standard deviation of social bonds in adulthood. Second, from the second column of Table \[tab:results\] we can see a strong total effect of early adversity on female baboon’s GCs levels across adulthood. Baboons who experienced at least one source of adversity had GCs concentrations that were approximately 9% higher than their peers who did not experience any adversity. Although the range of total effect sizes across all individual adversity sources varies from 4% to 14%, the point estimates are consistent toward higher GCs levels, even for the early adversity sources of which the credible interval includes zero. Among the individual sources of adversity, females who were born during a drought, into a high-density group, or to a low-ranking mother had particularly elevated GCs concentrations (12-14%) in adulthood, although the credible interval of high group density includes zero. \[tab:results\] Third, while female baboons who experienced harsh conditions in early life show higher GCs levels in adulthood, we found no evidence that these effects were significantly mediated by the absence of strong social bonds. Specifically, the mediation effect $\tau_{{\mbox{\tiny{ACME}}}}$ (third column in Table \[tab:results\]) is consistently small; the strength of females’ social bonds with other females accounted for a difference in GCs of only 0.85% when averaged across the six individual adversity sources, even though it had credible intervals that did not include zero for five of the six individual sources of adversity. On the other hand, the direct effects $\tau_{{\mbox{\tiny{ANDE}}}}$ (fourth column in Table \[tab:results\]) are much stronger than the mediation effect. When averaged across the six adversity sources, the direct effect of early adversity on GCs level was 11.6 times stronger than the mediation effect running through social bonds. For example, for females who experienced at least one source of early adversity, the direct effect can explain 8.4% increase in GCs level, while the mediation effect only takes up 0.7% for the increase in GCs. The above findings of the causal relationships between early adversity, social bonds, and GCs levels in wild baboons are compatible with observations in many other species that early adversity and weak relationships both give rise to poor health, and that early adversity predicts various forms of social dysfunction, including weaker relationships. However, they call into question the notion that social bonds play a major role in mediating the effect of early adversity on poor health. In wild female baboons, any such effect appears to be functionally biologically irrelevant, and what little exists is limited strictly to their relationships with other females. Discussion\[Sec\_Discussion\] ============================= We proposed a framework for conducting causal mediation analysis with sparse and irregular longitudinal data. We defined several causal estimands (total, direct and indirect effects) in such settings and developed assumptions to nonparametrically identify these effects. For estimation and inference, we combine functional principal component analysis (FPCA) techniques and the standard two structural-equation-model system. In particular, we use a Bayesian FPCA model to reduce the dimensions of the observed trajectories of mediators and outcomes. We applied the proposed method to analyze the causal effects of early adversity on adult social bonds and adult GC hormone level in a sample wild female baboons. We found that experiencing adversity before maturity generally hampers a baboon’s ability to build social bond with other baboons and increase the GC hormones level in adulthood, which in turn lead to stress and diseases. However, the effects on the GC concentration from early adversity is not mediated by social bonds. One limitation of our analysis is that the identification of mediation effects (ACME and ANDE) relies on strong assumptions, particularly sequential ignorability. Though not routinely performed in the literature of mediation analysis, a formal sensitivity analysis would shed light on how reliant the proposed method is on sequential ignorability. Given the complexity of mediation analysis, a model-based approach appears to be the most feasible for sensitivity analysis [@imai2010identification]. Alternatively, one could consider the framework developed by [@didelez2012direct; @vanderweele2014effect] and relax sequential ignorability to allow for observed treatment-induced mediator-outcome confounding. This framework targets at a different set of causal estimands, namely the interventional direct or indirect effect, instead of the natural direct or indirect effects considered in this paper. We proposed a framework for conducting causal mediation analysis with sparse and irregular longitudinal data. We defined several causal estimands (total, direct and indirect effects) in such settings and developed assumptions to nonparametrically identify these effects. For estimation and inference, we combine functional principal component analysis (FPCA) techniques and the standard two structural-equation-model system. In particular, we use a Bayesian FPCA model to reduce the dimensions of the observed trajectories of mediators and outcomes. We applied the proposed method to analyze the causal effects of early adversity on adult social bonds and adult GC hormone level in a sample wild female baboons. We found that experiencing adversity before maturity generally hampers a baboon’s ability to build social bond with other baboons and increase the GC hormones level in adulthood, which in turn lead to stress and diseases. However, the effects on the GC concentration from early adversity is not mediated by social bonds. One limitation of our analysis is that the identification of mediation effects (ACME and ANDE) relies on strong assumptions, particularly sequential ignorability. Though not routinely performed in the literature of mediation analysis, a formal sensitivity analysis would shed light on how reliant the proposed method is on sequential ignorability. Given the complexity of mediation analysis, a model-based approach appears to be the most feasible for sensitivity analysis [@imai2010identification]. Alternatively, one could consider the framework developed by [@didelez2012direct; @vanderweele2014effect] and relax sequential ignorability to allow for observed treatment-induced mediator-outcome confounding. This framework targets at a different set of causal estimands, namely the interventional direct or indirect effect, instead of the natural direct or indirect effects considered in this paper. An important extension of our method is to incorporate time-to-event outcomes, a common setting in longitudinal studies [@lange2012simple; @vanderweele2011causal]. For example, it is of much scientific interest to extend our application to investigate the causal mechanisms between early adversity, social bonds, GCs level and survival time. A typical complication in mediation analysis with time-to-event outcomes and time-varying mediators is that the mediators are undefined for the time period in which a unit was not observed [@didelez2019defining; @vansteelandt2019mediation]. Within our framework, we can bypass this problem by imputing the underlying process of the mediators in an identical range for every unit. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Surya Tokdar, Stacy Rosenbaum, Fernando Campos, and Georgia Papadogeorgou for helpful discussions. This research is supported by NIH grants 1R01AG053330-01A1 and 1R01 AG053308-01A1. Supplementary {#supplementary .unnumbered} ============= A. Proof of Theorem 1 {#a.-proof-of-theorem-1 .unnumbered} --------------------- For the first part, identification of total effect, we have for any $z\in \{0,1\}$, $$\begin{aligned} E(Y_{i}^{t}|Z_{i}=z,\mathbf{X}_{i}^{t}) =E(Y_{i}^{t}(z,\mathbf{M}_{i}(z))|Z_{i}=z,\mathbf{X}_{i}^{t}) =E(Y_{i}^{t}(z,\mathbf{M}_{i}(z))|\mathbf{X}_{i}^{t}).\end{aligned}$$ The second equality follows from Assumption \[A.1\]. Therefore, we prove the identification of $\tau_{{\mbox{\tiny{TE}}}}^{t}$, $$\begin{aligned} \tau_{{\mbox{\tiny{TE}}}}^{t}&=\int_{X} \{ E(Y_{i}^{t}(1,\mathbf{M}_{i}(1))|\mathbf{X}_{i}^{t})-E(Y_{i}^{t}(0,\mathbf{M}_{i}(0))|\mathbf{X}_{i}^{t})\}\textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t}),\\ &=\int_{X} \{E(Y_{i}^{t}|Z_{i}=1,\mathbf{X}_{i}^{t}=\mathbf{x}^{t})-E(Y_{i}^{t}|Z_{i}=0,\mathbf{X}_{i}^{t}=\mathbf{x}^{t})\}\textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t}),\\\end{aligned}$$ For the second part, identification of $\tau^{t}_{{\mbox{\tiny{ACME}}}}$, we make the following regularity assumptions. Suppose the potential outcomes $Y_{i}^{t}(z,\mathbf{m})$ as a function of $\mathbf{m}$ is Lipschitz continuous on $[0,T]$ with probability one. There exists $A<\infty$ $|Y_{i}^{t}(z,\mathbf{m})-Y_{i}^{t}(z,\mathbf{m'})|\leq A||\mathbf{m}-\mathbf{m'}||_{2}$, for any $z,t,\mathbf{m},\mathbf{m'}$ almost surely. For any $z,z'\in\{0,1\}$, we have $$\begin{aligned} \int_{\mathcal{X}} \int_{R^{[0,t]}} E(Y_{i}^{t}|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m}) \textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t})\times\textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\}\\ =\int_{\mathcal{X}} \int_{R^{[0,t]}}E(Y_{i}^{t}(z,\mathbf{m})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m})\times\textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\}.\end{aligned}$$ For any path $\mathbf{m}$ on the time span $[0,t]$, we make a finite partition into $H$ pieces at points $\mathcal{M}_{H}=\{t_{0}=0,t_{1}=t/H,t_{2}=2t/H,\cdots,t_{H}=t\}$. Now we consider using a step functions with jumps at points $\mathcal{M}_{H}$. Denote the step function as $\mathbf{m}_{H}$, which is: $$\begin{aligned} \mathbf{m}_{H}(x)=\begin{cases} \mathbf{m}(0)=m_{0} & 0\leq x< t/H, \\ \mathbf{m}(t/H)=m_{1} & t/H\leq x<2t/H, \\ \cdots \\ \mathbf{m}((H-1)t/H)=m_{H} & (H-1)t/H\leq x\leq t. \\ \end{cases} \end{aligned}$$ We wish to use this step function $\mathbf{m}_{H}(x)$ to approximate function $\mathbf{m}$. First, given $\mathbf{m}$ is Lipschitz continuous, there exists $B>0$ such that $|m(x_{1})-m(x_{2})|\leq B|x_{1}-x_{2}|$. Therefore, the step functions $\mathbf{m}_{H}$ approximates the original function $\mathbf{m}$ well in the sense that, $$\begin{aligned} ||\mathbf{m}_{H}-\mathbf{m}||_{2}\leq \sum_{i=1}^{H}\frac{t}{H} B^{2}\frac{t^{2}}{H^{2}} \asymp O(H^{-2}).\end{aligned}$$ As such we can approximate the expectation over a continuous process with expectation on a vector with values on the jumps, $(m_{0},m_{1},\cdots,m_{H})$. That is, $$\begin{aligned} \int_{\mathcal{X}} \int_{R^{[0,t]}}E(Y_{i}^{t}(z,\mathbf{m})&|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m})\times\textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\}\\ &\asymp\int_{\mathcal{X}} \int_{R^{[0,t]}}E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m}_{H})\\ &\times\textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m}_{H})\}+O(H^{-2}).\end{aligned}$$ This equivalence follows from the regularity condition that the potential outcome $Y_{i}^{t}(z,\mathbf{m})$ as a function of $\mathbf{m}$ is continuous with the $L_{2}$ metrics of $\mathbf{m}$. As the values of steps function $\mathbf{m}_{H}$ are completely determined by the values on finite jumps, we can further reduce to, $$\begin{aligned} \asymp\int_{\mathcal{X}} \int_{R^{H}}E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t},m_{0},m_{1},m_{2},\cdots m_{H})\\ \times\textup{d}\{\textup{F}_{m_{0},m_{1},\cdots,m_{H}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(m_{0},m_{1},m_{2},\cdots m_{H})\}+O(H^{-2}).\end{aligned}$$ With Assumption \[A.1\], we can show that $$\begin{aligned} &\textup{d}\{\textup{F}_{m_{0},m_{1},\cdots,m_{H}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(m_{0},m_{1},m_{2},\cdots m_{H})\}\\ &=\textup{d}\{\textup{F}_{m_{0}(z'),m_{1}(z'),\cdots,m_{H}(z')|\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(m_{0},m_{1},m_{2},\cdots m_{H})\},\\ &=\textup{d}\{\textup{F}_{\mathbf{m}_{H}(z')|\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m}_{H})\}.\end{aligned}$$ With a slightly abuse of notations, we use $\mathbf{m}_{H}(z)$ to denote the potential process induced by the original potential process $\mathbf{M}_{i}^{t}(z)$ and $m_{i}(z)$ to denote potential values of $\mathbf{M}_{i}^{t}(z)$ evaluated at point $x_{i}=it/H$. Also, with the assumption \[A.2\], we can choose a large $H$ such that $t/H\leq \varepsilon$. Then we have the following conditional independence conditions, $$\begin{aligned} Y_{i}^{0}(z,\mathbf{m}_{H}) {\protect\mathpalette{\protect\independenT}{\perp}}& m_{0}|Z_{i},\mathbf{X}_{i}^{t},\\ \{Y_{i}^{t/H}(z,\mathbf{m}_{H})- Y_{i}^{0}(z,\mathbf{m}_{H})\} {\protect\mathpalette{\protect\independenT}{\perp}}& (m_{1}-m_{0})|Z_{i},\mathbf{X}_{i}^{t},\mathbf{m}_{H}^{0},\\ \{Y_{i}^{2t/H}(z,\mathbf{m}_{H})- Y_{i}^{t/H}(z,\mathbf{m}_{H})\} {\protect\mathpalette{\protect\independenT}{\perp}}& (m_{2}-m_{1})|Z_{i},\mathbf{X}_{i}^{t},\mathbf{m}_{H}^{t/H},\\ \cdots& \\ \{Y_{i}^{t}(z,\mathbf{m}_{H})- Y_{i}^{t(H-1)/H}(z,\mathbf{m}_{H})\} {\protect\mathpalette{\protect\independenT}{\perp}}& (m_{H}-m_{H-1})|Z_{i},\mathbf{X}_{i}^{t},\mathbf{m}_{H}^{t(H-1)/H},\\\end{aligned}$$ where are equivalent to, $$\begin{aligned} Y_{i}^{0}(z,\mathbf{m}_{H}) {\protect\mathpalette{\protect\independenT}{\perp}}& m_{0}|Z_{i},\mathbf{X}_{i}^{t},\\ \{Y_{i}^{t/H}(z,\mathbf{m}_{H})- Y_{i}^{0}(z,\mathbf{m}_{H})\} {\protect\mathpalette{\protect\independenT}{\perp}}& (m_{1}-m_{0})|Z_{i},\mathbf{X}_{i}^{t},m_{0},\\ \{Y_{i}^{2t/H}(z,\mathbf{m}_{H})- Y_{i}^{t/H}(z,\mathbf{m}_{H})\} {\protect\mathpalette{\protect\independenT}{\perp}}& (m_{2}-m_{1})|Z_{i},\mathbf{X}_{i}^{t},m_{0},m_{1},\\ \cdots& \\ \{Y_{i}^{t}(z,\mathbf{m}_{H})- Y_{i}^{t(H-1)/H}(z,\mathbf{m}_{H})\} {\protect\mathpalette{\protect\independenT}{\perp}}& (m_{H}-m_{H-1})|Z_{i},\mathbf{X}_{i}^{t},m_{0},m_{1}\cdots,m_{H-1},\\\end{aligned}$$ as the step function $m_{H}^{it/H}$ is completely determined by values $m_{0},\cdots,m_{i}$. With the above conditional independence, we have, $$\begin{aligned} E(Y_{i}^{t}(z,\mathbf{m}_{H})&|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t},m_{0},m_{1},m_{2},\cdots m_{H})\\ &=E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}).\end{aligned}$$ With similar arguments, it also equals: $$\begin{aligned} &E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t})=E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}),\\ &=E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t},m_{0}=m_{0}(z'),\cdots m_{H}=m_{H}(z')),\\ &=E(Y_{i}^{t}(z,\mathbf{m}_{H})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t},\mathbf{m}_{H}(z')=\mathbf{m}_{H}),\\ &=E(Y_{i}^{t}(z,\mathbf{m}_{H})|\mathbf{X}_{i}^{t}=\mathbf{x}^{t},\mathbf{m}_{H}(z')=\mathbf{m}_{H}).\end{aligned}$$ As a conclusion, we have shown that, $$\begin{aligned} &\int_{\mathcal{X}} \int_{R^{[0,t]}}E(Y_{i}^{t}(z,\mathbf{m})|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m}) \times\textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\},\\ & \asymp \int_{\mathcal{X}} \int_{R^{[0,t]}}E(Y_{i}^{t}(z,\mathbf{m}_{H})|\mathbf{X}_{i}^{t}=\mathbf{x}^{t},\mathbf{m}_{H}(z')=\mathbf{m}_{H})\\ &\times \textup{d}\{\textup{F}_{\mathbf{m}_{H}(z')|\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m}_{H})\}+O(H^{-2}),\\ & \asymp \int_{\mathcal{X}} E(Y_{i}^{t}(z,\mathbf{m}_{H}(z'))|\mathbf{X}_{i}^{t}=\mathbf{x}^{t})+O(H^{-2}),\\ & \asymp \int_{\mathcal{X}} E(Y_{i}^{t}(z,\mathbf{m}(z'))|\mathbf{X}_{i}^{t}=\mathbf{x}^{t})+O(H^{-2}).\end{aligned}$$ The last equivalence comes from the regularity condition of $Y_{i}^{t}(z,\mathbf{m}(z'))$ as a function of $\mathbf{m}(z')$. Let $H$ goes to infinity, we have, $$\begin{aligned} \int_{\mathcal{X}} \int_{R^{[0,t]}} E(Y_{i}^{t}|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t} =\mathbf{m}) \textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t})\times\textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=z',\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\}\\=\int_{\mathcal{X}} E(Y_{i}^{t}(z,\mathbf{m}(z'))|\mathbf{X}_{i}^{t}=\mathbf{x}^{t})\textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t}).\end{aligned}$$ With this relationship established, it is straightforward to show that, $$\begin{aligned} \tau_{{\mbox{\tiny{ACME}}}}^{t}(z)&=\int_{\mathcal{X}} \{E(Y_{i}^{t}(z,\mathbf{m}(1))|\mathbf{X}_{i}^{t}=\mathbf{x}^{t})-E(Y_{i}^{t}(z,\mathbf{m}(0))|\mathbf{X}_{i}^{t}=\mathbf{x}^{t})\}\textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t}),\\ &=\int_{\mathcal{X}} \int_{R^{[0,t]}} E(Y_{i}^{t}|Z_{i}=z,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}, \mathbf{M}_{i}^{t}=\mathbf{m}) \textup{dF}_{\mathbf{X}_{i}^{t}}(\mathbf{x}^{t})\times\\ & \quad\quad \quad \textup{d}\{\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=1,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m}) -\textup{F}_{\mathbf{M}_{i}^{t}|Z_{i}=0,\mathbf{X}_{i}^{t}=\mathbf{x}^{t}}(\mathbf{m})\}, \end{aligned}$$ which completes the proof. B. Gibbs Sampler {#b.-gibbs-sampler .unnumbered} ---------------- In this section, we provide detailed descriptions on the Gibbs sampler for the model in Section 4. We only include the sampler for mediator process as the sampling procedure is essentially identical for the outcome process. For simplicity, we introduce some notations to represent vector values, $M_{i}=(M_{i1},M_{i2},\cdots,M_{in_{i}})\in \mathcal{R}^{T_{i}}$,$X_{i}=[X_{i1},X_{i2},\cdots,X_{in_{i}}]'\in \mathcal{R}^{T_{i}\times p}$, $\psi_{r}(\mathbf{t}_{i})=[\psi_{r}(t_{i1}),\cdots,\psi_{r}(t_{in_{i}})]\in\mathcal{R}^{T_{i}}$ 1. **Sample the eigen function $\psi_{r}(t),r=1,2\cdots,R$.** - (a)$\psi_{r}|\cdots \sim N(Q_{\phi_{rss^{-1}l_{\phi_{r}}}},Q_{\phi_{r}}^{-1})$ conditional on $C_{r}\psi_{r}=0$, $C_{r}$=\ $[\psi_{1},\psi_{2},\cdots,\psi_{r-1},\psi_{r+1},\cdots,\psi_{R}]'B_{G}=[\psi_{1},\cdots,\psi_{r-1},\psi_{r+1},\cdots, \psi_{R}]B_{G}'B_{G}$, $B_{G}$ is the basis functions evaluated at a equal spaced grids on \[0,1\],$\{t_{1},t_{2},\cdots,t_{G}\}$, $G=50$ for example, $B_{G}=[\mathbf{b}(t_{1}),\cdots ,\mathbf{b}(t_{G})]'\in R^{G\times (L+2)}$. The corresponding mean and covairance functions are, $$\begin{gathered} Q_{\psi_{r}}=\frac{\sum_{i=1}^{N}B_{i}'B_{i}\zeta_{r,i}^{2}}{\sigma_{m}^{2}}+h_{k}\Omega,\\ l_{\psi_{r}}=\frac{\sum_{i=1}^{N}B_{i}'\zeta_{i,r}(M_{i}-X_{i}\beta_{M}^{T}-\sum_{r'\neq r}^{R}\psi_{r}(\mathbf{t_{i}})\zeta_{r',i})}{\sigma_{m}^{2}}. \end{gathered}$$ Update the $\psi_{r}\leftarrow\psi_{r}/\sqrt{\psi_{r}'B_{G}'B_{G}\psi_{r}}=\psi_{r}/||\psi_{r}(t)||_{2}$ to ensure $||\psi_{r}(t)||_{2}=1$ and $\psi_{r}(t)=\mathbf{b}(t)\psi_{r}$ and update $\zeta_{r,i}=\rightarrow \zeta_{r,i}*||\psi_{r}(t)||_{2}$ to maintain likelihood function. - (b)$h_{k}|\cdots \sim \textup{Ga}((L+1)/2,\psi_{r}'\Omega\psi_{r})$ truncated on $[\lambda_{r}^{2},10^{4}]$. 2. **Sample the principal score $\zeta_{r,i}$**.$\zeta_{r,i}|\cdots \sim N(\mu_{r}/\lambda_{r}^{2 },\lambda_{r}^{2})$ $$\begin{gathered} \sigma_{r}^{2}=(||\psi_{r}(\mathbf{t}_{i})||_{2}^{2}/\sigma_{m}^{2}+\xi_{i,r}/\lambda_{r}^{2})^{-1}\\ \mu_{r}=\frac{(M_{i}-X_{i}\beta_{M}^{T}-(\sum_{r'\neq r}\phi_{r'}(\mathbf{t}_{i})\zeta_{r',i}))'\psi_{r}(\mathbf{t_{i}}) }{\sigma_{\varepsilon}^{2}}+\frac{(\tau_{0,r}(1-Z_{i})+\tau_{1,r}Z_{i})\xi_{i,r} }{\lambda_{r}^{2}} \end{gathered}$$ 3. **Sample the causal parameters $\chi_{0}^{r},\chi_{1}^{r}$**. Let $\chi_{z}=(\chi_{z}^{r},\cdots,\chi_{z}^{R}),z=0,1$,$\chi_{z}^{r}|\cdots \sim N(Q_{z,r}^{-1}l_{z,r},Q_{z,r}^{-1})$. $$\begin{gathered} Q_{z,r}=(\sum_{i=1}^{N} \xi_{r,i}\mathbf{1}_{Z_{i}=z}/\lambda_{r}^{2}+1/\sigma_{\chi_{r}}^{2})^{-1}\\ l_{z,r}=\sum_{i=1}^{N} \zeta_{r,i}\xi_{r,i}\mathbf{1}_{Z_{i}=z}/\lambda_{r}^{2} \end{gathered}$$ 4. **Sample the coefficients $\beta_{M}$**. The coefficients for covariates are $\beta_{M}|\cdots \sim N(Q_{\beta}^{-1}\mu_{\beta},Q_{\beta}^{-1})$, $$\begin{gathered} Q_{\beta}=X'X/\sigma_{m}^{2}+100^{2}I_{\textup{dim}(X)}\\ \mu_{\beta}=\sum_{i=1}^{N}X_{i}'(M_{i}-\sum_{r=1}^{R}\psi_{r}(\mathbf{t_{i}})\zeta_{i,r})/\sigma_{m}^{2} \end{gathered}$$ 5. **Sample the precision/variance parameters**. - \(a) $\sigma_{m}^{-2}|\cdots\sim \textup{Ga}(\sum_{i=1}^{N}T_{i}/2,\sum_{i=1}^{N}||M_{i}-X_{i}\beta_{M}'-\sum_{r=1}^{R}\psi_{r}(\mathbf{t_{i}})\zeta_{i,r}||_{2}^{2}/2)$ - \(b) $\sigma_{\chi_{r}}^{2}|\cdots$, $$\begin{gathered} \delta_{\chi_{1}}|\cdots \sim \textup{Ga}(a_{\chi_{1}}+R,1+\frac{1}{2}\sum_{r=1}^{R}\chi_{1}^{(r)}(\chi_{0}^{r 2}+\chi_{1}^{r 2})),\chi_{l}^{(r)}=\prod_{i=l+1}^{r}\delta_{\chi_{i}}\\ \delta_{\chi_{r}}|\cdots \sim \textup{Ga}(a_{\chi_{2}}+R+1-r,1+\frac{1}{2}\sum_{r'=r}^{R}\chi_{r'}^{(r)}(\tau_{0}^{r' 2}+\chi_{1}^{r' 2})),r\geq 2,\\ \sigma_{\chi_{r}}^{-2}=\prod_{r'=1}^{r}\delta_{\chi_{r'}}. \end{gathered}$$ - (c)$\lambda_{r}^{2}|\cdots$, $$\begin{gathered} \delta_{1}|\cdots\sim \textup{Ga}(a_{1}+RN/2,1+\frac{1}{2}\sum_{r=1}^{R} \chi_{1}^{(r)'}\xi_{i,r}(\zeta_{i,r}-(1-Z_{i})\chi_{0}^{r}-Z_{i}\chi_{1}^{r})^{2},\\ \chi_{l}^{(r)'}=\prod_{i=l+1}^{r}\delta_{i} \end{gathered}$$ $$\begin{aligned} \delta_{r}|\cdots \textup{Ga}(a_{2}+&(R-r+1)N/2,\\ &1+\frac{1}{2}\sum_{r'=r}^{R}\chi_{r'}^{(r)'}\xi_{i,r'}(\zeta_{i,r'}-(1-Z_{i})\chi_{0}^{r'}-Z_{i}\chi_{1}^{r'})^{2}),r\geq 2\\ &\lambda_{r}^{-2}=\prod_{r'=1}^{r}\delta_{r'}. \end{aligned}$$ - \(d) $\xi_{i,r}|\cdots\sim \textup{Ga}(\frac{v+1}{2},\frac{1}{2}(v+(\zeta_{i,r'}-(1-Z_{i})\chi_{0}^{r'}-Z_{i}\chi_{1}^{r'})^{2}/\lambda_{r}^{2}))$. - \(e) $a_{1},a_{2},a_{\chi_{1}},a_{\chi_{2}}$ can be sampled with Metropolis-Hasting algorithm. The sampling for the outcomes model $Y_{ij}$ is similar to that for the mediator model except that we added the imputed value of the mediator process $M(t_{ij})$ as a covariate. C. Imputed processes of the mediators and outcomes of eight subjects {#c.-imputed-processes-of-the-mediators-and-outcomes-of-eight-subjects .unnumbered} -------------------------------------------------------------------- Figure \[fig:imputation\] shows the posterior means of the imputed smooth processes of the mediators and the outcomes against their respective observed trajectories of eight randomly selected subjects in the sample. For social bonds (left panel of Figure \[fig:imputation\]), the imputed smooth process adequately captures the overall time trend of each subject while reduce the noise in the observations, evident in the subjects with code name HOK, DUI and LOC. ![The imputed underlying smooth process against the observed trajectories for social bonds (left panel) and GCs levels (right panel).[]{data-label="fig:imputation"}](DSI_F_individual_trajectory.pdf "fig:"){width="45.00000%"} ![The imputed underlying smooth process against the observed trajectories for social bonds (left panel) and GCs levels (right panel).[]{data-label="fig:imputation"}](GC_individual_trajectory.pdf "fig:"){width="45.00000%"} For the subjects with few observations or observations concentrating in a short time span, such as subject NEA, the imputed process matches the trend of the observations while extrapolating to the rest of the time span with little information. FPCA achieves this by borrowing information from other units when learning the principal component on the population level. Compared with social bonds, variation of the adult GCs levels across the lifespan is much smaller. In the right panel in Figure \[fig:imputation\], we can see the imputed processes for the GCs levels are much flatter than those for social bonds. It appears that most variation in the GCs trajectories is due to noise rather than intrinsic developmental trend.

--- abstract: 'In this review I briefly describe the nature of the three kinds of High-Mass X-ray Binaries (HMXBs), accreting through: (i) Be circumstellar disc, (ii) supergiant stellar wind, and (iii) Roche lobe filling supergiants. A previously unknown population of HMXBs hosting supergiant stars has been revealed in the last years, with multi-wavelength campaigns including high energy ([*INTEGRAL, Swift, XMM, Chandra*]{}) and optical/infrared (mainly ESO) observations. This population is divided between obscured supergiant HMXBs, and supergiant fast X-ray transients (SFXTs), characterized by short and intense X-ray flares. I discuss the characteristics of these types of supergiant HMXBs, propose a scenario describing the properties of these high-energy sources, and finally show how the observations can constrain the accretion models (e.g. clumpy winds, magneto-centrifugal barrier, transitory accretion disc, etc). Because they are the likely progenitors of Luminous Blue Variables (LBVs), and also of double neutron star systems, related to short/hard gamma-ray bursts, the knowledge of the formation and evolution of this HMXB population is of prime importance.' address: | Laboratoire AIM (UMR 7158 CEA/DSM-CNRS-Université Paris Diderot), Irfu/Service d’Astrophysique, Centre de Saclay, FR-91191 Gif-sur-Yvette Cedex, France\ Institut Universitaire de France, 103, boulevard Saint-Michel, FR-75005 Paris, France\ $^*$E-mail: [email protected]\ author: - 'Sylvain Chaty$^*$' title: 'High Mass X-ray Binaries: Progenitors of double neutron star systems' --- High-Mass X-ray Binaries ======================== High energy binary systems are composed of a compact object – neutron star (NS) or black hole (BH) – orbiting around and accreting matter from a companion star (see review Ref. ). The companion star is either a low-mass star ($\sim 1 \Msol$ or less, with a spectral type later than B, called LMXB for “Low-Mass X-ray Binary”), or a luminous early spectral type OB high-mass star ($\sim 10 \Msol$ or more, called HMXB for “High-Mass X-ray Binary”). $\sim 300$ high energy binary systems are now known in our Galaxy: 187 LMXBs and 114 HMXBs (respectively 62% and 38% of the total number, Ref.  & ). The number of HMXBs is an indicator of star formation rate and starburst activity (Ref. ). Accretion of matter is different for both types of sources. For LMXBs, the small and low-mass companion star fills and overflows its Roche lobe, and accretion of matter always occurs through the formation of an accretion disc. For HMXBs, while accretion can also occur through an accretion disc for Roche lobe filling systems, this is generally not the case, and there are two alternatives, that we now further describe. Be X-ray binaries ----------------- The first one concerns the case of HMXBs containing a main sequence early spectral type B0-B2e III/IV/V donor star, called in the following BeHMXBs. These rapidly rotating stars possess a circumstellar disc of gas created by a low velocity and high density stellar wind of $\sim 10^{-7} \Msol / yr$. This “decretion” disc is characterized by an H$\alpha$ emission line (whose width is correlated with the disc size) and a continuum free-free/free-bound emission, causing an infrared excess (Ref. ). In these systems, accretion periodically occurs, with transient and bright X-ray outbursts: [*i.*]{} “Type I” are regular and periodic outbursts each time the compact object –usually a NS on a wide and eccentric orbit– crosses the disc at periastron; [*ii.*]{} “Type II” are giant outbursts at any phase, with a dramatic expansion of the disc, enshrouding the NS; [*iii.*]{} “Missed” outbursts exhibit low H$\alpha$ emission (due to a small disc or a centrifugal inhibition of accretion), iv. “Shifting phase outbursts” are likely due to the rotation of density structures in the circumstellar disc[^1]. Supergiant X-ray binaries ------------------------- The second one concerns HMXBs with an early spectral type supergiant OB I/II donor star, later called sgHMXBs. These massive stars eject a steady, slow and dense wind, radially outflowing from the equator, and the compact object –usually a NS on a circular orbit– directly accretes the stellar wind through e.g. Bondy-Hoyle-Littleton process. We distinguish two groups: Roche lobe overflow and wind-fed systems[^2]. The former group constitutes the classical «bright» sgHMXBs with accreted matter flowing via inner Lagrangian point to the accretion disc, causing a high X-ray luminosity ($L_X \sim 10^{38} \ergs$) during outbursts. The later group concerns close systems ($P_\mathrm{orb} < 15$days) with a low eccentricity, the NS accreting from deep inside the strong steady radiative and highly supersonic stellar wind. These systems exhibit a persistent X-ray emission at regular low-level effect ($L_X \sim 10^{35-36} \ergs$), on which are superimposed large variations on short timescales, due to wind inhomogeneities. During their long term evolution, the orbits of sgHMXBs tend to circularize more rapidly with time, while the rate of mass transfer steadily increases (Ref. ). A milestone in the evolution of these binary systems takes place during the so-called “common envelope phase”. This phase initiates when the compact object penetrates inside the envelope of the companion star, in a rapidly inward spiralling orbit due to a large loss of orbital angular momentum. This phase has been invoked in Ref.  to explain how high energy binary systems with very short $P_\mathrm{orb}$ can be formed, while both components of these systems – large stars at their formation – would not have been able to fit inside a binary system with such a small orbital separation. This phase, while taken into account in population synthesis models, has never been observed yet, probably because it is short (models predict a maximum duration of common envelope phase of only $\sim 1000$years [@meurs:1989]) compared to the lifetime of a massive star ($\sim 10^{6-7}$years). It is a fundamental ingredient to understand the evolution of high energy binary systems [@tauris:2006]. The [*INTEGRAL*]{} supergiant revolution {#section:INTEGRAL} ======================================== The [*INTEGRAL*]{} observatory is an ESA satellite launched on 17 October 2002 by a PROTON rocket on an eccentric orbit. It hosts 4 instruments: 2 $\gamma$-ray coded-mask telescopes –imager IBIS and spectro-imager SPI, observing in the range 10 keV-10 MeV, with a resolution of $12\amin$ and a field-of-view of $19\adeg$–, a coded-mask telescope JEM-X (3-100 keV), and an optical telescope (OMC). The $\gamma$-ray sky seen by [*INTEGRAL*]{} is very rich, with 723 sources detected, reported in the $4^{th}$ IBIS/ISGRI soft $\gamma$-ray catalogue [@bird:2010], spanning nearly 7 years of observations in the 17-100 keV domain[^3]. Among these sources, there are 185 X-ray binaries (representing 26% of the whole sample, called “IGRs” in the following), 255 Active Galactic Nuclei (35%), 35 Cataclysmic Variables (5%), and $\sim 30$ sources of other type (4%): 15 SNRs, 4 Globular Clusters, 3 Soft $\gamma$-ray Repeaters, 2 $\gamma$-ray bursts, etc. 215 objects still remain unidentified (30%). X-ray binaries are separated into 95 LMXBs and 90 HMXBs, each category representing $\sim 13$% of IGRs. Among identified HMXBs, there are 24 BeHMXBs and 19 sgHMXBs (resp. 31% and 24% of HMXBs). It is interesting to follow the evolution of the ratio between BeHMXBs and sgHMXBs [@chaty:2013]. During the pre-[*INTEGRAL*]{} era, HMXBs were mostly BeHMXBs. In the catalogue of 110 HMXBs[@liu:2000], there were 52 BeHMXBs and 13 sgHMXBs (respectively 47% and 12% of HMXBs). Then, the situation drastically changed with the discovery by [*INTEGRAL*]{} of 24 sgHMXBs: in the catalogue of 114 HMXBs [@liu:2006], there were 60 BeHMXBs and 37 sgHMXBs (respectively 50% and 32% of HMXBs). Therefore, while the BeHMXB/HMXB ratio remained stable, the sgHMXB/HMXB ratio nearly tripled. The ISGRI energy range ($> 20$keV), immune to the absorption that prevented the discovery of intrinsically absorbed sources by earlier soft X-ray telescopes, allowed us to go from a study of individual sgHMXBs (such as GX301-2, 4U1700-377, VelaX-1, etc) to a comprehensive study of the characteristics of a whole population of HMXBs [@coleiro:2013a]. The most important result of [*INTEGRAL*]{} is the discovery of new X-ray wind-accreting pulsars, mainly sgHMXBs – concentrated towards tangential directions of Galactic arms, rich in star forming regions –, exhibiting common characteristics which previously had rarely been seen, with longer spin periods and higher absorption, compared to previously known sgHMXBs [@bodaghee:2007; @chaty:2008a; @rahoui:2008a; @coleiro:2013b]. The classical and obscured, the fast and eccentric ================================================== After an extensive multi-wavelength study of IGR sources by various groups, we have reached the consensus that sgHMXBs can be sub-divided into two main classes, which are likely connected, representing two distinct positions within the continuum of their general characteristics: Classical and obscured sgHMXBs ------------------------------ There are $\sim 16$ classical and persistent sgHMXBs, nearly half of them exhibit a substantial intrinsic and local extinction: there are $\sim 8$ such obscured sgHMXBs, the most extreme example being the highly absorbed source IGR J16318-4848[@chaty:2012a]. These systems share the following common properties: O8-B1 spectral type companion stars, $\nh \geq 10^{23} \cmmoinsdeux$, compact object on a short/circular orbit ($P_{orb} \sim 3.7-9.7$days), and luminous X-ray emission ($L_X = 10^{36-38} \ergs$). In transition to Roche Lobe Overflow, these systems are characterized by slow winds, causing a deep spiral-in, leading to Common Envelope Phase. A dichotomy exists in the orbital period: systems with P$_{orb} \leq 6$days[^4], and systems with P$_{orb} \geq 9$days[^5]. Fast and eccentric sgHMXBs -------------------------- Fast and transient X-ray outbursts –an unusual characteristic among HMXBs–, are the signature of the so-called Supergiant Fast X-ray Transients (SFXTs [@negueruela:2006a]). There are 17 (+5 candidate) SFXTs (thus representing a significant subclass of sgHMXBs)[^6]. These SFXTs are divided in three sub-classes: 7 classic-like systems, 4 fast transients reaching anomalously low luminosities, 3 eccentric systems (and 3 unclear sources). Their common characteristics are: a compact object on a short/circular orbit, transient and intense X-ray flares detected any time on the orbit, rising in tens of minutes ($L_X \sim 10^{35-37} \ergs$), lasting a few hours, and alternating with long ($\sim 70$days) quiescence ($L_X \sim 10^{32-34} \ergs$), with an impressive variability factor $\frac{L_{max}}{L_{min}}$ going from $15-50$ to $10^{2-5}$. They host OB supergiant companions, with P$_{orb} \sim 3.3 - 54$days; absorbed ($\nh \sim 10^{22} \cmmoinsdeux$) cut-off (10-30 keV) power-law X-ray spectrum; with X-ray pulsations (from a few to 1000s seconds) implying that they are young neutron star ($B \sim 10^{11-12}$G). Accretion processes ------------------- Various accretion processes have been proposed to account for X-ray properties of different kinds of sgHMXBs, we enumerate here the main mechanisms[^7]: i.) Clumpy stellar wind accretion, ii.) Magnetic/centrifugal gating mechanism, iii.) Hydrodynamic properties of accretion stream («breathing» of shock front), iv.) Formation/dissipation of temporary accretion disc, and v.) Cooling switch. Conclusions {#section:conclusion} =========== While the [*INTEGRAL*]{} satellite was not primarily designed for this, it allowed a great progress in the study of HMXBs in general, and of sgHMXBs in particular. Let us recall the [*INTEGRAL*]{} legacy on sgHMXBs. First, the [*INTEGRAL*]{} satellite tripled the total number of known sgHMXBs in our Galaxy, most of them being slow and absorbed wind-fed accreting X-ray pulsars. Second, the [*INTEGRAL*]{} satellite revealed the existence in our Galaxy of previously hidden populations of high energy binary systems: i.) a population of persistent and obscured sgHMXBs, exhibiting long $P_\mathrm{spin}$ ($\sim1$ks) and strong intrinsic absorption (large $\nh$, with the NS deeply embedded in the dense stellar wind). ii.) the SFXTs, hosting supergiant companion stars and exhibiting brief and intense X-ray flares (luminosity $L_X \sim 10^{36} \ergs$ at the peak, during a few ks every $\sim 7$days), which can be explained by various accretion processes. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to warmly thank Thomas Tauris, as an efficient organizer and chairman of the parallel session BN3 - Double Neutron Stars and Neutron Star-White Dwarf Binaries. I am lifelong indebted to my close collaborators on the study of [*INTEGRAL*]{} sources: A. Coleiro, P.A. Curran, Q.Z. Liu, I. Negueruela, L. Pellizza, F. Rahoui, J. Rodriguez, M. Servillat, J.A. Tomsick, J.Z. Yan, and J.A. Zurita Heras. This work, supported by the Centre National d’Etudes Spatiales (CNES), was based on observations obtained with MINE –Multi-wavelength [*INTEGRAL*]{} NEtwork–. [10]{} S. [Chaty]{}, [Optical/infrared observations unveiling the formation, nature and evolution of High-Mass X-ray Binaries]{}, [*Advances in Space Research*]{} [ **52**]{}, 2132 (December 2013). Q. Z. [Liu]{}, J. [van Paradijs]{} and E. P. J. [van den Heuvel]{}, [A catalogue of low-mass X-ray binaries in the Galaxy, LMC, and SMC (Fourth edition)]{}, [ *A&A*]{} [**469**]{}, 807 (July 2007). Q. Z. [Liu]{}, J. [van Paradijs]{} and E. P. J. [van den Heuvel]{}, [Catalogue of high-mass X-ray binaries in the Galaxy (4th edition)]{}, [*A&A*]{} [**455**]{}, 1165 (September 2006). A. [Coleiro]{} and S. [Chaty]{}, [Distribution of High-mass X-Ray Binaries in the Milky Way]{}, [*ApJ*]{} [**764**]{}, p. 185 (February 2013). I. [Negueruela]{}, [Populations of Massive X-ray binaries]{}, in [*Revista Mexicana de Astronomia y Astrofisica Conference Series*]{}, eds. G. [Tovmassian]{} and E. [Sion]{}July 2004. P. A. [Charles]{} and M. J. [Coe]{}, [*Compact stellar X-ray sources.*]{}, in [ *Compact stellar X-ray sources*]{}, ed. [Lewin, W. H. G. & van der Klis, M.]{} (Cambridge Astrophysics Series, Cambridge University Press, April 2006), ch. [Optical, ultraviolet and infrared observations of X-ray binaries]{}, pp. 215–265. P. [Podsiadlowski]{}, S. [Rappaport]{} and Z. [Han]{}, [On the formation and evolution of black hole binaries]{}, [*MNRAS*]{} [**341**]{}, 385 (May 2003). L. [Kaper]{}, A. [van der Meer]{} and A. H. [Tijani]{}, [High-mass X-ray binaries and OB runaway stars]{}, in [*Revista Mexicana de Astronomia y Astrofisica Conference Series*]{}, eds. C. [Allen]{} and C. [Scarfe]{}August 2004. B. [Paczynski]{}, [Common Envelope Binaries]{}, in [*Structure and Evolution of Close Binary Systems*]{}, eds. P. [Eggleton]{}, S. [Mitton]{} and J. [Whelan]{}, IAU Symposium, Vol. 731976. E. J. A. [Meurs]{} and E. P. J. [van den Heuvel]{}, [The number of evolved early-type close binaries in the Galaxy]{}, [*A&A*]{} [**226**]{}, 88 (December 1989). T. M. [Tauris]{} and E. P. J. [van den Heuvel]{}, [*Compact stellar X-ray sources*]{} (Cambridge Astrophysics Series, Cambridge University Press, April 2006), ch. [Formation and evolution of compact stellar X-ray sources]{}, pp. 623–665. A. J. [Bird]{}, A. [Bazzano]{}, L. [Bassani]{}, F. [Capitanio]{}, M. [Fiocchi]{}, A. B. [Hill]{}, A. [Malizia]{}, V. A. [McBride]{}, S. [Scaringi]{}, V. [Sguera]{}, J. B. [Stephen]{}, P. [Ubertini]{}, A. J. [Dean]{}, F. [Lebrun]{}, R. [Terrier]{}, M. [Renaud]{}, F. [Mattana]{}, D. [G[ö]{}tz]{}, J. [Rodriguez]{}, G. [Belanger]{}, R. [Walter]{} and C. [Winkler]{}, [The Fourth IBIS/ISGRI Soft Gamma-ray Survey Catalog]{}, [*ApJSS*]{} [**186**]{}, 1 (January 2010). Q. Z. [Liu]{}, J. [van Paradijs]{} and E. P. J. [van den Heuvel]{}, [A catalogue of high-mass X-ray binaries]{}, [*A&ASS*]{} [**147**]{}, 25 (November 2000). A. [Bodaghee]{}, T. J.-L. [Courvoisier]{}, J. [Rodriguez]{}, V. [Beckmann]{}, N. [Produit]{}, D. [Hannikainen]{}, E. [Kuulkers]{}, D. R. [Willis]{} and G. [Wendt]{}, [A description of sources detected by INTEGRAL during the first 4 years of observations]{}, [*A&A*]{} [**467**]{}, 585 (May 2007). S. [Chaty]{}, F. [Rahoui]{}, C. [Foellmi]{}, J. A. [Tomsick]{}, J. [Rodriguez]{} and R. [Walter]{}, [Multi-wavelength observations of Galactic hard X-ray sources discovered by INTEGRAL. I. The nature of the companion star]{}, [*A&A*]{} [ **484**]{}, 783 (June 2008). F. [Rahoui]{}, S. [Chaty]{}, P.-O. [Lagage]{} and E. [Pantin]{}, The most obscured high-energy sources of the galaxy brought to light by optical/infrared observations, [*A&A*]{} [**484**]{}, p. 801 (2008). A. [Coleiro]{}, S. [Chaty]{}, J. A. [Zurita Heras]{}, F. [Rahoui]{} and J. A. [Tomsick]{}, [Infrared identification of high-mass X-ray binaries discovered by INTEGRAL]{}, [*A&A*]{} [**560**]{}, p. A108 (December 2013). S. [Chaty]{} and F. [Rahoui]{}, [Broadband ESO/VISIR-Spitzer Infrared Spectroscopy of the Obscured Supergiant X-Ray Binary IGR J16318-4848]{}, [*ApJ*]{} [ **751**]{}, p. 150 (June 2012). I. [Negueruela]{}, D. M. [Smith]{}, P. [Reig]{}, S. [Chaty]{} and J. M. [Torrej[ó]{}n]{}, [Supergiant Fast X-ray Transients: a new class of high mass X-ray binaries unveiled by INTEGRAL]{}, in [*ESA Special Publication*]{}, ed. A. [Wilson]{}January 2006. L. J. [Pellizza]{}, S. [Chaty]{} and I. [Negueruela]{}, [IGR J17544-2619: a new supergiant fast X-ray transient revealed by optical/infrared observations]{}, [*A&A*]{} [**455**]{}, 653 (August 2006). [^1]: For more details about BeHMXBs, we warmly recommend the excellent review on optical/infrared emission of HMXBs: Ref. . [^2]: CygX-1 is the only sgHMXB with both Roche lobe overflow and stellar wind accretion, hosting a confirmed BH, probably a rare product of stellar evolution in X-ray binary systems (Ref. ). [^3]: Up-to-date list maintained by J. Rodriguez & A. Bodaghee: [*http://irfu.cea.fr/Sap/IGR-Sources*]{} [^4]: IGRJ16393-4611, IGRJ16418-4532, IGRJ18027-2016, 4U1538-522, 4U1700-37, 4U1909+07 and XTEJ1855-026 [^5]: IGRJ16320-4751, IGRJ19140+0951, EXO1722-363, GX301-2, OAO1657-415, 1A0114+650, 1E1145.1-6141 and VelaX-1 [^6]: IGRJ08408-4503, IGRJ11215-5952, IGRJ16195-4945, IGRJ16207-512, IGRJ16328-4726, IGRJ16418-4532, IGRJ16465-4507, IGRJ16479-4514, IGRJ17354-3255, IGRJ17544-2619 (their archetype [@pellizza:2006]), IGRJ18462-0223, IGRJ18483-0311, SAXJ1818.6-1703, XTEJ1739-302, AXJ1820.5-1434, AXJ1841.0-0536 and AXJ1845.0-0433 [^7]: All references are in Ref. .

--- abstract: 'A sequence of $S_n$-representations $\{V_n\}$ is said to be uniformly representation stable if the decomposition of $V_n = \bigoplus_{\mu} c_{\mu,n} V(\mu)_n$ into irreducible representations is independent of $n$ for each $\mu$—that is, the multiplicities $c_{\mu,n}$ are eventually independent of $n$ for each $\mu$. Church-Ellenberg-Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church-Ellenberg-Farb). We also explore some combinatorial consequences of this stability.' address: - 'Department of Mathematics, University of Wisconsin-Eau Claire, Eau Claire, WI, U.S.A.' - 'Department of Mathematics and Statistics, Smith College, Northampton, MA, U.S.A.' author: - Aba Mbirika - Julianna Tymoczko title: Representation stability of the cohomology of Springer varieties and some combinatorial consequences --- [^1] [^2] Introduction ============ Homological stability is a topological property of certain sequences of topological spaces: given a sequence $\{X_n\}_{n=1}^\infty$ of topological spaces with maps $\phi_n : X_{n} \rightarrow X_{n+1}$ for each $n$, then $\{X_n\}_{n=1}^\infty$ is homologically stable if there exists a positive integer $N$ so that the maps $(\phi_n)_*: H_i(X_{n}) \rightarrow H_i(X_{n+1})$ is an isomorphism whenever $n \geq N$. In other words the homology groups stabilize after a certain point in this sequence; however the topological spaces change later in the sequence, the changes do not affect the $i^{\mathrm{th}}$ homology group. Church and Farb [@CF], and later Church, Ellenberg, and Farb [@CEF] defined [*representation stability*]{} to mimic the topological definition. Informally, the sequence of $S_n$-representations $V_1 \stackrel{f_1}{\rightarrow} V_2 \stackrel{f_2}{\rightarrow} V_3 \stackrel{f_3}{\rightarrow} \cdots$ is representation-stable if there are $S_n$-equivariant linear injections $f_n: V_n \rightarrow V_{n+1}$ and if there exists an $N$ so that the multiplicities $c_{\mu,n}$ in the decomposition into irreducibles $$V_n = \bigoplus_\mu c_{\mu,n} V(\mu)_n$$ are independent of $n$ for all $n \geq N$. (Section \[sec: representation stability\] defines representation stability precisely.) One important problem is to find families of representations that are representation stable; these families often arise from geometric considerations. In their original work, Church, Ellenberg, and Farb identify a number of representation stable families arising from geometry/topology and classical representation theory [@CEF]. Indeed their example of the diagonal coinvariant algebra [@CEF Section 5] is the sequence of cohomology rings of flag varieties $\{H^*(GL_n/B_n)\}_{n=1}^\infty$ which in some sense is the springboard of this paper. Since then, others have demonstrated that representation stability arises naturally in many contexts, including arrangements associated to root systems [@Bibby], linear subspace arrangements [@Gadish], configuration spaces in $\mathbb{R}^d$ [@Hersh-Reiner], filtrations of Torelli groups [@Patzt], moduli spaces of Riemann surfaces of genus $g$ with $n$ labeled marked points [@Rolland2011], and others [@Duque-Rolland; @Rolland2016]. The central goal of this paper is to prove that an important family of $S_n$-representations called [*Springer representations*]{} is representation stable. The Springer representation is the archetypal geometric representation: in its most basic form, it arises when the symmetric group $S_n$ acts on the cohomology of a family of subvarieties of the flag variety called [*Springer fibers*]{}. The flag variety can be described as the set of nested vector subspaces $$V_1 \subseteq V_2 \subseteq \cdots \subseteq V_{n-1} \subseteq \mathbb{C}^n$$ where each $V_i$ is $i$-dimensional. Given a nilpotent matrix $X$, the Springer fiber consists of the flags that are fixed by $X$ in the sense that $XV_i \subseteq V_i$ for all $i$. Every nilpotent matrix is conjugate to one in Jordan form, which is determined by a partition of $n$ into Jordan blocks. Since Springer fibers associated to conjugate matrices $X$ and $gXg^{-1}$ are homeomorphic, the Springer fibers are parametrized by partitions of $n$. Springer first constructed a representation of the symmetric group $S_n$ on the cohomology of Springer fibers [@Spr78]. Since then, the representation has been recreated in many different ways [@dCP81; @Tani82; @BorMac83; @Lus84]. The geometry of Springer fibers encodes key data about representations of the symmetric group. For instance the top-dimensional cohomology is an irreducible $S_n$-representation [@Spr78]; and the ungraded representation on $H^*(Spr_{\lambda})$ is Young’s representation associated to the partition $\lambda$ [@GarPro92 Introduction]. Our analysis of Springer representations uses the [**co-FI**]{} category, which Church, Ellenberg, and Farb defined to concisely describe the compatibility conditions needed for representation stability. Theorem \[thm:the\_main\_theorem\] proves that for any increasing sequence of Young diagrams, the corresponding Springer representations form a graded co-FI-module of finite type. The main consequence from our point of view is that sequences of Springer representations are representation stable (see Corollary \[cor:springer\_rep\_is\_representation\_stable\] for a precise statement). From this many other properties follow: (1) for each fixed degree and $n$ large enough, the character is given by a polynomial that is independent of $n$, and in particular (2) the dimension of the Springer representation is eventually polynomial in $n$. It is this second consequence that we explore in Section \[sec:combinatorial\_consequences\]. \[rem:Hotta\] Curiously, the literature does not make a clear distinction between the Springer representation and its dual. Hotta first observed this and classified existing constructions of the Springer representation up to that point [@Hotta81]. However, the ambiguity persisted with subsequent constructions of the Springer representation. In this paper, we treat “the" Springer representation interchangeably with its dual. We prove that the Garsia-Procesi construction of the Springer representation is a co-FI-module (graded, of finite type) and its [*dual*]{} is representation stable. Properties like dimension and decomposition into irreducibles are well-behaved with respect to duality and apply to the original Garsia-Procesi construction, too. Kim proves a different kind of stability of Springer representations, giving conditions under which the lower-graded parts of the Springer representation for $\lambda$ coincide with those of $\lambda'$ for partitions $\lambda, \lambda'$ of the same $n$ [@Kim_Stability]. Kim recovers Theorem \[thm: at least k+1 rows gives a polynomial dimension formula\] using his notion of stability [@Kim_Stability Corollary 4.3]. This paper is organized as follows. In Section \[sec:FI\_and\_co-FI\], we define the conditions that guarantee the stability of a sequence of $S_n$-representations. In Section \[sec:Springer\_theory\], we describe Garsia-Procesi’s combinatorial description of the Springer representation, and in particular the so-called [*Tanisaki ideal*]{}. In Section \[sec:FI-module\_structure\], we prove there is no FI-module structure on sequences of Springer representations, except for the trivial representation. On the other hand, in Section \[sec:co-FI-module\_structure\] we prove that the sequence of Tanisaki ideals forms a co-FI-ideal and deduce that a co-FI-module structure exists on all sequences of Springer representations. In Section \[sec: representation stability\], we conclude with our main result on the stability of the Springer representations. We give some concrete combinatorial consequences of this stability in Section \[sec:combinatorial\_consequences\], and a collection of open questions in Section \[sec:open\_questions\] that probe the new combinatorial ideas raised by representation stability. [**[FI]{}**]{} and [**[co-FI]{}**]{} {#sec:FI_and_co-FI} ==================================== In this section we describe FI-modules and co-FI-modules, defined by Church, Ellenberg, and Farb [@CEF] to streamline and extend the essential features of Church and Farb’s earlier notion of representation stability [@CF]. We begin with the categories [**FI**]{} and [**co-FI**]{}, which carry actions of the permutation groups $S_n$ and are constructed to be compatible with inclusions. The key example of FI- and co-FI-modules for this manuscript is the sequence of polynomial rings $\{k[x_1,\ldots,x_n]\}$. We then list the properties about FI- and co-FI-modules that we will need to establish that sequences of Springer representations are graded co-FI-modules. This section provides only what is needed in this paper. The interested reader is referred to Church, Ellenberg, and Farb’s work for many other interesting results [@CEF]. We assume $k$ is a field of characteristic zero. Parts of Church-Ellenberg-Farb’s theory extends to other fields as well [@CEF]. Let $\mathbf{n}$ denote the set $\{1,\ldots,n\}$. [**FI**]{} is the category whose objects are finite sets and whose morphisms are injections. This is equivalent to the category whose objects are sets $\mathbf{n} = \{1,2,\ldots,n\}$ and whose morphisms are injections $\mathbf{m} \rightarrow \mathbf{n}$. An *FI-module* (*FI-algebra*, *graded FI-algebra*) over a commutative ring $k$ is a functor $V$ from [**FI**]{} to the category of modules over $k$ ($k$-algebras, graded $k$-algebras). We usually denote the $k$-module (respectively algebra) $V(\mathbf{n})$ by $V_n$. Modules that carry permutation actions compatible with the permutation action on integers provide a rich source of examples of FI-modules. The next example is the most important for our purposes. Define a functor $R$ by: - $R$ sends the object $\mathbf{n}$ to the polynomial ring $k[x_1,\ldots,x_n]$ and - $R$ sends the morphism $f: \mathbf{m} \rightarrow \mathbf{n}$ to the homomorphism $$f_*: k[x_1,\ldots,x_m] \rightarrow k[x_1,\ldots,x_n]$$ induced by the condition that $f_*(x_i)=x_{f(i)}$ for all $i \in \mathbf{m}$. Then $R$ is a graded FI-algebra. In particular, we have that $R_n$ is the polynomial ring $k[x_1,\ldots,x_n]$. The category [**co-FI**]{} is opposite to [**FI**]{}. The polynomial algebras also form a graded co-FI-algebra, as described below. Denote the opposite category of [**FI**]{} by [**co-FI**]{}. In particular the objects in [**co-FI**]{} are finite sets, without loss of generality the sets $\mathbf{n}$, and the morphisms in [**co-FI**]{} from $\mathbf{n}$ to $\mathbf{m}$ are the morphisms in [**FI**]{} from $\mathbf{m}$ to $\mathbf{n}$. A *co-FI-module* (*co-FI-algebra*, *graded co-FI-algebra*) over a commutative ring $k$ is a functor from [**co-FI**]{} to the category of $k$-modules ($k$-algebras, graded $k$-algebras). For instance if $V$ is an FI-module over $k$ then the dual $V^\vee$ forms a co-FI-module, and vice versa. Define a functor $R$ as follows: - $R$ sends the object $\mathbf{n}$ to the polynomial ring $k[x_1,\ldots,x_n]$ and - $R$ sends the FI-morphism $f: \mathbf{m} \rightarrow \mathbf{n}$ to the homomorphism $$f^*: k[x_1,\ldots,x_n] \rightarrow k[x_1,\ldots,x_m]$$ induced by the condition that $$f^*(x_i)=\left\{ \begin{array}{ll} x_{f^{-1}(i)} &\textup{ if } i \in \textup{Im}(f) \textup{ and} \\ 0 & \textup{ otherwise.} \end{array}\right.$$ Then $R$ is a graded co-FI-algebra. The module categories [**FI**]{} and [**co-FI**]{} are abelian and so admit many of the algebraic constructions that modules and algebras do. In particular we can consider co-FI-quotients of a co-FI-module and FI-submodules of an FI-module. \[example: ideal of symmetric polys is not FI-module\] For each $n$ let $I_n$ denote the ideal of symmetric polynomials with no constant term. The sequence of ideals $I$ is not an FI-submodule of the FI-module $R$ because the image of an $S_m$-symmetric polynomial under the inclusion map $\iota: \{1,\ldots,m\} \rightarrow \{1,\ldots,n\}$ is not symmetric under the larger group $S_n$. However the ideals $I$ do form a co-FI-submodule of the co-FI-module $R$. The next lemma proves that the image of a set of $S_m$-invariant polynomials under an arbitrary injection is the same as the image under the inclusion $\iota$ that is the identity on the integers $\{1,\ldots,m\}$. We use it to simplify later calculations. \[lemma: inclusion suffices\] Suppose that $m \leq n$ and that $f: \{1,\ldots,m\} \rightarrow \{1,\ldots,n\}$ is an injection. Let $\iota$ be the inclusion $\iota: \{1,\ldots,m\} \rightarrow \{1,\ldots,n\}$ that sends $\iota(i) = i$ for each $1 \leq i \leq m$. Consider the action of $S_n$ on $R_n$ under which for each $w \in S_n$ and $x_i \in R_n$ we have $w(x_i) = x_{w(i)}$. Let ${\mathcal{J}} \subseteq R_n$ be any set of polynomials that are preserved under the $S_{n}$-action, in the sense that $w {\mathcal{J}} \subseteq {\mathcal{J}}$ for all permutations $w \in S_{n}$. Then $$f^*(\mathcal{J}) = \iota^*(\mathcal{J}).$$ Our hypothesis means that the polynomial $p \in {\mathcal{J}}$ if and only if the image $w(p) \in {\mathcal{J}}$ for each $w \in S_{n}$. The map $f: \{1,\ldots,m \} \hookrightarrow \{1,\ldots,n\}$ is an injection so we can define a permutation $w_f \in S_{n}$ by $$w_f = \left\{ \begin{array}{ll} f(i) \mapsto i & \textup{ for all $i$ with } 1 \leq i \leq m \\ j_i \mapsto m+i & \textup{ for $j_i$ such that both } j_1<j_2< \cdots <j_{n-m} \textup{ and } \\ & \hspace{0.5in} \{ j_1, j_2, \cdots,j_{n-m}\} \cup \textup{Im}(f) = \{1,2,\ldots,n\}. \end{array} \right.$$ We know that for each $i$ $$\iota^* (w_fx_i) = \iota^* (x_{w_f(i)})$$ by definition of the $S_n$-action on permutations. By construction of $\iota^*$ we have $$\iota^* (x_{w_f(i)}) = \left\{ \begin{array}{ll} 0 & \textup{ if } i \in \{ j_1, j_2, \cdots,j_{n-m}\} \\ x_{f^{-1}(i)} & \textup{ otherwise}. \end{array} \right.$$ This is exactly $f^*(x_i)$. Thus $f^*(p) = \iota^* (w_fp)$ for all polynomials $p \in {\mathcal{J}}$ and hence as desired $f^*({\mathcal{J}}) = \iota^*({\mathcal{J}})$. It follows that if $I$ is a sequence of symmetric homogeneous ideals in the graded co-FI-algebra $R$ then we can prove $I$ is a co-FI-submodule simply by considering inclusions—or even just a subset of inclusions. \[corollary: simple inclusions\] Let $R$ be a graded co-FI-algebra and $I$ be a sequence of ideals $\{I_1,I_2,\ldots\}$ with each $I_n$ an $S_n$-invariant homogeneous ideal in $R_n$. The following are equivalent: 1. \[FI part\] $I$ is a co-FI-submodule of $R$. 2. \[general inclusion part\] For each $m<n$ and inclusion $\iota: \{1,\ldots,m\} \rightarrow \{1,\ldots,n\}$ defined by $\iota(i)=i$ for all $i$, the induced map satisfies $\iota^*(I_n)\subseteq I_m$. 3. \[step inclusion part\] For each $n$ and inclusion $\iota_n: \{1,\ldots,n-1\} \rightarrow \{1,\ldots,n\}$, the induced map satisfies $\iota_n^*(I_n)\subseteq I_{n-1}$. Part (\[FI part\]) is equivalent to Part (\[general inclusion part\]) by Lemma \[lemma: inclusion suffices\]. Part (\[general inclusion part\]) implies Part (\[step inclusion part\]) by definition. The composition of inclusions $\iota_n \circ \iota_{n-1} \circ \cdots \circ \iota_{m+1}$ is the inclusion $\iota: \{1,\ldots,m\} \hookrightarrow \{1,\ldots,n\}$. By functoriality if $\iota_{i}^*(I_i) \subseteq I_{i-1}$ for each $i\geq 2$ then $\iota^*(I_n) \subseteq I_m$ for each pair $n \geq m \geq 1$. So Part (\[step inclusion part\]) implies Part (\[general inclusion part\]). The definition of finitely-generated FI-modules is crucial to representation stability. It differs importantly from the corresponding definition for modules or rings because it incorporates the underlying $S_n$-action. As we see in Example \[example: finite generation\], this implies that FI-modules often have fewer generators than we might expect. In fact, though this does not appear explicitly in the literature, the category of [*finitely generated*]{} [**FI**]{} modules over Noetherian rings is also abelian (by the Noetherian property, proven over $\mathbb{C}$ in [@CEF] and over other Noetherian rings in [@CEFN]) [@C20]. This is the thrust of the arguments that we cite in this paper. \[def:finite\_generation\_and\_type\] An FI-module $V$ is *finitely generated* if there is a finite set $S$ of elements in $\coprod_i V_i$ so that no proper sub-FI-module of $V$ contains $S$. A graded FI-module $V$ has *finite type* if the $i^{\mathrm{th}}$ graded part $V^i$ is finitely generated for each $i$. A graded co-FI-module $W$ is of finite type if its dual $W^\vee$ is a graded FI-module of finite type [@CEF Co-FI-algebras in Section 4.2] \[example: finite generation\] The sequence $R = \{k[x_1,\ldots,x_n]\}$ of polynomial rings are not finitely generated as an FI-module; indeed no ring $k[x_1,\ldots,x_n]$ is a finite dimensional $k$-vector space. However when graded by polynomial degree, each graded part of the sequence $R$ is finitely generated as an FI-module. For instance the graded part of degree $3$ is generated by $x_1^3$, $x_1^2x_2$, and $x_1x_2x_3$ since every monomial of degree three is obtained by permuting indices of one of these three. More generally the graded part of degree $d$ is generated by all monomials of the form $x_1^{d_1}x_2^{d_2}\cdots x_n^{d_n}$ over partitions $d_1 \geq d_2 \geq \cdots \geq d_n \geq 0$ of $d$. For completeness, we define the quotient of an FI-algebra or co-FI-algebra. Let $R$ be a graded FI-algebra. If $I$ is a graded FI-submodule (respectively co-FI-submodule) for which each object $I_n$ is a homogeneous ideal in $R_n$ then $I$ is called an *FI-ideal* (respectively *co-FI-ideal*). The quotient FI-module $R/I$ is defined so that $(R/I)_n=R_n/I_n$ for each $n$ (respectively co-FI). The following proposition is our main tool, and sketches the main points of [@CEFN Theorem F]. This result implies that we only need to prove each sequence of Tanisaki ideals forms a co-FI-submodule; after that, a straightforward algebraic argument allows us allow us to conclude that sequences of Springer representations are representation-stable. \[prop: quotient is graded co-FI\] Let $R$ be a graded co-FI-module over a Noetherian ring and let $I$ be a co-FI-ideal. Then the dual $\left(R/I\right)^\vee$ is a graded FI-module. If $R^\vee$ has finite type then so does $\left(R/I\right)^\vee$. The dual of a graded co-FI-module is a graded FI-module by purely formal properties. Suppose further that $R^\vee$ has finite type and consider the $j^{\mathrm{th}}$ graded parts $\left(R_n\right)_j^\vee$ and $\left(R_n/I_n\right)_j$ for each $n$. The dual $\left(R_n/I_n\right)_j^\vee$ is the FI-submodule of $\left(R_n\right)_j^\vee$ consisting of those functionals that vanish on the $j^{\mathrm{th}}$ graded parts $\left(I_n\right)_j = I_n \cap \left(R_n\right)_j$ by definition of graded quotients. Each FI-submodule of an FI-module of finite type over a Noetherian ring is also of finite type [@CEFN Theorem A], thus proving the claim. In Section \[sec:combinatorial\_consequences\] and \[sec:open\_questions\] of this manuscript we analyze properties of FI-modules in the case of Springer representations. Springer theory {#sec:Springer_theory} =============== This section summarizes two key combinatorial tools: Biagioli-Faridi-Rosas’s description of generators for each Tanisaki ideal and Garsia-Procesi’s description of a basis for the Springer representation. The Tanisaki ideal and its generators ------------------------------------- We use Garsia-Procesi’s presentation of the cohomology of the Springer fiber, which describes the cohomology as a quotient of a polynomial ring analogous to the Borel construction of the cohomology of the flag variety [@Borel53]. Let $R_n$ be the polynomial ring $\mathbb{C}[x_1, \ldots, x_n]$. For each partition $\lambda$ of $n$ Tanisaki defined an ideal $I_{\lambda} \subseteq R_n$ that is now called the [*Tanisaki ideal*]{}. To describe the Tanisaki ideal, we define certain sets of elementary symmetric functions. Given a subset $S \subseteq \{x_1, \ldots, x_n\}$ the elementary symmetric function $e_i(S)$ is the polynomial $$e_i(S) = \sum_{\mbox{\scriptsize $\begin{array}{c}T \textup{ such that }\\ T \subseteq S \textup{ and } |T|=i \end{array}$}} \prod_{x_j \in T} x_j.$$ The set $E_{i,j}^n$ is the set of elementary symmetric functions $e_i(S)$ over all possible subsets $S\subseteq \{x_1, \ldots, x_n\}$ of cardinality $j$. Let $S = \{x_1, x_3, x_4\} \subseteq \{x_1, x_2, \ldots, x_5\}$. Then $e_2(S) = x_1x_3 + x_1x_4 + x_3x_4$. Letting $S$ run over all $\binom{5}{3}=10$ size three subsets of $\{x_1, x_2, \ldots, x_5\}$ gives the 10 elementary symmetric polynomials $e_2(S)$ comprising the set $E_{2,3}^5$. We follow Biagioli, Faridi and Rosas’s construction of the Tanisaki ideal [@BFR08 Definition 3.4]. Let $\lambda = (\lambda_1, \ldots, \lambda_k)$ be a partition of $n$. The *BFR-filling* of $\lambda$ is constructed as follows. From the leftmost column of $\lambda$ to the rightmost column, place the numbers $1, 2, \ldots, n-\lambda_1$ bottom to top skipping the top row. Finally fill the top row from right to left with the remaining numbers $n-\lambda_1+1, \ldots, n$. The *BFR-generators* are polynomials in the set ${\mathcal{G}}(\lambda)$ defined as the following union: If the box filled with $i$ has $j$ in the top row of its column, then include the elements of $E_{i,j}^n$ in the set ${\mathcal{G}}(\lambda)$. Consider the partition $\lambda = (4,2,2,1)$. Then the BFR-filling of $\lambda$ is $$\ytableausetup{centertableaux,boxsize=1.25em} \begin{ytableau} 9& 8 & 7 & 6 \\ 3 & 5 \\ 2 & 4 \\ 1 \end{ytableau}$$ and hence the set ${\mathcal{G}}(\lambda)$ is $${\mathcal{G}}(\lambda) = \bigcup_{i \in \{1,2,3,9\}} \!\!\! E^9_{i,9} \; \cup \; \bigcup_{i \in \{4,5,8\}} E^9_{i,8} \; \cup \; E^9_{7,7} \; \cup \; E^9_{6,6}.$$ The first union of sets arises from the first column, the second from the second, and so on. Each set of the form $E^9_{i,9}$ contains the single elementary symmetric function $e_i(x_1, x_2, \ldots, x_9)$. Given a partition $\lambda$ of $n$ the Tanisaki ideal $I_\lambda$ is generated by the set ${\mathcal{G}}(\lambda)$. Combinatorial presentation for the Springer representation and the Garsia-Procesi basis --------------------------------------------------------------------------------------- It turns out that the natural $S_n$-action on $R_n$ given by $$w \cdot p(x_1, \ldots, x_n) = p(x_{w(1)}, \ldots, x_{w(n)})$$ restricts to the Tanisaki ideal $I_{\lambda}$. Thus the $S_n$-action on $R_n$ induces an $S_n$-action on $R_n/I_{\lambda}$. The key result of Garsia-Procesi’s work (with others [@Kra80; @dCP81; @Tani82]) is that this quotient is the cohomology of the Springer variety. Let $R_n = \mathbb{C}[x_1, \ldots, x_n]$ and let $I_\lambda$ be the Tanisaki ideal corresponding to the partition $\lambda$. The quotient $R_n/I_{\lambda}$ is isomorphic to the cohomology of the Springer variety $H^*(Spr_{\lambda})$ as a graded $S_n$-representation. Moreover Garsia-Procesi describe an algorithm to compute a nice basis $\mathcal{B}(\lambda)$ of monomials for $R_n/I_{\lambda}$. Our exposition owes much to the presentation in the first author’s work [@Mbir10 Definition 2.3.2]. If $\lambda = (1)$ is the unique partition of $1$ then $\mathcal{B}(\lambda) = \{1\}$. If $n>1$ and $\lambda$ is a partition of $n$ with $k$ parts then: - Number the rightmost box in the $i^{\mathrm{th}}$ row of $\lambda$ with $i$ for each $i \in \{1,\ldots,k\}$. - For each $i \in \{1,\ldots,k\}$ construct the partition $\lambda_i$ of $n-1$ by erasing the box labeled $i$ and rearranging rows if needed to obtain a Young diagram once again. - Recursively define $\mathcal{B}(\lambda)$ as $$\mathcal{B}(\lambda) = \bigcup_{i=1}^k x_n^{i-1} \hspace{0.25em} \mathcal{B}(\lambda_i).$$ From the *GP-algorithm* above we construct the *GP-tree* as follows. Let $\lambda$ sit alone at Level $n$ in a rooted tree directed down. Since $\lambda$ has $k$ parts, create $k$ edges labeled $x_n^0, x_n^1, \ldots, x_n^{k-1}$ left-to-right to the $k$ subdiagrams $\lambda_i$ for each $i \in \{1, \ldots, k\}$ in Level $n-1$. For each of these subdiagrams and their descendants, recursively repeat this process. The process ends at Level $1$, whose diagrams all contain one single box. Multiplying the edge labels on any downward path gives a unique *GP-monomial* in the *GP-basis*. \[example: GP-tree\] We illustrate the first two steps of the GP-algorithm on the partition of 5 given by $\lambda = (2,2,1)$. Number the far-right boxes and branch down from Level 5 to Level 4 of the recursion as follows: $$\xymatrix{ \mathrm{Level \; 5} & & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {\mbox{\tiny 1}} \\ {} & {\mbox{\tiny 2}} \\ {\mbox{\tiny 3}} \end{ytableau}} \ar[dl]_{1} \ar[d]^{x_5} \ar[dr]^{x_5^2} \\ \mathrm{Level \; 4} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} \\ {} & {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} & {} \end{ytableau}} }.$$ After rearranging rows to obtain a Young diagram, we begin the recursion again on each of the three partitions to produce Level 3 of the tree. $$\xymatrixcolsep{.225in} \xymatrix{ \mathrm{Level \; 4} & & & & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \\ {} \end{ytableau}} \ar[dlll]|{1} \ar[dll]|{x_4} \ar[dl]|{x_4^2} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \\ {} \end{ytableau}} \ar[dl]|{1} \ar[d]|{x_4} \ar[dr]|{x_4^2} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} & {} \end{ytableau}} \ar[dr]|{1} \ar[drr]|{x_4} \\ \mathrm{Level \; 3} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} \\ {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} \\ {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \end{ytableau}} & {\ytableausetup{centertableaux,boxsize=.75em} \begin{ytableau} {} & {} \\ {} \end{ytableau}} }$$ By Level 1, there will be 30 diagrams, each equal to the partition $(1)$. Recovering the Garsia-Procesi basis from this tree is equivalent to multiplying the edge labels of the 30 paths. The reader can verify that we obtain the following basis $\mathcal{B}(\lambda)$: degree \# monomials in $\mathcal{B}(\lambda)$ -------- ---- ----------------------------------------------------------------------------------- 0 1 1 1 4 $x_i$ for $2 \leq i \leq 5$ 2 9 $x_i^2$ for $3 \leq i \leq 5$ and $x_ix_j$ for $2 \leq i<j \leq 5$ 3 11 $x_i x_j x_5$ for $2 \leq i<j \leq 4$, $x_i x_j^2$ for $2 \leq i < j \leq 5$, and $x_i^2 x_5$ for $3 \leq i \leq 4$ 4 5 $x_i x_j^2 x_5$ for $2 \leq i<j \leq 4$ and $x_i x_4 x_5^2$ for $2 \leq i \leq 3$ Garsia-Procesi bases have several nice containment properties that we use when analyzing the FI- and co-FI-structure of Springer representations. The first describes the relationship between dominance order and the Garsia-Procesi bases. \[proposition: dominance order\] Let $\lambda$ and $\lambda'$ be partitions of $n$ and suppose that $\lambda \unlhd \lambda'$ in dominance order, namely that we have $\lambda_1+\lambda_2+\cdots+\lambda_i \leq \lambda'_1+\lambda'_2+\cdots+\lambda'_i$ for all $i\geq 1$. Then $\mathcal{B}(\lambda') \subseteq \mathcal{B}(\lambda)$. The second describes the relationship between containment of Young diagrams and the Garsia-Procesi bases. \[lemma: containment and Garsia-Procesi basis\] If $\lambda \subseteq \lambda'$ then $\mathcal{B}(\lambda) \subseteq \mathcal{B}(\lambda')$. We prove the claim assuming that $\lambda'$ has exactly one more box than $\lambda$. Repeating the argument gives the desired result. Consider the subdiagrams $\lambda_1, \lambda_2, \ldots, \lambda_k$ obtained from $\lambda'$ in the recursive definition of the Garsia-Procesi algorithm. The Garsia-Procesi algorithm says $$\mathcal{B}(\lambda') = \bigcup_{i=1}^k x_n^{i-1} \hspace{0.25em} \mathcal{B}(\lambda_i) = \mathcal{B}(\lambda_1) \cup \bigcup_{i=2}^k x_n^{i-1} \hspace{0.25em} \mathcal{B}(\lambda_i)$$ so $\mathcal{B}(\lambda') \supseteq \mathcal{B}(\lambda_1)$. By construction $\lambda_i$ is obtained from $\lambda'$ by removing a box that is above and possibly to the right of the box removed for $\lambda_{i+1}$ for each $i \in \{1,\ldots,k-1\}$. In particular $\lambda_i \unlhd \lambda_{i+1}$ in dominance order. Proposition \[proposition: dominance order\] implies that $\mathcal{B}(\lambda_{i+1}) \subseteq \mathcal{B}(\lambda_i)$ for each $i$ and so $\mathcal{B}(\lambda') \supseteq \mathcal{B}(\lambda_i)$ for each $i$. Since $\lambda$ is obtained from $\lambda'$ by removing a single box, we know $\lambda = \lambda_i$ for some $i$. The claim follows. The Springer representations with FI-module structure {#sec:FI-module_structure} ===================================================== Recall that the sequence of polynomial rings carries an FI-module structure and that both the Tanisaki ideal and the quotients $R_n/I_{\lambda}$ carry an $S_n$-action. The question in this section is: do these fit together to give an FI-module structure on Springer representations? Lemma \[lemma: containment and Garsia-Procesi basis\] suggests that the answer could be yes, since it proved that if $\lambda \subseteq \lambda'$ then the Garsia-Procesi basis for $R_n/I_{\lambda}$ is contained in the Garsia-Procesi basis for $R_{n'}/I_{\lambda'}$. We prove that this is misleading: the inclusion $R_n/I_{\lambda} \hookrightarrow R_{n'}/I_{\lambda'}$ in no way preserves the $S_n$ action (and is not what Church and Farb call a consistent sequence [@CF pg.6]). In particular we prove that there is [*no*]{} FI-module structure on sequences of Springer representations, except for the trivial representation. Church, Ellenberg, and Farb observed that the sequence of ideals of symmetric functions with no constant term is not an FI-ideal (see Example \[example: ideal of symmetric polys is not FI-module\]); in our language, they study the special case of the sequence of Tanisaki ideals $\{I_{\lambda_1}, I_{\lambda_2}, \ldots\}$ where each $\lambda_i$ is a column with $i$ boxes. For each positive integer $n$, let $\lambda_n$ denote a Young diagram with $n$ boxes. The only sequence of Young diagrams $\lambda_1 \subseteq \lambda_2 \subseteq \lambda_3 \subseteq \cdots$ for which the sequence of Tanisaki ideals $\{I_{\lambda_1}, I_{\lambda_2}, I_{\lambda_3}, \ldots\}$ forms an FI-ideal is [1 $\yng(1) \rightarrow \yng(2) \rightarrow \yng(3) \rightarrow \cdots$]{}. Suppose that $\lambda$ is a partition of $n-1$ and $\lambda'$ is a partition of $n$ with $\lambda \subseteq \lambda'$. We prove that if $f: \{1,\ldots,n-1\} \rightarrow \{1,\ldots,n\}$ is an injection for which $f_*(I_\lambda) \subseteq I_{\lambda'}$ then $I_{\lambda'} = \langle x_1, x_2, \ldots, x_n\rangle$. By definition of FI-modules this suffices to prove our claim. We first show that $I_{\lambda'}$ contains one of the variables $x_1, \ldots, x_n$. Construct the BFR-fillings of both $\lambda$ and $\lambda'$ and compare the boxes labeled $1$. A Young diagram with at least two rows has label $1$ in the bottom box of the first column. A Young diagram with only one row has label $1$ in the rightmost box of its row. If either $\lambda$ or $\lambda'$ has just one row then the generating set for its Tanisaki ideal contains $e_1(x_1)$. Hence $I_{\lambda'} \supseteq I_{\lambda}$ contains the variable $x_1$. Otherwise the top-left boxes of the BFR-fillings for $\lambda$ and $\lambda'$ have the labels $n-1$ and $n$ respectively, so ${\mathcal{G}}(\lambda)$ contains $e_1(x_1, \ldots, x_{n-1})$ and ${\mathcal{G}}(\lambda')$ contains $e_1(x_1,\ldots,x_n)$. In this case $$e_1(x_1,\ldots,x_n) - f_*(e_1(x_1,\ldots, x_{n-1})) = x_j$$ lies in $I_{\lambda'}$ for the unique $j \in \{1,2,\ldots,n\} - \textup{Im}(f)$. Thus $I_{\lambda'}$ contains at least one of the variables $x_1, \ldots, x_n$. The Tanisaki ideal is invariant under the action of $S_n$ that permutes the variables $x_1, \ldots, x_n$ so in fact $I_{\lambda'} = \left\langle x_1, \ldots, x_n \right\rangle$. We conclude that $$R_n/I_{\lambda'} = R_n/\left\langle x_1, \ldots, x_n \right\rangle \cong {\mathbb{C}}$$ is trivial. Since $\lambda'$ gives the trivial representation, it consists of a single row, as desired. The Springer representation with the co-FI-module structure {#sec:co-FI-module_structure} =========================================================== In this section we prove that Springer representations admit a co-FI-module structure. Recall from Section \[sec:FI\_and\_co-FI\] that the sequence of polynomial rings forms a co-FI-module. This section shows that the natural restriction maps from $k[x_1, \ldots, x_n]$ to $k[x_1, \ldots, x_{n-1}]$ are defined on Tanisaki ideals, too. Our proof mimics a similar proof for the cohomology of flag varieties in the arXiv version of a paper by Church, Ellenberg, and Farb [@CEFa Theorem 3.4]. A surprising feature of our result is that this co-FI-module structure exists for Springer representations corresponding to [*every possible*]{} sequence $\lambda_1 \subseteq \lambda_2 \subseteq \lambda_3 \subseteq \cdots$ of Young diagrams. The main features of the proof were outlined in previous sections, especially Section \[sec:FI\_and\_co-FI\], which collected steps that reduce the proof that a sequence of Springer representations form a co-FI-module to proving that particular inclusions $\iota_n^*: R_n \rightarrow R_{n-1}$ preserve Tanisaki ideals. We prove the main theorem first and then prove the key lemma. \[thm:the\_main\_theorem\] Suppose $\lambda_1 \subseteq \lambda_2 \subseteq \cdots$ is a sequence of Young diagrams for which $\lambda_n$ has $n$ boxes for each $n$. Then the sequence $\{R_n/I_{\lambda_n}\}$ forms a graded co-FI-module of finite type. Definition \[def:finite\_generation\_and\_type\] states that to show $\{R_n/I_{\lambda_n}\}$ is a graded co-FI-module of finite type, we must show 1) it is a graded co-FI-module and 2) its dual is a graded FI-module of finite type. Assuming a field $k$ of characteristic zero, Proposition \[prop: quotient is graded co-FI\] states that if $R$ is a graded co-FI-module and $I$ is a co-FI-ideal then $R/I$ is a graded co-FI-module. The sequence of polynomial rings is a graded co-FI-algebra under the co-FI-algebra structure that sends the map $f: \{1,\ldots,m\} \rightarrow \{1,\ldots,n\}$ to the map with $f^*(x_i)$ equal to $x_{f^{-1}(i)}$ if $i \leq m$ and zero otherwise (see Example \[example: finite generation\]). Corollary \[corollary: simple inclusions\] proves that if $\{\lambda_1 \subseteq \lambda_2 \subseteq \cdots \}$ is a sequence of Young diagrams for which $\iota_{n}^*(I_{\lambda_n}) \subseteq I_{\lambda_{n-1}}$ for every $n \geq 2$ then the sequence of Tanisaki ideals $\{I_{\lambda_1}, I_{\lambda_2}, \ldots\}$ forms a co-FI-ideal. Lemma \[lemma: standard inclusion acts right on Tanisaki ideals\] proves that $\iota_{n}^*(I_{\lambda_n}) \subseteq I_{\lambda_{n-1}}$ for every $n \geq 2$ and for every sequence of Young diagrams $\{\lambda_1 \subseteq \lambda_2 \subseteq \cdots \}$. Thus $\{R_n/I_{\lambda_n}\}$ is a graded co-FI module. It is known that the dual ${ (R_n)^\vee }$ is a graded FI-module of finite type. The proof uses the facts that 0) the variables $x_1, \ldots, x_n \in R_n$ form a basis for the dual to $k^n$, 1) the $j^{\mathrm{th}}$ graded part $(R_n)^\vee_j$ is isomorphic to the $j^{\mathrm{th}}$ symmetric power of $k^n$, and 2) the $k$-modules $k, k^2, k^3, \ldots, k^n, \ldots$ are the parts of a natural FI-module $M(1)$ whose algebraic properties are well understood; for details, see [@CEFN proof of Theorem F]. Proposition \[prop: quotient is graded co-FI\] then implies that $\{\left(R_n/I_{\lambda_n}\right)^\vee\}$ is a graded FI-module of finite type, proving the claim. Sequences of Tanisaki ideals form a co-FI-ideal ----------------------------------------------- We now analyze the BFR-generators of the Tanisaki ideals to show that $\iota_n^*(I_{\lambda'}) \subseteq I_{\lambda}$ when $\lambda \subseteq \lambda'$. In an interesting twist, this co-FI-module structure exists for any path in the poset of Young diagrams. Our proof that these sequences of Tanisaki ideals $\{I_{\lambda_i}\}$ form a co-FI-ideal occurs over the course of the following lemmas. Each result actually analyzes a subset of homogeneous polynomials of the same degree inside $I_{\lambda}$ so our proofs preserve grading (though we don’t use this fact). We note the following well-known relation (e.g., [@Pro07 pg. 21], [@FGP97 Equation (3.1)]). Let $S \subseteq \{x_1,\ldots,x_{n}\}$. If $x_{n} \in S$ then $e_i(S)$ decomposes as $$e_i(S) = x_{n} \cdot e_{i-1}(S-\{x_{n}\}) + e_i(S-\{x_{n}\}).$$ This immediately implies the following. (Recall that $i$ is the degree of each function in the set $E_{i,j}^n$ while $j$ is the cardinality $|S|$ of the variable subset $S \subseteq \{x_1,x_2,\ldots,x_n\}$.) \[lemma: image of BFR-generators for a box\] $$\iota_n^*\left(E_{i,j+1}^{n}\right) = E_{i,j+1}^{n-1} \cup E_{i,j}^{n-1}$$ where $E_{i,k}^{n-1}$ is empty if $k > n-1$ or if $i > k$ or if $n-1 < 1$. The next lemma is our main tool: if an ideal contains $E_{i,j}^n$ then it contains $E_{i,j+1}^n$ as well. \[lemma: E\_ij contains E\_ij+1\] The $\mathbb{Q}$-linear span of $E_{i,j}^n$ contains $E_{i,j+1}^n$. Let $S' \subseteq \{x_1, \ldots, x_n\}$ be an arbitrary subset of cardinality $j+1$. We construct $e_i(S')$ explicitly in the $\mathbb{Q}$-linear span of $E_{i,j}^n$. Consider the polynomial $$p = \! \! \! \! \! \! \! \! \sum_{\mbox{\scriptsize $\begin{array}{c}S \textup{ such that } \\ S \subseteq S' \textup{ and } |S|=j \end{array}$}} \! \! \! \! \! \! \! \! e_i(S)$$ which is in $\langle E_{i,j}^n \rangle_{\mathbb{Q}}$ by definition. Let $T$ be a subset of $S'$ of cardinality $i$ and consider the coefficient of the monomial $\prod_{x_k \in T} x_k$ in $p$. Whenever $S \subseteq S'$ contains $T$ the monomial $\prod_{x_k \in T} x_k$ appears with coefficient $1$ in $e_i(S)$. There are $j+1$ subsets of $S'$ with cardinality $j$ and all but $i$ of them contain $T$. Hence the coefficient of $\prod_{x_k \in T} x_k$ in $p$ is exactly $j+1-i$. Thus the polynomial $\frac{1}{j+1-i}p = e_i(S')$ as desired. Consider the sets $E_{2,3}^5$ and $E_{2,4}^5$ and let $S' = \{x_1,x_2,x_4,x_5\}$. The polynomial $p$ from Lemma \[lemma: E\_ij contains E\_ij+1\] is $$\begin{aligned} p = (x_1x_2+x_1x_4+x_2x_4) &+ (x_1x_2+x_1x_5+x_2x_5)\\ &\hspace{-.75in} +(x_1x_4+x_1x_5+x_4x_5)+(x_2x_4+x_2x_5+x_4x_5)\end{aligned}$$ which simplifies as desired to $$p = 2(x_1x_2+x_1x_4+x_1x_5+x_2x_4+x_2x_5+x_4x_5)=(4-2)e_2(S').$$ Applying Lemma \[lemma: E\_ij contains E\_ij+1\] repeatedly gives the following. \[corollary: does the trick\] If an ideal $I \subseteq \mathbb{Q}[x_1, \ldots, x_n]$ contains the subset $E_{i,j}^n$ then it contains $E_{i,j+k}^n$ for all $k \in \{1, 2, \ldots, n-j\}$. We now use these combinatorial properties of symmetric functions together with the BFR-generators of the Tanisaki ideal to prove the main lemma of this section \[lemma: standard inclusion acts right on Tanisaki ideals\] Suppose that $\lambda$ is a Young diagram with $n-1$ boxes. If $\lambda'$ is obtained from $\lambda$ by adding one box then $\iota_n^*(I_{\lambda'}) \subseteq I_{\lambda}$. Below we give schematics for the relative configurations of $\lambda$, $\lambda'$, and the deleted box (shown in light grey). $$\begin{tikzpicture}[scale=.35] \draw (0,0)--(0,8)--(10,8)--(10,7)--(10,5)--(8,5)--(8,4)--(4,4)--(4,3)--(3,3)--(3,1)--(2,1)--(2,0)--(0,0); \draw [fill=gray] (0,7) rectangle (10,8); \draw [fill=gray] (4,4)--(4,7)--(10,7)--(10,5)--(8,5)--(8,4)--(4,4); \node [above] at (2,4) {\textbf{A}}; \node [above] at (6.75,6.05) {\textbf{B}}; \draw [white] (4,3)--(4,4)--(5,4);\draw [fill=lightgray, dashed] (4,3)--(4,4)--(5,4)--(5,3)--(4,3); \end{tikzpicture} \hspace{.75in} \begin{tikzpicture}[scale=.35] \draw (0,0)--(0,8)--(10,8)--(10,7)--(10,5)--(8,5)--(8,4)--(4,4)--(4,3)--(3,3)--(3,1)--(2,1)--(2,0)--(0,0); \node [above] at (5,5) {\textbf{A}}; \draw [fill=gray] (0,7) rectangle (10,8); \node [above] at (5,6.675) {\textbf{B}}; \draw [white] (10,7)--(10,8);\draw [fill=lightgray, dashed] (10,7)--(10,8)--(11,8)--(11,7)--(10,7); \end{tikzpicture}$$ We will compare the BFR-generators for $I_{\lambda'}$ and $I_{\lambda}$ for region A, region B, and the two different cases of grey boxes. Each box in region A corresponds to BFR-generators $E_{i,j}^{n-1}$ in $I_{\lambda}$ and $E_{i,j+1}^{n}$ in $I_{\lambda'}$. In each of these cases the image $$\iota_n^*(E_{i,j+1}^{n}) = E_{i,j+1}^{n-1} \cup E_{i,j}^{n-1}$$ by Lemma \[lemma: image of BFR-generators for a box\]. By construction $I_{\lambda}$ contains $E_{i,j}^{n-1}$ so by Corollary \[corollary: does the trick\] we know $I_{\lambda}$ contains $E_{i,j+1}^{n-1}$ as well. It follows that $I_{\lambda}$ contains $\iota_n^*(E_{i,j+1}^{n})$ for every box in region A. If the box in $\lambda' - \lambda$ is not on the first row, then it is labeled $E_{i,j+1}^{n}$ in $I_{\lambda'}$ and the box above it is labeled $E_{i,j}^{n-1}$ in $I_{\lambda}$. This is the case of the previous paragraph. If the box in $\lambda' - \lambda$ is on the first row, then $I_{\lambda'}$ contains $E_{i,i}^n$ while $I_{\lambda}$ contains $E_{i,i}^{n-1}$. Lemma \[lemma: image of BFR-generators for a box\] shows that $\iota_n^*(E_{i,i}^{n}) = E_{i,i}^{n-1} \cup E_{i,i-1}^{n-1}$. However $E_{i,i-1}^{n-1}$ is empty by definition. So $I_{\lambda}$ contains $\iota^*(E_{i,i}^{n})$ in this case too. Finally, each box in region B corresponds to BFR-generators $E_{i,j}^{n-1}$ in $I_{\lambda}$ and $E_{i+1,j+1}^{n}$ in $I_{\lambda'}$. As above we know $$\iota_n^*(E_{i+1,j+1}^{n}) = E_{i+1,j+1}^{n-1} \cup E_{i+1,j}^{n-1}$$ by Lemma \[lemma: image of BFR-generators for a box\]. The box labeled $i+1$ in $\lambda$ is either above but in the same column as in $\lambda'$ or in a column to the right of $i+1$ in $\lambda'$. In either case $E_{i+1,j-k}^{n-1}$ is a BFR-generator for $\lambda$ for some nonnegative integer $k$. Corollary \[corollary: does the trick\] implies that both $E_{i+1,j+1}^{n-1}$ and $E_{i+1,j}^{n-1}$ are in $I_{\lambda}$. Hence $\iota_n^*(E_{i+1,j+1}^{n})$ is in $I_{\lambda}$ for every box in region B, proving the claim. Representation stability of Springer representations {#sec: representation stability} ==================================================== We want to say that a sequence of representations $\{V_n\}$ is representation stable if, when decomposed into irreducible representations, the sequence of multiplicities of each irreducible representation eventually becomes constant. This doesn’t quite make sense because the irreducible representations of the symmetric group $S_n$ depend on $n$, and in fact correspond to the partitions of $n$. The next definition describes a particular family of irreducible representations whose multiplicities we use to define representation stability. Let $\mu = (\mu_1, \ldots, \mu_l)$ be a partition of $k$. For any $n \geq k + \mu_1$ define the *padded partition* to be $$\mu[n] := (n-k, \mu_1, \ldots, \mu_l).$$ Define $V(\mu)_n$ to be the irreducible $S_n$-representation $$V(\mu)_n := V_{\mu[n]}.$$ Note that every partition of $n$ can be written as $\mu[n]$ for a unique partition $\mu$ and hence every irreducible $S_n$-representation is of the form $V(\mu)_n$ for a unique partition $\mu$. We can now define representation stability precisely. A sequence of $S_n$-representations $V_1 \stackrel{f_1}{\rightarrow} V_2 \stackrel{f_2}{\rightarrow} V_3 \stackrel{f_3}{\rightarrow} \cdots$ is [*representation stable*]{} if 1. the linear maps $f_n: V_n \rightarrow V_{n+1}$ are $S_n$-equivariant, in the sense that for each $w \in S_n$ we have $f_n \circ w = w(n+1) \circ f_n$ where $w(n+1) \in S_{n+1}$ is the permutation that sends $n+1 \mapsto n+1$ while otherwise acting as $w$; 2. the maps $f_n: V_n \rightarrow V_{n+1}$ are injective; 3. the span of the $S_{n+1}$-orbit of the image $f_n(V_n)$ is all of $V_{n+1}$; and 4. if $V_n$ decomposes into irreducible representations as $$V_n = \bigoplus_\mu c_{\mu,n} V(\mu)_n$$ then there exists $N$ so that the multiplicities $c_{\mu,n}$ are independent of $n$ for all $n \geq N$. If $N$ is independent of $\mu$ then $V_1 \stackrel{f_1}{\rightarrow} V_2 \stackrel{f_2}{\rightarrow} V_3 \stackrel{f_3}{\rightarrow} \cdots$ is [*uniformly*]{} representation stable. FI-module structure corresponds to a sequence of representations being uniformly representation stable. Moreover the stability of the multiplicities $c_{\mu,n}$ corresponds to a kind of stabilization of the characters of the representations, and thus the dimensions of the representations. Since $\{R_n/I_{\lambda_n}\}$ forms a co-FI-module, we will actually show that the dual $\{\left(R_n/I_{\lambda_n}\right)^\vee\}$ is representation stable. The term [*Springer representation*]{} is applied interchangeably to these two dual representations (as described in the Introduction–see Remark \[rem:Hotta\]). Moreover, the key implications of representation stability (including dimensions and characters) apply to $\{R_n/I_{\lambda_n}\}$ by virtue of applying to its dual. More precisely we have the following. \[cor:springer\_rep\_is\_representation\_stable\] Suppose $\lambda_1 \subseteq \lambda_2 \subseteq \lambda_3 \subseteq \cdots$ is a sequence of Young diagrams for which $\lambda_n$ has $n$ boxes for each $n$. Then the following are all true: 1. Each graded part of the sequence $\{\left(R_n/I_{\lambda_n}\right)^\vee\}$ is uniformly representation stable. 2. For each $k$ and each $n$ let $\chi_{k,n}$ be the character in the $k^{\mathrm{th}}$ graded part of $R_n/I_{\lambda_n}$. The sequence $(\chi_{k,1},\chi_{k,2},\chi_{k,3},...)$ is eventually polynomial in the sense of [@CEF]. 3. For each $k$ and $n$ let $d_{k,n}$ be the dimension of the $k^{\mathrm{th}}$ graded piece of $R_n/I_{\lambda_n}$ as a complex vector space. Then the sequence $(d_{k,1},d_{k,2},d_{k,3},\ldots)$ is eventually polynomial, in the sense that there is an integer $s_k$ and polynomial $p_k(n)$ in $n$ so that for all $n \geq s_k$ the dimensions $d_{k,n}=p_k(n)$. This follows immediately from corresponding results of Church, Ellenberg, and Farb. Part (1) follows from Theorem \[thm:the\_main\_theorem\] together with the definition of representation stability (or, e.g., [@CEF Theorem 1.13]). Part (2) follows for $\left(R_n/I_{\lambda_n}\right)^\vee$ from [@CEF Theorem 1.5] and for the dual representation $R_n/I_{\lambda_n}$ because the characters of complex representations of $S_n$ respect the operation of taking duals. Part (3) is an easy corollary of Part (2) for $R_n/I_{\lambda_n}$ by [@CEF Theorem 1.5] and for its dual because they have the same dimension as complex representations. Combinatorial consequences {#sec:combinatorial_consequences} ========================== The results of Corollary \[cor:springer\_rep\_is\_representation\_stable\] open up new combinatorial questions about Springer fibers. Representation stability guarantees that for any sequence of Young diagrams $\{\lambda_n\}$, the $k^{\mathrm{th}}$ degree of the cohomology of the corresponding Springer fibers stabilizes as a polynomial $p_k(n)$. But what is this polynomial? For instance, given the sequence $\lambda_1 \subseteq \lambda_2 \subseteq \lambda_3 \subseteq \cdots$ of Young diagrams: - Is there an explicit formula for the dimensions $p_k(n)$ of the $k^{\mathrm{th}}$ graded part of $R_n/I_{\lambda_n}$? What if the sequence contains a particular family of Young diagrams (e.g. hooks, two-row, two-column, etc.)? - What is the minimal integer $s_k$ at which the dimension of $R_n/I_{\lambda_n}$ becomes polynomial? Given $k$ and a sequence of Young diagrams, can we find some $s_k$ (not necessarily minimal) after which the dimension of $R_n/I_{\lambda_n}$ is polynomial? - Can we show the Springer representations have polynomial dimension via the monomial bases $\mathcal{B}(\lambda_n)$ from Section \[sec:Springer\_theory\]? To the best of our knowledge, these are entirely new questions about Springer representations; they arise only because of the consequences of representation stability. Moreover, our preliminary answers to these questions suggest additional combinatorial structures that may undergird Springer representations. We give details and some concrete results in this section, followed in the next by (more) open questions. Minimal monomials {#subsec:min_mono} ----------------- Our first results suggest that the exponents that appear in the monomials within each $\mathcal{B}(\lambda)$ are more rigid than previously thought. Let $x^{\alpha}$ be a monomial with exponent vector $(\alpha_1, \ldots, \alpha_n)$. In other words, the exponent of the variable $x_i$ in $x^{\alpha}$ is $\alpha_i$. Let $(\alpha_{i_1}, \ldots, \alpha_{i_r})$ be the nonzero exponents in the order in which they appear in $x^{\alpha}$. This $r$-tuple is called the *monomial type* of $x^{\alpha}$. For instance both $x_2x_3^2x_4$ and $x_3x_9^2x_{11}$ have monomial type $(1,2,1)$. The following result says that if $x_I^{\alpha}$ is a monomial of a fixed monomial type in the GP-basis $\mathcal{B}(\lambda)$ then increasing the indices of the variables lexicographically while preserving the monomial type produces another monomial in the GP-basis $\mathcal{B}(\lambda)$. \[thm:shift\_the\_indices\] Fix a composition $\alpha = (\alpha_1, \ldots, \alpha_k)$. For each ordered sequence $I=(i_1,\ldots,i_k)$ let $x_I^{\alpha}$ denote the monomial $\prod_{j=1}^k x_{i_j}^{\alpha_j}$. If $x_I^{\alpha} \in \mathcal{B}(\lambda)$ and $I'$ satisfies $$I \leq I' \leq (n-k+1,\ldots,n-1,n)$$ in lexicographic order then $x_{I'}^{\alpha} \in \mathcal{B}(\lambda)$. First note that we may simply consider the case when $I$ and $I'$ differ by exactly one in exactly one entry. Indeed, for each $j$ with $1 \leq j \leq k+1$, define the sequences $I_j = (i_1, i_2, \ldots, i_{j-1}, i'_j, i'_{j+1}, \ldots, i'_k)$. Note that $I=I_{k+1}$ and $I' = I_1$. Thus it suffices to prove that if $x_{I_{j+1}}^{\alpha} \in \mathcal{B}(\lambda)$ then $x_{I_{j}}^{\alpha} \in \mathcal{B}(\lambda)$. Furthermore we may assume that $i'_j = i_j+1$ since repeating the argument would successively increment the index and imply the result for more general $i'_j$. Thus we prove the claim for $I_j$ and $I_{j+1}$ assuming that $i'_j = i_j+1$. The GP-algorithm proceeds the same for both monomials $x_{I_j}^{\alpha}$ and $x_{I_{j+1}}^{\alpha}$ in the variables $x_n, x_{n-1}, \ldots, x_{i_j+2}$. Let $\lambda^{(j+2)}$ be the Young diagram left after those steps. Implementing the GP-algorithm for $x_{I_{j+1}}^{\alpha}$ removes a box from the first row and then from the $\alpha_j+1^{\mathrm{th}}$ row of $\lambda^{(j+2)}$, whereas for $x_{I_j}^{\alpha}$ it removes a box from the $\alpha_j+1^{\mathrm{th}}$ row and then from the first row. Call the resulting Young diagrams $\lambda^{(j+1)}$ and $\lambda^{(j)}$ respectively. We now show that $\lambda^{(j+1)} \unrhd \lambda^{(j)}$ in dominance order. If the first row of $\lambda^{(j+2)}$ is strictly larger than the $\alpha_j+1^{\mathrm{th}}$ row of $\lambda^{(j+2)}$ then the result is clear, since then $\lambda^{(j+1)} = \lambda^{(j)}$. If not, then removing a box from the first row of $\lambda^{(j+2)}$ leaves a Young diagram whose shape is constrained: at least the first $\alpha_j$ rows have the same length and there is at least one row immediately after of length one less. Thus $\lambda^{(j+1)} \unrhd \lambda^{(j)}$ with equality in the case that exactly the first $\alpha_j+1$ rows of $\lambda^{(j+2)}$ are the same length, and any subsequent rows are smaller. By Proposition \[proposition: dominance order\], we have $\mathcal{B}(\lambda^{(j+1)}) \subseteq \mathcal{B}(\lambda^{(j)})$. Since $x_{i_1}^{\alpha_1}x_{i_2}^{\alpha_2} \cdots x_{i_{j-1}}^{\alpha_{j-1}}$ is a GP-monomial in $\mathcal{B}(\lambda^{(j+1)})$ it is also a GP-monomial in $\mathcal{B}(\lambda^{(j)})$ so the monomial $x_{I_j}^{\alpha}$ can be obtained from the GP-algorithm starting with the shape $\lambda^{(j+2)}$. Thus $x_{I_j}^{\alpha}$ is in the GP-basis for $\mathcal{B}(\lambda)$ as desired, and the claim follows. In fact we conjecture that there is a unique minimal monomial of each monomial type, in the following lexicographic sense. Fix a monomial type $\alpha$ and a partition $\lambda$ of $n$. The monomial $x_I^{\alpha} \in \mathcal{B}(\lambda)$ is a [*minimal monomial*]{} of type $\alpha$ if for any other $x_{I'}^{\alpha} \in \mathcal{B}(\lambda)$ of the same monomial type, the index sets satisfy $I \leq I'$ in lexicographic order. The next result proves that when $\lambda$ has exactly two rows then there is a unique minimal monomial of each monomial type in $\mathcal{B}(\lambda)$. The previous result then says that this minimal monomial produces all other monomials of that fixed monomial type within $\mathcal{B}(\lambda)$. \[thm:2\_row\_case\_min\_monomials\] Fix a two-row Young diagram $\lambda = (n-k,k)$. Then the basis $\mathcal{B}(\lambda)$ of $R_{n}/I_\lambda$ has the unique minimal monomial set $\left\{ \prod\limits_{j=1}^i x_{2j} \right\}_{i=1}^k$. Since $\lambda$ has two rows, the maximum exponent that appears in a monomial for $\lambda$ is $1$. Since $\lambda$ has $k$ boxes in the second row, the maximum degree in a monomial for $\lambda$ is $k$. So every monomial type that appears in $\mathcal{B}(\lambda)$ is the partition given by $j$ copies of $1$, where $j$ is any integer with $0 \leq j \leq k$. We claim that the minimal monomial of degree $j$ is $x_2 x_4 \cdots x_{2j}$. To see this, let $x_I \in \mathcal{B}(\lambda)$ have degree $j$ and index set $I = (i_1, \ldots, i_j)$. It suffices to show that $(2, 4, \ldots, 2j) \leq I$ in lexicographic order, namely that $i_r \geq 2r$ for each $r \in \{1,\ldots,j\}$. Recall that in the GP-algorithm, exactly one rightmost box in a row is removed when going down each level of the GP-tree. Going from Level $i$ to Level $i-1$, the box removed is in the first row of the Level $i$ diagram if $i \notin \{i_1, \ldots, i_j\}$ and the second row otherwise. After a box is removed from the first row, the rows of the resulting Level $i-1$ diagram are switched if necessary to maintain the shape of a Young diagram. Now suppose $r \in \{1,\ldots,j\}$. The path in the GP-tree from Level $i_r$ to Level $i_r - 1$ removes a box from the second row of the Level $i_r$ diagram $\lambda^{(i_r)}$. The diagram $\lambda^{(i_r)}$ must have at least $r$ boxes in the second row since the $r-1$ factors $x_{i_{r-1}}, x_{i_{r-2}}, \ldots, x_{i_1}$ remaining in $x_I$ correspond to $r-1$ more second-row boxes removed in the remaining $i_r - 2$ levels of the GP-tree below Level $i_r - 1$. Since all Level $i_r$ diagrams have $i_r$ boxes and $\lambda^{(i_r)}$ has at least $r$ boxes in its second row, we conclude $i_r \geq 2r$ as desired. We conjecture that for each $\mathcal{B}(\lambda)$ and each monomial type that appears in $\mathcal{B}(\lambda)$, there is a unique minimal monomial of that monomial type (see Section \[subsec:unique\_min\_mono\]). Polynomial dimension when the number of rows exceed a certain minimum {#subsec:poly_dim} --------------------------------------------------------------------- We can explicitly compute the polynomial dimensions described in Corollary \[cor:springer\_rep\_is\_representation\_stable\] in some cases, including when the Young diagram has “enough" rows. In this case, the dimensions in low degree coincide with the corresponding dimensions for the diagonal coinvariant algebra, as described in the next theorem. Kim’s version of stability coincides with ours in this case [@Kim_Stability Remark in Section 4]. \[thm: at least k+1 rows gives a polynomial dimension formula\] Let $\dim_i(R_n/I_{\lambda_n})$ denote the dimension of the $i^{\mathrm{th}}$-degree part of $R_n/I_{\lambda_n}$. Let $\{ \lambda_n \}$ be a nested sequence of Young diagrams such that $|\lambda_n| = n$ for all $n$. If for some $N$ the diagram $\lambda_N$ contains at least $k+1$ rows, then $\dim_i\left({{\raisebox{.2em}{$R_n$}\left/\raisebox{-.2em}{$I_{\lambda_n}$}\right.}}\right)$ agrees with the dimension of the $i^{\mathrm{th}}$-degree part of the diagonal coinvariant algebra ${{\raisebox{.2em}{$R_n$}\left/\raisebox{-.2em}{$\left\langle e_j \right\rangle_{j=1}^n$}\right.}}$ for all $i \leq k$ and $n \geq N$. In particular for all $i \leq k$ and $n \geq N$ we have $\dim_i\left({{\raisebox{.2em}{$R_n$}\left/\raisebox{-.2em}{$I_{\lambda_n}$}\right.}}\right) = p_i(n)$ where $p_i(n)$ is the polynomial that gives the number of permutations of the set $\{1,2,\ldots,n\}$ with exactly $i$ inversions. Let ${\mathrm{col}}_n = (1,\ldots,1)$ denote the column partition with exactly $n$ parts. The diagonal coinvariant algebra is $R_n/I_{{\mathrm{col}}_n}$. There is a unique monomial $x_2 x_3^2 x_4^3 \cdots x_n^{n-1} \in \mathcal{B}({\mathrm{col}}_n)$ of maximal degree; it is obtained by choosing the far-right edge at each level in the GP-tree. Garsia-Procesi proved that their basis elements form a lower-order ideal with respect to division [@GarPro92 Proposition 4.2]. It follows that the other elements of $\mathcal{B}({\mathrm{col}}_n)$ are precisely the divisors of this maximal monomial, so $$\label{basis for column partition of size n} \mathcal{B}({\mathrm{col}}_n) = \{ x_2^{\alpha_2} x_3^{\alpha_3} x_4^{\alpha_4} \cdots x_n^{\alpha_n} \; | \; 0 \leq \alpha_j \leq j-1 \}.$$ Now suppose that $\lambda_n$ has at least $k+1$ rows. Then $\lambda_n \supseteq {\mathrm{col}}_{k+1}$ and so $\mathcal{B}(\lambda_n) \supseteq \mathcal{B}({\mathrm{col}}_{k+1})$ by Lemma \[lemma: containment and Garsia-Procesi basis\]. By Theorem \[thm:shift\_the\_indices\] we can lexicographically increase the subscripts of each monomial in $\mathcal{B}({\mathrm{col}}_{k+1})$ to produce monomials $x_I^{\alpha}$ for each $k$-element subset $I \subseteq \{x_2, \ldots, x_n\}$ and exponents $\alpha$ which satisfy $0 \leq \alpha_j \leq j-1$ for each $j=1,2,\ldots,k+1$. This proves that $\mathcal{B}(\lambda_n)$ also contains all of those possible elements of degrees $0, 1, \ldots, k$ in the variable set $\{x_2, \ldots, x_n\}$. Trivially, these are the same as the monomials of degrees $0, 1, \ldots, k$ in the basis for $\mathcal{B}({\mathrm{col}}_n)$ above, so in these degrees the basis elements for $R_n/I_{\lambda_n}$ and the diagonal coinvariant algebra $R_n / I_{{\mathrm{col}}_n}$ coincide. The claim about $p_i(n)$ follows from the similar statement for the $i^{\mathrm{th}}$ graded part of $R_n / I_{{\mathrm{col}}_n}$. The polynomial function $p_i(n)$ can be found explicitly as an alternating sum of certain combinations. Example \[exam:poly\_dim\_closed\_form\] gives closed formulas for $p_i(n)$ for the first few values of $i$. Since the diagonal coinvariant algebra is the cohomology ring of the full flag variety, this result implies that the Garsia-Procesi bases for the cohomology of the Springer fibers coincide with the cohomology of the full flag variety in many degrees. More precisely if $\mathcal{S}_{\lambda_n}$ denotes the Springer fiber corresponding to the nilpotent of Jordan type $\lambda_n$ then we have an isomorphism $H^i(\mathcal{S}_{\lambda_n}) \cong H^*(GL_n(\mathbb{C})/B)$ for all $i \leq k$ and $n \geq N$, where $k$ and $N$ are as given in Theorem \[thm: at least k+1 rows gives a polynomial dimension formula\]. Below we give the basis elements for $\mathcal{B}({\mathrm{col}}_5)$ in degrees $0$ through $3$ and the cardinalities of the $k^{\mathrm{th}}$ degree parts for $4 \leq k \leq 10$, using the unimodality of the sequence of dimensions for $k\geq 6$. degree \# monomials in $\mathcal{B}({\mathrm{col}}_5)$ -------------------- -------------------------------------------- ---------------------------------------------- 0 1 1 1 4 $x_i$ for $2 \leq i \leq 5$ 2 9 $x_i^2$ for $3 \leq i \leq 5$ and $x_ix_j$ for $2 \leq i<j \leq 5$ 3 15 $x_i x_j x_k$ for $2 \leq i<j<k \leq 5$, $x_i x_j^2$ for $2 \leq i < j \leq 5$, $x_i^2 x_j$ for $3 \leq i < j \leq 5$, and $x_i^3$ for $4 \leq i \leq 5$ 4 20 5 22 $6 \leq k \leq 10$ same cardinality as $(10-k)^{\mathrm{th}} \mbox{ degree part}$ $k > 10$ 0 Compare this to the basis $\mathcal{B}(\lambda)$ for the partition $\lambda = (2,2,1)$ that we computed in Example \[example: GP-tree\]. The basis elements for $\mathcal{B}(\lambda)$ and $\mathcal{B}({\mathrm{col}}_n)$ coincide in degrees 0, 1, and 2, as expected. However in degree 3 the set $\mathcal{B}({\mathrm{col}}_5)$ contains four basis elements that are not in $\mathcal{B}(\lambda)$, namely $x_2 x_3 x_4$, $x_3^2 x_4$, $x_4^3$, and $x_5^3$. \[exam:poly\_dim\_closed\_form\] We give the following closed formulas for the dimensions $p_i(n)$ from Theorem \[thm: at least k+1 rows gives a polynomial dimension formula\] when $0 \leq i \leq 6$: - When $i=0$ we have $p_0(n) = 1$. - When $i = 1, 2, 3, 4$ we have $$\label{nice_closed_formula} p_i(n) = \binom{n-2+i}{i} - \binom{n-3+i}{i-2}.$$ - When $i=5$ we have $$p_5(n) = \binom{n+3}{5} - \binom{n+2}{3} + 1.$$ - When $i=6$ we have $$p_6(n) = \binom{n+4}{6} - \binom{n+3}{4} + n.$$ To prove these, let $i \in \{1,2,3,4\}$. Our convention is that $0 \in \mathbb{N}$. Define the set $$\begin{aligned} \widetilde{E_i} &= \left\{ (d_2,\ldots,d_n)\in \mathbb{N}^{n-1} \; \middle\vert \; \sum_{k=2}^n d_k = i \mbox{ and } d_k \leq i \right\}\end{aligned}$$ of nonnegative integer solutions to the equation $d_2 + \cdots + d_n = i$. We know $|\widetilde{E_i}| = \binom{(n-2)+i}{i}$. Define $E_i \subseteq \widetilde{E_i}$ to be the subset satisfying also $d_k \leq k-1$ for each $k$ so that $p_i(n) = |E_i|$. We now compute $|E_i| = |\widetilde{E_i}| - |E_i^c|$. For each $m$, define the set $F_{m,i} \subseteq \widetilde{E_i}$ by the condition that $d_m > m-1$. Observe that $E_i^c = \textstyle\bigcup\limits_{m=2}^{n} F_{m,i}$. Moreover the sets $F_{m,i}$ are pairwise disjoint. Indeed, suppose there were $m_1<m_2$ with an element $(d_2,\ldots,d_n) \in F_{m_1,i} \cap F_{m_2,i}$. Then $d_{m_1} > m_1-1 \geq 1$ and $d_{m_2} > m_2-1 \geq 2$ and so $d_2 + \cdots + d_n \geq 5$. This contradicts $i \in \{1,2,3,4\}$. So $|E_i^c| = \textstyle\sum\limits_{m=2}^n|F_{m,i}|$. We now find $|F_{m,i}|$. Since $d_m > m-1$ we write $d_m = m + d_m'$ for some $d_m' \geq 0$ and then count instead the solutions to $d_2 + \cdots + d_m' + \cdots + d_n = i-m$. Thus $$\begin{aligned} |E_i^c| = \sum_{m=2}^n |F_{m,i}| = \sum_{m=2}^n \binom{(n-2) + (i-m)}{i-m}.\end{aligned}$$ As long as $i \leq n$ this sum has the form $|E_i^c| = \cdots + \binom{r_m + 2}{2} + \binom{r_m + 1}{1} + \binom{r_m}{0}$ for some $r_m$. Using a combinatorial identity we write $|E_i^c| = \binom{(n-2) + (i-2) + 1}{i-2}$. This gives the formula for $p_i(n)$ in Equation (\[nice\_closed\_formula\]). When $i=5$ and $i=6$ we can use essentially the previous argument, since only a few pairs of $F_{m,i}$ share elements. When $i=5$ the sets $F_{2,5}$ and $F_{3,5}$ both contain $(2,3,0,0,0,\ldots,0)$. Inclusion-Exclusion gives the desired formula. Similarly if $i=6$ then $F_{2,6} \cap F_{3,6}$ contains the two elements $(3,3,0,0,\ldots,0)$ and $(2,4,0,0,\ldots,0)$ as well as the $n-3$ elements $(2,3,d_4,\ldots,d_n)$ where exactly one of the remaining entries $d_k$ is nonzero (and thus is 1). Additionally $F_{2,6} \cap F_{4,6}$ contains the unique element $(2,0,4,0,0,\ldots,0)$. Inclusion-Exclusion gives the desired formula. This process becomes more complicated when $i$ is larger. Kim gives equivalent but different versions of the formulas in Example \[exam:poly\_dim\_closed\_form\] [@Kim_Stability Example 3.3]. Open questions {#sec:open_questions} ============== This section describes a set of questions that arise from our analysis of representation stability, including how to characterize the polynomials into which the dimensions stabilize, and how to describe the monomials in $\mathcal{B}(\lambda)$. Minimal monomials and monomial types {#subsec:unique_min_mono} ------------------------------------ In Subsection \[subsec:min\_mono\] we discussed monomial types and minimal monomials. We pose several questions here. For each Young diagram $\lambda$ and monomial type $\alpha$, is there a unique monomial $x_I^{\alpha} \in \mathcal{B}(\lambda)$ that is minimal with respect to lexicographic ordering, in the sense that if $x_{I'}^{\alpha} \in \mathcal{B}(\lambda)$ then $I \leq I'$? All examples that we have computed do in fact have minimal monomials. There are several related questions about how the set of monomial types that appear in $\mathcal{B}(\lambda)$ are related to $\lambda$. The first essentially asks how to predict the monomial types that appear for a given $\lambda$, while the second asks which $\mathcal{B}(\lambda)$ contain a given monomial type $\alpha$. For each Young diagram $\alpha$, let $\mathcal{A}(\lambda)$ be the set of monomial types in $\mathcal{B}(\lambda)$. Is there a quick algorithm to construct $\mathcal{A}(\lambda)$ for arbitrary $\lambda$? We know there is an algorithm to construct $\mathcal{A}(\lambda)$, namely use the GP-algorithm to find all of $\mathcal{B}(\lambda)$ and then identify which monomial types appear. The previous question is asking for a direct algorithm, or for characterizations of the monomial types that do or do not appear. Fix a monomial type $\alpha$. For which $\lambda$ is there a monomial of type $\alpha$ in $\mathcal{B}(\lambda)$? For instance $\mathcal{B}(\lambda)$ contains a monomial of type $(k)$ if and only if $\lambda$ has at least $k+1$ rows. Polynomial dimension for particular families of Young diagrams -------------------------------------------------------------- In this section, we return to questions from Subsection \[subsec:poly\_dim\] to ask for concrete formulas for dimensions of the Springer representations, and especially for the polynomials $p_k(n)$ to which they stabilize. Representation stability relies on sequences of diagrams while the literature on Springer fibers instead uses families of Young diagrams like two-row, two-column, hook, and so on. To study families of Young diagrams, we choose a sequence $\{\lambda_n\}$ that characterizes particular shapes, as Example \[example: 2-column\] does for two-column Young diagrams. For what families of Young diagrams $\{\lambda_n\}$ can we find an explicit closed formula for the polynomial dimension $p_k(n)$? For instance, Kim gave a formula for two-row Young diagrams [@Kim_Euler_char], following ideas due to Fresse [@Fresse Example 4.5]. When we can identify minimal monomials (unique or not), strictly combinatorial techniques might then give information about the polynomial dimension formula. More precisely: Can we find minimal monomials for other shapes, and use them together with the index-incrementation tools in Theorem \[thm:shift\_the\_indices\] to describe the dimension polynomials, either partially or completely? For instance, Theorem \[thm:2\_row\_case\_min\_monomials\] gave the minimal monomials for the case of the two-row diagrams $\lambda = (n-k,k)$. In that case, the previous question sketches an alternate proof of Kim’s result for two-row Young diagrams. Stable shapes ------------- The central question of this section is to determine the integer $s_k$ after which the polynomial $p_k(n)$ counts the degree-$k$ monomials in $\mathcal{B}(\lambda_n)$. More colloquially, when does the dimension stabilize? We conjecture that there is a core shape such that as soon as $\lambda_n$ contains the core shape, the dimension stabilizes. The following describes one possible choice of a core shape, though we do not believe that it is in general minimal. Let $\lambda_1 \subseteq \lambda_2 \subseteq \cdots$ be a sequence of Young diagrams with $|\lambda_n| = n$ for each $n$. Denote by $\tau_{k+1}$ the staircase partition $(k+1, k, \ldots,1)$. The *$k$-stable shape* of $\{ \lambda_n \}$ is the partition $(\cup_n \, \lambda_n) \cap \tau_{k+1}$. \[example: 2-column\] Let $\{ \lambda_n \}$ be the nested sequence of Young diagrams defined by $\lambda_1 = (1)$, $\lambda_{2n} = (\underbrace{2, 2, \ldots, 2}_{n \; \mbox{\tiny times}})$, and $\lambda_{2n+1} = (\underbrace{2, 2, \ldots, 2}_{n \; \mbox{\tiny times}},1)$. Then $\{ \lambda_n \}$ is the following $$\mbox{\small $\Yvcentermath1 \yng(1) \rightarrow \yng(2) \rightarrow \yng(2,1) \rightarrow \yng(2,2) \rightarrow \yng(2,2,1) \rightarrow \yng(2,2,2) \rightarrow \yng(2,2,2,1) \rightarrow \yng(2,2,2,2) \rightarrow \cdots$}$$ The $1$-, $2$- and $3$-stable shapes are $\mbox{\tiny \Yvcentermath1 \yng(2,1)}$, $\mbox{\tiny \Yvcentermath1 \yng(2,2,1)}$ and $\mbox{\tiny \Yvcentermath1 \yng(2,2,2,1)}$, respectively. Let $\{ \lambda_i \}$ be a sequence of Young diagrams such that $\lambda_1 \subseteq \lambda_2 \subseteq \cdots$ with $|\lambda_i| = i$. Let $N$ be the smallest integer for which $\lambda_N$ contains the $k$-stable shape of $\{ \lambda_i \}$. Then there exists a polynomial $p_k(n)$ that gives the dimension of the $k^{\mathrm{th}}$ graded part of the Springer representation $R_n/I_{\lambda_n}$ for all $n \geq N$. If true, this conjecture would provide more tools to explicitly compute $p_k(n)$. The authors thank Tom Church, Jordan Ellenberg, and Benson Farb for useful conversations. We also thank the anonymous referees for their helpful comments and suggestions. R. Biagioli, S. Faridi, and M. Rosas: The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of [W]{}eyman. (2008) C. Bibby: Representation stability for the cohomology of arrangements associated to root systems. **48**, no. 1, 51–75 (2018) A. Borel: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de [L]{}ie compacts. (2) **57**, 115–207 (1953) W. Borho and R. MacPherson: Partial Resolutions of Nilpotent Varieties. *Astérisque* **101**-**102**, 23–74, Soc. Math. France, Paris (1983) T. Church. Personal communication, 2020. T. Church, J.S. Ellenberg, and B. Farb: [FI]{}-modules and stability for representations of symmetric groups. **164**, no. 9, 1833–1910 (2015) T. Church, J.S. Ellenberg, and B. Farb: [FI]{}-modules: a new approach to stability for $S_n$-representations. This is the 2012 version of the paper above. Available at <http://arxiv.org/pdf/1204.4533v2.pdf>. T. Church, J.S. Ellenberg, B. Farb, and R. Nagpal: [FI]{}-modules over Noetherian rings. **18**, no. 5, 2951–2984 (2014). T. Church and B. Farb: Representation theory and homological stability. **245**, 250–314 (2013) C. De Concini and C. Procesi: Symmetric functions, conjugacy classes and the flag variety. **64**(2), 203–219 (1981) F. De Mari, C. Procesi, and M.A. Shayman: Hessenberg varieties. **332**(2), 529–534 (1992) J. Duque and R. Rolland: Representation stability for the pure cactus group. Available at <https://arxiv.org/pdf/1501.02835>. B. Farb: Representation Stability. Available at <http://arxiv.org/pdf/1404.4065.pdf>. S. Fomin, S. Gelfand, and A. Postnikov: Quantum [S]{}chubert polynomials. **10**(3), 565–596 (1997) L. Fresse: Betti numbers of [S]{}pringer fibers in type [A]{}. **322**, 2566–2579 (2009) N. Gadish: Representation stability for families of linear subspace arrangements. **322**, 341–377 (2017) A. Garsia and C. Procesi: On certain graded $S_n$-modules and the $q$-[K]{}ostka polynomials. **94**, 82–138 (1992) P. Hersh and V. Reiner: Representation stability for cohomology of configuration spaces in $\mathbb{R}^d$. **5**, 1433–1486 (2017) R. Hotta.: On [S]{}pringer’s representations. **28** (1981), no. 3, 863–876 (1982) D. Kim: Euler characteristic of Springer fibers. . **24**, no. 2, 403–428 (2019) D. Kim: Stability of [S]{}pringer representations for type [A]{}. Available at <https://arxiv.org/pdf/1612.02381>. H. Kraft: Conjugacy classes and [W]{}eyl group representations. In: Young tableaux and [S]{}chur functors in algebra and geometry ([T]{}oruń, 1980), *Astérisque* **87**-**88**, 191–205, Soc. Math. France, Paris (1981) G. Lusztig: Intersection cohomology complexes on a reductive group. **75**, 205–272 (1984) A. Mbirika: A [H]{}essenberg generalization of the [G]{}arsia-[P]{}rocesi basis for the cohomology ring of [S]{}pringer varieties. **17**(1), Research Paper 153 (2010) P. Patzt: Representation stability for filtrations of [T]{}orelli groups. **372**, 257–298 (2018) C. Procesi: [*Lie groups – An approach through invariants and representations*]{}. Universitext. Springer, New York (2007) R. Rolland: Representation stability for the cohomology of the moduli space $M_g^n$. **11**, 3011–3041 (2011) R. Rolland: The cohomology of $\mathcal{M}_{0,n}$ as an [FI]{}-module. **14**, 313–323 (2016) T.A. Springer: A construction of representations of [W]{}eyl groups. **44**(3), 279–293 (1978) T. Tanisaki: Defining ideals of the closures of the conjugacy classes and representations of the [W]{}eyl groups. (2) **34**(4), 575–585 (1982) J. Wilson: $\mathrm{FI}_W$-modules and stability criteria for representations of classical [W]{}eyl groups. **420**, 269–332 (2014) [^1]: The first author was partially supported by the UWEC Department of Mathematics and Office of Research and Sponsored Programs. [^2]: The second author was partially supported by a Sloan Research Fellowship as well as NSF grants DMS-1248171 and DMS-1362855.

--- abstract: 'In distributed training, the communication cost due to the transmission of gradients or the parameters of the deep model is a major bottleneck in scaling up the number of processing nodes. To address this issue, we propose *dithered quantization* for the transmission of the stochastic gradients and show that training with *Dithered Quantized Stochastic Gradients (DQSG)* is similar to the training with unquantized SGs perturbed by an independent bounded uniform noise, in contrast to the other quantization methods where the perturbation depends on the gradients and hence, complicating the convergence analysis. We study the convergence of training algorithms using DQSG and the trade off between the number of quantization levels and the training time. Next, we observe that there is a correlation among the SGs computed by workers that can be utilized to further reduce the communication overhead without any performance loss. Hence, we develop a simple yet effective quantization scheme, nested dithered quantized SG (NDQSG), that can reduce the communication significantly *without requiring the workers communicating extra information to each other*. We prove that although NDQSG requires significantly less bits, it can achieve the same quantization variance bound as DQSG. Our simulation results confirm the effectiveness of training using DQSG and NDQSG in reducing the communication bits or the convergence time compared to the existing methods without sacrificing the accuracy of the trained model.' author: - Afshin Abdi - | Afshin Abdi & Faramarz Fekri [^1]\ School of Electrical and Computer Engineering\ Georgia Institute of Technology\ Atlanta, GA, USA\ `{abdi, fekri}@ece.gatech.edu` bibliography: - 'DitheredQSGD.bib' title: Nested Dithered Quantization for Communication Reduction in Distributed Training --- Introduction {#sec:introduction} ============ Preliminaries {#sec:prelimiary} ============= Distributed Training Using Dithered Quantization ================================================ Dithered Quantized Stochastic Gradient -------------------------------------- Reducing Communication Overhead by Nested Quantization ------------------------------------------------------ Experiments =========== Conclusion ========== In this paper, first, we introduced DQSG, dithered quantized stochastic gradient, and showed that how it can reduce communication bits per training iteration both theoretically and via simulations, without affecting the accuracy of the trained model. Next, we explored the correlation that exists among the SGs computed by workers in a distributed system and proposed NDQSG, a nested quantization method for the SGs. Using theoretical analysis as well as simulations, we showed that NDSQG performs almost the same as DQSG in terms of accuracy and training speed, but with much fewer number of communication bits. Finally, we would like to mention that although the simulations and analysis of the proposed distributed training method is done in synchronous training setup, it is applicable to the asynchronous training as well. Further, our nested quantization scheme can be easily extended to hierarchical distributed structures. [^1]: This work is supported by Sony Faculty Research Award.

--- abstract: 'We present a detailed spectroscopic analysis of the hot DA white dwarf G191-B2B, using the best signal to noise, high resolution near and far UV spectrum obtained to date. This is constructed from co-added *HST* STIS E140H, E230H, and *FUSE* observations, covering the spectral ranges of 1150-3145Å and 910-1185Å respectively. With the aid of recently published atomic data, we have been able to identify previously undetected absorption features down to equivalent widths of only a few mÅ. In total, 976 absorption features have been detected to $3\sigma$ confidence or greater, with 947 of these lines now possessing an identification, the majority of which are attributed to Fe and Ni transitions. In our survey, we have also potentially identified an additional source of circumstellar material originating from Si [iii]{}. While we confirm the presence of Ge detected by @vennes05a, we do not detect any other species. Furthermore, we have calculated updated abundances for C, N, O, Si, P, S, Fe, and Ni, while also calculating, for the first time, an NLTE abundance for Al, deriving Al [iii]{}/H=$1.60_{-0.08}^{+0.07}\times{10}^{-7}$. Our analysis constitutes what is the most complete spectroscopic survey of any white dwarf. All observed absorption features in the *FUSE* spectrum have now been identified, and relatively few remain elusive in the STIS spectrum.' author: - | S. P. Preval$^{1}$[^1], M. A. Barstow$^{1}$, J. B. Holberg$^{2}$ & N. J. Dickinson$^{1}$\ $^{1}$Department of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH\ $^{2}$Lunar and Planetary Laboratory, Sonett Space Sciences Building, University of Arizona, Tucson, AZ 85721 date: 'Accepted 2013 January 15. Received 2013 January 14; in original form 2013 January 11' title: 'A comprehensive near and far ultraviolet spectroscopic study of the hot DA white dwarf G191-B2B' --- \[firstpage\] stars: individual: G191-B2B, abundances, white dwarfs, circumstellar matter. Introduction ============ The archetype H rich white dwarf, G191-B2B (WD0501+527), has long been used both as a photometric standard due to its apparent brightness, and also as a spectroscopic “gold standard” when analysing other hot DA stars. This object has also been used as a flux standard at almost all wavelengths, beginning with the work of @oke74a to the absolute *Hubble Space Telescope* (HST) flux scale of @bohlin04a. In Table \[table:factfile\], we list the basic stellar parameters of G191-B2B, where the mass, absolute magnitude ($M_\nu$) and cooling time ($t_{\mathrm{cool}}$) have been determined using the photometric tables of @holberg06a [@kowalski06a; @tremblay11a; @bergeron11a][^2] (hereafter the Montreal photometric tables). The designation G191-B2B originates from the Giclas Lowell proper motion survey. While not formally identified as a white dwarf, @greenstein69a did identify it as being degenerate in nature, assigning it the spectral classification DAwk, and also designating it as a possible subdwarf. It was also listed by @eggen67a as a common proper motion pair with the K star G191-B2A, located 84" to the north. Such a proper motion pairing is now considered erroneous, as *Hipparcos* has shown the two stars to have distinctly different proper motions. G191-B2B is therefore an isolated star and efforts to determine a gravitational radial velocity from both stars (@reid88a and @bergeron95a) are moot. A seminal high dispersion UV observation of G191-B2B by @bruhweiler81a, using the *International Ultraviolet Explorer* (*IUE*), revealed the surprising discovery of many highly ionised features such as C [iv]{}, N [v]{}, and Si [iv]{} in a star that was thought to possess a pure H spectrum. The detection of such absorption features led to extensive studies of other white dwarf stars. A decade later, the first detections of an ionised heavy metal, Fe [v]{}, were made by @sion92a in the photospheric spectrum of G191-B2B using the *HST* Faint Object Spectrograph. This led to searches for additional heavy metals in white dwarf spectra, with the next Fe group metal, Ni, being discovered in G191-B2B and REJ2214-492 by @holberg94a, and Feige 24 and REJ0623-377 by @werner94a, using co-added *IUE* high dispersion spectra. Further spectroscopic surveys were conducted, with yet more new heavy metals being discovered by @vennes96a, who made detections of the resonant transitions of P [v]{} and S [vi]{} in ORFEUS spectra. A later survey by @bruhweiler99a used a spectrum from the *HST*, and discovered the presence of an additional component in the resonant lines of C [iv]{} 1548.203 and 1550.777Å (hereafter C [iv]{} 1548 and 1550Å respectively), that could not be attributed to either the photosphere of G191-B2B, or to the ISM. This feature was thought to be interstellar or “circumstellar” by @vennes01a, who in their survey of G191-B2B, showed that this feature was separated from the photospheric velocity by  15 km/s. This circumstellar material was not unique to G191-B2B, and similar high ionisation features were found in seven other DA white dwarfs by @bannister03a, corroborated by @dickinson12b. The heaviest metal detected in a DA white dwarf thus far was found by @vennes05a, who made detections of resonant absorption features of Ge [iv]{}. The *Extreme Ultraviolet Explorer* (*EUVE*), and the *Joint Astrophysical Plasma-dynamic Experiment* (*J-PEX*) [@bannister99a] have observed G191-B2B extensively, covering 70-770Å and 170-290Å respectively. *EUVE* observations of the star showed that the predicted flux far exceeded that observed [@kahn84a], meaning that model atmospheres for some effective temperature ($T_{\mathrm{eff}}$), gravity (log $g$), and composition could not simultaneously match the EUV, UV, and optical spectra. This was accomplished by @lanz96a, who showed that including additional opacity in model atmosphere calculations reduced the predicted flux to the correct level. However, this came at the cost of including additional He [ii]{} opacity in the photosphere in order to maintain good agreement below the He [ii]{} Lyman limit. An alternative was explored by @barstow98a, whereby they invoked a stratified H+He and homogeneous heavy metal atmosphere. While the model could also reproduce the EUV spectrum, it predicted He [ii]{} absorption features that descended far deeper than that observed. A substantial piece of work was done by @dreizler99a, who performed self consistent calculations including the effects of radiative levitation and gravitation settling for G191-B2B. By depleting the Fe and Ni abundance at the surface and having it increase at greater depths, they were able to reproduce the EUV spectrum without including other sources of opacity, mass loss, or accretion. A similar atmospheric configuration was used by @barstow99a, who stratified Fe, with increasing abundance with greater depth. They showed that this stratified configuration was preferred statistically over a homogeneous Fe distribution. Observations with *J-PEX* yielded further discoveries in the EUV spectrum of G191-B2B. @cruddace02a performed a spectroscopic analysis of the star using *J-PEX* data, and made detections of the He [ii]{} Lyman series. This would be significant if it was found to be interstellar, as this would imply an unusual ionization fraction. It was also proposed that this He may be associated with the additional absorption component observed in the C [iv]{} doublet, discussed by @bruhweiler99a and @vennes01a. A further spectroscopic survey using *J-PEX* was conducted by @barstow05a. They found that by splitting the He [ii]{} features into two components, one for the Local Interstellar Cloud, and the other as another interstellar feature, the He ionisation fraction agreed with values obtained from other lines of sight. Observations of the H Lyman/Balmer line series of a white dwarf also allows information on its physical parameters to be inferred. Pioneered by @holberg85a, a grid of theoretical spectra with differing values of $T_{\mathrm{eff}}$ and log $g$ can be used to fit either the H Lyman or Balmer absorption profiles to that of an observed spectrum, as such profiles are very sensitive to changes in these parameters. Such a method, however, appears to have limitations for white dwarfs whose $T_{\mathrm{eff}}>$40,000K. Dubbed the “Lyman-Balmer line problem”, measurements of $T_{\mathrm{eff}}$ made in DA white dwarfs using either the Lyman or Balmer line series yield differing values, some by a few 1,000K, and some even by 10,000K, getting larger for increasing values of $T_{\mathrm{eff}}$ [@barstow03a]. No similar appreciable effect is seen on the measurement of log $g$. A similar, more severe effect is seen in DAO stars [@good04a]. A study by @lajoie07a considered many different causes of this problem, ranging from atmospheric composition to unresolved binaries. One such cause was concluded to be due to some form of wavelength dependent extinction, however, the exact relation between the two was left open to debate. It was also suggested that atmospheric composition may play a part in causing the discrepancy, but results were inconclusive. It is not unreasonable to consider such an idea, however, as metal line blanketing dramatically effects a hot DA SED. @barstow98a have shown that using a model grid with a pure H atmosphere and a grid with a line blanketed heavy metal atmosphere yield $T_{\mathrm{eff}}$ measurements again differing by 1000s of K. @barstow03a also considered the Lyman-Balmer line problem, postulating that the temperature discrepancy may be due to inadequacies in the input atomic physics in generating the model atmospheres. Using G191-B2B and REJ2214-492, they calculated two model grids with varying $T_{\mathrm{eff}}$ and log $g$, but with 0.1 and 10 times the nominal abundances of these stars. They then measured $T_{\mathrm{eff}}$ and log $g$ using the Lyman and Balmer line series, finding that while the temperature difference decreased by 20-30%, a statistically significant discrepancy still remained. An accurate knowledge of a star’s photospheric composition is, therefore, paramount in accurately constraining $T_{\mathrm{eff}}$ and log $g$ in model atmosphere calculations. Determination of these parameters can provide useful insights into the white dwarf’s origin and evolution. For example, the cooling time of a degenerate object has a one to one correspondence with $T_{\mathrm{eff}}$, and hence gives a monotonic cooling age. Knowledge of log $g$ yields information on the mass of the white dwarf, and along with evolutionary models such as those from @wood98a or the Montreal Photometric Tables, an estimate of the radius of the star. Calculations by @chayer95a for various values of $T_{\mathrm{eff}}$ and log $g$ describe the predicted variation of metal abundances due to radiative levitation. However, the observational results do not agree very well with the predictions. In the case of G191-B2B for example, @barstow03b reported the Fe and Ni abundances to be $3.30_{-1.20}^{+3.10}\times{10^{-6}}$ and $2.40_{-0.24}^{+0.84}\times{10^{-7}}$ respectively, differing roughly by an order of magnitude. However, @chayer95a predict that these abundances should share a similar value. This disagreement is also present in other white dwarfs in the sample of @barstow03b. It should be noted, however, that @chayer94a reported significant differences in predicted atmospheric abundances of Fe and Ni dependent on the number of transitions included in their calculations. It is this variation in results that suggests that poor agreement between theory and observation may be dependent on the number of opacities included in these calculations. Parameter Value Reference ------------------------------------- --------------------- ------------- V $11.727\pm{0.016}$ @holberg06a $M_{v}$ $8.280\pm{0.164}$ This work $T_{\mathrm{eff}}$ (K) $52,500\pm{900}$ @barstow03b Log $g$ $7.53\pm{0.09}$ @barstow03b Mass ($M_{\odot}$) $0.52\pm{0.035}$ This work Radius ($R_{\odot}$) $0.0204\pm{0.0014}$ This work Distance (Pc) $48.9\pm{3.7}$ This work $t_{\mathrm{cool}}$ (Myr) $1.50\pm{0.08}$ This work $v_{\mathrm{phot}}$ (km s$^{-1}$) $23.8\pm{0.03}$ This work $v_{\mathrm{LIC}}$ (km s$^{-1}$) $19.4\pm{0.03}$ This work $v_{\mathrm{Hyades}}$ (km s$^{-1}$) $8.64\pm{0.03}$ This work : A summary of the physical parameters of G191-B2B. The velocities $v_{\mathrm{LIC}}$ and $v_{\mathrm{Hyades}}$ are calculated along the line of sight to the star. \[table:factfile\] In this paper we present and analyse a unique spectrum of G191-B2B with unprecedented signal to noise (S/N), over the wavelength range 910-3145Å. The spectrum is constructed using co-added *Far Ultraviolet Spectroscopic Explorer* (*FUSE*) LWRS (30$\times$30“, low resolution), MDRS (4.0$\times$20”, medium resolution), and HIRS (1.21$\times$20", high resolution) spectra, and co-added Space Telescope Imaging Spectrometer (STIS) E140H and E230H spectra. The resolution of our data sets range from 25,000 for the *FUSE* data to 144,000 for the *HST* spectra. The S/N exceeds 100 in many regions. We begin by describing the observational data used, and how co-addition of several data sets has allowed access to previously undetectable absorption lines. In section 3, we discuss the new atomic data releases provided by the Kurucz[^3] (@kurucz92a [@kurucz06a; @kurucz11a], hereafter Kurucz) and Kentucky[^4] (hereafter Kentucky) databases and the effect this has had on our ability to identify new lines. A comprehensive table of identifications has been included in Appendix B. We calculate the atmospheric abundances of C, N, O, Al, Si, P, S, Fe, and Ni. We discuss a new potential circumstellar identification, and the abundance pattern of the white dwarf. The potential significance of including additional opacities into model atmosphere calculations is also discussed, providing a tentative solution to the Lyman-Balmer line problem. In summary, this data set is a unique and invaluable record of the archetype white dwarf G191-B2B. It is the most complete, and highest S/N NUV and FUV spectrum available for any white dwarf observed. It will serve as a template for studies of other hot white dwarfs, as a test bed for model atmosphere calculations, and also for the improvement of atomic databases. Observational Data ================== Three detailed coadded spectra were used to analyse the NUV/FUV flux distribution of G191-B2B, spanning 910 to 3145Å. All data used in the coadded spectra are hosted on the Mikulski Archive for Space Telescopes[^5] (MAST). Using LWRS, MDRS, and HIRS exposures observed by *FUSE*, @barstow10a constructed a coadded spectrum spanning 910-1185Å with an exceptional S/N ratio. Observations by STIS aboard the *HST* covered the remainder of the spectrum. We made use of as many high resolution ($R\approx{144,000}$) observations with the echelle gratings E140H (centroid wavelength 1400Å) and E230H (centroid wavelength 2300Å) as possible, and produced a coadded spectrum spanning 1160-1680Å and 1625-3145Å respectively. *FUSE* ------ *FUSE* was launched in 1999, providing coverage from 910Å to 1185Å, making the Lyman series accessible to Ly $\beta$. While the satellite has been described many times (e.g. @moos00a), we provide a brief summary of the hardware here. *FUSE* utilised a Rowland Circle design with four separate channels or coaligned optical paths. The satellite employed two detectors, each of which had two independent segments, SiC and LiF, upon which the spectra from the four channels are recorded. *FUSE* has three different aperture configurations, LWRS, MDRS and HIRS. *FUSE* also has a pinhole aperture (RFPT), which was mainly used for calibration. To minimise the possibility that the target’s light did not fall on the detector, observations were conducted primarily using the LWRS aperture, with a spectral resolution of 15,000-20,000 for early observations, and 23,000 for later ones, where the mirror focusing had been adjusted [@sahnow00a]. Observations were also taken using both the TIMETAG (TTAG) and HISTOGRAM (HIST) modes. All *FUSE* data used here were reduced and processed using version 3.2 of [calfuse]{} [@dixon07a]. The data comprises exposures from each of the detector/segment combinations (eight in total). We used the FUSE spectrum of G191-B2B from @barstow10a, constructed with 48 observations listed in Table \[table:exposurelist\], and plotted in Figure \[fig:fuseplot\]. The different observations were coadded according to the process described by @barstow03a. Prior to coaddition, all spectra were rebinned to a common wavelength spacing to account for the difference in resolution between apertures. A consequence of coadding exposures from the different slits are discontinuities in the flux. It is for this reason we applied a correcting factor of 1.08 shortward of 1089Å to ensure continuity between the FUSE and STIS flux distributions. Observation ID Number of exposures Start time Exposure time (s) Aperture ---------------- --------------------- ---------------- ------------------- ---------- M1010201000 8 13/10/99 01:25 4164 LWRS M1030501000 1 12/11/99 07:35 266 MDRS M1030502000 1 20/11/99 07:22 900 MDRS M1030401000 1 20/11/99 09:02 1298 HIRS M1030603000 5 20/11/99 10:43 3664 LWRS M1030503000 3 21/11/99 06:43 1709 MDRS M1030504000 4 21/11/99 10:03 3212 MDRS M1030602000 5 21/11/99 11:39 2812 LWRS S3070101000 32 14/01/00 09:40 15456 LWRS M1010202000 7 17/02/00 06:10 3450 LWRS M1030604000 1 09/01/01 09:02 503 LWRS M1030506000 1 09/01/01 09:26 503 MDRS M1030605000 1 10/01/01 13:20 503 LWRS M1030507000 1 10/01/01 13:45 503 MDRS M1030403000 2 10/01/01 15:08 483 HIRS M1030606000 5 23/01/01 06:08 2190 LWRS M1030508000 5 23/01/01 07:55 2418 MDRS M1030404000 5 23/01/01 11:18 1853 HIRS M1030607000 5 25/01/01 04:46 1926 LWRS M1030509000 5 25/01/01 06:33 2417 MDRS M1030405000 5 25/01/01 09:53 2419 HIRS M1030608000 5 28/09/01 13:50 2728 LWRS M1030510000 4 28/09/01 15:35 1910 MDRS M1030406000 5 28/09/01 17:15 1932 HIRS M1030609000 5 21/11/01 09:54 2703 LWRS M1030511000 4 21/11/01 11:39 1910 MDRS M1030407000 5 21/11/01 13:19 1932 HIRS M1030610000 16 17/02/02 07:27 8639 LWRS M1030512000 11 17/02/02 12:34 4757 MDRS M1030408000 5 17/02/02 17:43 1932 HIRS M1030611000 8 23/02/02 02:05 3645 LWRS M1030513000 5 23/02/02 06:43 1797 MDRS M1030409000 5 23/02/02 08:23 1921 HIRS M1030612000 14 25/02/02 02:17 7004 LWRS M1030514000 4 25/02/02 06:59 1617 MDRS M1030613000 5 03/12/02 21:00 2358 LWRS M1030614000 3 06/12/02 02:30 702 LWRS M1030515000 4 06/12/02 05:16 2002 MDRS M1052001000 16 07/12/02 21:46 7061 LWRS M1030615000 4 08/12/02 22:36 1895 LWRS M1030516000 4 09/12/02 00:29 1911 MDRS M1030412000 4 09/12/02 03:53 1880 HIRS M1030616000 4 05/02/03 19:14 1980 LWRS M1030517000 4 05/02/03 21:07 1910 MDRS M1030413000 4 06/02/03 00:36 1932 HIRS M1030617000 8 23/11/03 20:16 4121 LWRS M1030519000 4 25/01/04 21:31 1887 MDRS M1030415000 4 26/01/04 00:51 1902 HIRS \[table:exposurelist\] {width="120mm"} STIS ---- G191-B2B has been extensively observed as a calibration standard by *HST*. In particular, observations of this star were conducted as part of Cycle 8 STIS calibration programs 8067, 8421 and 8915, which were designed to provide flux calibrations at the 1% level for all E140H and E230H primary and secondary echelle grating modes with a strong stellar continuum source. After the 2009 repair mission STS-125, an additional calibration program 11866 was proposed in Cycle 17 in order to evaluate the post-repair echelle blaze dependence on MSM position. The data were obtained in the ACCUM mode in four periods; 17th December 1998, 16th to 19th March 2000, 17th to 19th September 2001, and 28th November 2009 to 6th January 2010. Standard target acquisition procedures were used to acquire the source and centre it within the 0.2$\times$0.2" slit. The wavelength range 1140-3145Å was covered by using all nine STIS primary grating settings and 28 secondary grating settings (see Chapter 11 of @stishand12). We examined all of the available E140H and E230H datasets (39 and 77 respectively) from the MAST website in order to check for discontinuities and errors in the observations. We found that 32 E140H and 66 E230H observations were free of such problems, and were hence included in the final coadded product, with total exposure times of 53318 and 77743s respectively. We summarise the individual STIS spectra in Table \[table:2stis\]. After the extraction of the echelle orders, including ripple correction, the spectra were interpolated on to a single linear wavelength scale prior to exposure time weighted coaddition. The result was two single continuous spectra as shown in Figure \[fig:1stis\]. The distribution of S/N as a function of wavelength is shown in Figure \[fig:2stis\]. {width="120mm"} {width="120mm"} Observation ID Date Prog. ID Grating $\lambda_{c}$ (Å) Setting Exposure time (s) ---------------- ------------------ ---------- --------- ------------------- --------- ------------------- O57U01020 17/12/1998 08:17 8067 E140H 1416 P 2040 O57U01030 17/12/1998 09:34 8067 E140H 1234 P 2789 O57U01040 17/12/1998 11:14 8067 E140H 1598 P 2703 O5I010010 16/03/2000 23:31 8421 E140H 1234 P 2279 O5I010020 17/03/2000 00:52 8421 E140H 1234 P 3000 O5I010030 17/03/2000 02:29 8421 E140H 1234 P 3000 O5I011010 17/03/2000 04:21 8421 E140H 1598 P 2284 O5I011020 17/03/2000 05:42 8421 E140H 1598 P 3000 O5I011030 17/03/2000 07:18 8421 E140H 1598 P 3000 O5I014010 18/03/2000 02:52 8421 E230H 2513 P 2304 O5I014020 18/03/2000 04:14 8421 E230H 2513 P 3000 O5I014030 18/03/2000 05:50 8421 E230H 2513 P 3000 O5I015010 18/03/2000 22:10 8421 E230H 3012 P 2304 O5I015020 18/03/2000 23:32 8421 E230H 3012 P 3000 O5I015030 19/03/2000 01:09 8421 E230H 3012 P 3000 O5I013010 19/03/2000 03:00 8421 E230H 1763 P 2304 O5I013020 19/03/2000 04:22 8421 E230H 1763 P 3000 O5I013030 19/03/2000 05:59 8421 E230H 1763 P 3000 O6HB40080 12/09/2001 23:09 8915 E230H 2413 S 774 O6HB40090 13/09/2001 00:08 8915 E230H 3012 P 2228 O6HB10010 17/09/2001 13:49 8915 E140H 1234 P 867 O6HB10020 17/09/2001 14:05 8915 E140H 1234 P 867 O6HB10040 17/09/2001 14:54 8915 E140H 1271 S 640 O6HB10050 17/09/2001 15:11 8915 E140H 1307 S 654 O6HB10080 17/09/2001 16:30 8915 E140H 1380 S 719 O6HB10090 17/09/2001 16:48 8915 E140H 1416 P 851 O6HB100A0 17/09/2001 17:09 8915 E140H 1453 S 809 O6HB100B0 17/09/2001 18:07 8915 E140H 1453 S 229 O6HB100C0 17/09/2001 18:17 8915 E140H 1489 S 1263 O6HB100D0 17/09/2001 18:44 8915 E140H 1526 S 887 O6HB100E0 17/09/2001 19:43 8915 E140H 1526 S 749 O6HB100F0 17/09/2001 20:02 8915 E140H 1562 S 1996 O6HB20010 18/09/2001 15:32 8915 E230H 1763 P 1314 O6HB20020 18/09/2001 16:00 8915 E230H 1813 S 654 O6HB20030 18/09/2001 16:35 8915 E230H 1813 S 455 O6HB20040 18/09/2001 16:49 8915 E230H 1863 S 997 O6HB20050 18/09/2001 17:12 8915 E230H 1913 S 907 O6HB20060 18/09/2001 18:11 8915 E230H 1963 S 871 O6HB20070 18/09/2001 18:32 8915 E230H 2013 P 808 O6HB20080 18/09/2001 18:51 8915 E230H 2063 S 718 O6HB20090 18/09/2001 19:47 8915 E230H 2113 S 679 O6HB200A0 18/09/2001 20:05 8915 E230H 2163 S 640 O6HB200B0 18/09/2001 20:21 8915 E230H 2213 S 620 O6HB200C0 18/09/2001 20:38 8915 E230H 2263 P 101 O6HB200D0 18/09/2001 21:24 8915 E230H 2263 P 609.6 O6HB200E0 18/09/2001 21:40 8915 E230H 2313 S 734.3 O6HB200F0 18/09/2001 21:59 8915 E230H 2363 S 748.7 O6HB30010 19/09/2001 15:39 8915 E230H 2463 S 668 O6HB30020 19/09/2001 15:56 8915 E230H 2513 P 696 O6HB30030 19/09/2001 16:13 8915 E230H 2563 S 247 O6HB30040 19/09/2001 16:39 8915 E230H 2563 S 484 O6HB30050 19/09/2001 16:54 8915 E230H 2613 S 769 O6HB30060 19/09/2001 17:13 8915 E230H 2663 S 992.3 O6HB30070 19/09/2001 18:16 8915 E230H 2713 S 900 O6HB30080 19/09/2001 18:37 8915 E230H 2762 P 978 O6HB30090 19/09/2001 18:59 8915 E230H 2812 S 519 O6HB300A0 19/09/2001 19:52 8915 E230H 2812 S 578 O6HB300B0 19/09/2001 20:08 8915 E230H 2862 S 1232 O6HB300C0 19/09/2001 20:35 8915 E230H 2912 S 549 O6HB300D0 19/09/2001 21:28 8915 E230H 2912 S 873 \[table:2stis\] Observation ID Date Prog. ID Grating $\lambda_{c}$ (Å) Setting Exposure time (s) ---------------- ------------------ ---------- --------- ------------------- --------- ------------------- O6HB300E0 19/09/2001 21:49 8915 E230H 2962 S 1862 OBB002010 28/11/2009 08:36 11866 E230H 1863 S 1000 OBB002020 28/11/2009 08:59 11866 E230H 1963 S 870 OBB002030 28/11/2009 09:55 11866 E230H 1913 S 920 OBB002040 28/11/2009 10:16 11866 E230H 2013 P 810 OBB002050 28/11/2009 10:36 11866 E230H 2063 S 740 OBB002060 28/11/2009 11:31 11866 E230H 2263 P 800 OBB002070 28/11/2009 11:50 11866 E230H 2113 S 850 OBB002080 28/11/2009 12:10 11866 E230H 2163 S 800 OBB002090 28/11/2009 13:07 11866 E230H 1763 P 1800 OBB0020A0 28/11/2009 13:43 11866 E230H 2213 S 1000 OBB0020B0 28/11/2009 14:43 11866 E230H 1813 S 1160 OBB0020C0 28/11/2009 15:08 11866 E230H 2313 S 650 OBB0020D0 28/11/2009 15:25 11866 E230H 2363 S 650 OBB004080 29/11/2009 12:12 11866 E230H 2413 S 645 OBB004090 29/11/2009 13:05 11866 E230H 3012 P 2192 OBB001010 30/11/2009 06:58 11866 E140H 1271 S 696 OBB001020 30/11/2009 07:15 11866 E140H 1453 S 1038 OBB001030 30/11/2009 08:16 11866 E140H 1380 S 752 OBB001040 30/11/2009 08:34 11866 E140H 1234 P 867 OBB001050 30/11/2009 08:54 11866 E140H 1416 P 851 OBB001070 30/11/2009 10:09 11866 E140H 1526 S 2100 OBB001080 30/11/2009 11:27 11866 E140H 1562 S 2134 OBB001090 30/11/2009 12:09 11866 E140H 1307 S 654 OBB0010A0 30/11/2009 13:03 11866 E140H 1489 S 1200 OBB005010 01/12/2009 05:12 11866 E140H 1234 P 2200 OBB005020 01/12/2009 06:38 11866 E140H 1234 P 6200 OBB053010 06/01/2010 13:30 11866 E230H 2563 S 900 OBB053020 06/01/2010 13:51 11866 E230H 2613 S 950 OBB053030 06/01/2010 14:43 11866 E230H 2663 S 830 OBB053040 06/01/2010 15:03 11866 E230H 2463 S 670 OBB053050 06/01/2010 15:20 11866 E230H 2713 S 900 OBB053060 06/01/2010 16:19 11866 E230H 2762 P 1197 OBB053070 06/01/2010 16:45 11866 E230H 2862 S 1647 OBB053080 06/01/2010 17:55 11866 E230H 2513 P 1000 OBB053090 06/01/2010 18:18 11866 E230H 2912 S 1800 OBB0530A0 06/01/2010 19:31 11866 E230H 2812 S 1097 OBB0530B0 06/01/2010 19:55 11866 E230H 2962 S 1747 \[table:2stis\] Line survey =========== Atomic data ----------- We combined data from the Kurucz and Kentucky atomic databases to compile as complete a line list as possible. The Kurucz database has been updated several times, with major updates occurring in 1992, 2006, and 2011 [@kurucz92a; @kurucz06a; @kurucz11a]. Table \[table:lines\] shows how the number of transitions available for Fe [iv]{}-[vii]{} and Ni [iv]{}-[vii]{} have increased from the 1992 to the 2011 data releases; an order of magnitude increase in the number of transitions is seen for each ion. Figure \[fig:oldnewcom\] illustrates the improvement made in reproducing the 980-1020Å spectral region of G191-B2B using the 1992 Kurucz data release and some lines from the National Institute of Standards and Technology[^6] (NIST), and our combined line list, synthesised using the model atmosphere of [@barstow03b]. Some transitions in the Kentucky database lacked oscillator strengths ($f$-values). For Figure \[fig:oldnewcom\], we set the missing $f$-values to $1.00\times{10}^{-6}$ for illustrative purposes. We choose this value as weak Fe/Ni transitions have oscillator strengths $\sim{10^{-6}}$. The Kurucz and Kentucky databases often had records of the same transition. For the purposes of identification, we used the data from the Kentucky database, as this supplies errors on the laboratory wavelengths of transitions. To identify resonant transitions, we used the line list of @verner94a (hereafter V94), and found the error on the wavelength by cross-correlating this line list with the Kentucky database. Ion No of lines 1992 No of lines 2011 ------------ ------------------ ------------------ Fe [iv]{} 1,776,984 14,617,228 Fe [v]{} 1,008,385 7,785,320 Fe [vi]{} 475,750 9,072,714 Fe [vii]{} 90,250 2,916,992 Ni [iv]{} 1,918,070 15,152,636 Ni [v]{} 1,971,819 15,622,452 Ni [vi]{} 2,211,919 17,971,672 Ni [vii]{} 967,466 28,328,012 Total 10,420,643 111,467,026 : The number of lines present in the Kurucz linelist in 1992 and 2011 for Fe and Ni [iv]{}-[vii]{}. \[table:lines\] {width="160mm"} {width="160mm"} Absorption feature parameterisation ----------------------------------- For the purposes of identification, we need to extract the basic parameters of each absorption feature, such as wavelength centroid, velocity, and equivalent width. To parameterise the absorption feature as accurately as possible, we fit a Gaussian and Lorentzian profile (see Appendix A for the exact parameterisation) to the feature with a chi squared ($\chi^2$) minimisation technique, using the IDL routine [mpfit]{} [@markwardt09a]. The fit that achieved the lowest $\chi^2$ was assumed to be the best fit. Hence we used the calculated parameters from this fit. To differentiate between signal and noise, we adopted a $3\sigma$ threshold. In the cases where absorption features appeared to be blended, we used a double Gaussian absorption profile (see also Appendix A). Detections and identification ----------------------------- We detected 976 absorption features, successfully identifying 947 of them. Lines that could not be identified are listed in Table \[table:unidentified\]. Several measured lines in our survey were found to be the result of multiple blended features. Therefore, the velocity of each individual absorber was included in calculating the weighted velocity of each component. To calculate the uncertainty on the velocity measurements, the uncertainties in the measured wavelength ($\lambda_{\mathrm{tot}}$) were obtained by adding in quadrature the uncertainties in laboratory ($\delta\lambda_{\mathrm{lab}}$) and observed wavelength ($\delta\lambda_{\mathrm{obs}}$) calculated from the Gaussian/Lorentzian fit. Some velocity uncertainties were anomalously high due to large uncertainties from the Kentucky database, and it is for this reason that the average photospheric and interstellar velocities were calculated by weighting each line velocity by its inverse error: $$\bar{v}=\frac{\sum_{i=1}^{N}\frac{v_{i}}{\delta{v_i}^2}}{\sum_{i=1}^{N}\frac{1}{\delta{v_i}^2}}$$ With associated error: $$\delta\bar{v}=\sqrt{\frac{1}{\sum_{i=1}^{N}\frac{1}{\delta{v_i}^2}}}$$ Where $v_i$ and $\delta{v_i}$ are the line velocities and their respective errors, and $N$ is the number of lines used to calculate the mean. The velocities were calculated using only lines with wavelength errors from Kentucky, and were observed in the STIS data. In our line survey, we identified three definitive velocity populations, one photospheric, and two interstellar. We measured the photospheric velocity as $23.8\pm{0.03}$km s$^{-1}$, while the interstellar velocities were measured as $19.4\pm{0.03}$km s$^{-1}$ and $8.64\pm{0.03}$km s$^{-1}$ respectively. In Table \[table:velocities\], we compare our photospheric velocity with that obtained by @vennes01a, and our interstellar velocities with those from @redfield08a. Hereafter, we refer to the interstellar velocities by the name of the cloud from which they appear to originate, as named by @redfield08a, where the $19.4\pm{0.03}$km s$^{-1}$ velocity corresponds to the Local Interstellar Cloud (LIC), and the $8.64\pm{0.03}$km s$^{-1}$ velocity to the Hyades Cloud. The velocities determined in this study appear to be in excellent agreement with those obtained by previous authors. $\lambda_{\mathrm{Obs}}$(Å) $\delta\lambda_{\mathrm{Obs}}$(mÅ) $W_{\lambda}$(mÅ) $\delta{W_{\lambda}}$(mÅ) ----------------------------- ------------------------------------ ------------------- --------------------------- 1171.277 1.9730 2.584 0.793 1174.424 1.3261 5.417 0.843 1186.174 2.8080 3.046 0.962 1186.355 0.9279 9.953 0.927 1196.733 1.7920 3.897 0.730 1198.240 0.9257 6.856 0.636 1199.443 1.0994 2.369 0.572 1201.548 2.2597 1.689 0.543 1204.561 2.0988 3.064 0.613 1206.756 1.4513 3.354 0.686 1206.812 2.4298 2.788 0.761 1228.604 2.9613 3.232 0.663 1232.311 3.4623 1.813 0.554 1253.405 2.0101 1.727 0.472 1255.177 3.0864 2.695 0.329 1270.950 4.4021 4.710 0.558 1274.017 1.8103 1.920 0.386 1285.088 2.8810 1.148 0.359 1291.912 2.3123 3.007 0.586 1292.590 2.7298 2.880 0.578 1295.987 2.6044 1.651 0.311 1302.927 2.8485 3.624 0.653 1318.082 1.0165 2.999 0.353 1321.307 1.4181 2.564 0.403 1322.416 1.3813 2.398 0.375 1333.462 2.3978 2.265 0.613 1442.574 2.4808 1.673 0.401 1499.254 3.6779 3.035 0.827 1513.608 3.2663 2.266 0.730 : A list of absorption features that could not be identified in our survey, where $\lambda_{\mathrm{Obs}}$ is the observed wavelength with accompanying error $\delta\lambda_{\mathrm{Obs}}$, and $W_{\lambda}$ is the equivalent width, with error $\delta{W_{\lambda}}$. \[table:unidentified\] Origin $v_{\mathrm{Current}}$ (km s$^{-1}$) $v_{\mathrm{Previous}}$ (km s$^{-1}$) ------------- -------------------------------------- --------------------------------------- Photosphere $23.8\pm{0.03}$ $24.3\pm{1.7}$ LIC $19.4\pm{0.03}$ $19.19\pm{0.09}$ Hyades $8.64\pm{0.03}$ $8.61\pm{0.74}$ : A summary of the velocity populations identified in the spectrum. Each velocity was calculated using the weighted mean of the various line velocities measured in the STIS data. We also compare our determined photospheric velocity to that determined by @vennes01a, and our ISM velocities with that of @redfield08a. \[table:velocities\] Model atmospheres and abundance determination ============================================= To determine the abundances, we first used [tlusty]{} version 200 to calculate a model atmosphere based on the abundances given by @barstow03b (C, N, O, Si, Fe, Ni) and @vennes01a (P, S) in NLTE. We fixed He/H to $1.00\times{10^{-5}}$, and $T_{\mathrm{eff}}$ and log $g$ to 52,500K and 7.53 respectively. The model atoms used by [tlusty]{}, along with the number of levels included, are listed in Table \[table:modelatoms\]. Next, as Al had not been included in a full NLTE model atmosphere calculation for G191-B2B before, we introduced the metal into the solution, and calculated a grid of models with varying Al abundances, the values of which are given in Table \[table:abungrid\]. The model spectra were then synthesized using [synspec]{} version 49, and Kurucz’s 1992 line list as this has a complete set of oscillator strengths. It was interesting to note that even with the greatest abundance of Al, there was no noticable flux redistribution. We then determined the Al abundance using [xspec]{} [@arnaud96a], which we will describe shortly. Upon determining the abundance, we recalculated the atmosphere with this value using [tlusty]{}. As the abundances of C, N, O, Si, P, S, Fe and Ni are not so different from the values derived in @barstow03b and @vennes01a, any flux redistribution due to small abundance variations is likely to be a second order effect. Therefore, instead of calculating a model atmosphere for each metal, which is computationally expensive, we used [synspec]{} 49 to modify the abundances, creating a model grid for each metal as given in Table \[table:abungrid\], again using the 1992 Kurucz line list. As with Al, we then used [xspec]{} to determine the abundances. [xspec]{} calculates the abundances and associated errors by taking a model atmosphere grid and observational data as input, and interpolating between the different abundance values using a chi square ($\chi^{2}$) minimisation technique. [xspec]{} has difficulty in performing computations with spectra containing many data points. With 60,000 data points, the STIS spectrum can not, therefore, be analysed in its entirety without dividing it into segments. Therefore, we extracted small regions of spectra containing the absorption features that we wish to analyse, summarised in Table \[table:linerange\]. Isolating small sections of spectrum also has the advantage of minimising systematic errors due to poor normalisation of the continuum whilst fitting the abundances. We determined the abundance of each metal ionisation stage individually. In cases where there were multiple lines in an ionisation stage, we fixed the abundance to be the same for each feature. Photospheric features that appeared to be blended with additional components were modelled by including an additional Gaussian absorber using the [xspec]{} model [gabs]{} (see Appendix A for the exact parameterisation) as done by @dickinson12c. In all cases, we quote our formal errors to $1\sigma$ confidence, and assume one degree of freedom except where noted. We have tabulated our abundance determinations in Table \[table:obsabun\]. We have also tabulated the parameter values and their respective errors obtained from fitting the [gabs]{} component in Table \[table:circparam1\], along with the velocity of the absorber where relevant. Element Ion No of Levels --------- --------- -------------- H [i]{} 9 He [i]{} 24 He [ii]{} 20 C [iii]{} 23 C [iv]{} 41 N [iii]{} 32 N [iv]{} 23 N [v]{} 16 O [iv]{} 39 O [v]{} 40 O [vi]{} 20 Si [iii]{} 30 Si [iv]{} 23 P [iv]{} 14 P [v]{} 17 S [iv]{} 15 S [v]{} 12 S [vi]{} 16 Fe [iv]{} 43 Fe [v]{} 42 Fe [vi]{} 32 Ni [iv]{} 38 Ni [v]{} 48 Ni [vi]{} 42 : Model atoms used in calculating the initial model atmosphere. \[table:modelatoms\] C N O Al Si P S Fe Ni ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- $2.00\times{10}^{-8}$ $2.00\times{10}^{-8}$ $2.00\times{10}^{-8}$ $2.00\times{10}^{-8}$ $2.00\times{10}^{-8}$ $2.00\times{10}^{-9}$ $2.00\times{10}^{-8}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-8}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-9}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-8}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-8}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-7}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-8}$ $6.00\times{10}^{-7}$ $6.00\times{10}^{-6}$ $6.00\times{10}^{-7}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-7}$ $2.00\times{10}^{-6}$ $2.00\times{10}^{-5}$ $2.00\times{10}^{-6}$ \[table:abungrid\] Ion Wavelength (Å) $f$-value Spectral Region (Å) Ion Wavelength (Å) $f$-value Spectral Region (Å) ------------ ---------------- ----------- --------------------- ----------- ---------------- ----------- --------------------- C [iii]{} 1174.9327 0.114 1174-1178 Fe [iv]{} 1592.050 0.3341 1590-1605 C [iii]{} 1175.263 0.274 1174-1178 Fe [iv]{} 1601.652 0.3379 1590-1605 C [iii]{} 1175.5903 0.069 1174-1178 Fe [iv]{} 1603.177 0.2679 1590-1605 C [iii]{} 1175.7112 0.205 1174-1178 Fe [v]{} 1280.470 0.0236 1280-1290 C [iii]{} 1175.9871 0.091 1174-1178 Fe [v]{} 1287.046 0.0363 1280-1290 C [iii]{} 1176.3697 0.068 1174-1178 Fe [v]{} 1288.172 0.0541 1280-1290 C [iii]{} 1247.383 0.163 1245-1255 Fe [v]{} 1293.382 0.0330 1290-1300 C [iv]{} 1548.202 0.190 1545-1555 Fe [v]{} 1297.549 0.0440 1290-1300 C [iv]{} 1550.777 0.095 1545-1555 Fe [v]{} 1311.828 0.1710 1305-1315 N [iv]{} 1718.551 0.173 1715-1725 Fe [v]{} 1320.409 0.1944 1320-1333 N [v]{} 1238.821 0.156 1235-1245 Fe [v]{} 1321.489 0.0905 1320-1333 N [v]{} 1242.804 0.078 1235-1245 Fe [v]{} 1323.271 0.1930 1320-1333 O [iv]{} 1338.615 0.118 1334-1344 Fe [v]{} 1330.405 0.2085 1320-1333 O [iv]{} 1342.99 0.011 1334-1344 Fe [v]{} 1331.189 0.0774 1320-1333 O [iv]{} 1343.514 0.104 1334-1344 Fe [v]{} 1331.639 0.1867 1320-1333 Al [iii]{} 1854.716 0.556 1850-1860 Ni [iv]{} 1356.079 0.0963 1350-1360 Al [iii]{} 1862.79 0.277 1860-1870 Ni [iv]{} 1398.193 0.3837 1395-1405 Si [iii]{} 1206.4995 1.610 1200-1210 Ni [iv]{} 1399.947 0.2964 1395-1405 Si [iii]{} 1206.5551 1.640 1200-1210 Ni [iv]{} 1400.682 0.2950 1395-1405 Si [iv]{} 1122.4849 0.819 1120-1130 Ni [iv]{} 1411.451 0.3507 1410-1422 Si [iv]{} 1128.3248 0.0817 1120-1130 Ni [iv]{} 1416.531 0.1949 1410-1422 Si [iv]{} 1128.3400 0.736 1120-1130 Ni [iv]{} 1419.577 0.1309 1410-1422 Si [iv]{} 1393.7546 0.508 1390-1400 Ni [iv]{} 1421.216 0.2863 1410-1422 Si [iv]{} 1402.7697 0.252 1400-1410 Ni [iv]{} 1430.190 0.1794 1425-1435 P [iv]{} 950.657 1.470 945-955 Ni [iv]{} 1432.449 0.1253 1425-1435 P [v]{} 1117.977 0.467 1115-1125 Ni [iv]{} 1452.220 0.3596 1450-1460 P [v]{} 1128.008 0.231 1125-1135 Ni [iv]{} 1498.893 0.1618 1495-1505 S [iv]{} 1062.662 0.052 1060-1070 Ni [v]{} 1230.435 0.2649 1230-1240 S [iv]{} 1072.974 0.045 1070-1080 Ni [v]{} 1232.807 0.1764 1230-1240 S [vi]{} 933.378 0.433 930-940 Ni [v]{} 1233.257 0.1605 1230-1240 S [vi]{} 944.523 0.213 940-950 Ni [v]{} 1234.393 0.1334 1230-1240 Fe [iv]{} 1542.155 0.1386 1540-1550 Ni [v]{} 1235.831 0.1982 1230-1240 Fe [iv]{} 1542.697 0.2818 1540-1550 Ni [v]{} 1236.277 0.1094 1230-1240 Fe [iv]{} 1544.486 0.2511 1540-1550 Ni [v]{} 1239.552 0.1116 1230-1240 Fe [iv]{} 1546.404 0.2070 1540-1550 Ni [v]{} 1241.627 0.2003 1240-1250 Fe [iv]{} 1562.751 0.2032 1560-1572 Ni [v]{} 1243.504 0.0815 1240-1250 Fe [iv]{} 1568.276 0.3012 1560-1572 Ni [v]{} 1243.662 0.1194 1240-1250 Fe [iv]{} 1569.222 0.1291 1560-1572 Ni [v]{} 1244.027 0.0503 1240-1250 Fe [iv]{} 1570.178 0.3147 1560-1572 Ni [v]{} 1245.176 0.2348 1240-1250 Fe [iv]{} 1570.416 0.2741 1560-1572 \[table:linerange\] Ion Abundance $-1\sigma$ $+1\sigma$ ------------ ----------------------- ----------------------- ----------------------- C [iii]{} $1.72\times{10}^{-7}$ $0.02\times{10}^{-7}$ $0.02\times{10}^{-7}$ C [iv]{} $2.13\times{10}^{-7}$ $0.15\times{10}^{-7}$ $0.29\times{10}^{-7}$ N [iv]{} $1.58\times{10}^{-7}$ $0.14\times{10}^{-7}$ $0.14\times{10}^{-7}$ N [v]{} $2.16\times{10}^{-7}$ $0.04\times{10}^{-7}$ $0.09\times{10}^{-7}$ O [iv]{} $4.12\times{10}^{-7}$ $0.08\times{10}^{-7}$ $0.08\times{10}^{-7}$ Al [iii]{} $1.60\times{10}^{-7}$ $0.08\times{10}^{-7}$ $0.07\times{10}^{-7}$ Si [iii]{} $3.16\times{10}^{-7}$ $0.30\times{10}^{-7}$ $0.31\times{10}^{-7}$ Si [iv]{} $3.68\times{10}^{-7}$ $0.14\times{10}^{-7}$ $0.13\times{10}^{-7}$ P [iv]{} $8.40\times{10}^{-8}$ $1.18\times{10}^{-8}$ $1.18\times{10}^{-8}$ P [v]{} $1.64\times{10}^{-8}$ $0.02\times{10}^{-8}$ $0.02\times{10}^{-8}$ S [iv]{} $1.71\times{10}^{-7}$ $0.02\times{10}^{-7}$ $0.02\times{10}^{-7}$ S [vi]{} $5.23\times{10}^{-8}$ $0.13\times{10}^{-8}$ $0.10\times{10}^{-8}$ Fe [iv]{} $1.83\times{10}^{-6}$ $0.03\times{10}^{-6}$ $0.03\times{10}^{-6}$ Fe [v]{} $5.00\times{10}^{-6}$ $0.06\times{10}^{-6}$ $0.06\times{10}^{-6}$ Ni [iv]{} $3.24\times{10}^{-7}$ $0.05\times{10}^{-7}$ $0.13\times{10}^{-7}$ Ni [v]{} $1.01\times{10}^{-6}$ $0.03\times{10}^{-6}$ $0.03\times{10}^{-6}$ : Summary of the abundances determined in this work, along with $1\sigma$ errors. \[table:obsabun\] Ion Parameter Value $-1\sigma$ $+1\sigma$ ------------ ------------------------------ ----------- ------------ ------------ C [iv]{} $\lambda_{\mathrm{lab}}$ (Å) 1548.203 - - $E_l$ ($10^{-3}$keV) 8.008122 0.0000047 0.0000038 $\sigma_l$ ($10^{-7}$keV) 1.34141 0.03611 0.03689 Strength ($10^{-7}$) 1.19420 0.0426 0.0489 $\lambda_{\mathrm{obs}}$ (Å) 1548.246 0.000735 0.000909 $v$ (km s$^{-1}$) 8.26 0.14 0.18 C [iv]{} $\lambda_{\mathrm{lab}}$ (Å) 1550.777 - - $E_l$ ($10^{-3}$keV) 7.994829 0.0000036 0.0000040 $\sigma_l$ ($10^{-7}$keV) 1.26307 0.03597 0.02623 Strength ($10^{-7}$) 7.18607 0.20447 0.17263 $\lambda_{\mathrm{obs}}$ (Å) 1550.820 0.000776 0.000698 $v$ (km s$^{-1}$) 8.30 0.15 0.13 Si [iii]{} $\lambda_{\mathrm{lab}}$ (Å) 1206.4995 - - $E_l$ ($10^{-3}$keV) 10.276140 0.0000070 0.0000020 $\sigma_l$ ($10^{-7}$keV) 0.925806 0.026966 0.025814 Strength ($10^{-7}$) 4.79551 0.13501 0.14099 $\lambda_{\mathrm{obs}}$ (Å) 1206.537 0.000235 0.000822 $v$ (km s$^{-1}$) 9.24 0.06 0.20 Si [iv]{} $\lambda_{\mathrm{lab}}$ (Å) 1393.7546 - - $E_l$ ($10^{-3}$keV) 8.895525 0.0000277 0.0000208 $\sigma_l$ ($10^{-7}$keV) 0.926033 0.183263 0.293867 Strength ($10^{-7}$) 0.434293 0.072113 0.098237 $\lambda_{\mathrm{obs}}$ (Å) 1393.795 0.003259 0.004340 $v$ (km s$^{-1}$) 8.73 0.70 0.93 Si [iv]{} $\lambda_{\mathrm{lab}}$ (Å) 1402.7697 - - $E_l$ ($10^{-3}$keV) 8.838369 0.0000503 0.0000411 $\sigma_l$ ($10^{-7}$keV) 0.890117 0.341027 0.550783 Strength ($10^{-7}$) 0.206168 0.069738 0.087402 $\lambda_{\mathrm{obs}}$ (Å) 1402.809 0.006523 0.007984 $v$ (km s$^{-1}$) 8.31 1.39 1.71 \[table:circparam1\] Carbon ------ As discussed in the introduction, C [iv]{} 1548 and 1550Å in G191-B2B’s photospheric spectrum are well documented, each being blended with a circumstellar absorption feature. Including a Gaussian, we obtain C [iv]{}/H=$2.13_{-0.15}^{+0.29}\times{10^{-7}}$ (cf. Figures \[fig:civ1548circ\] and \[fig:civ1550circ\]. We can also see in Figures \[fig:civ1548circ\] and \[fig:civ1550circ\] that there is a small discrepancy between the predicted and observed absorption profiles. This “shelf” is not observed in @vennes01a’s analysis of the C [iv]{} profiles, however, this is likely due to the lower resolution of the data, which was obtained using the E140M grating. In both cases, we believe the shelf arises due to the presence of Ni [iv]{} transitions at 1548.220 and 1550.777Å that have poor oscillator strength determinations. {width="100mm"} {width="100mm"} As well as the C [iii]{} lines listed in Table \[table:linerange\], the C [iii]{} resonant transition at 977.0201Å (hereafter C [iii]{} 977Å) was available to fit, however, attempts to do so were unsuccessful, as the predicted profile could not descend deeply enough relative to the continuum, and increasing the C abundance resulted in large pressure broadening that was uncharacteristic of the observed profile. This does not come as a surprise, as the C [iii]{} 977Å line has been observed along the line of sight to G191-B2B by @lehner03a in the ISM. We attempted to add a Gaussian to account for this, but again, a satisfactory fit could not be obtained. Therefore, we determined the C [iii]{} abundance using only the lines in Table \[table:linerange\], obtaining C [iii]{}/H=$1.72_{-0.02}^{+0.02}\times{10^{-7}}$ (cf. Figure \[fig:ciii1174\] for the C [iii]{} sextuplet). ![Comparison between the model and observed spectra for the C [iii]{} sextuplet spanning 1174-1177Å with C [iii]{}/H=$1.72\times{10^{-7}}$.[]{data-label="fig:ciii1174"}](G191CIII1174.eps){width="80mm"} Our C [iii]{} and C [iv]{} abundances are in good agreement with @barstow03b’s C [iii]{} value of $1.99_{-0.88}^{+0.44}\times{10}^{-7}$, but not with their C [iv]{} value of $4.00_{-0.98}^{+0.44}\times{10}^{-7}$. We note, however, that @barstow03b’s C [iv]{} abundance was obtained without taking the circumstellar absorption into account. We also calculated the velocity of the circumstellar lines, obtaining $8.26_{-0.14}^{+0.18}$ and $8.30_{-0.15}^{+0.13}$ km s$^{-1}$ for the C [iv]{} 1548 and 1550Å lines respectively, both in relatively good agreement with our obtained velocity corresponding to the Hyades Cloud velocity as identified by @redfield08a. Nitrogen -------- Many N [iv]{} transitions exist in the *FUSE* spectrum of G191-B2B from 915-930Å, but we neglected using these lines in favour of N [iv]{} 1718.551Å (hereafter N [iv]{} 1718Å) due to the higher S/N. We obtained N [iv]{}/H=$1.58_{-0.14}^{+0.14}\times{10^{-7}}$. Using the resonant N [v]{} doublet lines at 1238.821 and 1242.804Å, we obtain N [v]{}/H=$2.16_{-0.04}^{+0.09}\times{10^{-7}}$. Our N [iv]{} abundance is in good agreement with that obtained by @barstow03b of $1.60_{-0.21}^{+0.41}\times{10^{-7}}$. Oxygen ------ Absorption features of O [iv]{}-[vi]{} were detected in our line survey. Given the temperature of G191-B2B, the equivalent widths of the resonant O [vi]{} transitions (1031.93 and 1037.62Å) are quite weak, and were neglected from our analysis. Futhermore, attempts to obtain an abundance with the well known O [v]{} 1371.296Å line were not possible. The predicted profile of the O [v]{} line did not appear in our synthesised spectrum unless O abundances of $\approx{10^{-6}}$ were specified. This issue may be related to that noted by @vennes00a, whereby the ionisation fraction for oxygen is poorly calculated for O [iv]{}/O [v]{}. @vennes00a ruled out $T_{\mathrm{eff}}$ and log $g$ variations as being the cause, suggesting reasons such as additional atmospheric consituents, stratification, or inadequate model atom treatment. This is addressed in @vennes01a, whereby they used the O abundance determined from the O [iv]{} lines. This may also explain the large uncertainty in the O abundance value calculated by @barstow03b ($3.51_{-2.00}^{+7.40}\times{10^{-7}}$). Our abundance determination therefore is based only upon the O [iv]{} triplet listed in Table \[table:linerange\], whereby we found O [iv]{}/H=$4.12_{-0.08}^{+0.08}\times{10^{-7}}$, in good agreement with @barstow03b. Aluminium --------- Al was first observed in G191-B2B by @holberg98b using *IUE*, however, an abundance was not calculated. @holberg03a, then considered the abundance of Al assuming LTE, arriving at an estimated value of $3.02\times{10}^{-7}$. We do not observe other isolated photospheric Al features. We obtained Al [iii]{}/H=$1.60_{-0.08}^{+0.07}\times{10^{-7}}$ (cf. Figure \[fig:alfigs\]). ![Comparison between the model (solid line) and observed (error bars) spectra for the Al [iii]{} 1854.716 (top plot) and 1862.79Å (bottom plot) lines, with Al [iii]{}/H=$1.60\times{10^{-7}}$.[]{data-label="fig:alfigs"}](G191AlIII1854.eps "fig:"){width="80mm"} ![Comparison between the model (solid line) and observed (error bars) spectra for the Al [iii]{} 1854.716 (top plot) and 1862.79Å (bottom plot) lines, with Al [iii]{}/H=$1.60\times{10^{-7}}$.[]{data-label="fig:alfigs"}](G191AlIII1862.eps "fig:"){width="80mm"} Silicon ------- For Si [iii]{} 1206.4995Å (hereafter Si [iii]{} 1206Å), we were unable to fit the absorption feature as the predicted line profile did not descend deeply enough relative to the continuum. Increases in Si abundance produced a line profile that was pressure broadened far beyond that observed. Including a Gaussian into the fit, we obtained Si [iii]{}/H=$3.16_{-0.30}^{+0.31}\times{10^{-7}}$ (see Figure \[fig:siiii1206circ\]). We also determined the velocity of the Gaussian to be $9.24_{-0.06}^{+0.20}$ km s$^{-1}$. While we note that this velocity is not encompassed by the Hyades cloud velocity, the two values are quite close. @holberg03a noted an asymmetry in the absorption profiles of Si [iv]{} 1393.7546 and 1402.7697Å (hereafter Si [iv]{} 1393Å and 1402Å respectively), observing a blue shifted component relative to the photospheric velocity. We confirmed this observation and included a Gaussian to account for its presence. We also included the excited Si IV transitions 1122.4849, 1128.3248, and 1128.3400Å in order to further constrain the abundance. We derived Si [iv]{}/H=$3.68_{-0.14}^{+0.13}\times{10^{-7}}$. The velocities of the two Gaussians (see Figures \[fig:siiv1393circ\] and \[fig:siiv1402circ\]) are $8.73_{-0.70}^{+0.93}$ and $8.31_{-1.39}^{+1.71}$ km s$^{-1}$ for the Si [iv]{} 1393 and 1402Ålines respectively, in good agreement with the Hyades cloud velocity. Both of our determined Si abundances are not encompassed by @barstow03b’s value of $8.65_{-3.50}^{+3.20}\times{10^{-7}}$, but the difference is not very large. {width="100mm"} {width="100mm"} {width="100mm"} Phosphorus ---------- Resonance absorption features of P have been observed in a handful of DAs, including GD394 [@chayer00a], GD71, REJ1918+595 and REJ0605-482 [@dobbie05a]. P was first observed in G191-B2B by @vennes96a using the *ORFEUS* telescope, deriving an LTE abundance of $2.51_{-0.93}^{+1.47}\times{10^{-8}}$. Determination of the P [iv]{} abundance was complicated due to the proximity of the line to the Lyman $\gamma$ centroid, where the continuum varies rapidly with wavelength. We determined P [iv]{}/H=$8.40_{-1.18}^{+1.18}\times{10^{-8}}$ and P [v]{}/H=$1.64_{-0.02}^{+0.02}\times{10^{-8}}$. Our P [v]{} abundance appears to be in closer agreement with that obtained by @vennes96a. Sulphur ------- This metal was also detected in G191-B2B by @vennes96a with the *ORPHEUS* telescope, deriving an LTE abundance of S/H=$3.16_{-1.58}^{+3.15}\times{10^{-7}}$. We calculated S [iv]{}/H=$1.71_{-0.02}^{+0.02}\times{10^{-7}}$ and S [vi]{}=$5.23_{-0.13}^{+0.10}\times{10^{-8}}$. Our S [iv]{} abundance is also in agreement with @vennes96a. Iron ---- While there have been many measurements of the Fe content in various white dwarf photospheric spectra [@barstow03b; @vennes06b], none have used the same selection of lines, due to the large choice available. Here, we chose to use 12 strong lines for each ion listed in Table \[table:linerange\]. We calculated Fe [iv]{}/H=$1.83_{-0.03}^{+0.03}\times{10^{-6}}$ and Fe [v]{}/H=$5.00_{-0.06}^{+0.06}\times{10^{-6}}$ (cf. Figure \[fig:fe1330\]), with our Fe [v]{} abundance in agreement with @barstow03b’s value of $3.30_{-1.20}^{+3.10}\times{10^{-6}}$. ![Comparison between the model (top line) and observed (bottom line) spectra for several Fe [v]{} lines, with Fe [v]{}/H=$5.00\times{10^{-6}}$. The synthetic spectrum is offset for clarity.[]{data-label="fig:fe1330"}](G191Fe1320-1330.eps){width="80mm"} Nickel ------ Like Fe, Ni is a complicated atomic system with many transitions. We use a similar method to Fe to determine the abundance, obtaining Ni [iv]{}/H=$3.24_{-0.05}^{+0.13}\times{10^{-7}}$ and Ni [v]{}/H=$1.01_{-0.03}^{+0.03}\times{10^{-6}}$ (cf. Figure \[fig:ni1230\], where our Ni [iv]{} abundance agrees with @barstow03b’s value of $2.40_{-0.24}^{+0.84}\times{10^{-7}}$. ![Comparison between the model (top line) and observed (bottom line) spectra for several Ni [v]{} lines, with Ni [v]{}/H=$1.01\times{10^{-6}}$. The synthetic spectrum is offset for clarity. The large absorption feature near 1239Å is from N [v]{}.[]{data-label="fig:ni1230"}](G191Ni1230-1240.eps){width="80mm"} Discussion ========== Our line survey has been successful in providing new line identifications, with over 95 percent of the absorption features detected in the STIS spectra and 100 percent of those in the *FUSE* data accounted for. Curiously, we confirmed the detection made by @vennes05a of Ge [iv]{}, but we made no detections of heavier metals, or metals with atomic number between S and Fe. In particular, we made no detections of Cr, Mn, or Co, for which @holberg03a gave an abundance limit of $10^{-8}$. However, as highlighted in the previous section, some discrepancies still remain regarding predicted line profiles and ionisation balance for particular metals. In our discussion of the results, we first consider the metals where we have potentially identified circumstellar absorption features. Next, we discuss the calculated abundances of G191-B2B, and compare them to the solar abundances of @asplund09a, and the predicted abundances of @chayer94a [@chayer95a; @chayer95b] from radiative levitation calculations (cf. Figure \[fig:abunplot\]). Finally, we consider the new atomic data, and the potential impact this may have on future model atmosphere calculations through the additional opacity that needs to be included. Circumstellar absorption ------------------------ As stated in section 4.1, the C [iv]{} 1548 and 1550Å lines are accompanied by circumstellar absorption features which have been thoroughly documented. We have also confirmed the detection of circumstellar absorption in Si [iv]{}. The method of including a Gaussian to account for the circumstellar component is not new. @dickinson12c accounted for the line shape of the photospheric profile (which may include re-emission) as well as the circumstellar component, giving a more accurate representation of the absorption. However, our analysis of the profile allowed us to derive velocities and to make a comparison between the Hyades cloud and the circumstellar velocities. @bannister03a noted the similarity between the ISM and circumstellar velocities along the line of sight to G191-B2B, and suggested that the circumstellar lines arose from material ionised by the strömgren sphere of the white dwarf. @redfield08a made the association between this ISM component with that of the Hyades cloud. This hypothesis was supported by @dickinson12b’s findings, and suggested that the Hyades cloud fell inside G191-B2B’s strömgren sphere. Our velocity measurements of the C [iv]{} and Si [iv]{} circumstellar features encompass the Hyades cloud velocity, and appear to support the strömgren sphere hypothesis. The velocity and uncertainty of the Si [iii]{} 1206Å line does not quite encompass the Hyades cloud velocity. However, the discrepancy is small. So we can reasonably associate the Si [iii]{} line with the Hyades cloud. The issue of the C [iii]{} 977Å line is more difficult owing to the low resolution of the data. While we can infer that there is additional absorption, we cannot derive any information about the absorption profile, and hence cannot derive a reliable velocity. The inability to fit a profile is most likely due to the resolution of the *FUSE* spectrometer. General abundance discussion ---------------------------- With the exceptionally high S/N of the *FUSE* and STIS spectra, we have been able to constrain the formal $1\sigma$ uncertainties on most of the metal abundances to within a very small range. However, in reality, the true uncertainty on the abundances can be much greater than this due to a number of factors. As demonstrated by Figure \[fig:abunplot\], uncertainties in the ionisation balance can lead to order of magnitude differences in the calculated abundance from different ionisation stages. The ionic partition function, which is used to calculate the ionisation balance, is dependent on the atomic data of energy levels, and uncertain data can lead to incorrect calculations of level populations. It is interesting to note that in Figure \[fig:abunplot\], the largest difference between ionisation stage abundances was for P and S, where P [iv]{}/H and P [v]{}/H differ by $\sim{0.8}$ dex. The best agreement between ion abundances was for C and Si. {width="120mm"} In Figure \[fig:abunplot\], we have plotted our measured abundances, comparing them to the solar abundances from @asplund09a, as well as the atmospheric abundances predicted by @chayer94a [@chayer95a; @chayer95b] due to radiative levitation. The only obvious agreement that can be observed in Figure \[fig:abunplot\] is between the radiative levitation prediction and our Fe [v]{} abundance. The largest discrepancy appears to be for Al, where radiative levitation grossly underestimates the abundance by more than two dex, this may, however, be due to the small number of bound-bound Al transitions included in the calculations. Opacity considerations ---------------------- The additional atomic data obtained for Fe and Ni has allowed all but a few lines to be identified. However, the number of heavy metal lines also presents a question regarding the efficacy of current methods in accounting for opacity in NLTE model atmosphere calculations. In @chayer95a, the authors describe calculations performed to predict the abundance of Fe and Ni in the atmosphere of hot white dwarfs due to radiative levitation, and compared the Fe abundances predicted from using different atomic datasets from different generations (see @chayer95a, Fig 11), concluding that the numbers of transitions, and the accuracy of their respective oscillator strengths varied the predicted Fe abundances. Such a result implies that using as complete a sample of transitions as possible is important in creating accurate models of radiative levitation. As shown in Table \[table:lines\], the number of calculated transitions for Fe [iv]{}-[vii]{} alone in 1992 was $\sim{10^7}$, while for 2011 it was $\sim{10^8}$. Radiative transfer will also likely be affected by including the additional calculated lines. As described by @barstow98a, the inclusion of Fe and Ni transitions to a pure H atmosphere results in $T_{\mathrm{eff}}$ decreasing by a few thousand K. In terms of the SED, additional opacity may result in flux attenuation in the EUV, producing better agreement with observation. The additional opacity also potentially offers a solution to the Lyman-Balmer line problem. [@barstow03a] constructed two model grids with 0.1 and 10 times a nominal abundance, and measured $T_{\mathrm{eff}}$ and log $g$ for G191-B2B and REJ2214-492 using both the Lyman and Balmer line series. They found that the temperature discrepancy was larger for the lower abundance grid, and smaller for the higher grid. This implies that including additional opacity into future calculations may help to resolve the Lyman-Balmer line problem. To include the additional opacity contributed by the new lines, [tlusty]{} requires information on the energy levels, transitions, and photoionisation cross sections. [tlusty]{} is able to calculate bound-bound cross section directly from data provided by the Kurucz data, however, the photoionisation cross sections need to be included separately. Future work will involve calculations of the photoionisation cross sections. Our model atmospheres currently include 2,005,173 Fe [iv-vi]{} transitions, and 2,751,277 Ni [iv-vi]{} transitions. Future calculations therefore will include $\sim{30,000,000}$ Fe and $\sim{47,000,000}$ Ni transitions. This work not only has applications to white dwarf stars, but can also be applied to hot O and B stars. Conclusions =========== We have presented the most detailed NUV and FUV spectroscopic survey of G191-B2B to date. Using all available high resolution observations with the E140H and E230H apertures with STIS, we have constructed a co-added spectrum whose S/N exceeds 100 in particular sections of the spectrum, and spans 1160-3145Å. The detail unveiled by such a high S/N spectrum far exceeds any observation of white dwarfs made thus far. Using the STIS spectrum, and the co-added *FUSE* spectrum from @barstow10a covering 910-1185Å, we have made detections of 976 absorption features. By combining the latest releases of the Kurucz and Kentucky databases, we have been able to identify 947 of the detected features, blended or otherwise. We have identified every single absorption feature present in the *FUSE* spectrum of G191-B2B at our given confidence limit. We have found that over 60% of the identifications made can be attributed to highly ionised Fe and Ni features. While the new Fe and Ni features may be inaccurate in terms of their wavelengths and oscillator strengths, the high S/N STIS spectrum presents the opportunity to perform high quality measurements of atomic transition parameters in order to improve atomic databases. Our survey confirmed the presence of previously observed circumstellar features and also potentially revealed a new, previously unconsidered Si [iii]{} circumstellar line. Further, exploiting the high S/N data from the STIS and *FUSE* data, we have made highly accurate measurements of the abundances of metals present in G191-B2B’s photosphere. Our abundance analysis has revealed areas for improvement with regards to ionisation balance calculations. S and P appear to have the largest discrepancies. We compared the calculated abundances from G191-B2B’s atmosphere to the solar photosphere, as well as predicted abundances due to radiative levitation. We highlighted the need for updated radiative levitation calculations using newly available atomic data in order to reconcile theory with observation. We discussed the potential consequences of including new Fe and Ni transitions into NLTE model atmosphere calculations, as well as radiative levitation calculations. While the exact effect on the NLTE calculations cannot be quantified at present, we plan to perform a full, detailed investigation into the inclusion of new transitions in line blanketing calculations. Acknowledgments {#acknowledgments .unnumbered} =============== We gratefully thank Jean Dupuis for providing a very thorough referee report, helping to improve the content and quality of this paper. SPP, MAB, and NJD gratefully acknowledge the support of the Science and Technology Facilities Council (STFC). JBH acknowledges the support of a visiting professorship at the University of Leicester, and the Space Telescope Science Institute Archive Grant AR9202. We gratefully acknowledge the work of Meena Sahu and Wayne Landsman for their work on the original STIS dataset. We also gratefully acknowledge the help of Peter Van Hoof for assistance in creating the line list used in this paper. This research used the ALICE High Performance Computing Facility at the University of Leicester. All of the spectroscopic data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts \[lastpage\] K. A., 1996, in [Jacoby]{} G. H., [Barnes]{} J., eds, Astronomical Data Analysis Software and Systems V Vol. 101 of Astronomical Society of the Pacific Conference Series, [XSPEC: The First Ten Years]{}. p. 17 Asplund M., Grevesse N., Sauval A. J., Scott P., [2009]{}, in [Blandford]{} ed., [Annual Review of Astronomy and Astrophysics]{}, Vol. [47]{} N. P., [Barstow]{} M. A., [Fraser]{} G. W., [Spragg]{} J. E., [Curddace]{} R., [Kowalski]{} M., [Fritz]{} G., [Yentis]{} D., [Lapington]{} J., [Tandy]{} J., [Sanderson]{} B., [Barbee]{} T., [Goldstein]{} W. H., [Kordas]{} 1999, in [Solheim]{} S. E., [Meistas]{} E. G., eds, 11th European Workshop on White Dwarfs Vol. 169 of Astronomical Society of the Pacific Conference Series, [The Joint Astrophysical Plasmadynamic Experiment (JPEX)]{}. p. 188 Bannister N. P., Barstow M. A., Holberg J. B., Bruhweiler F. C., 2003, MNRAS, 341, 477 Barstow M. A., Boyce D. D., Welsh B. Y., Lallement R., Barstow J. K., Forbes A. E., Preval S., 2010, ApJ, 723, 1762 Barstow M. A., Cruddace R. G., Kowalski M. P., Bannister N. P., Yentis D., Lapington J. S., Tandy J. A., Hubeny I., Schuh S., Dreizler S., Barbee T. W., 2005, MNRAS, 362, 1273 Barstow M. A., Good S. A., Burleigh M. R., Hubeny I., Holberg J. B., Levan A. J., 2003, MNRAS, 344, 562 Barstow M. A., Good S. A., Holberg J. B., Hubeny I., Bannister N. P., Bruhweiler F. C., Burleigh M. R., Napiwotzki R., 2003, MNRAS, 341, 870 Barstow M. A., Hubeny I., Holberg J. B., 1998, MNRAS, 299, 520 Barstow M. A., Hubeny I., Holberg J. B., 1999, MNRAS, 307, 884 Bergeron P., Liebert J., Fulbright M. S., 1995, ApJ, 444, 810 Bergeron P., Wesemael F., Dufour P., Beauchamp A., Hunter C., Saffer R. A., Gianninas A., Ruiz M. T., Limoges M. M., Dufour P., Fontaine G., Liebert J., 2011, ApJ, 737, 28 Bohlin R. C., Gilliland R. L., 2004, AJ, 127, 3508 Bruhweiler F. C., Kondo Y., 1981, ApJL, 248, L123 F., [Barstow]{} M., [Holberg]{} J., [Sahu]{} M., 1999, in American Astronomical Society Meeting Abstracts Vol. 31 of Bulletin of the American Astronomical Society, [The Unusual C IV Absorption in the Spectrum of the DA White Dwarf G191-B2B]{}. p. 1419 Chayer P., Fontaine G., Wesemael F., 1995, ApJS, 99, 189 Chayer P., Kruk J. W., Ake T. B., Dupree A. K., Malina R. F., Siegmund O. H. W., Sonneborn G., Ohl R. G., 2000, ApJL, 538, L91 Chayer P., LeBlanc F., Fontaine G., Wesemael F., Michaud G., Vennes S., 1994, ApJL, 436, L161 Chayer P., Vennes S., Pradhan A. K., Thejll P., Beauchamp A., Fontaine G., Wesemael F., 1995, ApJ, 454, 429 R. G., [Kowalski]{} M. P., [Yentis]{} D., [Brown]{} C. M., [Gursky]{} H., [Barstow]{} M. A., [Bannister]{} N. P., [Fraser]{} G. W., [Spragg]{} J. E., [Lapington]{} J. S., [Tandy]{} J. A., [Sanderson]{} B., [Culhane]{} J. L., [Barbee]{} T. W., [Kordas]{} J. F., [Goldstein]{} W., [Fritz]{} G. G., 2002, ApJL, 565, L47 Dickinson N. J., Barstow M. A., Hubeny I., 2012, MNRAS, 421, 3222 N. J., [Barstow]{} M. A., [Welsh]{} B. Y., 2012, MNRAS, 428, 1873 Dickinson N. J., Barstow M. A., Welsh B. Y., Burleigh M., Farihi J., Redfield S., Unglaub K., 2012, MNRAS, 423, 1397 Dixon W. V., Sahnow D. J., Barrett P. E., Civeit T., Dupuis J., Fullerton A. W., Godard B., Hsu J. C., Kaiser M. E., Kruk J. W., Lacour S., Lindler D. J., Massa D., Robinson R. D., Romelfanger M. L., Sonnentrucker P., 2007, PASP, 119, 527 Dobbie P. D., Barstow M. A., Hubeny I., Holberg J. B., Burleigh M. R., Forbes A. E., 2005, MNRAS, 363, 763 S., [Wolff]{} B., 1999, AAP, 348, 189 Eggen O. J., Greenstein J. L., 1967, ApJ, 150, 927 S. A., [Barstow]{} M. A., [Holberg]{} J. B., [Sing]{} D. K., [Burleigh]{} M. R., [Dobbie]{} P. D., 2004, MNRAS, 355, 1031 Greenstein J. L., 1969, ApJ, 158, 281 Hernandez, S., et al. 2012, “STIS Instrument Handbook”, Version 12.0, (Baltimore: STScI) Holberg J. B., Wesemael F., Wegner G., Bruhweiler F. C., 1985, ApJ, 293, 294 Holberg J. B., Hubeny I., Barstow M. A., Lanz T., Sion E. M., Tweedy R. W., 1994, ApJL, 425, L105 J. B., [Barstow]{} M. A., [Sion]{} E. M., 1998, ApJS, 119, 207 J. B., [Barstow]{} M. A., [Sion]{} E. M., 1999, in [Solheim]{} S. E., [Meistas]{} E. G., eds, 11th European Workshop on White Dwarfs Vol. 169 of Astronomical Society of the Pacific Conference Series, [Elemental abundances in hot white dwarfs]{}. p. 485 J. B., [Barstow]{} M. A., [Hubeny]{} I., [Sahu]{} M. S., [Bruhweiler]{} F. C., [Landsman]{} W. B., 2002, in American Astronomical Society Meeting Abstracts \#200 Vol. 34 of Bulletin of the American Astronomical Society, [A High S/N View of the Photosphere of the Hot White Dwarf G191-B2B from STIS]{}. p. 765 Holberg J. B., Barstow M. A., Hubeny I., Sahu M. S., Bruhweiler F. C., Landsman W. B., 2003 Vol. 291 of ASPCS, [A Detailed View of the Photosphere of the Hot White Dwarf G191-B2B from STIS]{}. p. 383 J. B., [Bergeron]{} P., 2006, AJ, 132, 1221 Hubeny I., 1988, Computer Physics Communications, 52, 103 Hubeny I., Lanz T., 2011, in Astrophysics Source Code Library, record ascl:1109.022 [Synspec: General Spectrum Synthesis Program]{}. p. 9022 S. M., [Wesemael]{} F., [Liebert]{} J., [Raymond]{} J. C., [Steiner]{} J. E., [Shipman]{} H. L., 1984, ApJ, 278, 255 P. M., [Saumon]{} D., 2006, ApJL, 651, L137 R. L., 1992, RMxAA, 23, 45 Kurucz R. L., 2006, in Stee P., ed., EAS Publications Series Vol. 18 of EAS Publications Series, [Including all the Lines]{}. pp 129–155 Kurucz R. L., 2011, Canadian Journal of Physics, 89, 417 Lajoie C. P., Bergeron P., 2007, ApJ, 667, 1126 T., [Barstow]{} M. A., [Hubeny]{} I., [Holberg]{} J. B., 1996, ApJ, 473, 1089 N., [Jenkins]{} E. B., [Gry]{} C., [Moos]{} H. W., [Chayer]{} P., [Lacour]{} S., 2003, ApJ, 595, 858 Markwardt C. B., 2009, in Bohlender D. A., Durand D., Dowler P., eds, Astronomical Data Analysis Software and Systems XVIII Vol. 411 of Astronomical Society of the Pacific Conference Series, [Non-linear Least-squares Fitting in IDL with MPFIT]{}. p. 251 H. W. et al, 2000, ApJL, 538, L1 Oke J. B., 1974, ApJS, 27, 21 Redfield S., Linsky J. L., 2008, ApJ, 673, 283 Reid N., Wegner G., 1988, ApJ, 335, 953 Sahnow D. J. et al, 2000, ApJL, 538, L7 Sion E. M., Bohlin R. C., Tweedy R. W., Vauclair G. P., 1992, ApJL, 391, L29 Tremblay P. E., Bergeron P., Gianninas A., 2011, ApJ, 730, 128 Vennes S., Chayer P., Hurwitz M., Bowyer S., 1996, ApJ, 468, 898 S., [Polomski]{} E. F., [Lanz]{} T., [Thorstensen]{} J. R., [Chayer]{} P., [Gull]{} T. R., 2000,\ apj, 544, 423 Vennes S., Lanz T., 2001, ApJ, 553, 399 Vennes S., Chayer P., Dupuis J., 2005, ApJL, 622, L121 Vennes S., Chayer P., Dupuis J., Lanz T., 2006, ApJ, 652, 1554 Verner D. A., Barthel P. D., Tytler D., 1994, A&A, 108, 287 Werner K., Dreizler S., 1994, AAP, 286, L31 M. A., 1998, Highlights of Astronomy, 11, 427 Parameterisation of absorption features ======================================= The Gaussian and Lorentzian profiles fitted were parameterised respectively as: $$F_{\lambda}=C_{1}\exp\left[-\frac{(\lambda-C_{2})^2}{2C_{3}^2}\right]$$ $$F_{\lambda}=\frac{C_{1}C_{3}^2}{(\lambda-C_{2})^2+C_{3}^2}$$ Where $C_1$, $C_2$ and $C_3$ are the height, centroid wavelength and line width respectively. The double Gaussian profile took the form: $$F_{\lambda}=C_{1}\exp\left[-\frac{(\lambda-C_{2})^2}{2C_{3}^2}\right]+C_{4}\exp\left[-\frac{(\lambda-C_{5})^2}{2C_{6}^2}\right]$$ The $C_{i}$ are the same as defined for a single Gaussian. The Gaussian profile used in [xspec]{} was parameterised as: $$F(E)=\exp\left[-\frac{\tau_l}{\sigma_{l}\sqrt{2\pi}}\exp\left[-\frac{(E-E_{l})^2}{2\sigma_{l}^{2}}\right]\right]$$ Where $E_l$ is the line energy in keV (line centroid), $\sigma$ is the line width (keV), and $\tau_{l}$ is the strength. This profile was multiplied by the model flux. Line identifications ==================== Included here is a small excerpt of the table of identifications made in our survey. The table in its entirety can be found with the online version of the journal. Wavelengths and their associated errors are given in Å, equivalent width and error in mÅ, and velocities and errors in km s$^{-1}$. $\lambda_{\mathrm{obs}}$ and $\delta\lambda_{\mathrm{obs}}$, are the observed wavelength centroid and it’s error, and $\lambda_{\mathrm{lab}}$ and $\delta\lambda_{\mathrm{lab}}$ are the reference wavelength and, where available, the error on said wavelength. The velocity $v$ comes with two errors $\delta{v}$ and $\delta{v_{\mathrm{tot}}}$. The former is the error on the velocity assuming no error on the lab wavelength, and the latter takes all errors into account. The List column states where the transition data came from, with KENTUCKY=Kentucky database, KURUCZ=Kurucz database, NIST=NIST website, and RESONANT=V94. The Origin column states where the transition originated. Where PHOT=Photosphere, ISM1=Local Interstellar cloud, and ISM2=Hyades cloud. As was stated previously, many lines had several possible identifications, meaning that a line will be the result of several blends of absorption features. Therefore, where there was more than one identification, the characteristics of the measured line is given, followed only by the additional identifications and their velocities. $\lambda_{\mathrm{obs}}$ $\delta\lambda_{\mathrm{obs}}$ $W_{\lambda}$ $\delta{W}_{\lambda}$ Ion $\lambda_{\mathrm{lab}}$ $\delta\lambda_{\mathrm{lab}}$ $v$ $\delta{v}$ $\delta{v_{\mathrm{tot}}}$ List Origin -------------------------- -------------------------------- --------------- ----------------------- ---------- -------------------------- -------------------------------- ------- ------------- ---------------------------- ---------- -------- 916.503 0.533 168.903 2.369 H [i]{} 916.429 0.004 24.18 0.17 0.17 KENTUCKY ISM1 917.254 0.600 170.652 2.643 H [i]{} 917.181 0.005 24.02 0.20 0.20 KENTUCKY ISM1 918.196 0.574 196.800 2.500 H [i]{} 918.129 0.007 21.78 0.19 0.19 KENTUCKY ISM1 918.964 7.564 10.266 2.198 N [iv]{} 918.893 7.500 23.16 2.47 3.48 KENTUCKY PHOT 919.428 0.360 178.807 1.626 H [i]{} 919.351 0.009 25.00 0.12 0.12 KENTUCKY ISM1 921.032 0.347 189.055 1.609 H [i]{} 920.963 0.012 22.45 0.11 0.11 KENTUCKY ISM1 922.090 2.485 10.595 1.404 N [iv]{} 921.994 7.600 31.22 0.81 2.60 KENTUCKY PHOT 922.607 2.409 10.460 1.358 N [iv]{} 922.519 7.600 28.60 0.78 2.59 KENTUCKY PHOT 923.220 0.352 209.245 1.687 H [i]{} 923.150 0.016 22.64 0.11 0.11 KENTUCKY ISM1 923.757 3.069 9.727 1.454 N [iv]{} 923.676 7.600 26.29 1.00 2.66 KENTUCKY PHOT \[table:photfuseident\] [^1]: E-mail: [email protected] [^2]: http://www.astro.umontreal.ca/$\sim$bergeron/CoolingModels [^3]: http://kurucz.harvard.edu [^4]: http://www.pa.uky.edu/$\sim$peter/newpage/ [^5]: http://archive.stsci.edu/ [^6]: See http://nova.astro.umd.edu/Synspec49/data/ for more information on this line list.

--- author: - 'Sina Shaham, Ming Ding, Bo Liu, Shuping Dang, Zihuai Lin, and Jun Li[^1] [^2][^3] [^4] [^5] [^6]' bibliography: - 'MLA\_Ref.bib' title: 'Privacy Preserving Location Data Publishing: A Machine Learning Approach' --- of data by different organizations and institutes is crucial for open research and transparency of government agencies. Just in Australia, since 2013, over 7000 additional datasets have been published on ’data.gov.au,’ a dedicated website for the publication of datasets by the Australian government. Moreover, the new Australian government data sharing legislation encourage government agencies to publish their data, and as early as 2019, many of them will have to do so [@aa]. Unfortunately, the process of data publication can be highly risky as it may disclose individuals’ sensitive information. Hence, an essential step before publishing datasets is to remove any uniquely identifiable information from them. However, such an operation is not sufficient for preserving the privacy of users. Adversaries can re-identify individuals in datasets based on common attributes called quasi-identifiers or may have prior knowledge about the trajectories traveled by the users. Such side information enables them to reveal sensitive information that can cause physical, financial, and reputational harms to people. One of the most sensitive sources of data is location trajectories or spatiotemporal trajectories. Despite numerous use cases that the publication of spatiotemporal data can provide to users and researchers, it poses a significant threat to users’ privacy. As an example, consider a person who has been using GPS navigation to travel from home to work every morning of weekdays. If an adversary has some prior knowledge about a user, such as the home address, it is possible to identify the user. Such an inference attack can compromise user privacy, such as revealing the user’s health condition and how often the user visits his/her medical specialist. Therefore, it is crucial to anonymize spatiotemporal datasets before publishing them to the public. The privacy issue gets even more severe if the adversary links identified users to other databases, such as the database of medical records. That is the very reason why nowadays most companies are reluctant to publish any spatiotemporal trajectory datasets without applying an effective privacy preserving technique. A widely accepted privacy metric for the publication of spatiotemporal datasets is $k$-anonymity. This metric can be summarized as ensuring that every trajectory in the published dataset is indistinguishable from at least $k-1$ other trajectories. The authors in [@medical], adopted the notion of $k$-anonymity for spatiotemporal datasets and proposed an anonymization algorithm based on generalization. Xu et al. [@xu2017trajectory] investigated the effects of factors such as spatiotemporal resolution and the number of users released on the anonymization process. Dong et al. [@dong2018novel] focused on improving the existing clustering approaches. They proposed an anonymization scheme based on achieving $k$-anonymity by grouping similar trajectories and removing the highly dissimilar ones. More recently, the authors in [@comparison] developed an algorithm called k-merge to anonymize the trajectory datasets while preserving the privacy of users from probabilistic attacks. Local suppression and splitting techniques were also considered to protect privacy in [@terrovitis2017local]. However, there are three major problems with the aforementioned approaches. - Lack of a well-defined method to cluster trajectories as there is not an easy way to measure the cost of clustering when considering the distances among trajectories rather than simply the locations. - The existing literature focuses on pairwise sequence alignment, which results in a high amount of information loss [@medical; @comparison; @nergiz2008towards; @gurung2014traffic; @yarovoy2009anonymizing]. - There is no unified metric to evaluate and compare the existing anonymization methods. In this paper, we address the mentioned problems by proposing an enhanced anonymization framework termed machine learning based anonymization (MLA) to preserve the privacy of users in the publication of spatiotemporal trajectory datasets. MLA consists of two interworking algorithms: clustering and alignment. We have summarized our main contributions in the following bullet points. - By formulating the anonymization process as an optimization problem and finding an alternative representation of the system, we are able to apply machine clustering algorithms for clustering trajectories. We propose to use $k'$-means [^7] algorithm for this purpose, as part of the MLA framework. - We propose a variation of $k'$-means algorithm to preserve the privacy of users in the publication of overly sensitive spatiotemporal trajectory datasets. - We enhance the performance of sequence alignment in clusters by considering multiple sequence alignment instead of pairwise sequence alignment. - We propose a utility metric to evaluate and compare the anonymization frameworks. MLA and all algorithms associated with it are applied on two real-life GPS datasets following different distributions in time and spatial domains. The experimental results indicate a significantly higher utility levels while maintaining $k$-anonymity of trajectories. The rest of this paper is organized as follows. First, a comprehensive review of the currently existing literature is presented in Section \[Related work\], followed by the system model used in Section \[System model\]. Next, the proposed framework is explained and analyzed in Sections \[approach\] and \[Experiments\], respectively. Several real-world applications of the framework are elaborated in Section \[applications\], and finally, the paper is concluded in Section \[conclusion\]. Related Work {#Related work} ============ Unfortunately, merely removing unique identifiers of users cannot protect their privacy, as databases can be linked to each other based on their quasi-identifiers. Doing so, adversaries can reveal sensitive information about the users and compromise their privacy. In this section, we review the existing approaches for the anonymization of spatiotemporal datasets. Generalization Technique ------------------------ Generalization is currently one of the mainstream approaches for the anonymization of spatiotemporal trajectory datasets. The generalization technique is predicated on two interrelated mechanisms: clustering and alignment. Clustering aims at finding the best grouping of trajectories that minimizes a predefined cost function, and the alignment process aligns trajectories in each group. The notion of k-anonymity was adopted in [@nergiz2008towards] for anonymization of spatiotemporal datasets . The authors proved that the anonymization process is NP-hard and followed a heuristic approach to cluster the trajectories. The use of ‘edit distance’ metric for anonymization of spatiotemporal datasets was proposed in [@gurung2014traffic]. In this work, the authors target grouping the trajectories based on their similarity and choose a cluster head for each cluster to represent the cluster. Also, dummy trajectories were added to anonymize the datasets further. Yarovoy et al. [@yarovoy2009anonymizing] proposed to use Hilbert indexing for clustering trajectories. The authors in [@dong2018novel; @liu2018location] chose to avoid alignment by selecting trajectories with the highest similarity as representatives of clusters. Poulis et al. [@poulis2017anonymizing] investigated applying restriction on the amount of generalization that can be applied by proposing a user-defined utility metric. Takahashi et al. [@takahashi2012cmoa] proposed an approach termed as CMAO to anonymize the real-time publication of spatiotemporal trajectories. The proposed idea is based on generalizing each queried location point with $k-1$ other queried location by other users, and hence, achieving $k$-anonmity. The current state-of-art technique for applying gerelization to spatiotemporal datasets is based on generalization hierarchy (DGH) trees. In essence, DGH can be seen as a coding scheme to anonymize trajectories. We have categorized types of DGHs in the literature as: - Full-domain generalization: This technique emphasizes on the level that each value of an attribute is located in the generalization tree. If a value of an attribute is generalized to its parent node, all values of that attribute in the dataset must be generalized to the same level [@f1; @f2; @f3]. - Subtree generalization: In this method, if a value of an attribute is generalized to its parent node, all other child nodes of that parent node need to be replaced with the parent node as well [@sub1; @sub2]. - Cell generalization: This generalization technique considers each cell in the table separately. One cell can be generalized to its parent node while other values of that attribute remain unchanged [@cell1; @cell2; @cell3]. Other Anonymization Techniques ------------------------------ Aside from the generalization technique, we have categorized the existing methods for the anonymization spatiotemporal datasets into three major groups: - **Perturbation** anonymizes location datasets by addition of noise to data; - **ID swapping** swaps user IDs in road junctions to anonymize location datasets; - **Splitting** divides trajectories into shorter lengths to anonymize location datasets. The authors in [@ding2015trajectory] proposed an algorithm that swaps the IDs of users in trajectories once they reach an intersection. Doing so, the algorithm prevents adversaries from identifying a particular user. Cicek et al. [@cicek2014ensuring] made a distinction between sensitive and insensitive location nodes of trajectories. Their proposed algorithm only groups the paths around the sensitive nodes and exploits generalization to create supernodes. Moreover, Cristina et al. [@romero2018protecting] shifted the burden of privacy preservation in data publishing to the user side. The authors attempted to anonymize the data on the mobile phones before storage on the database as they would have more control over their privacy. Instead of clustering trajectories for anonymization, Cicek et al. in [@cicek2014ensuring] focused on the obfuscation of underlying map for sensitive locations. Brito et al. [@brito2015distributed] minimized the information loss during the data anonymization by suppressing key locations. The Local suppression and splitting techniques were considered for trajectory anonymization in [@terrovitis2017local]. Although the proposed approach is useful for a predefined number of locations, it cannot be generalized to system models in which the users can make queries from an arbitrary location on the map. Naghizadeh et al. [@naghizade2014protection] focused on the stop points along trajectories. A sensitivity measure is introduced in this work, which relies on the amount of time users spend in different locations. Sensitive locations are replaced or displaced with a less sensitive location to preserve the privacy of users. Jiang et al. [@jiang2013publishing] considered the perturbation of locations by adding noise to preserve the privacy of users. Adding noise can generate fake trajectories that do not correspond to realistic scenarios. System Model {#System model} ============ We assume that a map has been discretized into an $\epsilon \times \epsilon$ grid and the time is discretized into bins with length $\epsilon_t$. Therefore, each point in the dataset represents a snapshot of a real-world location query including $x$-coordinate, $y$-coordinate, and time. The datasets with continuous time or space data can fit into our model using interpolation. The level of spatial-temporal granularity in discretization does not affect the effectiveness of the proposed model. In our model, we consider a spatiotemporal trajectory datasets denoted by $T$. The dataset consists of trajectories $tr_1,...,tr_n$ where $n$ represents the number of trajectories in the dataset ($T=\{ tr_1,...,tr_n\}\textrm{,} \ |T|=n$). The $i$-th trajectory $tr_i$ is an ordered set of $l_i$ spatiotemporal 3D points (i.e., $tr_i=\{ p_1,...,p_{l_i}\}\textrm{,} \ |tr_i|=l_i$). Each point $p_j$ is defined by a triplet $<x_j,y_j,t_j>$, where $x_j,y_j,t_j$ indicate the $x$-coordinate, $y$-coordinate, and the time of query, respectively. Generalization Model {#Hierarchical Tree Transformation} -------------------- Our proposed framework is based on the generalization technique to anonymize the spatiotemporal datasets. To apply this technique, we use the domain generalization hierarchy (DGH) trees and quantify the information loss accordingly. ### Domain Generalization Hierarchies DGH tree is defined formally in Definition \[def1\]. To clarify the construction of DGHs, an example of such a tree for spatiotemporal datasets is provided in Example \[ex1\]. In our model, we utilize three dimensions: x-coordinate, y-coordinate, and the time of queries in hours. \[def1\] A DGH tree for an attribute $\mathcal{A}$, denoted as $H_{\mathcal{A}}$, is a partially ordered tree structure, which maps specific and generalized values of the attribute $\mathcal{A}$. The root of the tree is the most generalized value and is returned by the function $RT$. \[ex1\] Consider an $4 \times 8$ map shown in Example \[ex1\]. As can be seen in the figure, the generalization technique is applied by three DGH trees, each of them corresponding to one of the attributes. For instance, the $x$-coordinate attribute can have $8$ possible values ($0,1,...,7$). At the lowest level of the tree, each coordinate needs three bits of information to be shown that indicates the maximum information bits. As we go higher up the DGH tree, more information loss incurs, and less number of bits are used to represent the coordinates. Each node on a DGH tree can be generalized by moving up one or multiple levels of the DGH. The process of generalizing $\textrm{node}_i$ to one of its parent nodes $\textrm{node}_j$ is denoted using $\textrm{node}_i \rightarrow \textrm{node}_j $. A special case of generalization, in which the node is generalized to the root of the DGH, is referred to as suppression.  For generalizing two nodes, it is necessary to find the lowest common ancestor (LCA). The LCA is a critical point in the generalization process due to its corresponding subtree that entails both the nodes and achieves the lowest information loss for the generalization of two nodes. The definition of LCA is given in Definition \[def3\]. \[def3\] The LCA of $\textrm{node}_i$ and $\textrm{node}_j$ in $H_{\mathcal{A}}$ is defined as the lowest common parent root of the two nodes. Function $LCA$ returns the LCA. For instance, in Example \[ex1\], if two leaf nodes ‘000’ and ‘010’ are to be generalized, their LCA corresponds to the parent node ‘0’. Hence, in the dataset, the x-coordinates ‘000’ and ‘010’ will be replaced by ‘0’ to prevent adversaries from distinguishing between these two nodes. ### Information Loss {#Information Loss} The information loss incurred by generalizing $\textrm{node}_i$ to $\textrm{node}_j$ in DGH $H_{\mathcal{A}}$ is defined as $$LS(\textrm{node}_i,\textrm{node}_j) = \log_{2}{LF(\textrm{node}_j)}-\log_{2}{LF(\textrm{node}_i)}\textrm{ bits} ,$$ where $LF(.)$ function returns the number of leaves in the subtree generated by a node, and $LS(.)$ function returns the loss incurred by the generalization of nodes. The calculation of information loss is elaborated in Example \[ex2\]. \[ex2\] Consider the x-coordinate DGH tree given in Fig. \[tree\], the information loss incurred by generalizing node ‘$10$’ to ‘$1$’ can be calculated as $ \log_{2}{4} - \log_{2}{2}=1 \textrm{ bits}$. Moreover, Lemma \[def33\] can be used to derive the total loss incurred by the generalization of two nodes to their LCA. \[def33\] The total loss incurred by generalizing $\textrm{node}_i$ and $\textrm{node}_j$ in $H_{\mathcal{A}}$ to their LCA, $\textrm{node}_p$, can be calculated as $$\begin{aligned} LS(\textrm{node}_i+\textrm{node}_j&,\textrm{node}_p ) =\nonumber \\ &LS(\textrm{node}_i,\textrm{node}_p) + LS(\textrm{node}_j,\textrm{node}_p).\end{aligned}$$ The total loss incurred during anonymization of a trajectory and a dataset are defined in Definitions \[def4\] and \[def6\], respectively. \[def4\] The total loss rendered by the generalization of trajectory $tr$ to achieve the anonymized trajectory $\overline{tr}$ with respect to attribute $\mathcal{A}$ can be calculated as $$\begin{aligned} LS(\overline{tr},\mathcal{A} ) = \sum_{i=1}^{|\overline{tr}|} LS(tr_i.\mathcal{A},\overline{tr}_i.\mathcal{A}).\end{aligned}$$ where $tr_i.\mathcal{A}$ indicates the $i$-th location of the trajectory $tr$ with respect to the attribute $\mathcal{A}$. Here, $\mathcal{A}$ could denote $x$-coordinate, $y$-coordinate, or time. \[def6\] The total loss with respect to an attribute $\mathcal{A}$ in an anonymized dataset $\overline{T}$ can be computed as $$\begin{aligned} LS(\overline{T},\mathcal{A} ) = \sum_{\overline{tr} \in \overline{T}}^{|\overline{T}|}LS(\overline{tr},\mathcal{A} )\end{aligned}$$ Privacy Model ------------- ### Adversary Model In our work, we consider coordinates and the time of queries both to be quasi-identifiers, as they can be linked to other databases and compromise the privacy of users. We also assume that no uniquely identifiable information is released while publishing the dataset. However, the adversary may: - already know about part of the released trajectory for an individual and attempt to identify the rest of the trajectory. For instance, the adversary is aware of the workplace of an individual and attempts to identify his or her home address. - already know the whole trajectory that an individual has traveled, but try to access other information released while publishing the dataset by identifying the user in the dataset. For instance, the published dataset may also include the type of services provided to users and if the adversary can identify a user by its trajectory, it can also know the services provided to that user. To this end, our aim is to protect users against the adversary’s attempt to access sensitive information that may compromise user privacy. ### Privacy Metric In this paper, we use a well-known metric called $k$-anonymity [@sweeney2002k] to ensure the privacy of users. The $k$-anonymity in our dataset implies that a given trajectory in the original dataset can at best be linked to $k-1$ other trajectories in the anonymized dataset. Definition \[def7\] formally defines the $k$-anonymity in the context of dataset. \[def7\] *k-anonymous dataset:* A trajectory dataset $\overline{T}$ is a $k$-anonymization of a trajectory dataset $T$ if for every trajectory in the anonymized dataset $\overline{T}$, there are at least $k-1$ other trajectories with exactly the same set of points, and there is a one to one mapping relation between the trajectories in $\overline{T}$ and $T$. ### Utility Metric K-anonymity metric ensures that the users are k-anonymous, and they can be identified from at least $k-1$ other users. Unfortunately, to achieve k-anonymity, significant loss of information can occur, which results in the much lower utility of published datasets. Moreover, as different anonymization techniques utilize various generalization schemes in the existing works, the information loss cannot be measured based on a unified metric as the one introduced in Section \[Information Loss\]. Therefore, it is necessary to develop a new metric to evaluate and compare the performance of anonymization schemes. In this work, we propose to use average released area per location to assess and compare various schemes. In the following, the calculation of average released area per location is explained. Any anonymization approach aims to maximize the utility while preserving the privacy of users. Utility in generalization techniques refers to the area released for locations in the dataset. Consider a location in the dataset $T$ with coordinates $<x_1, y_1,t_1>$ and an arbitrary generalization function $\mathcal{F}: T\rightarrow \overline{T}$. After the anonymization process, $<x_1, y_1,t_1>$ is generalized with a number of other locations $<x_2, y_2,t_2>$,..., $<x_a, y_a,t_a>$ in the dataset and an area $S$ would be released representing these locations. For instance, if generalization returns the minimum rectangle surrounding the locations. The generalized area is given by: $$\begin{aligned} \label{l1} S = (\underset{i}{\textrm{max}}\{ x_i\}-\underset{i}{\textrm{min}}\{ x_i\} )\times (\underset{i}{\textrm{max}}\{ y_i\} -\underset{i}{\textrm{min}}\{ y_i\} ).\end{aligned}$$ Once the anonymization is conducted, assume that $n_1$ locations are generalized to area $S_1$, $n_2$ locations are generalized to area $S_2$,..., $n_b$ locations are generalized to area $S_b$. In this case, the average released area per location can be calculated as $$\begin{aligned} \label{l2} (\mathlarger{\sum}_{i=1}^{b} n_i\times S_i)/(\mathlarger{\sum}_{i=1}^{b} n_i),\end{aligned}$$ in which no location belongs to more than one area. Average released area per location helps to understand how efficiently the data has been generalized and how much loss of utility has occurred by the generalization. Having $k$-anonymous locations, a smaller released area per location indicates a higher utility of data while preserving the privacy of users. Problem Formulation {#Architectural Overview} ------------------- The problem we seek to answer in this paper is formally presented in Problem 1 as follows. Given a trajectory dataset $T$, a privacy requirement $k$, quasi-identifiers $x$-coordinate, $y$-coordinate, and time, how to generate an anonymized dataset $\overline{T}$ which achieves the $k$-anonymity privacy metric and minimizes the total loss with respect to all quasi-identifiers, which can be explicitly formulated as $$\begin{aligned} \label{ee1} Minimize\{ LS(\overline{T},x) + LS(\overline{T},y)+ LS(\overline{T},t)\}. \end{aligned}$$ MLA {#approach} === In this section, we present our proposed framework, MLA, for anonymization of spatiotemporal datasets. Overview of the MLA Framework ----------------------------- ![Overview of our proposed MLA framework.[]{data-label="architecture33"}](Figures/architecture33.pdf) Fig. \[architecture33\] demonstrates the overview of our proposed framework. The original dataset and the value of $k$ are inputs of the framework, and the output is the anonymized dataset preserving the privacy of users. The MLA framework consists of three mechanisms working together to anonymize spatiotemporal datasets explained in the following: - **Generalization:** At the heart of MLA framework resides the generalization approach. The generalization process is conducted based on DGHs explained in Section \[Hierarchical Tree Transformation\]. - **Alignment:** For a given trajectory cluster, we propose to use progressive sequence alignment to find the arrangement of trajectories that results in the minimum information loss. Our approach for the alignment of trajectories is explained in Section \[Alignment\]. - **Clustering:** At the highest level of the MLA framework, clustering is applied to seek for the most suitable grouping of trajectories that minimizes information loss. We propose to use $k'$-means clustering algorithm and a variation of it for overly sensitive datasets. Moreover, to have a baseline for comparison purposes, we develop a heuristic approach to cluster datasets. Our proposed clustering approaches are elaborated in Section \[Clustering\]. Alignment {#Alignment} --------- The process of alignment is defined as finding the best match between two trajectories in order to minimize the overall cost of generalization and suppression. The process of alignment between two trajectories has been studied in different domains mostly referred to as sequence alignment (SA). In this paper, we adopt a multiple SA technique called progressive SA [@chowdhury2017review] for anonymization of spatiotemporal trajectories. ### Progressive Sequence Alignment {#ProgressiveAlignment} The progressive SA is commonly used for SA of a set of protein sequences. Progressive SA is a greedy approach for multiple SA. As a part of the algorithm, pairwise alignment of the trajectories is required. We use dynamic SA for this purpose. Dynamic SA is based on dynamic programming and commonly used in DNA SA [@chen2017cmsa; @le2017protein]. Fig. \[example\_figure1\] illustrates an example of how the progressive SA works for four hypothetical sequences $tr_a = \{a_1,\, a_2,\, a_3,\, a_4\}$, $tr_b = \{b_1,\, b_2\,\}$, $tr_c = \{c_1,\, c_2,\, c_3\}$ and $tr_d = \{d_1,\, d_2\}$ to generate the resultant aligned trajectory $tr_r = \{r_1,\, r_2,\, r_3,\, r_4\}$. The longest path $tr_a$ is chosen as the basis and it is aligned with a randomly chosen trajectory $tr_b$. The pairwise alignment process is implemented using dynamic SA. Then, the resultant trajectory is aligned with a third trajectory. The process continues until all trajectories are aligned. Instead of choosing the trajectories randomly during the progressive SA, the algorithm can choose the trajectory resulting in the lowest loss during the alignment. In Fig. \[example\_figure1\], the way trajectory elements are located with respect to the longest path is referred to as the structure of the shorter path, and also, the spaces indicate the suppression operation during the alignment. ![An overview of progressive SA for alignment of four trajectories and generating the anonymized trajectory.[]{data-label="example_figure1"}](Figures/TrajExa.pdf) The dynamic SA algorithm is formally represented in Algorithm \[DynamicSA\]. Dynamic SA is based on dividing the problem of finding the best SA to subproblems and storing the solutions of subproblems in a table or matrix referred to as $SAmatrix$ in the pseudocode. The objective is to achieve the minimal cost for SA. As before, the cost of alignment refers to the loss incurred during the alignment for different attributes of the sequence, which are $x$-coordinate, $y$-coordinate, and the time of the query. The algorithm starts by creating a $(m+1)\times (n+1)$ matrix ($SAmatrix$), where $m$ and $n$ denote the length of the trajectories. The matrix will be used to store the minimum cost of each cell of the grid. Moreover, a list called $code$ stores how cells have been reached. Cell $[j+1,i+1]$ can be reached from three cells $[j,i+1],\, [j+1,i],\, [j,i]$. Each path corresponds to one of the subproblems explained. After finding all values of the matrix and tracing back the list $code$, the outputs of the algorithm are the value of cell $[m,n]$ indicating the minimum value of the total loss ($TotLoss$) required for the dynamic SA, the aligned trajectory ($GenTraj$), and the structure of the shorter path compared to the longer path as $ShoTrajStr$. **Required variables:** $tr_1=\{ p_1,\,,p_2,..., p_m \}$, $tr_2=\{ q_1,\,,q_2,..., q_n \}$, $H_x$, $H_y$, $H_t$\ $SAmatrix \leftarrow \textrm{np.zeros}$($[m+1,n+1]$)\ $options \leftarrow \textrm{np.zeros}(3)$\ $code\leftarrow$ list()\ $TotLoss \leftarrow SAmatrix[m,n]$\ $GenTraj \leftarrow $ trace back the $code$ to generate the aligned trajectory\ $ShoTrajStr \leftarrow $ trace back the $code$ to find out structure of shorter trajectory while alignment\ **Return** $GenTraj, ShoTrajStr, TotLoss$ Clustering {#Clustering} ---------- Clustering can be seen as a search for hidden patterns that may exist in datasets. In simple words, it refers to grouping data entries in disjointed clusters so that the members of each cluster are very similar to each other. Clustering techniques are applied in many application areas, such as data analysis and pattern recognition. In this subsection, first, we develop a heuristic approach for clustering the spatiotemporal datasets. Heuristic approaches are widely used in the literature, as the problem is found to NP-hard. The heuristic approach is used as a baseline for comparison in our work, and it is not part of the MLA framework. Next, we propose a technique that enables the $k'$-means algorithm for clustering the spatiotemporal trajectories and extend our approach for sensitive locations by developing a variation of $k'$-means algorithm that guarantees privacy requirements for all users. ### Heuristic Approach {#Heuristic Approach} $NumOfClus \leftarrow \lceil \dfrac{|T|}{k} \rceil $\ $T \leftarrow OriginalDataset$\ Let $Clusters$ be a two-dimensional array storing the clusters and their corresponding trajectories $(\overline{T},\, Loss) \leftarrow $GenerateAnonymizedDataset($cluster$, $OriginalDataset$) **Return** $(\overline{T},\, Loss) $ The heuristic approach for clustering spatiotemporal trajectory datasets is detailed in Algorithm \[HeuristicClustering\] and its helper function in Algorithm \[GenerateAnonymizedDataset\]. The intuition behind the heuristic algorithm is to form the clusters by sequentially adding the most suitable trajectory that minimizes the total loss incurred by generalization and suppression for $x$-coordinate, $y$-coordinate, and the time of query, given their DGHs $H_x$, $H_y$, $H_t$. The algorithm starts by calculating the number of clusters that need to be generated and making a duplicate of the dataset called $T$. Moreover, a two-dimensional list is created, which holds the trajectory IDs for each cluster. For each cluster (i.e., cluster $c$), the algorithm appends a random trajectory from $T$. This trajectory is removed for $T$ and would be the first member of the cluster $c$. Then, given the privacy requirement $k$, $k-1$ other members of the cluster are chosen in a greedy approach. For every remaining trajectory in the dataset, the algorithm calculates the information loss incurred by applying dynamic alignment and determine the trajectory that results in the minimum loss. The chosen trajectory will be added to the cluster and removed from the dataset. The process continues until all members of the cluster are chosen. After clustering the trajectories, the helper function GenerateAnonymizedDataset is called in order to generate the anonymized dataset ($\overline{T}$) and the total incurred loss. The helper function (GenerateAnonymizedDataset) takes the original dataset and the two-dimensional list of clusters as inputs. The target of the algorithm is to find the total loss and anonymize the dataset. The algorithm starts by initializing the total loss to zero and creating an empty list ($\overline{T}$) to hold the generated anonymized dataset. Then, for each cluster, the progressive SA is applied to calculate the incurred loss in addition to the generalized trajectory. In the next step, the total loss is accumulated, and the generalized trajectory is appended to the anonymized dataset $\overline{T}$. Eventually, the anonymized dataset and the overall information loss happened due to alignment are returned. Let the $TotalLoss$ store the total loss incurred by applying progressive SA\ Let $\overline{T}$ be an empty set that will store the anonymized dataset\ **Return** $(\overline{T},\, TotalLoss) $ ### $k'$-means Clustering Approach $k'$-means algorithm [@macqueen1967some] is an attractive clustering algorithm currently used in many applications, especially in data analysis and pattern recognition [@pal2017genetic]. The main advantage of the $k'$-means algorithm is simplicity and fast execution. $$\begin{aligned} \label{n3} &\textrm{Total loss} = \underbrace{\mathlarger{\sum}_{i=1}^{|T|} (LS(tr_i.x,RT(H_{x}))+ LS(tr_i.y,RT(H_{y})) + LS(tr_i.t,RT(H_{t}))) }_\text{A}-\nonumber \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \underbrace{ (\mathlarger{\sum}_{i=1}^{|cluster|} \mathlarger{\sum}_{j=1}^{|cluster[i]|} (LS(h_j.x,RT(H_{x}))+LS(h_j.y,RT(H_{y}))+ LS(h_j.t,RT(H_{t})) ))}_\text{B}.\end{aligned}$$ The algorithm aims to partition the input dataset into $k'$ clusters. The only inputs to the algorithm are the number of clusters $k'$ and the dataset. Clusters are represented by adaptively-changing cluster centres. The initial values of the cluster centres are chosen randomly. In each stage, the algorithm computes the Euclidean distance of data from the centroids and partition them based on the nearest centroid to each data. More formally, representing the set of all centroids by $C= \{c_1,\, c_,...,\, c_{k'} \}$, each point in the dataset, denoted by $x$, is assigned to a centroid that has the shortest Euclidean distance to the point. This can be written as $$\begin{aligned} \label{e2} \underset{c_i \in C}{\textrm{argmin}} \,dist(x,c_i)^2,\end{aligned}$$ where the function $dist(.)$ returns the Euclidean distance between two points. Denoting the set of assigned data to the $i$-th cluster by $S_i$, new centroids are calculated in the second stage via $$\begin{aligned} c_i= \dfrac{1}{|S_i|} \mathlarger{\sum}_{x_i \in S_i} x_i.\end{aligned}$$ The algorithm continues the same process until the values of centroids no longer change. The $k'$-means algorithm is guaranteed to converge [@fischer2018convergence]. In the rest of this section, we first present a Lemma followed by explaining how the $k'$-means algorithm can be applied to trajectory datasets to reinforce the privacy preservation of users. \[def8\] The total loss incurred by generalizing $\textrm{node}_i$ and $\textrm{node}_j$ with respect to $H_{\mathcal{A}}$ can be calculated as $$\begin{aligned} LS(\textrm{node}_i ,&\textrm{node}_j ) =\nonumber \\ &|LS(\textrm{node}_i,RT(H_{\mathcal{A}})) - LS(\textrm{node}_j,RT(H_{\mathcal{A}}))|.\end{aligned}$$ \[ex22\] Lemma \[def8\] provides an alternative way to calculate the information loss by generalizing $\textrm{node}_i$ and $\textrm{node}_j$ in a given DGH. For instance, based on Lemma \[def8\], the information loss incurred by generalizing node ‘$10$’ to ‘$1$’ in Fig. \[tree\] (x-coordinate DGH), can be calculated as $|(\log_{2}{8} - \log_{2}{2})-(\log_{2}{8} - \log_{2}{4})| = 1$ bit. Lemma \[def8\] indicates that the loss incurred by generalizing two nodes is equal to the difference between losses incurred by their suppression. As before, for any clustering outcome of data, assume that $cluster$ is a two-dimensional list, in which the $j$-th element of the list returns the IDs of the trajectories in the $j$-th cluster. Moreover, we denote the $j$-th cluster head after generalization and suppression for all trajectories as $h_j$. Therefore, the total loss can be written as $$\begin{aligned} \label{n1} \textrm{Total loss} = LS(\overline{T},x)& + LS(\overline{T},y)+ LS(\overline{T},t)\nonumber \\ =\mathlarger{\sum}_{j=0}^{k-1} &\mathlarger{\sum}_{tr \in cluster[j]}(LS(h_j.x,tr.x)\nonumber \\ &+LS(h_j.y,tr.y)+LS(h_j.t,tr.t)).\end{aligned}$$ As explained in (\[ee1\]), the objective of clustering algorithms is to minimize this equation. Therefore, using Lemma \[def8\] the equation (\[n1\]) can be written as $$\begin{aligned} \label{n2} &\textrm{Total }\textrm{loss} =\\ &\mathlarger{\sum}_{j=0}^{k-1} \mathlarger{\sum}_{tr \in cluster[j]}(|LS(h_j.x,RT(H_{x})) - LS(tr.x,RT(H_{x})|\nonumber \\ &\quad \quad+|LS(h_j.y,RT(H_{y})) - LS(tr.y,RT(H_{y})|\nonumber \\&\quad \quad+|LS(h_j.t,RT(H_{t})) - LS(tr.t,RT(H_{t}))| .\end{aligned}$$ Rearranging (\[n2\]), the objective equation can be found by minimizing total loss formulated in (\[n3\]). This can be done by maximizing part B and minimizing part A. Since the cluster heads are generated based on the clustering algorithm, they cannot be used as part of the optimization process. Therefore, we aim at minimizing part A in (\[n3\]). Part A in the equation (\[n3\]) refers to finding the total distance of each trajectory from DGH root of the attributes. Therefore, for each trajectory, a three-dimensional vector $<d_x,\, d_y,\, d_t>$ is constructed, where $d_x$, $d_y$, $d_t$ store the loss incurred by generalizing the $x$-coordinate, $y$-coordinate, and time, respectively. Having distances of all points from the roots, we cluster the trajectories using the $k'$-means algorithm. The algorithm clusters trajectories with a similar loss from the root in the same group. This process is particularly important as trajectory datasets usually include trajectories as short as one query to trajectories with hundreds of queries. A major drawback of the $k'$-means algorithm is clustering the trajectories without any constraint on the minimum number of trajectories that needs to be in each cluster. Therefore, the algorithm might result in some of the clusters containing less than $k$ trajectories that violates the $k$-anonymity of trajectories. If the data is not extremely sensitive such as the data used in the military, it is usually acceptable to have a few trajectories below the $k$-anonymity criterion. As it will be demonstrated in Section \[Experiments\], the number of trajectories not achieving $k$-anonymity is close to or below $20\%$ of the trajectories based on the value of $k$ chosen for the privacy. To amend the naive $k'$-means algorithm for sensitive applications, we propose to use a variation of $k'$-means algorithm, which we call it iterative $k'$-means. The idea relies on running the $k'$-means algorithm iteratively to ensure that all clusters will achieve $k$-anonymity. Therefore, after each iteration of the $k'$-means algorithm, the clusters including at least $k$ trajectories are disbanded, and the trajectories are put back into the pool for the next iteration of the $k'$-means algorithm. This process continues until all clusters have at least $k$ members. Algorithm \[kmeans\] represents the pseudocode of the iterative $k'$-means. Experiments {#Experiments} =========== [0.3]{}  [0.3]{}  [0.3]{}  [0.3]{}  [0.3]{}  [0.3]{}  In our experiments, we use the data collected by Geolife project [@d1; @d2; @d3] in addition to the T-Drive dataset [@yuan2010t; @yuan2011driving], both published online by Microsoft. The Geolife and T-Drive datasets include the GPS trajectories of mobile users, and taxi drivers in Beijing (China), respectively. Each entry of the dataset is represented by the coordinates and the time of query. We have conducted our experiments on a $1\, km\times 1\, km$ central part of the Beijing map with the resolution of $0.01km\times 0.01km$ for each grid cell. The detailed statistics on the datasets are given in Table \[t1\]. The various location privacy requirements ($k$) of the users are investigated for values $2$, $5$, $10$, and $15$. The experiments were performed on a PC with a $3.40$ GHz Core-i7 Intel processor, $64$-bit Windows $7$ operating system, and an $8.00$ GB of RAM. The Python programming language was used to implement the algorithms. [|&gt;m[4cm]{} || &gt;m[1.5cm]{} | &gt;m[1.5cm]{} |]{} Dataset& Geolife & TDrive\ Total number of samples & 47581 & 27916\ Number of trajectories & 13561 & 301\ Average number of samples per trajectory & 3.5 & 92.74\ Average sampling time interval & 177 s & 1-5 s\ Average sampling distance interval & 623 m & 5-10 m\ \[t1\] We compare our work to prior methods as follows: - Many of prior approaches for the anonymization of spatiotemporal trajectory datasets use a greedy or so-called heuristic approach to anonymize datasets. In Section \[Heuristic Approach\], we explained and adopted this approach based on our system model. We use the heuristic approach in Section 5.1 as a baseline for comparison. - The full comparison of the MLA framework with the recent work in [@comparison] is provided in Section \[Comparison 2\]. The results are verified on both of the TDrive and Geolife datasets to ensure reliability. - As MLA and the proposed algorithm in [@comparison] seek to fulfill different objectives, we have further evaluated the two frameworks based on random clustering. Doing so shifts the focus to the alignment of trajectories in each cluster. The results are verified on both of the TDrive and Geolife datasets (Section 5.3). - We also compare our alignment approach with the widely used static algorithm in [@medical] (Section 5.3). Performance Evaluation ---------------------- Fig. \[main1\] presents the performance evaluation of MLA predicated on three clustering approaches developed in this paper. The algorithms have been investigated from three aspects: information loss, increase in trajectory length, and execution time. In all graphs, $x$-axis indicates $k$-anonymity requirement for the dataset. The total information loss and average information loss per cluster of algorithms are considered in Figs. \[a\] and \[b\], respectively. Information loss, shown in the $y$-axis, indicates the total loss incurred while applying generalization and suppression on $x$-coordinate, $y$-coordinate, and the time of the query. The maximum possible incurred information loss for the whole dataset by suppressing all trajectories is $474572$ bits. This value is the upper bound on all anonymization algorithms. Note that this constant changes for different datasets. The main existing trend in Figs. \[a\] and \[b\] is that by increasing the value of $k$, the total incurred loss increases. This outcome meets our expectation as increasing the value of $k$ indicates having larger cluster sets, which results in the alignment of a higher number of trajectories in each cluster, and thereby, a higher total loss by the alignment. Among our proposed algorithms, $k'$-means algorithm provides the best performance as it corresponds to minimum lost bits incurred by the generalization and suppression. The amount of information that $k'$-means algorithm preserves is higher than that of the heuristic approach, in which the most suitable trajectories are chosen to minimize the information. This trend can be seen for both of the total information loss of the dataset and the average information loss of dataset per cluster for different $k$ values. Such a trade-off exists, because some clusters contain a small number of trajectories not satisfying the $k$-anonymity requirement. The loss of privacy by $k'$-means algorithm is further analyzed in Fig. \[main\_figure2\] which will be explained later in this section. The iterative $k'$-means algorithm is constructed on top of the $k'$-means algorithm to ensure that all the trajectories satisfy the required privacy requirement. This is particularly important for sensitive applications, in which there are strict requirements for privacy preservation. The cost of having higher privacy for the iterative $k'$-means algorithm is a larger loss of information. [0.5]{}    [0.5]{}  Figs. \[c\] and \[d\] present the average increase in the length of trajectories for the whole dataset and per cluster. Due to the alignment process, shorter trajectories may need to be aligned with longer trajectories, which result in an increase in the length of trajectories in the anonymized released dataset. The best performance among the algorithms is yielded by the $k'$-means algorithm with the lowest increase in the lengths of trajectories. Compared to other two approaches, the heuristic strategy performs better than the iterative $k'$-means with a smaller $k$, but as the $k$ value increases, the average increase in trajectory length converges due to large cluster size. Figs. \[e\] and \[f\] compare the total and average per cluster execution time of the different algorithms. Note that since the heuristic algorithm requires a significantly higher amount of time to run, it is shown on top of the graphs as a flat line with the corresponding values shown below it. The execution time of the $k'$-means and iterative $k'$-means algorithms are significantly lower than that of the heuristic algorithm and as expected the iterative $k'$-means consumes slightly more execution time as it has additional steps to ensure the $k$-anonymity of all trajectories. Detailed Analysis of $k'$-means Algorithm ----------------------------------------- Overall, the detailed $k'$-means algorithm’s results in satisfactory performance in terms of information loss, execution time, and the average increase in the length of trajectories. Moreover, the complexity of $k'$-means algorithm is of an order of the number of data entries for large datasets, whereas the order of the heuristic algorithm is proportional to the square of this number. Therefore, the $k'$-means algorithm has several significant advantages compared to the heuristic approach. Hence, if it is acceptable for the datasets to have a few trajectories below the $k$-anonymity requirement, then, it is more beneficial to use the $k'$-means algorithm instead of the heuristic or the iterative $k'$-means algorithm. This is usually true for datasets not entailing classified information. Therefore, we further analyze the performance of this algorithm in the remaining of this section and compare it to the state-of-art algorithms recently proposed. Also, note that in the rest of this paper when MLA is mentioned, the $k'$-means algorithms is adopted for clustering by default. Fig. \[main\_figure2\] provides two graphs showing the details of the performance yielded by the $k'$-means algorithm. The first graph indicates the average value of $k$ achieved while applying the $k'$-means algorithm, and the second graph shows the percentage of trajectories that did not achieve the $k$-anonymity in the anonymization process with different values of $k$. In Fig. \[main\_figure2\](a), it is evident that despite some of the trajectories losing their $k$-anonymity during the anonymization, the average value of anonymity achieved is above the minimum requirement. The value of the average gets even better as the value of $k$ increases. Fig. \[main\_figure2\](b) shows the percentage of the trajectories not achieving the minimum required $k$-anonymity. This value is below $20\%$ on average, which means that over $80\%$ of the trajectories are guaranteed to at least have $k$-anonymity. The reason causing the uneven curves in the figure is because the number of clusters is divisible by $k$, which results in an additional cluster distorting the curves. Comparison {#Comparison 2} ---------- We compare MLA with the static algorithm proposed in [@medical], and recently published anonymization approach in [@comparison]. The idea behind the static alignment algorithm in [@medical] is that two trajectories are matched element by element without any shifts or spaces. In more details, the static algorithm attempts to match two sequences based on the same index. Therefore, each element of the first sequence $tr_1$ is aligned with an element having the same index in the other input trajectory $tr_2$. Based on our evaluation, the total incurred information loss is reduced by $7.2\%$ by using the proposed progressive SA algorithm. It must be noted that the dataset includes trajectories as large as hundreds of queries and as small as a single query from the location-based service provider. Therefore, matching these length-variant trajectories would impose a substantial information loss even for the best possible match of the sequences. [0.5]{}    [0.5]{}  [0.5]{}    [0.5]{}  Fig. \[comp1\] indicates the comparison result between our proposed anonymization technique and the recent generalization method proposed in [@comparison]. The authors in [@comparison] attempted to minimize the incurred loss of the anonymization by sorting out the spatiotemporal locations in the time domain and applying a heuristic approach for generalization. They also used a heuristic approach for clustering trajectories. Note that any anonymization approach aims to maximize utility while preserving the privacy of users. Utility in generalization techniques refers to the area released for locations in the dataset. Therefore, to have a fair comparison, we compare our work with the approach proposed in [@comparison] based on the average released area for locations. The metric is thoroughly explained in Section \[System model\]. It can be seen from the figure that our proposed algorithm can significantly increase the utility of the generalization approach. In other words, the anonymized dataset has on average smaller released area per location while preserving the privacy of users. To further compare alignment approaches, in Fig. \[comp1\], we applied random clustering to group the trajectories, and then, used the alignment approach in our proposed work and the previous work to generate anonymized trajectories. As can be seen in the figure, our alignment approach outperforms the previous work by a higher utility of anonymized dataset. Discussion {#Discussion} ---------- As can be seen in Fig. \[comp1\], the MLA framework has significantly improved the utility of data while achieving $k$-anonymity for the entries of datasets. A major reason for such an improvement is that MLA considers all three dimensions of time, $x$-coordinate, and $y$-coordinate together. Such consideration helps to minimize the overall cost and not just the utility in time or spatial domain. For instance, the Geolife datasets consists of sampling time interval of $177$ seconds with the average distance interval of $623$ meters, whereas the TDrive dataset has the average sampling interval of $1-5$ seconds and $5-10$ meters of sampling distance interval. Therefore, the two datasets enatail a highly different sparsity characteristic in time and spatial domain. However, as can be seen in Figs. \[comp1\] and \[comp2\], the MLA algorithm considers all three dimensions, and can significantly improve the utility in the process of anonymization. In essence, the performance improvement in our proposed model is predicated on both the clustering and alignment of trajectories. In terms of the alignment, progressive SA has resulted in significant improvement of the alignment process. Such an impact can be seen in Fig. \[comp1\], where we apply random clustering, and therefore, the focus is on the alignment. As the figure suggests, utilizing a multiple SA technique such as progressive SA used in MLA provides major improvements to the utility of the anonymized datasets. For clustering, as finding the optimal anonymization of the datasets is proven to be NP-hard, most of the literature has focused on following heuristic approaches to cluster the trajectories. We adopted such a heuristic approach for the system model of our paper and presented the results in Fig. \[main1\]. Note that in the heuristic approach used in the figure, we are applying progressive SA alignment; therefore, the results show the improved version of the previously existing algorithms. As it was revealed in Fig. \[main1\], the $k'$-means algorithm can outperform such heuristic approaches in addition to having a much lower implementation complexity and processing time. Applications ============ In this section, we introduce several applications that we believe our work has the most impact on. Location-Based Data ------------------- As the framework for anonymization presented in this paper considers location trajectories, one of the main applications of the framework is the privacy of location-based data. The use of location-based applications is more prevalent than any time before. Governments attempt to analyze the infrastructure using the location data and researchers use these data to investigate human behavior. Research has verified that even simple analytics on these published trajectory data would yield serious risk of users’ privacy and even be capable of identifying users of location-based applications. [@sun2017asa]. Therefore, applying anonymization techniques such as the one we have developed in this paper is necessary to preserve the privacy of the users. Medical Records --------------- The recent advances in medical information technology have enabled the collection of a detailed description of patients and their medical status [@asan2018preferences]. Such data is usually stored in electronic medical record systems [@gates2018electronic; @rahman2017systems; @colleti2018evaluation]. Similar to spatiotemporal trajectories, many of the medical records need to be published by agencies and organizations. Unfortunately, research has shown that solely relying on de-identification is insufficient to protect users’ privacy, as the medical records from multiple databases can be linked together to identify individual patients [@medical]. Therefore, there is an urgent need for viable algorithms to anonymize the medical data. The problem of anonymization in spatiotemporal trajectories is very similar to anonymization in longitudinal electronic medical records. This can be easily justified by the similar way, in which these data are stored. Assume a patient who has referred to medics several times in his or her lifetime. Each time the records of the patient are stored in a longitudinal dataset, in which the age and the diagnosed disease record are registered. These longitudinal records can be seen as a trajectory for the patient, and our proposed algorithms in this paper can be applied to anonymize a dataset of such longitudinal electronic medical records. Web Analytics ------------- Another important application of the framework developed in this paper is web analytics. Web analytics refers to analyzing online traces of users. Web analytics has become a competitive advantage for many companies due to the amount of detailed information that can be extracted from the data. Therefore, protecting the trajectories that the users explored on the Internet has become a major challenge for researchers. The similarity between spatiotemporal trajectories and web analytics can be well explained by the following example. For instance, Geoscience Australia is constantly recording and publishing the site logs users make on their website. The site log filename is composed of a four-digit station identifier, followed by a two-digit month and a two-digit year, e.g., ALIC0414 is the site log for the Alice Springs GNSS site that was updated in April 2014 [@aa]. Such a trajectory of logins to the website is analogous to a spatiotemporal trajectory with three attributes. Therefore, the framework developed in this paper can be used to anonymize the online traces of users before publishing web browsing data. Conclusion =========== In this paper, we have proposed a framework to preserve the privacy of users while publishing the spatiotemporal trajectories. The proposed approach is based on an efficient alignment technique termed as progressive sequence alignment in addition to a machine learning clustering approach that aims at minimizing the incurred loss in the anonymization process. We also devised a variation of $k'$-means algorithm for guaranteeing the $k$-anonymity in overly sensitive datasets. The experimental results on real-life GPS datasets indicate the superior utility performance of our proposed framework compared with the previous works. [Sina Shaham]{} received B.Eng (Hons) in Electrical and Electronic Engineering from the University of Manchester (with first class honors). He is currently an MPhil student at the University of Sydney. He has years of experience as a Data Scientist and Software Engineer in companies such as InDebted. His current research interests include applications of artificial intelligence in big data and privacy. -2plus -1fil [Ming Ding]{} (M’12-SM’17) received the B.S. and M.S. degrees (with first class Hons.) in electronics engineering from Shanghai Jiao Tong University (SJTU), Shanghai, China, and the Doctor of Philosophy (Ph.D.) degree in signal and information processing from SJTU, in 2004, 2007, and 2011, respectively. From April 2007 to September 2014, he worked at Sharp Laboratories of China in Shanghai, China as a Researcher/Senior Researcher/Principal Researcher. He also served as the Algorithm Design Director and Programming Director for a system-level simulator of future telecommunication networks in Sharp Laboratories of China for more than 7 years. Currently, he is a senior research scientist at Data61, CSIRO, in Sydney, NSW, Australia. He has authored over 80 papers in IEEE journals and conferences, all in recognized venues, and about 20 3GPP standardization contributions, as well as a Springer book “Multi-point Cooperative Communication Systems: Theory and Applications”. Also, he holds 16 US patents and co-invented another 100+ patents on 4G/5G technologies in CN, JP, EU, etc. Currently, he is an editor of IEEE Transactions on Wireless Communications. Besides, he is or has been Guest Editor/Co-Chair/Co-Tutor/TPC member of several IEEE top-tier journals/conferences, e.g., the IEEE Journal on Selected Areas in Communications, the IEEE Communications Magazine, and the IEEE Globecom Workshops, etc. He was the lead speaker of the industrial presentation on unmanned aerial vehicles in IEEE Globecom 2017, which was awarded as the Most Attended Industry Program in the conference. Also, he was awarded in 2017 as the Exemplary Reviewer for IEEE Transactions on Wireless Communications. -2plus -1fil [Bo Liu]{} (M’10) received the B.Sc. degree from the Department of Computer Science and Technology from Nanjing University of Posts and Telecommunications, Nanjing, China, in 2004 and then he received the and MEng. and PhD degrees from the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China, in 2007, and 2010 respectively. He was an assistant research professor at the Department of Electronic Engineering of Shanghai Jiao Tong University between 2010 and 2014, and a Postdoctoral Research Fellow at Deakin University, Australia, between November 2014 and September 2017. He is currently a lecturer in the Department of Engineering, La Trobe University, from October 2017. His research interests include wireless communications and networking, security and privacy issues in wireless networks. -2plus -1fil [Zihuai Lin]{} received the Ph.D. degree in Electrical Engineering from Chalmers University of Technology, Sweden, in 2006. Prior to this he has held positions at Ericsson Research, Stockholm, Sweden. Following Ph.D. graduation, he worked as a Research Associate Professor at Aalborg University, Denmark and currently at the School of Electrical and Information Engineering, the University of Sydney, Australia. His research interests include source/channel/network coding, coded modulation, MIMO, OFDMA, SC-FDMA, radio resource management, cooperative communications, small-cell networks, 5G cellular systems, etc. -2plus -1fil [Shuping Dang]{} (S’13–M’18) received B.Eng (Hons) in Electrical and Electronic Engineering from the University of Manchester (with first class honors) and B.Eng in Electrical Engineering and Automation from Beijing Jiaotong University in 2014 via a joint ‘2+2’ dual-degree program. He also received D.Phil in Engineering Science from University of Oxford in 2018. Dr. Dang joined in the R&D Center, Huanan Communication Co., Ltd. after graduating from University of Oxford and is currently working as a Postdoctoral Fellow with the Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST). He serves as a reviewer for a number of key journals in communications and information science, including [<span style="font-variant:small-caps;">[IEEE Transactions on Wireless Communications]{}</span>]{}, [<span style="font-variant:small-caps;">[IEEE Transactions on Communications]{}</span>]{} and [<span style="font-variant:small-caps;">[IEEE Transactions on Vehicular Technology]{}</span>]{}. His current research interests include artificial intelligence assisted communications, novel modulation schemes and cooperative communications. -2plus -1fil [Jun Li]{} (M’09-SM’16) received Ph. D degree in Electronic Engineering from Shanghai Jiao Tong University, Shanghai, P. R. China in 2009. From January 2009 to June 2009, he worked in the Department of Research and Innovation, Alcatel Lucent Shanghai Bell as a Research Scientist. From June 2009 to April 2012, he was a Postdoctoral Fellow at the School of Electrical Engineering and Telecommunications, the University of New South Wales, Australia. From April 2012 to June 2015, he is a Research Fellow at the School of Electrical Engineering, the University of Sydney, Australia. From June 2015 to now, he is a Professor at the School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, China. His research interests include network information theory, ultra-dense wireless networks, and mobile edge computing. [^1]: This work was submitted in part and accepted to appear in the proceedings of INFOCOM WORKSHOPS, 2019 [@shaham2019machine]. [^2]: S. Shaham and Z. Lin are with the Department of Engineering, The University of Sydney, Sydney, NSW, 2006 Australia (e-mail: sina.shaham, zihuai.lin}@sydney.edu.au). [^3]: M. Ding is with Data61, Sydney, NSW, 1435 Australia (email: [email protected]) [^4]: B. Liu is with Latrobe University, VIC, 3086 Australia (email: [email protected]) [^5]: S. Dang was with the R&D Center, Guangxi Huanan Communication Co., Ltd., Nanning 530007, China when completing the major work of this paper, and is now with Computer, Electrical and Mathematical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia (e-mail: [email protected]). [^6]: J. Li is with NJUST, Nanjing, China (email: [email protected]) [^7]: The prime notation on the top of variable “$k$” is to distinguish between the variable $k$ in the clustering algorithm and the variable $k$ used in the definition of $k$-anonymity.

--- abstract: 'This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other geometric optimality criteria need to be identified. To this end, we use an iteration of the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. We focus on the lowest degree case in which the embedding is quartic.' author: - 'Bernhard G. Bodmann and John I. Haas IV[^1]' bibliography: - 'tier2\_bib.bib' title: 'Low frame coherence via zero-mean tensor embeddings' --- Introduction ============ The construction of equiangular lines has a long history in the mathematical literature [@Haantjes1948; @Rankin1955; @vanLintSeidel1966; @LemmensSeidel1973; @Zauner1999; @MR1984549; @MR2021601; @XiaZhouGiannakis2005; @MR2890902; @MR2921716; @MR3150919; @Fickus:2015aa]. If the number of unit-norm vectors spanning these lines cannot be enlarged any more without changing the set of angles/distances between them, then these vectors constitute an example of an optimal packing. Such packings have applications ranging from coding, fiber-optic or wireless communications to phase retrieval and quantum information theory [@bod_cas_edi_bal_2008; @MR2142983; @MR836025]. An analytic formulation of equiangular lines as solutions of an optimization problem is the so-called Welch bound [@Welch1974]. It can be obtained by combining a mapping of Conway, Hardin and Sloane [@ConwayHardinSloane1996] with a spherical cap packing bound by Rankin  [@Rankin1955]. In an earlier work, the case of redundancy beyond the equiangular regime was addressed by combining the embedding by Conway, Hardin and Sloane with the orthoplex bound, which is saturated by the example of maximal sets of mutually unbiased bases. With the help of relative difference sets, previously unknown examples of Grassmannian packings could be constructed [@MR3557826]; for examples, see the tables of Refs.  for instances of Grassmannian frames with redundancies varying between that of maximal equiangular frames and maximal mutually unbiased bases. Here, we iterate the embedding to obtain higher degree polynomial maps that are used to embed specific unit-norm frames to simplices. As a result, we identify several cases of optimal packings. Preliminaries ============= Frame Theory ------------ Let $\{e_j\}_{j=1}^m$ denote the canonical orthonormal basis for the Hilbert space $\mathbb R^m$. A sequence of vectors $\mathcal F = \{f_j\}_{j=1}^n \subset \mathbb R^m$ is a [**(finite) frame**]{} for $\mathbb R^m$ if it spans the entire Hilbert space. From now on, we reserve the symbols $m$ and $n$ to refer to the dimension of the span of a frame and the cardinality of a frame, respectively. The [**redundancy**]{} of a given frame is the ratio $\frac n m$. A frame $\mathcal F = \{f_j\}_{j=1}^n$ is [**$a$-tight**]{} if $$\sum_{j=1}^n f_j f_j^*= a\, \mathbf{I}_m, \text{ for some } a>0$$ where $\mathbf{I}_m$ denotes the $m \times m$ identity matrix. The frame is [**unit-norm**]{} if each frame vector has norm $\|f_j\|=1$. Given a unit-norm frame $\mathcal F = \{ f_j \}_{j=1}^n$, its [**frame cosines**]{} are the elements of the set $$\Theta_\mathcal F : =\{ |\langle f_j, f_l \rangle | : j \neq l \},$$ and we say that $\mathcal F$ is [**$k$-angular**]{} if $|\Theta_\mathcal F| =k$ for some $k \in \mathbb N$. If $\Theta_{\mathcal F}$ has only one element and $\mathcal F$ is tight, then we speak of an [**equiangular tight frame**]{}. Let ${{\Omega}_{n,m} {(\mathbb {R})}}$ denote the space of unit-norm frames for $\mathbb R^m$ consisting of $n$ vectors. Given any set of unit vectors, $\mathcal F = \{ f_j \}_{j=1}^n \subset \mathbb F^m$, its [**coherence**]{} is defined by $$\mu(\mathcal F) = \max\limits_{j \neq l} |\langle f_j, f_l \rangle |.$$ We define and denote the [**Grassmannian constant**]{} as $${\mu_{n,m} (\mathbb R)} = \min\limits_{\scriptscriptstyle \mathcal F \in {{\Omega}_{n,m} {(\mathbb {R})}}} \mu(\mathcal F).$$ Correspondingly, a frame $\mathcal F \in {{\Omega}_{n,m} {(\mathbb {R})}}$ is a [**Grassmannian frame**]{} if $$\mu(\mathcal F) = {\mu_{n,m} (\mathbb R)}.$$ Zero-mean tensor embeddings =========================== Our path toward identifying certain optimal line packings involves a two step process. First, we apply a norm-preserving map to the frame vectors, thereby embedding the frame into a higher dimensional real sphere. For the second step, we interpret the embedded vectors as the centers of spherical caps (which we discuss below) and exploit the cap packing results of Rankin [@Rankin1955]. If a frame embeds into an optimal cap packing and certain additional conditions are satisfied, then the minimal coherence of the lifted frame is verified by the isometric nature of the the embedding. We begin by defining the aforementioned family of norm-preserving maps. We denote the unit sphere in $\mathbb R^m$ by ${{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle m}\right)}$ and we let ${\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}$ denote the real vector space of self-adjoint linear maps on ${{\mathbb R}^{m}}$. From here on, $\omega$ is a random vector with values in ${{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle m}\right)}$ and $\mathbb E$ denotes the expectation with respect to the underlying uniform probability measure on ${{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle m}\right)}$. \[def\_Qt\] The [**first zero-mean tensor embedding**]{} is defined and denoted by $${\mathcal{\bf Q}^{(1)}_{{m} } }: {{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle m}\right)} \rightarrow {\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}: x \mapsto x \otimes x^* - {\scriptstyle \frac{1}{m}} {\bf I}_m \, ,$$ and for $t \in \mathbb N$, the [**$(t+1)$-th zero-mean tensor embedding**]{}, ${\mathcal{\bf Q}^{(t+1)}_{{m} } }$, is defined recursively by $${\mathcal{\bf Q}^{(t+1)}_{{m} } }: {{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle m}\right)} \rightarrow {\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}^{\otimes 2^{t-1}} : x \mapsto \left( {\mathcal{\bf Q}^{(t)}_{{m} } }(x)\right)^{\otimes 2} - \mathbb E \left[ \left({\mathcal{\bf Q}^{(t)}_{{m} } }(\omega) \right)^{\otimes 2} \right] \, .$$ For brevity, we also refer the $t$-th zero-mean tensor embedding simply as [**the $t$-th embedding**]{}. The purpose of subtracting the expected value is that, just as $\mathbb E[\omega]=0$, the mean of the embedding vanishes, $$\mathbb E\left[ {\mathcal{\bf Q}^{(t)}_{{m} } }(\omega) \right] = 0 .$$ In comparison with the action of of taking simple tensor powers of $x\otimes x^*$, the dimension of the range of the embedding is reduced by subtracting the expectation, as we show in the next theorem. To simplify notation, for each $t \geq 2$, we write $${{\mathbf V}^{(t)}}:= \mathbb E\left[\left( {\mathcal{\bf Q}^{(t-1)}_{{m} } }(\omega)\right)^{\otimes 2}\right].$$ \[th\_orth\_cond\] If $t \ge 2$ and $x \in \mathbb F^m$, then $ \operatorname{tr}[ {\mathcal{\bf Q}^{(t)}_{{m} } }(x) {{\mathbf V}^{(t)}} ] = 0 \, . $ We note that for any unitary $U$ on ${{\mathbb R}^{m}}$, $$U^{\otimes 2^{t-1}} {{\mathbf V}^{(t)}} (U^*)^{\otimes 2^{t-1}} = \mathbb E\left[ {\mathcal{\bf Q}^{(t-1)}_{{m} } }(U \omega) \otimes {\mathcal{\bf Q}^{(t-1)}_{{m} } }(U\omega) \right] = {{\mathbf V}^{(t)}}$$ because $U\omega$ and $\omega$ are identically distributed. This implies that $$\begin{aligned} \operatorname{tr}\left[ {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x)\otimes {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x) {{\mathbf V}^{(t)}} \right] &= \operatorname{tr}\left[ {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x)\otimes {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x) U^{\otimes 2^{t-1}} {{\mathbf V}^{(t)}} (U^*)^{\otimes 2^{t-1}} \right] \\ &= \operatorname{tr}\left[ {\mathcal{\bf Q}^{(t-1)}_{{m} } }(U^*x)\otimes {\mathcal{\bf Q}^{(t-1)}_{{m} } }(U^*x) {{\mathbf V}^{(t)}} \right]\end{aligned}$$ and by averaging with respect to the choice of $U^*$ among all unitaries, $$\operatorname{tr}\left[ {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x)\otimes {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x) {{\mathbf V}^{(t)}} \right] = \operatorname{tr}\left[ \left({{\mathbf V}^{(t)}}\right)^2 \right] \, .$$ Consequently, $$\operatorname{tr}\left[ {\mathcal{\bf Q}^{(t)}_{{m} } }(x) {{\mathbf V}^{(t)}}\right] = \operatorname{tr}\left[ \left( {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x)\otimes {\mathcal{\bf Q}^{(t-1)}_{{m} } }(x)-{{\mathbf V}^{(t)}}\right) {{\mathbf V}^{(t)}} \right] = 0 \, .$$ The space of symmetric tensors in $({\mathbb R^d})^{\otimes 2}$ is of dimension $d(d+1)/2$, and with an additional orthogonality condition the range is reduced to a subspace of dimension $d(d+1)/2-1=(d+2)(d-1)/2$. Iterating this dimensionality bound yields a maximal dimension of the subspace containing the range of ${\mathcal{\bf Q}^{(t)}_{{m} } }$. Accordingly, we define and denote the [**first embedding dimension**]{} by $${{\mathbf d}^{(1)}_{{m}}} : = \frac{(m+2)(m-1)}{2} ,$$ and, for $t\in \mathbb N$, we define the [**$(t+1)$-th embedding dimension**]{} recursively by $${{\mathbf d}^{(t+1)}_{{m}}} : = \frac 1 2 ({{\mathbf d}^{(t)}_{{m}}}+2)({{\mathbf d}^{(t)}_{{m}}}-1) \, .$$ For example, the [**second embedding dimension**]{} is $${{\mathbf d}^{(2)}_{{m}}} =\frac{ \left(m^2+m+2\right)\left(m^2+m-4\right)}{8} ,$$ and the [**third embedding dimension**]{} is $${{\mathbf d}^{(3)}_{{m}}} = \frac{\left((m-1)m(m+1)(m+2)+8\right) \left((m-1)m(m+1)(m+2)-16\right)}{128} .$$ In particular, Theorem \[th\_orth\_cond\] yields the following corollary. The range of the map ${\mathcal{\bf Q}^{(t)}_{{m} } }$ is contained in a subspace of ${\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}^{\otimes 2^{t-1}}$, whose dimension is at most equal to $ {{\mathbf d}^{(t)}_{{m}}}\, . $ Rankin’s bound and achieving optimal coherence ============================================== In this section, we show how to exploit Rankin’s classical bounds for spherical cap packings [@Rankin1955] and deduce the optimality properties of certain frames. Given $d \in \mathbb N$, a unit vector $x\in{{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle d}\right)}$ and a real number $\theta \in (0, \pi]$, we define and denote the [**spherical cap of angular radius $\theta$ centered at $x$**]{} as $${\mathcal C_{x} \left(\theta \right) } :=\left\{ y\in {{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle d}\right)} : \langle x, y\rangle > \cos \theta \right\},$$ which is alternatively referred to as a [**$\theta$-cap**]{} when the center is arbitrary. Rankin considered following optimization problem. \[rankin\] Given a fixed dimension, $d$, and a fixed angle, $\theta$, what is the largest number, $n$, of $\theta$-caps, $\left\{ {\mathcal C_{x_j} \left(\theta \right) } \right\}_{j=1}^n$, that one can configure on the surface of ${{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle d}\right)}$ such that ${\mathcal C_{x_j} \left(\theta \right) } \cap {\mathcal C_{x_{j'}} \left(\theta \right) } = \emptyset$ for each $j, j' \in \{1,2,...,n\}$ with $j \neq j'$. As a partial solution, Rankin reformulated Problem \[rankin\] in terms of its inverse optimization problem, providing sharp upper bounds on the caps’ angular radii, completely solving the problem whenever $n\leq 2d$. We phrase his results in terms of the inner products between the caps’ centers. The first embedding ------------------- The first embedding has been used in conjunction with Rankin’s cap-packing results [@Rankin1955] to characterize and construct numerous families of Grassmannian frames [@ConwayHardinSloane1996; @Fickus:2015aa; @MR3557826; @Appleby2009]. In more detail, the vectors are mapped to a self-adjoint rank-one Hermitian and projected onto the orthogonal complement of the identity matrix. The inner product between the images of two unit vectors is a polynomial of the original inner product. As a consequence, under certain assumptions, optimal packings are equivalent to packings on a Euclidean sphere, and the Rankin bound can be applied. \[thm\_t1\]\[Conway, Hardin and Sloane; [@ConwayHardinSloane1996]\] If $\mathcal F$ is a unit-norm frame of $n$ vectors in $\mathbb{F}^m$ and $\{{\mathcal{\bf Q}^{(1)}_{{m} } }(f): f \in \mathcal F\}$ forms a simplex in a subspace of the real space of self-adjoint $m \times m$ matrices over $\mathbb F$, then $\mathcal F$ is a Grassmannian frame. Moreover, if $n > {{\mathbf d}^{(1)}_{{m}}} + 1$, and the Hilbert-Schmidt inner product of any pair from $\{{\mathcal{\bf Q}^{(1)}_{{m} } }(f): f \in \mathcal F\}$ is non-positive, then $\mathcal F$ is a Grassmannian frame. It is straightforward to verify that $$x \mapsto \tilde{T}^{(1)}(x) = \frac{m}{m-1} {\mathcal{\bf Q}^{(1)}_{{m} } }(x)$$ maps the unit sphere in ${\mathbb F}^m$ to the unit sphere in ${\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}$. More generally, the inner products of frame vectors $f_j$ and $f_l$ are related by $$\left\langle \tilde{T}^{(1)}(f_j), \tilde{T}^{(1)}(f_l) \right\rangle_{HS} = \frac{m}{(m-1)} \left(|\langle f_j, f_l \rangle |^2 - \frac 1 m \right) \, .$$ Now applying Rankin’s bound shows that if the Hilbert-Schmidt inner product assumes the constant value $\langle \tilde{T}^{(1)}(f_j), \tilde{T}^{(1)}(f_l) \rangle_{HS}=-\frac{1}{n-1}$ when $j \ne l$, then the maximal magnitude occurring among inner products of pairs of vectors from $\mathcal F$ is minimized. Moreover, if $n > {{\mathbf d}^{(1)}_{{m}}} + 1$ and the maximum $\max_{j \ne l} \left\langle \tilde{T}^{(1)}(f_j), \tilde{T}^{(1)}(f_l) \right\rangle_{HS} \le 0$, then by Rankin’s bound equality holds and the frame is Grassmannian. Converting between the (squared) inner product of the frame vectors and the Hilbert-Schmidt inner product of the embedded vectors gives the Welch and orthoplex bounds as consequence. If $\mathcal F$ is a unit-norm frame of $n$ vectors in $\mathbb{F}^m$, then $\max_{j \ne l} |\langle f_j , f_l \rangle | \ge \sqrt{\frac{n-m}{(n-1)m}}$ and if equality holds, $\mathcal F$ is an equiangular tight frame. Moreover, if $n > {{\mathbf d}^{(1)}_{{m}}}+1$, then $\max_{j \ne l} |\langle f_j , f_l \rangle | \ge \frac{1}{\sqrt m}$ and if equality holds, $\mathcal F$ is a Grassmannian frame. The second embedding -------------------- For the remainder of this work, we focus on the development of the analogous machinery corresponding to the second tensor embedding. In order to provide an explicit expression for second embedding, we must compute the expectation, $\mathbb E \left[ \left({\mathcal{\bf Q}^{(1)}_{{m} } }(\omega) \right)^{\otimes 2} \right]$, as given in Definition \[def\_Qt\]. To facilitate this, we define and denote the [**$t$-coherence tensor (for $\mathbb R^m$)**]{} by $${{\bf K}^{(t)}_{{m} } } := \int\limits_{{\mathcal O_{\scriptscriptstyle \mathbb {R}^{m}}}} \left( UPU^* \right)^{\otimes t} d\mu(U),$$ where ${\mathcal O_{\scriptscriptstyle \mathbb {R}^{m}}}$ denotes the matrix group of $m\times m$ orthogonal matrices, $\mu$ denotes the unique, left-invariant [*Haar-measure*]{} on ${\mathcal O_{\scriptscriptstyle \mathbb {R}^{m}}}$, and $P$ is any $m \times m$ orthogonal projection onto a one-dimensional subspace of $\mathbb R^m$. In the following proposition, we provide an analytic expression for the $2$-coherence tensor. In order to express its dependence on the underlying field and to simplify notation, we define the constants $${{\mathbf a}_{{m}}} := \frac{ {{\mathbf d}^{(1)}_{{m}}} + (m-1)^2 }{ m^2 {{\mathbf d}^{(1)}_{{m}}}} \, \text{ and } {{\mathbf b}_{{m}}} := \frac{{{\mathbf a}_{{m}}}}{3}, $$ and for each $j , j' \in \{1,2,...,n\}$, we denote the canonical matrix units for $\mathbb R^{m \times m}$ by $E_{j,j'} := e_j \otimes (e_{j'})^*.$ With these basic properties of the $2$-coherence tensor established, next we compute $\mathbb E \left[ \left({\mathcal{\bf Q}^{(1)}_{{m} } }(\omega) \right)^{\otimes 2} \right]$ and, in particular, provide the desired concrete expression of the second tensor embedding. \[prop\_Q2\] We have have ${{\mathbf V}^{(2)}} = {{\bf K}^{(2)}_{{m} } } - {{\frac{1}{m^2}} {\bf I}_m \otimes {\bf I}_m}$; in particular, the second embedding is given by $${\mathcal{\bf Q}^{(2)}_{{m} } }: {{\mathcal S}\left(\mathbb{R}^{\scriptscriptstyle m}\right)} \rightarrow {\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}^{\otimes 2}: x \mapsto \left({\mathcal{\bf Q}^{(1)}_{{m} } }(x)\right) \otimes \left({\mathcal{\bf Q}^{(1)}_{{m} } }(x)\right)^* - {{\bf K}^{(2)}_{{m} } } + {{\frac{1}{m^2}} {\bf I}_m \otimes {\bf I}_m}.$$ Upon the expansion of ${\mathcal{\bf Q}^{(1)}_{{m} } }(\omega)^{\otimes 2}$, $${\mathcal{\bf Q}^{(1)}_{{m} } }(\omega) \otimes {\mathcal{\bf Q}^{(1)}_{{m} } }(\omega) = (\omega \otimes \omega^*)^{\otimes 2} - {\frac{1}{m}} {\bf I}_m\otimes \omega \otimes \omega^* -{\frac{1}{m}} \omega \otimes \omega^* \otimes {\bf I}_m + {{\frac{1}{m^2}} {\bf I}_m \otimes {\bf I}_m},\,$$ computing the expectation term by term gives $${{\mathbf V}^{(t)}} = \mathbb E[ {\mathcal{\bf Q}^{(1)}_{{m} } }(\omega) \otimes {\mathcal{\bf Q}^{(1)}_{{m} } }(\omega) ] = {{\bf K}^{(2)}_{{m} } } -{\frac{2}{m^2}} {\bf I}_m \otimes {\bf I}_m +{{\frac{1}{m^2}} {\bf I}_m \otimes {\bf I}_m}\, ,$$ where we have used $\mathbb E[ \omega\otimes \omega^*] = {\scriptstyle\frac{1}{m}} {\bf I}_m$. Simplifying yields the claimed identity. The orthogonality condition implied by Theorem \[th\_orth\_cond\] and Corollary \[cor:QKorth\] for the second tensor embedding thus reads as follows. \[cor:QKorth\] For $t=2$, $${\left\langle {\mathcal{\bf Q}^{(2)}_{{m} } }(x), {{\mathbf V}^{(2)}}\right\rangle_{\scriptscriptstyle\! HS}} ={\left\langle {\mathcal{\bf Q}^{(2)}_{{m} } }(x), {{\bf K}^{(2)}_{{m} } } - {{\frac{1}{m^2}} {\bf I}_m \otimes {\bf I}_m}\right\rangle_{\scriptscriptstyle\! HS}}=0 .$$ In the following lemma, we compute the Hilbert Schmidt inner product between an arbitrary $1$-embedded vector, ${\mathcal{\bf Q}^{(1)}_{{m} } }(x)$, and the $2$-coherence tensor, ${{\bf K}^{(2)}_{{m} } }$ and show that ${\mathcal{\bf Q}^{(2)}_{{m} } }$ is indeed a norm-preserving embedding. Afterward, we show how the inner products between the embedded vectors relate to cosine set of the original frame. Finally, we present the desired equation, which governs the relationship between a frame’s cosine set and the corresponding set of signed angles between the higher-dimensional embedded vectors. We re-express this governing equation from Theorem \[thm\_tier2\] in terms of the ambient dimension, $m$. We conclude by combining this embedding with Rankin’s bound in order to characterize Grassmannian frames. \[thm:main\] Given a frame $\mathcal F$ of $n$ vectors in ${\mathbb R}^m$, with $m \ge 2$, $\frac{n-{{\mathbf d}^{(1)}_{{m}}}}{n-1} \ge \frac{{{\mathbf d}^{(1)}_{{m}}}}{(m-1)^2}$, and $\left\{{\mathcal{\bf Q}^{(2)}_{{m} } }(f): f \in {\mathcal F}\right\}$ forms a simplex in a subspace of ${\mathcal B_{\scriptscriptstyle\mathrm{SA}}( \mathbb{R}^{m})}^{\otimes 2}$, then $\mathcal F$ is Grassmannian. Moreover, if $n> {{\mathbf d}^{(2)}_{{m}}}+1$ and $(m-1)^2 \ge {{\mathbf d}^{(1)}_{{m}}}$, and the inner products between pairs of $\left\{{\mathcal{\bf Q}^{(2)}_{{m} } }(f): f \in {\mathcal F} \right\}$ are non-positive, then the frame is Grassmannian. If $\frac{n-{{\mathbf d}^{(1)}_{{m}}}}{n-1} \ge \frac{{{\mathbf d}^{(1)}_{{m}}}}{(m-1)^2} $, then by estimating ${{\mathbf d}^{(1)}_{{m}}} > (m-1)^2/2$, we have $n > 2{{\mathbf d}^{(1)}_{{m}}} - 1$ and with ${{\mathbf d}^{(1)}_{{m}}} \ge 2$, we get $n >{{\mathbf d}^{(1)}_{{m}}} + 1$, so the orthoplex bound holds, $\max_{j \ne l} |\langle f_j, f_l \rangle |^2 \ge \frac 1 m$. Because the polynomial $p(x) = \frac{m^2{{\mathbf d}^{(1)}_{{m}}}}{({{\mathbf d}^{(1)}_{{m}}}-1)(m-1)^2}\left((x- \frac 1 m)^2 - ({{\mathbf a}_{{m}}}- \frac{1}{m^2})\right)$ is decreasing on $[0, \frac 1 m]$ and increasing on $\left[\frac 1 m, \infty\right)$, it follows that $$\max_{j \ne l} p\left( |\langle f_j, f_l \rangle |^2\right) \le \max \left\{p(0), p(\max_{j \ne l} |\langle f_j, f_l \rangle |^2) \right\} \, .$$ If $\frac{n-{{\mathbf d}^{(1)}_{{m}}}}{n-1} \ge \frac{{{\mathbf d}^{(1)}_{{m}}}}{(m-1)^2} $, then $ p(0) \le - \frac{1}{n-1}$, and Rankin’s bound implies that $$p\left(\max_{j \ne l} |\langle f_j, f_l \rangle |^2\right) \ge - \frac{1}{n-1}.$$ Hence, if $\left\{{\mathcal{\bf Q}^{(2)}_{{m} } }(f): f \in {\mathcal F}\right\}$ forms a simplex then equality is achieved and the frame is Grassmannian. We continue with the more restrictive assumption $n>{{\mathbf d}^{(2)}_{{m}}}+1$, where we know Rankin’s strengthened bound holds. In this case, assuming $(m-1)^2 \ge {{\mathbf d}^{(1)}_{{m}}}$ implies $p(0) \le 0$, which, by Rankin’s bound, implies $p\left(\max_{j \ne l} |\langle f_j, f_l \rangle |^2\right) \ge 0$ and if all the Hilbert-Schmidt inner products between pairs of $\{{\mathcal{\bf Q}^{(2)}_{{m} } }(f): f \in {\mathcal F}\}$ are non-positive, then equality holds and the frame is Grassmannian. Examples of high redundancy frames with low coherence arising from the second embedding ======================================================================================= To conclude this work, we present examples of frames with low coherence that arise from second tensor embedding and discuss their interesting structural properties. A set of orthonormal bases, $\left\{\mathcal B_j\right\}_{j=1}^k$ for $\mathbb R^m$ are said to be [**mutually unbiased**]{} if $|\langle x, y \rangle|^2 = \frac 1 m$ for every $x \in \mathcal B_j, y \in \mathcal B_l$ with $j \neq l$. The existence of three such bases in $\mathbb R^4$ is well-known [@ConwayHardinSloane1996; @Appleby2009; @MR3557826] and it is also known that their union is a Grassmannian frame [@ConwayHardinSloane1996; @Appleby2009; @MR3557826], in accordance with the conditions of Theorem \[thm\_t1\]. Out of curiousity, we fed this system of vectors through equation from Corollary \[cor\_realtier2\] and discovered that the embedded bases, $\left\{ \mathcal B_j^{(2)} \right\}_{j=1}^3$, have the following peculiar property. Given any choice of $j \in \{1,2,3\}$ and any vector $T^{(2)}_j \in \mathcal B_j^{(2)}$, we observe that orthogonal vectors remain orthogonal when embedded and $$\Big\{ \left\langle T^{(2)}_j , T^{(2)}_l \right\rangle : T^{(2)}_l \in \mathcal B^{(2)}_l, l \in \{1,2,3\}, l\neq j \Big\} = \Bigg\{ - \frac 1 8 \Bigg\},$$ so that each embedded vector resembles the vertex of a 9-simplex relative to the eight vectors coming from the other two embedded bases. Assisted by Sloane’s database of putatively optimal packings [@sloanetbl], we confirmed numerically that the second embedding maps Sloane’s example of 16 vectors in $\mathbb R^3$ into a packing of 16 $\frac{\pi}{2}$-caps. After a helpful discussion with Dustin Mixon, we then ascertained that this example corresponds to the $16$ lines passing through the antipodal vertices of a [**biscribed pentakis dodecahedron**]{}. Analytic coordinate representations of the $32$ vertices of this polytope can be found at [*Visual Polyhedra*]{} [@vispoly], an online database of various exotic polyhedra in $\mathbb R^3$. After discarding antipodal points, the remaining $16$ vertices correpond to a $6$-angular unit norm tight frame, $\mathcal F$, for $\mathbb R^3$ with cosine set $$\Theta_{\mathcal F}= \left\{ \scriptstyle \sqrt{ \frac{1}{15} (5 - 2 \sqrt 5)}, \frac{1}{\sqrt 5}, \frac 1 3, \sqrt{\frac{7}{15}}, \frac{\sqrt 5}{3}, \sqrt{ \frac{1}{15} (5 + 2 \sqrt 5)} \right\}. $$ Passing these values through the equation from Corollary \[cor\_realtier2\] shows that the 16 frame vectors of $\mathcal F$ map to 16 vectors, $\mathcal T = \{T_j\}_{j=1}^{16}$, on the sphere in $\mathbb R^{14}$ for whose (signed) cosine set is $$\left\{ \langle T^{(2)}_j, T^{(2)}_l \rangle : j\neq l \right\} = \{-1/5, -1/9, 0\},$$ meaning the embedded vectors form the centers of an optimal cap-packing according to the second of Rankin’s conditions in Theorem \[thm\_rankin\]. Unfortunately, the parameters of this example do not satisfy the sufficiency conditions from Theorem \[thm:main\], so we may not state with certainty this is indeed a Grassmannian frame; however, curiously, we have also observed that the first embedding maps this frame into a set of vectors, $\left\{T^{(1)}_j\right\}_{j=1}^{16} \subset \mathbb R^5$, which forms a tight Grassmannian frame, characterized by the original orthoplex bound. The optimal incoherence of this intermediate frame has recently been observed by Fickus, Jasper, and Mixon [@2017arXiv170701858F]. Given the evidence, we find it reasonable to posit that $\mathcal F$ is likely a Grassmannian frame. If so, then it would follow that $$\mu_{16, 3}(\mathbb R)= \sqrt{ \frac{1}{15} (5 + 2 \sqrt 5)}.$$ The optimal coherence of this example has been verified via the Levenschtein bound [@lev_paper], but we recertify it here in terms of the second tensor embedding. After discarding antipodal vectors from the 240 shortest vectors of the E8 lattice, the remaining $120$ vectors form a unit norm frame, $\mathcal F$, for $\mathbb R^8$ with cosine set, $$\Theta_{\mathcal F} = \{0, 1/2 \}.$$ We verify the assumptions of Theorem \[thm:main\]: ${{\mathbf d}^{(1)}_{{8}}}=35$, so $\frac{120-{{\mathbf d}^{(1)}_{{8}}}}{119} = \frac{85}{119} \ge \frac{35}{49} = \frac{{{\mathbf d}^{(1)}_{{8}}}}{(8-1)^2}$. One can verify that the second tensor embedding maps this frame to a regular $119$-simplex, meaning the embedded vectors correspond to an optimal cap packing according to the first of Rankin’s conditions from Theorem \[thm\_rankin\], thereby verifying that $\mathcal F$ is a Grassmannian frame and $$\mu_{120,8} (\mathbb R)=\frac{1}{2}.$$ [^1]:  This work was partially supported by NSF grant DMS-1412524 and the AMS-Simons Travel grant.

--- abstract: 'We consider a multi-server queueing system under the power-of-two policy with Poisson job arrivals, heterogeneous servers and a general job requirement distribution; each server operates under the first-come first-serve policy and there are no buffer constraints. We analyze the performance of this system in light traffic by evaluating the first two light traffic derivatives of the average job response time. These expressions point to several interesting structural features associated with server heterogeneity in light traffic: For unequal capacities, the average job response time is seen to decrease for small values of the arrival rate, and the more diverse the server speeds, the greater the gain in performance. These theoretical findings are assessed through limited simulations.' author: - | A. Izagirre$^{a}$ and A.M. Makowski$^{b}$\  \ $^a$Univ. of the Basque Country (UPV/EHU), Spain\ $^b$ Department of Electrical and Computer Engineering, and Institute for\ Systems Research University of Maryland, College Park, MD 20742\ bibliography: - 'bibli\_PEVA16.bib' title: 'Light traffic behavior under the power-of-two load balancing strategy: The case of heterogeneous servers. ' --- Parallel servers, power-of-two scheduling, light traffic, heterogeneous servers Introduction {#sec:Introduction} ============ Systems of parallel servers are commonly used to model resource sharing applications. These queueing models have been adopted in classical performance studies for supermarket cashiers, bank tellers and toll booths; they have also appeared in the context of computer systems and communication networks. A basic design issue for such systems is the [*scheduling*]{} of incoming jobs, usually with an eye towards making the average job response time as small as can be. One possible choice is to randomly assign an incoming job to one of the available servers, a strategy which may lead to large delays but which has the advantage of requiring no state information. At the other extreme, the join-the-shortest-queue (JSQ) policy is known to possess certain optimality properties [@Whitt1986], but requires the queue length at each server to be available at the arrival epoch of every job. JSQ and its variants have been extensively studied [@MHB+Book] (and references therein) with most of the work focusing on the [*homogeneous*]{} case when servers have identical service speeds and use the same service discipline. In such cases it is known that job size variability greatly affects average job performance under the first-come first-serve (FCFS) service discipline [@MHB+Book Chapter 24]. However, the impact seems much reduced under the processor-sharing (PS) discipline, with near-insensitivity being reported by @GHBSW. Much work has also been done to explore the [*trade-off*]{} between the information overhead to implement job scheduling and the resulting performance. An interesting alternative which interpolates between random assignment and JSQ is the following policy $SQ(d)$ (for some integer $d\geq~2$): Upon arrival, an incoming job randomly selects $d$ servers from amongst the pool of available servers. The JSQ policy is then applied to these $d$ servers in isolation (with a random tiebreaker) – Here, shortest queue refers to the queue with the fewest jobs but other definitions (say in terms of workload) are possible. This queueing system, sometimes known as the supermarket model, has been studied for some time now with special attention given to the case $d=2$ (from which the terminology power-of-two derives); see the brief historical survey in [@M01 Section 1.1]. Analysis of the supermarket model is challenging because of the coupling between queues induced by local users of JSQ. This is so even when jobs arrive according to a Poisson process, servers are identical FCFS servers, and job requirements are exponentially distributed. In that setting, @M01 and @VDK96 (with $d=2$), independently, resorted instead to studying the limiting system obtained by letting the number of servers go to infinity. Together their results point to a substantial improvement in performance over the case $d=1$ (which corresponds to the random server assignment) [*without*]{} the full overhead of [*global*]{} JSQ. In view of these encouraging results it is natural to inquire whether the policy $SQ(d)$ still provides a performance advantage when servers have [*different*]{} capacities. With $d=2$, @MM took a step in that direction: Following the same limiting strategy as in [@M01; @VDK96] they discuss the average job response time for the $SQ(2)$ model under [*heterogeneous*]{} PS servers (but with a finite number of different server speeds), with Poisson arrivals and a general job requirement distribution. In this paper we consider $SQ(2)$ with [*heterogeneous*]{} FCFS servers, Poisson arrivals and a general job requirement distribution. Instead of looking at the many server asymptotics as in earlier papers, we focus instead on the [*light traffic*]{} regime under a [*fixed*]{} number of servers; this corresponds to the system operating with a very low traffic intensity. Using the framework developed by @RS89, we compute the first and second light-traffic derivatives of the average job response time; see Proposition \[prop:ExpectedR\_k=2\] in Section \[sec:MainResults\]. These derivatives already provide some crude [*structural*]{} insights into the impact that server heterogeneity may have on job performance; see Section \[sec:Discussion\] for a short discussion. For instance, at least in light traffic, the more diverse the server speeds, the greater the gain in performance. Moreover, job performance in $SQ(2)$ is [*not*]{} monotone in the traffic intensity (at least when this traffic intensity is small). A quadratic polynomial approximation” can be constructed on the basis of the first two light-traffic derivatives. While this [*local*]{} approximation cannot be accurate in moderate to heavy traffic regimes, we nevertheless use it as a benchmark against simulations to illustrate the structural features revealed through the light traffic calculations. In Section \[sec:LimitedSimulations\] we further explore some of the theoretical findings with the help of limited simulations. The paper is organized as follows: The model and assumptions are introduced in Section \[subsec:ModelAssumptions\], and the evaluation of the first two derivatives is presented in Section \[subsec:FirstTwoDerivatives\]. Various comments on and implications of the results are given in Section \[sec:Discussion\], while in Section \[sec:LimitedSimulations\] we illustrate some of the theoretical findings with the help of limited simulations. In Section \[sec:ReviewLightTrafficTheory\] we summarize the needed elements of the light traffic theory we use. In Section \[sec:n=0\] we evaluate the light-traffic response time of a tagged customer, the so-called $n=0$ case in the Reiman-Simon theory. We start the technical discussion in Section \[sec:AuxiliaryResult\] with an auxiliary result that greatly simplifies later computations of the first and second light-traffic derivatives. The first derivative is computed in Section \[sec:n=1\]. The calculations of the second derivative start in Section \[sec:n=2\], and are developed through Sections \[sec:ProofPropExpectedR\_2\]–\[sec:ProofLemmaCase2B+2\]. Additional calculations are given in the Appendices A–E. Main results {#sec:MainResults} ============ All random variables (rvs) under consideration in this paper are defined on the same sufficiently large probability triple $(\Omega , {\cal A} , \mathbb{P})$; its construction is standard and is omitted in the interest of brevity. Probabilistic statements are made with respect to this probability measure $\mathbb{P}$, and we denote the corresponding operator by $\mathbb{E}$. Throughout let $\sigma$ denote an $\mathbb{R}_+$-valued rv which is distributed according to some probability distribution function $F: \mathbb{R}_+ \rightarrow [0,1]$, so that $F(x) = {{\mathbb{P}}\left[{ \sigma \leq x }\right]}$ ($x \geq 0$). We assume at minimum that ${{\mathbb{E}}\left[{ \sigma }\right]} < \infty $. With any discrete set $S$ which is non-empty and finite (so $0 < |S| < \infty$), we write $U \sim \mathcal{U}(S)$ to indicate that the rv $U$ is uniformly distributed over $S$ (under $\mathbb{P}$), namely $${{\mathbb{P}}\left[{ U = u }\right]} = \frac{1}{|S|} , \quad u \in S.$$ Model and assumptions {#subsec:ModelAssumptions} --------------------- The system comprises $K\geq 2$ parallel servers labelled $k=1, \ldots , K$. Server $k$ has capacity $C_k $ (bytes/sec.), is attended by an infinite capacity buffer and operates in a FCFS manner. Jobs arrive according to a Poisson process $\{ A(t), \ t \geq 0 \}$ of rate $\lambda > 0$ with arrival epochs $\{ T_n, \ n=0,1, \ldots \} $ – By convention we take $T_0=0$. For each $n=0,1, \ldots $, we refer to the job arriving at time $T_n$ as the $n^{th}$ job; this job brings a random amount of work $\sigma_n$ (bytes). Upon arrival, the $n^{th}$ job is assigned to one of the $K$ servers according to the power-of-two load balancing scheme (with $d=2$): Specifically, this incoming customer randomly selects a pair $\Sigma_n$ of distinct servers from the pool of $K$ servers. The JSQ policy is then used in isolation with these two servers; ties are broken randomly (but other choices are possible). As usual, the Poisson arrival process $\{ A(t), \ t \geq 0 \}$, the sequence of job requirement rvs $\{ \sigma_n, \ n=0,1, \ldots \}$ and the sequence of server selection rvs $\{ \Sigma_n, \ n=0,1, \ldots \}$ are mutually independent collections of rvs. We also assume the following: (i) The rvs $\{ \sigma_n, \ n=0,1, \ldots \}$ are i.i.d. rvs distributed according to the probability distribution $F$ – The rv $\sigma$ introduced earlier is therefore a generic element of this sequence of i.i.d. rvs; and (ii) The server selection rvs $\{ \Sigma_n, \ n=0,1, \ldots \}$ are i.i.d. rvs, each of which is uniformly distributed over the collection of unordered pairs drawn from $\{1, \ldots , K \}$. Thus, with $\mathcal{P}_2(K)$ denoting the collection of unordered pairs drawn from $\{1, \ldots , K \}$, we have $\Sigma_n \sim \mathcal{U}(\mathcal{P}_2(K))$ with $${{\mathbb{P}}\left[{ \Sigma_n = T }\right]} = \frac{1}{ {K \choose 2 }}, \quad \begin{array}{c} T \in {\cal P}_2(K), \\ n =0,1, \ldots \\ \end{array}$$ Assuming the system to be initially empty (for sake of convenience), for each $n=0,1, \ldots $, let $R_{n,\lambda}$ denote the response time of the $n^{th}$ job when the arrival rate is $\lambda$. The stationary response time of a job when the arrival rate is $\lambda$ is denoted $R_\lambda$. The existence of $R_\lambda$, possibly as an $[0,\infty]$-valued rv, can be established through classical semi-Markovian methods; details are omitted in the interest of brevity. We set $$R(\lambda ) = {{\mathbb{E}}\left[{ R_\lambda}\right]}.$$ We expect ${{\mathbb{E}}\left[{R_\lambda}\right]} < \infty$ over some non-degenerate interval $(0, \lambda^\star)$ for some finite $\lambda^\star > 0$, in which case $R_{n,\lambda} \Longrightarrow_n R_\lambda$ with $\Longrightarrow_n$ denoting weak convergence (also known as convergence in distribution) with $n$ going to infinity; see also [@M01 Section 2.1, Lemma 1] and [@MM Lemma 6]. In what follows we shall not be concerned with this issue any further since we are mainly interested in the situation where $\lambda$ is very small (vanishingly so). Evaluating the first two derivatives {#subsec:FirstTwoDerivatives} ------------------------------------ Light-traffic analysis considers the performance of the system for small values of the arrival rate $\lambda > 0$. In that regime a so-called [*light-traffic approximation*]{} can often be constructed on the basis of the following Taylor series expansion argument: Assume that for some positive integer $L$, the $L$ first derivatives of the function $\lambda \rightarrow R(\lambda)$ all exist in a neighborhood $(0,\lambda_\star)$ with $\lambda_\star > 0$. Whenever $ 0 < x, \lambda < \lambda_\star $, Taylor’s formula $$R(x+\lambda) = R(x) + \lambda R^{(1)} (x) + \frac{\lambda^2}{2!} R^{(2)} (x) + \ldots + \frac{\lambda^L}{L!} R^{(L)} (x) + \mbox{Remainder}_L(x;\lambda)$$ holds where we use the notation $$R^{(\ell)} (x) = \frac{d^\ell R}{d\lambda^\ell}(\lambda) \Big |_{\lambda=x} , \quad \ell =1, \ldots , L .$$ It is not important for the discussion what is the exact form taken by the remainder term $\mbox{Remainder}_L(x;\lambda)$. Assume further that the limits $$R (0+) = \lim_{\lambda \downarrow 0} R(\lambda) \quad \mbox{and} \quad R^{(\ell)} (0+) = \lim_{\lambda \downarrow 0} \frac{d^\ell R}{d\lambda^\ell}(\lambda) , \quad \ell =1, \ldots , L \label{eq:LimitDerivatives}$$ were all to exist (in $\mathbb{R}$) – We refer to the quantities in the(\[eq:LimitDerivatives\]) as [*light-traffic derivatives*]{}. Then it is natural to use the polynomial $R_{\rm App}(\lambda): (0,\infty) \rightarrow \mathbb{R}$ given by $$R_{\rm App} (\lambda) = R(0+) + \lambda R^{(1)} (0+) + \frac{\lambda^2}{2} R^{(2)} (0+) + \ldots + \frac{\lambda^L}{L!} R^{(L)} (0+) , \quad \lambda > 0 \label{eq:LT_Approximation}$$ as a possible light-traffic approximation; this prompts us to write $$R(\lambda) \simeq R_{\rm App} (\lambda) , \quad \lambda \simeq 0. \label{eq:LT_Approximation2}$$ The light traffic analysis presented here uses an approach proposed by @RS89 to compute successive light-traffic derivatives in the sense of (\[eq:LimitDerivatives\]). It requires that some [*admissibility*]{} condition be satisfied. Following the discussion in [@RS89 Appendix A] we assume that the generic rv $\sigma$ satisfies the condition $${{\mathbb{E}}\left[{ e^{t \sigma} }\right]} < \infty \label{eq:ExponentialMoment}$$ for some $t > 0$. This finite exponential moment condition on $F$ entails admissibility; it is likely stronger than needed but its purpose here is to provide a convenient framework where calculations can be justified. In particular it ensures the requisite differentiability of $\lambda \rightarrow R(\lambda)$ where finite. We compute the first two derivatives in light traffic; these results were announced in the conference paper [@IzagirreMakowski] without proofs. [ *Under the enforced assumptions, the limit $\lim_{\lambda \downarrow 0} R(\lambda ) $ exists and is given by $$R(0+) \equiv \lim_{\lambda \downarrow 0} R(\lambda ) = \frac{\Gamma}{K} \cdot {{\mathbb{E}}\left[{ \sigma}\right]} \label{eq:ExpectedR_k=0}$$ with $$\Gamma = \sum_{k=1}^K \frac{1}{C_k} \label{eq:Gamma}$$* ]{} \[prop:ExpectedR\_k=0\] We now turn to the first derivative. [*Under the enforced assumptions, the function $\lambda \rightarrow R(\lambda)$ is differentiable in a small neighborhood of $\lambda = 0$. Furthermore, $$R^{\prime} (0+) \equiv \lim_{\lambda \downarrow 0} \frac{dR}{d\lambda} (\lambda ) = \frac{1}{K-1} \left( \left ( \frac{\Gamma}{K} \right )^2 - \frac{1}{K} \sum_{k=1}^K \dfrac{1}{C_k^2} \right) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^2 . \label{eq:ExpectedR_k=1}$$* ]{} \[prop:ExpectedR\_k=1\] The third result concerns the second derivative. [ *Under the enforced assumptions, the function $\lambda \rightarrow R(\lambda)$ is twice differentiable in a small neighborhood of $\lambda = 0$. Furthermore, $$R^{\prime \prime} (0+) \equiv \lim_{\lambda \downarrow 0} \frac{d^2R}{d\lambda^2} (\lambda ) = \dfrac{2}{K^2(K-1)^2} \left ( \dfrac{\Gamma^3}{K} - 2\Gamma\sum_{k=1}^K \dfrac{1}{C_k^2} + K\sum_{k=1}^K\dfrac{1}{C_k^3} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^3. \label{eq:ExpectedR_k=2}$$* ]{} \[prop:ExpectedR\_k=2\] Discussion {#sec:Discussion} ========== #### A probabilistic interpretation The results of Propositions \[prop:ExpectedR\_k=0\]-\[prop:ExpectedR\_k=2\] can be expressed more compactly with the help of the following probabilistic interpretation: Let $X \equiv X(C_1, \ldots , C_K)$ denote a rv uniformly distributed over the set of values $\frac{1}{C_1}, \ldots , \frac{1}{C_K}$, i.e., $X \sim \mathcal{U}(\{ \frac{1}{C_1}, \ldots , \frac{1}{C_K} \})$ with $${{\mathbb{P}}\left[{ X = \frac{1}{C_1} }\right]} = \ldots = {{\mathbb{P}}\left[{ X = \frac{1}{C_K} }\right]} = \frac{1}{K}.$$ With this notation it is easy to check that $${{\mathbb{E}}\left[{ X^p}\right]} = \frac{1}{K} \sum_{k=1}^K \frac{1}{C^p_k}, \quad p \geq 0.$$ The expressions (\[eq:ExpectedR\_k=0\]), (\[eq:ExpectedR\_k=1\]) and (\[eq:ExpectedR\_k=2\]) can now be rewritten more compactly as $$R(0+) = {{\mathbb{E}}\left[{X}\right]} \cdot {{\mathbb{E}}\left[{\sigma}\right]} , \label{eq:ExpectedR_k=0Alternate}$$ $$R^{\prime} (0+) = - \frac{1}{K-1} {\rm Var}[X] \cdot \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^2 \label{eq:ExpectedR_k=1Alternate}$$ and $$\begin{aligned} R^{\prime \prime} (0+) &=& \dfrac{2}{(K-1)^2} \left ( \left ( {{\mathbb{E}}\left[{X}\right]} \right )^3 - 2 {{\mathbb{E}}\left[{X}\right]} \cdot {{\mathbb{E}}\left[{X^2}\right]} + {{\mathbb{E}}\left[{X^3}\right]} \right ) \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^3 \nonumber \\ &=& \dfrac{2}{(K-1)^2} \left ( {{\mathbb{E}}\left[{X^3}\right]} - \left ( {{\mathbb{E}}\left[{X}\right]} \right )^3 - 2 {{\mathbb{E}}\left[{X}\right]} \cdot \rm{Var}[ X ] \right ) \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^3 , \label{eq:ExpectedR_k=2Alternate} \end{aligned}$$ respectively. #### Equal capacities From (\[eq:ExpectedR\_k=1Alternate\]) it follows that $R^{\prime} (0+) \leq 0$, with $R^{\prime} (0+) = 0$ if and only if ${\rm Var}[X] =~0$, or equivalently, $C_1 = \ldots = C_K $. In that case all $K$ servers have the same capacity, and we also have $R^{\prime \prime} (0+) = 0 $, whence $$R(\lambda) = \frac{{{\mathbb{E}}\left[{\sigma}\right]} }{C}+ o(\lambda^2)$$ assuming the existence of a third derivative (via either the Lagrange or Cauchy form of the remainder). #### Unequal capacities When the capacities are different, then $R^{\prime} (0+) < 0$ and $R(\lambda) $ is [*decreasing*]{} for small values of $\lambda $. This is a somewhat unexpected finding because most queueing systems are monotone” in the sense that increasing the traffic intensity $\lambda$ results in an increase in a performance metric such as the average job response time. This fact can be explained as follows: On the average, a job entering an empty system experiences a response time given by $R(0+)$ since the scheduling policy $SQ(2)$ assigns it to [*any*]{} of the $K$ servers with probability $\frac{1}{K}$. However, when the servers have different capacities, the assigned server may not have been the fastest, therefore making it possible for subsequent jobs to be served by faster servers by the luck of the draw. This will result in a decrease in the average job response time if the traffic intensity increases slightly but still allows for some faster server to be available with some non-negligible probability. #### How much of a decrease? We see from (\[eq:ExpectedR\_k=1Alternate\]) that the decrease in the average job response time will be more pronounced the larger the variance $\rm{Var}[X]$ of the rv $X$. It is therefore natural to wonder which set of capacity values $C_1, \ldots , C_K$ yield the largest value for this variance $\rm{Var}[X]$ under a given value for ${{\mathbb{E}}\left[{X}\right]}$, say ${{\mathbb{E}}\left[{X}\right]} = \frac{\Gamma}{K}$ for some $\Gamma > 0$. As we assess the range of $\rm{Var}[X]$ under this constraint on ${{\mathbb{E}}\left[{X}\right]}$ in Appendix A, we conclude that $$0 \leq \rm{Var}[X] < \frac{\Gamma^2}{K} - \left ( \frac{\Gamma}{K} \right )^2 = (K-1) \left ( \frac{\Gamma}{K} \right )^2 \quad \mbox{with ${{\mathbb{E}}\left[{X}\right]} = \frac{\Gamma}{K}$}. \label{eq:VAR}$$ As mentioned earlier, the lower bound is achievable by the vector of capacities given by $${{\mbox{\boldmath{$C$}}}}_\star = \left ( \frac{K}{\Gamma} , \ldots , \frac{K}{\Gamma} \right ). \label{eq:BalancedCapacities}$$ While the upper bound is [*not*]{} achievable by any vector of capacities satisfying the constraint, it is however tight in the following sense: For each $k=1, \ldots , K$, let the vector ${{\mbox{\boldmath{$e$}}}}_k$ denote the $K$-dimensional vector $(\delta_{k\ell})$ with all zero entries except in the $k^{th}$ position where it is one. The vectors of capacities given by $${{\mbox{\boldmath{$C$}}}}_{k,a} = \frac{1}{a\Gamma} {{\mbox{\boldmath{$e$}}}}_k + \frac{K-1}{(1-a) \Gamma} \sum_{\ell=1, \ \ell \neq k}^K {{\mbox{\boldmath{$e$}}}}_\ell, \quad \begin{array}{c} k=1, \ldots , K \\ 0 < a < 1 \\ \end{array} \label{eq:C_ak}$$ can approach the upper bound value arbitrarily close by letting $a$ go to $1$; this is shown in Appendix A. The lower bound is implemented by the most balanced capacity assignment (\[eq:BalancedCapacities\]) under the constraint ${{\mathbb{E}}\left[{X}\right]} = \frac{\Gamma}{K}$, whereas the upper bound is achieved, albeit asymptotically, by capacity assignments (\[eq:C\_ak\]) that are as imbalanced as they can be under the constraint. In the limit these assignments correspond to $K-1$ servers that are infinitely fast with the remaining slow" one with finite capacity. #### Only ${{\mathbb{E}}\left[{ \sigma }\right]}$ matters The two first derivatives at $\lambda~=~0+$ depend only on the first moment of $\sigma$, and could be read as a form of insensitivity in light traffic. This is in sharp contrast with other systems where the first light-traffic derivative depends on ${{\mathbb{E}}\left[{\sigma^2}\right]}$, e.g., $M|G|1$-like queues [@RS88] and the discriminatory processor sharing model [@IAM]. This is rather unexpected because the variance of $\sigma$ is known to be a key factor in shaping JSQ performance with homogeneous servers under FCFS scheduling [@MHB+Book Chapter 24]. See next item for a possible explanation. #### FCFS vs. PS Proposition \[prop:ExpectedR\_k=2\] was established under the assumption that the servers operate under the FCFS discipline. It is easy to see that both (\[eq:ExpectedR\_k=0\]) and (\[eq:ExpectedR\_k=1\]) (but not (\[eq:ExpectedR\_k=2\])) are still valid if the servers all use the PS discipline: This is because in the cases $n=0$ and $n=1$ the tagged job will not share a server with another job under either discipline; see Section \[sec:n=0\] and Section \[sec:n=1\]. However, this changes for the case $n=2$ that involves three customers. That the variability of $\sigma$ seems to play little role in light traffic is therefore consistent with the aforementioned fact that performance under the PS discipline is nearly insensitive to service variability [@GHBSW]. Limited simulations {#sec:LimitedSimulations} =================== [.24]{} {width="99.00000%"} [.24]{} {width="92.00000%"} [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} As explained in Section \[subsec:FirstTwoDerivatives\] the second order polynomial $$R_{\rm App} (\lambda) = R(0+) + \lambda R^{\prime } (0+) + \frac{\lambda^2}{2} R^{\prime \prime} (0+), \quad \lambda \geq 0 \label{SQ2 eq:LT_App_degree2}$$ can be used as a local approximation to $R(\lambda)$ for small $\lambda$. As already pointed out by @RS88 [@RS89], without additional information (e.g., heavy traffic information), we should not expect $R_{\rm App}(\lambda)$ to act as an accurate proxy for $R(\lambda)$ in medium to heavy traffic. This lack of accuracy is certainly apparent in the simulation results reported below. We have carried out simulations for different distributions of $\sigma$, all with unit mean, namely hyperexponential (obtained by mixing the exponential rvs ${\rm Exp}(1/2)$ and ${\rm Exp}(2)$ with probability $1/3$ and $2/3$, respectively), exponential ${\rm Exp(1)}$ (of parameter $1$), Weibull (with shape parameter $2$ and scale parameter $\Gamma(3/2)^{-1}$) and deterministic. The simulation results are based on averaging $10$ runs with each run comprising $10^5$ busy periods. A busy period is defined as the interval of time between two consecutive time epochs when the system becomes empty, such points being regenerative points for the stochastic process of interest. We have verified that the simulation results obtained for a system with $K=100$ homogeneous servers and exponential service requirements agree with those given by @M01 [Table 1]. [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} {width="99.00000%"} We are interested in the behavior of the average job response time in lightly loaded situations, and stability is therefore not a concern here as mentioned earlier. Three different scenarios were explored. In Scenarios 1 and 2 there are two types of servers, namely slow servers with capacity $C_{\text{slow}}$ bytes/sec and fast servers with capacity $C_{\text{fast}}$ bytes/sec. In Scenario 3 all the servers have the same capacity: - Scenario 1: $K=10$ servers. $5$ slow servers with capacity $C_{\text{slow}}=2$ bytes/sec and $5$ fast servers with capacity $C_{\text{fast}}=10$ bytes/sec. See Figure \[SQ2 fig:Sce1\]. - Scenario 2: $K=100$ servers. $50$ slow servers with capacity $C_{\text{slow}}=2$ bytes/sec and $50$ fast servers with capacity $C_{\text{fast}}=10$ bytes/sec. See Figure \[SQ2 fig:Sce2\]. - Scenario 3: $K=10$ servers with $C_1 = \ldots = C_{10} = 10$ bytes/sec. See Figure \[SQ2 fig:Sce3\]. In the figures we use $R_{\rm Sim}$ to denote the average job response time obtained by simulation. Also the subscript $p\star$ in the quantities $R_{p\star}(0+)$ and $R_{p\star}^{\prime} (0+) $ refers to Scenario $p$ under distribution $\star$ where $\star$ corresponds to the hyper-exponential (H), exponential (E), Weibull (W) or deterministic (D) distribution, respectively. Let ${\rm CV}_{p\star} $ denote the coefficient of variation corresponding to Scenario $p$ under distribution $\star = H, E,W,D$. Then ${\rm CV}_{iH}=1.4$, ${\rm CV}_{iE}=1$, ${\rm CV}_{iW}=0.52$ and ${\rm CV}_{iD}=~0$ for $i=1,2,3$. The simulations do confirm the structural insights gleaned from the light traffic derivatives for non-homogeneous servers; see Section \[sec:Discussion\]: (i) For all distributions, the average job response time decreases as $\lambda$ increases over a small neighborhood of $\lambda = 0$; (ii) Over that small interval, performance seems nearly insensitive to the variability of $\sigma$ (as measured by its coefficient of variation). Although in Scenario 1 and Scenario 2 there is an equal proportion of slow and fast servers, with $R_{p\star}(0+)=0.3000$ for $p=1,2$ and for $\star=H, E, W, D$, the impact of the variability in server speeds is seen to diminish with increasing $K$ since $R^{\prime}_{1\star} (0+)=-0.0044$ and $R^{\prime}_{2\star} (0+)= -4.0404\cdot 10^{-4}$ for $\star=H, E, W, D$. Figure \[SQ2 fig:Sce2\] is a zoom of Figure \[SQ2 fig:Sce2\_app\] that displays only this common approximation $R_{\rm App}(\lambda)$. Although in Figure \[SQ2 fig:Sce2\_app\] the response time seems to be a straight line in a small interval of $\lambda$, after a while it also increases. Since ${{\mathbb{E}}\left[{\sigma}\right]} = 1 $ for all four cases $\star=H,E,W,D$, and the approximation (\[SQ2 eq:LT\_App\_degree2\]) that we use depends only on the first moment, Figure \[SQ2 fig:Sce2\_app\] is the same for all distributions considered here. In Figure \[SQ2 fig:Sce3\] we observe the aforementioned property for homogeneous servers; $R_{\rm App} (\lambda) $ becomes a constant line while the simulation results show that the average response time of a job is increasing. [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} [.24]{} {width="99.00000%"} Review of the light traffic theory à la Reiman-Simon {#sec:ReviewLightTrafficTheory} ==================================================== The light traffic analysis presented here uses an ingenious approach proposed by @RS89 to compute successive light-traffic derivatives in the sense of (\[eq:LimitDerivatives\]): Imagine that the system starts at $t=-\infty$, so that its stationary regime will have been reached at time $t=0$. Enters a [*tagged*]{} job at time $t=0$ whose expected response time therefore coincides with the expected stationary response time. With this in mind, the $n^{th}$ derivative of the expected stationary response time at $\lambda = 0+$ (namely (\[eq:LimitDerivatives\])) is then shown to be computable in terms of the expected response time of the tagged customer in a [*scenario*]{} where [*exactly*]{} $n$ jobs (other than the tagged job) are allowed into the system. Details are outlined next. The framework {#subsec:Framework} ------------- On the way to describing the Reiman-Simon approach to light traffic we find it convenient to introduce the following terminology and notation: With $t$ in $\mathbb{R}$, a job arriving at time $t$, hereafter referred to as a $t$-job, has two rvs $\sigma_t$ and $\Sigma_t$ associated with it – The $\mathbb{R}_+$-valued rv $\sigma_t$ stipulates the amount of work (in bytes) requested by the $t$-job from the system, while the rv $\Sigma_t$ is an (unordered) pair of servers from amongst the $K$ available servers. The $t$-job is assigned to a server $\nu_t$ selected in $\Sigma_t$ according to the power-of-two policy (with a random tie-breaker). We shall refer to the rvs $(\sigma_t,\Sigma_t)$ as the characteristic pair of the $t$-job. As expected, we sometimes refer to the $0$-job with characteristics $(\sigma_0,\Sigma_0)$ as the [*tagged*]{} job. The Reiman-Simon approach to light traffic focuses on the performance of this tagged job under scenarios of increasing complexity. To define them, fix $n=0,1, \ldots $. Interpret every $n$-uple $(t_1, \ldots , t_n)$ in $\mathbb{R}^n$ as the arrival epochs of $n$ jobs into the system. For each $i=1, \ldots , n$, we lighten the notation by denoting the characteristic pair $(\sigma_{t_i}, \Sigma_{t_i})$ of the $t_i$-job arriving at time $t_i$ simply by $(\sigma_i, \Sigma_i)$. Throughout the following conditions are assumed to be enforced: 1. The rvs $\{ \sigma_0, \sigma_{1}, \ldots , \sigma_{n} \}$ are i.i.d. $\mathbb{R}_+$-valued rvs, each distributed according to the probability distribution $F$, namely $${{\mathbb{P}}\left[{ \sigma_{i} \leq x }\right]} = F(x), \quad \begin{array}{c} x \geq 0 \\ i=0,1, \ldots , n. \\ \end{array}$$ 2. The rvs $\{ \Sigma_0, \Sigma_1, \ldots , \Sigma_n \}$ are i.i.d. ${\cal P}_2(K)$-valued rvs, each of which is uniformly distributed on ${\cal P}_2(K) $ with $${{\mathbb{P}}\left[{ \Sigma_i = T }\right]} = \frac{1}{ {K \choose 2 }}, \quad \begin{array}{c} T \in {\cal P}_2(K) \\ i=0,1, \ldots , n. \\ \end{array}$$ 3. The collections of rvs $\{ \sigma_0, \sigma_1, \ldots , \sigma_n \}$ and $\{ \Sigma_0, \Sigma_1, \ldots , \Sigma_n \}$ are mutually independent We shall also have use for the rvs $\nu^\star_0, \nu^\star_1, \ldots , \nu^\star_n$ associated with the random pairs $\Sigma_0, \Sigma_1, \ldots , \Sigma_n $, and defined in the following manner: For each $i=0,1, \ldots , n$, [*conditionally*]{} on $\Sigma_i$, the rv $\nu^\star_i$ is an $\Sigma_i$-valued rv which is uniformly distributed on $\Sigma_i$ – We shall write $$[ \nu^\star_i | \Sigma_i ] \sim {\cal U}(\Sigma_i).$$ It is always understood that the rvs $\nu^\star_0, \nu^\star_1, \ldots , \nu^\star_n$ are conditionally mutually independent given the $2(n+1)$ rvs $\sigma_0, \sigma_1, \ldots , \sigma_n , \Sigma_0, \Sigma_1, \ldots , \Sigma_n $ with $$[ \nu^\star_i | \sigma_0, \ldots , \sigma_n, \Sigma_0 , \ldots , \Sigma_n ] \sim {\cal U}(\Sigma_i),. \quad i=0,1, \ldots , n.$$ Under the enforced assumptions, we readily conclude that the rvs $\nu^\star_0, \nu^\star_1, \ldots , \nu^\star_n$ are i.i.d. rvs, each of which is uniformly distributed on $\{ 1, \ldots , K \}$ (as shown in Proposition \[prop:ExpectedR\_0\]). Computing the derivatives ------------------------- Fix $n=1,2, \ldots $. For each $(t_1, \ldots , t_n)$ in $\mathbb{R}^n$, let the rv $R_n(t_1, \ldots , t_n)$ denote the response time of the tagged job under the scenario that [*in addition*]{} to the tagged job, only $n$ jobs are allowed to enter the system over $\mathbb{R}$, say at times $t_1, \ldots , t_n$, with characteristic pairs $(\sigma_1, \Sigma_1), \ldots , (\sigma_n,\Sigma_n)$ as defined earlier. Note that $R_n(t_1, \ldots , t_n)$ depends on the rvs $\{ \sigma_0, \sigma_1, \ldots , \sigma_n \}$, $\{ \Sigma_0, \Sigma_1, \ldots , \Sigma_n \}$ and $\{ \nu^\star_0, \nu^\star_1, \ldots , \nu^\star_n \}$ in a complicated manner through the scheduling policy used. We shall write $$\widehat R_n(t_1, \ldots , t_n ) = {{\mathbb{E}}\left[{ R_n(t_1, \ldots , t_n ) }\right]}. \label{eq:R_n}$$ Under some appropriate integrability conditions, Reiman and Simon show that the light-traffic derivatives in the sense of (\[eq:LimitDerivatives\]) can be expressed in terms of the quantities (\[eq:R\_n\]) – Here we consider the cases $n=0,1,2$: Using Theorems 1 and 2 in [@RS89 pp. 29-30] for $n=0,1,2$ we collect the expressions $$R (0+) = \lim_{\lambda \downarrow 0} R (\lambda ) = \widehat R_0 \label{eq:ExpressionFor_n=0}$$ with $\widehat R_0$ defined in Section \[sec:n=0\], $$R^{\prime} (0+) = \lim_{\lambda \downarrow 0} \frac{dR}{d\lambda} (\lambda ) = \int_{\mathbb{R}} \left ( \widehat R_1(t) - \widehat R_0 \right ) dt \label{eq:ExpressionFor_n=1}$$ and $$R^{\prime\prime} (0+) = \lim_{\lambda \downarrow 0} \frac{d^2R}{d\lambda^2} (\lambda ) = \int_{\mathbb{R}} \left ( \int_{\mathbb{R}} ( \widehat R_2(s,t) - \widehat R_1(s) - \widehat R_1(t) + \widehat R_0 ) dt \right ) ds \label{eq:ExpressionFor_n=2}$$ The case $n=0$ {#sec:n=0} ============== The case $n=0$ is slightly different and corresponds to the scenario when besides the tagged customer, no other job enters over the entire horizon $(-\infty, \infty)$. Let $R_0$ denote the response time of the tagged job under these circumstances. Obviously, under the power-of-two scheduling strategy, we have $$R_0 = \frac{ \sigma_0}{C_{\nu_0}} \quad \mbox{with $\nu_0 = \nu^\star_0$} \label{eq:n=0}$$ because in the absence of any other job in the system, the tagged job is necessarily assigned to server $\nu^\star_0$. Somewhat in analogy with earlier notation we write $$\widehat R_0 = {{\mathbb{E}}\left[{ R_0 }\right]}.$$ [*Under the enforced assumptions, the rv $\nu^\star_0$ is uniformly distributed over $\{1, \ldots , K \}$ with $${{\mathbb{P}}\left[{ \nu^\star_0 = k }\right]} = \frac{1}{K}, \quad k=1, \ldots , K \label{eq:UniformPMF}$$ and the relation $$\widehat R_0 = \left ( \frac{1}{K} \sum_{k=1}^K \frac{1}{C_k} \right ) \cdot {{\mathbb{E}}\left[{ \sigma}\right]} \label{eq:ExpectedR_0}$$ holds.* ]{} \[prop:ExpectedR\_0\] With $\Gamma$ given at (\[eq:Gamma\]) it will often be convenient to write (\[eq:ExpectedR\_0\]) more compactly as $$\widehat R_0 = \frac{ \Gamma}{K} \cdot {{\mathbb{E}}\left[{ \sigma }\right]}. \label{eq:ExpectedR_0Alternate}$$ [[\ **Proof.  **]{}]{}For each $k=1, \ldots , K$, the definition of $\nu_0^\star$ gives $$\begin{aligned} {{\mathbb{P}}\left[{ \nu^\star_0 = k }\right]} &=& \sum_{\ell=1, \ \ell \neq k }^K {{\mathbb{P}}\left[{ \Sigma_0 = \{ k, \ell \}, \nu^\star_0 = k }\right]} \nonumber \\ &=& \sum_{\ell=1, \ \ell \neq k }^K {{\mathbb{P}}\left[{ \nu^\star_0 = k | \Sigma_0 = \{ k, \ell \} }\right]} {{\mathbb{P}}\left[{ \Sigma_0 = \{ k, \ell \} }\right]} \nonumber \\ &=& (K-1) \cdot \frac{1}{2} \cdot \frac{2}{K(K-1)} = \frac{1}{K}.\end{aligned}$$ As pointed earlier, we necessarily have $\nu_0 = \nu^\star_0$. The rvs $\nu^\star_0$ and $\sigma_0$ being independent, we then obtain from (\[eq:n=0\]) that $$\widehat R_0 = {{\mathbb{E}}\left[{ \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} = {{\mathbb{E}}\left[{ \sigma_0 }\right]} \cdot {{\mathbb{E}}\left[{ \frac{1}{C_{\nu^\star_0}} }\right]},$$ and the conclusion (\[eq:ExpectedR\_0\]) readily follows from (\[eq:UniformPMF\]). [\ ]{} According to the Reiman-Simon theory, we have $R(0+) = \widehat R_0$ and Proposition \[prop:ExpectedR\_k=0\] is established with the help of (\[eq:ExpectedR\_0\]). An auxiliary result {#sec:AuxiliaryResult} =================== The cases $n=1$ and $n=2$ are computationally more involved. The technical result discussed next will simplify the presentation by isolating an evaluation which is repeatedly carried out during the analysis. This auxiliary result is given in a setting that mimics power-of-two scheduling with only two customers present: Fix $y < 0$. In addition to the tagged job arriving at time $t=0$ with characteristic pair $(\sigma_0, \Sigma_0)$, assume that another job arrives at time $y$ with (random) service requirement $\tau$. This $y$-job is then assigned to the server $\gamma$, with $\gamma$ being some $\{1, \ldots , K \}$-valued rv, while the tagged job is assigned to the server $\gamma_0$ (in $\Sigma_0$) in accordance with the power-of-two scheduling policy. Thus, if $y + \frac{\tau}{C_{\gamma}} \leq 0$, then $\gamma_0 = \nu^\star_0$. On the other hand, if $y + \frac{\tau}{C_{\gamma}} > 0$, then the operational rules of the power-of-two scheduling policy will preclude the tagged job to be assigned to server $\gamma$: Indeed, if $\gamma$ is not in $\Sigma_0$, then $\gamma_0 = \nu^\star_0$ again, while if $\gamma$ is an element of $\Sigma_0$, then $\gamma_0$ is necessarily the other server in the pair $\Sigma_0$, i.e., the one different from $\gamma$. In this scenario $\gamma$ is a rv given [*a priori*]{}, and should be thought as a place holder for a server assignment rv determined via power-of-two scheduling under various circumstances. On the other hand, $\gamma_0$ depends on $y$, $\tau$, $\gamma$ and $\Sigma_0$ (as well as $\nu^\star_0$). The explicit dependence on these quantities will be dropped from the notation. For reasons that will become apparent in subsequent developments, we also introduce an event $E$ (to be specified later). [*Given are the rvs ${{\bf 1}\left[E\right]}$, $\tau$, $\gamma$, $\sigma_0$, $\Sigma_0$ and $\nu^\star_0$. We assume that (i) the rv $\nu^\star_0$ is uniformly distributed on $\Sigma_0$ conditionally on all the other rvs ${{\bf 1}\left[E\right]}$, $\tau$, $\gamma$, $\sigma_0$ and $\Sigma_0$; (ii) the collections of rvs $\{ {{\bf 1}\left[E\right]}, \tau, \gamma \}$ and $\{ \nu^\star_0 , \Sigma_0 , \sigma_0\}$ are independent; and (iii) the rvs $\Sigma_0$ and $\sigma_0$ are independent. Then, for each $y<0$ and each $k=1, \ldots , K$, we have $$\begin{aligned} {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} &=& {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} \leq 0 }\right]} \cdot \widehat R_0 \nonumber \\ & & ~+ \frac{1}{K-1} {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \left ( \Gamma - \frac{1}{C_k} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:Auxiliary}\end{aligned}$$ with $\gamma_0$ as defined earlier. \[lem:Auxiliary\]* ]{} Recall that under the enforced assumptions, the rv $\sigma_0$ is independent of the collection of rvs $\{ \nu^\star_0 , \Sigma_0 \}$; see Section \[subsec:Framework\]. [[\ **Proof.  **]{}]{}Fix $k=1, \ldots , K $. We start with the natural decomposition $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} } & & \label{eq:Auxiliary1} \\ &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_{\gamma}} \leq 0\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} + {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_{\gamma}} > 0 \right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} . \nonumber\end{aligned}$$ For the first term, the definition of $\gamma_0$ leads to $$\begin{aligned} {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_{\gamma}} \leq 0\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} \leq 0\right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} \nonumber \\ &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} \leq 0\right]} }\right]} {{\mathbb{E}}\left[{ \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} \leq 0 }\right]} \cdot \widehat R_0 \label{eq:Auxiliary2}\end{aligned}$$ since the collections $\{ {{\bf 1}\left[E\right]}, \gamma, \tau \}$ and $\{ \nu^\star_0, \sigma_0 \}$ are independent under the enforced assumptions. We further decompose the second term in (\[eq:Auxiliary1\]) to obtain $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_{\gamma}} > 0\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} } & & \nonumber \\ &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y+ \frac{\tau}{C_k} > 0\right]}{{\bf 1}\left[ k \notin \Sigma_0\right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} \nonumber \\ & & ~+ {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]}{{\bf 1}\left[ k \in \Sigma_0\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]}. \label{eq:Auxiliary3}\end{aligned}$$ It is plain that $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y+ \frac{\tau}{C_k} > 0\right]}{{\bf 1}\left[ k \in \Sigma_0\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} } & & \nonumber \\ &=& \sum_{\ell=1, \ell \neq k }^K {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]} {{\bf 1}\left[ \Sigma_0 = \{ k, \ell \} \right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} \nonumber \\ &=& \sum_{\ell=1, \ell \neq k }^K {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]} {{\bf 1}\left[ \Sigma_0 = \{ k, \ell \} \right]} \frac{\sigma_0}{C_\ell} }\right]} \nonumber \\ &=& \frac{2}{K(K-1)} \sum_{\ell=1, \ell \neq k }^K {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \frac{{{\mathbb{E}}\left[{\sigma_0}\right]}}{C_\ell} \nonumber \\ &=& \frac{2}{K(K-1)} \left ( \sum_{\ell=1, \ell \neq k }^K \frac{{{\mathbb{E}}\left[{\sigma_0}\right]}}{C_\ell} \right ) {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \nonumber \\ &=& \frac{2}{K(K-1)} \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \cdot {{\mathbb{E}}\left[{ \sigma_0}\right]} \label{eq:Auxiliary4}\end{aligned}$$ since $\gamma_0=\ell$ if $\Sigma_0 = \{ k, \ell \}$ when $\gamma = k$ and $ y + \frac{\tau}{C_k} > 0 $. On the other hand, the definition of $\nu^\star_0$ implies $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]}{{\bf 1}\left[ k \notin \Sigma_0\right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} } & & \nonumber \\ &=& \sum_{ T \in {\cal P}_2(K) } {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]}{{\bf 1}\left[ k \notin \Sigma_0\right]} {{\bf 1}\left[ \Sigma_0 = T \right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k}^K \left (\sum_{b=1, b \neq k}^{a-1} {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]} {{\bf 1}\left[ \Sigma_0 = \{a,b\} \right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} \right ) \nonumber \\ &=& \sum_{a=1, a \neq k}^K \left (\sum_{b=1, b \neq k}^{a-1} {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} {{\mathbb{E}}\left[{ {{\bf 1}\left[ \Sigma_0 = \{a,b\} \right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} \right ) \nonumber \\ &=& \sum_{a=1, a \neq k}^K {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \left (\sum_{b=1, b \neq k}^{a-1} {{\mathbb{P}}\left[{ \Sigma_0 = \{a,b\} }\right]} \cdot \frac{1}{2} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{1}{K(K-1)} \sum_{a=1, a \neq k}^K {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \left (\sum_{b=1, b \neq k}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{1}{K(K-1)} \left ( \sum_{a=1, a \neq k}^K \left (\sum_{b=1, b \neq k}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) \right ) {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} . \label{eq:Auxiliary6}\end{aligned}$$ In Appendix B we show that $$\sum_{ a=1 , a \neq k }^K \left (\sum_{b=1, b \neq k}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) = (K-2) \left ( \Gamma - \frac{1}{C_k} \right ), \label{eq:Sigma_k}$$ so that (\[eq:Auxiliary6\]) can be written more compactly as $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_k} > 0\right]}{{\bf 1}\left[ k \notin \Sigma_0\right]} \frac{\sigma_0}{C_{\nu^\star_0}} }\right]} } & & \nonumber \\ &=& \frac{K-2}{K(K-1)} \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} . \label{eq:Auxiliary5}\end{aligned}$$ To conclude the proof, substitute (\[eq:Auxiliary4\]) and (\[eq:Auxiliary5\]) into (\[eq:Auxiliary3\]). It yields $${{\mathbb{E}}\left[{ {{\bf 1}\left[E\right]} {{\bf 1}\left[\gamma = k\right]}{{\bf 1}\left[y + \frac{\tau}{C_\gamma} > 0\right]} \frac{\sigma_0}{C_{\gamma_0}} }\right]} = \frac{1}{K-1} \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ E, \gamma = k, y + \frac{\tau}{C_k} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]},$$ and combining this last expression with (\[eq:Auxiliary2\]) we get the desired result (\[eq:Auxiliary\]) with the help of (\[eq:Auxiliary1\]). [\ ]{} The case $n=1$ {#sec:n=1} ============== The analysis of the first derivative is associated with the following scenario: The tagged job arrives at time $t=0$ with characteristic pair $(\sigma_0, \Sigma_0)$. With $t$ in $\mathbb{R}$, in addition to the tagged job, a single job arrives during the entire horizon $(-\infty, \infty)$, say at time $t$ with characteristic pair $(\sigma_t, \Sigma_t)$. The tagged job and this $t$-job are assigned to the servers $\nu_0$ (in $\Sigma_0$) and $\nu_t$ (in $\Sigma_t$), respectively, in accordance with the power-of-two scheduling policy. Evaluating $\widehat R_1(t)$ ---------------------------- For each $t$ in $\mathbb{R}$, in accordance with (\[eq:R\_n\]) we have $$\widehat R_1(t) = {{\mathbb{E}}\left[{ R_1(t) }\right]} \quad \mbox{with~} R_1(t) = \frac{ \sigma_0 }{ C_{\nu_0} } . \label{eq:n=1}$$ However, with the presence of the $t$-job, $\nu_0$ does not always coincide with $\nu^\star_0$, as the determination of $\nu_0$ may be affected by whether the $t$-job completed service at the time the tagged job arrives. First some notation: With $t$ arbitrary in $\mathbb{R}$, set $$H_k(t) = {{\mathbb{P}}\left[{ C_k t + \sigma_t \leq 0 }\right]} \cdot \widehat R_0 + \frac{1}{K-1} \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:H_k(t)}$$ for each $k=1, \ldots , K$. Note that $$\begin{aligned} H_k(t) &=& \widehat R_0 \cdot \left ( 1 - {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \right ) + \frac{1}{K-1} \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{E}}\left[{\sigma_0}\right]} \cdot {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \nonumber \\ &=& \widehat R_0 + \left ( \frac{1}{K-1} \left ( \Gamma - \frac{1}{C_k} \right ) - \frac{\Gamma}{K} \right ) {{\mathbb{E}}\left[{\sigma_0}\right]} \cdot {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \nonumber \\ &=& \widehat R_0 + \frac{1}{K-1} \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:H_k(t)_Alternate}\end{aligned}$$ as we make use of the expression (\[eq:ExpectedR\_0Alternate\]). [*Under the enforced independence assumptions, we have $\widehat R_1(t) = \widehat R_0$ if $t > 0$, while for $t < 0$ it holds that $$\begin{aligned} \widehat R_1(t) = \frac{1}{K} \sum_{k=1}^K H_k(t). \label{eq:ExpectedR_1B}\end{aligned}$$* ]{} \[prop:ExpectedR\_1\] [[\ **Proof.  **]{}]{}Fix $t$ in $\mathbb{R}$. As we seek to evaluate $\widehat R_1(t)$ as given by (\[eq:n=1\]), two cases need to be examined: If $ t > 0$, then $\nu_0 = \nu^\star_0$, whence $R_1(t) = R_0$, and the conclusion $\widehat R_1(t) = \widehat R_0$ follows. If $t < 0$, then $\nu_t = \nu^\star_t$ and we are in the setting of Lemma \[lem:Auxiliary\] with $y=t$, $E=\Omega$, $\tau = \sigma_t$ and $\gamma = \nu^\star_t$ (so that $\gamma_0=\nu_0$): For each $k=1, \ldots , K$, the expression (\[eq:Auxiliary\]) becomes $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[\nu^\star_t = k\right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_t = k, t + \frac{\sigma_t}{C_k} \leq 0 }\right]} \cdot \widehat R_0 + \frac{1}{K-1} {{\mathbb{P}}\left[{ \nu^\star_t = k, t + \frac{\sigma_t}{C_k} > 0 }\right]} \left ( \Gamma - \frac{1}{C_k} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{1}{K} \cdot H_k(t)\end{aligned}$$ since the rv $\nu^\star_t$ is independent of $\sigma_t$ and uniformly distributed on $\{1, \ldots , K \}$ (as pointed out in Proposition \[prop:ExpectedR\_0\]). The desired result (\[eq:ExpectedR\_1B\]) now follows from (\[eq:n=1\]) upon noting the decomposition $$\widehat R_1(t) = \sum_{k=1}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[\nu^\star_t = k\right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} .$$ [\ ]{} A proof of Proposition \[prop:ExpectedR\_k=1\] ---------------------------------------------- We can now complete the proof of Proposition \[prop:ExpectedR\_k=1\]: The expression (\[eq:ExpressionFor\_n=1\]) now takes the form $$\begin{aligned} R^{\prime} (0+) = \int_{\mathbb{R}} \left ( \widehat R_1(t) - \widehat R_0 \right ) dt = \int_{-\infty}^0 \left ( \widehat R_1(t) - \widehat R_0 \right ) dt \label{eq:IntegralFor_n=1}\end{aligned}$$ as we recall that $\widehat R_1(t) = \widehat R_0$ for $t > 0$. Next, for $t < 0$, with the help of (\[eq:H\_k(t)\_Alternate\]) and (\[eq:ExpectedR\_1B\]) we can rewrite the integrand as $$\begin{aligned} \widehat R_1(t) - \widehat R_0 = \frac{1}{K} \sum_{k=1}^K \frac{1}{K-1} \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k t + \sigma > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma}\right]} \label{eq:Integrand}\end{aligned}$$ as we recall that the rvs $\sigma_t$ and $\sigma_0$ are both distributed like $\sigma$. Inserting this expression back into (\[eq:IntegralFor\_n=1\]) we get $$\int_{-\infty}^0 \left ( \widehat R_1(t) - \widehat R_0 \right ) dt = \frac{1}{K} \sum_{k=1}^K \frac{1}{K-1} \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \cdot {{\mathbb{E}}\left[{\sigma}\right]} \label{eq:IntegratingDifference}$$ upon noting that $$\int_{-\infty}^0 {{\mathbb{P}}\left[{ C_{k}t + \sigma > 0 }\right]} dt = \frac{1}{C_k} \int_{0}^\infty {{\mathbb{P}}\left[{ \sigma > x }\right]} dx = \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k}, \quad k=1, \ldots , K \label{eq:AnIdentityExpectation}$$ by a simple change of variable. Uninteresting algebra on (\[eq:IntegratingDifference\]) readily yield (\[eq:ExpectedR\_k=1\]) with the help of (\[eq:ExpectedR\_0Alternate\]), and this completes the proof of Proposition \[prop:ExpectedR\_k=1\] . [\ ]{} The case $n=2$ {#sec:n=2} ============== The computation of the second derivative is given under the following scenario: The tagged job arrives at time $t=0$ with characteristic pair $(\sigma_0, \Sigma_0)$. With $s$ and $t$ in $\mathbb{R}$, in addition to the tagged job, exactly two jobs arrive over the entire horizon $(-\infty, \infty)$, say at times $s$ and $t$ with characteristic pairs $(\sigma_s, \Sigma_s)$ and $(\sigma_t, \Sigma_t)$, respectively. The tagged job, the $s$-job and the $t$-job are assigned to their respective servers $\nu_0$ (in $\Sigma_0$), $\nu_s$ (in $\Sigma_s$) and $\nu_t$ (in $\Sigma_t$) in accordance with the power-of-two load balancing scheduling policy. Evaluating $\widehat R_2(s,t)$ ------------------------------ For each $s$ and $t$ in $\mathbb{R}$, we have $$\widehat R_2(s,t) = {{\mathbb{E}}\left[{ R_2(s,t) }\right]} \quad \mbox{with~} R_2(s,t) = \frac{ \sigma_0 }{ C_{\nu_0} } . \label{eq:n=2}$$ The server assignment rvs $\nu_0$, $\nu_s$ and $\nu_t$ do not always coincide with $\nu^\star_0$, $\nu^\star_s$ and $\nu^\star_t$, respectively, because these rvs may be affected by whether earlier jobs have completed service by the time server selection needs to be determined. [*Under the enforced independence assumptions, we have $\widehat R_2(s,t) = \widehat R_0$ for $0 < s < t$ and $\widehat R_2(s,t) = \widehat R_1(s)$ for $s < 0 < t$, while for $ s < t < 0 $, it holds that $$\begin{aligned} \widehat R_2(s,t) &=& \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_k} \leq t }\right]} \right ) \cdot \widehat R_1 (t) \nonumber \\ & & ~ + \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot H_k(t) \nonumber \\ & & ~+ \frac{1}{K(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k}^K \left ( \Gamma - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{C_\ell s + \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_k t + \sigma_t \leq 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ & & ~+ \frac{1}{K^2(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K \Sigma_{k\ell} {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:ExpectedR_2C}\end{aligned}$$ with $$\Sigma_{k\ell} = (K+1) \Gamma - K \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ), \quad k,\ell =1, \ldots , K. \label{eq:Sigma_kl}$$* ]{} \[prop:ExpectedR\_2\] Before starting the proof of Proposition \[prop:ExpectedR\_2\] in Section \[sec:ProofPropExpectedR\_2\], we pause to give a more compact expression for (\[eq:ExpectedR\_2C\]). Towards a more compact expression for (\[eq:ExpectedR\_2C\]) ------------------------------------------------------------- As we focus on the last two terms in (\[eq:ExpectedR\_2C\]), interchange the dummy indices $k$ and $\ell$, and then change the order of summations in the resulting expression. We can readily check that $$\begin{aligned} & & \frac{1}{K(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k}^K \left ( \Gamma - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{C_\ell s + \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_k t + \sigma_t \leq 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ & & ~+ \frac{1}{K^2(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K \Sigma_{k\ell} \cdot {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{1}{K^2(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K G_{k\ell}(s,t) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]}\end{aligned}$$ with $$\begin{aligned} G_{k\ell}(s,t) &=& K \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{C_k s + \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t \leq 0 }\right]} \nonumber \\ & & ~+ \left ( (K+1) \Gamma - K \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) \right ) \cdot {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]} \nonumber \\ &=& K \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{C_k s + \sigma_s > 0 }\right]} \nonumber \\ & & ~+ K \left ( \frac{\Gamma}{K} - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]}\end{aligned}$$ for every $k,\ell =1, \ldots ,K$. Upon substitution into (\[eq:ExpectedR\_2C\]), we then conclude that $$\begin{aligned} \widehat R_2(s,t) &=& \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_k} \leq t }\right]} \right ) \cdot \widehat R_1 (t) \nonumber \\ & & ~ + \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot H_k(t) \nonumber \\ & & ~+ \frac{1}{K(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K \left ( \frac{\Gamma}{K} - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0 }\right]} \nonumber \\ & & ~+ \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]}. \label{eq:ExpectedR_2C+X}\end{aligned}$$ Next using (\[eq:H\_k(t)\_Alternate\]) we get $$\begin{aligned} & & \sum_{k=1}^K \left ( \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot H_k(t) \nonumber \\ &=& \sum_{k=1}^K \left ( \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \left ( \widehat R_0 + \frac{1}{K-1} \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \right ) \nonumber\end{aligned}$$ and the second term in (\[eq:ExpectedR\_2C+X\]) becomes $$\begin{aligned} & & \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot H_k(t) \nonumber \\ &=& \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \widehat R_0 \nonumber \\ & & ~+ \frac{1}{K(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} {{\mathbb{P}}\left[{ C_k t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]}. \nonumber\end{aligned}$$ Substituting this last expression into (\[eq:ExpectedR\_2C+X\]) we readily get the following more compact expression for (\[eq:ExpectedR\_2C\]). [*Under the enforced independence assumptions, for $ s < t < 0 $, it holds that $$\begin{aligned} \widehat R_2(s,t) &=& \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell} \leq t }\right]} \right ) \cdot \widehat R_1 (t) + \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot \widehat R_0 \nonumber \\ & & ~ + \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} + \frac{1}{K(K-1)^2} \cdot H(s,t) \nonumber $$ where we have set $$\begin{aligned} H(s,t) = \sum_{\ell=1}^K \sum_{k=1, k \neq \ell}^K \left ( \frac{\Gamma}{K} - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma_s }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]}. \nonumber\end{aligned}$$* ]{} \[prop:ExpectedR\_2MoreCompact\] A proof of Proposition \[prop:ExpectedR\_k=2\] ============================================== . Our point of departure is the expression (\[eq:ExpressionFor\_n=2\]). For notational simplicity we shall write $$R^\star (s,t) = \widehat R_2(s,t) - \widehat R_1(s) - \widehat R_1(t) + \widehat R_0, \quad s,t \in \mathbb{R}.$$ The integral to be evaluated ---------------------------- We start with $$\begin{aligned} R^{\prime\prime}(0+) &=& \int_{\mathbb{R}} \left ( \int_{\mathbb{R}} R^\star (s,t) dt \right ) ds \nonumber \\ &=& \int_{\mathbb{R}} \left ( \int_{-\infty}^s R^\star (s,t) dt \right ) ds + \int_{\mathbb{R}} \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds. \label{eq:DecompositionIntegral}\end{aligned}$$ The second term in this expression can be written as $$\int_{\mathbb{R}} \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds = \int_{-\infty}^0 \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds + \int_{0}^\infty \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds . \label{eq:DecompositionA}$$ Now, by Propositions \[prop:ExpectedR\_1\] and \[prop:ExpectedR\_2\] we have $ R^\star (s,t) = \widehat R_0 - \widehat R_0 - \widehat R_0 + \widehat R_0 = 0$ whenever $ 0 < s < t $, and the conclusion $$\int_{0}^\infty \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds = 0 \label{eq:DecompositionB}$$ follows. Next, consider the decomposition $$\int_{-\infty}^0 \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds = \int_{-\infty}^0 \left ( \int_{s}^0 R^\star (s,t) dt \right ) ds + \int_{-\infty}^0 \left ( \int_{0}^\infty R^\star (s,t) dt \right ) ds. \label{eq:DecompositionC}$$ On the range $s < 0 < t$, Propositions \[prop:ExpectedR\_1\] and \[prop:ExpectedR\_2\] yield $\widehat R_2(s,t) = \widehat R_1(s)$ and $\widehat R_1(t) = \widehat R_0 $, whence $R^\star (s,t) = \widehat R_1(s) - \widehat R_1(s) - \widehat R_0 + \widehat R_0 = 0$ again, so that $$\int_{-\infty}^0 \left ( \int_{0}^\infty R^\star (s,t) dt \right ) ds = 0.$$ Combining (\[eq:DecompositionA\]), (\[eq:DecompositionB\]) and (\[eq:DecompositionC\]), we conclude that the second term in (\[eq:DecompositionIntegral\]) reduces to $$\int_{\mathbb{R}} \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds = \int_{-\infty}^0 \left ( \int_{s}^0 R^\star (s,t) dt \right ) ds. \label{eq:n=2IntegralToBeComputed}$$ Finally, returning to the first term in the decomposition (\[eq:DecompositionIntegral\]) we get $$\begin{aligned} \int_{\mathbb{R}} \left ( \int_{-\infty}^s R^\star (s,t) dt \right ) ds &=& \int_{\mathbb{R}} \left ( \int_{t}^\infty R^\star (s,t) ds \right ) dt \nonumber \\ &=& \int_{\mathbb{R}} \left ( \int_{s}^\infty R^\star (t,s) dt \right ) ds \nonumber \\ &=& \int_{\mathbb{R}} \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds \label{eq:n=2IntegralToBeComputedFinal}\end{aligned}$$ as we note that $R^\star (t,s) = R^\star (s,t)$ for arbitrary $s,t$ in $\mathbb{R}$ since the symmetry $\widehat R_2(t,s) = \widehat R_2(s,t) $ holds under the enforced statistical assumptions. It follows from (\[eq:DecompositionIntegral\]) that $$R^{\prime\prime}(0+) = \int_{\mathbb{R}} \left ( \int_{\mathbb{R}} R^\star (s,t) dt \right ) ds = 2 \int_{\mathbb{R}} \left ( \int_{s}^\infty R^\star (s,t) dt \right ) ds = 2 \int_{-\infty}^0 \left ( \int_{s}^0 R^\star (s,t) dt \right ) ds \label{eq:n=2IntegralToBeComputed3}$$ as we combine (\[eq:DecompositionIntegral\]), (\[eq:n=2IntegralToBeComputed\]) and (\[eq:n=2IntegralToBeComputedFinal\]). Computing the integrand in (\[eq:n=2IntegralToBeComputed3\]) ------------------------------------------------------------ On the way to evaluating the integral (\[eq:n=2IntegralToBeComputedFinal\]) we consider $R^\star (s,t) $ for $s < t < 0$. On that range, applying (\[eq:Integrand\]) with $t$ replaced by $s$ yields $$\widehat R_1(s) - \widehat R_0 = \frac{1}{K(K-1) } \sum_{k=1}^K \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k s + \sigma_s > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:IntegrandAgain}$$ Using the expression for $\widehat R_2(s,t)$ in Proposition \[prop:ExpectedR\_2MoreCompact\] and recalling the expression for $\widehat R_0$ we then readily get $$\begin{aligned} R^\star (s,t) &=& \left ( \widehat R_0 - \widehat R_1(s) \right ) + \left ( \widehat R_2(s,t) - \widehat R_1(t) \right ) \nonumber \\ &=& - \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k s + \sigma_s > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ & & + \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot \widehat R_0 + \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell} \leq t }\right]} - 1 \right ) \cdot \widehat R_1 (t) \nonumber \\ & & ~ + \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \Gamma - \frac{1}{C_k} \right ) {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} + \frac{1}{K(K-1)^2} \cdot H(s,t) \nonumber \\ &=& \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot \widehat R_0 - \left ( \frac{1}{K} \sum_{\ell=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell} > t }\right]} \right ) \cdot \widehat R_1 (t) \nonumber \\ & & ~ + \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ C_k s + \sigma_s > 0 }\right]} \cdot \left ( \frac{\Gamma}{K} {{\mathbb{E}}\left[{\sigma_0}\right]} \right ) + \frac{1}{K(K-1)^2} \cdot H(s,t) \nonumber \\ &=& \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma_s }\right]} \right ) \cdot \widehat R_0 - \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_k} > t }\right]} \right ) \cdot \widehat R_1 (t) \nonumber \\ & & + \frac{1}{K(K-1)^2} \cdot H(s,t) \nonumber \\ &=& \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma_s }\right]} \right ) \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) + \frac{1}{K(K-1)^2} \cdot H(s,t) . \label{eq:For_n=2IntegrandToBeComputed}\end{aligned}$$ Evaluating (\[eq:n=2IntegralToBeComputed3\]) -------------------------------------------- Next, recall that in these expressions the rvs $\sigma_s$ and $\sigma_0$ are distributed like $\sigma$. Thus, after a change of order of integration and a change of variable, we note that $$\begin{aligned} & & \int_{-\infty}^0 \left ( \int_{s}^0 \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma }\right]} \right ) \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) dt \right ) ds \nonumber \\ &=& \frac{1}{K} \sum_{k=1}^{K} \int_{-\infty}^0 \left ( \int_{s}^0 {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma }\right]} \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) dt \right ) ds \nonumber \\ &=& \frac{1}{K} \sum_{k=1}^{K} \int_{-\infty}^0 \left ( \int_{-\infty}^t {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma }\right]} \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) ds \right ) dt \nonumber \\ &=& \frac{1}{K} \sum_{k=1}^{K} \int_{-\infty}^0 \left ( \int_{-\infty}^t {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma }\right]} ds \right ) \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) dt \nonumber \\ &=& \frac{1}{K} \sum_{k=1}^{K} \int_{-\infty}^0 \left ( \int_0^{\infty} {{\mathbb{P}}\left[{ C_k x < \sigma }\right]} dx \right ) \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) dt \nonumber \\ &=& \frac{1}{K} \sum_{k=1}^{K} \int_{-\infty}^0 \frac{ {{\mathbb{E}}\left[{\sigma}\right]} } {C_k} \cdot \left ( \widehat R_0 -\widehat R_1 (t) \right ) dt \nonumber \\ &=& - \left ( \int_{-\infty}^0 \left ( \widehat R_1 (t) - \widehat R_0 \right ) dt \right ) \cdot \frac{\Gamma}{K} {{\mathbb{E}}\left[{\sigma}\right]} \nonumber \\ &=& - \left ( \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \frac{\Gamma}{K} - \frac{1}{C_k} \right ) \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \cdot {{\mathbb{E}}\left[{\sigma}\right]} \right ) \cdot \frac{\Gamma}{K} {{\mathbb{E}}\left[{\sigma}\right]} \nonumber \\ &=& \left ( \frac{1}{K} \sum_{k=1}^K \frac{ 1 }{C^2_k} - \left ( \frac{ \Gamma}{K} \right )^2 \right ) \cdot \frac{\Gamma}{K(K-1)} \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3 \label{eq:For_n=2+A}\end{aligned}$$ where the step before last made used of the expression (\[eq:IntegratingDifference\]). In a similar vein, we find that $$\begin{aligned} \int_{-\infty}^0 \left ( \int_{s}^0 H (s,t) dt \right ) ds = \frac{1}{K(K-1)^2} \sum_{\ell=1}^K \sum_{k=1, k \neq \ell}^K \left ( \frac{\Gamma}{K} - \frac{1}{C_\ell} \right ) I_{k\ell} \cdot {{\mathbb{E}}\left[{\sigma}\right]} \label{eq:For_n=2+B}\end{aligned}$$ with $$\begin{aligned} I_{k\ell} &=& \int_{-\infty}^0 \left ( \int_{s}^0 {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma > 0 }\right]} dt \right ) ds \nonumber \\ &=& \int_{-\infty}^0 \left ( \int_{-\infty}^t {{\mathbb{P}}\left[{ C_k t < C_k s + \sigma }\right]} ds \right ) {{\mathbb{P}}\left[{ C_\ell t + \sigma > 0 }\right]} dt \nonumber \\ &=& \int_{-\infty}^0 \left ( \int_{0}^\infty {{\mathbb{P}}\left[{ C_k x < \sigma }\right]} dx \right ) {{\mathbb{P}}\left[{ C_\ell t + \sigma > 0 }\right]} dt \nonumber \\ &=& \left ( \int_{0}^\infty {{\mathbb{P}}\left[{ C_k x < \sigma }\right]} dx \right ) \left ( \int_{-\infty}^0 {{\mathbb{P}}\left[{ C_\ell t + \sigma > 0 }\right]} dt \right ) \nonumber \\ &=& \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \cdot \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_\ell}, \quad k,\ell =1, \ldots , K. \label{eq:For_n=2+Ba}\end{aligned}$$ Therefore, $$\int_{-\infty}^0 \left ( \int_{s}^0 H (s,t) dt \right ) ds = \sum_{\ell=1}^K \sum_{k=1, k \neq \ell}^K \left ( \frac{\Gamma}{K} - \frac{1}{C_\ell} \right ) \left ( \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \cdot \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{\sigma}\right]} \label{eq:For_n=2+Bb}$$ with $$\begin{aligned} \sum_{\ell=1}^K \sum_{k=1, k \neq \ell}^K \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \cdot \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_\ell} &=& \sum_{\ell=1}^K \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_\ell} \left ( \sum_{k=1, k \neq \ell}^K \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \right ) \nonumber \\ &=& \sum_{\ell=1}^K \frac{ 1 }{C_\ell} \left ( \Gamma - \frac{1}{ C_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^2 \nonumber \\ &=& \left ( \Gamma^2 - \sum_{\ell=1}^K \frac{ 1 }{C^2_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^2 \label{eq:For_n=2+Bc}\end{aligned}$$ and $$\begin{aligned} \sum_{\ell=1}^K \sum_{k=1, k \neq \ell}^K \frac{1}{C_\ell} \left ( \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \cdot \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_\ell} \right ) &=& \sum_{\ell=1}^K \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C^2_\ell} \left ( \sum_{k=1, k \neq \ell}^K \frac{ {{\mathbb{E}}\left[{\sigma}\right]} }{C_k} \right ) \nonumber \\ &=& \sum_{\ell=1}^K \frac{ 1 }{C^2_\ell} \left ( \Gamma - \frac{1}{ C_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^2 \nonumber \\ &=& \left ( \Gamma \sum_{\ell=1}^K \frac{1}{C^2_\ell} - \sum_{\ell=1}^K \frac{ 1 }{C^3_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^2. \label{eq:For_n=2+Bd}\end{aligned}$$ Substitute (\[eq:For\_n=2+Bc\]) and (\[eq:For\_n=2+Bd\]) into (\[eq:For\_n=2+Bb\]), and we find $$\begin{aligned} \lefteqn{ \int_{-\infty}^0 \left ( \int_{s}^0 H (s,t) dt \right ) ds } & & \nonumber \\ &=& \frac{1}{K(K-1)^2} \left ( \frac{\Gamma}{K} \left ( \Gamma^2 - \sum_{\ell=1}^K \frac{ 1 }{C^2_\ell} \right ) - \left ( \Gamma \sum_{\ell=1}^K \frac{1}{C^2_\ell} - \sum_{\ell=1}^K \frac{ 1 }{C^3_\ell} \right ) \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^3 \nonumber \\ &=& \frac{1}{K(K-1)^2} \left ( \frac{\Gamma^3}{K} - \frac{K+1}{K} \cdot \Gamma \sum_{\ell=1}^K \frac{ 1 }{C^2_\ell} + \sum_{\ell=1}^K \frac{ 1 }{C^3_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma}\right]} \right )^3. \label{eq:For_n=2+C}\end{aligned}$$ Finally, return to (\[eq:For\_n=2IntegrandToBeComputed\]) and collect (\[eq:For\_n=2+A\]) and (\[eq:For\_n=2+C\]): Uninteresting calculations show that $$\begin{aligned} \lefteqn{ \int_{-\infty}^0 \left ( \int_{s}^0 R^\star_2 (s,t) dt \right ) ds } & & \nonumber \\ &=& \left ( - \left ( \frac{ \Gamma}{K} \right )^2 + \frac{1}{K} \sum_{k=1}^K \frac{ 1 }{C^2_k} \right ) \cdot \frac{\Gamma}{K(K-1)} \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3 \nonumber \\ & & ~ + \frac{1}{K(K-1)^2} \left ( \frac{\Gamma^3}{K} - \frac{K+1}{K} \Gamma \sum_{\ell=1}^K \frac{ 1 }{C^2_\ell} + \sum_{\ell=1}^K \frac{ 1 }{C^3_\ell} \right ) \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3 \nonumber \\ &=& \left ( - \frac{1}{K^3(K-1)} + \frac{1}{K^2(K-1)^2} \right ) \Gamma^3 \cdot \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3 \nonumber \\ & & ~ + \left ( \frac{1}{K^2 (K-1)} - \frac{ K+1 }{ K^2(K-1)^2} \right ) \left ( \sum_{k=1}^K \frac{ 1 }{C^2_k} \right ) \Gamma \cdot \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3 \nonumber \\ & & ~ + \frac{1}{K(K-1)^2} \left ( \sum_{\ell=1}^K \frac{ 1 }{C^3_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3 \nonumber \\ &=& \frac{1}{(K-1)^2} \left ( \left ( \frac{\Gamma}{K} \right )^3 - 2 \left ( \frac{1}{K} \sum_{k=1}^K \frac{ 1 }{C^2_k} \right ) \left ( \frac{ \Gamma }{K} \right ) + \frac{1}{K} \sum_{\ell=1}^K \frac{ 1 }{C^3_\ell} \right ) \cdot \left ( {{\mathbb{E}}\left[{\sigma }\right]} \right )^3,\end{aligned}$$ and the expression (\[eq:ExpectedR\_k=2\]) now follows from (\[eq:n=2IntegralToBeComputed3\]). [\ ]{} A proof of Proposition \[prop:ExpectedR\_2\] {#sec:ProofPropExpectedR_2} ============================================= The cases $0 < s< t $ and $ s< 0 < t$ are straightforward by virtue of the operational assumptions of the power-of-two load balancing policy. Indeed, when $0 < s< t $, $\nu_0 = \nu^\star_0$, hence $R_2(s,t) = R_0$ and $\widehat R_2(s,t) = \widehat R_0$ holds. On the other hand, when $s< 0 < t$, the future $t$-job does not affect the selection of $\nu_0$, hence has no impact on the performance of the tagged customer. As only the $s$-job can possibily affect the choice of $\nu_0$, we get $R_2(s,t) = R_1(s)$ and this shows that $\widehat R_2(s,t) = \widehat R_1(s)$. From now on we assume $s < t < 0$, in which case we have $\nu_s = \nu^\star_s$. The selection of $\nu_t$ can in principle be affected by whether the $s$-job has completed its service by time $t$, while that of $\nu_0$ will be determined by whether the $s$-job and $t$-job have completed service by the time the tagged job enters the system. Therefore, as the $s$-job completes at time $ s + \frac{\sigma_s}{C_{\nu^\star_s}}$, several possibilities arise; they are captured in the decomposition $$\begin{aligned} {{\mathbb{E}}\left[{ R_2(s,t) }\right]} &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t\right]} R_2(s,t) }\right]} \nonumber \\ & & ~+ {{\mathbb{E}}\left[{ {{\bf 1}\left[ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0 \right]} R_2(s,t) }\right]} \nonumber \\ & & ~+ {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} \leq 0 \right]} R_2(s,t) }\right]} \nonumber \\ & & ~+ {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} > 0 \right]} R_2(s,t) }\right]}. \label{eq:DECOMPOSITION}\end{aligned}$$ These four terms are evaluated separately in the next four lemmas. [*With $s < t < 0$, we have $${{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t\right]} R_2(s,t) }\right]} = \left ( \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_k} \leq t }\right]} \right ) \cdot \widehat R_1 (t) \label{eq:Case1}$$* ]{} \[lem:Case1\] [[\ **Proof.  **]{}]{}When $s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t $, the $s$-job will have completed service by the time the $t$-job arrives. Therefore, conditionally on $s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t $, it holds that $R_2(s,t) =_{st} R_1(t)$, whence $${{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t\right]} R_2(s,t) }\right]} = {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t\right]} R_1(t) }\right]} = {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t }\right]}\cdot \widehat R_1 (t)$$ with $${{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq t }\right]} = \frac{1}{K} \sum_{k=1}^K {{\mathbb{P}}\left[{ s + \frac{\sigma}{C_k} \leq t }\right]}$$ by the usual arguments. This completes the proof of (\[eq:Case1\]). [\ ]{} [*With $s < t < 0$, we have $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0 \right]} R_2(s,t) }\right]} } & & \nonumber \\ &=& \frac{1}{K(K-1)} \sum_{k=1}^K \left ( \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot H_k(t) \label{eq:Case2A}\end{aligned}$$ with $H_k(t)$ given by (\[eq:H\_k(t)\]) for all $k=1, \ldots , K$.* ]{} \[lem:Case2A\] [[\ **Proof.  **]{}]{}When $t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0$, the $s$-job has not completed its service by time $t$, but will have completed it by the time the tagged job arrives. Thus, only the $t$-job can affect the definition of $\nu_0$ (through $\sigma_t$ and $\nu_t$). With this in mind, consider the decomposition $${{\mathbb{E}}\left[{ {{\bf 1}\left[ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0 \right]} R_2(s,t) }\right]} = \sum_{k=1}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0\right]} {{\bf 1}\left[ \nu_t = k \right]} \frac{ \sigma_0 }{C_{\nu_0}} }\right]} . \label{eq:BasicRelationForN=2PieceA}$$ Fix $k=1, \ldots , K$. We are in the setting of Lemma \[lem:Auxiliary\] with $y=t$, $E = [ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0 ]$, $\tau = \sigma_t$ and $\gamma = \nu_t$ so that $\gamma_0 = \nu_0$: The expression (\[eq:Auxiliary\]) becomes $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0\right]} {{\bf 1}\left[\nu_t = k\right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} } & & \nonumber \\ &=& {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0, \nu_t = k, t + \frac{\sigma_t}{C_k} \leq 0 }\right]} \cdot \widehat R_0 \nonumber \\ & & ~+ \frac{1}{K-1} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0, \nu_t = k, t + \frac{\sigma_t}{C_k} > 0 }\right]} \left ( \Gamma - \frac{1}{C_k} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0, \nu_t = k}\right]} H_k(t) \label{eq:AppB+0}\end{aligned}$$ with $H_k(t)$ defined at (\[eq:H\_k(t)\]). In the last step we used the fact that under the enforced independence assumptions, the rv $\sigma_t$ is independent of the rvs $\{ \sigma_s, \nu^\star_s, \nu_t \}$ when $\nu_t$ is generated by the power-of-two load balancing policy. In Appendix C we show that $$\begin{aligned} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0, \nu_t = k}\right]} = \frac{1}{K(K-1)} \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]}. \label{eq:AppB+1}\end{aligned}$$ Inserting (\[eq:AppB+1\]) back into (\[eq:AppB+0\]) yields $${{\mathbb{E}}\left[{ {{\bf 1}\left[ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0 \right]} {{\bf 1}\left[ \nu_t = k \right]} \frac{\sigma_0}{c_{\nu_0}} }\right]} = \left ( \frac{1}{K(K-1)} \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \right ) \cdot H_k(t),$$ and the desired result is now obtained by making use of (\[eq:BasicRelationForN=2PieceA\]). [\ ]{} The last two terms in the decomposition (\[eq:DECOMPOSITION\]) are more cumbersome to evaluate. Their expressions are given in the next two lemmas whose proofs can be found in Sections \[sec:ProofLemmaCase2B+1\] and \[sec:ProofLemmaCase2B+2\], respectively. [*With $s < t < 0$, we have $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} \leq 0 \right]} R_2(s,t) }\right]} } & & \nonumber \\ &=& \frac{1}{K(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k}^K \left ( \Gamma - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{C_\ell s + \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_k t + \sigma_t \leq 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:Case2B+1}\end{aligned}$$* ]{} \[lem:Case2B+1\] [*With $s < t < 0$, we have $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} > 0 \right]} R_2(s,t) }\right]} } & & \nonumber \\ &=& \frac{1}{K^2(K-1)^2} \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K \Sigma_{k\ell} \cdot {{\mathbb{P}}\left[{ C_k s+ \sigma_s > 0 }\right]} {{\mathbb{P}}\left[{ C_\ell t + \sigma_t > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:Case2B+2}\end{aligned}$$ with the constants $\Sigma_{k\ell}, \ k,\ell =1, \ldots,$ given by (\[eq:Sigma\_kl\]).* ]{} \[lem:Case2B+2\] A proof of Lemma \[lem:Case2B+1\] {#sec:ProofLemmaCase2B+1} ================================= We are in the situation when $s < t < 0$. If $s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 $ (hence $s + \frac{\sigma_s}{C_{\nu^\star_s}} > t$), then the $s$-job completes its service only after the tagged arrives, so that both the $s$-job and $t$-job can possibly affect the definition of $\nu_0$. If in addition we have $t + \frac{\sigma_t}{C_{\nu_t}} \leq 0$, then only the $s$-job can affect the selection $\nu_0$. In the usual manner we have the decomposition $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} \leq 0 \right]} R_2(s,t) }\right]} } & & \nonumber \\ &=& \sum_{k=1}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ s + \frac{\sigma_s}{C_{\nu^\star_s}} > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_k} \leq 0 \right]} {{\bf 1}\left[\nu_t = k\right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} \nonumber \\ &=& \sum_{k=1}^K \sum_{\ell=1, \ell \neq k}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = \ell \right]} {{\bf 1}\left[ s + \frac{\sigma_s}{C_\ell } > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_k} \leq 0 \right]} {{\bf 1}\left[\nu_t = k\right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} . \label{eq:PieceZD}\end{aligned}$$ Pick distinct $k,\ell =1, \ldots , K$. This time we apply Lemma \[lem:Auxiliary\] with $y=s$, $E =[ s + \frac{\sigma_s}{C_\ell} > 0, \nu_t = k, t + \frac{\sigma_t}{C_k} \leq 0 ]$, $\tau = \sigma_s $ and $\gamma = \nu^\star_s$ (so that $\gamma_0 = \nu_0$). This leads to $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ \left ( {{\bf 1}\left[ s + \frac{\sigma_s}{C_\ell } > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_k} \leq 0 \right]} {{\bf 1}\left[\nu_t = k\right]} \right ) {{\bf 1}\left[ \nu^\star_s = \ell \right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} } & & \nonumber \\ &=& {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell } > 0 , t + \frac{\sigma_t}{C_k} \leq 0 , \nu_t = k, \nu^\star_s = \ell , s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \cdot \widehat R_0 \nonumber \\ & & ~+ \frac{1}{K-1} {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell } > 0 , t + \frac{\sigma_t}{C_k} \leq 0 , \nu_t = k, \nu^\star_s = \ell , s + \frac{\sigma_s}{C_\ell} > 0 }\right]} \left ( \Gamma - \frac{1}{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{1}{K-1} {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell } > 0 , t + \frac{\sigma_t}{C_k} \leq 0 , \nu_t = k, \nu^\star_s = \ell }\right]} \left ( \Gamma - \frac{1}{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{1}{K-1} {{\mathbb{P}}\left[{ t + \frac{\sigma_t}{C_k} \leq 0 }\right]} {{\mathbb{P}}\left[{ \nu^\star_s = \ell , s + \frac{\sigma_s}{C_\ell } > 0 , \nu_t = k}\right]} \left ( \Gamma - \frac{1}{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \label{eq:PieceZC}\end{aligned}$$ since the rvs $\sigma_t$ is independent of the collection $\{ \nu^\star_s, \sigma_s, \nu_t \}$ under the enforced independence assumptions. Next, we write $$\begin{aligned} \lefteqn{ {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \nu_t = k }\right]} } & & \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \ell \in \Sigma_t, \nu_t = k }\right]} + {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \ell \notin \Sigma_t, \nu_t = k }\right]} \nonumber\end{aligned}$$ Taking terms in turn we first get $$\begin{aligned} {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \ell \in \Sigma_t, \nu_t = k }\right]} &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \Sigma_t = \{ \ell, k \}, \nu_t = k }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \Sigma_t = \{ \ell, k \} }\right]} \nonumber \\ &=& \frac{2}{K^2(K-1)} {{\mathbb{P}}\left[{ s + \frac{\sigma_s}{C_\ell } > 0 }\right]} \label{eq:PieceZA}\end{aligned}$$ since under the constraint $s + \frac{\sigma_s}{C_\ell } > 0$, the fact that $\nu^\star_s$ is an element of $\Sigma_t$ forces $\nu_t$ to be the other element in $\Sigma_t$. In a similar way, under the constraint $s + \frac{\sigma_s}{C_\ell } > 0$, $\nu^\star_s$ not being in $\Sigma_t$ implies $\nu_t = \nu^\star_t$, and this leads to $$\begin{aligned} \lefteqn{ {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \ell \notin \Sigma_t, \nu_t = k }\right]} } \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \ell \notin \Sigma_t, \nu^\star_t = k }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k, a \neq \ell}^K {{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \Sigma_t = \{a,k\}, \nu^\star_t = k }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k, a \neq \ell}^K {{\mathbb{P}}\left[{ \nu^\star_s = \ell}\right]} {{\mathbb{P}}\left[{s + \frac{\sigma_s}{C_\ell } > 0 }\right]} \frac{1}{2} \cdot \frac{2}{K(K-1)} \nonumber \\ &=& \frac{K-2} {K^2(K-1)} {{\mathbb{P}}\left[{s + \frac{\sigma_s}{C_\ell } > 0 }\right]}. \label{eq:PieceZB}\end{aligned}$$ Collecting (\[eq:PieceZA\]) and (\[eq:PieceZB\]) gives $${{\mathbb{P}}\left[{ \nu^\star_s = \ell, s + \frac{\sigma_s}{C_\ell } > 0 , \nu_t = k }\right]} = \frac{1} {K(K-1)} {{\mathbb{P}}\left[{s + \frac{\sigma_s}{C_\ell } > 0 }\right]},$$ and with the help of (\[eq:PieceZC\]) we conclude that $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ \left ( {{\bf 1}\left[ s + \frac{\sigma_s}{C_\ell } > 0 \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_k} \leq 0 \right]} {{\bf 1}\left[\nu_t = k\right]} \right ) {{\bf 1}\left[ \nu^\star_s = \ell \right]} \frac{\sigma_0}{C_{\nu_0}} }\right]} } & & \nonumber \\ &=& \frac{1}{K(K-1)^2} {{\mathbb{P}}\left[{ t + \frac{\sigma_t}{C_k} \leq 0 }\right]} {{\mathbb{P}}\left[{s + \frac{\sigma_s}{C_\ell } > 0 }\right]} \left ( \Gamma - \frac{1}{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} .\end{aligned}$$ Inserting this last expression into (\[eq:PieceZD\]) we obtain (\[eq:Case2B+1\]) as desired. [\ ]{} A proof of Lemma \[lem:Case2B+2\] {#sec:ProofLemmaCase2B+2} ================================= We are in the situation when $s < t < 0$. If $s+ \frac{\sigma_s}{C_{\nu^\star_s}} > 0$ and $t + \frac{\sigma_t}{C_{\nu_t}} > 0$, then $\nu_t$ is determined by the $s$-job and we must have $\nu^\star_s \neq \nu_t$. When the tagged job arrives, both $\nu_s (=\nu^\star_s)$ and $\nu_t$ would have already been selected, with both $s$-job and $t$-job still in service when $\nu_0$ needs to be selected. In order to establish (\[eq:Case2B+2\]), we begin with the observation that $$\begin{aligned} \lefteqn{ {{\mathbb{E}}\left[{ {{\bf 1}\left[s+ \frac{\sigma_s}{C_{\nu^\star_s}} > 0\right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} > 0 \right]} R_2(s;t) }\right]} } & & \nonumber \\ &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[s+ \frac{\sigma_s}{C_{\nu^\star_s}} > 0\right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\nu_t}} > 0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \sum_{k=1}^K \sum_{\ell=1, \ell \neq k }^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]}. \label{eq:ZZ0}\end{aligned}$$ To take advantage of this decomposition, pick distinct $k,\ell=1, \ldots ,K$. As we keep in mind whether $\nu^\star_s$ and $\nu_t$ are in $\Sigma_0$, we shall have to consider four possible cases: First, if both $\nu^\star_s$ and $\nu_t$ are in $\Sigma_0$, then $\nu_0 = \nu^\star_0$ and we have $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ k \in \Sigma_0 , \ell \in \Sigma_0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ \Sigma_0 = \{ k, \ell \} \right]} \frac{ \sigma_0 }{ C_{\nu^\star_0}} }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} {{\mathbb{P}}\left[{ \Sigma_0 = \{ k, \ell \} }\right]} \frac{1}{2} \cdot \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]} \nonumber \\ &=& \frac{1}{K(K-1)} {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]}. \label{eq:ZZ1}\end{aligned}$$ Next, if $\nu^\star_s$ is not in $\Sigma_0$ but $\nu_t$ is in $\Sigma_0$, then $\nu_0$ is the other element in $\Sigma_0$, and we get $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ k \notin \Sigma_0 , \ell \in \Sigma_0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k, a \neq \ell}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ \Sigma_0 = \{ \ell, a \} \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k, a \neq \ell}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ \Sigma_0 = \{ \ell, a \} \right]} \frac{ \sigma_0 }{ C_a} }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k, a \neq \ell}^K {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} {{\mathbb{P}}\left[{ \Sigma_0 = \{ \ell, a \} }\right]} \cdot \frac{{{\mathbb{E}}\left[{\sigma_0}\right]}}{C_a} \nonumber \\ &=& \frac{2}{K(K-1)} \left ( \sum_{a=1, a \neq k, a \neq \ell}^K \frac{1}{C_a} \right ) {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{2}{K(K-1)} \left ( \Gamma - \frac{1}{C_k} - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} . \label{eq:ZZ2}\end{aligned}$$ In a similar way, if $\nu^\star_s$ is in $\Sigma_0$ but $\nu_t$ is not in $\Sigma_0$, then $\nu_0$ is necessarily the other element in $\Sigma_0$, and we get $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ k \in \Sigma_0 , \ell \notin \Sigma_0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \sum_{b=1, b \neq k, b \neq \ell}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ \Sigma_0 = \{ k, b \} \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \sum_{b=1, b \neq k, b \neq \ell}^K {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ \Sigma_0 = \{ k, b \} \right]} \frac{ \sigma_0 }{ C_b} }\right]} \nonumber \\ &=& \sum_{b=1, b \neq k, b \neq \ell}^K {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} {{\mathbb{P}}\left[{ \Sigma_0 = \{ b, k \} }\right]} \cdot \frac{{{\mathbb{E}}\left[{\sigma_0}\right]}}{C_b} \nonumber \\ &=& \frac{2}{K(K-1)} \left ( \sum_{b=1, b \neq k, b \neq \ell}^K \frac{1}{C_b} \right ) {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]} \nonumber \\ &=& \frac{2}{K(K-1)} \left ( \Gamma - \frac{1}{C_k} - \frac{1}{C_\ell} \right ) {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{\sigma_0}\right]}. \label{eq:ZZ3}\end{aligned}$$ Finally, when neither $\nu^\star_s$ nor $\nu_t$ are in $\Sigma_0$, then $\nu_0 = \nu^\star_0$, whence $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ k \notin \Sigma_0 , \ell \notin \Sigma_0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ k \notin \Sigma_0 , \ell \notin \Sigma_0 \right]} \frac{ \sigma_0 }{ C_{\nu^\star_0}} }\right]} \nonumber \\ &=& \sum_{a=1, a \neq k, a \neq \ell}^K \sum_{b=1, b \neq a, b \neq k, b \neq \ell}^{a-1} \left ( \ldots \right )_{k\ell}\end{aligned}$$ with $$\begin{aligned} \left ( \ldots \right )_{k\ell} &=& {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ \Sigma_0 = \{ a,b \} \right]} \frac{ \sigma_0 }{ C_{\nu^\star_0}} }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} {{\mathbb{P}}\left[{ \Sigma_0 = \{ a, b \} }\right]} \frac{1}{2} \cdot \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]} \nonumber \\ &=& \frac{1}{K(K-1)} {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]}. \label{eq:ZZ4}\end{aligned}$$ It then follows that $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} {{\bf 1}\left[ k \notin \Sigma_0 , \ell \notin \Sigma_0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \frac{1}{K(K-1)} \cdot H_{k\ell} \cdot {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]} \label{eq:ZZ5}\end{aligned}$$ where we have set $$H_{k\ell} = \sum_{a=1, a \neq k, a \neq \ell}^K \left ( \sum_{b=1, b \neq a, b \neq k, b \neq \ell}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ). \label{eq:Defn+H_kl}$$ Collecting terms (\[eq:ZZ1\])-(\[eq:ZZ5\]), we conclude from (\[eq:ZZ0\]) that $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \frac{ 1 }{K(K-1)} \cdot H^\star_{k\ell} \cdot {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]}\end{aligned}$$ with $$H^\star_{k\ell} = H_{k\ell} + 4 \Gamma - 3 \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) .$$ In Appendix D we show that $${{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} = \frac{1}{K(K-1)} {{\mathbb{P}}\left[{ s+ \frac{\sigma_s}{C_{k}} > 0 }\right]} {{\mathbb{P}}\left[{t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \label{eq:ProbabilityZ}$$ and the conclusion $$\begin{aligned} & & {{\mathbb{E}}\left[{ {{\bf 1}\left[ \nu^\star_s = k\right]} {{\bf 1}\left[s+ \frac{\sigma_s}{C_{k}} > 0\right]} {{\bf 1}\left[\nu_t = \ell \right]} {{\bf 1}\left[ t + \frac{\sigma_t}{C_{\ell}} > 0 \right]} \frac{ \sigma_0 }{ C_{\nu_0}} }\right]} \nonumber \\ &=& \frac{ H^\star_{k\ell} }{K^2(K-1)^2} \cdot {{\mathbb{P}}\left[{ s+ \frac{\sigma_s}{C_{k}} > 0 }\right]} {{\mathbb{P}}\left[{t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \cdot {{\mathbb{E}}\left[{ \sigma_0 }\right]} \label{eq:ZZ6}\end{aligned}$$ follows. In Appendix E we also show that $$H_{k\ell} = (K-3) \left ( \Gamma - \frac{1}{C_k} - \frac{1}{C_\ell} \right ) \label{eq:H_kell}$$ so that $$\begin{aligned} H^\star_{k\ell} &=& H_{k\ell} + 4 \Gamma - 3 \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) \nonumber \\ &=& (K-3) \left ( \Gamma - \frac{1}{C_k} - \frac{1}{C_\ell} \right ) + 4 \Gamma - 3 \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) \nonumber \\ &=& (K+1) \Gamma - K \left ( \frac{1}{C_k} + \frac{1}{C_\ell} \right ) = \Sigma_{k\ell}\end{aligned}$$ with $\Sigma_{k\ell}$ given by (\[eq:Sigma\_kl\]). Inserting this last expression into (\[eq:ZZ6\]) yields the desired conclusion (\[eq:Case2B+2\]). [\ ]{} Acknowledgements {#acknowledgements .unnumbered} ================ This research was partially carried out while the authors were in residence under the Saiotek Program on “Virtual Machines for the Traffic Analysis in High Capacity Networks” was partially supported by grant SA-2012/00331 (Department of Industry, Innovation, Trade and Tourism, Basque Government). The work of Ane Izagirre was also supported by the grants of the Ecole Doctorale EDSYS and INP. References {#references .unnumbered} ========== Appendix A: A proof of (\[eq:VAR\]) {#App:A .unnumbered} =================================== We are interested in assessing the range of values for $\rm{Var}[X]$ under the constraint ${{\mathbb{E}}\left[{X}\right]} = \frac{\Gamma}{K}$ for some $\Gamma > 0$. This amounts to considering the expression $$\frac{1}{K} \sum_{k=1}^K \frac{1}{C^2_k} - \left ( \frac{1}{K} \sum_{k=1}^K \frac{1}{C_k} \right )^2, \quad {{\mbox{\boldmath{$C$}}}} = (C_1, \ldots , C_K) \in (0,\infty)^K$$ under the constraint $$\sum_{k=1}^K \frac{1}{C_k} = \Gamma.$$ Defining the mapping $g: (0,\infty)^K \rightarrow \mathbb{R}_+ $ as $$g({{\mbox{\boldmath{$C$}}}}) \equiv \sum_{k=1}^K \frac{1}{C^2_k} , \quad {{\mbox{\boldmath{$C$}}}} = (C_1, \ldots , C_K) \in (0,\infty)^K,$$ we need only focus on studying the range of $ \left \{ g({{\mbox{\boldmath{$C$}}}}) : \ {{\mbox{\boldmath{$C$}}}} \in \mathcal{C}(\Gamma) \right \} $ where the constraint set $\mathcal{C}(\Gamma)$ is given by $$\mathcal{C}(\Gamma) = \left \{ {{\mbox{\boldmath{$C$}}}} \in (0,\infty)^K: \ \sum_{k=1}^K \frac{1}{C_k} = \Gamma \right \} .$$ This issue is more easily understood with the help of the change of variables $T: (0,\infty)^K \rightarrow (0,\infty)^K$ given by $$T({{\mbox{\boldmath{$C$}}}} ) = \left ( \frac{1}{C_1} , \ldots , \frac{1}{C_K} \right ), \quad {{\mbox{\boldmath{$C$}}}} = (C_1, \ldots , C_K) \in (0,\infty)^K.$$ The transformation $T$ is a bijection from $(0,\infty)^K$ into itself (with inverse $T^{-1} = T$). Note that $T$ puts the set $\mathcal{C}(\Gamma)$ into one-to-one correspondence with the set $\mathcal{X}(\Gamma)$ given by $$\mathcal{X}(\Gamma) = \left \{ {{\mbox{\boldmath{$x$}}}} = (x_1, \ldots , x_K) \in (0,\infty)^K: \ \sum_{k=1}^K x_k = \Gamma \right \} .$$ If we define the mapping $h: \mathbb{R}_+^K \rightarrow \mathbb{R}_+ $ by $$h({{\mbox{\boldmath{$x$}}}}) \equiv \sum_{k=1}^K x_k^2 , \quad {{\mbox{\boldmath{$x$}}}} = (x_1, \ldots , x_K) \in \mathbb{R}_+^K ,$$ then we obviously have $$g({{\mbox{\boldmath{$C$}}}}) = h( T({{\mbox{\boldmath{$C$}}}})), \quad {{\mbox{\boldmath{$C$}}}} \in (0,\infty)^K. \label{eq:FromGtoH}$$ Moreover it holds that $ \left \{ g({{\mbox{\boldmath{$C$}}}}) : \ {{\mbox{\boldmath{$C$}}}} \in \mathcal{C}(\Gamma) \right \} = \left \{ h({{\mbox{\boldmath{$x$}}}}) : \ {{\mbox{\boldmath{$x$}}}} \in \mathcal{X}(\Gamma) \right \} $ since $T( \mathcal{C}(\Gamma) ) = \mathcal{X}(\Gamma)$. With $\prec$ denoting majorization [@MarshallOlkin+Book p. 7], whenever ${{\mbox{\boldmath{$x$}}}}_1 \prec {{\mbox{\boldmath{$x$}}}}_2$ in $\mathbb{R}_+^K$, we have $$h({{\mbox{\boldmath{$x$}}}}_1) \leq h( {{\mbox{\boldmath{$x$}}}}_2 ) \label{eq:SchurConvexityInequality}$$ by the Schur-convexity of the function $h: \mathbb{R}_+^K \rightarrow \mathbb{R}_+$ (inherited from the convexity of $t \rightarrow t^{2}$) [@MarshallOlkin+Book Prop. C.1, p. 64]; see also [@MarshallOlkin+Book p. 54] for the definition of Schur-convexity. The most balanced“ element of $\mathcal{X}(\Gamma)$ is the vector ${{\mbox{\boldmath{$x$}}}}_\star$ given by $${{\mbox{\boldmath{$x$}}}}_\star = \frac{\Gamma}{K} \left ( 1, \ldots , 1 \right ).$$ It represents the smallest” element in the constraint set in the sense of majorization [@MarshallOlkin+Book p. 7] – We have $ {{\mbox{\boldmath{$x$}}}}_\star \prec {{\mbox{\boldmath{$x$}}}}$ for any ${{\mbox{\boldmath{$x$}}}}$ in $\mathcal{X}(\Gamma)$, whence $h({{\mbox{\boldmath{$x$}}}}_\star) \leq h( {{\mbox{\boldmath{$x$}}}} )$ by (\[eq:SchurConvexityInequality\]). With ${{\mbox{\boldmath{$C$}}}}_\star$ given by (\[eq:BalancedCapacities\]) we see from (\[eq:FromGtoH\]) that $$g( {{\mbox{\boldmath{$C$}}}}_\star) \leq g( {{\mbox{\boldmath{$C$}}}}), \quad {{\mbox{\boldmath{$C$}}}} \in \mathcal{C}(\Gamma)$$ since ${{\mbox{\boldmath{$C$}}}}_\star = T( {{\mbox{\boldmath{$x$}}}}_\star)$. This establishes the lower bound in (\[eq:VAR\]) in agreement with the earlier discussion concerning the zero variance when all the capacities are identical. We now turn to the upper bound: For each $k=1, \ldots , K$, introduce the vector ${{\mbox{\boldmath{$x$}}}}^\star_k$ in $\mathbb{R}_+^K$ given by $${{\mbox{\boldmath{$x$}}}}^\star_k = \Gamma {{\mbox{\boldmath{$e$}}}}_k$$ with ${{\mbox{\boldmath{$e$}}}}_1, \ldots , {{\mbox{\boldmath{$e$}}}}_K$ as defined in Section \[sec:Discussion\]. It is well known [@MarshallOlkin+Book p. 7] that ${{\mbox{\boldmath{$x$}}}} \prec {{\mbox{\boldmath{$x$}}}}^\star_k$ for every ${{\mbox{\boldmath{$x$}}}}$ in $ \mathcal{X}(\Gamma) $, so that $h({{\mbox{\boldmath{$x$}}}}) \leq h({{\mbox{\boldmath{$x$}}}}^\star_k)$ for every ${{\mbox{\boldmath{$x$}}}}$ in $ \mathcal{X}(\Gamma) $. Although ${{\mbox{\boldmath{$x$}}}}^\star_k$ is [*not*]{} an element of $\mathcal{X}(\Gamma) $, we nevertheless have $$\sup \left \{ h({{\mbox{\boldmath{$x$}}}}) : \ {{\mbox{\boldmath{$x$}}}} \in \mathcal{X}(\Gamma) \right \} = \sup \left \{ h({{\mbox{\boldmath{$x$}}}}) : \ {{\mbox{\boldmath{$x$}}}} \in \overline{ \mathcal{X}(\Gamma) } \right \} = h({{\mbox{\boldmath{$x$}}}}^\star _k)$$ by the continuity of $h$; the closure $\overline{ \mathcal{X}(\Gamma) } $ of $\mathcal{X}(\Gamma)$ is given by $ \overline{ \mathcal{X}(\Gamma) } = \left \{ {{\mbox{\boldmath{$x$}}}} \in \mathbb{R}_+^K : \ \sum_{k=1}^K x_k = \Gamma \right \}$. In particular, we have $$\begin{aligned} \sup \left \{ g({{\mbox{\boldmath{$C$}}}}) : \ {{\mbox{\boldmath{$C$}}}} \in \mathcal{C}(\Gamma) \right \} = h({{\mbox{\boldmath{$x$}}}}^\star _k) = \Gamma^2.\end{aligned}$$ Note that for each $k=1, \ldots , K$ there is no vector ${{\mbox{\boldmath{$C$}}}}^\star_k$ in $\mathcal{C}(\Gamma) $ such that ${{\mbox{\boldmath{$x$}}}}^\star_k = T( {{\mbox{\boldmath{$C$}}}}^\star_k )$. However, there are vectors in $\mathcal{C}(\Gamma) $ whose value under $g$ will come arbitrarily close to $\Gamma^2$. For instance, consider the vectors $${{\mbox{\boldmath{$x$}}}}_{k,a} = a\Gamma {{\mbox{\boldmath{$e$}}}}_k + \frac{(1-a)\Gamma}{K-1} \sum_{\ell=1, \ell \neq k }^K {{\mbox{\boldmath{$e$}}}}_\ell , \quad \begin{array}{c} 0 < a < 1 \\ k=1, \ldots , K. \\ \end{array}$$ These vectors are elements of $\mathcal{X}(\Gamma) $ with $\lim_{a \uparrow 1 } {{\mbox{\boldmath{$x$}}}}_{k,a} = {{\mbox{\boldmath{$x$}}}}^\star_k$, hence $\lim_{a \uparrow 1 } h( {{\mbox{\boldmath{$x$}}}}_{k,a} ) = h( {{\mbox{\boldmath{$x$}}}}^\star_k)$ by continuity. As we recall the definition (\[eq:C\_ak\]) we check that $${{\mbox{\boldmath{$C$}}}}_{k,a} = \frac{1}{a\Gamma} {{\mbox{\boldmath{$e$}}}}_k + \frac{K-1}{(1-a)\Gamma} \sum_{\ell=1, \ell \neq k }^K {{\mbox{\boldmath{$e$}}}}_\ell = T( {{\mbox{\boldmath{$x$}}}}_{k,a} ), \quad \begin{array}{c} 0 < a < 1 \\ k=1, \ldots , K \\ \end{array}$$ so that $h( {{\mbox{\boldmath{$x$}}}}_{k,a} ) = g( {{\mbox{\boldmath{$C$}}}}_{k,a} )$ for all $0 < a < 1$. It follows that $$\lim_{a \uparrow 1 } g( {{\mbox{\boldmath{$C$}}}}_{k,a} ) = \lim_{a \uparrow 1 } h( {{\mbox{\boldmath{$x$}}}}_{k,a} ) = h( {{\mbox{\boldmath{$x$}}}}^\star_k) = \Gamma^2$$ and this completes the discussion of the upper bound at (\[eq:VAR\]). [\ ]{} Appendix B: A proof of (\[eq:Sigma\_k\]) {#App:B .unnumbered} ======================================== Fix $k=1,2, \ldots , K$. Elementary calculations give $$\begin{aligned} & & \sum_{ a=1 , a \neq k }^K \left (\sum_{b=1, b \neq k}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) \nonumber \\ &=& \sum_{a=1}^{k-1} \left (\sum_{b=1, b \neq k}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) + \sum_{a=k+1}^K \left (\sum_{b=1, b \neq k}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) \nonumber \\ &=& \sum_{a=1}^{k-1} \left (\sum_{b=1}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) + \sum_{a=k+1}^K \left ( - \frac{1}{C_a} - \frac{1}{C_k} + \sum_{b=1}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \right ) \nonumber \\ &=& \sum_{a=1}^{k-1} \frac{a-1}{C_a} + \sum_{a=k+1}^K \frac{a-1}{C_a} + \sum_{a=1}^{k-1} \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) + \sum_{a=k+1}^K \left ( - \frac{1}{C_a} - \frac{1}{C_k} + \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) \nonumber \\ &=& \sum_{a=1}^{K} \frac{a-1}{C_a} - \frac{k-1}{C_k} + \sum_{a=1}^{K} \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - \sum_{b=1}^{k-1} \frac{1}{C_b} - \sum_{a=k+1}^K \left (\frac{1}{C_a} + \frac{1}{C_k} \right ) \nonumber \\ &=& \sum_{a=1}^{K} \frac{a-1}{C_a} - \frac{k-1}{C_k} + \sum_{a=1}^{K} \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - \sum_{a=1}^{K} \frac{1}{C_a} - \frac{K-k-1}{C_k} \nonumber \\ &=& \sum_{a=1}^{K} \frac{a-1}{C_a} - \frac{K-2}{C_k} + \sum_{a=1}^{K} \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - \sum_{a=1}^{K} \frac{1}{C_a} \nonumber \\ &=& \sum_{a=1}^{K} \frac{a-1}{C_a} - \frac{K-2}{C_k} + \sum_{b=1}^{K-1} \left ( \sum_{a=b+1}^{K} \frac{1}{C_b} \right ) - \sum_{a=1}^{K} \frac{1}{C_a} \nonumber \\ &=& \sum_{a=1}^{K} \frac{a-1}{C_a} - \frac{K-2}{C_k} + \sum_{b=1}^{K-1} \frac{K-b}{C_b} - \sum_{a=1}^{K} \frac{1}{C_a} \nonumber \\ &=& (K-1) \Gamma - \frac{K-2}{C_k} - \Gamma \end{aligned}$$ and the proof of (\[eq:Sigma\_k\]) is complete. [\ ]{} Appendix C: A proof of (\[eq:AppB+1\]) {#App:C .unnumbered} ====================================== We are in the situation $s < t < 0$. Fix $k=1, \ldots , K$. Our point of departure is the obvious decomposition $${{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_{\nu^\star_s}} \leq 0, \nu_t = k }\right]} = \sum_{\ell=1, \ell \neq k}^K {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \nu_t = k}\right]} . \label{eq:PieceZ1}$$ Pick $\ell =1, \ldots, K$ distinct from $k$, and note that $$\begin{aligned} {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \nu_t = k}\right]} &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \ell \in \Sigma_t, \nu_t = k}\right]} \nonumber \\ & & ~+ {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \ell \notin \Sigma_t, \nu_t = k}\right]}. \label{eq:PieceZ2}\end{aligned}$$ We examine each term in turn: First, when $\ell$ belongs to $\Sigma_t$ with $\nu^\star_s = \ell$, then $\nu_t = k$ happens only if $\nu^\star_s = \ell $ and $\Sigma_t = \{k,\ell \}$, whence $$\begin{aligned} {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \ell \in \Sigma_t, \nu_t = k}\right]} &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \Sigma_t = \{ k,\ell \}, \nu_t = k}\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \Sigma_t = \{ k,\ell \}}\right]} \nonumber \\ &=& \frac{2}{K^2(K-1)} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0}\right]}. \label{eq:PieceX}\end{aligned}$$ Next, we have $\nu_t = \nu^\star_t$ when $\nu^\star_s$ is not in $\Sigma_t$, so that $$\begin{aligned} \lefteqn{ {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \ell \notin \Sigma_t, \nu_t = k}\right]} } & & \nonumber \\ &=& \sum_{a=1, a\neq k, a \neq \ell}^{K} {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \ell \notin \Sigma_t, \Sigma_t = \{k,a\}, \nu^\star_t = k}\right]} \nonumber \\ &=& \frac{1}{2} \sum_{a=1, a\neq k, a \neq \ell}^{K} {{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \Sigma_t = \{k,a\} }\right]} \nonumber \\ &=& \frac{1}{2} \sum_{a=1, a\neq k, a \neq \ell}^{K} \frac{1}{K} \frac{2}{K(K-1)} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \nonumber \\ &=& \frac{1}{K^2(K-1)} \sum_{a=1, a\neq k, a \neq \ell}^{K} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \nonumber \\ &=& \frac{K-2}{K^2(K-1)} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} . \label{eq:PieceY}\end{aligned}$$ Inserting (\[eq:PieceX\]) and (\[eq:PieceY\]) back into (\[eq:PieceZ2\]) we get $${{\mathbb{P}}\left[{ \nu^\star_s = \ell, t < s + \frac{\sigma_s}{C_\ell} \leq 0, \nu_t = k}\right]} = \frac{1}{K(K-1)} {{\mathbb{P}}\left[{ t < s + \frac{\sigma_s}{C_\ell} \leq 0 }\right]} \label{eq:PieceZ2withX+Y}$$ and the desired conclusion (\[eq:AppB+1\]) follows with the help of (\[eq:PieceZ1\]). [\ ]{} Appendix D: A proof of (\[eq:ProbabilityZ\]) {#App:D .unnumbered} ============================================ Recall that we are in the situation $s < t < 0$. Fix distinct $k,\ell =1, \ldots , K$. We need to show that $${{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} = \frac{1}{K(K-1)} {{\mathbb{P}}\left[{ s+ \frac{\sigma_s}{C_{k}} > 0 }\right]} {{\mathbb{P}}\left[{t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \label{eq:ProbabilityZAgain}$$ By arguments used earlier we get $$\begin{aligned} & & {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, k \in \Sigma_t , \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \Sigma_t = \{ k, \ell \} , \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \Sigma_t = \{ k, \ell \} , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& \frac{2}{K^2(K-1)} {{\mathbb{P}}\left[{ s+ \frac{\sigma_s}{C_{k}} > 0 }\right]} {{\mathbb{P}}\left[{t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \label{eq:ProbabilityZZZZ1}\end{aligned}$$ under the enforced independence assumptions. In a similar way, we find $$\begin{aligned} & & {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, k \notin \Sigma_t , \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& \sum_{ a=1, a \neq k, a \neq \ell}^K {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \Sigma_t = \{ a, \ell \} , \nu_t = \ell , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& \frac{1}{2} \sum_{ a=1, a \neq k, a \neq \ell}^K {{\mathbb{P}}\left[{ \nu^\star_s = k, s+ \frac{\sigma_s}{C_{k}} > 0, \Sigma_t = \{ a, \ell \} , t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& \frac{1}{K^2(K-1)} \sum_{ a=1, a \neq k, a \neq \ell}^K {{\mathbb{P}}\left[{ s+ \frac{\sigma_s}{C_{k}} > 0 }\right]} {{\mathbb{P}}\left[{t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \nonumber \\ &=& \frac{K-2}{K^2(K-1)} {{\mathbb{P}}\left[{ s+ \frac{\sigma_s}{C_{k}} > 0 }\right]} {{\mathbb{P}}\left[{t + \frac{\sigma_t}{C_{\ell}} > 0 }\right]} \label{eq:ProbabilityZZZZ2}\end{aligned}$$ under the enforced independence assumptions. Collecting (\[eq:ProbabilityZZZZ1\]) and (\[eq:ProbabilityZZZZ2\]) we conclude to the validity of (\[eq:ProbabilityZAgain\]). [\ ]{} Appendix E: A proof of (\[eq:H\_kell\]) {#App:E .unnumbered} ======================================= To show (\[eq:H\_kell\]) it suffices to establish this fact for $k=1$ and $\ell=2$ – This follows from the fact that the labeling of the servers is arbitrary. Thus, form the definition (\[eq:Defn+H\_kl\]) we get $$\begin{aligned} H_{12} &=& \sum_{a=3}^K \sum_{b=3}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) \nonumber \\ &=& \sum_{a=3}^K \left ( \sum_{b=1}^{a-1} \left ( \frac{1}{C_a} + \frac{1}{C_b} \right ) - \left ( \frac{1}{C_a} + \frac{1}{C_1} + \frac{1}{C_a} + \frac{1}{C_2} \right ) \right ) \nonumber \\ &=& \sum_{a=3}^K \left ( \frac{a-3}{C_a} + \sum_{b=1}^{a-1} \frac{1}{C_b} - \frac{1}{C_1} - \frac{1}{C_2} \right ) \nonumber \\ &=& \sum_{a=3}^K \frac{a-3}{C_a} + \sum_{a=3}^K \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - (K-2) \left ( \frac{1}{C_1} + \frac{1}{C_2} \right ) \nonumber \\ &=& \sum_{a=3}^K \frac{a-3}{C_a} + \sum_{a=1}^K \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - (K-2) \left ( \frac{1}{C_1} + \frac{1}{C_2} \right ) - \frac{1}{C_1} \nonumber \\ &=& \sum_{a=1}^K \frac{a-3}{C_a} + \sum_{a=1}^K \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - (K-2) \left ( \frac{1}{C_1} + \frac{1}{C_2} \right ) - \frac{1}{C_1} - \left ( - \frac{2}{C_1} - \frac{1}{C_2} \right ) \nonumber \\ &=& \sum_{a=1}^K \frac{a-3}{C_a} + \sum_{a=1}^K \left ( \sum_{b=1}^{a-1} \frac{1}{C_b} \right ) - \frac{K-3}{C_1} - \frac{K-3}{C_2} \nonumber \\ &=& \sum_{a=1}^K \frac{a-3}{C_a} + \sum_{b=1}^{K-1} \left ( \sum_{a=b+1}^{K} \frac{1}{C_b} \right ) - \frac{K-3}{C_1} - \frac{K-3}{C_2} \nonumber \\ &=& \sum_{a=1}^K \frac{a-3}{C_a} + \sum_{b=1}^{K-1} \frac{K-b}{C_b} - \frac{K-3}{C_1} - \frac{K-3}{C_2} \nonumber \\ &=& \sum_{a=1}^{K-1} \frac{K-3}{C_a} + \frac{K-3}{C_K} - \frac{K-3}{C_1} - \frac{K-3}{C_2}, \nonumber\end{aligned}$$ whence (\[eq:H\_kell\]) follows as announced. [\ ]{}

--- author: - | S. I. MUSLIH\ *[Department of Physics Al-Azhar University Gaza, Palestine]{}* title: 'The equivalence between the Hamiltonian and Lagrangian formulations for the parametrization invariant theories[^1]' --- 5 cm $\mathbf{Summary}$.- The link between the treatment of singular Lagraingians as field systems and the the canonical Hamiltonian approach is studied. It is shown that the singular Lagrangians as field systems are always in exact agreement with the canonical approach for the parametrization invariant theories. PACS 11.10- Field theory. PACS 11.10. Ef- Lagrangian and Hamiltonian approach. Introduction ============ In previous papers \[1-4\] the Hamilton-Jacobi formulation of constrained systems has been studied. This formulation leads us to obtain the set of Hamilton-Jacobi partial differential equations \[HJPDE\] as follows: &&H\^[’]{}\_(t\_, q\_a, ,) =0,\ &&, =0,n-r+1,...,n, a=1,...,n-r,where $$H^{'}_{\a}=H_{\a}(t_{\bt}, q_a, p_a) + p_{\a},$$ and $H_{0}$ is defined as &&H\_[0]{}= p\_[a]{}w\_[a]{}+ p\_ |\_[p\_=-H\_]{}- L(t, q\_i, , =w\_a),\ &&, =n-r+1,...,n. The equations of motion are obtained as total differential equations in many variables as follows: &&dq\_a=dt\_, dp\_a= -dt\_, dp\_= -dt\_.\ && dz=(-H\_+ p\_a )dt\_;\ &&, =0,n-r+1,...,n, a=1,...,n-rwhere $z=S(t_{\a};q_a)$. The set of equations (4,5) is integrable \[3,4\] if &&dH\^[’]{}\_[0]{}=0,\ &&dH\^[’]{}\_=0, =n-r+1,...,n. If condition (6,7) are not satisfied identically, one considers them as new constraints and again testes the consistency conditions. Hence, the canonical formulation leads to obtain the set of canonical phase space coordinates $q_a$ and $p_a$ as functions of $t_{\a}$, besides the canonical action integral is obtained in terms of the canonical coordinates.The Hamiltonians $H^{'}_{\a}$ are considered as the infinitesimal generators of canonical transformations given by parameters $t_{\a}$ respectively. In ref. \[5\] the singular Lagrangians are treated as field systems. The Euler-Lagrange equations of singular systems are proposed in the form $$\frac{\p }{\p t_{\a}}[\frac{\p L'}{\p (\p_{\a} q_{a})}]- \frac{\p L'}{\p q_{a}}=0,\;\;\; \p_{\a}q_{a} = \frac{\p q_{a}}{\p t_{\a}},$$ with constraints && dG\_[0]{}= -dt,\ &&dG\_= -dt, where && L’(t\_, \_q\_[a]{}, , q\_[a]{})= L(q\_[a]{}, q\_, = (\_ q\_[a]{})),= ,\ &&G\_= H\_(q\_[a]{}, t\_, p\_[a]{}= ). In order to have a consistant theory, one should consider the variations of the constraints (9), (10). In this paper we would like to study the link between the treatment of singular Lagrangians as field systems and the canonical formalism for the parametrization invariant theories. Prametrization invariant theories as singular systems ====================================================== In ref. \[3\] the canonical method treatment of the parametrization-invariant theories is studied and will be briefly reviewed here. Let us consider a system with th action integral as $$S(q_{i}) =\int dt {\cal L}(q_{i}, \dot{q_{i}}, t),\;\;\;\;i=1,...,n,$$ where $\cal L$ is a regular Lagrangian with Hessian $n$. Parameterize the time $t\rightarrow\tau(t)$, with $\dot{\tau} =\frac{d \tau}{dt}>0$. The velocities $\dot{q_{i}}$ may be expressed as $$\dot{q_{i}}= q_{i}^{'}\dot{\tau},$$ where $ q_{i}^{'}$ are defined as $$q_{i}^{'}= \frac{dq_{i}}{d\tau}.$$ Denote $t= q_{0}$ and $ q_{\m}=(q_{0}, q_{i}),\;\; \m=0, 1,...,n,$ then the action integral (13) may be written as $$S(q_{\m}) =\int d\tau \dot{t}{\cal L}(q_{\m}, \frac{ q_{i}^{'}}{\dot{t}}),$$ which is parameterization invariant since $L$ is homogeneous of first degree in the velocities $ q_{\m}^{'}$ with $L$ given as $$L(q_{\m}, \dot{q_{\m}}) = \dot{t}{\cal L}(q_{\m}, \frac{ q_{i}^{'}}{\dot{t}}).$$ The Lagrangian $L$ is now singular since its Hessian is $n$. The canonical method \[1-4\] leads us to obtain the set of Hamilton-Jacobi partial differential equations as follows: \_[0]{}=&& p\_ -L(q\_[0]{}, q\_[i]{}, , = w\_[i]{}) + p\_[i]{}\^q\_[i]{}\^[’]{} +\ && p\_[t]{}\_[p\_[t]{}= -H\_[t]{}]{}=0, p\_= ,\ [H’]{}\_[t]{} =&& p\_[t]{} + H\_[t]{}=0, p\_[t]{}=, where $H_{t}$ is defined as $$H_{t}= -{\cal L}(q_{i}, w_{i}) + p_{i}^{\tau} w_{i}.$$ Here, $p_{i}^{\tau}$ and $p_{t}$ are the generalized momenta conjugated to the generalized coordinates $q_{i}$ and $t$ respectively. The equations of motion are obtained as total differential equations in many variables as follows: &&dq\^[i]{}= d+ dq\^[0]{}= dq\^[0]{},\ &&dp\^[i]{}=- d+ dq\^[0]{}= - dq\^[0]{},\ &&dp\_[t]{}=- d+ dq\^[0]{}= 0. Since $$d{H'}_{t} = dp_{t} + H_{t},$$ vanishes identically, this system is integrable and the canonical phase space coordinates $q_{i}$ and $p_{i}$ are obtained in terms of the time $(q_{0}=t)$. Now, let us look at the Lagrangian (17) as a field system. Since the rank of the Hessian martix is $n$, this Lagrangian can be be treated as a field system in the form $$q_{i}= q_{i}(\tau, t),$$ thus, the expression $$q_{i}^{'} = \frac{\p q_{i}}{\p \tau} + \frac{\p q_{i}}{\p t}{\dot t},$$ can be replaced in eqn. (17) to obtain the modified Lagrangian $L'$: $$L'= \dot{t}{\cal L}(q_{\m}, \frac{1}{\dot t}(\frac{\p q_{i}}{\p \tau} + \frac{\p q_{i}}{\p t}{\dot t})).$$ Making use of eqn (8), we have $$\frac{\p L'}{\p q_{i}} - \frac{\p}{\p t}(\frac{\p L'}{\p (\frac{\p q_{i}}{\p t})})- \frac{\p}{\p \tau}(\frac{\p L'}{\p (\frac{\p q_{i}}{\p \tau})})=0.$$ Calculations show that eqn. (28) leads to well-known Lagrangian equation as $$\frac{\p {\cal{L}}}{\p q_{i}} - \frac{d}{dt}(\frac{\p {\cal{L}}}{\p (\frac{dq_{i}}{dt})})=0.$$ Using eqn. (20), we have $$H_{t}=- {\cal L} + \frac{\p \cal L}{\p \dot{q_{i}}}\dot{q_{i}},$$ In order to have a consistent theory, one should consider the total variation of $H_{t}$. In fact $$dH_{t}=-\frac{\p \cal{L}}{\p t} dt.$$ Making use of eq. (10), one finds $$dH_{t}=- \frac{\p L'}{\p t}d\tau.$$ Besides, the quantity $H_{0}$ is identically satisfied and does not lead to constriants. One should notice that equations (21,22) are equivalent to equations (28,29). Classical fields as constrained systems ======================================= In the following sections we would like to study the Hamiltonian and the Lagrangian formulations for classical field systems and demonstrating the equivalence between these two formulations for the reparametrization invariant fields. A classical relativistic field $ \f_{i}= \f_{i}(\vec{x}, t)$ in four space-time dimensions may be described by the action functional $$S(\f_{i})= \int dt \int d^{3}x\{{\cal{L}}(\f_{i}, \p_{\m}\f_{i})\},\;\;\m= 0, 1, 2, 3,\;\;i = 1, 2,...,n,$$ which leads to the Euler-Lagrange equations of motion as $$\frac{\p {\cal{L}}}{\p {\f}_{i}} -\p_{\m}[\frac{\p {\cal{L}}}{\p(\p_{\m}\f_{i})}]=0.$$ One can go over from the Lagrangian description to the Hamiltonian description by using the definition $$\pi_{i}=\frac{\p \cal{L}}{\p{\dot{\f_{i}}}},$$ then canonical Hamiltonian is defined as $$H_{0}=\int d^{3}x(\pi_{i} {\dot{\f_{i}}} -\cal{L}).$$ The equations of motion are obtained as && = -,\ &&= . Reparametrization invariant fields ================================== In analogy with the finite dimensional systems, we introduce the reparametrization invariant action for the field system as $$S=\int d\tau\int {\cal{L}}_{R} d^{3}x,$$ where $${\cal{L}}_{R}= {\dot{t}}{\cal{L}}(\f_{i}, \p_{\m}\f_{i}).$$ Following the canonical method \[1-4\], we obtain the set of \[HJPDE\] as && [H’]{}\_[0]{}=\_+ \_[i]{}\^[()]{} +\_[t]{} -\_[R]{}= 0, \_= ,\ &&[H’]{}\_[t]{}= \_[t]{} + H\_[t]{} = 0, \_[t]{}= , where $H_{t}$ is defined as $$H_{t}= -{\cal{L}}(\f_{i}, \p_{\m}\f_{i}) + \pi_{i}^{(\tau)}\frac{d {\f}_{i}}{dt},$$ and $\pi_{i}^{(\tau)}$, $\pi_{t}$ are the generalized momenta conjugated to the generalized coordinates $\f_{i}$ and $t$ respectively. The equations of motion are obtained as &&d\_[i]{}= d+ dt= dt,\ &&d\^[i]{}=- d- dt= - dt,\ &&d\_[t]{}=- d- dt= 0. Now the Euler-Lagrangian equation for the field system reads as $$\frac{\p {\cal{L}}}{\p \f_{i}} - \frac{\p}{\p x^{\m}}(\frac{\p {\cal{L}}}{\p (\frac{\p \f_{i}}{\p x_{\m}})})=0.$$ Again as for the finite dimensional systems, equations (44,45) are equivalent to equations (47) for field systems. Conclusion =========== As it was mentioned in the introduction, if the rank of the Hessian matrix for discrete systems is $(n-r)$; $0< r< n$, then the systems can be treated as field systems \[5\]. The treatment of Lagrangians as field systems is always in exact agreement with the Hamilton-Jacobi treatment for reparametrization invariant theories. The equations of motion (21, 22) are equivalent to the equations of motion (28, 29). Besides the the variations of constraints (31) and (32) are identically satisfied and no further constraints arise. In analogy with the finite dimensional systems, it is observed that the Lagrangian and the Hamilton-Jacobi treatments for the reparametrization invariant fields are in exact agreement. [widest-label]{} GULER Y., Nuovo Cimento B, $\mathbf{107 }$(1992)1389. GULER Y., Nuovo Cimento B, $\mathbf{107}$ (1992)1143. MUSLIH S. I. and GULER Y., Nuovo Cimento B, $\mathbf{110}$ (1995) 307. MUSLIH S. I. and GULER Y., Nuovo Cimento B, $\mathbf{113}$ (1998)277 FARAHAT N. I. and GULER Y., Phys. Rev. A, 51 (1995) 68. [^1]: e-mail: $sami_{-}[email protected]

--- abstract: 'We present the results of a long-term variability survey of the old open cluster NGC 6791. The $BVI$ observations, collected over a time span of 6 years, were analyzed using the ISIS image subtraction package. The main target of our observations were two cataclysmic variables B7 and B8. We have identified possible cycle lenghts of about 25 and 18 days for B7 and B8, respectively. We tentatively classify B7 as a VY Scl type nova-like variable or a Z Cam type dwarf nova. B8 is most likely an SS Cygni type dwarf nova. We have also extracted the light curves of 42 other previously reported variable stars and discovered seven new ones. The new variables show long-period or non-periodic variability. The long baseline of our observations has also allowed us to derive more precise periods for the variables, especially for the short period eclipsing binaries.' author: - 'B. J. Mochejska, K. Z. Stanek' - 'J. Kaluzny' title: 'Long-term variability survey of the old open cluster NGC 6791.' --- [Introduction]{} ================ The open cluster NGC 6791 is unique in many ways. At an age of about 8 Gyr it is believed to be the oldest open cluster in the Galaxy (Chaboyer et al. 1999, Kaluzny & Rucinski 1995). It is also probably the most metal-rich, with \[Fe/H\] estimates ranging from 0.1 to 0.5 dex (Friel et al. 2002, Chaboyer et al. 1999, Peterson & Green 1998, Kaluzny & Rucinski 1995). NGC 6791 possesses two of the three cataclysmic variables (CVs) found in open clusters (Kaluzny et al.1997), with the third residing in M67 (Gilliland et al.1991)[^1]. Variability in NGC 6791 was first studied by Kaluzny & Rucinski (1993), who discovered 17 variable stars, among them one CV candidate. Rucinski, Kaluzny & Hilditch (1996) reported additional 5 variables, including another potential CV. The CV candidates were confirmed spectroscopically by Kaluzny et al. (1997). Mochejska et al. (2002; hereafter M02) discovered additional 47 variables, bringing the total to 69. In this paper we present a long-term variability study of the open cluster NGC 6791. Over a time span of nearly 6 years we have collected 465 observations in $V$, 229 in $I$ and 72 in $B$. Our main motivation was to study the long-term behavior of the two cataclysmic variables. This dataset also offered us the unique possibility of investigating long period and non-periodic variables in the cluster. The paper is organized as follows: Section 2 describes the observations, §3 summarizes the data reduction and variable selection procedures and §4 contains the variable star catalog. Concluding remarks are found in §5. [Observations]{} ================ The data analyzed in this paper were obtained between September 1996 and May 2002 at three telescopes, using four CCD cameras. The observations were collected during: 1. 28 nights between 8 Sep and 23 Oct 1996 on the 1.3 m McGraw-Hill Telescope at the Michigan-Dartmouth-MIT (MDM) Observatory, equipped with the front-illuminated, Loral $2048^2$ CCD “Wilbur” (Metzger, Tonry & Luppino 1993). This dataset will hereafter be referred to as the $MDM$ dataset. 2. 33 nights between 26 Oct 1996 and 9 Oct 1997 on the 1.2 m telescope at the F. L. Whipple Observatory (FLWO), equipped with the thinned, back-illuminated, AR coated Loral $2048^2$ CCD “AndyCam” (Szentgyorgyi et al. 2002), hereafter $AndyCam$ dataset. 3. 47 nights between 19 Sep 1998 and 22 May 2002 on the 1.2 m telescope at FLWO, equipped with the “4Shooter” CCD mosaic with four thinned, back-illuminated, AR coated Loral $2048^2$ CCDs (Szentgyorgyi et al. 2002), hereafter $4Shooter$ dataset. Only data from chip 3, centered on the cluster, were analyzed here. 4. 15 nights between Sep 30 and Nov 7, 1999 and between Oct 14 and 21, 2001 on the the Kitt Peak National Observatory[^2] (KPNO) 2.1 m telescope, equipped with the Tektronix $2048^2$ CCD “T2KA”, hereafter $KPNO$ dataset. Typical exposure times for the 1 m/2 m telescopes, respectively, were 450/600 s in $B$, 450/300 s in $V$ and 300/300 s in $I$. Some short exposures were also collected to investigate the variability of bright stars, which are normally saturated on the longer exposures. Typical 1 m/2 m telescope short exposure times were 45/30 s in $V$ and 30/30 s in $I$. Short $B$ exposures were only taken at KPNO and their exposure times varied between 25 and 180 s. Over 6 years of monitoring of this cluster we have collected 391 long and 74 short exposures in $V$, 140/89 in $I$ and 65/7 in $B$. The number of $BVI$ images in each dataset is listed in Table \[tab:log\]. [Data Reduction]{} ================== [*Photometry*]{} ---------------- The preliminary processing of the CCD frames was performed with the standard routines in the IRAF ccdproc package.[^3] The KPNO data were corrected for CCD non-linearity at this stage, as described by Mochejska et al. (2001). Photometry was extracted using the ISIS image subtraction package (Alard & Lupton 1998, Alard 2000). A brief outline of the applied reduction procedure is presented here. For a more detailed description the reader is referred to M02. The ISIS reduction procedure consists of the following steps: (1) transformation of all frames to a common $(x,y)$ coordinate grid; (2) construction of a reference image from several best exposures; (3) subtraction of each frame from the reference image; (4) selection of stars to be photometered and (5) extraction of profile photometry from the subtracted images. In further discussion we will refer to images called “template” and “reference”. By “template” we mean a single best quality exposure in a filter and dataset combination. A reference frame is a high S/N image constructed from the template and several other high quality exposures processed to match the template point-spread function (PSF) and background level. A template is chosen and a reference image constructed for each of the 19 filter and dataset combinations. All computations were performed with the frames internally subdivided into four sections ([sub\_x=sub\_y=2]{}). Differential brightness variations of the background were fit with a first degree polynomial ([deg\_bg=1]{}). A convolution kernel varying quadratically with position was used ([deg\_spatial=2]{}). The psf width ([psf\_width]{}) was set to 33 pixels for all chips, except for the KPNO data, where it was set to 15 pixels. We used a photometric radius ([radphot]{}) of 5 pixels for 4Shooter and AndyCam, 4 pixels for MDM and 3 pixels for KPNO data. [*Calibration*]{} ----------------- The transformations of instrumental magnitudes to the standard system were derived from observations of 67 stars in 21 Landolt (1992) standard fields, collected with the 4Shooter on 4 November 1999. The following transformations were adopted: $$\begin{aligned} \label{eq:vbv} v = V + 2.1326 + 0.0439 (B-V)+ 0.1480 (X-1.25)\\ \label{eq:bv} b-v = 0.1275 + 0.9151 (B-V)+ 0.1074 (X-1.25)\\ \label{eq:vvi} v = V + 2.1318 + 0.0385 (V-I)+ 0.1478 (X-1.25)\\ \label{eq:vi} v-i = -0.7390 + 1.0240 (V-I)+ 0.0880 (X-1.25)\end{aligned}$$ Figure \[fig:res\] shows the $V$, $B-V$ and $V-I$ transformation residuals of the standard stars as a function of color. The calibrated 4Shooter reference image photometry is shown in V/V-I and V/B-V color-magnitude diagrams (CMDs) in Figure \[fig:cmd\]. The zero points of the light curves of variable stars from the other datasets were adjusted to the 4Shooter data by adding an offset. It was computed as the mean offset for stars within a radius of 300 pixels around the variable, with the rejection of $>3$ sigma outliers. The color term was not taken into account. Additional offsets, if necessary, were applied between overlapping datasets for all variables, ie. MDM-FLWO and 4Shooter-KPNO, and between all datasets in case of periodic variables. The offsets are probably due in large part to the difference in color terms between the chips. Such offsets were added in 14% of the cases and typically were $\leq 0.06$ mag. For some variables the offsets were large (up to 0.5 mag) because of blending with nearby stars in the datasets with inferior seeing. [*Astrometry*]{} ---------------- The transformation from rectangular to equatorial coordinates was derived using 997 transformation stars from the USNO A-2 catalog (Monet et al. 1998) for the 4Shooter $V$-band template. The average difference between the catalog and the computed coordinates for the transformation stars was $0\farcs 14$ in right ascension and $0\farcs 12$ in declination. [*Variability Search*]{} ------------------------ We used the standard ISIS procedure to search for variable stars. For every pixel it computes a median of absolute deviations on all subtracted images and performs a simple rejection of cosmic rays and defects. The result is stored as an image, where the variables are then identified as bright peaks. We used this method to search for new variables in all filter and dataset combinations. We also extracted the light curves for all stars detected by DAOphot (Stetson 1987) on the template frames and searched them for variability using the index $J$ (Stetson 1996), as described in more detail by M02. To search for periodicity we used the analysis of variance statistic (Schwarzenberg-Czerny 1996). \[lcm\] [Variable Star Catalog]{} ========================= We have confirmed the variability of 37 out of 44 previously known variables within our field of view. Of the 7 variables not recovered, three are detached eclipsing binaries with no eclipses observed by us. Many of the variables (V29-V67) were discovered in higher S/N 900 s $R$-band observations (M02), hence it was anticipated that some fainter and/or lower amplitude objects might not be recovered. We have also discovered seven new variables. The properties of the 51 variable stars are discussed in subsections 4.1 - 4.5. Tables \[tab:misc\]-\[tab:pul\] list the following parameters for the variables: the identification number, right ascension, declination, period and its error estimate (for eclipsing and periodic variables), $V$, $I$ and $B$-band magnitude (maximum for non-periodic and eclipsing variables, flux weighted mean for periodic), variability amplitudes in each band (semiamplitudes for periodic variables) and preliminary cluster membership probabilities, kindly provided to us by Dr. Kyle Cudworth. Table \[tab:misc\] contains an additional column with comments on the variability type. The variables are plotted on the CMD in Figure \[fig:cmd\]. Non-periodic variables are indicated by triangles, eclipsing binaries by squares and other periodic variables by circles. Filled symbols indicate membership probability above 75%, open symbols – below 75%, and skeletal – no membership probability estimate. The light curves, mostly in $V$, for a selection of variables are shown in Figures \[lcp\]-\[lcm\]. The $BVI$ light curves and finder charts for all variables and machine-readable versions of tables  \[tab:misc\]-\[tab:pul\] are available for download via ftp from [cfa-ftp.harvard.edu]{} in the [/pub/bmochejs/NGC6791]{} directory. The distribution of the cluster membership probabilities for the variables is shown in Figure \[pro\]. Proper motion data, available for 32 out of 51 variables, are in most cases conclusive in resolving the question of their cluster membership. Of those variables, 21 are likely cluster members and 8 are not, with probabilities in excess of 75% and below 25%, respectively. There are only three variables for which the membership is very uncertain ($25\%<P<75\%$). *Cataclysmic variables* ----------------------- Our main interest was to study the long-term behavior of the two cataclysmic variables B7 (a.k.a. V15) and B8. These are unique objects, as only three such stars have been found and confirmed in all open clusters. B7 is a cluster member, as shown by proper motion data, while for B8 no estimate is available. Both were spectroscopically confirmed to be CVs by Kaluzny et al. (1997). During our monitoring the brighter cataclysmic variable B7 underwent a large drop in brightness, by 3 mag in $V$ and 2 mag in $I$, and three outbursts of about 0.5-1 mag. Smaller amplitude variations, as well as long-term trends are also present in the light curve. The large decline in the MDM data and intermediate brightness at other times would suggest that B7 may be an VY Scl type nova-like (NL) variable, which vary little about their mean magnitude, but occasionally fall in brightness by $>1$ mag (Warner 1995). Another possibility is that it is a Z Cam type dwarf nova (DN), which exhibit periods of standstill at irregular intervals. One notable difference between these two types of CVs is that Z Cam DNe are brighter by 0.5-1 mag in outburst than in the standstill phase, while the apparent outbursts in VY Scl type NL variables reach at most the high state brightness. We have observed the variable at minimum light only once, when it fell from the mean brightness by 3 magnitudes, so we are unable to classify B7 based on this criterion. The 4Shooter and KPNO data phase rather well with a period of $\sim25.4$ days, which is compatible with the typical cycle length of 10-30 days for a Z Cam type DN (Sterken & Jaschek 1996). The other cataclysmic variable, B8, displayed several large outbursts about 3 magnitudes in amplitude, as well as a few smaller ones. The MDM and AndyCam data display incoherent variability, possibly due to flickering. The variability of B8 is reminiscent of an SS Cygni type dwarf nova (Sterken & Jaschek 1996). The 4Shooter and KPNO light curve phases reasonably with a period of $\sim17.73$ days, which if real, is shorter than the typical range of 30-100 days for these variables. *Newly Discovered Variables* ---------------------------- We have identified a total of seven new variables in the field of NGC 6791. V68, located 4 magnitudes above and 0.3 mag to the blue of the turnoff, is a periodic variable, with a period of $\sim15$ days. It may be an RV Tau variable with twice the period. V69 and V70 are very red, bright variables. V69 seems to be periodic, with a period of about 43 or 86 days. The period determinations are very uncertain and different for the $V$ and $B$ bands due to the star’s long period and sparse sampling – the star was unsaturated only on some of the short exposures. V70 is an irregular variable with an amplitude of $\sim0.4$ mag in $V$. The preliminary proper motion data are inconclusive as to the cluster membership of these variables, yielding probabilities of 47% for V69 and 62% for V70. V71 is also irregular, with a $V$-band amplitude of $\sim0.3$ mag. On the CMD it is located near the base of RGB. Proper motion data gives a membership probability of 55% for this variable. V72 and V73 are located about 2 mag above the cluster turnoff. They exhibit irregular variability with $V$-band amplitudes of about $0.04$ and $0.08$ mag, respectively. When the observations between HJD 2452099 and 2452204 are excluded, the $V$-band light curve for V73 can be phased with a period of $\sim34$ days (Figure \[lcp\]). Its membership probability is low (20%). V74 exhibits irregular brightness variations with an amplitude of $0.04$ mag in $V$. On the CMD it is located just blueward of the red clump. It is very likely a cluster member. *Previously known long period and non-periodic variables* --------------------------------------------------------- During previous investigations the cluster was found to possess six other long period and non-periodic variables: V13, V19, V62 and V65-V67. V13 and V19 show irregular variability with sudden rises in brightness by 0.1-0.2 mag in $V$. V13 is a cluster member. We confirm the variability of V62, but due to its faintness and poor quality light curve, we are unable to fathom its nature. V65 seems to be periodic, with a period of $\sim 11.3$ days, when the MDM dataset and May 2002 4Shooter run are excluded (Figure \[lcp\]). It does not belong to the cluster. Of the two variables on the RGB, V66 and V67 (confirmed member), the first is more or less periodic, with a period of $\sim49$ or $\sim99$ days (Figure \[lcp\]), while the second seems to be irregular. Of the five variables classified in previous investigations as detached eclipsing binaries with unknown periods, V10, V18, V20, V21 and V60, we have detected eclipses only in V18 and V20, but were unable to determine convincing periods. V10 seems to have decreased its brightness by 0.1 mag between the 1999 and 2001 seasons. V60 also displays a long-term trend, especially visible in the 4Shooter data. Based on proper motions, V18 and V60 are most likely cluster members. *Previously known eclipsing binaries* ------------------------------------- Thanks to the long time span of our observations, we have derived more precise periods for the eclipsing binaries B4, V1, V3-V7, V9, V11, V12, V16 and V38. We have confirmed the variability of V2 and V14 with the previously reported periods, but were unable to improve their determinations. Variables V3-V5, V9, V16, V38, V12 are likely cluster members, while B4, V6 and V14 are non-members. Figure \[WUMa\] shows a plot of apparent distance modulus $(m-M)_V$ as a function of period for W UMa type contact binaries V1-V7. The absolute $V$ magnitude was derived from the calibration of Rucinski (2000), based on B-V and V-I colors (left and right panel, respectively). A reddening of E(B-V)=0.1 (Chaboyer et al. 1999) was adopted. Filled circles indicate membership probability above 75%, open symbols – below 75%, and skeletal – no membership probability estimate. Of the three variables, which are high probability cluster members based on proper motions, V3 and V4 are within 0.7 mag of the cluster distance modulus $(m-M)_V = 13.42$ (dotted line; Chaboyer et al. 1999) and V5 is about a magnitude brighter than predicted from the calibration. V6, which is a non-member, based on its proper motion, is located $\sim2.5$ magnitudes in front of the cluster. Of the three variables with no membership probability estimate, V2 and V7 are probably located at the distance of the cluster, while V1 seems closer by $\sim2$ magnitudes. V29, which was classified in M02 as an eclipsing binary with a period of $\sim0.4$ days, shows a long-term rise in brightness with an amplitude of $\sim0.5$ mag during our observations (the large amplitude in Table \[tab:ecl\] is mainly due to scatter in the AndyCam data). The eclipses were not recovered. For V31 and V32 we did not detect any coherent variability at the periods determined in M02. These periods are also not confirmed by the additional extensive $R$-band photometry for this cluster, collected by the authors (Mochejska et al. 2003, in preparation), where the variables only show weak long-term trends. V31 is a probable cluster member and V32 is most likely not. V33, previously classified as an EB with a period of $\sim2.37$ days, exhibits long term brightness variations. In the 4Shooter data we have observed a recovery by 0.15 mag to its maximum observed brightness. It also displays quasi-periodic variations with roughly the period reported in M02. This variable is most likely not a member of the cluster. The light curve for V37 is very noisy and we were not able to recover the previously determined period. In the M02 data the variable underwent a 0.1 mag rise in brightness. It appears to have experienced a rapid increase in brightness by 0.6 mag at the beginning of the AndyCam run. *Previously Known Other Periodic Variables* ------------------------------------------- We have confirmed the previously determined periods of V41, V48, V54, V56 and V58. We have confirmed the variability of V17, V45, V46 and V52 with previously derived periods, but were unable to improve their determinations. V45 seems to exhibit long term brightness variations, besides short term variability. We were unable to confirm the periods of V42, V51 and V53, determined by M02. V42 seems to have steadily decreased its brightness by $\sim0.6$ in $V$. Variables V17, V41, V42, V48, V53, V56, V58 are high probability cluster members, and V45 is most likely a non-member. Based on its period and location on the color-magnitude diagrams, V17 might be a member of the newly proposed class of variable stars termed “red stragglers” (Albrow et al. 2001) or “sub-subgiant stars” (Mathieu et al. 2003). To date, six such stars have been found in 47 Tuc (Albrow et al. 2001) and two in M67 (Mathieu et al. 2003). Thus far, the origin and evolutionary status of these stars remains unknown. [Conclusions]{} =============== In this paper we have presented $BVI$ photometry for 51 variable stars in the field of the open cluster NGC 6791. Over 6 years of monitoring of this cluster we have collected 391 long and 74 short exposures in $V$, 140/89 in $I$ and 65/7 in $B$. The light curves of the variables were extracted using the ISIS image subtraction package (Alard & Lupton 1998). The main target of our observations were the two cataclysmic variables B7 and B8. Their light curves show outbursts characteristic of CVs. We have identified possible cycle lenghts of 25 and 18 days for B7 and B8, respectively. We tentatively classify B7 as a VY Scl type nova-like variable or a Z Cam type dwarf nova. More observations of the variable at minimum brightness would help elucidate its nature. B8 is most likely an SS Cygni type dwarf nova. We have also discovered seven new variables: two periodic and five long-period or non-periodic ones. In addition, we have extracted the light curves of 42 other previously reported variable stars. For 31 of them we confirm the variability period, if applicable, and type, given by M02 and for 17 we have derived new, more precise periods. Alard, C., Lupton, R. 1998, ApJ, 503, 325 Alard, C. 2000, A&AS, 144, 363 Albrow, M. D., Gilliland, R. L., Brown, T. M., Edmonds, P. D., Guhathakurta, P., & Sarajedini, A. 2001, , 559, 1060 Chaboyer, B., Green, E. M., Liebert, J. 1999, AJ, 117, 1360 Friel, E. D., Janes, K. A., Tavarez, M., Scott, J., Katsanis, R., Lotz, J., Hong, L., & Miller, N. 2002, , 124, 2693 Gilliland, R. L. et al. 1991, , 101, 541 Kaluzny, J., Ruci[ń]{}ski, S. M. 1995, A&AS, 114, 1 (KR95) Kaluzny, J., Ruci[ń]{}ski, S. M. 1993, MNRAS, 265, 34 Kaluzny, J., Stanek, K. Z., Garnavich, P. M. & Challis, P.  1997, ApJ, 491, 153 Landolt, A. U. 1992, , 104, 340 Mathieu, R. D., van den Berg, M., Torres, G., Latham, D., Verbunt, F., & Stassun, K. 2003, , 125, 246 Mochejska, B. J. & Kaluzny, J. 1999, Acta Astronomica, 49, 351 Mochejska, B. J., Kaluzny, J., Stanek, K. Z., Sasselov, D. D., & Szentgyorgyi, A. H. 2001, , 121, 2032 Mochejska, B. J., Stanek, K. Z., Sasselov, D. D., & Szentgyorgyi, A. H. 2002, , 123, 3460 (M02) Monet, D., et al. 1998, USNO-A2.0, (U.S. Naval Observatory, Washington DC). Peterson, R. C. & Green, E. M. 1998, , 502, L39 Ruci[ń]{}ski, S. M., Kaluzny, J., Hilditch, R. W. 1996, MNRAS, 282, 705 (RKH) Schwarzenberg-Czerny, A. 1996, ApJ, 460, L107 Sterken, C. & Jaschek, C. 1996, Light Curves of Variable Stars, A Pictorial Atlas, ISBN 0521390168, Cambridge University Press, 1996 Stetson, P. B. 1996, PASP, 108, 851 Stetson, P. B. 1987, PASP, 99, 191 Szentgyorgyi, A. H. et al. 2002, in preparation Warner, B. 1995, Cataclysmic Variable Stars (Cambridge: Cambridge University Press) [^1]: Another unconfirmed as of yet candidate was reported by Mochejska & Kaluzny (1999) in NGC 7789. [^2]: Kitt Peak National Observatory is a division of NOAO, which are operated by the Association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation. [^3]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the NSF.

--- abstract: 'We consider the nucleon-nucleon potential in quenched and partially-quenched QCD. The leading one-meson exchange contribution to the potential is found to fall off exponentially at long-distances, in contrast with the Yukawa-type behaviour found in QCD. This unphysical component of the two-nucleon potential has important implications for the extraction of nuclear properties from lattice simulations.' address: | Department of Physics, University of Washington,\ Seattle, WA 98195. author: - '[**Silas R. Beane**]{} and [**Martin J. Savage**]{}' title: 'Nucleon-Nucleon Interactions on the Lattice' --- \#1 8.0cm It is hoped that lattice methods will ultimately lead to an understanding of nuclear physics directly from QCD. To date there exist two distinct approaches to investigating the nucleon-nucleon (NN) system using lattice QCD. The first approach makes use of Lüscher’s finite-volume algorithm, which expresses the energy of a two-particle state as a perturbative expansion in the scattering length divided by the size of the box [@Luscher]. Practical computations require a sufficiently small box, which in turn should be larger than the scattering length. Hence Lüscher’s method is ideal for systems with “natural” scattering lengths – with size of order a characteristic physical length scale – and has been used to study $\pi\pi$ scattering [@Gupta:1993rn]. Of course in the NN system, the scattering lengths are much larger than the characteristic physical length scale given by $\Lambda_{\scriptstyle QCD}^{-1}$. Nevertheless one may expect that at the unphysical values of the pion mass currently used in lattice simulations, the scattering lengths relax to natural values, thus allowing their determination from the lattice. Attempts have been made to compute NN scattering parameters in lattice QCD; Ref. [@fuku] computes the $\si$ and $\siii$ scattering lengths in quenched QCD (QQCD) using Lüscher’s method. A second, less-ambitious approach to the NN system on the lattice is to study the simplified problem of two interacting heavy-light particles [@Richards:1990xf; @Mihaly:1996ue; @Stewart:1998hk; @Michael:1999nq; @Richards:2000ix]. Here the hope is to extract information about the long-distance tail of the potential, which in nature is provided by one-pion exchange (OPE). At present computational efforts are limited by the use of quark masses that are significantly larger than those of nature, typically $m_\pi^{\rm latt.}\sim 500~{\rm MeV}$. In order to make a connection between present (and near-future) lattice calculations and nature, an extrapolation to the physical values of the quarks masses is required. The technology which allows a systematic extrapolation has been developed for QQCD [@Sharpe90; @S92; @BG92; @LS96; @S01a] and partially quenched QCD (PQQCD) [@Pqqcd; @SS01; @CS01a; @BS02a]. In this paper we will consider the potential between two nucleons at leading order (LO) in quenched chiral perturbation theory (Q$\chi$PT) and in partially-quenched chiral perturbation theory (PQ$\chi$PT), the low-energy effective field theories that describe QQCD and PQQCD, respectively. We will be considering the extension from two-flavor QCD, where the three $\pi$’s are pseudo-Goldstone bosons associated with the breaking of chiral symmetry and the $\eta$ is the $SU(2)$ singlet meson whose mass arises mainly from the $U(1)_A$ axial anomaly, to QQCD and PQQCD. Essential to our discussion will be the modifications to the $\eta$ propagator which result from quenching and partial quenching. In QQCD the quarks, $Q=(u,d,{\tilde u},{\tilde d})^T$, live in the fundamental representation of $SU(2|2)$, with the bosonic quarks, ${\tilde u},{\tilde d}$, accounting for the graded structure of the symmetry group. The low-energy effective field theory (EFT) is Q$\chi$PT, whose degrees of freedom are constrained by the chiral group $SU(2|2)_L\otimes SU(2|2)_R$. In the isospin limit the $\eta$ propagator in Q$\chi$PT is given by [@Sharpe90; @S92; @BG92] $$\begin{aligned} G_{\eta\eta} (q^2) & = & {i\over {(q^2- m_\pi^2 +i\epsilon )}} \ +\ {{i(M_0^2-\alpha_\Phi q^2)}\over{(q^2- m_\pi^2 +i\epsilon )^2}} \ \ . \label{eq:etaqqcd}\end{aligned}$$ The hairpin parameters $M_0$ and $\alpha_\Phi$ characterize the dynamics of the singlet field and do not vanish in the chiral limit. The double-pole, or hairpin, structure is an unphysical artifact of quenching. Unfortunately QQCD is not related to QCD by any smooth variation of parameters. This is not the case in PQQCD where the quarks, (four fermionic and two bosonic) $Q=(u,d,j,l,{\tilde u},{\tilde d})^T$ live in the fundamental representation of the graded symmetry group $SU(4|2)$ and the low-energy EFT is PQ$\chi$PT, whose degrees of freedom are constrained by chiral $SU(4|2)_L\otimes SU(4|2)_R$. As the $j,l$ sea quarks become degenerate with the $u,d$ valence quarks, PQQCD smoothly maps to QCD. If the sea quark masses are taken to infinity, PQQCD formally goes to QQCD. In the isospin limit for both valence and sea quarks the $\eta$ propagator in PQ$\chi$PT is $$\begin{aligned} G_{\eta\eta} (q^2) & = & {{i (m_{\eta_j}^2-m_\pi^2)}\over{(q^2- m_\pi^2 +i\epsilon )^2}} \ \ , \label{eq:etapqqcd}\end{aligned}$$ where $m_{\eta_j}$ is the mass of the pseudo-Goldstone bosons composed of sea quarks. Clearly this unphysical double pole vanishes in the QCD limit. The hairpin structure resulting from (partial) quenching will have important implications for lattice simulations of heavy-heavy systems, such as the two-nucleon system or systems composed of two (or more) hadrons containing heavy quarks. [=4.0in ]{} 0.15in .2in The EFT power-counting for NN scattering has been developed in Ref. [@We90; @KSWb; @Beane:2001bc]. While details of the power-counting are channel dependent, LO for the low partial waves consists of OPE and a four-nucleon contact interaction with no derivatives or insertions of the quark mass matrix. This contact operator encodes information about short-distance physics. The EFT power-counting is readily generalized to QQCD and PQQCD, and there is no direct[^1] (partial) quenching of the NN potential at LO in the EFT. This is easily seen by assigning a conserved charge to the various quark types. We will consider PQQCD but the same argument applies to QQCD. For instance, consider the charge assignments $q_V=(1/3,0,0)$, $q_S=(0,1/3,0)$, $q_G=(0,0,1/3)$ for valence-, sea- and ghost-quark number, respectively. Here $q_V$ plays the role of ordinary baryon number. The NN initial state carries total quark-number charge $(2,0,0)$. Baryon states with two valence quarks and either one ghost quark or one sea quark carry charge $(2/3,0,1/3)$ or $(2/3,1/3,0)$, respectively, while meson states with one valence quark and either one ghost or one sea anti-quark carry charge $(1/3,0,-1/3)$ or $(1/3,-1/3,0)$, respectively. It follows that the NN potential must have NN in the initial and in the final state. Of course the potential is directly (partially) quenched, but only through loops, which appear beyond LO in the EFT, as in fig. (\[fig:NNpotential2\]b). Direct (partial) quenching modifies the tree-level potential through higher fock components. For instance, one can have a final state of one nucleon and one baryon composed of two valence quarks and one sea or ghost quark and one meson composed of one valence quark and one sea or ghost anti-quark. This contribution can arise either from meson-exchange graphs, as in fig. (\[fig:NNpotential2\]a), or from four-nucleon contact operators with one insertion of the quark mass matrix or one insertion of a chiral covariant derivative. In either case, the external meson legs at the level of the potential must be reabsorbed by the final state nucleons. In contrast with the “potential” pions exchanged between the baryons, these “radiation” pions generate contributions which are subleading in the EFT. Hence at LO there is no direct (partial) quenching and the NN potential is given by the one meson exchange graphs of fig. (\[fig:NNpotential\]). However, there are hairpin contributions to the potential at LO. [=4.5in ]{} 0.15in .2in For simplicity we work in the isospin limit for both the valence and the sea quarks. The diagrams in fig. (\[fig:NNpotential\]) give the LO meson-exchange contributions to the NN potential. In coordinate space the NN potential is $$\begin{aligned} V^{(Q)} (r)\ =\ {1\over 8\pi f^2}\ {\bf\sigma}_1\cdot {\bf\nabla} {\bf\sigma}_2\cdot {\bf\nabla} \left( g_A^2\ {{\bf\tau}_1\cdot{\bf\tau}_2\over r} \ +\ g_0^2\ {1-\alpha_\Phi\over r} \ -\ g_0^2\ {M_0^2 - \alpha_\Phi m_\pi^2\over 2 m_\pi} \right) \ e^{-m_\pi r} \ \ ,\end{aligned}$$ in QQCD, where the axial coupling to the pions is $g_A$, while the coupling to the $\eta$ is $g_0$. However, one should keep in mind that there is no relation between axial couplings in QQCD and those of QCD. In PQQCD, we find that $$\begin{aligned} V^{(PQ)} (r)\ =\ {1\over 8\pi f^2}\ {\bf\sigma}_1\cdot {\bf\nabla} {\bf\sigma}_2\cdot {\bf\nabla} \left( g_A^2\ {{{\bf\tau}_1\cdot{\bf\tau}_2}\over r} - g_0^2\ {{\left(m_{\eta_j}^2-m_\pi^2\right)}\over{2 m_\pi}} \right) \ e^{-m_\pi r} \ \ ,\end{aligned}$$ where the couplings $g_A$ and $g_0$ in this potential should be perturbatively close to those of QCD. Notice that effects due to partial-quenching (but not quenching) disappear in the short-distance limit, and are irrelevant to the renormalization of the potential. This in turn implies that the four-nucleon contact operator, that appears in PQQCD at the same order in the EFT as OPE, is that of QCD [@We90; @KSWb; @Beane:2001bc]. In contrast, the long-distance behaviour of the potentials are dominated by the hairpin contributions, $$\begin{aligned} V (r)\ \rightarrow\ -{g_0^2\over 16\pi f^2}\ {\bf\sigma}_1\cdot \hat{\bf r} {\bf\sigma}_2\cdot \hat{\bf r}\ m_\pi \ e^{-m_\pi r} \ \cases{ \left( M_0^2 - \alpha_\Phi m_\pi^2 \right)\ , & {\rm QQCD};\cr \left(m_{\eta_j}^2-m_\pi^2\right)\ ,& {\rm PQQCD} \ \ \ ,\cr} \end{aligned}$$ which has both a central piece and a tensor piece. Both quenching and partial-quenching lead to an NN potential which falls off [*exponentially*]{} at long distances, as opposed to the physical Yukawa fall-off. In nature there is a fine-tuning between the long-distance Yukawa part of the potential due to OPE, and the short-distance part which is encoded in the contact operator which appears at the same order in the EFT. It is this fine-tuning which leads, for instance, to a shallowly bound deuteron. It is clear that this fine-tuning does not exist for QQCD, and quickly disappears in PQQCD as one moves away from the QCD limit. In conclusion, multi-nucleon quenched and partially-quenched simulations on the lattice are complicated by the presence of a long-distance exponential tail, that disturbs a fine-tuning which is a key feature of nuclear physics. It has been shown in Ref. [@Beane:2001bc] that an understanding of the quark-mass dependence of the deuteron binding energy requires the matrix element of an operator that is not constrained by NN scattering data but which in principle can be obtained from lattice QCD. Any meaningful determination of this matrix element from a partially-quenched simulation will have to account for the unphysical, long-distance behavior highlighted in this work. This work is supported in part by the U.S. Dept. of Energy under Grant No. DE-FG03-97ER4014. M. Lüscher, [*Commun. Math. Phys.*]{} [**105**]{}, 153 (1986). R. Gupta, A. Patel and S. R. Sharpe, [*Phys. Rev.*]{} [**D48**]{}, 388 (1993). M. Fukugita, Y. Kuramashi, M. Okawa, H. Mino and A. Ukawa, [*Phys. Rev.*]{} [**D52**]{}, 3003 (1995). D. G. Richards, D. K. Sinclair and D. W. Sivers, [*Phys. Rev.*]{} [**D42**]{}, 3191 (1990). A. Mihaly, H. R. Fiebig, H. Markum and K. Rabitsch, [*Phys. Rev.*]{} [**D55**]{}, 3077 (1997). C. Stewart and R. Koniuk, [*Phys. Rev.*]{} [**D57**]{}, 5581 (1998). C. Michael and P. Pennanen \[UKQCD Collaboration\], [*Phys. Rev.*]{} [**D60**]{}, 054012 (1999). D. G. Richards, [nucl-th/0011012]{}. S. R. Sharpe, [*Nucl. Phys.*]{} [**B17**]{} (Proc. Suppl.), 146 (1990). S. R. Sharpe, [*Phys. Rev.*]{} [**D46**]{}, 3146 (1992). C. Bernard and M. F. L. Golterman, [*Phys. Rev.*]{} [**D46**]{}, 853 (1992). J. N. Labrenz and S. R. Sharpe, [*Phys. Rev.*]{} [**D54**]{}, 4595 (1996). M. J. Savage, [nucl-th/0107038]{}. S. R. Sharpe and N. Shoresh, [*Phys. Rev.*]{} [**D62**]{}, 094503 (2000); [*Nucl. Phys. Proc. Suppl.*]{} [**83**]{}, 968 (2000); M. F. L. Golterman and K.-C. Leung, [*Phys. Rev.*]{} [**D57**]{}, 5703 (1998). S. R. Sharpe, [*Phys. Rev.*]{} [**D56**]{}, 7052 (1997); C. W. Bernard and M. F. L. Golterman, [*Phys. Rev.*]{} [**D49**]{}, 486 (1994). S. R. Sharpe and N. Shoresh, [hep-lat/0011089]{}; [*Phys. Rev.*]{} [**D64**]{}, 114510 (2001). J. -W. Chen and M. J. Savage, [hep-lat/0111050]{}. S. R. Beane and M. J. Savage, [*in preparation*]{}. S. Weinberg, [*Phys. Lett.*]{} [**B251**]{}, 288 (1990); [*Nucl. Phys.*]{} [**B363**]{}, 3 (1991). D. B. Kaplan, M. J. Savage, and M. B. Wise, [*Phys. Lett.*]{} [**B424**]{}, 390 (1998); [*Nucl. Phys.*]{} [**B534**]{}, 329 (1998). S. R. Beane, P. F. Bedaque, M. J. Savage and U. van Kolck, [nucl-th/0104030]{}. [^1]: We define direct (partial) quenching to be (partial) quenching beyond modifications to the singlet propagator.

--- abstract: 'A $c$-coloring of ${G_{n,m}}=[n]\times [m]$ is a mapping of ${G_{n,m}}$ into $[c]$ such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed via the internet to find a 4-coloring of $G_{17,17}$. This attracted considerable attention from the popular mathematics community. A coloring was produced; however, finding it proved to be difficult. The question arises: is the problem of grid coloring is difficult in general? We present three results that support this conjecture: (1) Given a partial $c$-coloring of an ${G_{n,m}}$ grid, can it be extended to a full $c$-coloring? We show this problem is NP-complete. (2) The statement [*${G_{n,m}}$ is $c$-colorable*]{} can be expressed as a Boolean formula with $nmc$ variables. We show that if the ${G_{n,m}}$ is not $c$-colorable then any tree resolution proof of the corresponding formula is of size $2^{\Omega(c)}$. (We then generalize this result for other monochromatic shapes.) (3) We show that any tree-like cutting planes proof that $c+1$ by $c\binom{c+1}{2}+1$ is not $c$-colorable must be of size $2^{\Omega(c^3/\log^2 c)}$. Note that items (2) and (3) yield statements from Ramsey Theory which are of size polynomial in their parameters and require exponential size in various proof systems.' author: - | [Daniel Apon]{} [^1]\ [Univ. of MD at College Park]{} - | [William Gasarch]{} [^2]\ [Univ. of MD at College Park]{} - | [Kevin Lawler]{} [^3]\ [Permanent]{} title: The Complexity of Grid Coloring --- Introduction ============ If $x\in{{\mathbb{N}}}$ then $[x]$ denotes the set $\{1,\ldots,x\}$. ${G_{n,m}}$ is the set $[n]\times[m]$. If $X$ is a set and $k\in{{\mathbb{N}}}$ then $\binom{X}{k}$ is the set of all size-$k$ subsets of $X$. On November 30, 2009 the following challenge was posted on Complexity Blog [@17posedpost]. [**BEGIN EXCERPT**]{} The $17\times17$ challenge: worth \$289.00. I am not kidding. A *rectangle* of ${G_{n,m}}$ is a subset of the form $\{(a,b),(a+c_1,b),(a+c_1,b+c_2),(a,b+c_2)\}$ for some $a, b, c_1, c_2 \in {{\mathbb{N}}}$. A grid ${G_{n,m}}$ is *$c$-colorable* if there is a function $\chi: {G_{n,m}}\rightarrow [c]$ such that there are no rectangles with all four corners the same color. [**The $17\times 17$ challenge:**]{} The first person to email me a 4-coloring of $G_{17,17}$ in LaTeX will win \$289.00. ($289.00$ is chosen since it is $17^2$.) [**END EXCERPT**]{} There are two motivations for this kind of problem. (1) The problem of coloring grids to avoid rectangles is a relaxations of the classic theorem (a corollary of the Gallai-Witt theorem) which states that for a large enough grid any coloring yields a monochromatic square, and (2) grid-coloring problems avoiding rectangles are equivalent to finding certain bipartite Ramsey Numbers. For more details on these motivations, and why the four-coloring of $G_{17,17}$ was of particular interest, see the post [@17posedpost]. or the paper by Fenner, et al [@grid]. Brian Hayes, the Mathematics columnist for Scientific American, publicized the challenge [@17hayes]. Initially there was a lot of activity on the problem. Some used SAT solvers, some used linear programming, and one person offered an exchange: [*buy me a \$5000 computer and I’ll solve it*]{}. Finally in 2012 Bernd Steinbach and Christian Posthoff [@17solvedpaper; @17solvedpost] solved the problem. They used a rather clever algorithm with a SAT solver. They believed that the solution was close to the limits of their techniques. Though this particular instance of the problem was solved, the problem of grid coloring in general seems to be difficult. In this paper we formalize and prove three different results that indicate grid coloring is hard. Grid Coloring Extension is NP-Complete -------------------------------------- Between the problem being posed and resolved the following challenge was posted [@17NP] though with no cash prize. We paraphrase the post. [**BEGIN PARAPHRASE**]{} Let $c,N,M\in{{\mathbb{N}}}$. 1. A mapping $\chi$ of $G_{N,M}$ to $[c]$ is a [*$c$-coloring*]{} if there are no monochromatic rectangles. 2. A partial mapping $\chi$ of $G_{N,M}$ to $[c]$ is [*extendable to a $c$-coloring*]{} if there is an extension of $\chi$ to a total mapping which is a $c$-coloring of $G_{N,M}$. We will use the term [*extendable*]{} if the $c$ is understood. Let $$GCE=\{ (N,M,c,\chi)\mid\chi \hbox{ is extendable} \} .$$ $GCE$ stands for [*Grid Coloring Extension.*]{} [**CHALLENGE:**]{} Prove that $GCE$ is NP-complete. [**END PARAPHRASE**]{} In Section \[se:npc\] we show that $GCE$ is indeed NP-complete. This result may explain why the original $17\times 17$ challenge was so difficult. Then again—it may not. In Section \[se:fpt\] we show that $GCE$ is fixed-parameter tractable. Hence, for a fixed $c$, the problem might not be hard. In Section \[se:open\] we state open problems. There is another reason the results obtained may not be the reason why the $17\times 17$ challenge was hard. The $17\times 17$ challenge can be rephrased as proving that $(17,17,4,\chi)\in GCE$ where $\chi$ is the empty partial coloring. This is a rather special case of $GCE$ since [*none*]{} of the spots are pre-colored. It is possible that $GCE$ in the special case where $\chi$ is the empty coloring is easy. While we doubt this is true, we note that we have not eliminated the possibility. One could ask about the problem $$GC = \{ (n,m,c) \mid {G_{n,m}}\hbox{ is $c$-colorable } \}.$$ However, this does not quite work. If $n,m$ are in unary, then $GC$ is a sparse set. By Mahaney’s Theorem [@Mahaney; @sparseeasy] if a sparse set is NP-complete then $\rm P={{\rm NP}}$. If $n,m$ are in binary, then we cannot show that $GC$ is in NP since the obvious witness is exponential in the input. This formulation does not get at the heart of the problem, since we believe it is hard because the number of possible colorings is large, not because $n,m$ are large. It is an open problem to find a framework within which a problem like $GC$ can be shown to be hard. Grid Coloring is Hard for Tree Res ---------------------------------- The statement [*$G_{n,m}$ is $c$-colorable*]{} can be written as a Boolean formula (see Section \[se:treeres\]). If $G_{n,m}$ is not $c$-colorable then this statement is not satisfiable. A [*Resolution Proof*]{} is a formal type of proof that a formula is not satisfiable. One restriction of this is [*Tree Resolution*]{}. In Section \[se:treeres\] we define all of these terms. We then show that any tree resolution of the Boolean Formula representing [*$G_{n,m}$ is $c$-colorable*]{} requires size $2^{\Omega(c)}$. A Particular Grid Coloring Problem is Hard for Tree-Like Cutting Planes Proofs ------------------------------------------------------------------------------ The statement [*$G_{n,m}$ is $c$-colorable*]{} is equivalent to the statement [*$A\vec x \le \vec b$ has no 0-1 solution*]{} for some matrix $A$ and vector $\vec b$. (Written as $A\vec x \le \vec b \notin SAT$.) It is known [@grid] that $G_{n,m}$ is not $c$-colorable when $n=c+1$ and $m=c\binom{c}{2}+1$. A [*Cutting Planes Proof*]{} is a formal type of proof that $A\vec x \le \vec b \notin SAT$. One restriction of this is [*Tree-like Cutting Plane Proofs*]{}. In Section \[se:cptreeres\] we define all of these terms. We then show that any tree-like CP proof of $A\vec x \le \vec b\notin SAT$, where this is equivalent to $G_{c+1,c\binom{c}{2}+1}$ not being $c$-colorable, requires size $2^{\Omega(c^3/\log^2 c)}$. This lower bound on tree-like CP proofs yields a lower bound on tree-res proofs of the statement that $G_{c+1,c\binom{c}{2}+1}$ is not $c$-colorable of $2^{\Omega(c^3/\log^2 c)}$. This is not too far away from the upper bound of $O(c^4)$. $GCE$ is NP-complete {#se:npc} ==================== \[th:npc\] $GCE$ is NP-complete. Clearly $GCE\in{{\rm NP}}$. Let $\phi(x_1,\ldots,x_n)=C_1\wedge \cdots\wedge C_m$ be a 3-CNF formula. We determine $N,M,c$ and a partial $c$-coloring $\chi$ of $G_{N,M}$ such that $$(N,M,c,\chi)\in GCE \hbox{ iff } \phi\in 3{\hbox{-}}{{\rm SAT}}.$$ The grid will be thought of as a main grid with irrelevant entries at the left side and below, which are only there to enforce that some of the colors in the main grid occur only once. The colors will be $T,F$, and some of the $(i,j)\in G_{N,M}$. We use $(i,j)$ to denote a color for a particular position. The construction is in four parts. We summarize the four parts here before going into details. 1. We will often need to define $\chi(i,j)$ to be $(i,j)$ and then never have the color $(i,j)$ appear in any other cell of the main grid. We show how to color the cells that are not in the main grid to achieve this. While we show this first, it is actually the last step of the construction. 2. The main grid will have $2nm+1$ rows. In the first column we have $2nm$ blank spaces and the space $(1,2nm+1)$ colored with $(1,2nm+1)$. The $2nm$ blank spaces will be forced to be colored $T$ or $F$. We think of the column as being in $n$ blocks of $2m$ spaces each. In the $i$th block the coloring will be forced to be $$\begin{array}{rl} T & \cr F & \cr \vdots & \cr T & \cr F & \cr \end{array}$$ if $x_i$ is to be set to $T$, or $$\begin{array}{rl} F & \cr T & \cr \vdots & \cr F & \cr T & \cr \end{array}$$ if $x_i$ is to be set to $F$. 3. For each clause $C$ there will be two columns. The coloring $\chi$ will be defined on most of the cells in these columns. However, the coloring will extend to these two columns iff one of the literals in $C$ is colored $T$ in the first column. 4. We set the number of colors properly so that the $T$ and $F$ will be forced to be used in all blank spaces. 1\) [**Forcing a color to appear only once in the main grid.**]{} Say we want the cell $(2,4)$ in the main grid to be colored $(2,4)$ and we do not want this color appearing anywhere else in the main grid. We can do the following: add a column of $(2,4)$’s to the left end (with one exception) and a row of $(2,4)$’s below. Here is what we get: $$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|} \hline (2,4) & & & & & & & & \cr \hline (2,4) & && & & & & & \cr \hline T & & (2,4) & & & & & & \cr \hline (2,4) & & & & & & & & \cr \hline (2,4) & & & & & & & & \cr \hline (2,4) & & & & & & & & \cr \hline \hline (2,4) & (2,4) &(2,4)& (2,4) & (2,4) &(2,4) &(2,4)&(2,4)&(2,4) \cr \hline \end{array}$$ (The double lines are not part of the construction. They are there to separate the main grid from the rest.) It is easy to see that in any coloring of the above grid the only cells that can have the color $(2,4)$ are those shown to already have that color. It is also easy to see that the color $T$ we have will not help to create any monochromatic rectangles since there are no other $T$’s in its column. The $T$ we are using [*is*]{} the same $T$ that will later mean TRUE. We could have used $F$. If we used a new special color we would need to be concerned whether there is a monochromatic grid of that color. Hence we use $T$. What if some other cell needs to have a unique color? Lets say we also want to color cell $(5,3)$ in the main grid with $(5,3)$ and do not want to color anything else in the main grid $(5,3)$. Then we do the following: $$\begin{array}{|c|c||c|c|c|c|c|c|c|c|c|c|} \hline (5,3) & (2,4) & & & & & & & & \cr \hline (5,3) & (2,4) & && & & & & & \cr \hline (5,3) & T & & (2,4) & & & & & & \cr \hline T & (2,4) & & & & & (5,3) & & & \cr \hline (5,3) & (2,4) & & & & & & & & \cr \hline (5,3) & (2,4) & & & & & & & & \cr \hline \hline (5,3) & (2,4) & (2,4) &(2,4)& (2,4) & (2,4) &(2,4) &(2,4)&(2,4)&(2,4) \cr \hline (5,3) & (5,3) & (5,3) &(5,3)& (5,3) & (5,3) &(5,3) &(5,3)&(5,3)&(5,3) \cr \hline \end{array}$$ It is easy to see that in any coloring of the above grid the only cells that can have the color $(2,4)$ or $(5,3)$ are those shown to already have those colors. For the rest of the construction we will only show the main grid. If we denote a color as $D$ (short for [*Distinct*]{}) in the cell $(i,j)$ then this means that (1) cell $(i,j)$ is color $(i,j)$ and (2) we have used the above gadget to make sure that $(i,j)$ does not occur as a color in any other cell of the main grid. Note that we when we have $D$ in the $(2,4)$ cell and in the $(5,3)$ cell they denote different colors. 2\) [**Forcing $(x,{\overline{x}})$ to be colored $(T,F)$ or $(F,T)$.**]{} There will be one column with cells labeled by literals. The cells are blank, uncolored. We will call this row [*the literal column*]{}. We will put to the left of the literal column, separated by a triple line, the literals whose values we intend to set. These literals are not part of the construction; they are a visual aid. The color of the literal-labeled cells will be $T$ or $F$. We need to make sure that all of the $x_i$ have the same color and that the color is different than that of ${\overline{x}_i}$. Here is an example which shows how we can force $(x_1,{\overline{x}_1})$ to be colored $(T,F)$ or $(F,T)$. $$\begin{array}{c||c|c|c|} \hline {\overline{x}_1}&\hbox{\ \ } & T & F \cr \hline x_1 & & T & F \cr \hline \end{array}$$ We will actually need $m$ copies of $x_1$ and $m$ copies of ${\overline{x}_1}$. We will also put a row of $D$’s on top which we will use later. We illustrate how to do this in the case of $m=3$. $$\begin{array}{c|||c|c|c|c|c|c|c|c|c|c|c|} \hline &D& D & D &D &D &D & D & D & D & D & D \cr \hline {\overline{x}_1}& & D & D &D &D &D & D & D & D & T & F\cr \hline x_1 & & D & D &D &D &D & D & T & F & T & F\cr \hline {\overline{x}_1}& & D & D &D &D &T & F & T & F & D & D\cr \hline x_1 & & D & D &T &F &T & F & D & D & D & D\cr \hline {\overline{x}_1}& & T & F &T &F & D &D & D & D & D & D \cr \hline x_1 & & T & F &D &D & D &D & D & D & D & D\cr \hline \end{array}$$ We leave it as an exercise to prove that - If the bottom $x_1$ cell is colored $T$ then (1) all of the $x_1$ cells are colored $T$, and (2) all of the ${\overline{x}_1}$ cells are colored $F$. - If the bottom $x_1$ cell is colored $F$ then (1) all of the $x_1$ cells are colored $F$, and (2) all of the ${\overline{x}_1}$ cells are colored $T$. Note that (1) if we want one literal-pair (that is $x_1,{\overline{x}_1}$) then we use two columns, (2) if we want two literal-pairs then we use six columns, and (3) if we want three literal-pairs then we use ten columns. We leave it as an exercise to generalize the construction to $m$ literal-pairs using $2+4(m-1)$ columns. We will need $m$ copies of $x_2$ and $m$ copies of ${\overline{x}_2}$. We illustrate how to do this in the case of $m=2$. We use double lines in the picture to clarify that the $x_1$ and the $x_2$ variables are not chained together in any way. $$\begin{array}{c|||c|c|c|c|c|c|c||c|c|c|c|c|c|} \hline &D& D & D &D &D &D & D & D &D &D &D & D & D \cr \hline {\overline{x}_2}& & D & D &D &D &D & D & D &D &D &D & T & F \cr \hline x_2 & & D & D &D &D &D & D & D &D &T &F & T & F \cr \hline {\overline{x}_2}& & D & D &D &D &D &D & T & F &T &F & D & D \cr \hline x_2 & & D & D &D &D &D &D & T & F & D & D & D & D\cr \hline \hline {\overline{x}_1}& & D & D &D &D &T &F & D & D & D & D & D& D \cr \hline x_1 & & D & D &T &F &T &F & D & D & D & D & D & D \cr \hline {\overline{x}_1}& & T & F &T &F &D &D & D & D & D & D & D & D \cr \hline x_1 & & T & F &D &D &D &D & D & D & D & D & D & D \cr \hline \end{array}$$ We leave it as an exercise to prove that, for all $i\in \{1,2\}$: - If the bottom $x_i$ cell is colored $T$ then (1) all of the $x_i$ cells are colored $T$, and (2) all of the ${\overline{x}_1}$ cells are colored $F$. - If the bottom $x_i$ cell is colored $F$ then (1) all of the $x_i$ cells are colored $F$, and (2) all of the ${\overline{x}_1}$ cells are colored $T$. An easy exercise for the reader is to generalize the above to a construction with $n$ variables with $m$ literal-pairs for each variable. This will take $n(2+4(m-1))$ columns. For the rest of the construction we will only show the literal column and the clause columns (which we define in the next part). It will be assumed that the $D$’s and $T$’s and $F$’s are in place to ensure that all of the $x_i$ cells are one of $\{T,F\}$ and the ${\overline{x}_i}$ cells are the other color. 3\) [**How we force the coloring to satisfy ONE clause**]{} Say one of the clauses is $C_1=L_1 \vee L_2 \vee L_3$ where $L_1,L_2$, and $L_3$ are literals. Pick an $L_1$ row, an $L_2$ row, and an $L_3$ row. We will also use the top row, as we will see. For other clauses you will pick other rows. Since there are $m$ copies of each variable and its negation this is easy to do. The two $T$’s in the top row in the next picture are actually in the very top row of the grid. We put a $C_1$ over the columns that will enforce that $C_1$ is satisfied. We put $L_1$, $L_2$, and $L_3$ on the side to indicate the positions of the variables. These $C_1$ and the $L_i$ outside the triple bars are not part of the grid. They are a visual aid. $$\begin{array}{c|||c|c|c|c|c|c|} & & C_1 & C_1 \cr \hline \hline \hline & D & T &T\cr \hline L_3 & & D & F \cr \hline L_2 & & & \cr \hline L_1 & & F & D \cr \hline \end{array}$$ [**Claim 1:**]{} If $\chi'$ is a 2-coloring of the blank spots in this grid (with colors $T$ and $F$) then it CANNOT have the $L_1,L_2,L_3$ spots all colored $F$. [**Proof of Claim 1:**]{} Assume, by way of contradiction, that that $L_1, L_2,L_3$ are all colored $F$. Then this is what it looks like: $$\begin{array}{c|||c|c|c|c|c|c|} & & C_1 & C_1 \cr \hline \hline \hline & D & T &T\cr \hline L_3 & F & D & F \cr \hline L_2 & F & & \cr \hline L_1 & F & F & D \cr \hline \end{array}$$ The two blank spaces are both FORCED to be $T$ since otherwise you get a monochromatic rectangle of color $F$. Hence we have $$\begin{array}{c|||c|c|c|c|c|c|} & & C_1 & C_1 \cr \hline \hline \hline & D & T &T\cr \hline L_3 & F & D & F \cr \hline L_2 & F & T & T \cr \hline L_1 & F & F & D \cr \hline \end{array}$$ This coloring has a monochromatic rectangle which is colored $T$. This contradicts $\chi'$ being a 2-coloring of the blank spots. [**End of Proof of Claim 1**]{} We leave the proof of Claim 2 below to the reader. [**Claim 2:**]{} If $\chi'$ colors $L_1, L_2, L_3$ anything except $F,F,F$ then $\chi'$ can be extended to a coloring of the grid shown. [**Upshot:**]{} A 2-coloring of the grid is equivalent to a satisfying assignment of the clause. Note that each clause will require 2 columns to deal with. So there will be $2m$ columns for this. 4\) [**Putting it all together**]{} Recall that $\phi(x_1,\ldots,x_n)=C_1\wedge \cdots \wedge C_m$ is a 3-CNF formula. We first define the main grid and later define the entire grid and $N,M,c$. The main grid will have $2nm+1$ rows and $n(4m-2)+2m+1$ columns. The first $n(4m-2)+1$ columns are partially colored using the construction in Part 2. This will establish the literal column. We will later set the number of colors so that the literal column must use the colors $T$ and $F$. For each of the $m$ clauses we pick a set of its literals from the literals column. These sets-of-literals are all disjoint. We can do this since we have $m$ copies of each literal-pair. We then do the construction in Part 3. Note that this uses two columns. Assuming that all of the $D$’s are colored distinctly and that the only colors left are $T$ and $F$, this will ensure that the main grid is $c$-colorable iff the formula is satisfiable. The main grid is now complete. For every $(i,j)$ that is colored $(i,j)$ we perform the method in Part 1 to make sure that $(i,j)$ is the only cell with color $(i,j)$. Let the number of such $(i,j)$ be $C$. The number of colors $c$ is $C+2$. An Example ========== We can make the construction slightly more efficient (and thus can actually work out an example). We took $m$ pairs $\{x_i,{\overline{x}_i}\}$. We don’t really need all $m$. If $x_i$ appears in $a$ clauses and ${\overline{x}_i}$ appears in $b$ clauses then we only need $\max\{a,b\}$ literal-pairs. If $a\ne b$ then we only need $\max\{a,b\}-1$ literal-pairs and one additional literal. (This will be the case in the example below.) With this in mind we will do an example- though we will only show the main grid. $$(x_1\vee x_2 \vee {\overline{x}_3}) \wedge ({\overline{x}_2}\vee x_3 \vee x_4) \wedge ({\overline{x}_1}\vee {\overline{x}_3}\vee {\overline{x}_4})$$ We only need - one $(x_1,{\overline{x}_1})$ literal-pair, - one $(x_2,{\overline{x}_2})$ literal-pair, - one $(x_3,{\overline{x}_3})$ literal-pair, - one additional ${\overline{x}_3}$, - one $(x_4,{\overline{x}_4})$ literal-pair. $$\begin{array}{c|||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & & & & & & & & & & & & C_1&C_1& C_2&C_2&C_3& C_3 \cr \hline \hline \hline & D & D & D &D &D &D &D &D &D &D &D &T &T &T & T &T &T \cr \hline {\overline{x}_4}& & D & D &D &D &D &D &D &D &T &F &D &D &D & D &D &F \cr \hline x_4 & & D & D &D &D &D &D &D &D &T &F &D &D &D &F &D & D\cr \hline {\overline{x}_3}& & D & D &D &D &D &D &T &F &D &D &D &D &D & D &D &D\cr \hline x_3 & & D & D &D &D &T &F &T &F &D &D &D &D & & &D &D\cr \hline {\overline{x}_3}& & D & D &D &D &T &F &D &D &D &D &D &F &D & D & & \cr \hline {\overline{x}_2}& & D & D &T &F &D &D &D &D &D &D &D &D &F &D &D &D \cr \hline x_2 & & D & D &T &F &D &D &D &D &D &D & & &D & D & D & D \cr \hline {\overline{x}_1}& & T & F &D &D &D &D &D &D &D &D &D &D &D & D &F & D\cr \hline x_1 & & T & F &D &D &D &D &D &D &D &D &F &D &D & D & D& D\cr \hline \end{array}$$ Fixed Parameter Tractability {#se:fpt} ============================ The $17\times 17$ problem only involved 4-colorability. Does the result that $GCE$ is NP-complete really shed light on the hardness of the $17\times 17$ problem? What happens if the number of colors is fixed? Let $c\in{{\mathbb{N}}}$. Let $$GCE_c=\{ (N,M,\chi)\mid\chi \hbox{ can be extended to a $c$-coloring of $G_{N,M}$ } \} .$$ Clearly $GCE_c \in DTIME(c^{O(NM)})$. Can we do better? Yes. We will show that $GCE$ is in time $O(N^2M^2+2^{O(c^4)})$. \[le:dyn\] Let $n,m,c$ be such that $c\le 2^{nm}$. Let $\chi$ be a partial $c$-coloring of $G_{n,m}$. Let $U$ be the uncolored grid points. Let $|U|=u$. There is an algorithm that will determine if $\chi$ can be extended to a full $c$-coloring that runs in time $O(cnm2^{2u})=2^{O(nm)}$. For $S\subseteq U$ and $1\le i\le c$ let $$f(S,i)= \begin{cases} YES & \text{ if $\chi$ can be extended to color $S$ using only colors $\{1,\ldots,i\}$; } \\ NO & \text{ if not.} \\ \end{cases}$$ We assume throughout that the coloring $\chi$ has already been applied. We are interested in $f(U,c)$; however, we use a dynamic program to compute $f(S,i)$ for all $S\subseteq U$ and $1\le i\le c$. Note that $f({\emptyset},i)=YES$. We describe how to compute $f(S,i)$. Assume that for all $S'$ such that $|S'|<|S|$, for all $1\le i\le c$, $f(S',i)$ is known. 1. For all nonempty $1$-colorable $T\subseteq S$ do the following (Note that there are at most $2^u$ sets $T$.) 1. If $f(S-T,i)=NO$ then $f(S,i)=NO$. 2. If $f(S-T,i-1)=YES$ then determine if coloring $T$ with $i$ will create a monochromatic rectangle. If not then $f(S,i)=YES$. Note that this takes $O(nm)$. 2. We now know that for all 1-colorable $T\subseteq S$ (1) $f(S-T,i)=YES$, and (2) either $f(S-T,i-1)=NO$ or $f(S-T,i-1)=YES$ and coloring $T$ with $i$ creates a monochromatic rectangle. We will show that in this case $f(S,i)=NO$. Assume that, for all 1-colorable sets $T\subseteq S$: (1) $f(S-T,i)=YES$, and (2) either $f(S-T,i-1)=NO$ or $f(S-T,i-1)=YES$ and coloring $T$ with $i$ creates a rectangle with $\chi$. Also assume, by way of contradiction, that $f(S,i)=YES$. Let $COL$ be an extension of $\chi$ to $S$. Let $T$ be the set colored $i$. Clearly $f(S-T,i-1)=YES$. Hence the second clause of condition (2) must hold. Hence coloring $T$ with $i$ creates a monochromatic rectangle. This contradicts $COL$ being a $c$-coloring. The dynamic program fills in a table that is indexed by the $2^u$ subsets of $S$ and the $c$ colors. Each slot in the table takes $O(nm2^u)$ to compute. Hence to fill the entire table takes $O(cnm2^{2u})$ steps. \[le:bounds\] Assume $c+1\le N$ and $c\binom{c+1}{2} < M$. Then $G_{N,M}$ is not $c$-colorable. Hence, for any $\chi$, $(N,M,\chi)\notin GCE_c$. Assume, by way of contradiction, that there is a $c$-coloring of $G_{N,M}$. Since every column has at least $c+1$ elements the following mapping is well defined: Map every column to the least $(\{i,j\},a)$ such that the $\{i,j\}\in \binom{[c+1]}{2}$ and both the $i$th and the $j$th row of that column are colored $a$. The range of this function has $c\binom{c+1}{2}$ elements. Hence some element of the range is mapped to at least twice. This yields a monochromatic rectangle. \[le:c\] Assume $N\le c$ and $M\in {{\mathbb{N}}}$. If $\chi$ is a partial $c$-coloring of $G_{N,M}$ then $(N,M,\chi)\in GCE_c$. The partial $c$-coloring $\chi$ can be extended to a full $c$-coloring as follows: for each column use a different color for each blank spot, making sure that all of the new colors in that column are different from each other. $GCE_c \in DTIME(N^2M^2+2^{O(c^6)})$ time. 1. Input $(N,M,\chi)$. 2. If $N\le c$ or $M\le c$ then test if $\chi$ is a partial $c$-coloring of $G_{N,M}$. If so then output YES. If not then output NO. (This works by Lemma \[le:c\].) This takes time $O(N^2M^2)$. Henceforth we assume $c+1\le N,M$. 3. If $c\binom{c+1}{2} < M$ or $c\binom{c+1}{2} < N$ then output NO and stop. (This works by Lemma \[le:bounds\].) 4. The only case left is $c+1\le N,M\le c\binom{c+1}{2}$. By Lemma \[le:dyn\] we can determine if $\chi$ can be extended in time $O(2^{NM})=O(2^{c^6})$. Step 2 takes $O(N^2M^2)$ and Step 4 takes time $2^{O(c^6)})$, hence the entire algorithm takes time $O(N^2M^2 + 2^{O(c^6)})$. Can we do better? Yes, but it will require a result from [@grid]. \[le:better\] Let $1\le c'\le c-1$. 1. If $N\ge c+c'$ and $M > \frac{c}{c'}\binom{c+c'}{2}$ then $G_{N,M}$ is not $c$-colorable. 2. If $N\ge 2c$ and $M>2\binom{2c}{2}$ then $G_{N,M}$ is not $c$-colorable. (This follows from a weak version of the $c'=c-1$ case of Part 1.) $GCE_c \in DTIME(N^2M^2+2^{O(c^4)})$ time. 1. Input $(N,M,\chi)$. 2. If $N\le c$ or $M\le c$ then test if $\chi$ is a partial $c$-coloring of $G_{N,M}$. If so then output YES. If not then output NO. (This works by Lemma \[le:c\].) This takes time $O(N^2M^2)$. 3. For $1\le c'\le c-1$ we have the following pairs of cases. 1. $N=c+c'$ and $M> \frac{c}{c'}\binom{c+c'}{2}$ then output NO and stop. (This works by Lemma \[le:better\].) 2. $N=c+c'$ and $M\le \frac{c}{c'}\binom{c+c'}{2}$. By Lemma \[le:dyn\] we can determine if $\chi$ can be extended to a total $c$-coloring in time $2^{O(NM)}$. Note that $MN\le (c+c')\frac{c}{c'}\binom{c+c'}{2}$. On the interval $1\le c'\le c-1$ this function achieves its maximum when $c'=1$. Hence this case takes $2^{O(c^4)}$. Henceforth we assume $2c\le N,M$. 4. If $M>2\binom{2c}{2}$ or $N>2\binom{2c}{2}$ then output NO and stop. (This works by Lemma \[le:better\].) 5. The only case left is $2c\le N,M\le 2\binom{2c}{2}$. By Lemma \[le:dyn\] we can determine if $\chi$ can be extended in time $2^{O(NM)}\le 2^{O(c^4)}$. Step 2 and Step 4 together take time $O(N^2M^2 + 2^{O(c^4)})$. Even for small $c$ the additive term $2^{O(c^4)}$ is the real timesink. A cleverer algorithm that reduces this term is desirable. By Theorem \[th:npc\] this term cannot be made polynomial unless ${\hbox{P$=$NP}}$. Lower Bound on Tree Res {#se:treeres} ======================= For $n,m,c$ we define a Boolean formula ${GRID(n,m,c)}$ such that $$G_{n,m} \hbox{ is $c$-colorable iff } {GRID(n,m,c)}\in SAT.$$ - The variables are $x_{ijk}$ where $1\le i\le n$, $1\le j\le m$, $1\le k\le c$. The intention is that, for all $(i,j)$, there is a $k$ such that $x_{ijk}$ is true. We interpret $k$ to be the color of $(i,j)$. - For all $(i,j)$ we have the clause $$\bigvee_{k=1}^c x_{ijk}.$$ These clauses ensure that every $(i,j)$ has at least one color. - For all $1\le i < i'\le n$ and $1\le j < j'\le m$ we have the clause $$\bigvee_{k=1}^c \neg x_{ijk} \vee \neg x_{i'jk} \vee \neg x_{ij'k} \vee \neg x_{i'j'k}.$$ These clauses ensure there are no monochromatic rectangles. We do not use clauses to ensure that every $(i,j)$ has at most one color. This is because if the formula above is satisfied then one can extract out of it a $c$-coloring of $G_{n,m}$ by taking the color of $(i,j)$ to be the [*least*]{} $k$ such that $x_{ijk}$ is true. We show that if $G_{n,m}$ is not $c$-colorable then any tree resolution proof of ${GRID(n,m,c)}\notin SAT$ requires size $2^{\Omega(c)}$. Background on Tree Resolution and the Prover-Delayer Game --------------------------------------------------------- The definitions of Resolution and Tree Resolution are standard. Prover-Delayer games were first defined in [@proverdelayer], however we use the asymmetric version which was first defined in [@delayer]. See also [@bfc]. Let $\varphi=C_1\wedge \cdots \wedge C_L$ be a CNF formula. A [*Resolution Proof that $\varphi\notin SAT$*]{} is a sequence of clauses such that on each line you have either 1. One of the $C$’s in $\varphi$ (called an AXIOM). 2. $A\vee B$ where on prior lines you had $A\vee x$ and $B\vee \neg x$. 3. The last line has the empty clause. It is easy to see that if there is a resolution proof that $\varphi\notin SAT$ then indeed $\varphi\notin SAT$. The converse is also true though slightly harder to prove. A [*Tree Resolution*]{} proof is one whose underlying graph is a tree. The [*Prover-Delayer Game*]{} has parameters (1) $a,b\in (1,{\infty})$, such that $\frac{1}{a} + \frac{1}{b} = 1$, (2) $p\in{{\mathbb R}^+}$, and (3) a CNF-formula $$\varphi = C_1 \wedge \cdots \wedge C_L\notin SAT.$$ The game is played as follows until a clause is proven false: 1. The Prover picks a variable $x$ that was not already picked. 2. The Delayer either 1. Sets $x$ to $T$ or $F$. 2. Defers to the Prover. 1. If the Prover sets $x$ to $F$ then the Delayer gets $\lg a$ points. 2. If the Prover sets $x$ to $T$ then the Delayer gets $\lg b$ points. When some clause has all of its literals set to false the game ends. At that point, if the Delayer has $p$ points then he WINS; otherwise the Prover WINS. We assume that the Prover and the Delayer play perfectly. 1. [*The Prover wins*]{} means [*the Prover has a winning strategy*]{}. 2. [*The Delayer wins*]{} means [*the Delayer has a winning strategy*]{}. \[le:link\] Let $a,b\in (1,{\infty})$ such that $\frac{1}{a}+\frac{1}{b}=1$, $p\in {{\mathbb R}^+}$, $\varphi\notin SAT$, $\varphi$ in $CNF$-form. If the Delayer wins then [*EVERY*]{} Tree Resolution proof for $\varphi$ has size $\ge 2^p$. Note that the lower bound in Lemma \[le:link\] is $2^p$, not $2^{\Omega(p)}$. Lower Bound on Tree Resolution ------------------------------ \[th:lowerres\] Let $n,m,c$ be such that $G_{n,m}$ is not $c$-colorable and $c\ge 9288$. Any tree resolution proof of ${GRID(n,m,c)}\notin SAT$ requires size $2^{Dc}$ where $D=0.836$. By Lemma \[le:link\] it will suffice to show that there exists $a,b\in (1,{\infty})$ with $\frac{1}{a}+\frac{1}{b}=1$, such that the Delayer wins the Prover-Delayer game with parameters $a,b,Dc,$ and ${GRID(n,m,c)}$. We will determine $a,b$ later. We will also need parameter $r\in (0,1)$ to be determined. Here is the Delayers strategy: Assume $x_{ijk}$ was chosen by the Prover. 1. If coloring $(i,j)$ with color $k$ will create a monochromatic rectangle then the Delayer will NOT let this happen—he will set $x_{ijk}$ to $F$. The Delayer does not get any points but he avoids the game ending. (Formally: if there exists $i',j'$ such that $x_{i'jk}=x_{ij'k}=x_{i'j'k}=T$ then the Delayer sets $x_{ijk}$ to $F$.) Otherwise he goes to the next step of the strategy. 2. If there is a danger that all of the $x_{ij*}$ will be false for some $(i,j)$ then the Delayer will set $x_{ijk}$ to $T$. The Delayer does not want to panic and set $x_{ijk}$ to $T$ unless he feels he has to. He uses the parameter $r$. If there are at least $rc$ values $k'$ where the Prover has set $x_{ijk'}$ to $F$, and there are no $x_{ijk'}$ that have been set to $T$ (by anyone) then Delayer sets $x_{ijk}$ to $T$. Note that this cannot form a monochromatic rectangle since in step 1 of the strategy $x_{ijk}$ would have been set to $F$. 3. In all other cases the Delayer defers to the Prover. For the analysis we need two real parameters: $q\in (0,1)$ and $s\in (0,3-3q)$. Since we need $\frac{1}{a}+\frac{1}{b}=1$ we set $b=\frac{a}{a-1}$. We now show that this strategy guarantees that the Delayer gets at least $Dc$ points. Since the Delayer will [*never*]{} allow a monochromatic rectangle the game ends when there is some $i,j$ such that $$x_{ij1} = x_{ij2} = \cdots = x_{ijc} = F.$$ Who set these variables to $F$? Either at least $qc$ were set to $F$ by the Prover or at least $(1-q)c$ were set to $F$ by the Delayer. This leads to several cases. 1. At least $qc$ were set to $F$ by the Prover. The Delayer gets at least $qc\lg a$ points. 2. At least $(1-q)c$ were set to $F$ by the Delayer. For every $k$ such that the Delayer set $x_{ijk}$ to $F$ there is an $(i',j')$ (with $i\ne i'$ and $j\ne j'$) such that $x_{i'jk}$, $x_{ij'k}$, and $x_{i'j'k}$ were all set to $T$ (we do not know by who). Consider the variables we know were set to $T$ because Delayer set $x_{ijk}$ to $F$. These variables all have the last subscript of $k$. Therefore these sets-of-three variables associated to each $x_{ijk}$ are disjoint. Hence there are at least $3(1-q)c=(3-3q)c$ variables that were set to $T$. There are two cases. Recall that $s\in (0,3-3q)$. 1. The Prover set at least $sc$ of them to $T$. Then the Delayer gets at least $sc\lg(b)=sc\lg(a/(a-1))$ points. 2. The Delayer set at least $(s-(3-3q))c=(s+3q-3)c$ of them to $T$. If the Delayer is setting some variable $x_{i'j'k}$ to $T$ it’s because the Prover set $rc$ others of the form $x_{i'j'k'}$ to $F$. These sets-of-$rc$-variables are all disjoint. Hence the Prover set at least $(s+3q-3)rc^2$ variables to $F$. Therefore the Delayer gets at least $(s+3q-3)rc^2\lg a$ points. We need to set $a\in (1,{\infty})$, $q,r\in (0,1)$, and $s\in (0,3-3q)$ to maximize the minimum of 1. $qc\lg a$ 2. $sc\lg(a/(a-1))$ 3. $(s+3q-3)rc^2$ We optimize our choices by setting $qc\lg a = sc\lg(a/(a-1))$ (approximately) and thinking (correctly) that the $c^2$ term in $(s+3q-3)rc^2$ will force this term to be large when $c$ is large. To achieve this we take - $q=0.56415$. Note that $3-3q=1.30755$. - $s=1.30754$. Note that $s\in (0,3-3q)$. - $r=0.9$. Note that $(s+3q-3)r= (0.00001)*0.9=0.00009$. (Any value of $r\in (0,1)$ would have sufficed.) - $a=2.793200$ - $b=a/(a-1) = 1.557662$ (approximately) Using these values we get $qc\lg a, sc\lg(a/(a-1)) \ge 0.836$. We want $$(0.00009c^2) \ge 0.836c$$ $$(0.00009c) \ge 0.836$$ $$c \ge 9288$$ With this choice of parameters, for $c\ge 9288$, the Delayer gets at least $0.836c$ points. Hence any tree resolution proof of ${GRID(n,m,c)}$ must have size at least $2^{0.836c}$. Lower Bounds on Tree Res for Other Shapes ========================================= We did not use any property specific to rectangles in our proof of Theorem \[th:lowerres\]. We can generalize our result to any other shape; however, the constant in $2^{\Omega(c)}$ will change. First we give a definition of rectangle that will help us to generalize it. Let $c,N,M\in{{\mathbb{N}}}$. A (full or partial) mapping of $G_{N,M}$ to $\{1,\ldots,c\}$ is a [*$c$-coloring*]{} if there does not exists a set of points $\{(a,b), (a+t,b), (a,b+s), (a+t,b+s)\}$ that are all the same color. Look at the points $$\{ (a,b), (a+t,b), (a,b+s), (a+t,b+s)\}.$$ We can view them as $$\{ (s\times 0,t\times 0)+(a,b), (s\times 1,t\times 0)+(a,b), (s\times 0,t\times 1)+(a,b), (s\times 1,t\times 1)+(a,b) \}.$$ Informally, the set of rectangles is generated by $\{(0,0), (0,1), (1,0), (1,1)\}$. Formally we can view the set of rectangles on the lattice points of the plane (upper quadrant) as the intersection of ${{\mathbb{N}}}\times{{\mathbb{N}}}$ with $$\bigcup_{s,t,a,b\in{{\sf Q}}} \{ \{ (s\times 0,t\times 0)+(a,b), (s\times 1,t\times 0)+(a,b), (s\times 0,t\times 1)+(a,b), (s\times 1,t\times 1)+(a,b) \} \}$$ Note that the pair of curly braces is not a typo. We are looking at sets of 4-sets of points. We generalize rectangles. Let $$S=\{(x_1,y_1),\ldots,(x_L,y_L)\}$$ be a set of lattice points in the plane. Let $${{\rm stretch}}(S) = \bigcup_{s,t\in{{\sf Q}}} \{ \{ (sx_1,ty_1),\ldots,(sx_L,ty_L)\} \}$$ and $${{\rm translate}}(S)=\bigcup_{a,b\in{{\sf Q}}}\{ \{(x_1+a,y_1+b),\ldots,(x_L+a,y_L+b)\}\}.$$ These are the sets of points we will be trying to avoid making monochromatic. Hence let $${{\rm avoid}}(S)={{\rm translate}}({{\rm stretch}}(S)).$$ We can now generalize the rectangle problem. Let $N,M\in{{\mathbb{N}}}$ and $S$ be a set of lattice points. A (partial or full) mapping $\chi$ from $G_{N,M}$ into $[c]$ is a [*$(c,S)$-coloring*]{} if there are no monochromatic sets in ${{\rm avoid}}(S)$. Let $N,M\in{{\mathbb{N}}}$ and $S$ be a set of lattice points. Let $GRID(n,m,c;S)$ be the Boolean formula that can be interpreted as saying that $G_{n,m}$ is $(c,S)$-colorable. We omit details. The following theorems have proof similar to those in Section \[se:treeres\]. \[th:lowerresS\] Let $n,m\in{{\mathbb{N}}}$ and $S$ be a set of lattice points. Let $n,m,c$ be such that $G_{n,m}$ is not $(c,S)$-colorable. Any tree resolution proof of $GRID(n,m,c;S)\notin SAT$ requires size $2^{\Omega(c)}$. The constant in the $\Omega(c)$ depends only on $|S|$ and not the nature of $S$. One could look at other ways to move the points in $S$ around. There is one we find particular interesting. We motivate our definition. What if we wanted to look at colorings that avoided a monochromatic [*square*]{}? The square $$\{(a,b),(a+s,b), (a,b+s), (a+s,b+s)\}$$ can be viewed as $$\{ (0,0)+(a,b), (s,0)+(a,b), (0,s)+(a,b), (s,s)+(a,b) \}.$$ We generalize this. Let $$S=\{(x_1,y_1),\ldots,(x_L,y_L)\}$$ be a set of lattice points in the plane. Let $${{\rm halfstretch}}(S) = \bigcup_{s\in{{\sf Q}}}\{ \{ (sx_1,sy_1),\ldots,(sx_L,sy_L)\}\}$$ These are the sets of points we will be trying to avoid making monochromatic. We would like to call it “${{\rm avoid}}$” but that name has already been taken; hence we call it ${{\rm avoid}}_2$. $${{\rm avoid}}_2(S)={{\rm translate}}({{\rm halfstretch}}(S)).$$ (Note that the 2 has no significance. It is just there to distinguish ${{\rm avoid}}$ and ${{\rm avoid}}_2$.) Let $N,M\in{{\mathbb{N}}}$ and $S$ be a set of lattice points. A (partial or full) mapping $\chi$ from $G_{N,M}$ into $[c]$ is a [*$(c,S)_2$-coloring*]{} if there are no monochromatic sets in ${{\rm avoid}}_2(S)$. (Note that the 2 has no significance. It is just there to distinguish $(c,S)$ and $(c,S)_2$.) Let $N,M\in{{\mathbb{N}}}$ and $S$ be a set of lattice points. A (partial or full) mapping $\chi$ from $G_{N,M}$ into $[c]$ is a [*$(c,S)_2$-coloring*]{} if there are no monochromatic sets in ${{\rm avoid}}_2(S)$. We can now generalize the square problem. Let $N,M\in{{\mathbb{N}}}$ and $S$ be a set of lattice points. Let $GRID(n,m,c;S_2)$ be the Boolean formula that can be interpreted as saying that $G_{n,m}$ is $(c,S)_2$-colorable. We omit details. The following theorems have proof similar to those in Section \[se:treeres\]. Let $n,m\in{{\mathbb{N}}}$ and $S$ be a set of lattice points such that $|S|\ge 2$. Let $n,m,c$ be such that $G_{n,m}$ is not $(c,S)_2$-colorable. Any tree resolution proof of $GRID(n,m,c)_2\notin SAT$ requires size $2^{\Omega(c)}$. Lower Bound on CP-Tree Res for ${GRID(c+1,c\binom{c}{2}+1,c)}$ {#se:cptreeres} ============================================================== By Lemma \[le:bounds\] the formula ${GRID(c+1,c\binom{c}{2}+1,c)}\notin SAT$. Note that it’s just barely not satisfiable since ${GRID(c+1,c\binom{c}{2},c)}\in{{\rm SAT}}$. In this section we show that any Cutting Plane Tree Resolution proof that ${GRID(c+1,c\binom{c}{2}+1,c)}\notin{{\rm SAT}}$ requires size $2^{\Omega(c^3/\log^2 c)}$. Let $A$ be an integer valued matrix and $\vec b$ be an integer valued vector such that there is no 0-1 vector $\vec x$ with $A\vec x\le \vec b$. We refer to this as $A\vec x \le \vec b \notin SAT$. Any CNF-formula can be phrased in this form with only a linear blowup in size. For every variable $x$ we have variables $x$ and $\overline{x}$ and the inequalities $$\begin{array}{rl} x + \overline{x}& \le 1\cr -x - \overline{x}& \le -1\cr \end{array}$$ If $C$ is a clause with literals $L_1,\ldots,L_k$ then we have the inequality $$L_1 + \cdots + L_k \ge 1$$ In particular, the formulas ${GRID(n,m,c)}$ can be put in this form. Background on CP-Tree Resolution and Link to Communication Complexity --------------------------------------------------------------------- The definitions of Cutting Plane Proofs and Tree Cutting Plane Proofs are standard. The connection to communication complexity (Lemma \[le:linkcp\]) is from [@cptreeres] (see also [@bfc] Lemmas 19.7 and 19.11). A [*Cutting Planes Proof that $A\vec x\le \vec b\notin SAT$*]{} (henceforth CP Proof) is a sequence of linear inequalities such that on each line you have either 1. One of the inequalities in $A\vec x\le \vec b$ (called an AXIOM). 2. If $\vec a_1\cdot \vec x\le c_1$ and $\vec a_2\cdot \vec x\le c_2$ are on prior lines then $(\vec a_1+ \vec a_2)\cdot \vec x\le c_1+c_2$ can be on a line. 3. If $\vec a \cdot \vec x\le c$ is on a prior line and $d\in{{\mathbb{N}}}$ then $d(\vec a\cdot \vec x) \le dc$ can be on a line. (Also if $d\in{{\mathbb Z}}-{{\mathbb{N}}}$ then reverse the inequality.) 4. If $c(\vec a\cdot \vec x) \le d$ is on a prior line then $\vec a \cdot \vec x \le {\left\lfloor{\frac{d}{c}}\right\rfloor}$ can be on a line. 5. The last line is an arithmetically false statement (e.g., $1\le 0$). It is easy to see that if there is a cutting planes proof that $A\vec x \le \vec b\notin SAT$ then indeed $A\vec x \le \vec b \notin SAT$. The converse is also true though slightly harder to prove. A [*Tree-like CP proof*]{} is one whose underlying graph is a tree. Let $A$ be an integer valued matrix and $\vec b$ be an integer valued vector such that $A\vec x\le \vec b\notin SAT$. Let $P_1,P_2$ be a partition of the variables in $\vec x$. The Communication Complexity problem $FI(A,\vec b,P_1,P_2)$ is as follows. 1. For every variable in $P_1$ Alice is given a value (0 or 1). 2. For every variable in $P_2$ Bob is given a value (0 or 1). 3. These assignments constitute an assignment to all of the variables which we denote $\vec x$. 4. Alice and Bob need to determine an inequality in $A\vec x\le \vec b$ that is not true. \[le:linkcp\] Let $A$ be an integer valued matrix and $\vec b$ be an integer valued vector such that $A\vec x\le \vec b\notin SAT$. Let $n$ be the number of variables in $\vec x$. If there is a partition $P_1,P_2$ of the variables such that, for all ${\epsilon}$, ${{\rm R}}_{\epsilon}(FI(A,\vec b,P_1,P_2))=\Omega(t)$ then any tree-like CP proof of $A\vec x \le \vec b$ requires size $2^{\Omega(t/\log^2 n)}$. Lemmas on Communication Complexity ----------------------------------   1. The *Hamming weight* of a binary string $x$, denoted $w(x)$, is the number of 1’s in $x$. 2. The *Hamming distance* between two, equal-length, binary strings $x$ and $y$, denoted $d(x, y)$, is the number of positions in which they differ. 3. For a communication problem $\mathcal{P}$, $D(\mathcal{P})$ denotes the deterministic communication complexity of $\mathcal{P}$ and ${{\rm R}}_{\epsilon}(\mathcal{P})$ denotes the randomized public coin communication complexity of $\mathcal{P}$ with error $\le {\epsilon}$. We define several communication complexity problems. 1. ${{{\rm PHPstr}}}_n$: Alice gets a string $x\in\Sigma^n$, and Bob gets a string $y\in\Sigma^n$ with $|\Sigma|=2n-1$. They are promised that for all $i\ne j$, the letters $x_i$ and $x_j$ (resp. $y_i$ and $y_j$) are distinct. By the PHP, there must exist at least one $(i, j)\in [n]\times [n]$ such that $x_i = y_j$. They are further promised that $(i, j)$ is *unique*. *The goal is to find $(i, j)$.* (Alice learns $i$, and Bob learns $j$.) 2. ${{{\rm PHPset}}}_n$: Alice gets a set $x\in\binom{\Sigma}{n}$, and Bob gets a set $y\in\binom{\Sigma}{n}$ with $|\Sigma|=2n-1$. By the PHP, there must exist at least one $\sigma\in\Sigma$ such that $\sigma\in x\cap y.$ They are further promised that $\sigma$ is *unique*. *The goal is to find $\sigma$.* (Both learn $\sigma$.) 3. ${{{\rm PrMeet}}}_n$: Alice gets a string $x\in\{0,1\}^n$, and Bob gets a string $y\in\{0,1\}^n$ with $n=2m-1$. They are promised that (1) $w(x)=w(y)=m$, that (2) there is a *unique* $i\in [n]$ such that $x_i=y_i=1$, and that (3) for all $j\ne i,$ $(x_j, y_j) \in \{(0, 1), (1, 0)\}$. *The goal is to find $i$.* (Both learn $i$.) 4. ${{{\rm UM}}}_n$: (called the *universal monotone relation*) Alice is given $x\in\{0, 1\}^n$, and Bob is given $y\in\{0, 1\}^n$. They are promised that there exists $i$ such that $x_i=1$ and $y_i=0$. *The goal is to find some such $i$.* (Both learn $i$.) 5. ${{{\rm PrUM}}}_n$: This is a restriction of ${{{\rm UM}}}_n$. They are additionally promised (1) $n=2m-1$ is odd, (2) $w(x)=m$, (3) $w(y)=m-1$, and (4) $d(x, y)=1$. Hence (a) there is a *unique* index $i\in [n]$ such that $x_i=1$ and $y_i=0$, (b) for all $j\ne i$, $(x_j, y_j)\in\{(0, 0), (1, 1)\}$, and moreover (c) these $(0, 0)$’s and $(1, 1)$’s occur in an equal number. *The goal is to find $i$.* (Both learn $i$.) 6. ${{\rm DISJ}}_n$: Alice gets a string $x\in{\{0,1\}^{{n}}}$, and Bob gets a string $y\in{\{0,1\}^{{n}}}$. They need to *decide* if $x$ and $y$ intersect ($\exists i$ where $x_i = y_i$). 7. ${{\rm PrDISJ}}_n$: $n=2m+1$ is odd. Alice gets a string $x\in{\{0,1\}^{{n}}}$, and Bob gets a string $y\in{\{0,1\}^{{n}}}$. They are promised that $x$ and $y$ have exactly $m+1$ 1’s and $m$ 0’s and intersect at most once. They need to *decide* if $x$ and $y$ intersect ($\exists i$ where $x_i = y_i$). We will need the following notion of reduction. Let $f,g$ be a communication problem. It can be a decision, a function, and/or a promise problem. 1. $f{\le_{\rm cc}}g$ if there exists a protocol for $f$ that has the following properties. 1. The protocol may invoke a protocol for $g$ once on an input of length $O(n)$. 2. Before and after the invocation, the players may communicate polylog bits. The following lemma is obvious. \[le:lecc\]  If $f{\le_{\rm cc}}g$ and $(\forall{\epsilon}_)[{{\rm R}}_{\epsilon}(f)=\Omega(n)]$ then $(\forall {\epsilon})[{{\rm R}}_{\epsilon}(g)=\Omega(n)]$. \[le:pru\] For all ${\epsilon}$ ${{\rm R}}_{\epsilon}({{{\rm PrUM}}}_n) =\Omega(n)$. In  [@commcomp] it was shown that ${{\rm DISJ}}_n {\le_{\rm cc}}{{{\rm UM}}}_n$. A closer examination of the proof shows that it also shows ${{\rm PrDISJ}}_n {\le_{\rm cc}}{{{\rm PrUM}}}_n$. Kalyanasundaram and Schnitger [@disj] showed that, for all ${\epsilon}$, ${{\rm R}}_{\epsilon}({{\rm DISJ}}_n)=\Omega(n)$. Razborov [@Raz90] has a simpler proof where he only looks at inputs that satisfy the promise of ${{\rm PrDISJ}}_n$. Hence he showed ${{\rm R}}_{\epsilon}({{\rm PrDISJ}}_n) = \Omega(n)$. From ${{\rm PrDISJ}}_n {\le_{\rm cc}}{{{\rm PrUM}}}_n$, ${{\rm R}}_{\epsilon}({{\rm PrDISJ}}_n) = \Omega(n)$, and Lemma \[le:lecc\] the result follows. \[le:phpstr\]  1. $${{{\rm PrUM}}}_n {\le_{\rm cc}}{{{\rm PrMeet}}}_n {\le_{\rm cc}}{{{\rm PHPset}}}_{(n+1)/2}.$$ (The last reduction only holds when $n$ is odd.) 2. $${{{\rm PHPset}}}_n {\le_{\rm cc}}{{{\rm PHPstr}}}_n.$$ 3. For all ${\epsilon}$ ${{\rm R}}_{\epsilon}(PHPstr_n) = \Omega(n).$ (This follows from parts 1,2 and Lemmas \[le:lecc\], \[le:pru\].) : Alice gets $x\in {\{0,1\}^{{n}}}$, Bob gets $y\in {\{0,1\}^{{n}}}$ so that $(x,y)$ satisfies the promise of ${{{\rm PrUM}}}_n$. Let $n=2m-1$. We show that $(x,\overline{y})$ satisfies the promise of ${{{\rm PrMeet}}}_n$ and that ${{{\rm PrUM}}}_n(x,y)={{{\rm PrMeet}}}_n(x,\overline{y})$. Since $w(y)=m-1$, $w(\overline{y}) = n-(m-1)=m$. We still have $w(x)=m$ so $w(x)=w(y)=m$. Since there is a unique $i\in [n]$ such that $x_i = 1$ and $y_i = 0$, then must be a unique $i\in [n]$ (the same one) such that $x_i = \overline{y}_i = 1$. (This establishes ${{{\rm PrUM}}}_n(x,y)={{{\rm PrMeet}}}_n(x,\overline{y})$.) Since for all all $j\ne i, (x_j, y_j)\in\{(0, 0), (1, 1)\}$, for all $j\ne i, (x_j, \overline{y}_j)\in\{(0, 1), (1, 0)\}.$ : Alice gets $x\in {\{0,1\}^{{n}}}$, Bob gets $y\in {\{0,1\}^{{n}}}$ so that $(x,y)$ satisfies the promise of ${{{\rm PrMeet}}}_n$. Let $m=(n+1)/2$. Note that $w(x)=w(y)=m$. Let $\Sigma$ be an alphabet of size $n$. Both Alice and Bob agree on an ordering of $\Sigma$ ahead of time. Alice views her $n$-bit string $x$ (resp. Bob views his string $y$) as the bit vector of an $m$-element subset of $\Sigma$. We denote this subset $a$ (and for Bob $b$). Clearly $(a,b)$ satisfies the promise of ${{{\rm PHPset}}}_{(n+1)/2}$ and ${{{\rm PrMeet}}}_n(x,y)={{{\rm PHPset}}}_{(n+1)/2}(a,b)$. : $\Sigma$ is an alphabet of size $2n-1$. Alice and Bob agree on an ordering of $\Sigma$ ahead of time. Alice gets $x\in \binom{\Sigma}{n}$, Bob gets $y\in \binom{\Sigma}{n}$. The sets $x,y$ satisfy the promise of ${{{\rm PHPset}}}_n$. Alice (Bob) forms the string $x'\in \Sigma^n$ ($y'\in \Sigma^n)$ which is the elements of $x$ ($y$) written in order. Clearly $x',y'$ satisfy the promise of ${{{\rm PHPstr}}}_n$. Alice and Bob run the protocol for ${{{\rm PHPstr}}}_n$ on $(x',y')$. Alice obtains $i$, Bob obtains $j$. The $i$th element of $x'$ is the same as the $j$th element of $y'$. This element is $\sigma$ which is promised in ${{{\rm PHPset}}}_n(x,y)$. Lower Bound on CP-Tree Resolution for ${GRID(c+1,c\binom{c}{2}+1,c)}$ --------------------------------------------------------------------- Let $A\vec x\le \vec b$ be the translation of ${GRID(c+1,c\binom{c}{2}+1,c)}$ into an integer program. Any Tree-CP proof that $A\vec x\le \vec b\notin SAT$ requires $2^{\Omega(c^3/\log^2 c)}$ size. We do the case where $c\binom{c}{2}+1$ is even (so $c\equiv 3 \pmod 4$. The other cases are similar but require slight variants of Lemma \[le:phpstr\]. Split the $(c+1)\times c\binom{c}{2}+1$ evenly into two halves, both of size $((c+1)\times c\binom{c}{2}+1))/2$. Let $P_1,P_2$ be the partition of the variables so that Alice gets all of the variables involved in coloring the left half, and Bob gets all of the variables involved in coloring the right half. We show that $D(CC(A,\vec b,P_1,P_2))=\Omega(c^3)$. Note that the number of variables is $\Theta(c^4)$. Hence, by Lemma \[le:linkcp\] we obtain that the size of any Tree-CP proof of $A\vec x\le \vec b\notin SAT$ requires size $2^{\Omega(c^3/\log^2 c)}$. We restrict the problem to the case where every column has $c-1$ colors occurring once and the remaining color occurring twice. Hence one can view a coloring as string of length $2m=c\binom{c}{2}+1$ over an alphabet of size $n=c\binom{c}{2}$. Note that Alice and Bob each get a string of length $m$ over an alphabet of size $n=2m-1$. In order to find which inequality is violated Alice and Bob need to find which column they agree on (e.g., Alice’s column $i$ is the same as Bob’s column $j$). This is precisely the problem ${{{\rm PHPstr}}}_n$. Hence, by Lemma \[le:phpstr\] this problem has communication complexity $\Omega(n)=\Omega(c^3)$. Therefore, by Lemma \[le:linkcp\], any Tree-CP proof of $A\vec x \le \vec b$ requires $2^{c^3/\log^2 c}$. Lower bounds on Tree-CP proofs yield lower bounds on Tree-Resolution (with a constant factor loss) (see Prop 19.4 of [@bfc]). Hence we have the following. \[co:gcc\] Any Tree-resolution proof of ${GRID(c+1,c\binom{c}{2}+1,c)}\notin SAT$ requires $2^{\Omega(c^3/\log^2 c)}$ size. Open Problems {#se:open} ============= Open Problems Related to NP-Completeness ---------------------------------------- [**Open Problem 1:**]{} For which sets of lattice points $S$ is the following problem NP-complete? $$\{ (N,M,c,\chi)\mid\chi \hbox{ can be extended to a $(c,S)$-coloring of $G_{N,M}$} \}$$ [**Open Problem 2:**]{} For which sets of lattice points $S$ is the following problem NP-complete? $$\{ (N,M,c,\chi)\mid\chi \hbox{ can be extended to a $(c,S)_2$-coloring of $G_{N,M}$} \}$$ [**Open Problem 3:**]{} Improve our FPT algorithm. Develop an FPT algorithm for the variants we have discussed. [**Open Problem 4:**]{} Prove that grid coloring problems starting with the empty grid are hard. This may need a new formalism. Open Problems Related to Lower Bounds on Tree Resolution -------------------------------------------------------- If $\phi$ is a Boolean formula on $v$ variables then it has a Tree Resolution proof of size $2^{O(v)}$. Hence there is a tree resolution proof of ${GRID(n,m,c)}$ of size $2^{O(nmc)}$. For particular values of $m,n$ (functions of $c$) can we do better? We have already obtained this kind of result for ${GRID(c+1,c\binom{c}{2}+1,c)}$ (see Corollary \[co:gcc\]). [**Open Problem 1:**]{} For various $n$ and $m$ that are functions of $c$ such that $G_{n,m}$ is not $c$-colorable, obtain a better lower bound on Tree Resolution than $2^{\Omega(c)}$. There are unsatisfiable Boolean formulas for which Tree Resolution requires exponential size, but there are polynomial size resolution proofs. [**Open Problem 2:**]{} Determine upper and lower bounds for the size of Resolution proofs of ${GRID(n,m,c)}$. Open Problems Related to Lower Bounds on Tree-CP Refutations ------------------------------------------------------------ We showed that Tree-CP for ${GRID(c+1,c\binom{c}{2}+1,c)}\notin SAT$ require exponential size. For other families of non-$c$-colorable grids either show that tree CP proof requires exponential size or show that there are short tree CP proofs. For other families of non-$c$-colorable grids either show that (non-tree) CP proofs requires exponential size or show that CP proofs are short. Acknowledgments =============== We would like to thank Clyde Kruskal for proofreading and discussion. We would like to thank Stasys Jukna whose marvelous exposition of the Prover-Delayer games and the tree-CP proofs inspired the second and third parts of this paper, and for some technical advice (translation: he found a bug and helped us fix it). We would like to thank Daniel Marx for pointing out an improvement in the fixed parameter algorithm which we subsequently used. We would like to thank Wing-ning Li for pointing out that the case of $n,m$ binary, while it seems to not be in NP, is actually unknown. We would also like to thank Tucker Bane, Richard Chang, Peter Fontana, David Harris, Jared Marx-Kuo, Jessica Shi, and Marius Zimand, for listening to Bill present these results and hence clarifying them. [10]{} O. Beyersdorr, N. Galesi, and M. Lauria. A lower bound for the pigeonhole principle in the tree-like resolution asymmetric prover-delayer games. , 110, 2010. The paper and a talk on it are here: <http://www.cs.umd.edu/~gasarch/resolution.html>. S. Fenner, W. Gasarch, C. Glover, and S. Purewal. Rectangle free colorings of grids, 2012. <http://arxiv.org/abs/1005.3750>. W. Gasarch. The 17$\times$17 challenge. [W]{}orth \$289.00. [T]{}his is not a joke, 2009. [http://blog.computationalcomplexity.org/ 2009/11/17x17-challenge-worth-28900-this-is-not.html](http://blog.computationalcomplexity.org/ 2009/11/17x17-challenge-worth-28900-this-is-not.html). W. Gasarch. A possible [NP]{}-intermediary problem. [http://blog.computationalcomplexity.org/ 2010/04/possible-np-intermediary-problem.html](http://blog.computationalcomplexity.org/ 2010/04/possible-np-intermediary-problem.html), 2010. W. Gasarch. The 17$\times$17 [SOLVED]{}! (also $18\times 18$[http://blog.computationalcomplexity.org/ 2012/02/17x17-problem-solved-also-18x18.html](http://blog.computationalcomplexity.org/ 2012/02/17x17-problem-solved-also-18x18.html), 2012. B. Hayes. The 17$\times$17 challenge, 2009. <http://bit-player.org/2009/the-17x17-challenge>. S. Homer and L. Longpre. On reductions of [NP]{} sets to sparse sets. , 48, 1994. Prior version in [*S*TRUCTURES 1991]{}. R. Impagliazzo and T. P. and. Upper and lower bounds for tree-like cutting planes proofs. In [*Proceedings of the Ninth Annual IEEE Symposium on Logic in Computer Science, [Paris, France]{}*]{}, 1994. <http://www.cs.toronto.edu/~toni>. S. Jukna. . Algorithms and Combinatorics Vol 27. Springer, 2012. B. Kalyanasundaram and G. Schnitger. The probabilistic communication complexity of set intersection. , 5:545–557, 1992. Prior version in [*C*onf. on Structure in Complexity Theory, 1987 (STRUCTURES)]{}. E. Kushilevitz and N. Nisan. . Cambridge University Press, 1997. S. Mahaney. Sparse complete sets for [NP]{}: Solution to a conjecture of [B]{}erman and [H]{}artmanis. , 25:130–143, 1982. P. Pudlka and R. Impagliazzo. A lower bound for [DLL]{} algorithms for [SAT]{}. In [*Eleventh Symposium on Discrete Algorithms: Proceedings of SODA ’00*]{}, 2000. A. Razborov. On the distributional complexity of disjointness. , 106:385–390, 1992. Prior version in ICALP 1990. Available online at <http://people.cs.chicago.edu/~razborov/research>. B. Steinbach and C. Posthoff. Extremely complex 4-colored rectangle-free grids: Solution of an open multiple-valued problem. In [*Proceedings of the Forty-Second IEEE International Symposia on Multiple-Valued Logic*]{}, 2012. <http://www.cs.umd.edu/~gasarch/PAPERSR/17solved.pdf>. [^1]: University of Maryland, College Park, MD  20742. `[email protected]` [^2]: University of Maryland, College Park, MD  20742. `[email protected]` [^3]: Permanent, Berkeley, CA  94710. `[email protected]`

--- author: - '<span style="font-variant:small-caps;">Óscar João Campos Dias</span>' bibliography: - 'BibThese.bib' nocite: - '[@*]' - '[@*]' - '[@*]' title: --- <span style="font-variant:small-caps;">**UNIVERSIDADE TÉCNICA DE LISBOA\ 0.5cm INSTITUTO SUPERIOR TÉCNICO**</span> -0.5cm **Black Hole Solutions\ and\ Pair Creation of Black Holes\ in\ Three, Four and Higher Dimensional Spacetimes** \ 0.5cm (PhD Thesis) <span style="font-variant:small-caps;">**Jury:**</span> 0.5cm **** ----------------------------------------------------------------------------- ----------------------- Luis <span style="font-variant:small-caps;">Bento</span> Examinator Stanley <span style="font-variant:small-caps;">Deser</span> Examinator Jorge <span style="font-variant:small-caps;">Dias de Deus</span> President of the Jury Alfredo Barbosa <span style="font-variant:small-caps;">Henriques</span> Examinator José Pizarro de Sande e <span style="font-variant:small-caps;">Lemos</span> Supervisor Jorge <span style="font-variant:small-caps;">Romão</span> Examinator ----------------------------------------------------------------------------- ----------------------- <span style="font-variant:small-caps;">**UNIVERSIDADE TÉCNICA DE LISBOA\ 0.5cm INSTITUTO SUPERIOR TÉCNICO**</span> -0.5cm **Black Hole Solutions\ and\ Pair Creation of Black Holes\ in\ Three, Four and Higher Dimensional Spacetimes** \ 0.5cm (PhD Thesis) <span style="font-variant:small-caps;">**Jury:**</span> 0.5cm [**Preface**]{} The research included in this thesis has been carried out at Centro Multidisciplinar de Astrofísica (CENTRA) in the Physics Department of Instituto Superior Técnico. I declare that this thesis is not substantially the same as any that I have submitted for a degree or diploma or other qualification at any other University and that no part of it has already been or is being concurrently submitted for any such degree or diploma or any other qualification. Chapter \[chap:BTZ family\] was done in collaboration with Professor José Lemos and Dr. Carlos Herdeiro. Chapters \[chap:3D Dilaton BH\]-\[chap:Pair creation\] are the outcome of collaborations with Professor José Lemos. Chapter \[chap:Black holes in higher dimensions\] was done in collaboration with Professor José Lemos, Vitor Cardoso and Nuno Santos. Chapters \[chap:Pair creation in higher dimensions\] and \[chap:Grav Radiation\] were done in collaboration with Professor José Lemos and Vitor Cardoso. All these chapters have been submitted with minor modifications for publication. A list of the works published included in this thesis are listed below. [-]{} O. J. C. Dias, J. P. S. Lemos, [*Rotating magnetic solution in three dimensional Einstein gravity*]{}, JHEP [**0201**]{}: 006 (2002); (Chapter 2). [-]{} V. Cardoso, S. Yoshida, O. J. C. Dias, J. P. S. Lemos, [*Late-time tails of wave propagation in higher dimensional spacetimes*]{}, Phys. Rev. D [**68**]{}, 061503 (2003) \[Rapid Communications\]; (mentioned in Chapter 2). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Static and rotating electrically charged black holes in three-dimensional Brans-Dicke gravity theories*]{}, Phys. Rev. D [**64**]{}, 064001 (2001); (Chapter 3). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Magnetic point sources in three dimensional Brans-Dicke gravity theories*]{}, Phys. Rev. D [**66**]{}, 024034 (2002); (Chapter 3). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Magnetic strings in anti-de Sitter general relativity*]{}, Class. Quantum Grav. [**19**]{}, 2265 (2002); (mentioned in Chapter 4). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Pair of accelerated black holes in a anti-de Sitter background: the AdS C-metric*]{}, Phys. Rev. D [**67**]{}, 064001 (2003); (Chapter 6). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Pair of accelerated black holes in a de Sitter background: the dS C-metric*]{}, Phys. Rev. D [**67**]{}, 084018 (2003); (Chapter 6). [-]{} O. J. C. Dias, J. P. S. Lemos, [*The extremal limits of the C-metric: Nariai, Bertotti-Robinson and anti-Nariai C-metrics*]{}, Phys. Rev. D[**68**]{} (2003) 104010; (Chapter 7). [-]{} O. J. C. Dias, J. P. S. Lemos, [*False vacuum decay: Effective one-loop action for pair creation of domain walls*]{}, J. Math. Phys. [**42**]{}, 3292 (2001); (Chapter 8). [-]{} O. J. C. Dias, [*Pair creation of particles and black holes in external fields*]{}, (Chapter 8). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Pair creation of de Sitter black holes on a cosmic string background*]{}, Phys. Rev. D[**69**]{} (2004) 084006; (Chapter 9). [-]{} O. J. C. Dias, [*Pair creation of anti-de Sitter black holes on a cosmic string background*]{}, Phys. Rev. D[**70**]{} (2004) 024007 (Chapter 9). [-]{} V. Cardoso, O. J. C. Dias, J. P. S. Lemos, [*Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions*]{}, Phys. Rev. D[**70**]{} (2004) 024002 (Chapter 10). [-]{} N. L. Santos, O. J. C. Dias, J. P. S. Lemos, [*Global embedding Minkowskian spacetime procedure in higher dimensional black holes: Matching between Hawking temperature and Unruh temperature*]{}, Phys. Rev. D, in press (2004) (mentioned in Chapter 10). [-]{} O. J. C. Dias, J. P. S. Lemos, [*Pair creation of higher dimensional black holes on a de Sitter background*]{}, Phys. Rev. D, submitted (2004); (Chapter 11). [-]{} V. Cardoso, O. J. C. Dias, J. P. S. Lemos, [*Gravitational radiation in D-dimensional spacetimes*]{}, Phys. Rev. D [**67**]{}, 064026 (2003); (Chapter 12). \[part1\] \[part2\] \[part3\] [10]{} V. P. Frolov, I. D. Novikov, [*Black Hole Physics: Basic concepts and new developments*]{}, Kluwer Academic, Dordrecht, Netherlands (1998). S. Chandrasekhar, [*The Mathematical Theory of Black Holes*]{}, Oxford University Press, New York (1992). B. K. Harrison, K. S. Thorne, M. Wakano, J. A. Wheeler, [*Gravitation Theory and Gravitational Collapse*]{}, University of Chicago Press, Chicago (1965). J. Maldacena, [*The large N limit of superconformal field theories and supergravity*]{}, Adv. Theor. Math. Phys. [**2**]{}, 231 (1998).; O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, [*Large N field theories, string theory and gravity*]{}, Phys. Rep. [**323**]{}, 183 (2000). S. W. Hawking, Commun. Math. Phys. [**25**]{}, 152 (1972); S. W. Hawking, G. F. R. Ellis, [*The large scale structure of space-time*]{} (Cambridge University Press, England, 1973); G. J. Galloway, K. Scheich, D. M. Witt, E. Woolgar, Phys. Rev. D[**60**]{}, 3840 (1996). J. P. S. Lemos, Class. Quant. Grav. [**12**]{}, 1081 (1995);[*Cylindrical black hole in general relativity*]{}, Phys. Lett. B[**353**]{}, 46 (1995). J. P. S. Lemos and V. T. Zanchin, [*Rotating charged black strings in general relativity*]{}, Phys. Rev. D[**54**]{}, 3840 (1996). O. J. C. Dias, J. P. S. Lemos, [*Magnetic strings in anti-de Sitter general relativity*]{}, Class. Quantum Grav. [**19**]{}, 2265 (2002). D. Klemm, V. Moretti, L. Vanzo, Phys. Rev. D[**57**]{}, 6127 (1998). C. G. Huang, C-B. Liang, Phys. Lett. A[**201**]{}, 27 (1995). S. Åminnenborg, I. Bengtsson, S. Holst, P. Peldán, [*Making anti-de Sitter black holes*]{}, Class. Quant. Grav. [**13**]{}, 2707 (1996); D. R. Brill, [*Multi-black holes in 3D and 4D anti-de Sitter spaces*]{}, Helv. Phys. Acta [**69**]{}, 249 (1996); S. L. Vanzo, [*Black holes with unusual topology*]{}, Phys. Rev. D [**56**]{}, 6475 (1997); D. R. Brill, J. Louko, P. Peldán, [*Thermodynamics of (3+1)-dimensional black holes with toroidal or higher genus horizons*]{}, Phys. Rev. D [**56**]{}, 3600 (1997); R. B. Mann, [*Black holes of negative mass*]{}, Class. Quant. Grav. [**14**]{}, 2927 (1997); S. Holst, P. Peldan, [*Black holes and causal structure in anti-de Sitter isometric spacetimes*]{}, Class. Quant. Grav. [**14**]{}, 3433 (1997). J. P. S. Lemos, Phys. Rev. D[**57**]{}, 4600 (1998); Phys. Rev. D[**59**]{}, 044020 (1999). J. P. S. Lemos, Class. Quantum Grav. [**12**]{}, 1081 (1995); Phys. Lett. B[**353**]{}, 46 (1995). J. R. Gott, M. Alpert, Gen. Rel. Grav. [**16**]{}, 243 (1984). S. Giddings, J. Abbot, K. Kuchař, Gen. Rel. Grav. [**16**]{}, 751 (1984). S. Deser, R. Jackiw, G. t’ Hooft, [*Three-dimensional einstein gravity: Dynamics of flat spacetime*]{}, Ann. Phys. (N.Y.) [**152**]{}, 220 (1984). J. D. Barrow, A. B. Burd, D. Lancaster, Class. Quant. Grav. [**3**]{}, 551 (1986). R. Jackiw, Nucl. Phys. B[**252**]{}, 343 (1985). R. Jackiw, [*Diverse topics in theoretical and mathematical physics*]{} (World Scientific, 1988). J. D. Brown, [*Lower dimensional gravity*]{} (World Scientific, 1988). A. Staruszkiewicz, Acta. Phys. Polon. [**24**]{}, 734 (1963). G. Clément, Int. J. Theor. Phys. [**24**]{}, 267 (1985). S. Deser, P. O. Mazur, Class. Quant. Grav. [**2**]{}, L51 (1985). J. R. Gott, J. Z. Simon, M. Alpert, Gen. Rel. Grav. [**18**]{}, 1019 (1986). M. A. Melvin, Class. Quant. Grav. [**3**]{}, 117 (1986). S. Deser, R. Jackiw, Ann. Phys. (N.Y.) [**192**]{}, 352 (1989). G. Grignani, C. Lee, Ann. Phys. (N.Y.) [**196**]{}, 386 (1989). G. Clément, Ann. Phys. (N.Y.) [**152**]{}, 241 (1990). P. Menotti, D. Seminara, Ann. Phys. (N.Y.) [**208**]{}, 449 (1991); Nucl. Phys. B[**376**]{}, 411 (1992). S. Deser, R. Jackiw, Ann. Phys. (N.Y.) [**153**]{}, 405 (1984). J. D. Brown, M. Henneaux, Commun. Math. Phys. [**104**]{}, 207 (1986). J. R. Gott, [*Closed timelike curves produced by pairs of moving cosmic strings: Exact solutions*]{}, Phys. Rev. Lett. [**66**]{}, 1126 (1991). C. Cutler, [*Global structure of Gott’s two string spacetime*]{}, Phys. Rev. D[**45**]{}, 487 (1992). S. M. Carrol, E. Fahri, A. H. Guth, [*An obstacle to building a time machine*]{}, Phys. Rev. Lett. [**68**]{}, 263 (1992). Carrol, Fahri, Guth, Olum, [*Energy-momentum restrictions on the creation of Gott-time machine*]{}, Phys. Rev. D[**50**]{}, 6190 (1994). P. Menotti, D. Seminara, [*Energy theorem for 2+1 dimensional gravity*]{}, gr-qc/9406016 (1994). S. Holst, [*Gott time machines in the Anti-de Sitter space*]{}, Gen. Relativ. Gravit. [**28**]{}, 387 (1996). G. t’Hooft, [*Causuality in (2+1)-dimensional gravity*]{}, Class. Quant. Grav. [**9**]{}, 1335 (1992); [*Classical N-particle cosmology in 2+1 dimensions*]{}, Class. Quant. Grav. [**10**]{}, S79 (1993). M. Bañados, C. Teitelboim, J. Zanelli, Phys. Rev. Lett. [**69**]{}, 1849 (1992). M. Bañados, M. Henneaux, C. Teitelboim, J. Zanelli, Phys. Rev. D[**48**]{}, 1506 (1993). R. B. Mann, S. F. Ross, [*Gravitationally collapsing dust in (2+1)-dimensions*]{}, Phys. Rev. D[**47**]{}, 3319 (1993). H.-J. Matschull, [*Black hole creation in 2+1 dimensions*]{}, Class. Quant. Grav. [**16**]{}, 1069 (1999). S. Holst, H.-J. Matschull, [*The anti-de Sitter Gott universe: a rotating BTZ wormwhole*]{}, Class. Quant. Grav. [**16**]{}, 3095 (1999). D. Birmingham, S. Sen, [*Gott time machines, BTZ black hole formation, and Choptuik scaling*]{}, Phys. Rev. Lett. [**84**]{}, 1074 (2000). S. Carlip, C. Teiltelboim, [*Aspects of black hole quantum mechanics and thermodynamics in 2+1 dimensions*]{}, Phys. Rev. D[**51**]{}, 622 (1995). N. Cruz, J. Zanelli, [*Stellar equilibrium in (2+1)-dimensions*]{}, Class. Quant. Grav. [**12**]{}, 975 (1995). M. Lubo, M. Rooman, Ph. Spindel, [*(2+1) dimensional stars*]{}, Phys. Rev. D[**59**]{}, 044012 (1999). G. Clément, Class. Quant. Grav. [**10**]{}, L49 (1993). C. Martínez, C. Teitelboim and J. Zanelli, [*Charged rotating black hole in three spacetime dimensions*]{}, Phys. Rev. [**D61**]{}, 104013 (2000). M. Cataldo, [*Azimuthal electric field in a static rotationally symmetric (2+1)-dimensional spacetime*]{}, Phys. Lett. B[**529**]{}, 143 (2002). E. W. Hirschmann, D. L. Welch, Phys. Rev. D[**53**]{}, 5579 (1996). M. Cataldo, P. Salgado, Phys. Rev. D[**54**]{}, 2971 (1996). O. J. C. Dias, J. P. S. Lemos, JHEP [**0201:006**]{} (2002). M. Kamata, T. Koikawa, Phys. Lett. B[**353**]{}, 196 (1995). M. Kamata, T. Koikawa, Phys. Lett. B[**391**]{}, 87 (1997). M. Cataldo, P. Salgado, Phys. Lett. B[**448**]{}, 20 (1999). K. C. K. Chan, Phys. Lett. B[**373**]{}, 296 (1996). S. Deser, R. Jackiw, S. Templeton, Ann. Phys. (N.Y.) [**140**]{}, 372 (1982); (E) [**185**]{} , 406 (1988); Phys. Rev. Lett. [**48**]{}, 975 (1982). I. Vuori, [*Parity violation and the effective gravitational action in three dimensions*]{} Phys. Lett. B[**175**]{}, 176 (1986); J. van der Bij, R. Pisarski, S. Rao, [*Topological mass term for gravity induced by matter*]{}, Phys. Lett. B[**179**]{}, 87 (1986); M. Gonni, M. Valle, [*Massless fermions and (2+1)-dimensional gravitational massive action*]{} Phys. Rev. D[**34**]{}, 648 (1986). S. Deser, Yang Z., [*Is topologically massive gravity renormalizable?*]{} Class. Quant. Grav. [**7**]{}, 1603 (1990); B. Keszthelyi, G. Kleppe, [*Renormalizibity of $D=3$ topologically massive gravity*]{}, Phys. Lett. B[**281**]{}, 33 (1992). S. Carlip, Nucl. Phys. B[**324**]{}, 106 (1989). P. Gerbert, Nucl. Phys. B[**346**]{}, 440 (1990). S. Deser, A. R. Steif, [*Gravity theories with light sources in $D=3$*]{}, hep-th/9208018. G. Clément, Phys. Lett. B[**367**]{}, 70 (1996). S. Fernando and F. Mansouri, Indian J. Commun. Math. and Theor. Phys. [**1**]{}, 14 (1998); [gr-qc/9705015]{}. T. Dereli and Yu. N. Obukhov, Phys. Rev. D[**62**]{}, 024013 (2000). R. V. Wagoner, [*Scalar-tensor theory and gravitational waves*]{}, Phys. Rev. D[**1**]{}, 3209 (1970). C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. [**262**]{}, 593 (1985). P. M. Sá, A. Kleber, J. P. S. Lemos, Class. Quant. Grav. [**13**]{}, 125 (1996). P. M. Sá, J. P. S. Lemos, Phys. Lett. B[**423**]{}, 49 (1998). O. J. C. Dias, J. P. S. Lemos, [*Static and rotating electrically charged in three-dimensional Brans-dicke gravity theories*]{}, Phys. Rev. D[**64**]{}, 064001 (2001). O. J. C. Dias, J. P. S. Lemos, [*Magnetic point sources in three dimensional Brans-dicke gravity theories*]{}, Phys. Rev. D[**66**]{}, 024034 (2002). K. C. K. Chan and R. B. Mann, Phys. Rev. [**D50**]{}, 6385 (1994);(E) Phys. Rev. D[**52**]{}, 2600 (1995). S. Fernando, Phys. Lett. B[**468**]{}, 201 (1999). Y. Kiem, D. Park, Phys. Rev. D[**55**]{}, 6112 (1997). D. Park, J. K. Kim, J. Math. Phys. [**38**]{}, 2616 (1997). T. Koikawa, T. Maki, A. Nakamula, Phys. Lett. B[**414**]{}, 45 (1997). C. -M. Chen, Nucl. Phys. [**B544**]{}, 775 (1999). S. Carlip, [*Quantum gravity in 2+1 dimensions*]{}, (Cambridge Univ. Press: Cambridge, 1998). S. Carlip, [*The (2+1)-dimensional black hole*]{}, Class. Quant. Grav. [**12**]{}, 2853 (1995); [*Lectures on (2+1)-dimensional gravity*]{}, [gr-qc/9503024]{}; Mann, R. B. [*Lower dimensional black holes: inside and out*]{}, [gr-qc/9501038]{}. D. Kramer, H. Stephani, M. MacCallum, E. Herlt, [*Exact solutions of Einstein’s Field Equations*]{} (Cambridge University Press, 1980). T. Levi-Civita, [*$ds^2$ einsteiniani in campi newtoniani*]{}, Rend. Acc. Lincei [**27**]{}, 343 (1918). H. Weyl, [*Bemerkung über die axissymmetrischen lösungen der Einsteinschen gravitationsgleinchungen*]{}, Ann. Phys. (Germany) [**59**]{}, 185 (1919). W. Kinnersley, M. Walker, [*Uniformly accelerating charged mass in General Relativity*]{}, Phys. Rev. D[**2**]{}, 1359 (1970). F. J. Ernst, [*Removal of the nodal singularity of the C-metric*]{}, J. Math. Phys. [**17**]{}, 515 (1976). J. F. Plebański, M. Demiański, [*Rotating, charged and uniformly accelerating mass in general relativity*]{}, Annals of Phys. (N.Y.) [**98**]{}, 98 (1976). O. J. C. Dias, J. P. S. Lemos, [*Pair of accelerated black holes in a anti-de Sitter background: the AdS C-metric*]{}, Phys. Rev. D [**67**]{}, 064001 (2003). O. J. C. Dias, J. P. S. Lemos, [*Pair of accelerated black holes in de Sitter background: the dS C-metric*]{}, Phys. Rev. D [**67**]{}, 084018 (2003). O. J. C. Dias, J. P. S. Lemos, [*The extremal limits of the C-metric: Nariai, Bertotti-Robinson and anti-Nariai C-metrics*]{}, Phys. Rev. D[**68**]{} (2003) 104010; [hep-th/0306194]{}. W. Israel, K. A. Khan, [*Collinear particles and dipoles in general relativity*]{}, Nuovo Cimento [**33**]{}, 331 (1964). R. Bach, H. Weyl, [*Nene Lösungen der Einsteinschen Gravitationsgleichungen*]{}, Math. Z. [**13**]{}, 134 (1922). M. Aryal, L. H. Ford, A. Vilenkin, [*Cosmic strings and black holes*]{}, Phys. Rev. D[**34**]{}, 2263 (1986). M. S. Costa, M. J. Perry, [*Interacting Black Holes*]{}, Nucl. Phys. B[**591**]{} (2000) 469. A. Tomimatsu, [*Condition for equilibrium of two Reissner-Nordtröm black holes*]{}, Prog. Theor. Phys. [**71**]{}, 409 (1984). W. B. Bonnor, Phys. Lett. A[**83**]{}, 414 (1981). T. Ohta, T. Kimura, Prog. Theor. Phys. [**68**]{}, 1175 (1982). P. Ginsparg, M. J. Perry, [*Semiclassical perdurance of de Sitter space*]{}, Nucl. Phys. B [**222**]{}, 245 (1983). A. Vilenkin, [*Gravitational field of vacuum domain walls*]{}, Phys. Lett. B[**133**]{}, 177 (1983); [*Cosmic strings and domain walls*]{}, Phys. Rep. [**121**]{}, 263 (1985); A. Vilenkin, E. P. S. Shellard, [*Cosmic strings and other topological defects*]{}, (Cambridge University Press, 2000). J. Ipser, P. Sikivie, [*Gravitationally repulsive domain wall*]{}, Phys. Rev. D[**30**]{}, 712 (1984). J. Ehlers, W. Kundt, [*Exact solutions of the gravitational field equations*]{}, in [*Gravitation: an introduction to current research*]{}, edited by L. Witten (Wiley, New York, London, 1962). J. D. Brown, [*Black hole pair creation and the entropy factor*]{}, Phys. Rev. D [**51**]{}, 5725 (1995). M. A. Melvin, Phys. Lett. [**8**]{}, 65 (1964). H. Farhoosh, R. L. Zimmerman, [*Stationary charged C-metric*]{}, J. Math. Phys. [**20**]{}, 2272 (1979); [*Killing horizons and dragging of the inertial frame about a uniformly accelerating particle*]{}, Phys. Rev. D [**21**]{}, 317 (1980); [*Interior C-metric*]{}, Phys. Rev. D [**23**]{}, 299 (1981). A. Ashtekar, T. Dray, [*On the existence of solutions to Einstein’s equation with non-zero Bondi-news*]{}, Comm. Phys. [**79**]{}, 581 (1981). W. B. Bonnor, [*The sources of the vacuum C-metric*]{}, Gen. Rel. Grav. [**15**]{}, 535 (1983); [*The C-metric with $m=0,\ e \not=0$*]{}, Gen. Rel. Grav. [**16**]{}, 269 (1984). F. H. J. Cornish, W. J. Uttley, [*The interpretation of the C metric. The vacuum case*]{}, Gen. Rel. Grav. [**27**]{}, 439 (1995); [*The interpretation of the C metric. The charged case when $e^2 \leq m^2$*]{}, Gen. Rel. Grav. [**27**]{}, 735 (1995). W. Yongcheng, [*Vacuum C-metric and the metric of two superposed Schwarzschild black holes*]{}, Phys. Rev. D [**55**]{}, 7977 (1997). C. G. Wells, [*Extending the black hole uniqueness theorems I. Accelerating black holes: The Ernst solution and C-metric*]{}, [gr-qc/9808044]{}. V. Pravda, A. Pravdova, [*Co-accelerated particles in the C-metric*]{}, Class. Quant. Grav. [**18**]{}, 1205 (2001). J. Podolsk' y, J. B. Griffiths, [*Null limits of the C-metric*]{}, Gen. Rel. Grav. [**33**]{}, 59 (2001). J. Bič' ak, B. G. Schmidt, [*Asymptotically flat radiative space-times with boost-rotation symmetry: The general structure*]{}, Phys. Rev. D [**40**]{}, 1827 (1989). J. Bičák, [*Gravitational radiation from uniformly accelerated particles in general relativity*]{}, Proc. Roy. Soc. A [**302**]{}, 201 (1968). V. Pravda, A. Pravdova, [*Boost-rotation symmetric spacetimes - review*]{}, Czech. J. Phys. [**50**]{}, 333 (2000); [*On the spinning C-metric*]{}, in [*Gravitation: Following the Prague Inspiration*]{}, edited by O. Semerák, J. Podolský, M. Zofka (World Scientific, Singapore, 2002), [gr-qc/0201025]{}. H. F. Dowker, J. P. Gauntlett, D. A. Kastor, J. Traschen, [*Pair creation of dilaton black holes*]{}, Phys. Rev. D [**49**]{}, 2909 (1994). H. Farhoosh, R. L. Zimmerman, [*Surfaces of infinite red-shift around a uniformly accelerating and rotating particle*]{}, Phys. Rev. D [**21**]{}, 2064 (1980). P. S. Letelier, S. R. Oliveira, [*On uniformly accelerated black holes*]{}, Phys. Rev. D [**64**]{}, 064005 (2001). J. Bič' ak, V. Pravda, [*Spinning C-metric as a boost-rotation symmetric radiative spacetime*]{}, Phys. Rev. D [**60**]{}, 044004 (1999). H. F. Dowker and S. Thambyahpillai, Class. Quantum Grav. [**20**]{}, 127 (2003). K. Hong and E. Teo, Class. Quantum Grav. [**20**]{}, 3269 (2003); gr-qc/0410002. J. Podolský, J.B. Griffiths, [*Uniformly accelerating black holes in a de Sitter universe*]{}, Phys. Rev. D [**63**]{}, 024006 (2001). R. Emparan, G. T. Horowitz, R. C. Myers, [*Exact description of black holes on branes*]{}, JHEP [**0001**]{} 007 (2000); [*Exact description of black Holes on branes II: Comparison with BTZ black holes and black strings*]{}, JHEP [**0001**]{} 021 (2000). J. Podolsk' y, [*Accelerating black holes in anti-de Sitter universe*]{}, Czech. J. Phys. [**52**]{}, 1 (2002). A. Chamblin, [*Capture of bulk geodesics by brane-world black holes*]{}, Class. Quant. Grav. [**18**]{}, L17 (2001). R. Emparan, (private communication). H. Nariai, [*On some static solutions of Einstein’s gravitational field equations in a spherically symmetric case*]{}, Sci. Rep. Tohoku Univ. [**34**]{}, 160 (1950); [*On a new cosmological solution of Einstein’s field equations of gravitation*]{}, Sci. Rep. Tohoku Univ. [**35**]{}, 62 (1951). H. Nariai, H. Ishihara, [*On the de Sitter and Nariai solutions in General Relativity and their extension in higher dimensional spacetime*]{}, in [*A Random Walk in Relativity and Cosmology*]{}, edited by P. N. Dadhich, J. K. Rao, J. V. Narlikar, C. V. Vishveshwara (John Wiley $\&$ Sons, New York). B. Bertotti, [*Uniform electromagnetic field in the theory of general relativity*]{}, Phys. Rev. [**116**]{}, 1331 (1959); I. Robinson, Bull. Acad. Polon. [**7**]{}, 351 (1959). M. Caldarelli, L. Vanzo, Z. Zerbini, [*The extremal limit of D-dimensional black holes*]{}, in [*Geometrical Aspects of Quantum Fields*]{}, edited by A. A. Bytsenko, A. E. Gonçalves, B. M. Pimentel (World Scientific, Singapore, 2001); [hep-th/0008136]{}; N. Dadhich, [*On product spacetime with 2-sphere of constant curvature*]{}, [gr-qc/0003026]{}. R. Bousso, [*Proliferation of de Sitter space*]{}, Phys. Rev. D [**58**]{}, 083511 (1998); [*Adventures in de Sitter space*]{}, [hep-th/0205177]{}. J. Schwinger, Phys. Rev. [**82**]{}, 664 (1951). O. J. C. Dias, J. P. S. Lemos, [*Pair creation of de Sitter black holes on a cosmic string background*]{}, Phys. Rev. D[**69**]{} (2004) 084006; . O. J. C. Dias, [*Pair creation of anti-de Sitter black holes on a cosmic string background*]{}, Phys. Rev. D[**70**]{} (2004) 024007. S. W. Hawking, S. F. Ross, [*Pair production of black holes on cosmic strings*]{}, Phys. Rev. Lett. [**75**]{}, 3382 (1995). D. M. Eardley G. T. Horowitz, D. A. Kastor, J. Traschen, [*Breaking cosmic strings without black holes*]{}, Phys. Rev. Lett. [**75**]{}, 3390 (1995). A. Achúcarro, R. Gregory, K. Kuijken, [*Abelian Higgs hair for black holes*]{}, Phys. Rev. D [**52**]{}, 5729 (1995). R. Gregory, M. Hindmarsh, [*Smooth metrics for snapping strings*]{}, Phys. Rev. D [**52**]{}, 5598 (1995). J. Preskill, A. Vilenkin, Phys. Rev. D [**47**]{}, 2324 (1993). D. J. Gross, M. J. Perry, L. G. Yaffe, [*Instability of flat space at finite temperature*]{}, Phys. Rev. D [**25**]{}, 330 (1982). J.S. Langer, Ann. Phys. [**41**]{}, 108 (1967). S. Coleman, Phys. Rev. D [**15**]{}, 2629 (1977); Phys. Rev. D [**16**]{}, 1248 (1977)(E). C.G. Callan, S. Coleman, Phys. Rev. D [**16**]{}, 1762 (1977). S. Coleman, F. De Luccia, Phys. Rev. D [**21**]{}, 3305 (1980). M.B. Voloshin, I.Yu. Kobzarev, L.B. Okun Yad. Fiz. [**20**]{}, 1229 (1974) \[Sov. J. Nucl. Phys. [**20**]{}, 644 (1975)\]. M. Stone, Phys. Lett. B [**67**]{}, 186 (1977). V.G. Kiselev, K.G. Selivanov, Pisma Zh. Eksp. Teor. Fiz. [**39**]{}, 72 (1984) \[JETP Lett. [**39**]{}, 85 (1984)\]. V.G. Kiselev, K.G. Selivanov, Yad. Fiz. [**43**]{}, 239 (1986) \[Sov. J. Nucl. Phys. [**43**]{}, 153 (1986)\]. V.G. Kiselev, Phys. Rev. D [**45**]{}, 2929 (1992). M.B. Voloshin, Sov. J. Nucl. Phys. [**42**]{}, 644 (1985). M.B. Voloshin, in [*International School of Subnuclear Physics*]{}, (Ettore Majorana Centre for Scientific Culture, Italy, 1995). O. J. C. Dias, J. P. S. Lemos, [*False vacuum decay: effective one-loop action for pair creation of domain walls*]{}, J. Math. Phys. [**42**]{}, 3292 (2001); O. J. C. Dias, [*Pair creation of particles and black holes in external fields*]{} in: Proceedings of Xth Portuguese Meeting on Astronomy and Astrophysics, edited by J. P. S. Lemos, A. Mourão, et al (World Scientific, Singapore, 2001), [gr-qc/0106081]{}. J. H. Miller Jr, G. Cardenas, A. Garcia-Perez, W. More, A. W. Beckwith, J. P. McCarten, [*Quantum pair creation of soliton domain walls*]{}, J. Phys. A [**36**]{}, 9209 (2003); [*Soliton tunneling transistor*]{}, [cond-mat/0105409]{}. I.K. Affleck, N.S. Manton, Nucl. Phys. B [**194**]{}, 38 (1982). I.K. Affleck, O. Alvarez, N.S. Manton, Nucl. Phys. B [**197**]{}, 509 (1982). S. W. Hawking , in [*Black holes: an Einstein Centenary Survey*]{}, edited by S. W. Hawking, W. Israel (Cambridge University Press, 1979); [*Euclidean Quantum Gravity*]{}, edited by G. W. Gibbons, S. W. Hawking (Cambridge University Press, 1993). Z. C. Wu, [*Quantum creation of a black hole*]{}, Int. J. Mod. Phys. D [**6**]{}, 199 (1997); [*Real tunneling and black hole creation*]{}, Int. J. Mod. Phys. D [**7**]{}, 111 (1998). R. Bousso, S. W. Hawking, [*Lorentzian condition in quantum gravity*]{}, Phys. Rev. D [**59**]{}, 103501 (1999); [**60**]{}, 109903 (1999) (E). G. W. Gibbons, in [*Fields and Geometry*]{}, Proceedings of the 22nd Karpacz Winter School of Theoretical Physics, edited by A. Jadczyk (World Scientific, Singapore, 1986). D. Garfinkle, S. B. Giddings, [*Semiclassical Wheeler wormhole production*]{}, Phys. Lett. B[**256**]{}, 146 (1991). D. Garfinkle, S. B. Giddings, A. Strominger, [*Entropy in black hole pair production*]{}, Phys. Rev. D [**49**]{}, 958 (1994) H. F. Dowker, J. P. Gauntlett, S. B. Giddings, G. T. Horowitz, [*Pair creation of extremal black holes and Kaluza-Klein monopoles*]{}, Phys. Rev. D [**50**]{}, 2662 (1994). S. Ross, [*Pair creation rate for $U(1)^2$ black holes*]{}, Phys. Rev. D [**52**]{}, 7089 (1995). P. Yi, [*Toward one-loop tunneling rates of near-extremal magnetic black hole pair creation*]{}, Phys. Rev. D [**52**]{}, 7089 (1995). J. D. Brown, [*Duality invariance of black hole pair creation rates*]{}, Phys. Rev. D [**51**]{}, 5725 (1995). S. W. Hawking, G. T. Horowitz, S. F. Ross, [*Entropy, area, and black hole pairs*]{}, Phys. Rev. D [**51**]{}, 4302 (1995). S. W. Hawking, G. T. Horowitz, [*The gravitational hamiltonian, action, entropy and surface terms*]{}, Class. Quant. Grav. [**13**]{}, 1487 (1996). R. Emparan, [*Correlations between black holes formed in cosmic string breaking*]{}, Phys. Rev. D [**52**]{}, 6976 (1995). F. Mellor, I. Moss, [*Black holes and quantum wormholes*]{}, Phys. Lett. B [**222**]{}, 361 (1989); [*Black holes and gravitational instantons*]{}, Class. Quant. Grav. [**6**]{}, 1379 (1989). L. J. Romans, [*Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory*]{}, Nucl. Phys. B [**383**]{}, 395 (1992). R. B. Mann, S. F. Ross, [*Cosmological production of charged black hole pairs*]{}, Phys. Rev. D [**52**]{}, 2254 (1995). R. Bousso, S. W. Hawking, [*The probability for primordial black holes*]{}, Phys. Rev. D [**52**]{}, 5659 (1995); [*Pair production of black holes during inflation*]{}, Phys. Rev. D [**54**]{}, 6312 (1996); S. W. Hawking, [*Virtual black holes*]{}, Phys. Rev. D [**53**]{}, 3099 (1996). R. Garattini, Nucl. Phys. B (Proc. Suppl.) [**57**]{}, 316 (1997); Nuovo Cimento B[**113**]{}, 963 (1998); Mod. Phys. Lett. A [**13**]{}, 159 (1998); Class. Quant. Grav. [**18**]{}, 571 (2001). M. Volkov, A. Wipf, [*Black hole pair creation in de Sitter space: a complete one-loop analysis*]{}, Nucl. Phys. B[**582**]{}, 313 (2000). I. S. Booth, R. B. Mann, [*Complex instantons and charged rotating black hole pair creation*]{}, Phys. Rev. Lett. [**81**]{}, 5052 (1998); [*Cosmological pair production of charged and rotating black holes*]{}, Nucl. Phys. B [**539**]{}, 267 (1999). R. Bousso, [*Charged Nariai black holes with a dilaton*]{}, Phys. Rev. D [**55**]{}, 3614 (1997). R. Emparan, [*Pair production of black holes joined by cosmic strings*]{}, Phys. Rev. Lett. [**75**]{}, 3386 (1995). R. R. Caldwell, A. Chamblin, G. W. Gibbons, [*Pair creation of black holes by domain walls*]{}, Phys. Rev. D [**53**]{}, 7103 (1996). R. Bousso, A. Chamblin, [*Patching up the no-boundary proposal with virtual Euclidean wormholes*]{}, Phys. Rev. D [**59**]{}, 084004 (1999). R. Mann, [*Pair production of topological anti-de Sitter black holes*]{}, Class. Quantum Grav. [**14**]{}, L109 (1997); [*Charged topological black hole pair creation*]{}, Nucl. Phys. B [**516**]{}, 357 (1998). R. Parentani, S. Massar, [*The Schwinger mechanism, the Unruh effect and the production of accelerated black holes*]{}, Phys. Rev. D [**55**]{}, 3603 (1997); J. Garriga, M. Sasaki, [*Brane-world creation and black holes*]{}, Phys. Rev. D [**62**]{}, 043523 (2000); R. Emparan, R Gregory, C. Santos, [*Black holes on thick branes*]{}, Phys. Rev. D [**63**]{}, 104022 (2001). Z. C. Wu, [*Pair creation of black holes in anti-de Sitter space background (I)*]{}, Gen. Rel. Grav. [**31**]{}, 223 (1999); [*Pair creation of black hole in anti-de Sitter space background*]{}, Phys. Lett. B [**445**]{}, 274 (1999); [*Quantum creation of topological black hole*]{}, Mod. Phys. Lett. A [**15**]{}, 1589 (2000). S. W. Hawking, S. F. Ross, [*Duality between electric and magnetic black holes*]{}, Phys. Rev. D [**52**]{}, 5865 (1995). S. W. Hawking, [*Breakdown of predictability in gravitational collapse*]{}, Phys. Rev. D[**14**]{}, 2460 (1976). D. N. Page, [*Black hole information*]{}, Int. J. Mod. Phys. D[**3**]{}, 93 (1994); T. Banks, [*Lectures on black holes and information loss*]{}, [hep-th/9412131]{}; L. Thorlacius, [*Black hole evolution*]{}, [hep-th/9411020]{}; S. B. Giddings, [*Quantum mechanics of black holes*]{}, [ hep-th/9412138]{}; P. C. Argyres, A. O. Barvinski, V. Frolov, S. B. Giddings, D. A. Lowe, A. Strominger, L. Thorlacius, [*Quantum aspects of gravity*]{}, [ astrp-ph/9412046]{}; Y. Kazama, [*On quantum black holes*]{}, [ hep-th/941224]{}; S. B. Giddings, [*The black hole information paradox*]{}, [hep-th/9508151]{}; [*Why aren’t black holes infinitely produced?*]{}, Phys. Rev. D [**51**]{}, 6860 (1995); L. Susskind, [*Trouble for remnants*]{}, [hep-th/9501106]{}. S. Weinberg, [*Gravitation and Cosmology*]{} (Wiley, New York, 1972). V. Cardoso, O. J. C. Dias, J. P. S. Lemos, [*Gravitational radiation in D-dimensional spacetimes*]{}, Phys. Rev. D [**67**]{}, 064026 (2003). C. S. Wang Chang and D. L. Falkoff, Phys. Rev. [**76**]{}, 365 (1949). J. D. Jackson, [*Classical Electrodynamics*]{}, (J. Wiley, New York 1975). J. Bičák, P. Krtouš, [*Accelerated sources in de Sitter spacetime and the insuffiency of retarded fields*]{}, Phys. Rev. D [**64**]{}, 124020 (2001); [*The fields of uniformly accelerated charges in de Sitter spacetime*]{}, Phys. Rev. Lett. [**88**]{}, 211101 (2002). P. Krtouš, J. Podolsk' y, [*Radiation from accelerated black holes in de Sitter universe*]{}, [gr-qc/0301110]{}. J. Podolsk' y, M. Ortaggio, P. Krtouš, [*Radiation from accelerated black holes in an anti-de Sitter universe*]{}, [gr-qc/0307108]{}. J. M. Overduin, P. S. Wesson, [*Kaluza-Klein gravity*]{}, [gr-qc/9805018]{}. S. B. Giddings, [*Black holes at accelerators*]{}, [hep-th/0205027]{}; G. Landsberg, [*Black holes at future colliders and beyond: a review*]{}, [hep-th/0211043]{}. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [**B429**]{}, 263 (1998); Phys. Rev. D[**59**]{}, 086004 (1999); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [**B436**]{}, 257 (1998). P. C. Argyres, S. Dimopoulos, J. March-Russell, [**]{}, Phys. Lett. B[**441**]{}, 96 (1998). Banks, Fischler, [**]{}, JHEP [**9906**]{}, 014 (1999). Emparan, Horowitz, Myers, [**]{}, Phys. Rev. Lett. [**85**]{}, 499 (2000). S. B. Giddings, S. Thomas, [*High energy colliders as black hole factories: The end of short distance physics*]{}, Phys. Rev. D[**65**]{}, 056010 (2002). S. Dimopoulos, G. Landsberg, [*Black holes at the LHC*]{}, Phys. Rev. Lett. [**87**]{}, 161602 (2001). P. D. D’Eath, P. N. Payne, [*Gravitational radiation in high speed black hole collisions. 1. Perturbation treatment of the axisymmetric speed of light collision*]{}, Phys. Rev. D[**46**]{}, 658 (1992); [*Gravitational radiation in high speed black hole collisions. 2. Reduction to two independent variables and calculation of the second order news function Perturbation treatment of the axisymmetric speed of light collision*]{}, Phys. Rev. D[**46**]{}, 675 (1992); [*Gravitational radiation in high speed black hole collisions. 3. Results and conclusions*]{}, Phys. Rev. D[**46**]{}, 694 (1992). H. Yoshino and Y. Nambu, [*High-energy head-on collisions of particles and hoop conjecture*]{}, Phys. Rev. [**D67**]{}, 024009 (2003). D. M. Eardley and S. B. Giddings, [*Classical black hole production in high energy collisions*]{}, Phys. Rev. [**D66**]{}, 044011 (2002). F. R. Tangherlini, Nuovo Cim. [**27**]{}, 636 (1963). R. C. Myers and M. J. Perry, [*Black holes in higher dimensional space-times*]{} Annals Phys. [**172**]{}, 304 (1986). Boulware, S. Deser, Phys. Rev. Lett. [**55**]{}, 2656 (1985). M. Cvetic, D. Youm, Nucl. Phys. [**B476**]{}, 118 (1996). M. Cvetic, F. Larsen, Phys. Rev. D[**56**]{}, 4994 (1997). G. W. Gibbons, C. A. R. Herdeiro, Class. Quant. Grav. [**16**]{}, 3619 (1999). C. A. R. Herdeiro, Nucl. Phys. [**B582**]{}, 363 (2000). R. Emparan, H. S. Reall, Phys. Rev. Lett. [**88**]{} 101101 (2002); Phys. Rev. D[**65**]{} 084025 (2002). N. L. Santos, O. J. C. Dias and J. P. S. Lemos, [*Global embedding Minkowskian spacetime procedure in higher dimensional black holes:\ Matching between Hawking temperature and Unruh temperature*]{}, Phys. Rev. D, submitted (2004). V. Cardoso, O. J. C. Dias, J. P. S. Lemos, [*Nariai, Bertotti-Robinson and anti-Nariai solutions in higher dimensions*]{}, Phys. Rev. D[**70**]{} (2004) 024002. O. J. C. Dias, J. P. S. Lemos, [*Pair creation of higher dimensional de Sitter black holes*]{}, Phys. Rev. D, submitted (2004); hep-th/0410279. B. F. Schutz, Class. Quant. Grav. [**16**]{}, A131 (1999); B. F. Schutz and F. Ricci, in [*Gravitational Waves*]{}, eds I. Ciufolini et al, (Institute of Physics Publishing, Bristol, 2001). S. A. Hughes, astro-ph/0210481. K. Danzmann et al., in [*Gravitational Wave Experiments*]{}, eds. E. Coccia, G. Pizzella and F. Ronga (World Scientific, Singapore, 1995). A. Abramovici et al., Science [**256**]{}, 325 (1992). C. Bradaschia et al., in [*Gravitation*]{} 1990, Proceedings of the Banff Summer Institute, Banff, Alberta, 1990, edited by R. Mann and P. Wesson (World Scientific, Singapore, 1991). K. S. Thorne, Rev. Mod. Phys, [**52**]{}, 299 (1980); K. S. Thorne, in [*General Relativity: An Einstein Centenary Survey*]{}, eds. S. W. Hawking and W. Israel (Cambridge University Press, 1989). T. Damour, in [*Gravitational Radiation*]{}, eds Nathalie Deruelle and Tsvi Piran (North Holland Publishing Company, New York, 1983). V. Cardoso and J. P. S. Lemos, “Gravitational Radiation from the radial infall of highly relativistic point particles into Kerr black holes” gr-qc/0211094. E. Berti, M. Cavaglià, L. Gualtieri, [*Gravitational energy loss in high energy particle collisions: ultrarelativistic plunge into a multidimensional black hole*]{}, [hep-th/0309203]{}. V. Cardoso and J. P. S. Lemos, Phys. Lett. [**B 538**]{}, 1 (2002); Gen. Rel. Gravitation (in press), gr-qc/0207009. D. N. Page, [*Particle emissions rates from a black hole: massless particles from an uncharged, nonrotating hole*]{}, Phys. Rev. D[**13**]{}, 198 (1976); [*Particle emissions rates from a black hole. II. Massless particles from a rotating hole*]{}, Phys. Rev. D[**14**]{}, 3260 (1976). V. Frolov and D. Stojkovic, [*Quantum radiation from a 5-dimensional rotating black hole*]{}, Phys. Rev. Lett. [**89**]{}, 151302 (2002). D. Ida, Kin-ya Oda, S. C. Park, [*Rotating black holes at future colliders: Greybody factors for brane fields*]{}, Phys. Rev. D[**67**]{}, 064025 (2003). P. Kanti, J. March-Russell, [*Calculable corrections to brane black hole decay. I: The scalar case*]{}, Phys. Rev. D[**66**]{}, 024023 (2002); [*Calculable corrections to brane black Hole decay II: Greybody factors for spin 1/2 and 1* ]{}, Phys. Rev. D[**67**]{}, 104019 (2003). A. R. Steif, [*Supergeometry of three dimensional black holes*]{}, Phys. Rev. D[**53**]{}, 5521 (1996). H. Soodak, M. S. Tiersten, Am. J. Phys. [**61**]{}, 3955 (1993). V. Cardoso, S. Yoshida, O. J. C. Dias, J. P. S. Lemos, [*Late-time tails of wave propagation in higher dimensional spacetimes*]{}, Phys. Rev. D [**68**]{}, 061503 (2003) \[Rapid Communications\]. C. A. R. Herdeiro, O. J. C. Dias, unpublished work. J. H. Horne, G. T. Horowitz, Nucl. Phys. B[**368**]{}, 444 (1992). G. T. Horowitz, D. L. Welch, Phys. Rev. Lett. [**71**]{}, 328 (1993). J. Stachel, Phys. Rev. D[**26**]{}, 1281 (1982). T. Regge, C. Teitelboim, Ann. Phys. (NY) [**88**]{}, 286 (1974). W. A. Moura-Melo, J. A. Helayel-Neto, Phys. Rev. D[**63**]{}, 065013 (2001). A. García, [hep-th/9909111]{}. Y. Kiem and D. Park, Phys. Rev. [**D55**]{}, 6112 (1997). R. H. Boyer, R. W. Lindquist, J. Math. Phys. [**8**]{}, 265 (1967). B. Carter, Phys. Rev. [**174**]{}, 1559 (1968). E. Witten, Phys. Rev. D [**44**]{}, 314 (1991). G. Mandal, A. M. Sengupta and S. R. Wadia, Mod. Phys. Lett. A [**6**]{}, 1685 (1991). D. Park and S. Yang, Gen. Rel. Grav. [**31**]{}, 1343 (1999). J. D. Brown, C. Teitelboim, Nucl. Phys. B[**297**]{}, 297 (1988). S. G. Ghosh, Phys. Rev. D[**62**]{}, 127505 (2000). J. L. Friedman, K. Schleich, D. W. Witt, Phys. Rev. D[**71**]{}, 1486 (1993); (E) [**75**]{}, 1872 (1995); T. Jacobson, S. Venkatararamani, Class. Quantum Grav. [**12**]{}, 1055 (1995); G. J. Galloway, K. Schleich, D. W. Witt, E. Woolgar, Phys. Rev. D[**60**]{}, 104039 (1999). A. Ghosh, P. Mitra, Phys. Rev. Lett. [**78**]{}, 1858 (1997); D. R. Brill, J. Louko, P. Peldán in [@topological]; A. Chamblin, R. Emparan, C. V. Johnson, R. C. Myers, Phys. Rev. D[**60**]{}, 104026 (1999); C. Peça, J. P. S. Lemos, Phys. Rev. D[**59**]{}, 124007 (1999); C. Peça, J. P. S. Lemos, J. Math. Physics [**41**]{}, 4783 (2000). A. DeBenedictis, Class. Quantum Grav. [**16**]{}, 1955 (1999); Gen. Rel. Grav. [**31**]{}, 1549 (1999). M. M. Caldarelli, D. Klemm, Nucl. Phys. B[**545**]{}, 434 (1999); J. P. S. Lemos, Nucl. Phys. B[**600**]{}, 272 (2001). V. Cardoso, J. P. S. Lemos, Class. Quant. Grav. [**18**]{}, 5257 (2001). J. P. S. Lemos, in: [*Astronomy and Astrophysics: Recent Developments*]{} (World Scientific, 2001) , [gr-qc/0011092]{}. L. Witten, in: [*Gravitation: an introduction to current research*]{} (Wiley, 1962). J. P. S. Lemos, [*Thermodynamics of the two-dimensional black hole in the Teitelboim-Jackiw theory*]{}, Phys. Rev. D [**54**]{}, 6206 (1996). S. Deser, O. Levin, [*Accelerated detectors and temperature in (anti-) de Sitter spaces*]{}, Class. Quant. Grav. [**14**]{}, L163 (1997). G. Gibbons, S. Hawking, [*Cosmological event horizons, thermodynamics, and particle creation*]{}, Phys. Rev. D [**15**]{}, 2738 (1977); K. Lake, R. Roeder, [*Effects of a nonvanishing cosmological constant on the spherically symmetric vacuum manifold*]{}, Phys. Rev. D [**15**]{}, 3513 (1977). F. Mellor, I. G. Moss, Phys. Rev. D [**41**]{}, 403 (1990); Class. Quant. Grav. [**9**]{}, L43 (1992); P. R. Brady, E. Poisson, Class. Quant. Grav. [**9**]{}, 121 (1992); P. R. Brady, D. Nunez, S. Sinha, Phys. Rev. D [**47**]{}, 4239 (1993). L. M. Burko, A. Ori, in [*Internal structure of black holes and spacetime singularities*]{}, edited by L. M. Burko and A. Ori (IOP, Bristol, 1997). G. T. Horowitz, H. J. Sheinblatt, Phys. Rev. D [**55**]{}, 650 (1997). S. F. Hawking, G. F. R. Ellis, [*The Large Scale Structure of Space-Time*]{}, (Cambridge University Press, 1976). B. Carter, [*General theory of statinary black holes*]{}, in [*Black holes*]{} edited by B. DeWitt and C. DeWitt (Gordon and Breach, New York, 1973). M. Ortaggio, [*Impulsive waves in the Nariai universe*]{}, Phys. Rev. D [**65**]{}, 084046 (2002). R. Bousso, S. W. Hawking, [*(Anti-)evaporation of Schwarzschild-de Sitter black holes*]{}, Phys. Rev. D [**57**]{}, 2436 (1998); R. Bousso, [*Proliferation of de Sitter space*]{}, Phys. Rev. D [**58**]{}, 083511 (1998). S. Nojiri, S. D. Odintsov, [*Quantum evolution of Schwarzschild-de Sitter (Nariai) black holes*]{}, Phys. Rev. D [**59**]{}, 044026 (1999); [*De Sitter space versus Nariai black hole: stability in d5 higher derivative gravity*]{}, Phys. Lett. B [**523**]{}, 165 (2001). L. A. Kofman, V. Sahni, A. A. Starobinski, [*Anisotropic cosmological model created by quantum polarization of vacuum*]{}, Sov. Phys. JETP [**58**]{}, 1090 (1983). A. J. M. Medved, [*Nearly degenerated dS horizons from a 2-D perpective*]{}, [hep-th/0302058]{}. A. S. Lapedes, [*Euclidean quantum field theory and the Hawking effect*]{}, Phys. Rev. D [**17**]{}, 2556 (1978). O. B. Zaslavsky, [*Geometry of nonextreme black holes near the extreme state*]{}, Phys. Rev. D [**56**]{}, 2188 (1997); [*Entropy of quantum fields for nonextreme black holes in the extreme limit*]{}, Phys. Rev. D [**57**]{}, 6265 (1998). R. B. Mann, S. N. Solodukhin, [*Universality of quantum entropy for extreme black holes*]{}, Nucl. Phys. B [**523**]{}, 293 (1998). D. J. Navarro, J. Navarro-Salas, P. Navarro, [*Holography, degenerated horizons and entropy*]{}, Nucl. Phys. B [**580**]{}, 311 (2000); A. J. M. Medved, [*Reissner-Nordström Near extremality from a Jackiw-Teiltelboim perpective*]{}, [hep-th/0111091]{}. A. Barvinsky, S. Das, G. Kunstatter, [*Quantum mechanics of charged black holes*]{}, Phys. Lett. B [**517**]{}, 415 (2001). M. Ortaggio, J. Podolský, [*Impulsive waves in electrovac direct product spacetimes with Lambda*]{}, Class. Quant. Grav. [**19**]{}, 5221 (2002). J. Maldacena, J. Michelson, A. Strominger, [*Anti-de Sitter fragmentation*]{}, JHEP [**9902:11**]{}, (1999); [*Vacuum states for $AdS_2$ black holes*]{}, M. Spradlin, A. Strominger, JHEP [**9911:021**]{}, (1999). G. T. Horowitz, H. J. Sheinblatt, [*Tests of cosmic censorship in the Ernst spacetime*]{}, Phys. Rev. D [**55**]{}, 650 (1997). K. Hong, E. Teo, [*A new form of the C-metric*]{}, [gr-qc/0305089]{}. S. Coleman, Phys. Rev. D [**11**]{}, 2088 (1975). S.P. Gavrilov, D.M. Gitman, Phys. Rev. D [**53**]{}, 7162 (1996). R. Soldati, L. Sorbo, Phys. Lett. B [**426**]{}, 82 (1998). Q.-G. Lin, J. Phys. G [**25**]{}, 17 (1999); J. Phys. G [**25**]{}, 1793 (1999); J. Phys. G [**26**]{}, L17 (2000). B. Körs, M.G. Schmidt, Eur. Phys. J. C [**6**]{}, 175 (1999). C. Montonen, D.I. Olive, Phys. Lett. B [**72**]{}, 117 (1977). N.S. Manton, Nucl. Phys. B [**150**]{}, 397 (1978). R. Rajamaran, “Solitons and Instantons", (Holland, Amsterdam, 1989). J. Garriga, Phys. Rev. D [**49**]{}, 6327 (1994). G. Lavrelashvili, Nucl. Phys. Proc. Suppl. [**88**]{}, 75 (2000). J.B. Hartle, S.W. Hawking, [*Wave function of the Universe*]{}, Phys. Rev. D [**28**]{}, 2960 (1983). G. W. Gibbons, S. W. Hawking, M. J. Perry, [*Path integral and the indifiniteness of the gravitational action*]{}, Nucl. Phys. B [**138**]{}, 141 (1978). G. W. Gibbons, M. J. Perry, [*Quantizing gravitational instantons*]{}, Nucl. Phys. B [**146**]{}, 90 (1978). S. M. Christensen, M. J. Duff, [*Quantizing gravity with cosmological constant*]{}, Nucl. Phys. B [**146**]{}, 90 (1978). R. E. Young, [*Semiclassical stability of asymptotically locally flat spaces*]{}, Phys. Rev. D [**28**]{}, 2420 (1983). R. E. Young, [*Semiclassical instability of gravity with positive cosmological constant*]{}, Phys. Rev. D [**28**]{}, 2436 (1983). J. D. Brown, J. W. York, [*Quasilocal energy and conserved charges derived from the gravitational action*]{}, Phys. Rev. D [**47**]{}, 1407 (1993); , [*Microcanonical functional integral for the gravitational field*]{}, Phys. Rev. D [**47**]{}, 1420 (1993). G. W. Gibbons, S. W. Hawking, [*Action integrals and partition functions in quantum gravity*]{}, Phys. Rev. D [**15**]{}, 2752 (1977). T. Regge, Nuovo Cimento [**19**]{}, 558 (1961); G. W. Gibbons, M. J. Perry, Phys. Rev. D [**22**]{}, 313 (1980). M. Abramowitz, I. A. Stegun, [*Handbook of Mathematical Functions*]{}, (Dover, New York, 1970). H. F. Goenner, [*General Relativity and Gravitation*]{}, edited by A. Held (Plenum, New York, 1980), p. 441. S. Deser, Orit Levin, [*Mapping Hawking into Unruh thermal properties*]{}, Phys. Rev. D [**59**]{}, 064004 (1999). D. Birmingham, [*Topological black holes in anti-de Sitter space*]{}, Class. Quant. Grav. [**16**]{}, 1197 (1999). A. M. Awad, [*Higher dimensional charged rotating solutions in (A)dS space-times*]{}, [hep-th/0209238]{} (2002). S. D. H. Hsu, hep-ph/0203154; H. Tu, hep-ph/0205024; A. Jevicki and J. Thaler, Phys. Rev. [**D66**]{}, 024041 (2002); Y. Uehara, hep-ph/0205199. G. N. Watson, [*A Treatise on the Theory of Bessel Functions*]{} (Cambridge University Press, 1995) R. Courant and D. Hilbert, [*Methods of Mathematical Physics*]{}, Chapter VI (Interscience, New York, 1962). J. Hadamard, [*Lectures on Cauchy’s Problem in Linear Partial Differential Equations*]{} (Yale University Press, New Haven, 1923). J. D. Barrow and F. J. Tipler, [*The Anthropic Cosmological Principle*]{} (Oxford University Press, Oxford, 1986). D. V. Gal’tsov, Phys. Rev. [**D66**]{}, 025016 (2002). S. Hassani, [*Mathematical Physics*]{}, (Springer-Verlag, New York, 1998). P. O. Kazinski, S. L. Lyakhovich and A. A. Sharapov, Phys. Rev. [**D66**]{}, 025017 (2002); B. P. Kosyakov, Theor. Math. Phys. [**119**]{}, 493 (1999). B. Chen, M. Li and Feng-Li Lin, JHEP [**0211**]{}, 050 (2002). J. M. Weisberg and J. H. Taylor, astro-ph/0211217. P. C. Peters, Phys. Rev. [**136**]{}, B1224 (1964); P. C. Peters and J. Mathews, Phys. Rev. [**131**]{}, 435 (1963). B. F. Schutz, [*A First Course in General Relativity*]{}, (Cambridge University Press, 1985). F. Zerilli, Phys. Rev. Lett. [**24**]{}, 737(1970); F. Zerilli, Phys. Rev. [**D2**]{}, 2141 (1970). M. Davis, R. Ruffini, W. H. Press, and R. H. Price, Phys. Rev. Lett. [**27**]{}, 1466 (1971). S. L. Smarr (ed.), [*Sources of Gravitational Radiation*]{}, (Cambridge University Press, 1979). P. Anninos, D. Hobill, E. Seidel, L. Smarr, and W. M. Suen, Phys. Rev. Lett. [**71**]{}, 2851 (1993); R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Phys. Rev. Lett. [**77**]{}, 4483 (1996). R. Ruffini, Phys. Rev. [**D7**]{}, 972 (1973); M. J. Fitchett, [*The Gravitational Recoil Effect and its Astrophysical Consequences*]{}, (PhD Thesis, University of Cambridge, 1984). T. Regge, J. A. Wheeler, Phys. Rev. [**108**]{}, 1063 (1957). S. Weinberg, Phys. Lett. [**9**]{}, 357 (1964); Phys. Rev. [**135**]{}, B1049 (1964). L. Smarr, Phys. Rev. [**D15**]{}, 2069 (1977); R. J. Adler and B. Zeks, Phys. Rev. [**D12**]{}, 3007 (1975). K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. [**2**]{}, 2(1999). G. T. Horowitz and V. E. Hubeny Phys. Rev. [**D62**]{}, 024027 (2000); V. Cardoso and J. P. S. Lemos, Phys. Rev. [**D63**]{}, 124015 (2001); Phys. Rev. [**D64**]{}, 084017 (2001); Class. Quantum Grav. [**18**]{}, 5257 (2001); D. Birmingham, I. Sachs and S. N. Solodukhin, Phys. Rev. Lett. [**88**]{}, 151301 (2002); R. A. Konoplya, hep-th/0205142; S. F. J. Chan and R. B. Mann, Phys. Rev. [**D55**]{}, 7546 (1997); I. G. Moss and J. P. Norman, Class. Quant. Grav. [**19**]{}, 2323 (2002); R. Aros, C. Martinez, R. Troncoso and J. Zanelli, hep-th/0211024; D. T. Son, A. O. Starinets, JHEP [**0209**]{}:042, (2002). F. Mellor and I. Moss, Phys. Rev. [**D41**]{}, 403(1990); E. Abdalla, B. Wang, A. Lima-Santos and W. G. Qiu, Phys. Lett. [**B538**]{}, 435 (2002). J. M. Maldacena, Adv. Theor. Math. Phys. [**2**]{}, 253 (1998). B. F. Schutz and C. M. Will, Astrophys. Journal [**291**]{}, L33 (1985); C. M. Will and S. Iyer, Phys. Rev. [**D35**]{}, 3621 (1987); S. Iyer, Phys. Rev. [**D35**]{}, 3632 (1987). V. Cardoso and J. P. S. Lemos, Phys. Rev. [**D66**]{}, 064006 (2002). O. Dreyer, gr-qc/0211076; G. Kunstatter, gr-qc/0211076. I. S. Gradshteyn and I. M. Ryzhik, [*Table of Integrals, Series and Products*]{}, (Academic Press, New York, 1965).

--- abstract: 'We show that a right artinian ring $R$ is right self-injective if and only if $\psi(M)=0$ (or equivalently $\phi(M)=0$) for all finitely generated right $R$-modules $M$, where $\psi , \phi : {{\rm mod\,}}R \to \mathbb N$ are functions defined by Igusa and Todorov. In particular, an artin algebra $\Lambda$ is self-injective if and only if $\phi(M)=0$ for all finitely generated right $\Lambda$-modules $M$.' author: - | François Huard,\ Marcelo Lanzilotta title: 'Self-injective right artinian rings and Igusa Todorov functions[^1]' --- [^2] In their paper [@IT], Igusa and Todorov introduce two functions $\phi$ and $\psi$ in order to show that the finitistic dimension of an artin algebra with representation dimension at most three is finite. It turns out that these invariants also characterise self-injective right artinian rings. We start by recalling the definitions of $\phi$ and $\psi$. In what follows, $R$ is a right artinian ring and ${{\rm mod\,}}R$ is the category of finitely generated right $R$-modules. Let $K$ be the free abelian group generated by all symbols $[M]$ with $M\in{{\rm mod\,}}R$ modulo the subgroup generated by: - $[A]-[B]-[C]$ if $A\cong B\oplus C$, - $[P]$ if $P$ is projective. Then $K$ is the free abelian group generated by all isomorphism classes of finitely generated indecomposable non projective $R$-modules. The syzygy functor $\Omega$ then gives rise to a group homomorphism $\Omega: K \to K$. For any $M\in{{\rm mod\,}}R$, let $\langle M\rangle$ denote the subgroup of $K$ generated by all the indecomposable non projective summands of $M$. Since the rank of $\Omega (\langle M\rangle)$ is less or equal to the rank of $\langle M\rangle$ which is finite, it follows from the well ordering principle that there exists a non-negative integer $n$ such that the rank of $\Omega^n(\langle M\rangle)$ is equal to the rank of $\Omega^i(\langle M\rangle)$ for all $i\geq n$. We let $\phi (M)$ denote the least such $n$. The main properties of $\phi$ are summarized below. \[lem:itphi\][[@IT; @HLM1]]{} Let $R$ be a right artinian ring and $M,N\in{{\rm mod\,}}R$. - If the projective dimension of $M$, ${{\rm pd\,}}M$, is finite, then ${{\rm pd\,}}M=\phi(M)$, - If $M$ is indecomposable of infinite projective dimension, then $\phi(M)=0$, - $\phi(N\oplus M)\geq \phi(M)$, - $\phi(M^k)=\phi(M)$ if $k \geq 1$, - $\phi(M) \leq \phi(\Omega M) +1$. The function $\psi:{{\rm mod\,}}R \to \mathbb N$ is then defined as follows. For any $M\in{{\rm mod\,}}R$, $\psi(M)=\phi(M)+\max\{{{\rm pd\,}}X | X \hbox{ is a summand of } \Omega^{\phi(M)}M \hbox{ and } {{\rm pd\,}}X < \infty\}. $ \[lem:itpsi\][[@IT; @HLM1]]{} Let $R$ be a right artinian ring and $M,N\in{{\rm mod\,}}R$. - If the projective dimension of $M$ is finite, then ${{\rm pd\,}}M=\psi(M)$, - If $M$ is indecomposable of infinite projective dimension, then $\psi(M)=0$, - $\psi(N\oplus M)\geq \psi(M)$, - $\psi(M^k)=\psi(M)$ if $k \geq 1$, - $\psi(M) \leq \psi(\Omega M) +1$, - If $0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is a short exact sequence in ${{\rm mod\,}}R$, and ${{\rm pd\,}}C$ is finite, then $\Psi(C)\leq \Psi(A\oplus B) +1$. We introduce the natural concepts of $\phi$-dimension and $\psi$-dimension for a right artinian ring $R$. For a right artinian ring $R$, ${{\phi\,\rm{dim}}}(R)=\sup\{\phi(M) | M \in{{\rm mod\,}}R\} $ and ${{\psi\,\rm{dim}}}(R)=\sup\{\psi(M) | M \in{{\rm mod\,}}R\} .$ Another invariant for $R$ is its finitistic dimension, fin.dim$(R)$, defined as the supremum of the projective dimensions of the finitely generated right $R$-modules of finite projective dimension (see [@ZH]). It follows from Lemma \[lem:itphi\](a) and Lemma \[lem:itpsi\](a) that if ${{\phi\,\rm{dim}}}R$ or ${{\psi\,\rm{dim}}}R$ is finite, then the finitistic dimension of $R$ is also finite. Recall that a right artinian ring $R$ is right self-injective if the module $R_R$ is injective. Note that every indecomposable module over a right self-injective right artinian ring is either projective or has infinite projective dimension. However, this property does not characterize right self-injective rings. Consider $\Lambda$ the bound quiver algebra $kQ/J^2$ where $k$ is a field, $J$ is the ideal of $KQ$ generated by the arrows and $Q$ is given by $$\xymatrix{1\ar[r]& 2\ar[r]& 3\ar[r] & \cdots \ar[r] & n-1\ar[r] &n \ar@(ur,dr) }$$ In this case, the projective dimension of each finitely generated right $\Lambda$-module is either zero or infinite. Therefore the finitistic dimension of $R$ is equal to zero. However, ${{\psi\,\rm{dim}}}(\Lambda) ={{\phi\,\rm{dim}}}(\Lambda)=\phi(S_1 \oplus S_n)=n-1$ where $S_1$ and $S_n$ denote the simple modules at the vertices $1$ and $n$ respectively. Note that $\Lambda$ is not self-injective since the indecomposable projective at the vertex $n$ is not injective. For any right artinian ring $R$, we have fin.dim$(R)\leq {{\phi\,\rm{dim}}}(R) \leq$ gl.dim$(R)$, where gl.dim$(R)$ denotes the global dimension of $R$. As the example above shows, these inequalities can be strict. We can now state and prove our main result. For a right artinian ring $R$, the following are equivalent. - ${{\phi\,\rm{dim}}}(R)=0$. - ${{\psi\,\rm{dim}}}(R)=0$. - $R$ is right self-injective. Clearly, (b) implies (a). On the other hand, if ${{\phi\,\rm{dim}}}(R)=0$, then for each $R$-module $M$, either ${{\rm pd\,}}M=0$ or ${{\rm pd\,}}M=\infty$. Thus $\phi(M)=\psi(M)$ for all modules $M$ and hence ${{\psi\,\rm{dim}}}(R)=0$. We will now prove that (a) and (c) are equivalent. Assume that ${{\phi\,\rm{dim}}}(R)=0$. We start by showing that each indecomposable projective has simple socle. Let $P$ be an indecomposable projective module and assume that $P$ has two nonisomorphic simples $S_1$ and $S_2$ in its socle. This yields the short exact sequences: $0\rightarrow S_1 \rightarrow P \rightarrow M_1 \rightarrow 0,$ $0\rightarrow S_2 \rightarrow P \rightarrow M_2 \rightarrow 0,$ $0\rightarrow S_1\oplus S_2 \rightarrow P \rightarrow M_3 \rightarrow 0,$ with $M_1$, $M_2$, $M_3$ nonisomorphic indecomposable (since they have simple top) modules. But then the rank of $\langle M_1 \oplus M_2 \oplus M_3\rangle$ is 3 while the rank of $\Omega(\langle M_1 \oplus M_2 \oplus M_3\rangle)$ is 2, implying that $\phi(M_1\oplus M_2 \oplus M_3) \geq 1$, a contradiction. Assume now that $P$ has two isomorphic simples $S$ in its socle. This yields the short exact sequences: $0\rightarrow S \rightarrow P \rightarrow M_1 \rightarrow 0,$ $0\rightarrow S\oplus S \rightarrow P \rightarrow M_2 \rightarrow 0,$ with $M_1$ and $M_2$ nonisomorphic indecomposable (since they have simple top) modules. The rank of $\langle M_1 \oplus M_2\rangle$ is 2 while the rank of $\Omega(\langle M_1 \oplus M_2\rangle)$ is 1, thus $\phi(M_1\oplus M_2) \geq 1$, a contradiction. Thus each indecomposable projective module has simple socle. Given an indecomposable projective $P\in{{\rm mod\,}}R$, let $I$ be its injective hull. Since $P$ has simple socle, $I$ must be indecomposable. Note that since $R$ is right artinian, $I$ is not necessarily finitely generated. We will show that $P$ is injective. If not, we have the following commutative diagram $$\xymatrix{ 0 \ar[r] & P \ar[d] \ar[r]& U \ar[d]\ar[r]& S\ar[d] \ar[r] & 0\\ 0 \ar[r] & P\ar[r] & I \ar[r] & C \ar[r] & 0 }$$ where $C\neq 0$, $S$ is a simple $R$-module lying in the socle of $C$, and the upper sequence is obtained by lifting the monomorphism $S\to C$. Note that since $P$ and $S$ are finitely generated, so is $U$. Moreover, the map $U\to I$ is a monomorphism and hence $U$ is indecomposable since it has simple socle. This implies that $S$ is not projective, and that a fortiori neither is $U$ since otherwise we would have ${{\rm pd\,}}S=1=\phi(S)$, a contradiction. Let $P(S)$ be the projective cover of $S$. We have the following commutative diagram $$\xymatrix{ {} & {} & 0 \ar[d] & 0\ar[d] & {}\\ {} & 0 \ar[d] & \Omega(U)\oplus P'' \ar[d] \ar[r]_\cong & \Omega(S) \ar[d] & {}\\ 0 \ar[r] & P \ar[d] \ar[r]& P' \ar[d]\ar[r]& P(S)\ar[d] \ar[r] & 0\\ 0 \ar[r] & P\ar[d] \ar[r] & U \ar[d]\ar[r] & S \ar[d]\ar[r] & 0\\ {} & 0 & 0 & 0 & {} }$$ where the isomorphism follows from the snake lemma. In $K$, we have $[\Omega(S)]=[\Omega(U) \oplus P'']=[\Omega(U)]$. Therefore, since $U$ and $S$ are indecomposable, we have $\Omega(\langle U\rangle)= \Omega(\langle S\rangle)$, so that $\Omega(\langle U\oplus S\rangle)= \Omega(\langle U\rangle)+\Omega(\langle S\rangle)=\Omega(\langle S\rangle)$. Now $S$ is not a summand of $U$ and neither of them are projective hence we have ${\rm rk\,}\langle U\oplus S\rangle > {\rm rk\,}\langle S \rangle = {\rm rk\,}\Omega(\langle S \rangle) ={\rm rk\,}\Omega(\langle U\oplus S\rangle)$ where the first equality follows from the hypothesis that ${{\phi\,\rm{dim}}}(R)=0$. Thus $\phi(U\oplus S)>0$, a contradiction. Hence $P$ is injective. Since this holds for every indecomposable $R$-projective $P$, $R$ is right self-injective. Assume now that $R$ is right self-injective and let $M\in{{\rm mod\,}}R$. Since we wish to compute $\phi(M)$, we can assume that all summands of $M$ are non projective. Hence for each summand $M'$ of $M$, ${{\rm pd\,}}M'=\infty$. Let $M_1, M_2$ be direct summands of $M$. We claim that $\Omega^n M_1 \cong \Omega^n M_2 \iff M_1 \cong M_2$. Indeed, if we consider the n-th sysygy of $M_1$ and $M_2$, we have $0\rightarrow \Omega^nM_1\rightarrow P_{n-1} \rightarrow P_{n-2} \rightarrow \ldots\rightarrow P_0\rightarrow M_1\rightarrow 0,$ $0\rightarrow \Omega^nM_2\rightarrow Q_{n-1} \rightarrow Q_{n-2} \rightarrow \ldots\rightarrow Q_0\rightarrow M_2\rightarrow 0,$ where the $P_i$’s and the $Q_i$’s are projective and hence injective $R$-modules. The given resolutions are then injective resolutions. Consequently if $\Omega^n M_1 \cong \Omega^n M_2$, then $M_1\cong \Omega^{-n}(\Omega^n M_1) \cong \Omega^{-n}(\Omega^n M_2)\cong M_2$. But then ${\rm rk\,}\langle M \rangle = {\rm rk\,}\Omega^i(\langle M \rangle)$ for each $i\geq 0$, implying that $\phi(M)=0$. Since this holds for all $M\in{{\rm mod\,}}R$, we showed that ${{\phi\,\rm{dim}}}(\Lambda)=0$. Recall that a ring $R$ is self-injective if $_RR$ and $R_R$ are injective left and right $R$-modules respectively. Using the duality of an artin algebra and the fact that the number of isomorphism classes of indecomposable projective and indecomposable injective modules are the same, the following corollary is immediate. Let $\Lambda$ be an artin algebra. Then $\Lambda$ is self-injective if and only if ${{\phi\,\rm{dim}}}(\Lambda)=0$. [20]{} F. Huard, M. Lanzilotta, O. Mendoza. An approach to the Finitistic Dimension Conjecture. [*J. of Algebra*]{} 319, 3918-3934, (2008). K. Igusa, G. Todorov. On the finitistic global dimension conjecture for artin algebras. [*Representation of algebras and related topics,*]{} 201-204. Field Inst. Commun., 45. Amer. Math. Soc., Providence, RI, (2005). B. Zimmerman-Huisgen. The finitistic dimension conjecture- a tale of 3.5 decades. in: Abelian groups and modules (Padova, 1994) 501-517. Math. Appl. 343, Kluwer Acad. Publ. Dordrecht, (1995). François Huard:\ Department of mathematics, Bishop’s University,\ Sherbrooke, Québec, CANADA, J1M1Z7.\ [[email protected]]{} Marcelo Lanzilotta:\ Instituto de Matemática y Estadística Rafael Laguardia,\ J. Herrera y Reissig 565, Facultad de Ingeniería, Universidad de la República. CP 11300, Montevideo, URUGUAY.\ [[email protected]]{} [^1]: [^2]: The authors thank the financial supports received from Proyecto FCE-ANII 059 URUGUAY and NSERC Discovery Grants Program CANADA

--- abstract: 'Gas in protostellar disks provides the raw material for giant planet formation and controls the dynamics of the planetesimal-building dust grains. Accurate gas mass measurements help map the observed properties of planet-forming disks onto the formation environments of known exoplanets. Rare isotopologues of carbon monoxide (CO) have been used as gas mass tracers for disks in the Lupus star-forming region, with an assumed interstellar CO/H$_2$ abundance ratio. Unfortunately, observations of T-Tauri disks show that CO abundance is not interstellar—a finding reproduced by models that show CO abundance decreasing both with distance from the star and as a function of time. Here we present radiative transfer simulations that assess the accuracy of CO-based disk mass measurements. We find that the combination of CO chemical depletion in the outer disk and optically thick emission from the inner disk leads observers to underestimate gas mass by more than an order of magnitude if they use the standard assumptions of interstellar CO/H$_2$ ratio and optically thin emission. Furthermore, CO abundance changes on million-year timescales, introducing an age/mass degeneracy into observations. To reach factor of a few accuracy for CO-based disk mass measurements, we suggest that observers and modelers adopt the following strategies: (1) select the low-$J$ transitions; (2) observe multiple CO isotopologues and use either intensity ratios or normalized line profiles to diagnose CO chemical depletion; and (3) use spatially resolved observations to measure the CO abundance distribution.' author: - Mo Yu - 'Neal J. Evans II' - 'Sarah E. Dodson-Robinson' - Karen Willacy - 'Neal J. Turner' bibliography: - 'references.bib' - 'nje.bib' title: 'Disk masses around solar-mass stars are underestimated by CO observations' --- Introduction {#sec: intro} ============ Solar System formation models that allow the giant planets to form within observed protostellar disk lifetimes of a few million years [@haisch01] often require density enhancements up to an order of magnitude above the minimum-mass solar nebula (MMSN) [@pollack96; @hubickyj05; @thommes08; @lissauer09; @Dodson-Robinson_2010_UN; @dangelo14]. Indeed, planet accretion may be a fundamentally inefficient process, with both collisional fragmentation [e.g., @stewart12] and planetesimal scattering [e.g., @ida04] contributing to mass loss during solid embryo growth. Yet disk masses inferred from dust emission in (sub)millimeter often do not reach the MMSN mass of $0.01 M_{\odot}$ [@weidenschilling77; @hayashi81], and are more commonly of order 1-10 Jupiter masses [@andrews07; @Williams_Cieza_2011]. Reporting on a survey of T-Tauri stars in Lupus, @Ansdell_2016_ALMA_Lupus suggested that 80% of the disks had dust-derived total masses of $< 0.01 M_{\odot}$. However, dust-based disk mass estimates may be systematically low, because dust continuum observations lose sensitivity to solids that are much larger than the observing wavelength [@Williams_Cieza_2011]. In addition, the standard assumptions that the dust has a single temperature and that the sub-mm emission is optically thin everywhere in the disk may not be correct. Finally, gas masses may not be related to the dust masses by the usual interstellar ratio of 100. It is essential to measure the gas mass of disks directly. One such gas mass measurement came from @Bergin_HD_2013, who used [*Herschel*]{} observations of the HD $(J = 1 \rightarrow 0)$ transition to calculate a mass of $0.06 M_{\odot}$, or 6 MMSN, for the disk surrounding TW Hydra—a surprisingly large mass given the star age of $\sim 10$ Myr. The HD lines have since been detected in two more disks [@2016ApJ...831..167M], and are consistent with gas masses of 1-4.7 MMSN (DM Tau) and 2.5-20.4 MMSN (GM Aur). Although the GREAT instrument on the far-IR observatory SOFIA[^1] covers the frequency of the HD transition, it is not sensitive enough to observe HD in nearby disks. In the absence of the capability to observe the HD 10 transition in disks, CO has been the standard tracer of the gas mass because it is believed to have simple chemistry and to stay in the gas phase wherever $T > 20$ K in disks around Sunlike stars [@Oberg_2011_CtoO; @Qi13_TWHya], a region that includes the entire vertical column in the inner 30 AU and the warm surface layers of the outer disk. While the emission from $^{12}$C$^{16}$O is typically optically thick, it has been suggested [@VanZadelhoff_2001; @Dartois_2003] that rare isotopologues of CO could be used to probe the disk midplanes. @Yu_2016_COchem (hereafter Paper 1) have shown that the vertical optical depth of low-J rotational emission lines of C$^{17}$O is around unity in the inner $\sim 20$ AU of a 1.5-MMSN disk, meaning observers could see emission from the disk midplane—where most of the mass is concentrated—using C$^{17}$O lines. Unfortunately, Paper 1 also revealed some complexities in the CO chemistry that would interfere with disk mass measurements. First, the CO abundance varies with distance from the star within the planet-forming region. Second, the CO/H$_2$ ratio drops to an order of magnitude below the interstellar value well inside the CO freeze-out radius because of chemical depletion of CO. Finally, the CO abundance is a function of time, which introduces an age-mass degeneracy into the interpretation of the observations. Recent attempts to calculate gas-to-dust mass ratios using observations of rare CO isotopologues have also revealed problems. In their survey of disks in the Lupus star-forming region, @Ansdell_2016_ALMA_Lupus found gas-to-dust ratios, calculated assuming a constant CO/H$_{2}$ ratio of 10$^{-4}$, to be much lower than the interstellar value of 100. Yet the stars in the Lupus sample are still accreting, indicating that abundant gas is present. Studies of a more massive disk and star (2.3 ) also found very low gas-to-dust ratios [@2016MNRAS.461..385B]. Correcting for the joint effects of freeze-out and isotope-selective photodissociation still does not bring the estimated gas-to-dust ratios up to 100, according to the chemical models of @Miotello_2017. Either the disks are in the process of dispersing—unlikely given the rapid depletion timescale of $\sim 10^5$ years once photoevaporation dominates the disk dynamics [e.g., @hollenbach00; @alexander14; @gorti16]—or additional pathways to remove gaseous CO (hereafter referred to as chemical depletion) become important in disks, as suggested by @dutrey03, @favre13, @Miotello_2017, and @2016ApJ...831..167M. Some disks around HAeBe stars also appear to have CO/H$_2 < 10^{-4}$ [e.g. @chapillon08; @bruderer_2012]. Also challenging the assumption of simple CO chemistry and a constant CO/H$_{2}$ ratio in regions where CO is not frozen out are the TW Hya observations of @Schwarz_TWHya_2016, which show a drop in CO column density at $\sim 20$ AU, while the dust column density remains roughly constant with radius. In Paper 1, we found that CO chemical depletion due to dissociation by He$^+$ and subsequent complex organic molecule (COM) formation causes the CO/H$_2$ abundance ratio to drop far below the interstellar value of $10^{-4}$ at $r > 20 {\,{\rm AU}}$, well inside the CO freeze-out radius in the model (see figure \[fig: CO\_contour\]). For disks with similar chemistry to the Paper 1 models, using interstellar abundance ratios to extrapolate from CO to H$_2$ column density would result in large underestimates of disk mass. In this work we use radiative transfer models of CO rotational emission to assess the usefulness of rare CO isotopologues as disk mass indicators, given the possibility of chemical CO chemical depletion. We summarize the key results from our chemical evolution models in section \[sec: DiskModel\], including a new model for a more massive disk. We present the setup for and results of our line radiative transfer models in section \[sec: lineRTmodel\]. In section \[sec:measure\_mass\], we demonstrate that standard gas-mass measurement methods—based on integrated fluxes of lines assumed to be optically thin—fail when applied to the simulated CO emission from our model disk. We also evaluate the performance of published optical-depth correction methods [@Williams_Best_2014; @Miotello_2016], showing that they are inadequate for disks with non-uniform CO abundance. In section \[sec:agemass\] we further highlight the age-mass degeneracy problem caused by CO chemical depletion. Next we show that (1) combining observations of multiple isotopologues, (2) using information on line profiles, and (3) examining the spatial distribution of CO can diagnose CO chemical depletion (section \[sec:results\_agemass\]). In section \[sec: obs\], we explore whether our predicted CO/H$_2$ abundance ratios can increase observed masses up to MMSN or higher when applied to the Lupus disk sample of @Ansdell_2016_ALMA_Lupus. Section \[sec:conclusion\] summarizes our results. Disk Model {#sec: DiskModel} ========== We adopt the chemical-dynamical model from Paper 1 as the basis for our line radiative transfer models. Paper 1 presented the chemical evolution of a $0.015{\,{M_\odot}}$ disk around a Solar-type star for $3$ Myr. In this paper, we add a chemical evolution model for a disk that is twice as massive, at $0.03{\,{M_\odot}}$. The mass distributions and accretion temperatures of both model disks were presented by @Landry_2013, and we find the stellar contribution to the disk heating from the dust radiative transfer code RADMC[^2]. Because our goal is to measure disk masses in giant planet-forming regions, we focus our study on the region covered in @Landry_2013 – the inner $70$ AU of the disk. (As the disk ages, the amount of mass in the inner $70$ AU decreases due to viscous spearing and accretion – all disk masses reported in this paper are calculated as $\int 2\pi r \Sigma ~dr$, where $\Sigma$ is the column density, out to 70 AU.) The chemical reaction network is run locally at each independent (r, z) grid point for $3$ Myr, under the assumption that the chemical reaction timescale is much shorter than the viscous timescale—an assumption that is true for freezeout, desorption, and grain-surface reactions, but which may fail for gas-phase reactions. Each disk gridpoint starts with gas and ice abundances resulting from a 1 Myr simulation of the chemical evolution of a parent molecular cloud; as a result, a substantial fraction of the carbon is tied up in CO$_2$ and other ices at the start of disk evolution. The chemical evolution models include C, H, O, N based on the UMIST database RATE06 [@Woodall_UMIST_2007]. @Woods_Willacy_2009 extended the network to include C isotopes, and we included both C and O isotopes in Paper I. The chemical models follow the chemistry of $588$ species, $414$ gas-phase and $174$ ices, for $3{\,{\rm Myr }}$ from the beginning of the T-Tauri phase. The reaction network contains gas-phase reactions (including those that lead to C and O fractionation), grain-surface reactions, freezeout, thermal desorption, and reactions triggered by UV, X-rays and cosmic rays, such as isotope-selective photodissociation. To make our simulations computationally tractable, we include only molecules with two or fewer carbon atoms, which limits the network to 11316 reactions. Some grain-surface hydrogenation reactions, such as $(\mathrm{C}_2 \mathrm{H}_5 \; {\rm ice}) + H \rightarrow (\mathrm{C}_2 \mathrm{H}_6 \; {\rm ice})$, are not included because they only lead to sinks; we can save computational time by not computing the associated reaction rate and letting $\mathrm{C}_2 \mathrm{H}_5$ ice be the sink instead of $\mathrm{C}_2 \mathrm{H}_6$ ice. The simplifications we have made to the reaction network do not artificially remove carbon from the gas phase. In Paper 1 we demonstrated that using a range of chemical networks, input chemical abundances and dust properties does not change the evolution of CO abundance significantly. We take the temperatures and abundances from the fiducial model in Paper 1 as the foundation for this study and refer readers to Paper 1 for detailed discussions of disk model assumptions. The most debatable assumption from Paper 1 is that abundances in gas and icy components are inherited from the molecular cloud without modification as they enter the disk. Indeed, @drozdovskaya16 suggest that disk midplane composition is largely determined by the conditions during cloud collapse. @visser09 find that CO ice formed in molecular clouds desorbs during cloud collapse, though it re-freezes without significant abundance modification where the disk temperature is $< 18$ K. ----------------------------------------------- -------------------------------------------------- {width="40.00000%"} {width="40.00000%"} ----------------------------------------------- -------------------------------------------------- ------------------------------------------------------------------------ --------------------------------------------------------------------------- {width="40.00000%"} {width="40.00000%"} ------------------------------------------------------------------------ --------------------------------------------------------------------------- The disk is primarily heated passively from stellar irradiation; accretion contributes very little to the energy budget (@Landry_2013, Paper 1). For the temperature calculation, we use the dust opacities computed by [@Semenov_Henning_2003], adjusted for our gas/dust ratio. We assume the dust has already grown and aggregated at the start of the T-Tauri phase of disk evolution [@Oliveira_2010; @Perez_2012_AS209; @Birnstiel_graingrowth_2011; @Garaud_graingrowth_2013], and 90% of the solids have grown to still larger sizes (pebbles, rocks, etc.), that contribute very little submillmeter emission. We distinguish these [*solids*]{} from [*dust*]{} and take a gas/dust mass ratio of 1000. In Paper 1, we demonstrated that the CO abundance is only weakly sensitive to evolution of the gas/dust ratio from 100 to 1000. Our grain model assumes that there is a constant replenishment of small grains by collisions between larger objects (see @Dullemond_Dominik_2005_smallgrains [@Brauer_2008; @Birnstiel_fragmentation_2009; @Wada_2008; @Wada_2009; @Windmark_2012]; though [@Zsom_2010] argues against replenishing micron-size dust grains through collisions) and the size distribution of dust does not evolve. The theory that most of the solid mass has aggregated into pebbles and larger objects invisible to submillimeter wave observations is the second major, non-standard assumption in our model (though @andrews12 and @cleeves15 modeled the TW Hya disk with two distinct grain populations, small grains that provide the visible/infrared opacity and large grains with a maximum size of 1 mm that provide the submillimeter opacity). Following @cleeves15, we assume the 90% of solid mass that has grown to pebble and larger sizes provides negligible surface area for chemical reactions and exclude it from the chemical model. As the central star dims during its time on the Hayashi track, the disk cools and its scale height shrinks. The computational surface of our model grid, defined as the layer where the optical depth to the disk’s own radiation is $\tau = 0.2$ [@Landry_2013], moves from about two pressure scale heights to about one scale height above the midplane as the disk becomes cooler and thinner over time. The chemical model therefore contains fewer grid cells at the end of evolution than at the beginning, as grid layers high above the midplane begin to empty out. In appendix \[app: modelpars\], we show that our line profile and intensity calculations are only weakly sensitive to the changing disk surface. Stellar heating is efficient enough to prevent CO from freezing out in our modeled region—the inner 70 AU of the disk—at all vertical heights and at any time in the $3$ Myr of evolution. However, CO is depleted beyond $20$ AU from the central star due to the formation of complex organic molecules. The CO chemical depletion is driven by ionization of helium from X-rays and cosmic rays and happens over a million-year time scale. As a result, the CO abundance changes both with location in the disk and with time. We show the color map of CO abundance at $2$ Myr of the disk evolution in Fig. \[fig: CO\_contour\] for both the Paper 1 disk ($0.015$ ; left) and the new model of the more massive disk ($0.03$ ; right). The abundance is defined as the ratio of the number density of CO with respect to the number density of hydrogen nuclei (n$_{\rm H}+2\rm n_{\rm H_{2}}$). After 2 Myr, the CO chemical depletion front moves inward to 20 AU, with the surface layers more depleted than the disk midplane. Fig.  \[fig: chem\_time\_evolution\] shows the abundances of major carbon-bearing species as a function of time for the midplane at 40 AU in each disk, demonstrating the gradual chemical depletion of CO and subsequent sequestration of carbon into organic ices. We note that our network is not extensive enough to determine the exact end-product of organic ice formation. The most complex hydrocarbon we include is C$_2$H$_5$, which in reality should hydrogenate to form ethane, and radicals of the form C$_2$H$_{\rm x}$ should react with carbon atoms or hydrides to form longer carbon chains. However, these complex products of organic grain surface chemistry are all less volatile than the C$_2$H$_{\rm x}$ species and would stay on the grain surfaces unless the grains drift radially inward (e.g. Birnstiel & Andrews 2014) or experience a transient heating event (e.g. Cody et al. 2017, Cieza et al. 2016, Vorobyov & Basu 2015). Since almost all resolved T-Tauri disks detected in dust continuum emission have radii much larger than the $\sim 40$ AU C$_2$H$_5$ ice line (Paper 1) in our models[^3], we do not expect radial drift to deposit a significant amount of organic gas that could be recycled to form CO in the inner disk. To the extent that there are real astrophysical disks that evolve quiescently during the T-Tauri phase, our conclusion that carbon liberated by CO chemical depletion becomes sequestered in ices is robust. The more massive disk shows a similar pattern of CO chemical depletion to the disk presented in Paper 1, though the depletion timescale is a bit longer because the higher column density decreases the ionization fraction—and thus the abundance of ionized helium—in the midplane. The net result of all chemical models presented here and in Paper 1 is that CO becomes severely depleted well inside the CO freeze-out radius in disks with masses above the minimum needed to form planetary systems. Similar effects have been seen in other chemical evolution calculations (). @Aikawa_1997 and @1999ApJ...519..705A first pointed out that CO can react to form less volatile molecules such as CO$_2$, HCN, H$_2$CO, CH$_4$ and larger hydrocarbons over Myr timescales. @Aikawa_1997 used a static disk (not evolving), and @1999ApJ...519..705A used a vertically isothermal model and found a larger number of simple molecules as products (as opposed to a few complex molecules), but the essence of the chemical depletion of CO is the same as found here and in Paper 1. simulated the composition of complex organic molecules in a disk with no temperature evolution for about $1$ Myr and found the formation of complex organic molecules in the disk midplane via grain-surface reactions, while @Bergin14 found gas-phase organic formation and subsequent freezeout onto grain surfaces. @Furuya_carbon_2014 investigated the carbon and nitrogen chemistry during turbulent mixing, and found that volatile transport enhances COM formation near the surface and suppresses it in the disk midplane. It is difficult to compare our results directly with the earlier calculations because our models have different central star properties and less accretion heating, but it is clear that a variety of chemical processes can force a disk’s CO/H$_2$ abundance ratio far from the interstellar value. In Figure \[fcovst\] we show the fraction of our model disk’s carbon atoms contained in CO gas () as a function of time. The fraction is initially low because evolution in the molecular cloud has sequestered carbon in ices; for a while these evaporate, increasing , but then CO chemical depletion and formation of icy organics cause  to decrease. To complicate matters, the abundance of CO actually increases with time at small radii, as CO$_2$ ice desorbs and dissociates to form CO gas. Although the $0.015 {\,{M_\odot}}$ disk suffers more CO chemical depletion at larger radii, the re-formation of CO gas in the inner disk also proceeds at a higher rate than in the $0.03 {\,{M_\odot}}$ disk, leading to a higher disk-averaged  for the $0.015 {\,{M_\odot}}$ disk. These facts will compromise attempts to measure disk masses using CO isotopes (§\[sec:measure\_mass\]). Worse yet, the CO abundance is a function of time, leading to an age-mass degeneracy in interpreting observations (§\[sec:agemass\]). {width="40.00000%"} {width="40.00000%"} Line radiative transfer model and mass estimates {#sec: lineRTmodel} ================================================ In order to understand the effects of chemical evolution on CO emission, we set up radiative transfer models with the publicly available code LIME [LIne Modeling Engine; @Brinch_Hogerheijde_LIME_2010]. LIME calculates either non-LTE or LTE line excitation and solves the radiative transfer problem for molecular gas in arbitrary 3D geometries. For recent examples of studies using LIME, see @Walsh_methanol_2016 and @Oberg_2015_cyanides. We adopt the energy levels from the Leiden Atomic and Molecular Database (LAMDA)[^4]. As a first order approximation, we do not consider the hyperfine splitting in C$^{17}$O emission. As in Paper 1, we model emission from within a radius of $70{\,{\rm AU}}$ of the central star, which corresponds to a $1\arcsec$ beam diameter for an assumed distance of $140{\,{\rm pc}}$ from the Sun. We consider line broadening due to Keplerian rotation, thermal velocity and micro-turbulence. Thermal velocities are calculated assuming a Maxwell-Boltzmann speed distribution based on the disk’s temperature structure from Paper 1. For the micro-turbulence, we again assume an isotropic Maxwell-Boltzmann speed distribution with RMS of $100{\,{\rm m/s}}$, consistent with the upper limit to microturbulent speed found by @flaherty15 in a fit to multiple CO emission lines in the HD 163296 disk. For computing the level populations, we set a minimum scale of $0.07{\,{\rm AU}}$ to guarantee sub-pixel sampling of both Keplerian speeds and CO abundance gradients. We first generate the synthetic datacube of intensity as a function of $x$, $y$, and velocity for a disk around a $0.95 {\,{M_\odot}}$ star at $30 \degr$ inclination, similar to the disk surrounding AS 209. In velocity space, the spectra have $300$ channels of $125{\,{\rm m/s}}$ resolution. At any specific velocity, the synthetic image contains $600 \times 600$ pixels of $0.003\arcsec \times 0.003\arcsec$ in size. Finally, we generate the synthetic spectra presented here by integrating each velocity component over a square with $1.2\arcsec$ sides ($400 \times 400$ pixels), larger than the angular size of the disk. The pixels not covered by the disk contribute no flux and are included simply for ease of integration—here we assume that the sky background contains negligible flux compared with the disk at all wavelengths. Our current models assume that the gas temperature is the same as the dust temperature. This is a valid assumption in estimating the optical depth of C$^{17}$O and C$^{18}$O, as done in Paper 1, because the emission primarily comes from disk midplane and the midplane is strongly shielded from UV radiation. However, hot gas on the disk surface is more emissive, and we would need to consider the difference between the gas and dust temperature in order to use our models to fit high-J spectral lines emitted from the disk surface. Similarly, we would need to revisit the temperature calculation for a disk surrounding a star with a stronger UV field, which could decouple the gas and dust temperatures. In Appendix \[app: modelpars\], we demonstrate that the LTE approximation is adequate for computing the energy level populations, and show that the decreasing disk scale height has little effect on the computed line profiles (see §\[sec: DiskModel\]). For our purposes, we have adequately modeled the CO rare isotopologue emission, even though our models do not extend vertically to a large number of scale heights. For the rest of the paper, we focus our line profile discussion on 32  and 21, which are the most commonly observed transitions. ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ {width="30.00000%"} {width="30.00000%"} {width="30.00000%"} {width="30.00000%"} {width="30.00000%"} {width="30.00000%"} ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ ----------------------------------------- -------------------------------------------------   ----------------------------------------- ------------------------------------------------- Figure \[fig: line\_evo\] shows one set of results from our line radiative transfer models—the time evolution of the 32 line profiles for three isotopologues and two model disks. Line profiles in Janskys are plotted as well as [*normalized*]{} profiles, which better show the evolving [*shape*]{} of the lines (see §\[sec: lineprofiles\] for more on diagnosing CO chemical depletion by comparing line profiles from multiple isotopologues). Figure \[fig: time\_evolution\_intensities\] shows the total intensities (integrated in velocity space) of 32 emission from all CO isotopologues as a function of time. The decline with time is quite dramatic, especially for the rarer isotopologues where it exceeds an order of magnitude over 3 Myr. The emission from rarer isotopologues is very weak at later times, which can be expensive to observe, especially if one wants to achieve enough signal-to-noise to study the line profiles. Finally, we use standard observational procedures to try to recover the apparent masses of the disks, as an observer would do. Here we will use the 21 line, as it is commonly used to obtain disk masses (e.g.,@Williams_Best_2014, though @Ansdell_2016_ALMA_Lupus use the 32 line). We show in Appendix \[sec:rotdiag\] that higher $J$ transitions underestimate the mass more severely than 21 and 32. From our model line profiles we calculate level populations, total number of CO molecules, and finally gas mass. The equations used to derive the number of CO molecules in each energy level ($\mathcal{N}_J$) from integrated line flux, assuming optically thin emission, are given in Appendix \[sec:rotation\_diagram\_eqs\]. Values of $\mathcal{N}_J$ can then be translated to mass with the following equations. The total number of CO molecules in the disk $\mathcal{N}$ is given by $$\mathcal{N} = \frac{\mathcal{N}_J}{g_J} \times Q \times e^{{\mbox{$E_{\rm up}$}}/T} \label{eq:nco}$$ where  is the upper-state energy of the transition in K and $Q$ is the partition function, generally approximated by $Q = kT/hB$ (with $B$ in Hz). Note that the temperature of the CO reservoir appears in both the exponential and the partition function in Equation \[eq:nco\]; in Appendix \[sec: Testimates\] we show that no single temperature describes the CO reservoir. We then compute the mass of gas from $$M = \frac{\mathcal{N} {\mbox{$\mu_{\rm H}$}}{\mbox{$m_{\rm H}$}}{\mbox{$f_{\rm iso}$}}}{{\mbox{$f_{\rm C}$}}{\mbox{$f_{\rm CO}$}}} = 1.668 \times 10^{-53} {\mbox{M$_\odot$}}\ \mathcal{N} {\mbox{$f_{\rm iso}$}}/{\mbox{$f_{\rm CO}$}}\label{masseq}$$ where  is the mean atomic weight of the ISM including He,  is the mass of a hydrogen atom,  is the ratio of the mass of the most common isotopologue ($^{12}$C$^{16}$O) to that of the one being used,  is the abundance ratio of carbon to hydrogen nuclei, and  is the fraction of C in CO, averaged over the disk. ${\mbox{$\mu_{\rm H}$}}= 1.43$ and our chemical model has ${\mbox{$f_{\rm C}$}}= 7.21 \times 10^{-5}$. In the next section we describe how equations \[eq:nco\] and \[masseq\] to obtain the correct mass of our model disk. Even rare CO isotopologues underestimate disk mass {#sec:measure_mass} ================================================== $^{12}$C$^{16}$O and $^{13}$C$^{16}$O will be optically thick in disks, so the usual approach is to observe rarer isotopologues. At first glance, the CO chemical depletion we predict might be expected to ameliorate optical depth problems, but the concentration of CO to small radii increases optical depths there. At the same time, the chemical depletion of CO at larger radii causes mass underestimates by lowering the value of . The net result is that simple analysis fails for the following reasons. First, the usual assumption is that ${\mbox{$f_{\rm CO}$}}= 1$ inside the CO ice line, but our models challenge that assumption. Figure \[fcovst\] shows how  varies with time, never rising above 0.3 ($0.015$  disk) or $0.25$ ($0.03$ disk) and approaching 0.12 by 3 Myr. Since our disk does not include radii where CO itself would freeze, the low  is due to what we call chemical depletion. Alone, with no other error sources in equations \[eq:nco\] and \[masseq\], it will cause underestimates of disk mass by factors of 3 to 8. Second, the concentration of CO toward the inner disk means that temperatures, and hence partition functions, are higher than usually assumed for much of the CO reservoir. The most common assumption for temperature is $T = 20$ K (e.g., @Ansdell_2016_ALMA_Lupus). A temperature averaged over the the model disk and weighted by the number density of CO molecules ranges from 50 to 70 K, decreasing as the star and disk evolve, but stabilizing around 55 K for the 0.015  disk after about 1 Myr because the increasing concentration of CO in the inner disk counteracts the dropping luminosity of the star (Figure \[fig: Tvst\]). If CO chemical depletion is operating in the outer disk, assuming $T = 20$ K for the gas where CO is concentrated is never a good choice; it will systematically underestimate masses by a factor of about 2. In disks around T-Tauri stars that are less luminous than the proto-sun at 3 Myr, or that receive a low cosmic-ray flux so that cosmic ray-induced photons do not generate CO from CO$_2$ gas in the inner disk, 20 K may be appropriate, but we recommend constructing model-based temperature estimates for the CO reservoir rather than making any assumptions. As a further complication, we show in Appendix \[sec:rotdiag\] that rotation diagrams from multiple transitions are not effective in determining the best temperature to assume. --------------------------------------------- ---------------------------------------------------- {width="40.00000%"} {width="40.00000%"} --------------------------------------------- ---------------------------------------------------- Third, the increase in CO abundance in the inner disk causes optical depth effects where most of the CO actually resides (see Fig. 10 of Paper 1). Even if we correct for the mean  and somehow find an accurate value of $T$ averaged over the CO reservoir, the standard analysis still underestimates the mass. Figure \[massplot\] shows the mass estimates from different isotopes and analysis procedures versus time, along with the actual disk mass, which decreases slightly due to disk spreading and accretion onto the star. The estimates with black symbols are computed with equations \[eq:nco\] and \[masseq\] and based on the assumption of optically thin emission in the 21 line of the listed isotopologues. They all use the correct  and the mass-weighted CO temperature for our model—information that would not be available for astronomical sources—yet they all still underestimate the gas mass substantially. The rarer isotopologues perform best, suggesting that optical depth is the main culprit. The recent analysis of the Lupus disks [@Ansdell_2016_ALMA_Lupus] by @Miotello_2017 also suggests optical depth effects, even after ratios of [$^{13}$CO]{} and [C$^{18}$O]{} were used to correct for optical depth. The apparent gas to dust ratio declines with increasing dust mass in their Figure 6; if the dust mass is a reasonable proxy for disk mass, a more massive disk will have larger (and more complicated) effects from optical depth. Our results indicate that models that do not account for the radial dependence of  will underestimate the gas mass more severely for more massive disks, as observed in Figure 6 of @Miotello_2017. Correcting for optical depth using multiple isotopologues {#sec:multiso} --------------------------------------------------------- One approach to dealing with optical depth is to observe the same rotational transition in multiple isotopologues [e.g., @Williams_Best_2014; @Miotello_2016]. We tried a simple correction for our $^{13}$CO-based mass estimate by comparing with C$^{18}$O, using the C$^{18}$O/$^{13}$CO line intensity ratios plus the initial isotope ratios from the chemical model. This correction yielded the hollow green hexagons in Figure \[massplot\], which still underestimate disk mass badly, performing only as well as C$^{18}$O observations would on their own. More sophisticated optical depth corrections can be made with fitting formulae that relate the 21 line intensity of either $^{13}$CO or C$^{18}$O (or their ratio) to the total disk mass, based on a suite of models [e.g., @Williams_Best_2014; @Miotello_2016]. We placed our model line luminosities on Figure \[massplot\] (left panel) of the Williams and Best models to estimate mass, producing the red points, which have no correction for . They do well for early times when the CO is more uniformly distributed over the disk, but underestimate the mass badly at later times when  drops especially at larger radii. Most of our points lie near the top of their distribution of models, and interpolation between mass models is uncertain by a factor of three. Second, we used the formulae in equation 2 with coefficients in Table A.1 of @Miotello_2016 to estimate disk mass from luminosities of the 21 lines of [$^{13}$CO]{} and [C$^{18}$O]{}. Those from [$^{13}$CO]{} underestimated the mass badly, but those from [C$^{18}$O]{} did better (blue octagons). In these comparisons, we adjusted their mass estimates to be consistent with our assumption about the atomic carbon abundance, but not for the fraction of carbon in CO. In a recent paper, @Miotello_2017 find the same result; applying their models to the Lupus data [@Ansdell_2016_ALMA_Lupus] data leads to very low gas to dust ratios (less than 10 in most cases) unless CO is depleted. The Miotello et al. grid of chemical models was optimized to treat CO self-shielding and isotopic fractionation, which our chemical models also include; the low gas/dust ratios implied by C$^{18}$O and $^{13}$CO observations must result from other CO-depletion pathways. Clearly, both optical depth and CO chemical depletion must be accounted for when measuring disk masses: Figure \[massplot\] shows that measurements incorporating one correction, but not the other, will fail. For masses based on a single emission line, so that no optical depth correction is possible, the isotopotologue that performs the best is C$^{17}$O, which underestimates the mass by a factor of 2-3 if the correct value of  is known (which it will not be in real astrophysical situations). In Appendix \[sec:rotdiag\], we show that the 10 line does much better than the more commonly used 21 or 32 lines. We believe that this is primarily due to its lower optical depth and lesser sensitivity to temperature. The combination of the models of @Miotello_2016 and our calculations of  would improve the mass estimates. For example, if we use our “insider information” (i.e. model-derived knowledge) about  as a function of time for our 0.015  disk to further correct the masses from the formulae in @Miotello_2016, the resulting masses are accurate to within a factor of two at all ages. Unfortunately, to apply this method to observations, one needs to know the disk age. We explore the issues arising from that fact in the next section. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![ The mass of the disk inferred from the simulated observations is plotted versus time. The actual mass is shown as a solid line, while the masses inferred from equation \[masseq\] and the simulated emission from different isotopes and using different analysis methods are shown as points. Colored points show mass estimates corrected for optical depth but not , while black points show masses corrected for  but not optical depth. Labels are explained in the text. []{data-label="massplot"}](masswtau.pdf "fig:") ![ The mass of the disk inferred from the simulated observations is plotted versus time. The actual mass is shown as a solid line, while the masses inferred from equation \[masseq\] and the simulated emission from different isotopes and using different analysis methods are shown as points. Colored points show mass estimates corrected for optical depth but not , while black points show masses corrected for  but not optical depth. Labels are explained in the text. []{data-label="massplot"}](masswtau_p03Msun.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The Age-Mass Degeneracy {#sec:agemass} ======================= The time dependence of the fraction of C in CO (see Fig. \[fcovst\]) implies that one needs to know the disk age to apply a mass correction factor. Ages of host stars are not generally known to better than $\pm 1$ Myr. Worse yet, the “age" in our models is in some sense a chemical age because the speed of chemical evolution will depend on the ionization sources. EXor outbursts and X-ray flares may also introduce brief periods of intense ionization [e.g. @audard14]. If we don’t know the chemical age, there is an age-mass degeneracy—a massive, old disk may look similar to a young, less massive one in CO rare isotopologue emission. Figure \[fig: time\_line\_ratios\] shows an example of the age-mass degeneracy. The more massive disk reaches the same C$^{18}$O/$^{13}$CO total intensity ratio about 0.5 Myr later than the less massive disk for both 32 and 21. CO-based mass measurements are further complicated by the fact that the depletion occurs in the outer disk, where most of the mass resides. The decline in CO isotopologue line intensity as a function of time is precipitous: in the $0.015 {\,{M_\odot}}$ model, the intensity of $^{13}$CO J=3-2 emission drops from $2.72 {\mbox{${\rm Jy-km\ s^{-1}}$}}$ by $82\%$ to $0.48 {\mbox{${\rm Jy-km\ s^{-1}}$}}$ over the $3$ Myr disk evolution, and the intensity of C$^{17}$O J=3-2 drops by $95\%$ from $1.15{\mbox{${\rm Jy-km\ s^{-1}}$}}$ to $0.06 {\mbox{${\rm Jy-km\ s^{-1}}$}}$ (Fig. \[fig: time\_evolution\_intensities\]). For comparison, the total disk mass within 70 AU of the star drops by about 12% over 3 Myr and  drops by a factor of 2-3. Clearly, a simple correction for  does not capture all of the decrease in line emission with time. Without a good model of CO abundance as a function of radius and time, line-intensity measurements yield disk mass estimates that are accurate to factor of two at best and are systematically underestimated. Diagnosing CO chemical depletion {#sec:results_agemass} ================================= In section \[sec: DiskModel\] we demonstrated that the CO abundance decreases with both radius and time. In Section \[sec:measure\_mass\], we showed that other effects add to the underestimation of disk mass and that the underestimate gets worse with age. Given that star ages can be uncertain by over 1 Myr [e.g., @soderblom14]—the timescale over which we observe CO depletion in our model—interpreting CO observations requires a more direct CO chemical depletion indicator than the star age. The need for a disk-based CO chemical depletion indicator is especially evident given the likelihood that disks with different masses or incident cosmic-ray fluxes may evolve at different speeds. We explore here three types of observations that can diagnose CO chemical depletion . First, isotopologue intensity ratios for a transition change with time, revealing departures from the ISM CO/H$_2$ ratio. Second, line profile shapes for the most optically thin isotopologues, C$^{17}$O and C$^{18}$O, widen over time as the outer disk loses CO, while $^{13}$CO and CO profile shapes evolve very little. Third, spatially resolved observations will reveal CO chemical depletion patterns. While using intensity ratios to diagnose CO chemical depletion requires less observing time, high signal-to-noise line profiles or spatially resolved observations contain valuable information that could allow observers to reconstruct the radial distribution of CO gas. We present all three strategies here, beginning with intensity ratios. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Ratio of total intensities of C$^{18}$O and $^{13}$CO lines from for the $0.015 {\,{M_\odot}}$ and the $0.03 {\,{M_\odot}}$disks. We show the results for the J=$2 - 1$ lines on the left, and those for J = $3-2$ lines on the right.[]{data-label="fig: time_line_ratios"}](line_ratios_21.png "fig:"){width="45.00000%"} ![Ratio of total intensities of C$^{18}$O and $^{13}$CO lines from for the $0.015 {\,{M_\odot}}$ and the $0.03 {\,{M_\odot}}$disks. We show the results for the J=$2 - 1$ lines on the left, and those for J = $3-2$ lines on the right.[]{data-label="fig: time_line_ratios"}](line_ratios_32.png "fig:"){width="45.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- How intensity ratios reveal CO chemical depletion {#sec: lineratios} -------------------------------------------------- The relative strengths of emission lines for different isotopologues change significantly over time. This is because different isotopologues react to the change in CO abundance differently due to their various optical depths. Re-examining Figure \[fig: time\_line\_ratios\], we see that after $1{\,{\rm Myr }}$, the [C$^{18}$O]{}/[$^{13}$CO]{} intensity ratios decrease steadily as the disk-averaged CO abundance declines monotonically (Fig. \[fcovst\]). One could use Figure \[fig: time\_line\_ratios\] to estimate the chemical age of an observed disk. However, the predicted intensity ratios are higher for the more massive disk, so there is still a 0.5 Myr uncertainty in age, which propagates into a mass uncertainty. Intensity ratios are the most rudimentary diagnostic of CO chemical depletion. @Williams_Best_2014, @Miotello_2014, and @Miotello_2016 show how isotope-selective photodissociation, varying disk radii, and CO freezeout (or lack thereof) can affect intensity ratios, leading to a wide dispersion of C$^{18}$O/$^{13}$CO for disks of the same mass. How line profiles reveal CO chemical depletion {#sec: lineprofiles} ----------------------------------------------- The line profiles contain information not available in the integrated intensities. Here we demonstrate how a comparison of normalized line profiles from $^{13}$CO and C$^{18}$O or C$^{17}$O can diagnose CO chemical depletion, even over a factor of two range in disk mass. Figure \[fig: line\_evo\] shows that the normalized line profiles of the rarer isotopologues become broader than those of the more common isotopologues as time proceeds. Since CO chemical depletion happens primarily in the outer part of the disk where Keplerian velocities are small, the fraction of radiation in the high-velocity line wings increases with time for optically thin lines. Optically thick lines can mask CO chemical depletion: the CO column density has to first decline to the $\tau \approx 1$ threshold (where $\tau$ is the optical depth in the line center) before intensity changes start to track abundance changes. For the $0.015 {\,{M_\odot}}$ disk, the optically thin C$^{17}$O and C$^{18}$O emission tracks the changes in CO column densities well, so that the line profile gets significantly broader after $1$ Myr. $^{13}$CO ($J = 3 \rightarrow 2$) emission is optically thick well beyond the CO chemical depletion radius of $\sim 20 {\,{\rm AU}}$, so does not reveal reductions in CO column density with time. Similar trends are found for the $0.03 {\,{M_\odot}}$ disk. Because of the relatively high column densities, the change in C$^{18}$O line profile is not obvious until $3{\,{\rm Myr }}$, but C$^{17}$O still tracks the reductions in CO column density very well. Clearly, line profiles of rare isotopologues can diagnose CO chemical depletion . However, emission line profiles reflect not only the CO/H$_2$ abundance ratio, but the temperature and density structure of the disk as well. The temperature of our model disk decreases with time due to the young star’s dimming as it moves down the Hayashi track, and the density structure changes as the disk viscously evolves. To isolate the effect of CO depletion from density and temperature effects, we set up control models with a constant CO/H$_2$ ratio throughout the disk and compare the emission line profiles produced by these with those produced by our fiducial, CO-depleted models. In the constant CO/H$_2$ models, all atomic carbon available for gas-phase reactions is assumed to be in CO, and relative abundances of CO isotopologues are determined only by the abundance ratios of the isotopes (see Table 2 of Paper 1). The abundances normalized to the total proton density are $7.21\times 10^{-5}$ for CO, $9.34\times 10^{-7}$ for $^{13}$CO, $1.44\times 10^{-7}$ for C$^{18}$O, and $3.13\times 10^{-8}$ for C$^{17}$O. {width="70.00000%"} Figure \[constco\] shows side-by-side normalized line profiles for the $0.015$  disk with evolving chemistry and the one with constant . By 2 Myr, the profiles of rare isotopologues are noticeably wider than those of more common isotopologues and by 3 Myr, they are quite distinctive. These line profiles comparisons would provide clear evidence for ongoing CO chemical depletion and a potential way to correct for it (though there may be multiple rings of gas in some disks, which would complicate the line profile analysis; see @cleeves16). However, the actual (un-normalized) lines are very weak (Fig. \[fig: line\_evo\]), so diagnosing CO chemical depletion by comparing line profiles would be an expensive method in terms of observing time. Furthermore, variation in turbulent speeds between different layers of the disk could also produce different line profiles for $^{13}$CO, C$^{18}$O, and C$^{17}$O [@flaherty15; @simon15], an effect we have not explored here. Spatial distributions of CO isotopologues {#sec: spatial} ----------------------------------------- Our chemical evolution models also predict a characteristic radial dependence of the CO abundance. To translate this dependence into observables, we average the velocity-integrated, continuum-subtracted emission from our model disks in rings of $2$ AU in radius, and compare the spatial distributions of the 21 emission line from the fiducial models to that from the constant CO models in Figure \[fig: spatial\_comparison\]. The models shown in Figure \[fig: spatial\_comparison\] were run in face-on geometry for simplicity; all other simulated emission in the paper is calculated for $30^{\circ}$ inclination. The optical depth effects are enhanced in the face-on disks because the lines are not spread out by rotation. If a disk inclination angle is known, models could be run for that situation. Results from the fiducial $0.015 {\,{M_\odot}}$ disk are presented in the upper-left (early time), upper-right (2 Myr), and lower-left (3 Myr) panels. At the beginning of the evolution, the integrated intensities are lower in the fiducial model but the profiles are very similar to those in the constant CO disk for all isotopes. This is because part of the carbon is locked in CO$_{2}$ in the fiducial model, whereas the constant CO model has all available carbon in CO. As the disk evolves, the intensities beyond 20 AU decrease dramatically in the fiducial model due to the depletion of CO, and the differences between the fiducial model and the constant CO model are the greatest for C$^{17}$O lines, which have the lowest optical depth. We see sharp drops of intensity in models with $0.03 {\,{M_\odot}}$ (lower right) as well. However because the chemical depletion happens more slowly in the $0.03 {\,{M_\odot}}$ disk, the intensity profiles at 3 Myr of the $0.03 {\,{M_\odot}}$ disk resemble those of the $0.015 {\,{M_\odot}}$ disk at 2 Myr. ----------------------------------------------------- ---------------------------------------------------- {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} ----------------------------------------------------- ---------------------------------------------------- The effect of age is detectable with ALMA given enough integration time. We plot the time evolution of $^{13}$CO and C$^{18}$O lines for the $0.015 {\mbox{M$_\odot$}}$ disk in Figure \[fig: spatial\_time\], and the detection limit as a horizontal line at log $= -6.27$. At resolution, we get to 540 mJy in 1 km$/$s resolution in 10 hours. At $0\farcs03$ spatial resolution, with a velocity channel of width 1 km$/$s , we reach a sensitivity of 540 mJy per beam in 10 hours. The line width for a perfectly face-on disk could be small, but it only takes a small inclination angle to significantly broaden the lines (for example, it takes only 6 degrees for Keplerian velocity to broaden the line to 1 km/s at 10 AU around a solar mass star). $0\farcs03$ will resolve 4.2 AU at 140 AU and matches our resolution in the figure. For the purpose of illustration, we use the ${\mbox{$J = 2\rightarrow1$}}$ lines from a disk of $30^{\circ}$ inclination. The apparent edge of the disk in $^{13}$CO moves from beyond 70 AU to about 55 AU, and from beyond 70 AU to about 20 AU in C$^{18}$O emission, well within the condensation front of CO in both cases. The migration of the “fake” snowline is a combined effect of the drop of CO emission intensities from the outer disk and the detection limit of the observation. ---------------------------------------------------- ---------------------------------------------------- {width="40.00000%"} {width="40.00000%"} ---------------------------------------------------- ---------------------------------------------------- In disks where spatially resolved imaging is possible, a comparison between the C$^{18}$O and the CO or $^{13}$CO spatial distribution should reveal CO chemical depletion, and may even provide enough information to re-construct the CO distribution with the help of a chemical model. Finally, after 3 Myr of evolution, we see that two rings of C$_2$H have formed in the surface layers of the disk: a narrow ring at $1-3$ AU and a broader ring at $10-20$ AU. @Bergin_2016 found that C$_2$H formation is possible only with a strong UV field and C/O$ > 1$, conditions which are replicated in our model disk surface layers due in part to the breakdown of CO molecules. We will explore the detectability of the C$_2$H rings and the degree to which they may indicate CO depletion in future work. Consequences for Observations {#sec: obs} ============================= Our main result is that current observations and interpretations of CO isotopologues toward disks around solar-mass stars likely underestimate gas masses by substantial amounts. The surveys so far have emphasized shallow observations of many disks with modest spatial resolution. For example, @Ansdell_2016_ALMA_Lupus surveyed $89$ protoplanetary disks in Lupus with ALMA and found very low gas masses when assuming an ISM CO abundance. The analysis of those observations has led to the conclusion that almost no disks have masses that exceed the MMSN. This result would appear to conflict with the high fraction of stars with evidence for planetary systems. Our predicted line intensities are weak, and deriving correct masses requires deeper integrations to obtain either line profiles of weak lines or very high spatial resolution. Until those observations are available, we can offer only rough estimates of how much more massive the disks may be. As an example, we ask the following question: if we apply a simple correction for , how many of the disks in Table 3 of @Ansdell_2016_ALMA_Lupus might contain the mass of $0.01$ , the MMSN. If the mean age of stars in Lupus is $3\pm2$ Myr, we should use ${\mbox{$f_{\rm CO}$}}= 0.136$, the value at 3 Myr for the 0.015 disk, the closest to a MMSN. After applying this correction factor, 7 of 36 best guess masses exceed the MMSN, and 26 of 36 maximum masses exceed the MMSN, where best guess and maximum masses are defined by Ansdell et al. Within the uncertainties, a significant fraction of the disks in Lupus could contain enough mass to make a planetary system like ours. Recent analysis of similar observations toward disks in Cha I indicate a deficit of gas mass similar to, or worse than, that seen in the Lupus disks (Feng et al. 2017). For an age of 2 Myr and the 0.015  disk, the value of  is 0.18, which could increase the masses by a factor of about 5, leaving almost all disks in Cha I still short of the MMSN. Almost all these stars are lower in mass than our model star; models tuned to these stellar parameters would be needed to draw further conclusions. CO chemical depletion also affects estimates of gas/dust mass ratios. Assuming an ISM-like CO/H$_2$ abundance ratio, @Ansdell_2016_ALMA_Lupus found a wide range of gas/dust ratios, with a median ratio of $\sim 15$, which is much smaller than the canonical value of $100$ observed in the ISM. @Miotello_2017 have shown that this problem persists when their models with radiative transfer and isotope-selective photodissociation are used. We argue that the low gas-to-dust ratios measured in @Ansdell_2016_ALMA_Lupus are likely a result of a low CO/H$_{2}$ ratio due to CO chemical depletion. In our 0.015 disk model, the closest to a MMSN, only 13.6$\%$ of carbon is contained in CO gas by 3 Myr. If the mean age of stars in Lupus is $3\pm2$ Myr, and we apply the simple correction for CO fraction (= 0.136) to the Lupus data with both $^{13}$CO and C$^{18}$O detections, we obtain gas-to-dust ratios of 53 to 1700, with a median gas-to-dust ratio of 108. The gas-to-dust ratios correcting for the fraction of C in CO in our models are plotted in Fig.  \[fig: ansdell\_g2d\_corrected\]. {width="70.00000%"} The high values of the gas-to-dust ratios in the ${\mbox{$f_{\rm CO}$}}$ corrected gas-to-dust ratios could be a combined result from grain growth, weak-ionizing environments, and young disk ages. Gas/dust mass ratios are not necessarily the same as gas/[*solid*]{} mass ratios: the apparent dust mass is typically calculated from sub-mm dust emission, which is mostly sensitive to dust grains around sub-mm size or smaller. Once dust grains grow into cm-size pebbles, the reduction of observable dust mass could drive the gas/dust mass ratio to [*larger*]{} than the canonical molecular cloud value of $100$. If only $10\%$ of the solid mass is in the form of dust observable in $890$ , as we assume in our models, one would have to correct for the “dust fraction" (${\mbox{$f_{\rm dust}$}}$) when measuring the gas-to-dust ratios. We apply the correction for dust fraction on the ${\mbox{$f_{\rm CO}$}}$ corrected data, and plot the gas-to-dust ratios as blue squares in Fig.  \[fig: ansdell\_g2d\_corrected\]. Most the disks with high gas-to-dust ratios from the ${\mbox{$f_{\rm CO}$}}$ corrections can be come consistent with the ISM value of 100 if additional corrections for grain growth are included. Instead of assuming that dust evolution or disk clearing drives the variation in observed Lupus gas/dust ratios, we point to the large age spread of pre-main-sequence stars in Lupus. Ansdell et al. quote 1-3 Myr, while @mortier11 find either 0.1 to $>15$ Myr or 0.3 to $>15$ Myr depending on the model isochrones (though the $>10$ Myr objects were not detected by Ansdell et al. and are therefore not represented in Figure \[fig: ansdell\_g2d\_corrected\]). The large spread of gas/dust ratios calculated by Ansdell et al. is likely the result of the variation of CO/H$_2$ abundance ratio with protostellar age. Given the uncertainties in both gas and solid mass, the data so far do not rule out ISM gas/dust ratios. There are a few caveats to bear in mind regarding these simple mass correction factors. First, real disks evolve at different rates: a disk with a high incident cosmic-ray flux loses CO more quickly than a disk shielded by a “T-Tauriosphere” that deflects cosmic rays [@Cleeves_2013_CR; @cleeves15]. Even if star ages could be measured perfectly, the ages alone do not give enough information about the chemical evolutionary stage of the disk to compute a CO/H$_2$ ratio. In other words, our line profile diagnostics are measuring a “chemical age.” Comparison of $^{13}$CO and C$^{18}$O line profiles gives an empirical diagnostic of the level of CO chemical depletion . Second, chemical depletion of CO operates on a million-year timescale. If episodic accretion of the type observed in FU Orionis outbursts happens during the T-Tauri phase at intervals smaller than $\sim 1 {\,{\rm Myr }}$ [e.g., @dunham10; @kim12; @martin12; @green13], desorption of CO$_2$ ice followed by CO$_2$ dissociation from cosmic ray-induced photons could raise the CO gas abundance throughout the disk, not just in the inner 20 AU as in our current model disk. @cieza16 have already proposed chemical alteration in the disk surrounding V883 Ori, which is now mid-outburst. We have not modeled FU Orionis outbursts—our model T-Tauri star evolves smoothly on the Hayashi track. Other processes may further deplete CO in the outer regions of disks. For example, @Xu_Bai_2017 describe a process of “runaway freeze-out", which considers vertical transport at fixed radii. They find that higher layers, too warm for freeze-out, can become depleted by transport to the colder, lower layers. @kama16 suggest the same mechanism to explain the low atomic C and O abundances in the TW Hya disk. Applied to CO, the vertical transport/freezeout effect would primarily act at larger radii than we consider and affect primarily the emission from the more common, hence more optically thick, isotopes. Applied to the complex organics that sequester the carbon in our models, runaway freezeout could remove even more carbon from the gas phase, allowing for even higher H$_2$ masses to be consistent with observed gas-phase carbon abundances. Our model’s growing CO abundance in the inner disk and CO chemical depletion in the outer disk would be observationally almost identical to a disk with CO frozen onto grain surfaces in the outer disk, followed by inward radial drift of the grains and desorption of CO in the warm inner disk. The radial-transport effect, first applied to water ice, has been suggested on theoretical [@Ciesla_Cuzzi_2006; @Du_2015] and observational [@Hogerheijde_2011; @Kama_2016] grounds, diagnosed from the presence of rings of small hydrocarbon chains [@Bergin_2016] and ammonia gas [@Salinas_2016] in the outer disk of TW Hya, and inferred in cases where C/O$ >1$ [@Bergin_2016]. Robust temperature measurements in the outer disk would be required to distinguish between chemical CO chemical depletion and CO freezeout followed by grain inspiral. Conclusions {#sec:conclusion} =========== Our key findings and suggestions for observing strategies are summarized below. 1. CO abundance varies both with distance from the star and as a function of time as CO is dissociated and the carbon gets sequestered in organic molecules that freeze onto grain surfaces (chemical depletion) on a million year time scale (§\[sec: DiskModel\]). CO chemical depletion will cause very large underestimates in gas mass and gas-to-dust ratios when CO observations are used. One would need to correct for the chemical depletion of CO in order to correctly estimate the disk gas mass. The CO abundance correction factor ranges from 3 to 8 for the models we have run (§\[sec:measure\_mass\]). 2. Even though CO is destroyed by ionized helium throughout most of the disk, it has a higher-than-interstellar abundance in the inner 20 AU of the disk, where the temperature is relatively high. Adopting a constant $T = 20$ K for the CO reservoir will underestimate gas masses by a further factor of about 2 (§\[sec:measure\_mass\]). 3. The high CO abundance in the inner disk also results in high optical depth even for the mostly optically thin isotopologue we investigated (C$^{17}$O), and one could underestimate the disk gas mass due to the optical depth effect even after correcting for the CO chemical depletion in the outer disk and using the correct CO-averaged temperature (§\[sec:measure\_mass\]). 4. The CO-abundance time evolution also introduces a disk age-mass degeneracy—a massive, old disk may look similar to a young, less massive one in CO rare isotopologue emission (§\[sec:agemass\]). 5. One can diagnose CO chemical depletion by comparing the line intensity ratios (§\[sec: lineratios\]) or emission line profiles of multiple isotopologues (§\[sec: lineprofiles\]). If the disk is spatially resolved, one can also use spatial distribution of CO beyond $20$ AU to probe the chemical depletion of CO. (§\[sec: spatial\]). 6. Very different CO optical depths in different parts of the disk produce a complicated rotation diagram that does not probe the disk temperature well. One would underestimate the average temperature of CO molecules by deriving it from the lowest two J values (§\[sec: Testimates\]). 7. Higher-$J$ lines underestimate the disk mass by more than do lower-$J$ lines. The higher-$J$ lines miss low temperature gas. We suggest using low excitation lines (e.g., 10) to estimate the disk mass to minimize the temperature and optical depth effects. (§\[sec: transitions\]). 8. The strategy that comes closest to recovering the correct mass for our models is to use the formulae from @Miotello_2016 [*and*]{} to divide by the  from our models (§\[sec:measure\_mass\]). 9. If we correct the “best-guess” gas masses in @Ansdell_2016_ALMA_Lupus by the smallest value of  (the value at 3 Myr for the 0.015  disk), 7 of the disks could have masses of the MMSN; if we apply our  correction to the maximum masses, 26 of 36 could reach the MMSN (§\[sec: obs\]). The age of Lupus is $3\pm2$ Myr , so this correction is reasonable. A recent survey of disks in Cha I suggest almost no disks with MMSN masses. While still problematic for planet formation models, correcting for CO chemical depletion suggests better numbers than finding that no disks contain the MMSN. 10. Given reasonable uncertainties in both gas and solid masses, many observed disks can have gas/dust ratios consistent with the ISM value of 100 (§\[sec: obs\]). Work by MY, KW, SDR and NJT was supported by NASA grant NNX10AH28G and further work by MY and SDR was supported by NSF grant 1055910. This work was performed in part at the Jet Propulsion Laboratory, California Institute of Technology. NJT was supported by grant 13-OSS13-0114 from the NASA Origins of Solar Systems program. MY was supported by a Continuing Fellowship from the University of Texas at Austin. We acknowledge helpful input from L. Cleeves, E. Bergin, E. F. van Dishoeck, K. Öberg, J. Huang, M. Ansdell, and A. Kraus. We are grateful for Feng Long et al. sharing their paper in advance of publication, and Huang et al. sharing processed ALMA data for our model comparison. Sensitivity to model parameters {#app: modelpars} =============================== The effect of our assumption of LTE in the CO level populations and the fixed vertical grid are explored in the next two sub-sections. LTE vs. NLTE {#app: NLTE} ------------ LIME is capable of computing energy level populations either in local thermodynamic equilibrium (LTE) or in the more complex non-LTE case, where the kinetic temperature and excitation temperature are different. In Paper 1, the lowest density found in the modeled region within 70 AU of the star is about $10^9$ hydrogen molecules per cubic centimeter. Even in the most diffuse regions of our model disk, the density should be high enough for collisions to dominate CO excitation so that LTE is a good approximation for the energy level population. We demonstrate that the LTE approximation is appropriate for our model disk by comparing the results of LTE and non-LTE models for different CO isotopologues at multiple epochs. For each emission line of each isotopologue, the line profiles from the two models are indistinguishable. In figure \[fig: LTE\_NLTE\] we show the worst case example of C$^{17}$O J=$6-5$ emission at the beginning of the disk evolution. The high excitation energy of the J=$6-5$ transition means the LTE approximation requires a high collision rate and is hardest to satisfy, yet the differences between the LTE and non-LTE line profiles are still negligible. ![C$^{17}$O J=$2-1$ rotational emission at the beginning of the evolution.[]{data-label="fig: LTE_NLTE"}](c17O_100yr_LTE_NLTE_65.png) Definition of the disk surface {#subsec:surface} ------------------------------ Another parameter that can influence the computed line profiles is the placement of the disk surface. In the disk models of @Landry_2013, on which the chemical models in Paper 1 are based, the top surface of the computational grid is placed at the layer where the Rosseland mean optical depth to the disk’s own radiation, integrated downward from infinity, is 0.2. The $\tau = 0.2$ surface is located between one and two pressure scale heights above the disk midplane, depending on the distance from the star and time. Since most of the low-J CO emission comes from the disk interior where the bulk of the disk mass is located, the height of the grid surface should have minimal effect on the line profiles. For higher values of J, more of the emission would come from the warm surface and modeling line profiles accurately would require the computational grid to extend well into the tenuous disk atomsphere. Here we assess the effects of grid surface placement on our model emission line intensities and profiles. The chemical model in Paper 1 uses a static grid that does not evolve with time. Grid cells are spaced logarithmically in radius $r$ and linearly in aspect ratio $z/r$. At the beginning of evolution, which roughly corresponds to the start of the T-Tauri phase, the disk has a high scale height due to heating from the luminous protostar. The scale height decreases throughout the $3{\,{\rm Myr }}$ evolution due to both protostellar dimming and viscous dissipation. As the disk flattens, grid layers with high $z/r$ begin to empty out. To test the effect of surface placement on the radiative transfer calculation, we construct a new disk model by artificially removing the top three [*filled*]{} (not empty) $z/R$ layers from each time snapshot of our fiducial model from Paper 1. At six different epochs, we compute line profiles for the $1\rightarrow0$, $2\rightarrow1$, $3\rightarrow2$, $4\rightarrow3$, $5\rightarrow4$, and $6\rightarrow5$ transitions for all CO isotopologues using both the new “remove-top” model and the fiducial model. Removing emitting layers affects the peak intensities of the optically thin C$^{17}$O emission lines more than any other isotopologue. To verify that our radiative transfer models capture essentially all of the disk emission, we demonstrate how the grid surface placement affects C$^{17}$O. At the beginning of disk evolution, the top three grid layers contain very little mass and have little effect on the emission line intensity or profile (Figure \[fig: fiducial\_removetop\], left panel). As the the disk cools and the top grid layers empty out, the mass contained in the filled grid layers increases. Line profiles from the fiducial and remove-top models differ the most after 3 Myr of evolution (Figure \[fig: fiducial\_removetop\], right panel). Yet even at 3 Myr, the difference in the $J = 3 \rightarrow 2$ peak intensity between the remove-top and the fiducial model would be difficult to distinguish in observations, and the normalized line profiles are almost identical. The differences in peak intensities between the fiducial and remove-top models become significant for higher-J emission due to the higher energy needed to populate the upper state. For example, the peak intensity for $J=6 \rightarrow 5$ differs by roughly $33\%$ between the fiducial and remove-top models at 3 Myr of evolution. However, lower J lines ($J= 1\rightarrow 0$ to $3 \rightarrow 2$) are more commonly used for disk studies. In this paper, we focus our line profile discussion on 32 and 21 transitions, which are the most observationally relevant transitions. For our purposes, we have adequately modeled the CO rare isotopologue emission, even though our models do not extend vertically to a large number of scale heights. ----------------------------------------------------- ---------------------------------------------------- {width="40.00000%"} {width="40.00000%"} ----------------------------------------------------- ---------------------------------------------------- Measuring disk mass from the integrated intensities of CO isotopologues {#sec:rotation_diagram_eqs} ======================================================================= For optically thin emission lines, the total number of molecules per degenerate sublevel in the upper state can be written simply as $$\frac{\mathcal{N}_J}{g_J} = \frac{L_J}{g_J A_J h \nu}$$ where $L_J$ is the line luminosity, $g_J$ is the degeneracy, $A_J$ is the Einstein A value for the transition from upper state with quantum number $J$, $h$ is Planck’s constant, and $\nu$ is the frequency of the transition. $L_J$ is related to the integrated line flux $F_l$ as $$\label{eq: lj} L_J = F_l({\rm cgs}) \times 4 \pi D_{\rm cm}^2 = 1.1964\times 10^{20} \times {\mbox{$F_l({\rm Jy-km\ s^{-1}})$}}\ (\nu/c) D_{\rm pc}^2,$$ where $D_{\rm cm}$ and $D_{\rm pc}$ are the star’s distance in cm and pc, respectively, and $c$ is the speed of light. For the last part of Eq. \[eq: lj\] and Eq. \[eq: njgj\], F$_{l}$ is in the unit of $ {\mbox{${\rm Jy-km\ s^{-1}}$}}$, D$_{\rm pc}$ is in the unit of pc, and all the other values are in cgs units. The Einstein A value for rotational transitions from level $J$ to level $J-1$ is given by $$\label{eq: aj} A_J = \frac{64 \pi^4}{3h} \times (\nu/c)^3 \times |\mu(J,J-1)|^2 = 3.13613\times 10^{-7} \times (\nu/c)^3 \frac{J}{(2J+1)} {\mbox{$\mu_{\rm D}$}}^2$$ where $|\mu(J, J-1)|$ is the electric dipole matrix element and ${\mbox{$\mu_{\rm D}$}}$ is the dipole moment measured in Debye ($10^{-18}$ esu-cm). Combining these, we can write $$\label{eq: njgj} \frac{\mathcal{N}_J}{g_J} = 6.468 \times 10^{45} \frac{{\mbox{$F_l({\rm Jy-km\ s^{-1}})$}}{\mbox{$D_{\rm pc}$}}^2}{{\mbox{$B_{\rm GHz}$}}^3 J^4 {\mbox{$\mu_{\rm D}$}}^2}$$ where  is the rotation constant in GHz. The values of  vary slightly with the isotopologue and are easily obtained from on-line sources, but for reference, ${\mbox{$B_{\rm GHz}$}}\approx 55$ to 58 GHz for the isotopes discussed here. Does a single temperature characterize the CO emission? {#sec:rotdiag} ======================================================= Temperature Estimates from Rotation Diagrams {#sec: Testimates} -------------------------------------------- A temperature estimate is required in order to evaluate the partition function that will translate observed intensities into gas column densities. While the disk temperature can reach $\sim 1400$ K at the dust sublimation front [@muzerolle03], @andrews05 and @andrews13 suggest that most submillimeter emission comes from dust at $\sim 20-25$ K and a fixed temperature of 20 K for the gas is often used in the simplest methods to measure mass. Here we will use plots of level populations versus energy of the level above ground—called rotation diagrams for rotational transitions—to test whether CO in our model disk is well characterized by a single temperature. We calculate the rotation diagrams for the $0.015 {\,{M_\odot}}$ disk and the $0.03 {\,{M_\odot}}$ disk using the equations in Appendix \[sec:rotation\_diagram\_eqs\], following the method of @green13. Note that the emission must be optically thin (which is often assumed for CO isotopologues) for luminosity to be directly proportional to the upper state population. Figure \[plotrot2\] shows the $^{13}$CO rotation diagram of our $0.015 {\,{M_\odot}}$ model disk (left, circles) and our $0.03 {\,{M_\odot}}$ model disk (right), both after 2 Myr of evolution. A gas reservoir with a single temperature would yield a straight line in $\ln (N_J/g_J)$ as a function of $E_{\rm up}$. Unfortunately, Figure \[plotrot2\] shows that no single temperature characterizes the CO level populations, consistent with [*Herschel*]{} observations of Herbig Ae/Be and T-Tauri disks by @meeus13, @Fedele_2013, @vanderwiel14 and @Fedele_2016. Even when we construct an artificial disk with the same density and abundance structure as our 2 Myr models, but with a constant temperature enforced at all points (Figure \[plotrot2\], squares and triangles), the rotation diagram still appears to come from a disk with a range of temperatures because of the varying optical depth of different transitions. If we derive a temperature from the lowest two J values, the temperature is quite low—about 10 K. CO rotation diagrams, even for rare isotopologues, are not reliable ways to measure protostellar disk temperatures. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![ Rotation diagrams for $^{13}$CO for the disks with mass of $0.015 {\,{M_\odot}}$ (left) and $0.03 {\,{M_\odot}}$ (right) at 2 Myr. The circles are the values of the number of molecules per sublevel in the full model. The squares show values for the same model except that the gas temperature has been fixed at 20 K and the triangles show a model with $T = 55$ K. []{data-label="plotrot2"}](plotrot2.pdf "fig:") ![ Rotation diagrams for $^{13}$CO for the disks with mass of $0.015 {\,{M_\odot}}$ (left) and $0.03 {\,{M_\odot}}$ (right) at 2 Myr. The circles are the values of the number of molecules per sublevel in the full model. The squares show values for the same model except that the gas temperature has been fixed at 20 K and the triangles show a model with $T = 55$ K. []{data-label="plotrot2"}](rotplot_p03Msun.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Inferred Mass Depends on the Transition Used {#sec: transitions} -------------------------------------------- Because the temperature and optical depth are different in various parts of the disk, the mass estimation also depends on which isotopologue and which emission line is used. We show the mass estimated by different transitions and isotopologues at $100$ yr, $2$ Myr, and $3$ Myr of the disk evolution in Figure \[mass\_excitation\]. All models assume optically thin emission for simplicity. The estimated mass decreases as we use higher-J lines with higher excitation energy for the mass estimation. This is partially contributed by the large optical depth in the inner hot regions of the disk, partially because we are missing the low temperature CO that does not emit much at higher-J. Our models suggest that it is best to use lower $J$ transitions to measure disk mass. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The mass estimated from various transitions and different isotopologues. Excitation energies are for the upper excitation state of each transition. The populations at the zero excitation energy are extrapolated from the higher energy populations with a three degree polynomial function. The actual disk mass is marked by the blue line in the 100 yr diagram (on the left), and is above the chart in plots for the 2 Myr and 3 Myr disks because the mass is hugely underestimated with the shown method. The actual disk masses in the input models at those three epochs are $0.0144 {\,{M_\odot}}$, $0.0114 {\,{M_\odot}}$ and $0.0107 {\,{M_\odot}}$. []{data-label="mass_excitation"}](mass_excitation_100yr.png "fig:") ![The mass estimated from various transitions and different isotopologues. Excitation energies are for the upper excitation state of each transition. The populations at the zero excitation energy are extrapolated from the higher energy populations with a three degree polynomial function. The actual disk mass is marked by the blue line in the 100 yr diagram (on the left), and is above the chart in plots for the 2 Myr and 3 Myr disks because the mass is hugely underestimated with the shown method. The actual disk masses in the input models at those three epochs are $0.0144 {\,{M_\odot}}$, $0.0114 {\,{M_\odot}}$ and $0.0107 {\,{M_\odot}}$. []{data-label="mass_excitation"}](mass_excitation_2Myr.png "fig:") ![The mass estimated from various transitions and different isotopologues. Excitation energies are for the upper excitation state of each transition. The populations at the zero excitation energy are extrapolated from the higher energy populations with a three degree polynomial function. The actual disk mass is marked by the blue line in the 100 yr diagram (on the left), and is above the chart in plots for the 2 Myr and 3 Myr disks because the mass is hugely underestimated with the shown method. The actual disk masses in the input models at those three epochs are $0.0144 {\,{M_\odot}}$, $0.0114 {\,{M_\odot}}$ and $0.0107 {\,{M_\odot}}$. []{data-label="mass_excitation"}](mass_excitation_3Myr.png "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [^1]: https://www.sofia.usra.edu/ [^2]: http://www.ita.uni-heidelberg.de/ dullemond/software/radmc-3d/; developed by C. Dullemond [^3]: See the catalog of resolved disk images at [`w`ww.circumstellardisks.org]{} [^4]: http://home.strw.leidenuniv.nl/ moldata/

--- abstract: | A $d^{\{n\}}$-cage $\mathsf K$ is the union of $n$ groups of hyperplanes in $\Bbb P^n$, each group containing $d$ members. The hyperplanes from the distinct groups are in general position, thus producing $d^n$ points, where hyperplanes from all groups intersect. These points are called the nodes of $\mathsf K$. We study the combinatorics of nodes that impose independent conditions on the varieties $X \subset \Bbb P^n$ containing them. We prove that if $X$, given by homogeneous polynomials of degrees $\leq d$, contains the points from such a special set $\mathsf A$ of nodes, then it contains all the nodes of $\mathsf K$. Such a variety $X$ is very special: in particular, $X$ is a complete intersection. We generalize the notion of a $d^{\{n\}}$-cage in $\Bbb P^n$ to include cages in $n$-dimensional projective varieties $Y$: such cages $\mathsf K$ are formed by special configurations of positive and “completely $d$-reducible" divisors in $Y$. For subvarieties $X \subset Y$ that contain a special set $\mathsf A \subset \mathsf K$ of nodes, we prove results that analogues of our results for cages in $\Bbb P^n$. Then we study the reducible varieties form the cage families and the symmetric cages and symmetric varieties that are attached to their nodes. address: 'MIT, Department of Mathematics, 77 Massachusetts Ave., Cambridge, MA 02139, U.S.A.' author: - Gabriel Katz title: 'VARIETIES IN CAGES: a Little Zoo of Algebraic Geometry ' --- Introduction ============ This paper is an extension and generalization of [@K], which dealt with algebraic curves in *plane cages*, to algebraic varieties in the multidimensional cages (see Definition \[cage\]). In this text, we use the term “variety" as a synonym of “algebraic set". Our tools here are elementary[^1] (they do not go beyond some “Fubini’s-flavored" versions of the Bésout Theorem) and mostly *combinatorial*. We tried to make this text friendly to readers who, as the author himself, are not practitioners of Algebraic Geometry. Consider two groups of lines in the plane[^2], each group comprising three lines. We call such a configuration of six lines a $3\times 3$-*cage*, or $3^{\{2\}}$-cage for short. Let us label the lines of the first group with *red*, and of the second group with *blue*. Assume that there are exactly $9$ points where the blue lines intersect the red lines. We call them the *nodes* of the cage. Our original motivation for studying the varieties in cages comes from the following classical result in Algebraic Geometry. [**(The Cage Theorem for Plane Cubics)**]{}\[3x3\] Any plane cubic curve $\mathcal C$, passing through eight nodes of a $3\times 3$-cage, will automatically pass through the ninth node. At the first glance, this claim appears to be an esoteric fact. However, it reflects a deep intrinsic algebraic structure that non-singular cubic curves carry (such curves are called elliptic). It turns out that, in disguise, any elliptic curve $\mathcal C$ is an *abelian group*. From this angle, Cage Theorem \[3x3\] becomes a statement about the *associativity* of the binary group operation “$+$“ on $\mathcal C$! It’s a relatively subtle interpretation. Here is a sketch of the construction of the operation ”$+$" on elliptic curves: by definition, any three distinct *collinear* points $x, y, z$ on a cubic $\mathcal C$, satisfy the relation $x + y + z = 0$. This calls for designating one point $e$ on $\mathcal C$ as the neutral element $0$. With this choice in place, $z$ must play the role of $-(x + y)$ (see Figure \[asso\]). ![[]{data-label="asso"}](associativity.pdf){width="50.00000%"} \[cage\] A $d^{\{n\}}$-*cage* $\mathsf K$ is a configuration of $n$ distinctly colored groups of $d$ hyperplanes each[^3], located in the $n$-space (projective or affine) in such a way, that $\mathsf K$ generates exactly $d^n$ points where the hyperplanes of all $n$ distinct colors $\a_1, \dots, \a_n$ intersect transversally[^4]. These points are called the *nodes* of the cage. $\diamondsuit$ Hyperplanes in $\P^n$ form a dual projective space $\P^{n \ast}$, the space of linear homogeneous functions, considered up to proportionality. Therefore $d$ hyperplanes of the same color from a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$ represent an *unordered* configuration of $d$ points in $\P^{n \ast}$, a point in the symmetric product $\mathsf{Sym}^d(\P^{n \ast})$. So the color-ordered collection of $n$ such points from $\mathsf{Sym}^d(\P^{n \ast})$ is a point of the space $\big(\mathsf{Sym}^d(\P^{n \ast})\big)^n$. By Definition \[cage\], any set of $n$ hyperplanes of *distinct* colors has a single intersection point. The requirement that some set of $n$ hyperplanes of distinct colors has multiple intersection points in $\P^n$ puts algebraic constraints on the coefficients of the $dn$ homogeneous linear polynomials (in $n+1$ variables) that define the hyperplanes. Similarly, the requirement in Definition \[cage\] that all transversal $n$-colored intersections are distinct, and thus numbering $d^n$, produces a Zariski open set. Therefore, we get: \[space\_of\_cages\] The $d^{\{n\}}$-cages form a Zariski open set $\mathcal K$ in the $(dn^2)$-dimensional space $\big(\mathsf{Sym}^d(\P^{n \ast})\big)^n$. The group of projective transformations $\mathsf{PGL}_\A(n+1)$ acts naturally on $\big(\mathsf{Sym}^d(\P^{n \ast})\big)^n$, and thus on the set $\mathcal K$. $\diamondsuit$ The problem we address in this paper is to describe the varieties that contain all $d^n$ nodes of a given cage $\mathsf K$. It turns out, that every variety $V$, defined by polynomials of degrees $\leq d$ and containing the node set $\mathsf N$, is very special indeed. In particular, $V$ must be a complete intersection of the type $(\underbrace{d, \dots , d}_{s})$, where $s = n- \dim V$. Furthermore, the requirements that a hypersurface of degree $\leq d$ will pass through the nodes of a $d^{\{n\}}$-cage are very much *redundant*. In the paper, we describe the combinatorics of the nodes that impose *independent* constraints on the hypersurface in question. We call such maximal set $\mathsf A$ of “independent" nodes *supra-simplicial* (see Definition \[def2.1\] and Figure \[supra\]). Crudely, the proportion of cardinalities $\frac{\#\mathsf A}{\# \mathsf N}$ declines as $\sim 1/n$ with the growth of $d$. Our results are of the same flavor as some well-known theorems of Algebraic Geometry, operating within a much less restrictive environment than the one of the $d^{\{n\}}$-cages. However, the results about varieties in cages are more geometrical, transparent, and easy to state. Still, to provide a point of reference, let us describe briefly the classical results. Let $\Z_+$ denote non-negative integers. Recall that the Hilbert functions $h_X: \Z_+ \to \Z_+$ of a variety $X$ over a field $\A$ associates with a non-negative integer $k$ the dimension of the $k$-graded portion of the quotient ring $\A[x_0, \dots, x_N] / \mathcal I_X$, where $\mathcal I_X$ denotes the zero ideal of the polynomial ring that defines $X$. Since the node set $\mathsf N$ of a $(d\times d)$-cage is the intersection locus of $d$ red and $d$ blue lines, Theorem \[3x3\] is a special case of the Cayley-Bacharach Theorem ([@C], [@B]), stated below. For a complete intersection $X\subset \P^2$, Theorem \[Bach\] connects the Hilbert functions $h_X: \Z_+ \to \Z_+$, $h_{X_1}: \Z_+ \to \Z_+$, and $h_{X_2}: \Z_+ \to \Z_+$ of a finite set $X$, its subset $X_1$, and its complement $X_2:= X \setminus X_1$. Recall that, for a $0$-dimensional variety $X$ and all sufficiently big $k$, $h_X(k) = |X|$, the cardinality of $X$. [**(Cayley-Bacharach)**]{}\[Bach\] Let $\mathcal D$ and $\mathcal E$ be two projective plane curves of degrees $d$ and $e$, respectively, and let the finite set $X = \mathcal D \cap \mathcal E$ be a complete intersection in $\P^2$. Assume that $X$ is the disjoint union of two subsets, $X_1$ and $X_2$. Then for any $k \leq d+e-3$, the Hilbert functions $h_\sim(m)$ of $X$, $X_1$, and $X_2$ are related by the formula: $$h_X(k) - h_{X_1}(k) = |X_2| - h_{X_2}((d + e - 3) - k).\footnote{The RHS of this formula describes the failure to impose independent constrains by the points of the set $X_2$ on the polynomials of degree $k$.} \qquad \qquad \qquad \qquad \hfill \diamondsuit$$ In turn, Theorem \[Bach\] admits a comprehensive generalization by Davis, Geramita, Orecchia [@DGO], and by Geramita, Harita, Shin (see [@GHS1], and especially [@GHS2], Theorem 3.13). It is a “Fubini-type" theorem for the Hilbert function of a finite subset $X \subset \P^n$ that is contained in the union of a family of hypersurfaces $\{H_i \}_{1 \leq i \leq s}$, whose degrees $\{d_i\}$ add up to the degree of $X$. Under some subtle hypotheses that regulate the interaction between $X$ and the hypersurfaces $\{H_i \}_{1 \leq i \leq s}$ (they include the hypotheses “$X = \coprod_i (X \cap H_i)"$), a nice formula for the Hilbert functions $\{h_{X \cap H_i}: \Z_+ \to \Z\}_{1 \leq i \leq s}$ of $H_i$-slices of $X$ emerges: $$h_X(k) = h_{X \cap H_1}(k) + h_{X\cap H_2}(k - d_1)+ \dots +h_{X\cap H_s}(k- (d_1 + \dots +d_{s-1})).$$ A clear beautiful overview of the research, centered on the Cayley-Bacharach type theorems, can be found in [@EGH]. Now let us describe *the main results of the paper and its structure* in some detail. The paper is divided in five sections, including the Introduction. The main results of [*Section 2*]{} are: Theorem \[main\_th\], Theorem \[smooth\], and Theorem \[unique\_hyper\]. Here is a summary of their claims. Any variety $X \subset \P^n$ that is the zero set of homogeneous polynomials of degrees $\leq d$ and contains a supra-simplicial set $\mathsf A$ of nodes of a given $d^{\{n\}}$-cage $\mathsf K \subset \P^n$ contains all the nodes of $\mathsf K$. Such $X$ is a complete intersection of the multi-degree $(\underbrace{d, \dots , d}_{s})$, where $s = \textup{codim}(X, \P^n)$. Moreover, $X$ is smooth in the vicinity of the node set $\mathsf N$. The variety $X$ is completely determined by $\mathsf A$ and the tangent to $X$ space $\tau_p$ at any of the nodes $p$. Conversely, any subspace $\tau_p \subset T_p(\P^n)$ of codimension $s$, where $p \in \mathsf N$, with the help of $\mathsf A$, produces such $X$. In [*Section 3*]{}, we generalize the notion of a $d^{\{n\}}$-cage from cages in $\P^n$ to cages on projective varieties. The notion of a supra-simplicial set $\mathsf A$ of nodes is also generalized. The main result of this section is Theorem \[main\_thA\], an the analogue of Theorem \[main\_th\], Theorem \[smooth\], and Theorem \[unique\_hyper\], for cages on a given projective variety $X$. [*Section 4*]{} is concerned with the $d^{\{n\}}$-cages $\mathsf K$ that admit varieties $X$ (given by polynomials of degrees $\leq d$) which are attached to the nodes of $\mathsf K$ and are *reducible*. Of course, every cage contains at least $n$ completely reducible hypersurfaces of degree $d$, the unions of hyperplanes of a particular color. The challenging issue is whether there are other reducible members of the cage family. Here, we set up a framework for addressing these questions. We formulate some natural conjectures, related to the reducible varieties in a generic cage family. We get only partial results in this direction like Corollary \[cone\_Pascal\], Theorem \[reducible\_3\_cages\], Corollary \[spatial\_Pascal\], and Corollary \[spacial\_Pascal\_A\] (the spacial Pascal Theorem). These propositions describe seemingly new facts about the elementary projective geometry of $3D$-spaces. In [*Section 5*]{}, we study symmetric cages and symmetric varieties that are attached to their nodes. The symmetry group $G$ is finite. Here the main results are Theorem \[G-invariant\] and Theorem \[V\^G = N\], which describe constructions for generating interesting equivariant cages and equivariant varieties that contain their nodes. These examples produce $G$-equivariant varieties $V$ whose $G$-fixed point sets $V^G$ coincide with the nodes of appropriately designed cages. The projective varieties $V$ realize several copies of a given $G$-representation $\Psi$ as the normal $G$-bundle $\nu(V^G, V)$ over the finite base $V^G$. In all the figures, we restrict ourselves to depictions of cages in the space $\R^3$. Some of the figures are produced with the help of the *Graphing Calculator* application, and some of them are drawings. For technical reason, in most of the figures, the nodes of cages are invisible. Although the images depict real surfaces in only in $3^{\{3\}}$- and $4^{\{3\}}$-cages, the entire exhibition looks surprisingly rich. Despite being very special, the zoo of varieties in cages is a microcosmos of the old Italian style Algebraic Geometry. Visiting this zoo may also bring back memories of the good old days of Projective Geometry. A Multidimensional Zoo ====================== As a default, we choose the number field $\A$ to be the field of real or complex numbers. We suspect that our main result may be valid over any infinite field. In the notations, we do not emphasize the dependence of our constructions on the choice of a field. Let $\mathcal L_j$ be the degree $d$ homogeneous polynomial whose zero set is the union of $d$ hyperplanes of a particular color $\a_j$ ($\mathcal L_j$ is a product of $d$ linear forms). Since $\deg(\mathcal L_j) = d$, Bézout’s Theorem implies that the solution set $\mathsf N$ of the system $\{\mathcal L_j =0\}_{j \in [1, n]}$ consists of $d^n$ points at most, provided that $\mathsf N$ is finite. Thus Definition \[cage\] implies that each node $p \in \mathsf N$ of the cage belongs to a single hyperplane of a given color and the hyperplanes of distinct colors are in general position at $p$, and thus in the ambient $n$-space. It follows that the node locus $\mathsf N \subset \P^n$ is a $0$-dimensional complete intersection of degree $d^n$. [**Example 2.1.**]{} Consider the complex Fermat curve $\mathcal F \subset \C\P^2$, given by $\{\tilde x^d + \tilde y^d = \tilde z^d\}$ in the homogeneous coordinates $[\tilde x : \tilde y : \tilde z]$. In the affine coordinates $(x, y)$, its equation may be written as $x^d + y^d = 1$, or as $\prod_\xi (x - \xi) + \prod_\eta (y- \eta) = 0$, where $\xi, \eta$ run over the set of complex $d$-roots $\sqrt[d]{-1/2}$. Therefore $\mathcal F$ passes trough the nodes of the $d \times d$-cage $\mathsf K := (\bigcup_\xi \{x = \xi\})\,\bigcup \,(\bigcup_\eta \{y = \eta\}) \subset \C^2$. $\diamondsuit$ Let $\mathbf I^n(d)$ be the subset $\{I = (i_1, i_2, \dots, i_n)\}$ of the lattice $\Z^n_+$, such that each $i_j \in [1, d]$. So $\mathbf I^n(d)$ is a $n$-dimensional “cube" of the size $d$. By definition, $\| I\| = \sum_{j =1}^n i_j$. If we introduce *some order* among the hyperplanes of the same color $\a_j$ ($j = 1, \dots , n$), then each node $p_I$ of $\mathsf K$ will be marked with a unique multi-index $I \in \mathbf I^n(d)$. \[def2.1\] A set of nodes $\mathsf T$ from $d^{\{n\}}$-cage $\mathsf K$ is called *simplicial* if, with respect to some orderings of the hyperplanes in each group, it is comprised of the nodes $\{p_I\}_{I \in \mathbf I^n(d)}$, subject to the constraints $\| I\| \leq d + 1$. A set of nodes $\mathsf A$ from a cage $\mathsf K$ is called *supra-simplicial* if, with respect to some orderings of the hyperplanes in each group, it is comprised of the nodes $\{p_I\}_{I \in \mathbf I^n(d)}$, subject to the constraints $\| I\| \leq d + 2$. (see Figure \[supra\], where the grid corner is located at $(1, 1, 1)$). $\diamondsuit$ ![[]{data-label="supra"}](3D-cage.pdf){width="40.00000%"} [**Example 2.2.**]{} For $d =2$, the $2^{\{n\}}$-cage is modeled after the union of the hyperplanes in $\R^n$ that extend the faces of a $n$-cube. The cardinality of the node locus $\mathsf N$ is $2^n$, the cardinality of the simplicial set $\mathsf T$ is $n+1$, while the cardinality of the supra-simplicial set $\mathsf A$ is $C_n^2 + n +1 = \frac{1}{2}(n^2 + n +2)$. $\diamondsuit$ [**Example 2.3.**]{} The famous $K3$-surface is given by the equation $\{y_0^4 + y_1^4+ y_2^4 + y_3^4= 0\}$ in $\C\P^3$, or by the equation $\{x_1^4 + x_2^4+ x_3^4 +1 = 0\}$ in $\C^3$. Using the partition $\{1 = 1/3 +1/3 +1/3\}$, the latter equation may be written in the form $$\prod_\a(x_1- \a)+ \prod_\b(x_2- \b) + \prod_\g(x_3 -\g) = 0,$$ where $\a, \b, \g$ each runs over the four complex roots of the equation $\{z^4 = -1/3\}$. Therefore, the $K3$-surface contains all the $64$ nodes of a $4^{\{3\}}$-cage $\mathsf K$, defined by the equations $$\big\{\prod_\a(y_1- \a \cdot y_0) =0\big\} \bigcup \big\{\prod_\b(y_2- \b\cdot y_0) =0\big\} \bigcup \big\{\prod_\g(y_3 -\g\cdot y_0) = 0\big\}.$$ In fact, the $K3$-surface is nailed to the notes of a 2-dimensional variety of cages, produced in similar ways by writing down $1$ as a sum of three complex numbers, all different from $0$. The previous construction was based on the composition $\{1 = 1/3 +1/3 +1/3\}$. We notice that the nodes of this cage $\mathsf K$ are “invisible" in $\R\P^3$. The permutation group $\mathsf S_4$ of order $24$ acts on $\C\P^3$ by permuting the coordinates $(y_0, y_1, y_2, y_3)$. Under this $\mathsf S_4$-action, the $K3$-surface is invariant. In contrast, the cage $\mathsf K$ is invariant only under the $\mathsf S_3$-action that permutes the coordinates $(y_1, y_2, y_3)$. (This action does not preserve the colors of the cage!) Thus, using the $\mathsf S_4$-action on $\mathsf K$, the $K3$-surface contains the nodes of at least *four* distinct $4^{\{3\}}$-cages in $\C\P^3$. $\diamondsuit$ [**Example 2.4.**]{} Recall a remarkable Cayley-Salmon Theorem [@C1]: any smooth complex cubic surface $X$ contains exactly $27$ lines. If $X \subset \C\P^3$ is given by the equation $\{z_0^3 + z_1^3 + z_2^3 + z_3^3 = 0\}$ (this surface is called Fermat cubic surface), then putting $\omega := e^{2\pi \mathbf i/3}$, each of these $27$ lines is given by $2$ linear constraints (see [@M], Corollary (8.20)): $$\begin{aligned} \{z_0 + \omega^i z_1 = 0, \; z_2 + \omega^j z_3 =0\}, \;\; i, j \in [0, 2], \nonumber \\ \{z_0 + \omega^i z_2 = 0, \; z_1 + \omega^j z_3 =0\}, \;\; i, j \in [0, 2], \nonumber \\ \{z_0 + \omega^i z_3 = 0, \; z_1 + \omega^j z_2 =0\}, \;\; i, j \in [0, 2].\end{aligned}$$ As in the previous examples, using the composition $\{1 = 1/3 +1/3+ 1/3\}$, we notice that $X$ is inscribed in a $3^{\{3\}}$-cage $\mathsf K$, given by the formula $$\bigcup_{j =1}^3 \Big\{\prod_{k=0}^2 \big(z_j + \frac{1}{\sqrt[3]{3}}\, \omega^k \, z_0\big) = 0\Big\}.$$ As in Example 2.3, there exists a $2$-parameter family of cages in which $X$ is inscribed (it corresponds to different ways one can represent $1$ as a sum of three non-vanishing complex numbers). The symmetric group $\mathsf S_4$ acts on the Fermat surface $X$ by permuting the coordinates in $\C\P^3$. This action must preserve the configuration of $27$ lines in $X$ since these lines are the only ones residing in $X$. The subgroup $\mathsf S_3 \subset \mathsf S_4$ that permutes the coordinates $(z_1, z_2, z_3)$ evidently preserves the cage $\mathsf K$, but not its colors. Thus $X$ contains the nodes of at least $4$ distinct cages in $\C\P^3$, obtained from $\mathsf K$ by the $\mathsf S_4$-action. Consider the $27$ lines, contained the $3^{\{3\}}$-cage $\mathsf K$, where two planes of distinct colors intersect (this locus is the “$1$-skeleton" of $\mathsf K$), and compare them with the $27$ lines on a smooth cubic surface $X$. Although their equations are somewhat similar, we could not see if there is any relation between these two configurations (see [@H], Chapter V, Section 4, for the explicit description of the configuration the $27$ lines on $X$). $\diamondsuit$ For a smooth complex cubic surface $X \subset \C\P^3$ that contains all the nodes of a given $3^{\{3\}}$-cage $\mathsf K$, how to describe, in terms of $\mathsf K$, the pattern of $27$ lines that belong to $X$? Is there anything special about the locus where the $27$ lines in $X$ hit the nine planes that form the cage? Perhaps, within the family of cubic surfaces $X$ that are inscribed in $\mathsf K$, the $27$ bicolored lines of the cage are “the limits"[^5] of $27$ lines on $X$, as $X$ degenerates into the completely reducible variety of $3$ planes of a particular color? $\diamondsuit$ ![[]{data-label="3X3"}](3X3cageTWO.pdf){width="40.00000%"} ![[]{data-label="3X3A"}](3X3cageNEW.pdf){width="40.00000%"} ![[]{data-label="4X4A"}](4X4.pdf){width="50.00000%"} By examining the diagonal lines in the Pascal Triangle, we get the following useful combinatorial fact. \[cardinality\] Each simplicial set of nodes $\mathsf T$ in a $d^{\{n\}}$-cage is of the cardinality $C_{d+n-1}^n$. Each supra-simplicial set of nodes $\mathsf A$ in a $d^{\{n\}}$-cage is of cardinality $C_{d+n}^n - n$. $\diamondsuit$ Let $H_{j, i}$ be the $i$-th hyperplane of the color $\a_j$, and let $L_{j, i}$ be a homogeneous linear polynomial in the coordinates $(y_0, y_1, \dots , y_n)$ on the space $\A^{n+1}$ that defines $H_{j, i}$. Each $L_{j, i}$ is determined, up to proportionality, by $H_{j, i}$. In what follows, we fix a particular linear form $L_{j, i}$. Put $\mathcal L_j := \prod_{i \in [1, d]} L_{j, i}$. For any nonzero vector $\vec \l = (\l_1, \dots, \l_n) \in \A^n$, we consider the homogeneous polynomial of degree $d$ $$\begin{aligned} \label{P_Lambda} \mathcal P_{\mathsf K,\, \vec\l} \; := \sum_{j \in [1, n]} \l_j \cdot \mathcal L_j.\end{aligned}$$ Evidently, each polynomial $\mathcal P_{\mathsf K,\, \vec\l}$ vanishes at all the nodes of the cage $\mathsf K$. \[main\_th\] Consider a subvariety[^6] $V \subset \P^n$, given by one or several homogeneous polynomial equations of degrees $\leq d$. - If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ of a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, then $V$ contains all $d^n$ nodes of the cage. Moreover, any such variety $V$ is given by polynomial equations of the form $\{\mathcal P_{\mathsf K, \vec\l} = 0\}_{\vec\l}$ for an appropriate choice of $\vec\l$’s (see (\[P\_Lambda\])). - In contrast, no such variety $V$ contains all the nodes from a simplicial set $\tilde{\mathsf T}$ of any $(d+1)^{\{n\}}$-cage $\tilde{\mathsf K} \subset \P^n$. As in the case of encaged plane curves [@K], the argument is based on a combinatorial similarity between the Newton’s diagram of a generic polynomial of degree $d$ in $n$ variables and a simplicial set $\tilde{\mathsf T}$ of nodes of any $(d+1)^{\{n\}}$-cage. Also the cardinality of such a Newton’s diagram exceeds the cardinality of a supra-simplicial set $\mathsf A$ of nodes of a $d^{\{n\}}$-cage $\mathsf K$ by $n$. In other words, the dimension of the variety of hypersurfaces of degree $d$ in the space $\P^n$ exceeds $\#\mathsf A$ by $n-1$. Indeed, the monomials in the affine variables $x_1, \dots, x_n$ of degree $\leq d$ (equivalently, the homogeneous monomials in the variables $y_0, \dots, y_n$ of degree $d$) are in one-to-one correspondence with the set $\mathsf B$ none-negative integral $n$-tuples $I \in \Z^n$, subject to the inequality $\|I\| \leq d$. At the same time, the nodes $\{p_I\}$ of an supra-simplicial set $\mathsf A$ satisfy the inequality $\| I\| \leq d + 2$ together with $\{1 \leq i_s \leq d\}_{s \in [1, n]}$. Shifting by the vector $(-1, \dots , -1)$ embeds $\mathsf A$ into $\mathsf B$ so that only the $n$ corners $(d, 0, \dots , 0), (0, d, \dots, 0), \dots (0, 0, \dots, d)$ of the Newton diagram remain outside of the shifted $\mathsf A$. Finally, proportional polynomials define the same hypersurface. The following proof is recursive in nature. The induction is carried in $n$, the dimension of the cage. We assume that the first bullet of the theorem is valid for all $d^{\{k\}}$-cages of any size $d$ in spaces of dimension $k < n$, and the second bullet is valid for all cages of any size $d+1$ in spaces of dimension $k < n$. Our argument relies on slicing $\mathsf K \supset \mathsf A$ by the hyperplanes $\{H_{1, i} = 0\}_{i \in [1, d]}$ of the first color $\a_1$, thus reducing the argument to families of cages in $(n-1)$-dimensional affine or projective spaces. This leads to a “Fubini-type cage theorem” in the spirit of [@GHS2] (see Figure \[supra\] for guidance). For any integer $s \in [1, d-1]$, we consider the $(d-s+1)^{\{n-1\}}$ sub-cage $\mathsf K^{[s]} \subset \mathsf K\, \cap \, H_{1, s} $, formed by the hyperplanes $H_{1, s}\, \bigcap \, (\bigcup_{j \in [2,\,n],\; i \in [1,\, d-s+1]} H_{j, i})$ in $H_{1, s} \approx \P^{n-1}$. In the hyperplane $H_{1, s}$, the cage $\mathsf K^{[s]}$ is given by the equation $$\big\{\mathcal L\mathcal L^{[s]} \, := \prod_{j \in [2 ,n],\; i \in [1, d-s+1]} L_{j, i} \;= \;0\big\}.$$ We denote by $\mathsf T^{[s]}$ the simplicial set of nodes in $\mathsf T \cap \mathsf K^{[s]}$ and by $\mathsf A^{[s]}$—the set of nodes from the supra-simplicial set $\mathsf A \cap \mathsf K^{[s]}$. Note that the set $\mathsf T^{[s]}$ can serve as a simplicial set and $\mathsf A^{[s]}$— as an supra-simplicial set for the cage $\mathsf K^{[s]}$. We start with a given homogeneous degree $d$ polynomial $P$ in the projective coordinates $[y_0: y_1: \dots y_n]$, which vanishes at all the nodes of an supra-simplicial set $\mathsf A$ of a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$. Consider the restriction of $P$ to the first hyperplane $H_{1,1}$ of the color $\a_1$. Then $P$ vanishes at the supra-simplicial set $\mathsf A^{[1]} := \mathsf A \cap H_{1,1}$ of the induced $d^{\{n-1\}}$-cage $\mathsf K^{[1]} := \mathsf K \cap H_{1,1}$, the zero set of the polynomial $\mathcal L_2 \cdot \mathcal L_3 \cdot\; \dots \; \cdot \mathcal L_n$ in $H_{1, 1}$. By induction on $n$, the restriction $P|_{H_{1,1}}$ must be of the form $\mathcal P_1 := \sum_{j \in [2, n]} \l_j^{[1]} \cdot \mathcal L_j$ (being restricted to $H_{1,1}$) for some choice of the coefficients $\l_2^{[1]}, \dots \l_n^{[1]}$. For this special choice of $(\l_2^{[1]}, \dots \l_n^{[1]})$, the difference $P - \mathcal P_1$ is identically zero on $H_{1,1}$. By Lemma \[divisible\], if a homogeneous polynomial $R$ vanishes on a hyperplane, given by a homogeneous linear polynomial $L$, then $R$ is divisible by $L$. Therefore $P - \mathcal P_1$ is divisible by the liner polynomial $L_{1,1}$. So $P = \mathcal P_1 + L_{1,1}\cdot P_1$, where $P_1$ is a homogeneous polynomial of degree $d-1$. Next, we consider the restrictions of $P$ and $P_1$ to the hyperplane $H_{1, 2} = \{L_{1,2} = 0\}$ of color $\a_1$. Since both $P$ and $\mathcal P_1$ vanish at the set $\mathsf A \cap H_{1, 2}$ and, by Definition \[cage\], $L_{1,1} \neq 0$ at the points of $\mathsf A \cap H_{1, 2}$, we conclude that $P_1$ (of degree $d-1$) must vanish at the set $\mathsf A \cap H_{1, 2}$ as well. Note that $\mathsf A \cap H_{1, 2} = \mathsf A^{[2]}$ is a *simplicial set* for the induced $d^{\{n-1\}}$-cage $\mathsf K^{[2]} \subset \mathsf K \cap H_{1,2}$. So by induction, any homogeneous polynomial of degree $d-1$ that vanishes at a simplicial set $\mathsf A^{[2]}$ of the $d^{\{n-1\}}$-cage $\mathsf K^{[2]}$ must vanish at $H_{1,2}$. Hence $P_1 = L_{1,2} \cdot P_2$ for some homogeneous polynomial $P_2$ of degree $d-2$. So we get $P = \mathcal P_1 + L_{1,1}\cdot L_{1,2} \cdot P_2$. Similarly, we argue that of $P_2$ of degree $d-2$ vanishes on the simplicial set $\mathsf A^{[3]} \subset \mathsf A \cap H_{1,3}$ of the $(d-1)^{\{n-1\}}$-cage $\mathsf A^{[3]}$. Therefore $P_2|_{H_{1,3}}$ is zero, and $P_2 = L_{1,3}\cdot P_3$ for a homogeneous polynomial $P_3$ of degree $d-3$. As a result, $P = \mathcal P_1 + L_{1,1}\cdot L_{1,2} \cdot L_{1,3}\cdot P_3$. Continuing this reasoning, we get eventually $$P = \mathcal P_1 + \l(L_{1,1}\cdot L_{1,2}\cdot \dots \cdot L_{1,n}) = \sum_{j \in [2, n]} \l_j^{[1]} \cdot \mathcal L_j + \l \mathcal L_1,$$ where $\l$ is a constant. Therefore, $P = \l\cdot \mathcal L_1 + \sum_{j \in [2, n]} \l_j^{[1]} \cdot \mathcal L_j $ is of the form $\mathcal P_{\mathsf K, \vec\l}$ and must vanish at every node of the $d^{\{n\}}$-cage $\mathsf K \subset \P^n$. By a similar reasoning, we will validate the second bullet of the theorem. So we take any polynomial $P$ of degree $d$ that vanishes at a simplicial set $\tilde{\mathsf T}$ of a $(d+1)^{\{n\}}$-cage $\tilde{\mathsf K} \subset \P^n$. As before, we slice $\tilde{\mathsf K}$ by the hyperplanes $\{H_{1, s }\}_{i \in [1, d+1]}$ of the color $\a_1$. Now all the slices $\tilde{\mathsf T}^{[s]}$ (including the first one) are simplicial sets in $\tilde{\mathsf K}^{[s]}$. The latter locus $\tilde{\mathsf K}^{[s]}$ is given by the equations $$\big\{\tilde{\mathcal L\mathcal L}^{[s]} \, := \prod_{j \in [2 ,n],\; i \in [1,\, d-s+2]} L_{j, i} \;= \;0\big\}.$$ Since $P$ vanishes at $\tilde{\mathsf T}^{[1]}$, by the induction hypotheses, $P|_{H_{1,1}} = 0$. This implies that $P = L_{1,1}\cdot P_1$, where $P_1$ is a homogeneous polynomial of degree $d_1$. The set $\tilde{\mathsf T}^{[2]}$ is simplicial in the cage $d^{\{n-1\}}$-cage. Since $L_{1,1}|_{\tilde{\mathsf T}^{[2]}} \neq 0$, we get that $P_1$ must vanish at the nodes from $\tilde{\mathsf T}^{[2]}$. By induction, this implies that $P_1|_{H_{1,1}} = 0$ and thus is divisible by $L_{1,1}$. So $P= L_{1,1}\cdot L_{1,2}\cdot P_2$ for a homogeneous polynomial $P_2$ of degree $d-2$. Continuing this process, we get $P= L_{1,1}\cdot L_{1,2}\cdot\, \dots, \cdot L_{1,d} \cdot \l$ must vanish at the unique node of the set $\tilde{\mathsf T}^{[d+1]}$. This forces $\l = 0$, and so $P$ is identically zero. Finally, the validity of the basis of induction “$n=1$" is obvious for univariate polynomials of any degree $d$. In fact, Theorem \[main\_th\] has been proven in [@K] for $n=2$. Since the varieties $V$ we consider in the theorem are defined by polynomials of degrees $\leq d$, the claim follows. [**Remark 2.1.**]{} Note that the assumption that $\mathsf A$ is supra-simplicial set in Theorem \[cage\] is essential: not any subset of nodes of the cardinality $\#\mathsf A$ from a $d^{\{n\}}$-cage imposes independent relations on the set of homogeneous polynomials of degree $d$ in $n+1$ variables! For example, in a $4\times 4$-cage, $\#\mathsf A = 13$. However, if $\mathsf B$ is the complement to the set of four nodes $\mathsf C := \{p_{42}, p_{43}, p_{44}\}$, then not every curve of degree $4$ that contains $\mathsf B$ will contain $\mathsf C$. In fact, $\mathsf B$ is contained in the union of three red and one blue lines from the cage; they all miss $\mathsf C$. $\diamondsuit$ [**Example 2.5.**]{} Consider any curve $C$ in $\P^3$, given by homogeneous polynomial equations of degree $\leq 3$ (typically, $C$ is of degree $9$). If $C$ passes through $17$ nodes of a supra-simplicial set $\mathsf A$ of nodes of a $3^{\{3\}}$-cage, then it passes through all the $27$ nodes of the cage. A similar conclusion holds for any surface of degree $3$ in $\P^3$ that passes through the $17$ nodes from $\mathsf A$. $\diamondsuit$ \[d\_exactly\] Consider a subvariety $V \subset \P^n$, given by one or several homogeneous polynomial equations of degrees $\leq d$. If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ (of cardinality $C_{d+n}^n - n$) in a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, then all the polynomials that define $V$ are exactly of degree $d$. By Theorem \[main\_th\], if a homogeneous polynomial $P$ of degree less than $d$, which vanishes at $V$, also vanishes at the simplicial set $\mathsf T \subset \mathsf A$ of the $d^{\{n\}}$-cage $\mathsf K$, then $P = 0$ identically. Thus $\deg P = d$, provided that $P$ is nontrivial. Combining Theorem \[main\_th\] with the Bezout Theorem, leads instantly to the following claim: Let $V_1, V_2$ be two varieties in $\P^n$, both given by systems of homogeneous polynomial equations of degree $\leq d$. Assume that $V_1 \cap V_2$ contains all the nodes from a supra-simplicial set $\mathsf A$ of a $d^{\{n\}}$-cage $\mathsf K$. Then $V_1 \cap V_2$ contains all the nodes of $\mathsf K$. If $V_1 \cap V_2$ is a finite set, then $\deg(V_1) \cdot \deg(V_2) = d^n$. The first claim follows directly from Theorem \[main\_th\], while the second one follows from the first claim, being combined with the Bezout Theorem (cf., Section 2.3 in [@F]). \[smooth\] Let $V \subset \P^n$ be a subvariety of codimension $s$, given by one or several homogeneous polynomial equations of degrees $\leq d$. If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ of a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, then $V$ is a complete intersection of the multi-degree $(\underbrace{d, \dots , d}_{s})$, which is smooth at each node of the cage $\mathsf K$. Thus, $\deg V = d^{n-\dim V}$. We start with the case of $V$ being a hypersurface. By Theorem \[main\_th\], such a hypersurface $V$ is given by the equation $$\mathcal P_{\mathsf K,\, \vec\l} \; := \sum_{j \in [1, n]} \l_j \cdot \mathcal L_j = 0.$$ Any node $p \in \mathsf N \subset \mathsf K$ is the intersection of $n$ hyperplanes $H_{1, i_1}, \dots H_{j, i_j}, \dots , H_{n, i_n}$ of $n$ distinct colors. We may choose an affine chart $U_p \subset \P^n$ that is centered on $p$. In this chart, we replace the homogeneous linear forms $L_{j, i_j}$ by linear polynomial functions in $n$ variables. Abusing notations, we still denote them by $L_{j, i_j}$. Next, we choose the linear functions $L_{1, i_1}, \dots L_{j, i_j}, \dots , L_{n, i_n}$ as the new local affine coordinates at $p$ so that $L_{j, i_j}(p) = 0$. We need to verify that the differentials $d \mathcal L_1, \dots , d \mathcal L_n \in T^\ast\P^n$ are linearly independent at $p$. We represent each $\mathcal L_j$ as the product $L_{j, i_j}\cdot \mathcal M_{j, i_j}$, where $\mathcal M_{j, i_j} := \prod_{k \neq i_j} L_{j, k}$. Then $$d_p(\mathcal L_j) = \mathcal M_{j, i_j}(p) \cdot d_p(L_{j, i_j}) + L_{j, i_j}(p) \cdot d_p(\mathcal M_{j, i_j}) = \mathcal M_{j, i_j}(p) \cdot d_p(L_{j, i_j}),$$ where $\mathcal M_{j, i_j}(p) \neq 0$ by the definition of a cage. Thus the differential $1$-forms $d_p \mathcal L_1, \dots , d_p\mathcal L_n$ are linearly independent at $p$ since, by the definition of a cage, so are the $n$ differentials $\{d_p(L_{j, i_j})\}_j$. As a result, $d_p\mathcal P_{\mathsf K,\, \vec\l} = \sum_j \lambda_j d_p(\mathcal L_j) \neq 0$ for any $\vec\l \neq \vec 0$. So the hypersurface $V$ is nonsingular at each node $p$. Consider now the general case. If a variety $V \subset \P^n$, which contains supra-simplicial set $\mathsf A$, is given by homogeneous polynomials $P_1, \dots P_s$ of degrees $\leq d$, then by Lemma \[d\_exactly\], $\deg P_k = d$ for all $k$. By Theorem \[main\_th\], each $P_k = \mathcal P_{\mathsf K,\, \vec\l^{(k)}}$ for some choice of the nonzero vector $\vec\l^{(k)} = (\lambda_1^{(k)}, \dots , \lambda_n^{(k)}) \in \A^{n}$. For each node $p$, by the previous argument, $d_p P_k = \sum_j \lambda_j^{(k)} d_p\mathcal L_j$, where the differential $1$-forms $d_p \mathcal L_1, \dots , d_p\mathcal L_n$ on the affine chart $U_p$ are linearly independent at $p$. Therefore, when the vectors $\vec\l^{(1)}, \dots , \vec\l^{(s)}$ are linearly independent, so are the differentials $d_p P_1, \dots d_p P_s$ for all $p \in \mathsf N$. On the other hand, any dependence between $\vec\l^{(1)}, \dots , \vec\l^{(s)}$ leads to a *linear* dependence between the polynomials $P_1, \dots P_s$ and thus between their differentials $d_p P_1, \dots d_p P_s$. So we may drop all the linearly dependent polynomials from the list $\{P_1, \dots P_s\}$ to get a regular subsequence of degree $d$ elements for the ring $\A[y_0, \dots, y_n]/\mathcal I(V)$, where $\mathcal I(V)$ is the ideal of polynomials that vanish on $V$. The regularity of the new sequence follows from the $\A$-linear independence of elements of the same degree. Abusing notations, we denote the reduced list by $\{P_1, \dots P_s\}$. Thus $V$ is a complete intersection, and the hypersurfaces $H_1 :=\{P_1 = 0\}, \dots H_s :=\{P_s = 0\}$ are transversal at all the nodes of $\mathsf K$. As a result, $\dim(\mathsf{span}\{\vec\l^{(1)}, \dots , \vec\l^{(s)}\}) = \dim{T^\ast_pV}$ for any node $p$. Therefore $V$ is smooth in the vicinity of each node. The property of $V$ being a complete intersection of $H_1, \dots , H_s$, by the Bezout Theorem, implies that $\deg V = d^{n-\dim V}$. Theorem \[smooth\] forces the following obvious logical conclusion. \[not\_complete\] If a variety $V \subset \P^n$, given by homogeneous polynomials of degrees $\leq d$, is not a complete intersection, then it cannot be trapped in any $d^{\{n\}}$-cage in $\P^n$. $\diamondsuit$ [**Example 2.6.**]{} Since the twisted cubic curve $\mathcal C: [s: t] \to [s^3: s^2t : st^2: t^3]$ is not a complete intersection in $\P^3$, by Corollary \[not\_complete\], $\mathcal C$ does not contain the nodes of any $3^{\{3\}}$-cage $\mathsf K$ in $\P^3$, or even the nodes from a supra-simplicial set $\mathsf A \subset \mathsf K$. $\diamondsuit$ [**Example 2.7.**]{} Despite looking diverse, all the figures in this paper depict varieties, attached to the nodes of $d^{\{n\}}$-cages $\mathsf K(\mathbf Q)$ that are produced following a very simple recipe. It starts with a small set $\mathbf Q \subset \A^n$ of “nodes in the making" and uses the product structure in $\A^n$. Consider $d$ points $q_1, \dots, q_d \in \A^n$ such that, for each coordinate function $z_j: \A^n \to \A$, their $z_j$-coordinates are distinct. Let us denote by $\A(n, d)$ the space of such configurations $\mathbf Q := (q_1, \dots, q_d)$. Then each $\mathbf Q \in \A(n, d)$ produces a $d^{\{n\}}$-cage $\mathsf K(\mathbf Q) \subset \A^n$, formed by the hyperplanes $\{H_{j, i} := z_j^{-1}(z_j(q_i))\}_{j \in [1, n],\, i \in [1, d]}$. By Theorem \[main\_th\] and Theorem \[smooth\], for any $s \leq n$, the cage $\mathsf K(\mathbf Q)$ supports the family of varieties $X$ of the multi-degree $(\underbrace{d, \dots, d}_{s})$ and dimension $n-s$ that contain the node set $\mathsf N(\mathbf Q)$ of $\mathsf K(\mathbf Q)$. By Theorem \[unique\_hyper\] below, the family is parametrized by points of the Grassmanian $\mathsf{Gr}_\A(n, n-s)$. Over $\C$, we can enhance this cage construction. Consider the complex Viète map $\Sigma: \C^n \to \mathsf{Sym}^n\C \approx \C^n$, given by the elementary symmetric polynomials in $z_1, \dots, z_n$. It takes the “roots" $z_1, \dots, z_n \in \C$ to the coefficients of the monic polynomial $\prod_{j=1}^n (x - z_j)$ in the variable $x$. The complex Viète map is a smooth homeomorphism. We denote by $\mathcal D$ the hypersurface in $\mathsf{Sym}^n\C$, formed by the $x$-polynomials with multiple roots. It is called the *discriminant variety*. Remarkably, the $\Sigma$-images of the hyperplanes $\{H_{j, i} \subset \C^n \}$ are hyperplanes, *tangent* to $\mathcal D$; moreover, the normal vector to $\Sigma(H_{j, i})$, whose $n^{th}$ coordinate is $1$, has its $(n-1)^{st}$ coordinate equal to $z_j(q_i)$ ([@K2], Corollary 6.1)! Therefore, $\{\Sigma(H_{j, i})\}_{j, i}$ form a new $d^{\{n\}}$-cage $\Sigma(\mathsf K(\mathbf Q))$ in $\mathsf{Sym}^n\C \approx \C^n$, whose hyperplanes are tangent to $\mathcal D$. The nodes of $\Sigma(\mathsf K(\mathbf Q))$ reside in $\C^n$. Via the tangency property, the cage $\Sigma(\mathsf K(\mathbf Q))$ is completely determined by the configuration $\Sigma(\mathbf Q)$ of $d$ points in $\C^n \setminus \mathcal D$, since any point $p \in \C^n \setminus \mathcal D$ belongs to exactly $n$ hyperplanes that are tangent to $\mathcal D$ [@K2]. As a result, any generic (that is, of the form $\Sigma(\C(n, g))$) configuration $\mathbf P$ of points $p_1, \dots, p_d \in \C^n \setminus \mathcal D$ produces a $d^{\{n\}}$-cage $\mathsf K(\mathbf P)$ in $\C^n$, whose hyperplanes are alined with the tangent cones of $\mathcal D$. Again, for any $s \leq n$, the cage $\mathsf K(\mathbf P)$ supports a family of complex varieties $V$ of the multi-degree $(\underbrace{d, \dots, d}_{s})$ and dimension $n-s$ that contain the node set $\mathsf N(\mathbf P)$ of $\mathsf K(\mathbf P)$. By Theorem \[unique\_hyper\], the family is parametrized by points of the Grassmanian $\mathsf{Gr}_\C(n, n-s)$. Thus we got an effective device for producing varieties in cages. A configuration $\mathbf Q \in \C(n,d)$ or a configuration $\mathbf P \in \Sigma(\C(n,d))$, together with a choice of a $(n-s)$-dimensional affine subspace $\tau \subset \C^n$ at a point $q_1\in \mathbf Q$, or a $(n-s)$-dimensional affine subspace $\tilde\tau \subset \C^n$ at a point $p_1 \in \mathbf P$, produce unique varieties $X(\mathbf Q, \tau) \subset \C^n$ and $Y(\mathbf P, \tilde\tau) \subset \C^n_{\mathsf{coef}}$ of the dimension $n-s$ that are attached to the nodes of the two cages, respectively. Over the real numbers, the outcome is similar, if we consider only the camber $\mathcal C$ in the space $\R^n_{\mathsf{coef}}$ of monic real polynomials with all real roots; $\mathcal C$ is one of many chambers in which the real discriminant hypersurface $\mathcal D_\R$ divides $\R^n_{\mathsf{coef}}$. So, over $\R$, the cage-generating configuration $\mathbf P$ must be chosen in $\mathcal C$. The construction $(\mathbf Q, \tau) \Rightarrow X(\mathbf Q, \tau)$ has one pleasing property: if the configuration $\mathbf Q$ consists of $d$ points with all the coordinates in $\Z$ or $\Q$, then the variety $X(\mathbf Q, \tau)$ contains at least $d^n$ integral or rational points. Since the Viète map $\Sigma$ is given by elementary symmetric polynomials with integer coefficients, the same property holds for any variety $Y(\Sigma(\mathbf Q), \tilde\tau))$ that is attached to the nodes of the cage $\mathsf K(\Sigma(\mathbf Q)) \subset \mathsf{Sym}^n\C$. $\diamondsuit$ Let us recall few basic facts about the topology of complex projective spaces. The homology groups $H_i(\C\P^n; \Z) \approx \Z$ for all even $i \leq 2n$ and zero otherwise. Also, the homotopy groups $\pi_1(\C\P^n) \approx 0$, $\pi_2(\C\P^n) \approx \Z$, and $\pi_i(\C\P^n) \approx \pi_i(S^{2n+1})$ for all $i > 2$. Thus, $\pi_i(\C\P^n) = 0$ for all $i \in [3, 2n]$. Let $V \subset \C\P^n$ be a smooth subvariety, given by several homogeneous polynomial equations of degrees $\leq d$. If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ of a $d^{\{n\}}$-cage $\mathsf K \subset \C\P^n$, then - the homology groups $H_i(V;\Z) \approx H_i(\C\P^n; \Z)$ for all $i < \dim_\C V -1$, - the natural homomorphism $H_i(V;\Z) \to H_i(\C\P^n;\Z)$ is surjective for $i = \dim_\C V -1$, - $H_i(V;\Z) \approx H^{2n - 2 - i}(\C\P^n;\Z)$ for $i > n$, - the homotopy groups $\pi_i(V) \approx \pi_i(\C\P^n)$ for all $i < \dim_\C V$. By Theorem \[smooth\], $V$ is a complete intersection, defined by $s= \textup{codim}_\C(V,\, \C\P^n)$ linearly independent homogeneous polynomials $\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}}\}_{k \in [1, s]}$ of degree $d$. Put $N := C^d_{n+d} -1$. We apply the Veronese’s embedding $\beta: \C\P^n \to \C\P^N$ (see [@M], the text that follows Definition (6.9)) to $V$. Recall that $\b$ is given by the natural diagonal map $\P(\C^{n+1}) \to \P(\mathsf{Sym}^d(\C^{n+1}))$. Note that $\b(\C\P^n)$ is homeomorphic to $\C\P^n$, so their homology and homotopy are isomorphic. The image $\beta(V)$ is the intersection of $\b(\C\P^n)$ with $s$ hyperplane sections in $\C\P^N$, given by the polynomials $\big\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}}\big\}_{k \in [1, s]}$, the hyperplanes being in general position by the linear independence of $\big\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}}\big\}_{k \in [1, s]}$. Now, by applying the Lefschetz Hyperplane Theorems (see Corollary 7.3 and Theorem 7.4 in [@Mi]) iteratively to $\b(V)$, the claim follows. The application relies on $V$ being nonsingular. In order to validate the third bullet, we first apply the Poincaré duality $H_i(V; \Z) \approx H^{2\dim_\C(V) - i}(V; \Z)$, and then use the Lefschetz Hyperplane Theorems. Let us consider a $n$-dimensional polyhedron $\mathcal P$ in $\R^n$, whose combinatorics is modeled after the combinatorics of a $n$-cube. The opposite faces of $\mathcal P$ are labeled with the same color; so the total pallet has $n$ colors. We wish to place the vertexes of $\mathcal P$ on a given variety $V \subset \R^n$ that is defined as the zero set of several quadratic polynomials (think about $V$ as being an ellipsoid or a hyperboloid). The next corollary testifies that in order to accomplish this task, one needs to place just few vertexes of $\mathcal P$ on $V$, the rest of the vertexes will reside in $V$ automatically. Actually the following direct corollary of Theorem \[smooth\] makes sense over any infinite field $\A$. [**(Varieties in the Cube Cage)**]{} Let a variety $V \subset \P^n$ be given by homogeneous polynomials of degree $2$ (for big $n$, this is a weak restriction on $V$) and contains all $\frac{1}{2}(n^2 + n + 2)$ nodes of a supra-simplicial set $\mathsf A$ in a $2^{\{n\}}$-cage $\mathsf K \subset \P^n$. Then $V$ is a complete intersection of degree $2^s$, where $s = n - \dim(V)$. Moreover, $V$ contains all $2^n$ nodes of $\mathsf K$. $\diamondsuit$ [**Example 2.8.**]{} If a smooth curve $C \subset \P^3$ is given by two homogeneous quadratic forms and contains $7$ nodes of a $2^{\{3\}}$-cage $\mathsf K \subset \P^3$, then it contains the $8^{th}$ node of the cage. Moreover, $C$ is a complete intersection of degree $4$. In fact, such a curve $C$ is *elliptic* (i.e., smooth and of genus $1$). $\diamondsuit$ In particular, Theorem \[smooth\] claims that any variety $V \subset \P^n$ that is defined by polynomials of degrees $\leq d$ and contains the nodes of a $d^{\{n\}}$-cage is smooth in their vicinity! Therefore we get the following immediate corollary. \[distribution\] Let $V \subset \P^n$ be a subvariety, given by one or several homogeneous polynomial equations of degrees $\leq d$. If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ in a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, then $V$ defines a *distribution* $\tau_V$ of vector subspaces of dimension $\dim V$ in the tangent bundle $T(\P^n)$, being restricted to the node locus $\mathsf N$. $\diamondsuit$ \[unique\_hyper\] Consider a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$ and a vector subspace $\tau_p$ of dimension $n-s$ in the tangent space $T_p(\P^n)$, where $p$ is a node of $\mathsf K$. Then there exists a unique complete intersection $V \subset \P^n$ of the multi-degree $(\underbrace{d, \dots , d}_{s})$ and of dimension $n-s$ that contains all the nodes of $\mathsf K$ and whose tangent space $T_p(V) = \tau_p$. As a result, a supra-simplicial node set $\mathsf A \subset \mathsf K$ and a $(n-s)$-dimensional subspace $\tau_p \subset T_p(\P^n)$,[^7] where $p \in \mathsf N$, determines the variety $V$ and the *distribution* of $(n-s)$-subspaces $\tau_V$ in $T(\P^n)|_{\mathsf N}$ it produces. In other words, the cage $\mathsf K$, with the help of the inscribed $V$’s, defines canonically a “diagonal" embedding of Grassmanians $$\mathcal D_\mathsf K:\; \mathsf{Gr}_\A\big(T_p(\P^n),\, n-s\big) \to \prod_{q \in \mathsf N \setminus p} \mathsf{Gr}_\A\big(T_q(\P^n),\, n-s\big).$$ By the proof of Theorem \[smooth\], any variety $V$ that contains the supra-simplicial set $\mathsf A \subset \mathsf N$ and is defined by homogeneous polynomials of degrees $\leq d$ is actually defined by some linear independent polynomials $$\big\{P_k := \mathcal P_{\mathsf K,\, \vec\l^{(k)}} \; = \sum_{j \in [1, n]} \l_j^{(k)} \cdot \mathcal L_j\big\}_{k \in [1, s]},$$ where $s = n - \dim(V)$. In turn, such a collection of polynomials from the $d^{\{n\}}$-cage family is described by the linearly independent vectors $\{\vec\l^{(k)} := (\l_1^{(k)}, \dots \l_n^{(k)}) \in \A^n\}_{k \in [1, s]}$. Conversely, any such collection of polynomials $\{P_1, \dots, P_s\}$ produces a variety $V$ which, by Theorem \[smooth\], is complete intersection that contains all the nodes of the cage. We notice that any other choice of a basis $\{Q_1, \dots , Q_s\}$ in the space $W_{\mathsf K} := \mathsf{span}(P_1, \dots P_s)$ leads to the same variety $V = \{Q_1 = 0, \dots, Q_s = 0\}$. So, for any new basis $\{\vec\mu^{(1)}, \dots , \vec\mu^{(s)}\}$ of the space $\Lambda := \mathsf{span}_\A(\vec\l^{(1)}, \dots , \vec\l^{(s)})$, the two systems $\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}} = 0\}_{k \in [1, s]}$ and $\{\mathcal P_{\mathsf K,\, \vec\mu^{(k)}} = 0\}_{k \in [1, s]}$ share the same solution space $V \subset \P^n$. Therefore $V$ depends only on $\Lambda$, or rather, on its projectivization $\P(\Lambda)$. Let $\ell_p$ be the line in $\A^{n+1}$ that corresponds to a point $p \in \P^n$, and let $\ell_p^\perp$ be the complementary to $\ell_p$ subspace of $\A^{n+1}$. We may identify the cotangent space $T^\ast_p\P^n$ with $(\ell_p^\perp)^\ast$. Since the hyperplanes of distinct colors from $\mathsf K$ are in general position at the nodes $p \in \mathsf N$, the differentials $\{d \mathcal L_j \big |_{\ell_p^\perp}\}_{k \in [1, s]}$ are linearly independent on $\ell_p^\perp$. Thus $\P(\Lambda)$ is determined by the subspace $\mathsf{span}\{d P_k = \sum_j \lambda_j^{(k)} d \mathcal L_j \big |_{\ell_p^\perp}\}_{k \in [1, s]}$ of $T^\ast_p\P^n \approx (\ell_p^\perp)^\ast$ which is dual to the given subspace $\tau_p \subset T_p\P^n$, tangent to $V$ at $p$. Theorem \[unique\_hyper\] leads directly to the following special case. [**(The Cage Croquet[^8] Theorem)**]{} Given a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$ and a direction $\tau \in \P(T_p\P^n) \approx \P^{n-1}$ at one of the nodes $p \in \mathsf K$, there exists an algebraic curve $C$, a complete intersection of the multi-degree $(\underbrace{d, \dots , d}_{n-1})$, that passes through all the nodes of $\mathsf K$ and has $\tau$ as its tangent line at $p$. Moreover, $C$ is unique among the curves, given by homogeneous polynomials of degrees $\leq d$, that pass through the nodes of a supra-simplicial set $\mathsf A \subset \mathsf K$ in the direction of $\tau$. The curve $C$ is given by the equations $\big\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}} \; = 0 \big\}_{k \in [1, n-1]}$, where the vector space $\mathsf{span}\big\{d_p\mathcal P_{\mathsf K,\, \vec\l^{(k)}}\big |_{\ell_p^\perp}\big\}_{k \in [1, n-1]}$ is dual to the line $\tau \subset T_p(\P^n)$. $\diamondsuit$ Let us glance at $3$-dimensional cages and at the polyhedra that have their verticies at the nodes of these cages. A *tricolored* polyhedron $\mathcal P \subset \R^3$ is a polyhedron whose faces are colored with three colors. We say that a vertex $v$ of $\mathcal P$ is *trivalent* if exactly three distinctly colored faces join at $v$. Finally, a tricolored polyhedron is *trivalent* if all its vertices are. A *perfect* trivalent polyhedron is a trivalent polyhedron with equal number of faces, colored with each of the three colors. A cube is an example of a perfect trivalent polyhedron. Perfect trivalent polyhedra are rare. It seems that the only convex perfect trivalent polyhedra are combinatorially modeled after a cube. However, if we allow for *polyhedral surfaces* that are built from non-simply connected polygons, then numerous examples of perfect trivalent surfaces of high genus are available. The idea is to start with a perfect trivalent polyhedron $\mathcal P$ and to change it (in its perfect trivalent class) by a sequence of well-controlled surgery. One can use two basic elementary operations: [**(1)**]{} erecting a a prism with a quadrangular base from the interior of a face, and [**(2)**]{} connecting two similarly colored faces by a $1$-handle with a quadrangular section. To preserve trivalency, certain rules of coloring of the new appendices are forced upon us. For example, if we erect a prism from a red face, we have to color its top in red and its surface in blue and orange, following an alternating pattern. In the process, one red, two blue and two orange faces will be added to the original list of faces. Of course, this will violate the equilibrium between the number of red, blue and orange faces. To restore the balance of color, we erect one prism from a blue and one from an orange face. Now the new polyhedron is again perfect and trivalent. Note, that some faces of the new polyhedron are not simply-connected polygons! Similarly, if a $1$-handle with quadrangular section connects two red faces, we have to color its surface in blue and orange, following an alternating pattern. Again, to restore the balance of color, in addition, we attach one handle, connecting two blue faces, and one handle, connecting two orange ones. We notice that a generic perfect trivalent polyhedral surface with $3d$ faces determines a $d^{\{3\}}$-cage in the space. For example, a generic (disjoint) union of $k$ tricolored cubes in $\R^3$ is a perfect trivalent polyhedron that gives rise to a $(2k)^{\{3\}}$-cage in $\R^3$. In view of these observations, Theorem \[unique\_hyper\] leads to the following claim. Let $\Sigma \subset \R^3$ be a perfect trivalent polyhedral surface with $3d$ faces that generate a $d^{\{3\}}$-cage $\mathsf K_\Sigma$ in $\R^3$. Given a plane $\tau$ through one of vertices $v \in \Sigma$, there exists a unique surface $S$ of degree $d$ such that: - all the verticies of $\Sigma$ lie on $S$ (i.e., $\Sigma$ is inscribed in $S$), - $S$ contains all the nodes of the $d^{\{3\}}$-cage $\mathsf K_\Sigma$, - $S$ is tangent to the plane $\tau$ at $v$ and is smooth in the vicinity of all verticies of $\Sigma$. $\diamondsuit$ [**Example 2.9.**]{} The surface $\Sigma$ of a tricolored cube with three quadrangular wormholes that connect pairs of similarly colored opposite faces ($\Sigma$ is a surface of genus $3$) has $18 = 6 + 3\times 4$ faces (6 of which are not simply-connected polygons). It can be inscribed in a surface $S$ of degree $6 = 18/3$. In addition to the 32 vertices of $\Sigma$, lying on $S$, the rest of the nodes (numbering 184) of the $6^{\{3\}}$-cage $\mathsf K_\Sigma$ also belongs to $S$. Such a surface $S$ with a prescribed tangent plane $\tau$ at one vertex of $\Sigma$ is unique. $\diamondsuit$ cages on projective varieties and projective varieties in cages =============================================================== Let $X$ be a projective $n$-dimensional variety, equipped with a regular embedding $f: X \hookrightarrow \P^N$. Using $f$, we are going now to generalize the notion of a cage from a hyperplane configurations in projective spaces $\P^n$ to special configurations of positive codimension one divisors in $X$. \[cages\_on\_varieties\] Let us fix a regular embedding $f: X \hookrightarrow \P^N$ of a projective $n$-dimensional variety $X$ and a natural number $d$. We assume that the hyperplanes $\{H_{j, i}\}_{j \in [1, n],\, i \in [1,d]}$ in $\P^N$, given by linear homogeneous polynomials $\{L_{j, i}\}_{j \in [1, n],\, i \in [1,d]}$ in $N+1$ variables, are such that: - $\{H_{j, i} \cap f(X)\}_{j \in [1, n],\, i \in [1,d]}$ are subvarieties of $f(X)$ of dimension $n-1$, - in a Zariski open neighborhood of each node $x \in f(X)\, \bigcap \, \big(\bigcap_{j \in [1, n]} H_{j, i_j}\big)$, the variety $X$ is smooth, and the hyperplanes $\{H_{j, i_j}\}_{j \in [1, n]}$ are in general position (are maximally transversal) relative to each other and to $f(X)$, - the total number of such nodes is $\deg(f) \cdot d^n$.[^9] For each $j$, we consider the degree $d$ homogeneous polynomials $\mathcal L_j := \prod_{i \in [1,d]} L_{j, i}$ and denote by $\mathcal H_j$ the zero locus of $\mathcal L_j$ in $f(X)$. As before, we associate a distinct color $\a_j$ with each group of divisors $\{f(X) \cap H_{j, i}\}_{i \in [1,d]}$. We define $d^{\{n\}}$-*cage* $\mathsf K$ in $f(X)$ as the “colored" divisor $\sum_j \mathcal H_j$. By definition, its *nodes* are points of the locus $\mathsf N := f(X) \bigcap \big(\bigcap_{j \in [1, n]}\{\mathcal L_j = 0\}\big)$. $\diamondsuit$ For each $j$, we order the hyperplanes $\{H_{j, i}\}_{i \in [1,d]}$ of a particular color $\a_j$. Then there is a map $\e: \mathsf N \to \Z_+^n$ that associates the multi-index $\mathsf I = (i_1, \dots, i_n)$ to each node that belongs to the intersection $f(X) \cap H_{1, i_1} \cap \dots \cap H_{n, i_n}$. By Definition \[cages\_on\_varieties\], the image of $\e$ is the “cube" $[1, d]^n \cap \Z_+^n$, and the $\e$-fibers all are of the cardinality $\deg(f)$, since each projective subspace $H_{1, i_1} \cap \dots \cap H_{n, i_n} \subset \P^N$ is in general position with respect to $f(X)$ at their intersections; so by the transversality assumptions, $$\#\big\{f(X) \cap (H_{1, i_1} \cap \dots \cap H_{n, i_n})\big\} = \deg(f).$$ Now, with the help of $\e$, the definition of a *simplicial* and *supra-simplicial* sets of nodes of the cage $\sum_j \mathcal H_j \subset X$ are similar to Definition \[def2.1\]. \[new\_def\] A set of nodes $\mathsf T$ from $d^{\{n\}}$-cage $\mathsf K \subset X$ is called *simplicial* if, with respect to some orderings of the hyperplanes in each group $\{H_{j, i}\}_{i \in [1,d]}$, it is comprised of nodes $x \in \mathsf N$ whose $\e$-images are subject to the constraints $\|\e(x)\| \leq d + 1$. A set of nodes $\mathsf A$ from a cage $\mathsf K$ is called *supra-simplicial* if, with respect to some orderings of the hyperplanes in each group $\{H_{j, i}\}_{i \in [1,d]}$, it is comprised of nodes $x \in \mathsf N$, subject to the constraints $\| \e(x)\| \leq d + 2$. $\diamondsuit$ \[divisible\] We denote by $\mathcal I(f)\, \triangleleft \, \A[z_0, z_1 , \dots , z_{N}]$ the zero ideal of a projective variety $f: X \hookrightarrow \P^N$. Let $L(z_0, z_1 , \dots , z_{N})$ be a linear homogeneous polynomial. If a homogeneous polynomial $P(z_0, z_1 , \dots , z_{N})$ vanishes at the zero set $Z(L) \cap f(X)$ of $L$ in $f(X)$, then $P$ is divisible by $L$ in $\A[z_0, z_1 , \dots , z_{N}]/ \mathcal I(f)$. Let $(z_0: z_1 : \dots : z_{N})$ be homogeneous coordinates for $\P^{N}$. We may transform these coordinates linearly so that, in the new coordinates $(x_0: x_1 : \dots : x_{N})$, $L = x_0$. Then we may view $P$ as an element from the ring $\A[x_1, \dots, x_N][x_0]$. In fact, we may write $P = x_0 \cdot Q + R$, where $Q \in \A[x_0, x_1, \dots, x_N]$ is a homogeneous polynomial of degree $\leq \deg P -1$, and $R \in \A[x_1, \dots, x_N]$ is a homogeneous polynomial of degree $\leq \deg P$. By the hypotheses, $P|_{Z(L) \cap f(X)} =0$, which implies that $R|_{Z(L) \cap f(X)} =0$. However, $R$ is $x_0$-independent. Thus $R \in \mathcal I(f)$. So $P$ is divisible by $L$ in $\A[z_0: z_1 : \dots : z_{N}]/ \mathcal I(f)$. \[main\_thA\] Let $f: X \hookrightarrow \P^N$ be a regular embedding of a $n$-dimensional variety $X$. Assume that the homogeneous linear functions $\{L_{j, i}\}_{j \in [1, n],\, i \in [1,d]}$ in $N+1$ variables satisfy the properties from Definition \[cages\_on\_varieties\] and produce a $d^{\{n\}}$-cage $\mathsf K \subset f(X)$. Let a subvariety $V \subset f(X)$ be given as the common zero set of several homogeneous polynomials of degrees $\leq d$ in $N+1$ variables. - If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ in $\mathsf K$, then $V$ contains all the $\deg(f)\cdot d^n$ nodes of the cage. Moreover, any such subvariety $V \subset f(X)$ of codimension $s$ is the zero locus in $f(X)$ of the degree $d$ polynomials of the form $$\big\{\sum_j \lambda_{j, k} \cdot \mathcal L_j = 0\big\}_{\vec\l_k}$$ for an appropriate choice of linearly independent vectors $\vec\l_1, \dots, \vec\l_s \in \A^n$. In fact, $V$ is smooth in the vicinity of the node locus $\mathsf N \subset f(X)$. - In contrast, no variety $V \subset f(X)$, given as the intersection of $f(X)$ with the zero sets of homogeneous polynomials of degrees $\leq d$, contains all the nodes from a simplicial set $\tilde{\mathsf T}$ of any $(d+1)^{\{n\}}$-cage $\tilde{\mathsf K} \subset f(X)$. The general flow of the arguments follows closely the proof of Theorem \[main\_th\]. We assume that the statements of Theorem \[main\_thA\] have been validated for all regular embeddings $f: X \hookrightarrow \P^N$, where $\dim X < n$, and aim to justify the inductive step “$n-1 \Rightarrow n$". We adopt the notations from Definition \[cages\_on\_varieties\]. [*The base case*]{} $n =1$. Consider a projective curve $f: X \hookrightarrow \P^N$ of degree $\delta$. Let us choose some homogeneous linear polynomials $L_1, \dots L_d$ in $N+1$ variables so that their zeros $H_1, \dots H_d \subset \P^N$ will satisfy the following properties: the hyperplane $H_i$ misses the singularities of the curve $f(X)$, $H_i$ and $f(X)$ are transversal at $H_i \cap f(X)$ for all $i$, and $(f(X) \cap H_i) \cap (f(X) \cap H_{i'}) = \emptyset$ for all $i' \neq i$. Then the $d^{\{1\}}$-cage $\mathsf K$ is the $0$-divisor $\coprod_{i \in [1, d]} (f(X) \cap H_i)$. Its node locus $\mathsf N = \mathsf K$ is of cardinality $d \cdot \deg(f)$. The map $\e: \mathsf N \to \Z_+$ takes each node $x \in f(X) \cap H_i$ to $i \in \Z_+$. Thus the simplicial subset $\mathsf T$ and the supra-simplicial subset $\mathsf A$ both coincide with the node locus $\mathsf N$. Let $V$ be a $0$-dimensional subvariety of $f(X)$, given as the intersection of a degree $k$ hypersurface $\mathcal H \subset \P^N$ with $f(X)$, where $k \leq d$. Equivalently, $V$ is the zero set of a homogeneous polynomial $P$ of degree $k$ in $N+1$ variables, being restricted to the curve $f(X)$. By the Bézout Theorem (see [@R]), no such $V$ can contain more than $\deg(f) \cdot k$ points. Thus no such $V$ can contain $\mathsf T = \mathsf N$, unless $k = d$. In the latter case, if $V \supset \mathsf N$, then the two must coincide. Moreover, both polynomials, $P$ and $\mathcal L := \prod_{i=1}^d L_i$, of degree $d$ vanish at the same set of $\delta\cdot d$ nodes in $f(X)$. Therefore, $P|_{f(X)} = \lambda\cdot \mathcal L|_{f(X)}$, where $\lambda \neq 0$. $\diamondsuit$ [*The inductive step*]{} $n-1 \Rightarrow n$. We denote by $\mathcal I(f)$ the zero ideal in $\A[x_0, x_1, \dots , x_N]$ of the variety $f(X)$. We consider the slices of the subvariety $V \subset f(X)$ and of the $d^{\{n\}}$-cage $\mathsf K \subset f(X)$ by the hyperplanes $H_{1,1}, H_{1, 2}, \dots H_{1, d}$ in $\P^N$, labeled with the first color $\a_1$ and given by the linear homogeneous polynomials $L_{1,1}, L_{1, 2}, \dots L_{1, d} \in \A[z_0, z_1, \dots, z_N]$. Put $\mathcal L_j : = \prod_{i \in [1,d]} L_{j, i}$ and let $$X_i := f(X) \cap H_{1, i}, \quad V_i := V \cap H_{1, i} \subset X_i, \;\, \text{and} \;\,\mathsf K_i := X_i \cap \big(\sum_{j \geq 2} \{\mathcal L_j = 0\}\big).$$ By the general position (transversality) hypotheses that govern the intersections of the hyperplanes $H_{j, i}$ with $f(X)$ in the vicinity of the $\mathsf K$-nodes, each $\mathsf K_i$ is a $d^{\{n-1\}}$-cage for $X_i$. So we may consider the node sets $\mathsf N_i \subset \mathsf K_i$, together with its simplicial and supra-simplicial subsets $\mathsf T_i$ and $\mathsf A_i$. However, in general, $\mathsf A \cap H_{1, i}$ is smaller than $\mathsf A_i$! So, as in the proof of Theorem \[main\_th\], we will work with the slices $\mathsf A^{[i]} := \mathsf A \cap H_{1, i}$ and $\mathsf T^{[i]} := \mathsf T \cap K^{[i]}$. Note that the set $\mathsf T^{[i]}$ can serve as a simplicial set, and $\mathsf A^{[i]}$— as an supra-simplicial set for for the subcage $\mathsf K^{[i]}$. For any integer $i \in [1, d-1]$, we consider the $(d-i+1)^{\{n-1\}}$ sub-cage $\mathsf K^{[i]} \subset \mathsf K\, \cap \, X_i $, formed by the divisors $H_{1, i}\, \bigcap\, (\bigcup_{j \in [2,\,n],\; k \in [1,\, d-i+1]} H_{j, k})$ in $X_i$ (see Figure \[cage\]). We start with a homogeneous polynomial $P \in \A[z_0, z_1, \dots, z_N]$ of degree $\leq d$ that vanishes on $V$ and at all the nodes of a *supra-simplicial* set $\mathsf A$ of a $d^{\{n\}}$-cage $\mathsf K \subset X$. Consider the restriction of $P$ to the first slice $X_1$. Then $P$ vanishes at the supra-simplicial set $\mathsf A^{[1]} := \mathsf A \cap X_1$ of the induced $d^{\{n-1\}}$-cage $\mathsf K^{[1]} := \mathsf K \cap H_{1,1}$. We apply the $(n-1)^{st}$ induction assumption to $X_1 \subset H_{1,1} \approx \P^{N-1}$ to get the following conclusion: for some choice of the coefficients $\l_2^{[1]}, \dots \l_n^{[1]} \in \A$, the restriction $P|_{X_1}$ must be of the form $\mathcal P_1 := \sum_{j \in [2, n]} \l_j^{[1]} \cdot \mathcal L_j,$ a polynomial of degree $d$, being restricted to $X_1$. For this special choice of $(\l_2^{[1]}, \dots \l_n^{[1]})$, the difference $P - \mathcal P_1$ is identically zero on $X_1$. By Lemma \[divisible\], $P = \mathcal P_1 + L_{1,1}\cdot P_1 + R_1$, where $P_1$ is a homogeneous polynomial of degree $\leq d-1$ and $R_1 \in \mathcal I(f)$. In other words, $P = \mathcal P_1 + L_{1,1}\cdot P_1$ in $\A[z_0, z_1, \dots, z_N]/\mathcal I(f)$. Next, we consider the restrictions of $P$ and $P_1$ to the hyperplane $H_{1, 2} = \{L_{1,2} = 0\}$ of color $\a_1$. Since both $P$ and $\mathcal P_1$ vanish at the set $\mathsf A \cap X_2$ and, by Definition \[cage\], $L_{1,1} \neq 0$ at the points of $\mathsf A \cap X_2$, we conclude that $P_1|_{X_2}$ must vanish at the set $\mathsf A \cap X_2$ as well. Note that $\mathsf A \cap X_2 = \mathsf A^{[2]}$ is a *simplicial set* for the induced $d^{\{n-1\}}$-cage $\mathsf K^{[2]} \subset \mathsf K \cap X_2$. So by induction, any polynomial of degree $\leq d- 1$ that vanishes at a simplicial set $\mathsf A^{[2]}$ of the $d^{\{n-1\}}$-cage $\mathsf K^{[2]}$ must vanish at $X_2 := H_{1,2} \cap f(X)$. Applying Lemma \[divisible\] again, we get $P_1 = L_{1,2} \cdot P_2 \mod \mathcal I(f)$ for some homogeneous polynomial $P_2$ of degree $d-2$. So we get $$P = \mathcal P_1 + L_{1,1}\cdot L_{1,2} \cdot P_2 \mod \mathcal I(f).$$ Similarly, we argue that $P_2$ vanishes on the simplicial set $\mathsf A^{[3]} \subset \mathsf A \cap X_3$ of the $(d-1)^{\{n-1\}}$ -cage $\mathsf A^{[3]}$. Therefore $P_2|_{X_3}$ is zero, and $P_2 = L_{1,3} \cdot P_3 \mod \mathcal I(f)$ for a homogeneous polynomial $P_3$ of degree $d-3$. As a result, $P = \mathcal P_1 + L_{1,1}\cdot L_{1,2} \cdot L_{1,3} \cdot P_3 \mod \mathcal I(f)$. Continuing this reasoning, we get eventually $$P = \mathcal P_1 + \l(L_{1,1}\cdot L_{1,2}\cdot \dots \cdot L_{1,n}) \mod \mathcal I(f),$$ where $\l$ is a constant. Therefore, $P = \l\cdot \mathcal L_1 + \sum_{j \in [2, n]} \l_j^{[1]} \cdot \mathcal L_j \mod \mathcal I(f)$ is of the form $\mathcal P_{\mathsf K, \vec\l} \mod \mathcal I(f)$ and must vanish at every node of the $d^{\{n\}}$-cage $\mathsf K \subset X$. This completes the validation of the inductive step for the first bullet of the theorem. Now we will validate the second bullet of the theorem by a similar reasoning. So we take any homogeneous degree $\leq d$ polynomial $P \in \A[z_0, z_1, \dots, z_N]$ that vanishes on $V \subset f(X)$ and on a *simplicial* set $\tilde{\mathsf T}$ of a $(d+1)^{\{n\}}$-cage $\tilde{\mathsf K} \subset f(X)$. As before, we slice $f(X)$ and $\tilde{\mathsf K}$ by the hyperplanes $\{H_{1, i }\}_{i \in [1, d+1]}$ of the color $\a_1$. Now all the slices $\tilde{\mathsf T}^{[i]}$ (including the first one!) are simplicial sets in $\tilde{\mathsf K}^{[i]}$. The latter locus $\tilde{\mathsf K}^{[i]}$ is the union of the zero sets of the degree $d-i+2$ homogeneous polynomial $\tilde{\mathcal L}^{[i]}_j \, := \prod_{k \in [1,\, d-i+2]} L_{j, k}.$ Since $P$ vanishes at $\tilde{\mathsf T}^{[1]}$, by the $(n-1)^{st}$ induction hypotheses, $P|_{X_1} = 0$. By Lemma \[divisible\], this implies that $P = L_{1,1}\cdot P_1 \mod \mathcal I(f)$, where $P_1$ is a homogeneous polynomial of degree $d-1$. The set $\tilde{\mathsf T}^{[2]}$ is simplicial in the cage $d^{\{n-1\}}$-cage. Since $L_{1,1}|_{\tilde{\mathsf T}^{[2]}} \neq 0$, we get that $P_1$ must vanish at the nodes from $\tilde{\mathsf T}^{[2]}$. By induction, this implies that $P_1|_{X_1} = 0$ and thus $P_1$ is divisible by $L_{1,1}$ modulo $\mathcal I(f)$. So $P= L_{1,1}\cdot L_{1,2} \cdot P_2 \mod \mathcal I(f)$ for a polynomial $P_2$ of degree $d-2$. Continuing this process, we get that $P= \l \cdot L_{1,1}\cdot L_{1,2}\cdot\, \dots, \cdot L_{1,d}$ must vanish at the nodes of the set $\tilde{\mathsf T}^{[d+1]}$. Since $L_{1,1}, \dots, \ L_{1,d}$ do not vanish at $\tilde{\mathsf T}^{[d+1]}$, this forces $\l = 0$, and so $P$ is identically zero on $X$. This completes the validation of the second bullet from Theorem \[main\_thA\]. Therefore any subvariety $V \subset f(X)$ of codimension $s$ that is given by homogeneous polynomials of degree $\leq d$ contains the nodes of a supra-simplicial set $\mathsf A \subset f(X)$ is the intersection of $f(X)$ with the zero locus of polynomials of the form $\{\sum_j \lambda_{j, k} \cdot \mathcal L_j = 0\}_{\vec\l_k}$, $k \in [1, s]$, for an appropriate choice of linearly independent vectors $\vec\l_1, \dots, \vec\l_s \in \A^n$. So $V$ is a “complete intersection" in $X$ of the “inner" multi-degree $(\underbrace{d, \dots, d}_{s})$. The smoothness of $V$ in the vicinity of each node $p$ follows from a local calculation (in an affine chart on $f(X)$ that contains $p$) as in the proof of Theorem \[main\_th\]. ![[]{data-label="Clebsch"}](Clebsch_surface.pdf){width="50.00000%"} The next example illustrates the contrast between varieties *in* cages and cages *on* varieties. [**Example 3.1.**]{} The Clebsch surface $X \subset \C\P^4$ is defined by two homogeneous equations: $$\{x_0 + x_1 + x_2 + x_3 + x_4 = 0,\quad x_0^3 + x_1^3 + x_2^3 + x_3^3 + x_4^3 = 0\}.$$ The symmetric group $\mathsf S_5$ of order $120$, acting by permutations of the coordinates in $\C\P^4$, acts on $X$ as well. Since $X$ contains *only* $27$ lines, their configuration is $\mathsf S_5$-invariant[^10]. Moreover, all $27$ lines reside in the real slice $X \cap \R\P^4$ of $X$! Eliminating $x_0$ shows that $X$ is also isomorphic to the cubic surface $Y \subset \C\P^3$, given by $$x_1^3+x_2^3+x_3^3+x_4^3=(x_1+x_2+x_3+x_4)^3.$$ Consider the $\mathsf S_5$-invariant hypersurface $\mathbf S \subset \C\P^4$, given by the equation $\{x_0^3 + x_1^3 + x_2^3 + x_3^3 + x_4^3 = 0\}$. Using the algebraic trick from Example 2.3, we see that $\mathbf S$ is attached to the nodes of the $\mathsf S_4$-invariant $3^{\{4\}}$-cage $\mathsf K \subset \C\P^4$, given by $$\big\{\mathcal L_1 = 0\big\}\, \bigcup \, \big\{\mathcal L_2 = 0\big\}\, \bigcup\, \big\{\mathcal L_3 = 0\big\}\, \bigcup \, \big\{\mathcal L_4 = 0\big\},$$ where $$\mathcal L_1 := \prod_\a(x_1- \a x_0),\; \mathcal L_2 := \prod_\b(x_2- \b x_0), \; \mathcal L_3 := \prod_\g(x_3- \g x_0),\;\mathcal L_4 := \prod_\delta(x_4- \delta x_0)$$ and $\a, \b, \g, \delta$ run independently over the triple of complex degree $3$ roots of $-1/4$. The $\mathsf S_4$-action on the cage is induced by permutations of the coordinates $(x_1, x_2, x_3, x_4)$. The $\mathsf S_5$-invariant hypersurface $\mathbf S$ contains all the $81$ nodes of the $\mathsf S_4$-invariant cage $\mathsf K$ (so $\mathbf S$ is inscribed in at least $5 = (5!)/(4!)$ such cages). However, the Clebsch surface $X$ does not contain any node of $\mathsf K$, since such nodes would have to reside in the hyperplane $\{x_0 + x_1 + x_2 + x_3 + x_4 = 0\}$ and would be labeled by the $4$-tuples of roots $(\a, \b, \g, \delta)$ that satisfy the equation $\{\a+\b+\g+\delta = -1\}$. The latter equation has no solutions among the complex cubic roots of $-1/4$. So $X$ is not inscribed in $\mathsf K$. On the other hand, any *pair* of cubic polynomials from the set $\{\mathcal L_1, \mathcal L_2, \mathcal L_3, \mathcal L_4\}$ does define a $\mathsf S_2$-invariant $3^{\{2\}}$-cage $\mathsf K^\bullet$ (in the sense of Definition \[cages\_on\_varieties\]) *on* the surface $X$. Since $\deg X =3$, the cage $\mathsf K^\bullet$ has $27\, (= 9 \times 3)$ nodes. By Theorem \[main\_thA\], any curve $C \subset X$, given by an additional polynomial of degree $3$ and such that $C$ contains $24\, (= 8 \times 3)$ nodes of a supra-simplicial set $\mathsf A^\bullet \subset \mathsf K^\bullet$, will contain all $27$ nodes of $\mathsf K^\bullet$. $\diamondsuit$ It is possible to introduce the notion of a cage from Definition \[cages\_on\_varieties\] and to reformulate Theorem \[main\_thA\] in intrinsic terms of a variety $X$, equipped with some special line bundle $\xi$, a bundle that facilitates a regular embedding $f: X \hookrightarrow \P^N$. Let $\Xi$ be the Hopf line bundle (commonly denoted as $\mathcal O(1)$) over $\P^N$. The total space of $\Xi$ is $\P^{N+1} \setminus \infty$. Let $\xi$ be a *very ample* line bundle over an algebraic variety $X$. As a working definition, this means that, for some regular embedding $f: X \hookrightarrow \P^N$, $\xi : = f^\ast(\Xi)$, the pull-back of the Hopf line bundle $\Xi$ under $f$. We denote by $\xi^{\otimes k}$ the tensor product $\underbrace{\xi \otimes \dots \otimes \xi}_{k}$ of the line bundle $\xi$. In what follows, we rely on the basic relation between homogeneous polynomials $P$ of degree $d$ in the variables $(z_0, \dots, z_N)$, the positive divisors $Z(P)$—the zero loci of $P$ in $\P^N$—, and the $P$-induced sections $\s_P$ of the line bundle $\mathcal O(d) = \mathcal O(1)^{\otimes d} \;(= \Xi^{\otimes d})$ over $\P^N$. In the affine charts $\{U_i := \{z_i \neq 0\} \subset \P^N\}_{i \in [0, N]}$, the bundle $\mathcal O(d)$ is produced by the cocycle $\{f_{ij}: U_i \cap U_j \to \C^\ast\}_{ij}$, given by the meromorphic functions $f_{ij} = (z_i/z_j)^d$. Then, in the chart $U_i$, the section $\s_P: U_i \to \C$ is given by the formula $\s_P(z_0/z_i, \dots, z_N/z_i) = z_i^{-d}P(z_0, \dots, z_i, \dots z_N)$. The zero locus of $\s_P$ coincides with the zero locus of $P$. \[cages\_on\_varieties\_B\] We assume that a very ample line bundle $\xi$ over a $n$-dimensional variety $X$ admits sections $\{\s_{j, i}\}_{j \in [1, n],\, i \in [1,d]}$ such that: - the zero loci of $\{\s_{i, j}\}$ define subvarieties $\{H_{j, i} \subset X\}_{j \in [1, n],\, i \in [1,d]}$ of dimension $n-1$, - the $d^{\{n\}}$-cage $\mathsf K \subset X$ is the $j$-ordered union $\bigcup_{j \in [1, n]} \big(\bigcup_{i \in [1,d]} H_{j, i}\big)$, - the nodes of $\mathsf K$ are the points of $X$ that belong to the intersections of any subset of $n$ divisors $\{H_{j, i_j}\}_{j \in [1, n]}$, labeled with distinct $j$’s, - in a Zariski open neighborhood of each node $p \in \bigcap_{j \in [1, n]} H_{j, i_j}$, the variety $X$ and its subvarieties $\{H_{j, i_j}\}_{j \in [1, n]}$ are smooth and in general position (are maximally transversal), - the total number of such nodes is $\delta \cdot d^n$, where $\delta := \#(\bigcap_{j \in [1, n]} H_{j, i_j})$ is independent on the choice of $\{i_j\}_i$ for each $j$.[^11] For each $j$, we consider the product $\mathcal L_j := \bigotimes_{i \in [1,d]} \s_{j, i}\in \Gamma(\xi^{\otimes d})$, a section of the line bundle $\xi^{\otimes d}$, and denote by $\mathcal H_j$ its $(n-1)$-dimensional positive divisor $\sum_{i} H_{j, i}$ in $X$, the zero set of the section $\mathcal L_j$. With these notations in place, the $d^{\{n\}}$-cage $\mathsf K$ in $X$ is the “$j$-colored" divisor $\sum_j \mathcal H_j$. By definition, its nodes are points of the locus $\mathsf N := \bigcap_{j \in [1, n]} \mathcal H_j$. $\diamondsuit$ With the help of $\e: \mathsf N \to \Z_+^n$, the notions of a simplicial and supra-simplicial subsets of $\mathsf N$ are the same as in the paragraph that follows Definition \[cages\_on\_varieties\]. We may now restate Theorem \[main\_thA\] in the new intrinsic terms: \[main\_thB\] Assume that a a very ample line bundle $\xi$ over a $n$-dimensional variety $X$ admits sections $\{\s_{j, i}\}_{j \in [1, n],\, i \in [1,d]}$ that satisfy the properties from Definition \[cages\_on\_varieties\_B\] and produce a $d^{\{n\}}$-cage $\mathsf K \subset X$. Let a subvariety $V \subset X$ be given as the common zero set of several sections of the line bundles $\{\xi^{\otimes k}\}_k$, where $k \leq d$. - If $V$ contains all the nodes from a supra-simplicial set $\mathsf A$ in $\mathsf K$, then $V$ contains all the $\delta\cdot d^n$ nodes of the cage[^12]. Moreover, any such subvariety $V \subset X$ of codimension $s$ is the zero locus of $s$ sections of the form $\{\sum_j \lambda_{j, k} \cdot \mathcal L_j = 0\}_{\vec\l_k}$ of the bundle $\xi^{\otimes d}$ for an appropriate choice of linearly independent vectors $\vec\l_1, \dots, \vec\l_s \in \A^n$. In fact, $V$ is smooth in the vicinity of the node locus $\mathsf N \subset X$. - In contrast, no such variety $V$ contains all the nodes from a simplicial set $\tilde{\mathsf T}$ of any $(d+1)^{\{n\}}$-cage $\tilde{\mathsf K} \subset X$, produced by sections $\{\tilde\s_{j, i}\}_{j \in [1, n],\, i \in [1,d+1]}$. $\diamondsuit$ We conclude this section with an observation which is a bit in the spirit of Theorem \[main\_thA\], although it deals not with $d^{\{\dim X\}}$-cages (as the theorem does), but with $d^{\{\dim X +1\}}$-cages in the ambient to $X$ space $\P^{\,\dim X +1}$. The observation connects the special varieties $X$, containing all the nodes of a cage $\mathsf K$, which resides *in* an ambient to $X$ projective space, and the several cages $\{\mathsf K_k\}$ *on* $X$ that are formed as “partial traces" of $\mathsf K$ in $X$. Thus, the next lemma is a bridge between the varieties in cages and the cages on varieties (see Example 3.1 that illustrates this distinction). \[in\_and\_on\] Let $\mathsf K \subset \P^N$ be a $d^{\{N\}}$-cage, that is, the union of zero sets $\{\mathcal H_j\}_{j \in [1, N]}$ of some completely factorable degree $d$ homogeneous polynomials $\{\mathcal L_j\}_{j \in [1, N]}$ in $N+1$ variables, so that the hypersurfaces $\{\mathcal H_j\}$ are smooth and in general position at the locus $\mathsf N := \bigcap_{j \in [1, N]} \mathcal H_j$. Let $\kappa$ denote a subset $\{j_1, \dots , j_n\} \subset \{1, \dots , N\}$[^13]. If a variety $X \subset \P^N$ of dimension $n$ is given by homogeneous polynomials of degrees $\leq d$ and contains the all the nodes of a supra-simplicial set $\mathsf A \subset \mathsf N$, then the cage $\mathsf K$ induces $C_N^n$ cages $\{\mathsf K_\kappa^X\}_{\kappa}$ on $X$. Each such $d^{\{n\}}$-cage $\mathsf K_\kappa^X$, labeled by $\kappa$, is formed as $X \bigcap \big(\bigcup_{j \in \{j_1, \dots , j_n\}}\mathcal H_j\big)$. Moreover, all the cages $\{\mathsf K_\kappa^X\}_{\kappa \subset \{1, \dots , N\}}$ share the same set of nodes with $\mathsf K$. By Theorem \[main\_th\], if $X$ contains $\mathsf A$, it contains all the nodes of $\mathsf K$. Consider two sets of indices, $\kappa :=\{j_1, \dots , j_n\} \subset \{1, \dots , N\}$ and $\nu := \{i_{j_1}, \dots , i_{j_n}\} \subset [1, d]^n$. For each $\kappa, \nu$, we form $Q_{\kappa, \nu} := \bigcap_{j \in \{i_1, \dots , i_n\}} H_{j, i_j}$, a $(N-n)$-dimensional projective subspace of $\P^N$. Consider the intersections $X_{\kappa, \nu} := X \cap Q_{\kappa, \nu}$ and $\mathsf N_{\kappa, \nu} := \mathsf N \cap \mathcal Q_{\kappa, \nu}$. Since $X \supset \mathsf N$, we get $X_{\kappa, \nu} \supset \mathsf N_{\kappa, \nu}$. Our goal is to show that $X_{\kappa, \nu} = \mathsf N_{\kappa, \nu}$. Counted with multiplicities, the intersection $X \odot Q_{\kappa, \nu} = \deg X$. By Theorem \[smooth\], $X$ is transversal to $H_{\kappa, \nu}$ at the locus $\mathsf N_{\kappa, \nu}$. Thus $X \odot Q_{\kappa, \nu} = \deg X \geq |\mathsf N_{\kappa, \nu}|$. Again, by Theorem \[smooth\], $X$ is a complete intersection of the multi-degree $(\underbrace{d, \dots, d}_{N-n})$. (Moreover, $X$ is smooth in the vicinity of $\mathsf N$.) Hence, $d^{N-n} \geq |\mathsf N_{\kappa, \nu}|$. On the other hand, the nodes of $\mathsf N_{\kappa, \nu}$ are formed by intersecting $Q_{\kappa, \nu}$ with various projective subspaces $Q_{\mathbf c\kappa,\, \mu}$, where $\mathbf c\kappa$ denotes the complementary to $\kappa$ in $\{1, \dots , N\}$ set of indexes, and $\mu \in [1, d]^{N-n}$. So $|\mathsf N_{\kappa, \nu}| = d^{N-n}$. As a result, $X_{\kappa, \nu} = \mathsf N_{\kappa, \nu}$. Therefore all the cages $\{\mathsf K_\kappa^X\}_{\kappa \subset \{1, \dots , N\}}$ share the same set of nodes with $\mathsf K$! Applying Theorem \[smooth\] and Theorem \[main\_thB\], we see that the general position requirements from Definition \[cages\_on\_varieties\] are satisfied for all the cages $\{\mathsf K_\kappa^X\}_{\kappa \subset \{1, \dots , N\}}$. [**Example 3.1.**]{} Consider a $d^{\{4\}}$-cage $\mathsf K \subset \P^4$, and a surface $X \subset \P^4$ that is given by homogeneous polynomial equations of degrees $\leq d$ in $5$ variables. Assume that $X$ contains all the nodes of $\mathsf K$. By Theorem \[smooth\], the surface $X$ is a complete intersection of the multi-degree $(d, d)$. Then $X$ is attached to the nodes of *six* $d^{\{2\}}$-cages $\{\mathsf K_\kappa^X\}_{\kappa}$ (in the sense of Definition \[cages\_on\_varieties\]) that are induced by $\mathsf K$, each one containing the same set of $d^2 \times \deg X = d^4$ nodes. $\diamondsuit$ Lemma \[in\_and\_on\] leads instantly to the following claim. \[in\_and\_on\_for\_hyper\] Let $\mathsf K \subset \P^N$ be a $d^{\{N\}}$-cage, defined as the union of zero sets $\{\mathcal H_j\}_{j \in [1, N]}$ of the completely factorable homogeneous polynomials $\{\mathcal L_j\}_{j \in [1, N]}$ of degree $d$ in $N+1$ variables. If a hypersurface $X \subset \P^N$ of degree $d$ contains all the nodes of a supra-simplicial set $\mathsf A$ of the cage $\mathsf K$, then $\mathsf K$ induces $N$ cages $\{\mathsf K_k^X\}_{k \in [1, N]}$ on $X$; each such $d^{\{N-1\}}$-cage $\mathsf K_k^X$ on $X$ is defined as the union over $j \neq k$ of the zero sets of degree $d$ completely factorable polynomials $\{\mathcal L_j\}_{j \neq k}$, being restricted to $X$. Moreover, all the cages $\{\mathsf K_k^X\}_{k \in [1, N]}$ share the same set of nodes with $\mathsf K$.$\diamondsuit$ So a smooth generic hypersurface $X \subset \P^n$ of degree $d$, trapped in a $d^{\{n\}}$-cage $\mathsf K$, inherits a remarkable web $\mathsf W_{\mathsf K}$ of subvarieties (residing in $X$). The web $\mathsf W_{\mathsf K}$ is produced by intersecting $X$ with the $nd$ hyperplanes, forming the cage. (A similar pattern of subspaces is produced by intersecting the grid of coordinate hyperplanes in the first quadrant of the $n$-dimensional integral lattice $\Z_+^n$ with the hyperplane $\{x_1 + \dots + x_n = d\}$.) Let us describe the web $\mathsf W_{\mathsf K}$ in some detail when $X$ is a surface (see Figure \[web\]). The next proposition is a direct reformulation of Corollary \[in\_and\_on\_for\_hyper\]. ![[]{data-label="web"}](web.pdf){width="42.00000%"} A an algebraic surface $X \subset \P^3$ of degree $d$ that contains a supra-simplicial set of nodes of a $d^{\{3\}}$-cage $\mathsf K\subset \P^3$ carries a web $\mathsf W_{\mathsf K}$ of $d$ “red", $d$ “blue" and $d$ “orange" algebraic curves with the following properties: - each curve from $\mathsf W_{\mathsf K}$ is the intersection of the surface $X$ with a plane $\P^2 \subset \P^3$, so that each curve carries exactly $d^2$ tricolored intersections (nodes), - every two curves of distinct colors intersect each other transversally at exactly $d$ points, which all happen to be the tricolored, - each triple of curves of distinct colors has a single intersection point, and the whole web $\mathsf W_{\mathsf K}$ has $d^3$ tricolored intersections, - each curve is smooth in vicinity of each node. $\diamondsuit$ How algebraic beasts degenerate in cages ======================================== In 1640, Blaise Pascal, 16 years old, discovered a remarkable property of a hexagon inscribed in a circle. Pascal’s Theorem (see [@Ki], [@R]) was one of the first fundamental results in geometry, unknown to the Classical Greek school. Pascal called his theorem “*Mystic Hexagon Diagram*". In the words of Fermat, “*We learned that the ancient Greeks didn’t know everything about geometry*". Pascal’s Theorem and the Desargues’s Theorem (discovered four years earlier) gave birth to a new branch of non-metric geometry which we now call Projective Geometry. The following theorem (Theorem 3.3 from [@K]) animates our considerations in this section. It is a generalization of the Pascal Theorem about hexagons, inscribed in a plane quadric, to $2d$-gons, inscribed in a plane quadric. \[2d-polygon\][**(Mystic $2d$-gon Diagram)**]{} Let $\mathcal P$ be a polygon with $2d$ sides[^14], colored with two alternating colors, and inscribed into a quadratic curve $Q$ that resides in an affine or projective plane, so that a $d^{\{2\}}$-cage $\mathsf K$ is generated. Then all $d^2 - 2d$ new nodes of $\mathsf K$ lie on a plane curve $Q^\ast$ of degree $d-2$. $\diamondsuit$ Therefore if a $d^{\{2\}}$-cage $\mathsf K$ is such that its $2d$ nodes reside on a quadratic curve, then the family of degree $d$ plane curves through the nodes of $\mathsf K$ contains an interesting *reducible* curve: one of its component is quadratic. That curve is distinct from the two completely reducible curves, represented by the unions of $d$ lines of the same color that form the cage. Contemplating about Theorem \[2d-polygon\], one might wonder which plane curves $Q_\star$ of degree $d - 2$ can be produced via the Mystic $2d$-gram construction from a $2d$-gons, inscribed in a given quadratic $Q$? Clearly, for big $d$, such curves $Q_\star$ will be very exceptional. However, for a few small $d$, we might have a chance to manufacture almost any plane curve of degree $d - 2$ as a $Q_\star$ from Theorem \[2d-polygon\]. A naive dimensional analysis will do a crude selection. The space of plane curves of degree $k$ is $[k(k+3)/2]$-dimensional. For $d = 3$, the space of hexagons, inscribed in a given quadratic, is $6$-dimensional, and the space of lines is $2$-dimensional. For $d = 4$, the space of octagons, inscribed in a given quadratic, is $8$-dimensional, and the space of quadratic curves is $5$-dimensional. For $d = 5$, the space of dodecagons, inscribed in a given quadratic, is $10$-dimensional, and the space of cubics is $9$-dimensional. So far, so good! Already for $d = 6$, the dimension of the inscribed $12$-gones is $12$, and the dimension of quartics is $14$. So, not any quartic can be of the form $Q_\star$ for a *fixed* $Q$. However, if we allow to vary the quadratic curve $Q$ as well, we gain $5$ extra-degrees of freedom. This might take us just through the next case $d = 7$: the space of quintics is $19$-dimensional, and $19 = 14 + 5$. Already for $d > 7$, the realizable $Q_\star$’s will form a subvariety in the space of curves of degree $d - 2$. Of course, we do not claim that this dimensional count proves the realization theorems for small $d$’s. It is curious to notice a very special role, played by quadratic curves in Theorem \[2d-polygon\]. As another dimensional analysis shows, *generically*, nothing can be claimed about $2d$-gones inscribed in curves of degrees $u$, when $2 < u < d$; no valid Pascal’s Theorems inhabit that range! Nevertheless, the situation is better than one might think, if we are willing to abandon the inscribed $2d$-gones in favor of more intricate subsets of cages. ![[]{data-label="4x4degeneration"}](deg4by4.pdf){width="28.00000%"} Let us move now towards multidimensional generalizations of Theorem \[2d-polygon\]. Given a variety $V \subset \P^k \subset \P^n$, and a point $p \in \P^n \setminus \P^k$, we form a new variety $\mathsf{con}(V, p)$, the union of all lines $\P^1$ in $\P^n$ that contain the tip $p$ and a point of $V$. For a given a $d^{\{k\}}$-cage $\mathsf K \subset \P^k \subset \P^n$, let us form two cones, $\mathsf{con}(\mathsf K, p)$ and $\mathsf{con}(\P^k, p) \approx \P^{k+1}$. We denote by $\mathsf N$ the node set of $\mathsf K$. Let us add $k$ hyperplanes $H_1, \dots, H_k \subset \mathsf{con}(\P^k, p)$ so that, together with the hyperplane $\P^k \subset \mathsf{con}(\P^k, p)$ and the hyperplanes from $\mathsf{con}(\mathsf K, p)$, a $d^{\{k+1\}}$-cage $\mathsf K^\#$ in $\mathsf{con}(\P^k, p)$ is formed (see Figure \[pyramid\]). The hyperplanes from $\mathsf{con}(\mathsf K, p)$ inherit the $k$ colors from $\mathsf K$, while the hyperplanes $\P^k, H_1, \dots, H_k$ are labeled with a new color. The choice $H_1, \dots, H_k \subset \mathsf{con}(\P^k, p)$ that generate $\mathsf K^\#$ is generic. More accurately, it can be described as follows: first, we pick $H_1$ that is transversal to the $k$-dimensional hyperplanes from $\mathsf{con}(\mathsf K, p)$ and does not contain any nodes from $\mathsf K$. Such a choice of $H_1$ adds $d^k$ new nodes to the cage $\mathsf K^\#$ under construction. Let us call them $\mathsf N_1$. Then we pick $H_2$ that is transversal to the $k$-dimensional hyperplanes from $\mathsf{con}(\mathsf K, p)$ and does not contain any nodes from $\mathsf N \coprod \mathsf N_1$. Such a choice of $H_1$ adds another $d^k$ new nodes to $\mathsf K^\#$ under construction. Proceeding in this way, we produce the desired $d^{\{k+1\}}$-cage $\mathsf K^\#$ in $\mathsf{con}(\P^k, p)$. We call the cage $\mathsf K^\#$ *a tower with the base* $\mathsf K$. This construction leads instantly to the following simple lemma and its corollaries. \[con\] For a variety $V \subset \P^k$ that contains the node locus $\mathsf N$ of a cage $\mathsf K \subset \P^k$, the variety $\mathsf{con}(V, p)$ contains the node locus $\mathsf N^\#$ of the cage tower $\mathsf K^\# \subset \mathsf{con}(\P^k, p) \approx \P^{k+1}$. If $V$ is a reducible variety, so is $\mathsf{con}(V, p)$. $\diamondsuit$ For any variety $V \subset \P^k$ that is given by homogeneous polynomials of degree $\leq d$ and contains the nodes of a supra-simplicial set $\mathsf A$ of a $d^{\{k\}}$-cage $\mathsf K \subset \P^k$, the variety $\mathsf{con}(V, p)$ contains the entire node locus $\mathsf N^\#$ of the cage tower $\mathsf K^\# \subset \mathsf{con}(\P^k, p) \approx \P^{k+1}$. Moreover, any degree $d$ hypersuface $W \subset \mathsf{con}(\P^k, p)$ that contains a supra-simplicial set $\mathsf A^\#$ of the cage tower $\mathsf K^\# $ and is tangent to the cone $\mathsf{con}(V, p)$ at some node of $\mathsf K^\# $ coincides with the cone $\mathsf{con}(V, p)$. If $V$ contains all the nodes of $\mathsf K$, by the nature of the tower construction, $\mathsf{con}(V, p)$ contains all the nodes of $\mathsf K^\#$. Now, by Theorem \[main\_th\], the first claim follows. The second claim of the corollary follows from Theorem \[unique\_hyper\]. Lemma \[con\] provides us with a recipe for generating an vast array of very special high dimensional cages that admit interesting reducible hypersurfaces (different from the unions of monocolored hyperplanes), nailed to their nodes. Unfortunately, I do not know any other mechanism that have the same property. By iterating the cone construction $\{V \Rightarrow \mathsf{con}(V, p)\}$ $s$ times, we may produce a $(\dim V +s)$-dimensional variety $\mathsf{con}^s(V, \vec p) \subset \P^{k+s}$ of degree $\deg V$. \[cone\_Pascal\] Let $\mathcal P$ be a $2d$-gone, inscribed in a quadratic plane curve $Q \subset \P^2$ and let $\mathsf K$ be the $d^{\{2\}}$-cage that $\mathcal P$ generates. (By Theorem \[2d-polygon\], the remaining $d^2 - 2d$ nodes of $\mathsf K$ lie of a curve $Q^\ast$ of degree $d-2$.) Then the $s$ times iterated $d^{\{2+s\}}$ tower cage $\mathsf K^{s\#} \subset \P^{2+s}$ contains a reducible hypersurface of degree $d$. It is a union of a $s$ times iterated quadratic cone $\mathsf{con}^s(Q, \vec p)$ and a $s$ times iterated cone $\mathsf{con}^s(Q^\ast, \vec p)$, a hypersurface of degree $d-2$. $\diamondsuit$ A special case of Corollary \[cone\_Pascal\] leads to the first appearance of the *spatial* Pascal Theorem, shown in Figure \[pyramid\][^15]: ![[]{data-label="pyramid"}](cone.pdf){width="40.00000%"} [**(Mystic Hexagonial Pyramid Diagram)**]{}\[spatial\_Pascal\] Let $\mathcal P$ be a hexagon, inscribed in a quadratic plane curve $Q \subset \P^2$, and let $\mathsf K \subset \P^2$ be the $3^{\{2\}}$-cage that $\mathcal P$ generates. Consider a quadratic cone $\mathsf{con}(Q, p) \subset \P^3$ with the apex $p \in \P^3 \setminus \P^2$ and a hexagonial pyramid $\mathsf{con}(\mathcal P, p)$ with the base $\mathcal P$ and apex $p$, inscribed in $\mathsf{con}(Q, p)$. Let $\mathsf K^{\#} \subset \P^{3}$ be any tower $3^{\{3\}}$-cage, associated with $\mathsf{con}(\mathsf K, p)$ (its 18 nodes lie on $\mathsf{con}(\mathcal P, p) \cap \mathsf{con}(Q, p)$). Then the nine nodes[^16] of $\mathsf K^{\#}$, that do not belong to the cone $\mathsf{con}(Q, p)$, are coplanar. $\diamondsuit$ [**Example 4.1.**]{} Consider an *octagon*, inscribed in a quadratic plane curve $Q \subset \P^2$, and the $4^{\{2\}}$-cage $\mathsf K$ in the plane it generates. Then, by Theorem \[2d-polygon\], the remaining $8$ nodes of $\mathsf K$ will lie on another quadratic curve $Q^\star$. Forming a tower $4^{\{3\}}$-cage $\mathsf K^\#$ with the base $\mathsf K$ and apex $p$, each of the quadratic cones, $\mathsf{con}(Q, p)$ and $\mathsf{con}(Q^\star, p)$, will contain two complimentary set of nodes from $\mathsf K^\#$, each one of cardinality $32$. So the family of degree $4$ surfaces that are nailed to the $64$ nodes of the cage $\mathsf K^\#$ contains the reducible surface, a union of two quadratic cones which share the apex $p$. $\diamondsuit$ Now let us formulate few observations about reducible varieties in cages. Let $\mathcal W_\mathsf K(n, d)$ be the $(n-1)$-dimensional variety of hypersurfaces $H \subset \P^n$ of degree $d$ that contain all the nodes of a given $d^{\{n\}}$-cage $\mathsf K$. It is given by linear constraints, imposed on the coefficients of homogeneous degree $d$ polynomials in $n+1$ variables. By Theorem \[main\_th\], $\mathcal W_\mathsf K(n, d)$ admits a biregular map from $\P^{n-1}$. In fact, $\mathcal W_\mathsf K(n, d)$ is a base of a fibration $\pi: \mathcal E_\mathsf K(n, d) \to \mathcal W_\mathsf K(n, d)$ whose fibers are the hypersurfaces $H \subset \P^n$ of degree $d$ that contain all the nodes of $\mathsf K$. So $\mathcal E_\mathsf K(n, d)$ may be viewed as a codimension one subvariety of $\P^n \times \P^{n-1}$, and $\pi$ as being induced by the obvious projection $\P^n \times \P^{n-1} \to \P^{n-1}$. Evidently, $\mathcal W_\mathsf K(n, d)$ contains $n$ points $\theta_1, \dots , \theta_n$ that represent the completely reducible unions of the hyperplanes from $\mathsf K$ of a particular color $\a_1, \dots , \a_n$. Thus $\mathcal E_\mathsf K(n, d)$ contains at least $n$ completely reducible fibers. \[Q\] For which cages $\mathsf K$ the space $\mathcal E_\mathsf K(n, d)$ contains other reducible $\pi$-fibers? Unfortunately, our understanding of this natural problem is very limited (see Theorem \[2d-polygon\], Lemma \[numerics\], Theorem \[reducible\_3\_cages\], and Theorem \[reducible\_cages\]). Figure \[4x4degeneration\] demonstrates the phenomenon we are after. It also indicates that the answer does not depend on the combinatorics of the cage *only*: that is, two $d^{\{n\}}$-cages, $\mathsf K_1$ and $\mathsf K_2$, that produce isomorphic colored posets $\mathcal S(\mathsf K_1)$ and $\mathcal S(\mathsf K_2)$ (as the hyperplane arrangements) may support different types of reducible varieties from $\mathcal W_{\mathsf K_1}(n, d)$ and $\mathcal W_{\mathsf K_2}(n, d)$. Indeed, a small perturbation of the cage $\mathsf K$ in Figure \[4x4degeneration\] destroys the collinearity of the $4$ diagonal nodes, but does not change the associated poset $\mathcal S(\mathsf K)$. As a result, the perturbation eliminates the union of a line and a cubic curve from the perturbed cage family. Let us set the stage for tackling Question \[Q\]. Put $N := C^n_{d+n} -1$. Let $\mathcal V(n, d)$ be the $N$-dimensional variety of hypersurfaces $H \subset \P^n$ of degree $d$. We may identify $\mathcal V(n, d)$ with $\P^N$. The space $\mathcal V(n, d)$ has a natural stratification $\{\mathcal V(n, \om)\}_\om$ by the (unordered) *partitions* $\om = \{d = \sum_i d_i\}$ of the natural number $d$: $H \in \mathcal V(n, \om)$ if the homogeneous polynomial $P_H$ that defines $H$ is a product over $\A$ of some polynomials of degrees $\{d_i\}_{i \in [1, m]}$, prescribed by $\om$. So, for $d =\sum_{i=1}^m d_i$, we get $$\dim(\mathcal V(n, \om)) = -1 + \sum_{i=1}^m C^n_{d_i+n},$$ and $$\begin{aligned} \label{eq4.1} \text{codim}(\mathcal V(n, \om), \mathcal V(n, d)) = C^n_{d+n} - \sum_{i=1}^m C^n_{d_i+n } \nonumber \\= \frac{1}{n!}\big[(d+1)\dots (d+n) - \sum_{i=1}^m (d_i+1)\dots (d_i+n)\big].\end{aligned}$$ The image of $\mathcal V(n, \om)$ in $\mathcal V(n, d)$ is given by the generalized Veronese Map $$\begin{aligned} \label{eq4.2} \mathcal{V}er_\om :\; \P\big(\prod_{i=1}^m \mathsf{Sym}^{d_i}(\A^{n+1})\big) \to \P\big(\mathsf{Sym}^{d}(\A^{n+1})\big),\end{aligned}$$ where $\mathsf{Sym}^{k}(V)$ stands for the $k^{th}$ symmetric product of a vector space $V$. For example, for $\om_\star = (\underbrace{1, \dots , 1}_{d})$, we get $\text{codim}(\mathcal V(n, \om_\star), \mathcal V(n, d)) = C^n_{d+n}-d(n+1)$, which grows rapidly as a polynomial of degree $n$ in the variable $d$. Evidently, the $(n-1)$-dimensional cage variety $\mathcal W_\mathsf K(n, d) \approx \P^{n-1}$ embeds regularly into $\mathcal V(n, d) \approx \P^{N}$. By Theorem \[main\_th\], this image is *the minimal linear subspace* of $\P^N$ that contains the given “completely reducible" points $\theta_1, \dots , \theta_n$, and hence is determined by $\theta_1, \dots , \theta_n$. The linear subspace $\mathcal W_\mathsf K(n, d) \subset \mathcal V(n, d) \approx \P^N$ intersects the stratification $\{\mathcal V(n, \om)\}_\om$, thus producing the stratification of the space $\mathcal W_\mathsf K(n, d)$: $$\{\mathcal W_\mathsf K(n, \om) := \mathcal W_\mathsf K(n, d) \cap \mathcal V(n, \om)\}_\om.$$ Evidently, the cage $\mathsf K$ is defined by the $n$ points $\theta_1, \dots , \theta_n$ from the stratum $\mathcal W_\mathsf K(n, \om_\star) \subset \mathcal V(n, \om_\star)$, where $\om_\star := (\underbrace{1, \dots, 1}_d)$. For simplicity, let us assume that the cage has the property $\#(\mathcal W_\mathsf K(n, \om_\star)) = n$; that is, the only *completely* reducible hypersurfaces in the cage family are the $\{\theta_i\}_{i \in [1, n]}$ that generate the cage[^17]. Then by Theorem \[main\_th\], the variety $\mathcal W_\mathsf K(n, d)$ is determined by $\mathcal W_\mathsf K(n, \om_\star)$. So we may rephrase Question \[Q\]: *“How to describe all the strata $\{\mathcal W_\mathsf K(n, \om)\}_\om$ in terms of the stratum $\mathcal W_\mathsf K(n, \om_\star) \subset \mathcal V(n, d)$?"* For $d > 2$ and a *generic*[^18] $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, every reducible hypersurface $H \in \mathcal W_\mathsf K(n, d)$ is the union of the $d$ planes from $\mathsf K$ that are labeled with a particular color. $\diamondsuit$ [**Remark 4.1.**]{} The “genericity" hypothesis in the conjecture seems to be essential. For example, for $d=3,\, n=3$, and $\om = \{1+2\}$, using equation (\[eq4.1\]), we get $\text{codim}(\mathcal V(3, \om), \mathcal V(3, 3))\hfill \break = 6$, while the family $\mathcal W_\mathsf K(3, 3)$ of cubic surfaces in the $3^{\{3\}}$-cage $\mathsf K$ is $2$-dimensional. So one might expect that $\mathcal W_\mathsf K(3, \{1+2\}) = \emptyset$. However, Corollary \[spatial\_Pascal\] tells us that some special $3^{\{3\}}$-cage family contains a cubic surface that is a union of a quadratic cone and a plane. Perhaps, this phenomenon is due to the very special “ruled" geometry of the Veronese map (\[eq4.2\]). In fact, the dimension of the subspace of $\P\big(\mathsf{Sym}^{d}(\A^{n+1})\big)$ that is generated by the projective lines (“chords") through the pairs of points from the image $\mathsf{im}(\mathcal{V}er_\om)$ is smaller than the dimension of a similar chords’ subspace for a generic subvariety of $\P\big(\mathsf{Sym}^{d}(\A^{n+1})\big)$ of the same dimension as $\mathsf{im}(\mathcal{V}er_\om)$. $\diamondsuit$ The next lemma is only a small step towards answering the questions above. \[numerics\] For a given $d^{\{n\}}$-cage $\mathsf K \subset \P^n$ and a number $k \in [1, d-1]$, if at least $$C_{n+d}^n - C^n_{n+d-k} - n + 1$$ nodes from a supra-simplicial set $\mathsf A \subset \mathsf K$ lie on a hypersurface of degree $ k$, then the rest of the nodes of the cage lie on a hypersurface of degree $d-k$. Under these assumptions, the cage family $\mathcal W_\mathsf K(n, d)$ contains a reducible hypersurface of degree $d$ that is the union of a hypersurface of degree $k$ and a hypersurface of degree $d-k$. The space of homogenous polynomials of degree $m$ in $n+1$ variables has the dimension $C^n_{n+m}$. Therefore the linear constraints on the coefficients of such a polynomial $P$ that require $P$ to vanish at any set of $C^n_{n+m} - 1$ points in $\P^n$ must have a nontrivial solution. We consider the nodes of a supra-simplicial set $\mathsf A \subset \mathsf K$ (by Lemma \[cardinality\] $\#\mathsf A = C_{d+n}^n - n$) and a subset $\mathsf B \subset \mathsf A$ of cardinality at least $(C_{n+d}^n - n) - (C^n_{n+d-k} - 1)$. If we arrange for the nodes from $\mathsf B$ to lie on a hypersurface of degree $k$, the rest of the nodes from $\mathsf A \setminus \mathsf B$ (numbering $C^n_{n+d-k} - 1$) will impose realizable constraints on the polynomials of degree $d-k$. So, under the lemma hypotheses, there exit a homogeneous polynomial $S$ of degree $d-k$ that vanishes on $\mathsf A \setminus \mathsf B$ and a homogeneous polynomial $T$ of degree $k$ that vanishes on $\mathsf B$. Their product $S\cdot T$ of degree $d$ vanishes of $\mathsf A$. By Theorem \[main\_th\], $S\cdot T$ vanishes on all the nodes of the cage. Thus the hypersurface, defined by $S\cdot T$, belongs to the variety $\mathcal W_\mathsf K(n, d)$ and is reducible. [**Remark 4.2.**]{} Of course, the real issue is when one can place at least $C_{n+d}^n - C^n_{n+d-k} - n + 1$ nodes of a supra-simplicial set $\mathsf A$ of a given $d^{\{n\}}$-cage $\mathsf K$ on a hypersurface of degree $k$ (this is a constraint imposed on $\mathsf K$). As $n$ grows, this task becomes more and more challenging... So it is unclear when the basic hypothesis of Lemma \[numerics\] is realizable! $\diamondsuit$ [**Example 4.2.**]{} - Let us consider the case: $n=2, d=3, k =2$. Then Lemma \[numerics\] claims that, if one can place $6$ nodes of a supra-simplicial set $\mathsf A$ of a $3^{\{2\}}$-cage $\mathsf K \subset \P^2$ on a quadric curve $Q \subset \P^2$, then the rest of the nodes, numbering $3$, will reside in a line. This is exactly the Pascal’s Mystic Diagram Theorem. - Let us consider the case: $n=3, d=3, k =2$. Then Lemma \[numerics\] claims that, if one can place *at least* $14$ nodes of a supra-simplicial set $\mathsf A$ of a $3^{\{3\}}$-cage $\mathsf K \subset \P^3$ on a quadratic surface $Q\subset \P^3$, then the rest of the nodes, numbering $\leq 13$, will reside in a plane $\Pi \subset \P^3$. To accommodate exactly $13$ nodes of a $3^{\{3\}}$-cage on a single plane seems to be an impossible task ... At the same time, Corollary \[spatial\_Pascal\] delivers an example of a $3^{\{3\}}$-cage $\mathsf K^\# \subset \P^3$ that places $18$ nodes on a quadratic surface $Q^\# \subset \P^3$ and the remaining $9$ nodes on a plane! This example suggests that the numerical hypothesis of Lemma \[numerics\] is not sharp. $\diamondsuit$ Let us glance at the numerics from Lemma \[numerics\] in the case of surfaces. Homogeneous polynomials in four variables of degree $d$ form a vector space of dimension $$C_{d+3}^3 = (d^3 + 6d^2 + 11d + 6)/6.$$ Thus the projective space of surfaces in $\P^3$ of degree $d$ is $[(d^3 + 6d^2 + 11d)/6]$-dimensional. At the same time, a supra-simplicial set of a $d^{\{3\}}$-cage consists of $$C^3_{d+3} - 3 = (d^3 + 6d^2 + 11d - 12)/6$$ nodes. As this calculation shows, for each $k \in [1, d-1]$, every $[(k^3 + 6k^2 + 11k)/6]$ points in $\P^3$ lie on a surface of degree $\leq k$. Thus, by Lemma \[numerics\], we get the following corollary. If $[(d^3 - k^3 ) + 6(d^2 - k^2 ) + 11(d - k) - 12]/6 $ nodes of a supra-simplicial set of a $d^{\{3\}}$-cage $\mathsf K \subset \P^3$ lie on a surface $Q$ of degree $d - k$, then the rest of the nodes must lie on a surface $Q_\star$ of degree $k$. $\diamondsuit$ The next two theorems are based on what looks like a stronger but esthetically more pleasing hypotheses than the ones in Lemma \[numerics\] or in the previous corollary. They both do not rely on the intricate combinatorics of supra-simplicial sets and share the same conclusion. \[reducible\_3\_cages\] Let $k \in [1, d-1]$, and let $\mathsf K$ be a $d^{\{3\}}$-cage in $\P^3$. If an irreducible surface $Q \subset \P^3$ of degree $k$ contains $k d^2$ nodes of $\mathsf K$, then there exists a surface $Q^\star \subset \P^3$ of degree $d-k$ that contains the rest of the nodes, numbering $(d-k)d^{2}$. Thus, for a such a cage $\mathsf K$, the family of surfaces $\mathcal W_\mathsf K(3, d)$ has the reducible member $Q \cup Q^\star \in \mathcal W_\mathsf K(3,\, \{2+(d-2)\})$. Consider a $2$-plane $R \subset \mathsf K$ and the union $\mathcal R$ of planes that share the same color with $R$. The intersection $Q \cap R$ is $1$-dimensional at least. So there is a point $a \in Q \cap R$ that does not belong to the node set $\mathsf N \subset \mathsf K$. Next we claim that there is a surface $X$ of degree $d$ that contains all the nodes and the point $a$. Indeed, the difference between the dimension of the space $\mathcal V(3, d)$ of degree $d$ surfaces in $\P^3$ and the number of elements in a supra-simplicial set $\mathsf A$ of $\mathsf K$ is $2$. Therefore, there exists a homogeneous polynomial $P$ of degree $d$ that vanishes on $a \cup \mathsf A$. By Theorem \[main\_th\], $P$ must vanish on $a \cup \mathsf N$. We choose the zero set of $P$ for $X$. Consider the intersection $C : = X \cap Q$. Since $Q$ is irreducible, we face an alternative: [**(1)**]{} either $Q \subset X$, [**(2)**]{} or $C$ is a curve. In the first case, the polynomial $P$ is divisible by the polynomial $P_Q$ of degree $k$ that defines $Q$. The quotient $T^\star := P/P_Q$ of degree $d-k$ must vanish at all the nodes where $P_Q$ does not. So the zero set $Q^\star$ of $T^\star$ is a variety of degree $d-k$ that contains all the nodes from $\mathsf N \setminus (Q \cap \mathsf N)$. So, in case [**(1)**]{}, we are done. In the second case, we consider the intersection of $C$ with $\mathcal R$. Note that $\mathcal R \supset \mathsf N$. By the theorem hypotheses and the construction of $X$, the curve $C$ of degree $ \leq kd$ contains all the nodes from $Q \cap \mathsf N \subset C \cap \mathcal R$ and the point $a \in C \setminus (C \cap \mathsf N) \subset \mathcal R$. Therefore $C \cap \mathcal R$ must contain $kd^{2} +1$ points. On the other hand, by the Bezout Theorem, $\deg(C \cap \mathcal R) \leq (kd)(d) = kd^{2}$. This contradiction rules out the second case. \[reducible\_cages\] Let $k \in [1, d-1]$, and let $\mathsf K$ be a $d^{\{n\}}$-cage in $\P^n$. Assume that an irreducible hypersurface $Q \subset \P^n$ of degree $k$ contains $k d^{n-1}$ nodes of $\mathsf K$. Moreover, assume that the hyperplanes from $\mathsf K$ of all the colors, but a single one, induce a $d^{\{n-1\}}$-cage $\mathsf K^\dagger$ on $Q$ (in the sense of Definition \[cages\_on\_varieties\]). Then there exists a hypersurface $Q^\star \subset \P^n$ of degree $d-k$ that contains the rest of the nodes, numbering $(d-k)d^{n-1}$. For a such a cage $\mathsf K$, the family of hypersurfaces $\mathcal W_\mathsf K(n, d)$ has the reducible member $Q \cup Q^\star \in \mathcal W_\mathsf K(n,\, \{k+(d-k)\})$. Let $Q$ be a hypersurface of degree $k < d$ that contains $k d^{n-1}$ nodes of $\mathsf K$. Assume that the homogeneous polynomials $\mathcal L_2, \dots, \mathcal L_n$ from the set $\{\mathcal L_1, \mathcal L_2, \dots, \mathcal L_n\}$ that defines $\mathsf K$ induce a $d^{\{n-1\}}$-cage $\mathsf K^\dagger$ on $Q$. We consider the set of $d^{n-1}$ lines $\{\ell_{\b} \subset \P^n\}_\b$, each of which is the intersection of the hyperplanes from $\mathsf K$ of the distinct colors $\a_2, \dots, \a_n$. The hypotheses that $\mathsf K^\dagger$ is a cage on $Q$ is equivalent to the assumption that $Q$ is smooth in the vicinity of each intersection point $b \in Q \cap \ell_\b$ and $Q$ is transversal to $\ell_\b$ at $b$. Since $\deg(Q) = k$, each intersection $\ell_\b \cap Q$ may contain $k$ points at most. On the other hand, $Q \cap \mathsf N \subset Q \cap (\cup_\b \ell_\b)$, the latter set being of cardinality $k d^{n-1}$ at most. By the hypothesis, $\#(Q \cap \mathsf N) = k d^{n-1}$. Therefore, $Q \cap \mathsf N = Q \cap (\cup_\b \ell_\b)$; each line $\ell_\b$ hits $Q$ only at the nodes of the cage. Next we pick a node $a \in Q \cap \mathsf K$. Let $\tau_a$ be the hyperplane tangent to $Q$ at $a$. By Theorem \[unique\_hyper\], there exists a hypersurface $X = \{P = 0\}$ of degree $d$ that contains $\mathsf N$ and has $\tau_a$ as its tangent hyperplane at $a$. Consider the intersection $C : = X \cap Q$. Since $Q$ is irreducible, once more we face an alternative: [**(1)**]{} either $Q \subset X$, or [**(2)**]{} $C$ is a subvariety of $Q$. In the first case, the polynomial $P$ is divisible by the polynomial $P_Q$ of degree $k$ that defines $Q$. The quotient $T^\star := P/P_Q$ of degree $d-k$ must vanish at all the nodes where $P_Q$ does not. Thus the zero set $Q^\star$ of $T^\star$ is a hypersurface of degree $d-k$ that contains all the nodes from $\mathsf N \setminus (Q \cap \mathsf N)$. So, in case [**(1)**]{}, we are done. In the second case, let us consider an affine chart in the vicinity of $a$. In that chart, $dP|_a \sim dP_Q|_a$ since $\tau_a$ is tangent to both $X$ and $Q$. Therefore $dP|_{\tau_a}= 0$, and $a$ is a critical point of $P$ in the chart. So the subvariety $C$ is singular at $a$. However, by Theorem \[main\_thA\], $C$ must be smooth at the nodes of the induced $d^{\{n-1\}}$-cage $\mathsf K^\dagger$ on $Q$. This contradiction rules out the second case. The assumption in Theorem \[reducible\_cages\] that $\mathsf K$ induces a $d^{\{n-1\}}$-cage $\mathsf K^\dagger$ on $Q$ seems to be superfluous: for example, the argument in Theorem \[reducible\_3\_cages\] is free from it. The following conjecture, an obvious generalization of Theorem \[reducible\_3\_cages\] and Theorem \[reducible\_cages\], reflects this shortcoming. Let $k \in [1, d-1]$ and let $\mathsf K$ be a $d^{\{n\}}$-cage in $\P^n$. If an irreducible hypersurface $Q \subset \P^n$ of degree $k$ contains $k d^{n-1}$ nodes of $\mathsf K$, then there exists a hypersurface $Q^\star \subset \P^n$ of degree $d-k$ that contains the rest of the nodes, numbering $(d-k)d^{n-1}$. For a such a cage $\mathsf K$, the family of hypersurfaces $\mathcal W_\mathsf K(n, d)$ has the reducible member $Q \cup Q^\star \in \mathcal W_\mathsf K(n,\, \{k+(d-k)\})$. $\diamondsuit$ Let us compare the next corollary of Theorem \[reducible\_3\_cages\] with Corollary \[spatial\_Pascal\] and Example 4.2. \[spacial\_Pascal\_A\][**(Spacial Pascal’s Theorem).**]{} Let $\mathsf K$ be a $3^{\{3\}}$-cage in $\P^3$. If an irreducible quadratic surface $Q \subset \P^3$ contains $18$ nodes of $\mathsf K$, then there exists a plane $Q^\star \subset \P^3$ that contains the rest of the nodes, numbering $9$. Conversely, if $9$ nodes of $\mathsf K$ are coplanar, then the rest of the nodes belong to a quadratic surface. As a result, the family $\mathcal W_\mathsf K(3, 3)$ of cubic surfaces that contain the nodes of a $3^{\{3\}}$-cage $\mathsf K$ has a reducible member $Q \cup Q^\star \in \mathcal W_\mathsf K(3, \{2+1\})$ if and only if some $9$ nodes of $\mathsf K$ are coplanar. $\diamondsuit$ [**Remark 4.3.**]{} At least over the real numbers, it looks that the quadratic surface $Q$ in Corollary \[spacial\_Pascal\_A\] could not be anything, but a cone. $\diamondsuit$ Symmetry behind Bars ==================== In previous examples we have noticed that a variety, containing the nodes of a particular cage, must inherit *at least* the the symmetry of the cage (that takes all the hyperplanes of the same color $\a$ to hyperplanes of the same color $\b$, possibly different from the original $\a$). At the same time, we also have encountered varieties that are more symmetric than the cages in which they are inscribed. For instance, in Example 2.3, we saw that the $K3$-surface has $\mathsf S_4$-symmetry, while its cage is just $\mathsf S_3$-symmetric. These kind of observations are valid for any finite group $G$ of projective transformations that preserve a given cage: the cage-generated varieties inherit the symmetry of the cage. The obvious Lemma \[sym\] below is motivated by the question: *For a given projective variety $V \subset \P^n$ that contains all the nodes of a $d^{\{n\}}$-cage $\mathsf K^\bullet \subset \P^n$, “how many" $d^{\{n\}}$-cages $\mathsf K$ have all their nodes contained in $V$?* \[sym\] Let $G \subset \mathsf{PGL}_\A(n+1)$ be a subgroup. If a projective variety $V \subset \P^n$ is invariant under the natural $G$-action on $\P^n$, and $V$ contains all the nodes of a $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, then $V$ contains the all nodes of the cage $g(\mathsf K)$ for any $g \in G$. $\diamondsuit$ Thus, at least projectively symmetric varieties from a given cage family have the potential to be inscribed in many cages... For any subgroup $\tilde G \subset \mathsf{GL}_\A(n+1)$, we denote by $G$ its image in the projective linear group $\mathsf{PGL}_\A(n+1)$. The group $\tilde G$ acts on the space of homogeneous polynomials in $n+1$ variables of a given degree $d$. At the same time, $G$ acts on $\P^n$ by projective transformations, and thus on the space of subvarieties in $\P^n$. Let us fix a character $\mu: \tilde G \to \A^\ast$, where $\A^\ast := \A \setminus \{0\}$. The next lemma testifies that a color-preserving symmetry of a $d^{\{n\}}$-cage is shared by all the varieties that contain its nodes and are defined by polynomials of degrees $\leq d$. \[equi\] Let $\tilde G$ be a finite subgroup of $\mathsf{GL}(n+1)$, and let $\mu: G \to \A^\ast$ be a character. Consider a $d^{\{n\}}$-cage $\mathsf K = \bigcup_{j \in [1,n],\, i \in [1, d]} H_{j, i} \subset \P^n$, given by the polynomials $\mathcal L_j := \prod_{i \in [1, d]} L_{j, i}$ such that $\tilde g^\ast(\mathcal L_j) = \mu(\tilde g) \cdot \mathcal L_j$ for each $\tilde g \in \tilde G$ and all $j \in [1, n]$.[^19] Then any variety $V \subset \P^n$ that is defined by some homogeneous polynomials of degrees $\leq d$ and contains all the nodes of a supra-simplicial set $\mathsf A \subset \mathsf K$ is $G$-invariant. By Theorem \[main\_th\], any such variety $V$ of a codimension $s$ is given by polynomial equations of the form $\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}} = 0\}_{k \in [1, s]}$. By the proof of Theorem \[smooth\], we may assume that the vectors $\{\vec\l^{(k)}\}_{k \in [1, s]}$ are linearly independent. The lemma’s hypotheses imply that each $d$-polynomial $\mathcal L_j$ (whose zero set is $\bigcup_{i} H_{j, i}$) is mapped by any $\tilde g \in \tilde G$ to a $\mu(\tilde g)$-proportional polynomial $\tilde g^\ast(\mathcal L_j)$ with the same zero set as $\mathcal L_j$. Thus the group $\tilde G$ acts on the polynomials $\{\mathcal P_{\mathsf K,\, \vec\l^{(k)}}\}_{k \in [1, s]}$ by acting on their ingredients $\{\mathcal L_j\}_{j \in [1, n]}$. Since all the polynomials $\{\mathcal L_j\}_{j \in [1, n]}$ are eigenvectors for the $\tilde g$-action on the $d$-graded subspace of $\A[y_0, \dots , y_n]$ with the same eigenvalue $\mu(\tilde g)^d$, we get that $\tilde g^\ast(\mathcal P_{\mathsf K,\, \vec\l^{(k)}}) = \mu(\tilde g)^d \cdot \mathcal P_{\mathsf K,\, \vec\l^{(k)}}$ for all $\tilde g$ and $k$. Therefore $V$ is $G$-invariant. On the other hand, starting with a given finite symmetry group $\tilde G \subset \mathsf{GL}_\A(n+1)$, due to the next theorem, we can manufacture quite easily $G$-symmetrical cages and varieties, nailed to their nodes. \[G-invariant\] Let a finite subgroup $\tilde G \subset \mathsf{GL}_\A(n+1)$ of order $d$ be such that the intersection of $\tilde G$ with the kernel of the homomorphism $\pi: \mathsf{GL}_\A(n+1) \to \mathsf{PGL}_\A(n+1)$ is the unit element. For every integer $s \in [1, n]$, there is a $G$-invariant complete intersection $V \subset \P^n$ of dimension $n-s$ and of the multi-degree $(\underbrace{d, \dots , d}_{s})$ such that: 1. there exists a $G$-invariant $d^{\{n\}}$-cage $\mathsf K \subset \P^n$, 2. the $G$-action is *free* on the set $\mathsf N$ of its nodes, 3. $V$ contains all the nodes[^20] and is smooth in their vicinity, 4. for any such cage $\mathsf K$, the variety $V$ is unique among the varieties that contain a supra-simplicial set of nodes $\mathsf A \subset \mathsf K$ and have a given tangent space $\tau \subset T_p\P^n$ of dimension $n-s$ at an arbitrary chosen node $p$ of the cage, 5. the quasi-projective set $\mathcal V_{\mathsf A}$ of such $G$-invariant varieties $V \subset \P^n$ (i.e., of the multi-degree $(\underbrace{d, \dots , d}_{s})$ and that contain $\mathsf A$) is of dimension $s(n-s)$. We consider the natural left $\tilde G$-action on the space $(\A^{n+1})^\ast$ of homogeneous linear forms on the space $\A^{n+1}$. Since the representation $\tilde G \hookrightarrow \mathsf{GL}_\A(n+1)$ is faithful, the main orbit-type of this action is $[\tilde G]$. So for a *generic* choice of $L \in (\A^{n+1})^\ast$, the orbit $\tilde G L$ of $L$ contains $d:= |\tilde G|$ elements. Let us order the elements of $\tilde G$, so that the unit element is the first. Under this ordering, $\tilde G = \{\tilde g_1, \tilde g_2, \dots , \tilde g_d\}$ as sets. We start with a generic linear form $L_1$ and produce the linear forms $\{L_{1,i} :=\tilde g^\ast_i(L_1)\}_{i \in [1, d]}$, the $\tilde G$-orbit of $L_1$ of cardinality $d$. We denote by $H_{1,i}$ the hyperplane $\{L_{1,i} =0\} \subset \P^n$. Next, we pick a generic linear form $L_2$ so that the hyperplane $\{L_2 = 0\}$ is in general position with respect to all the hyperplanes $\{H_{1,i}\}_{i \in [1,d]}$. Let $L_{2,i} :=\tilde g^\ast_i(L_2)$ and $H_{2, i} := \{L_{2,i} =0\}$. Then the hyperplanes from the family $\{H_{1,i}\}_{i \in [1,d]}$ and the hyperplanes from the family $\{H_{2,k}\}_{k \in [1,d]}$ will automatically be in general position mutually. Now, we pick a generic linear form $L_3$ so that the hyperplane $\{L_3 = 0\}$ is in general position with respect to all the hyperplanes $\{H_{1,i}\}_{i \in [1,d]}\, \bigcup\, \{H_{2,k}\}_{k \in [1,d]}$. Let $L_{3, \ell} :=\tilde g^\ast_\ell(L_3)$ and $H_{3, \ell} := \{L_{3,\ell} =0\}$. Then the hyperplanes from the family $\{H_{3,\ell}\}_{\ell \in [1,d]}$ and the hyperplanes from the family $\{H_{1,i}\}_{i \in [1,d]}\, \bigcup \, \{H_{2,k}\}_{k \in [1,d]}$ will automatically be in general position. Continuing this way, we will pick the linear forms $\{L_{j,i}\}_{j\in [1, n],\, i \in [1, d]}$ and the hyperplanes $\{H_{j,i}\}_{j\in [1, n],\, i \in [1, d]}$ in $\P^n$ that will form a $G$-*invariant* $d^{\{n\}}$-cage $\mathsf K$ in $\P^n$. By this construction, the $G$-action is free on the set of nodes, since $G$ permutes freely the hyperplanes of each color and every node is characterized by the hyperplanes of $n$ distinct colors to which it belongs. By forming the $\tilde G$-invariant polynomials $\{\mathcal L_j := \prod_{i \in [1, d]} \, L_{j, i})\}_{j \in [1,n]}$ of degree $d$, we have produced a setting to which Lemma \[equi\] is applicable (in this case, the character $\mu: \tilde G \to \A^\ast$ is trivial). Combining this lemma with Theorem \[main\_th\] and Theorem \[smooth\], we validate the claims (1)-(4). In particular, any variety, containing the nodes of this $G$-invariant cage $\mathsf K$, is $G$-invariant. Note that the set $\mathcal V_{\mathsf A}$ of varieties $V \subset \P^n$ of the multi-degree $(\underbrace{d, \dots , d}_{s})$ that contain the supra-simplicial set $\mathsf A$ of the cage $\mathsf K$ is a projective algebraic set. Therefore the claim (5) follows from the claim (4), since the choice of the subspace $\tau_p \subset T_p(\P^n)$ of the dimension $n-s$ is equivalent to a choice of a point in the Grassmanian $\mathsf{Gr}_\A(n, s)$ of dimension $s(n-s)$. [**Example 5.1.**]{} We take the icosahedral group[^21] $\mathsf I_{120} \subset \mathsf{SU}(2)$ for the role of $\tilde G$ in Theorem \[G-invariant\] and consider the representation $\Psi: \mathsf I_{120} \hookrightarrow \mathsf{GL}_\C(3)$ that is induced by the direct sum of the obvious embedding $\mathsf{SU}(2) \subset \mathsf{GL}_\C(2)$ with the trivial complex $1$-dimensional representation of $\mathsf{SU}(2)$. Since the $\mathsf{SU}(2)$-action on $\C\P^1$ is faithful, it follows that $\mathsf I_{120} \cap \ker(\mathsf{GL}_\C(3) \to \mathsf{PGL}_\C(3)) = 1$. Applying Theorem \[G-invariant\], we produce a $\mathsf I_{120}$-invariant plane curve $V \subset \C\P^2$ of degree $120$ that contains all the nodes of a $120^{\{2\}}$-cage $\mathsf K \subset \C\P^2$, the $\mathsf I_{120}$-action on the nodes being free, and thus the main orbit-type of the $\mathsf I_{120}$-action on $V$ is $[\mathsf I_{120}]$. The curve $V$ is smooth in the vicinity of the node set $\mathsf N$. In fact, by the Hironaka’s Desingularization Theorem 7.1, [@Hi], $V$ admits an equivariant resolution $O: \tilde V \to V$ which is a biregular map over vicinity of $\mathsf N \subset V$. $\diamondsuit$ In Theorem \[G-invariant\], the $G$-action on the set of nodes is free, and $V$ is nonsingular in their vicinity. Thus the $H$-fixed loci $\{V^H\}_{H \subset G,\; H \neq 1}$— the “singularities" of the $G$-action— and the singularities of $V$ itself both stay away from the set of nodes. Now we will produce $G$-invariant varieties $V \subset \P^m$, whose fixed point sets $V^G$ *form* the node sets of cages in a lower dimensional projective space $\P^s \subset \P^m$. In these examples, $V^G$, the strongest singularities of the $G$-action, and the singularities of $V$ itself are separated. To state Theorem \[V\^G = N\] below, we need to introduce some notations. Let a finite group $\tilde G \subset \mathsf{GL}_\A(n+1)$ of order $d$ be such that its intersection with the kernel of the homomorphism $\pi: \mathsf{GL}_\A(n+1) \to \mathsf{PGL}_\A(n+1)$ is the unit element. So we may identify $\tilde G$ with its image $G$ in $\mathsf{PGL}_\A(n+1)$. Let $\mathcal R(n+1, d)$ be the space of homogeneous polynomials of degree $d$ in $n+1$ variables. Consider the $\tilde G$-representation $\Phi_d: \tilde G \to \mathsf{GL}_\A(\mathcal R(n+1, d))$, induced by the tautological representation $\Psi: \tilde G \hookrightarrow \mathsf{GL}_\A(n+1)$. Let $\mathcal R(n+1, d)^{\tilde G}_\mu \subset \mathcal R(n+1, d)$ be the subspace of polynomials $P$ in $n+1$ variables, subject to the constraint $\{\tilde g^\ast(P) = \mu(\tilde g)^d \cdot P\}$ for all $\tilde g \in \tilde G$ and a character $\mu: \tilde G \to \A^\ast$. By taking $n+1$ generic linear forms $\{L_i: \A^{n+1} \to \A\}_{i \in [1, n+1]}$ and producing the degree $|\tilde G|$ polynomials $\mathcal L_i := \prod_{g \in \tilde G} g^\ast(L_i)$, we see that at least $\dim_\A\big(\mathcal R(n+1, |\tilde G|)^{\tilde G}_\mu\big) \geq n+1$, where $\mu = 1$. In the next theorem, we pick $\mu$ to be the trivial character and drop $\mu$ from the notations. The arguments in Theorem \[V\^G = N\] follow closely the arguments in [@K1], Theorem A. \[V\^G = N\] Let a finite group $\tilde G \subset \mathsf{GL}_\A(n+1)$ be such that $\tilde G \, \cap \, \ker(\pi) = 1$. Assume that no line in $\A^{n+1}$ is invariant under the $\tilde G$-action[^22]. Put $$s := \dim_\A\big(\mathcal R(n+1, d)^{\tilde G}\big).\footnote{By the previous remark, for $d= |\tilde G|$, we get $s \geq n+1$.}$$ Then, for any $k \in [1, s]$, there exists a $G$-invariant variety $V \subset \P^{n+k+1}$ such that: 1. $V$ is a complete intersection of the multi-degree $(\underbrace{d, \dots, d}_{k})$ and dimension $n+1$, which contains all the nodes of a $d^{\{k\}}$-cage $\mathsf K \subset \P^{k} \subset \P^{n+k+1}$, 2. $V^G = \mathsf N$, the node locus of $\mathsf K$ (of cardinality $d^k$), 3. $V$ is smooth in the vicinity of $V^G$ and transversal to the subspace $\P^{k}$, 4. all the normal $G$-representations on $V$ at the points of $V^G$ are isomorphic to the representation $\Psi: \tilde G \hookrightarrow \mathsf{GL}_\A(n+1)$, 5. the orbit-types of the $G$-action on $V$ are drawn from the lists of the orbit-types of the $\tilde G$-action on $\A^{n+1}$ and on $\P(\A^{n+1})$. 6. For $\A = \C$, $V$ admits an $G$-equivariant resolution $O: \tilde V \to V$, where $\tilde V$ is a nonsingular complex projective $G$-variety such that the equivariant morphism $O$ is biregular in the vicinity of $\tilde V^G$ (note that, $O(\tilde V^G) = V^G$). In particular, all the normal $G$-representations in $\nu(\tilde V^G, \tilde V)$ are isomorphic to the representation $\Psi$, and all the orbit-types of the $G$-action on $\tilde V$ are not smaller than[^23] the orbit-types of the $\tilde G$-action on $\A^{n+1}$ and on $\P(\A^{n+1})$. We take the trivial character for the role of $\mu: \tilde G \to \A^\ast$. Consider the vector space $\A^{n+1} \times \A^{k+1}$ on which $\tilde G$ acts via the representation $\Psi \oplus Id$. We pick some linear independent degree $d$ homogeneous polynomials $P_1, \dots P_k \in \mathcal R(n+1, d)^{\tilde G}$ in the variables $z_0, \dots, z_n$, where $k \leq \dim_\A \mathcal R(n+1, d)^{\tilde G}$. Then we choose $k$ homogeneous polynomials $Q_1, \dots Q_k$ in $k+1$ variables $y_0, \dots, y_k$ such that each $Q_\ell$ is a product of $d$ linear forms, and the union $\bigcup_{\ell \in [1,k]}Z(Q_\ell)$ of their zero sets $Z(Q_\ell)$ forms a $d^{\{k\}}$-cage $\mathsf K \subset \P^k := \P(\A^{k+1})$. In particular, the nodes $\mathsf N$ of $\mathsf K$ form a $0$-dimensional complete intersection in $\P^k$ of the multi-degree $(\underbrace{d, \dots , d}_{k})$. Let $\P^n$ be the projectivization of the first factor $\A^{n+1}$ in the sum $\A^{n+1} \oplus \A^{k +1}$, and $\P^k$ the projectivization of the second factor. Now we define the variety $V \subset \P(\A^{n+1} \times \A^{k+1})$ by the homogeneous degree $d$ equations $$\begin{aligned} \label{eq5.1} \big\{P_\ell(z_0, \dots, z_n) + Q_\ell(y_0, \dots, y_k) = 0\big\}_{\ell \in [1, k]}. \end{aligned}$$ By the choice of $P_\ell$’s, $V$ is invariant under the $G$-action, defined via the representation $\Psi \oplus Id: \tilde G \to \mathsf{GL}_\A(n+k +2)$. Since no line in $\A^{n+1}$ is $\Psi(\tilde G)$-invariant, we conclude that the $G$-fixed locus in the subspace $\P^n \subset \P^{n+k +1}$ is empty. As a result, $(\P^{n+k +1})^G = \P^k$. So $V^G = V \cap \P^k = \mathsf N$, a complete intersection in $\P^k$ of the multi-degree $(\underbrace{d, \dots , d}_{k})$. By the choice of $\{Q_\ell\}_\ell$, their differentials are independent at each node from $\mathsf N = V^G$. As a result, $V$ is transversal to $\P^k$ and smooth in the vicinity of $V^G$. The group $G$ acts in the fibers of the normal bundle $\nu(\P^k, \P^{n+k+1}) \approx \P^{n+k+1} \setminus \P^n$ via $\Psi(\tilde G)$. The normal $G$-bundle $\nu(\P^k, \P^{n+k+1})$ restricts to the normal $G$-bundle $\nu(V^G, V)$ over the finite base $V^G$. So the normal $G$-representations in $\nu(V^G, V)$ all may be identified with the representation $\Psi$. Since $\tilde G \cap \ker(\pi) = 1$, the orbit-types of $G$-action on $\P^{n+k+1}$ are the same as the orbit-types of $\tilde G$-action on $\A^{n+1}$ and on $\P^n$. So the orbit-types of the $G$-action on $V$ are drawn from these two lists. In particular, the orbit-type $[G/G]$ comes from the orbit-type of the origin in $\A^{n+1}$. The last statement (6) of the theorem is based on the deep Hironaka’s Desingularization Theorem 7.1, [@Hi] (see also [@Hi1] for the non-equivariant version). The theorem claims that, in the category of complex projective varieties, there is a nonsingular resolution $O: \tilde V \to V$ such that any biregular map $\g: V \to V$ lifts uniquely to a biregular map $\tilde\g: \tilde V \to \tilde V$. Since, in our case, the complex projective variety $V$ is nonsingular in a Zariski open neighborhood of $V^G$, this equivariant resolution $O$ is a biregular morphism over such a neighborhood. Therefore $d^k$ copies of $\Psi$ are realizable on a nonsingular complex projective $G$-variety $\tilde V$ as the normal data $\nu(\tilde V^G, \tilde V)$. In turn, this fact may be interpreted as providing the estimate $d^k$ for the order of the unit $G$-sphere $S(\C^{n+1}, \Psi)$ in the appropriately chosen $G$-equivariant complex bordism group $\mathbf\Omega_{2n+1}(G, \mathcal F)$. Here $\mathcal F$ denotes the family of orbit-types of the $G$-space $\C^{n+1} \setminus \{\vec 0\}$ and the $G$-space $\P(\C^{n+1})$. [**Example 5.2.**]{} We define a $n$-*weighted composition* of a natural number $d$ as an ordered sequence of nonnegative integers $d_2, \dots , d_{n+1}$ such that $\sum_{i=2}^{n+1} i\cdot d_i = d$. Let $\a(n, d)$ denote the cardinality of the set of such $n$-weighted compositions of $d$. Consider the permutation representation $\tilde\Phi: \mathsf S_{n+1} \to \mathsf{GL}_\C(n+1)$. It is a direct sum of the trivial $1$-dimensional representation with the representation $\Phi: \mathsf S_{n+1} \to \mathsf{GL}_\C(n)$, whose space is given by a linear constraint $\{\s_1 := x_0 + x_1 + \dots + x_n = 0\}$. The composition of $\Phi$ with the homomorphism $\pi: \mathsf{GL}_\C(n) \to \mathsf{PGL}_\C(n)$ has a trivial kernel, and no line in $\C^n$ is $\Phi(\mathsf S_{n+1})$-invariant. Note that the ring $\mathcal P(n)^{\mathsf S_{n+1}}$ of $\mathsf S_{n+1}$-invariant polynomials in $n+1$ variables, being restricted to the hyperplane $\{\s_1 = 0\}$, is generated by the elementary symmetric polynomials $\s_1, \dots , \s_{n+1}$ in $n+1$ variables, modulo the relation $\{\s_1 = 0\}$. So we may chose $\{\s_2 \mod \langle \s_1\rangle,\; \dots , \; \s_{n+1} \mod \langle \s_1\rangle\}$ for the multiplicative generators of $\mathcal P(n)^{\mathsf S_{n+1}}$. Therefore $\dim_\C \mathcal P(n, d)^{\mathsf S_{n+1}} = \a(n, d),$ the number of $n$-weighted compositions of $d$. Applying Theorem \[V\^G = N\], we construct a $\mathsf S_{n+1}$-invariant variety $V \subset \P^{n + \a(n, d)}$ of dimension $n$ and of the multi-degree $(\underbrace{d, \dots , d}_{\a(n, d)})$, together with a $d^{\{\a(n, d)\}}$-cage $\mathsf K \subset \P^{\a(n, d)} \subset \P^{n + \a(n, d)}$, so that the fixed point set $V^{\mathsf S_{n+1}} = \mathsf N$, the node set of $\mathsf K$. Moreover all the normal $\mathsf S_{n+1}$-representations in $\nu(V^{\mathsf S_{n+1}}, V)$ are isomorphic to $\Phi$. $\diamondsuit$ Consider a finite set $\{P_1, \dots , P_k\} \subset \mathcal R(n+1, d)^{\tilde G}$ of $\tilde G$-invariant linearly independent polynomials such that the system of homogeneous equations $\{P_1 =0, \dots , P_k= 0\}$ has only the trivial solution $\vec 0 \in \A^{n+1}$. We call such collection $\{P_1, \dots , P_k\}$ a $d$-*regular system*. Note that $k > n$ for any regular system over $\C$. [**Example 5.3.**]{} Let $\mathsf I_{120} \subset \mathsf{SU}(2)$ be the icosahedral group of order $120$. The ring $\C[z_1, z_2]^{\mathsf I_{120}}$ is generated by the homogeneous polynomials $X, Y, Z$ of degrees $12, 20$, and $30$ in $(z_1, z_2)$. These three polynomials satisfy the identity $\{X^5 + Y^3 + Z^2 \equiv 0\}$. Then the pairs $\{X, Y\}$ and $\{X, Z\}$ form regular systems; i.e., the two systems of equations, $\{X=0,\, Y=0\}$ and $\{X=0,\, Z=0\}$, each has only the trivial solution. Let $\Psi: \mathsf I_{120} \stackrel{\Delta}{\rightarrow} \mathsf{SU}(2) \times \mathsf{SU}(2) \subset \mathsf{GL}_\C(4)$ be a representation, produced by the diagonal map $\Delta$ and then by taking the direct sum $A \oplus A$ of a $2\times2$-matrix $A \in \mathsf{SU}(2)$. With the help of $\Psi$, the group $\mathsf I_{120}$ acts faithfully on $\C\P^3$ with the homogeneous coordinates $[z_1, z_2, z_3, z_4]$. Moreover, no line in $\C^4$ is invariant under the $\mathsf I_{120}$-action. We denote by $\bar X, \bar Y, \bar Z \in \C[z_3, z_4]^{\mathsf I_{120}}$ the “doubles" of $X, Y, Z$ in the coordinates $(z_3, z_4)$. Let $s(d) := \dim_\C\big(\mathcal R(4, d)^{\Psi(\mathsf I_{120})}\big)$. In particular, since the four degree $60$ polynomials $X^3, Y^5, \bar X^3, \bar Y^5 \in \mathcal R(4, 60)^{\Psi(\mathsf I_{120})},$ we conclude that $s(60) \geq 4$. (The same conclusion follows from the fact that $X^5, Z^2, \bar X^5, \bar Z^2 \in \mathcal R(3, 60)^{\Psi(\mathsf I_{120})}$.) Therefore, letting $k = s(60) =4$ and picking the invariant polynomials $X^3, Y^5, \bar X^3, \bar Y^5$ for the role of $P_\ell$’s in (\[eq5.1\]), by Theorem \[V\^G = N\], we construct a $\mathsf I_{120}$-invariant complex $4$-dimensional complete intersection $V \subset \C\P^8$ of the multi-degree $(60, 60, 60, 60)$ such that $V^{\mathsf I_{120}} = \mathsf N$, the node set of a $60^{\{4\}}$-cage $\mathsf K \subset \C\P^4 \subset \C\P^8$. The normal representation in $\nu(V^{\mathsf I_{120}}, V)$ at each point of $V^{\mathsf I_{120}}$ is the $4$-dimensional representation $\Psi$. Since $\{X, Y, \bar X, \bar Y\}$ form a regular system in the $60$-graded portion of $\C[z_1, z_2, z_3, z_4]^{\mathsf I_{120}}$, we get $V \cap \C\P^3 = \emptyset$, where $\C\P^3 \subset \C\P^8$ is the projective space with the projective coordinates $[z_1, z_2, z_3, z_4]$. Therefore, the orbit-types of $\mathsf I_{120}$-action on $V$ are among the orbit-types of the $\Psi(\mathsf I_{120})$-action on $\C^4$ (see Corollary \[supplement\] for a generalization of this argument). This example gives a ridiculously high upper bound on the number of copies of the normal representation $\Psi$ that are realizable on a complex $4$-dimensional projective variety $V$. If one does not care about alining the fixed point set $V^{\mathsf I_{120}}$ with the nodes of a cage, a much lower upper estimate is available (cf. [@K1]). $\diamondsuit$ Reviewing the proof of Theorem \[V\^G = N\], we get its supplement. \[supplement\] Under the notations and hypotheses of Theorem \[V\^G = N\], and assuming the existence of $d$-regular system $\{P_1, \dots , P_k\} \subset \mathcal R(n+1, d)^{\tilde G}$, the equivariant variety $V$ is contained in $\P^{n+k +1} \setminus \P^n$, the space of the normal bundle $\nu(\P^k, \P^{n+k +1})$. Thus the statement (5) in Theorem \[V\^G = N\] can be strengthen: the orbit-types of the $G$-action on the variety $V$ are the same as the orbit-types of the $\tilde G$-action on $\A^{n+1}$. Since $\{P_1, \dots , P_k\} \subset \mathcal R(n+1, d)^{\tilde G}$ is a regular system, the homogeneous polynomials that define $V$ in (\[eq5.1\]) have no solution in $\P(\A^{n+1})$. Thus $V \cap \P(\A^{n+1}) = \emptyset$. So $V \subset \nu(\P^k, \P^{n+k +1})$. The group $G$ acts trivially on the base $\P^k$ of this normal bundle. Hence the orbit-types of the $G$-action on the variety $V$ are the same as the orbit-types of the $\tilde G$-action on $\A^{n+1}$, the fiber of $\nu(\P^k, \P^{n+k +1})$. We consider now the special case of $\tilde G$-action on the space $\A^{n+1} \times \A^{n+1}$ via the representation $\Psi \oplus Id$. We denote by $\P^n_1$ the projectivization of the first factor $\A^{n+1}$, and by $\P^n_2$ of the second one. So we have the obvious embedding $\b: \P^n_1 \coprod \P^n_2 \hookrightarrow \P^{2n+1}$. \[two G-cages\] Under the assumptions of Theorem \[V\^G = N\], and for $d = |G|$ and $k =n$, there exists a $G$-invariant variety $V$ which satisfies the properties (1)-(5), described in the theorem. In addition, 1. $V$ contains all the nodes $\mathsf N_1$ of a $G$-invariant $d^{\{n\}}$-cage $\mathsf K_1$ in $\P^n_1 \subset \P^{2n+1}$, 2. the $G$-action is free on $\mathsf N_1$, 3. $V\cap \P^n_1 = \mathsf N_1$, 4. $V$ is transversal to $\P^n_1$ and nonsingular in the vicinity of $\mathsf N_1$. As a result, the $G$-invariant variety $V$ of the multi-degree $(\underbrace{|G|, \dots, |G|}_{n})$ and dimension $n+1$ “interpolates" between the $G$-*free* set of nodes of the $|G|^{\{n\}}$-cage $\mathsf K_1 \subset \P_1^n$ and the $G$-*fixed* set of nodes of the $|G|^{\{n\}}$-cage $\mathsf K_2 \subset \P_2^n$.[^24] Let $d = |G|$. By choosing the $\tilde G$-invariant polynomials $\{P_\ell \in \mathcal R(n+1, d)\}_{\ell \in [1, n]}$ from the proof of Theorem \[V\^G = N\] according the recipe from Theorem \[G-invariant\] (so that each $P_\ell$ is a product of $d$ linear forms, and the entire collection $\{P_\ell\}_{\ell \in [1, n]}$ produces an $G$-invariant $d^{\{n\}}$-cage $\mathsf K_1$ in $\P^n_1$), the claim follows. Corollary \[two G-cages\] suggests a notion of cobordism between cages and varieties that are inscribed in them. However, the development of such theories belong to a different paper. So our trip to the Little Zoo of Algebraic Geometry has come to its completion. [30]{} Bacharach, I., [*ÜberdenCayleyÕschenSchnittpunktsatz*]{}, Math.Ann. 26(1886), 275-299. Cayley, A., [*On the Intersection of Curves*]{}, serialized in Cambridge Math. J., 25-27, and published by Cambridge University Press, Cambridge, 1889. Cayley, A., “On the triple tangent planes of surfaces of the third order”, Cambridge and Dublin Math. J., 4: 118Ð138, (1849). Davis, E. D., Geramita, A.V., and Orecchia, F., [*Gorenstein algebras and Cayley-Bacharach theorem*]{}, Proceedings Amer. Math. Soc. 93 (1985), 593-597. Eisenbud, D., Green, M., and Harris, J., [*Cayley-Bacharach theorems and conjectures*]{}, Bull. Amer. Math. Soc. 33 (1996), 295-324. Fulton, W., [Introduction to Intersection Theory in Algebraic Geometry]{}, Regional Conference Series in Mathematics, Number 54, published by American Mathematical Society, 1983. Geramita, A. V., Harima, T., and Shin, Y.S., [*Extremal point sets and Gorenstein ideals*]{}, Adv. Math 152 (2000) 78-119. Geramita, A. V., Harima, T., and Shin, Y.S., [*An alternative to the Hilbert function for the ideal of a finite set of points in $\P^n$*]{}, Illinois J. 45 (2001) 1-23. Geramita, A. V., Harima, T., and Shin, Y.S., [*Decompositions of the Hilbert Function of a Set of Points in $\P^n$*]{}, Canad. J. Math. 53 (2001) 925-943. Hartshorne, R., [*Algebraic Geometry*]{}, Springer-Verlag, New York, 1977. Hironaka, H., [*Bimeromorphic smoothing of a complex analytic space*]{}, Acta Mathematica Vietnamica, tom 2, no. 2 (1977), 103-168. Hironaka, H., [*Resolution of singularities of an algebraic variety over a field of characteristic zero. I*]{}, Ann. of Math., 2, 79 (1): 109-203, (1964). Katz, G., [*Curves in Cages: An Algebro-Geometric Zoo*]{}, The American Mathematical Monthly, 113 : 9 (2006), 777-791. Katz, G., [*On normal representations of $G$-actions on projective varieties*]{}, Comment. Math. Helvetici, 63 (1988), 169-183. Katz, G., [*How Tangents Solve Algebraic Equations, or a Remarkable Geometry of Discriminant Varieties*]{}, Expositiones Math., 21 (2003), 219-261. Kirwan, M. [*Complex Algebraic Curves, London Mathematical Society, Student Texts 23*]{}, (1992) Cambridge University Press. Mamford, D., [*Algebraic Geometry I, Complex Projective Varieties*]{}, Springer-Verlag, Berlin Heidelberg New York 1976. Milnor, J., [*Morse Theory*]{}, Annals of Mathematical Studies, no. 51, Princeton University Press, 1963. Ried, M. [*Undergraduate Algebraic Geometry, London Mathematical Society, Student Texts 12.*]{} (1998) Cambridge University Press. [^1]: with the exception of claim (6) in Theorem \[V\^G = N\] [^2]: real or complex [^3]: The entire hyperplane configuration $\mathsf K$ consists of $nd$ hyperplanes. [^4]: It follows that any group of $n$ distinctly colored hyperplanes is in general position in the ambient $n$-space. [^5]: Say, the limit line from $\mathsf K$ and the appropriate line from $X$ belong to the same ruled surface in $\P^3$. [^6]: Here and on, we use the term “variety" as synonym of “algebraic set"; so varieties may be reducible. [^7]: equivalently, a point in the Grassmanian $\mathsf{Gr}_\A(n, n-s)$ [^8]: In our game, the nodes are the gates, and the curve represent the desired trajectory of the ball. [^9]: This hypotheses frees us from the tyranny of the requirement on the ground field $\A$ to be algebraically closed. [^10]: We do not know how exactly $\mathsf S_5$ acts on the $27$ lines’ configuration, but the equations of these lines below should tell the story... [^11]: $\delta = \deg(f)$ for the $\xi$-generating embedding $f: X \hookrightarrow \P^N$. [^12]: See Definition \[cages\_on\_varieties\_B\], where $\delta$ is introduced. [^13]: The number of such $\kappa$’s is $C_N^n$. [^14]: $\mathcal P$ may be a union of several even-sided polygons. [^15]: The hexagon in the Figure \[pyramid\] is not convex. [^16]: (the points $a, b, c, d, e, f, g, h, i$ in Figure \[pyramid\]) [^17]: There are simple examples of $2^{\{2\}}$-cages that violate this assumption. [^18]: for a Zariski open set in the space $\big(\mathsf{Sym}^d(\P^{n \ast})\big)^n$ of all $d^{\{n\}}$-cages [^19]: Of course, this implies that $\mathsf K$ is $G$-invariant. [^20]: Therefore $[G]$ is the main orbit-type of the $G$-action on $V$. [^21]: Recall that $\mathsf I_{120} \approx \mathsf A_5 \times \Z_2$, where $\mathsf A_5$ is the alternating subgroup of $\mathsf S_5$. [^22]: i.e., the $\tilde G$-representation on $\A^{n+1}$ has no $1$-dimensional direct summand. [^23]: That is, “more free" [^24]: In Theorem \[V\^G = N\], the cage $\mathsf K_2$ is denoted “$\mathsf K$".